mit radiaton lab series v25 theory of servomechanisms

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MAS SA CHUS ETTS INS T ITUTE OF TECHA’OLOGY

RADIATION LABORATORY SERIES

Boa rd of E dit or s

Lor es h “. RIDENOUR, Editor-in-Chief

GEORGEB. COLLINS,Depu t~ Ed i tor -in -Ch ief

BIUTTONCHANCE,S. A. GOUDSMIT, R. G. HERB, HUBERTM. JAMES,JULIANK. h-m~l,

JAMES L. LAWSON,LEON Il. LINFORD,CAROL G. MONTGOMERY, C. NEWTON, ALBERT

NI. STONE, LOIJISA. TURNER,GEORGEE. VALLEY,J IL, HERBERTH. WHEATON

1.

2.3.

4.

5.

6.

7.

8.

9.

10.

11.

12,

13.

14.

15,

16.

17.

18.

1920.

21

22.

23.

24.

25

26

27

28

RADAR SYSTEM ExGINEER1?JG-Ridenour ’

RADAR AIDS TO hTAVIGATION—HallRADAR BmicoNs—Roberts

Lormiv—Pierce, McKen zie, and Woad ward

pLTLSE GENERATORS<JGSCW ~nd Ldraqz

TvIIcROwAvE kTAGNETRoNs—COllins

KLYSTRONS AND MICROWAVE TRIoDEs—Ham d&on, Knipp, and Kuper

PRINCIPLES OF MICROWAVE CIRCUITS—.~f071~g0mP7Y, I)icke, and Purcel l

~IICROWAVE TRANSM[SS1ON C1Rcu 1T5— Ragan

WAVEGUIDE HANDBOOK—~fa7 C7W&

TECRNIQUE OF MICROWAVE MEASUREMENTS—&107@ome?~

?VfICROWAVEANTENNA THEORY AND DESIGN—SdV~7

PROPAGATION OF SHORT RADIO WAVES-—KWT

k ~lcn owAvl? Du PLE xE Rs -S rn zd lin a nd .!fOn k7 0?7 U?!

CRYSTAI. RECTIFIERS— T orrey an d Whitm er

k~lCROWAVE ~I1xERs—Poun d

COMPONENTS HANDBooR—~/ ackb u rn

~ACUUM TUBE AMPLIFIERS—~U~/ e~ an d w at~m an

W’AVX FORM S-’chanCt?,Hughes , MacN i chol, s ayre, a nd Willia ms~LECTRONIC TIME kIEAsu REMENTs-Chan ce, Hu ls iz er , MacN i chol,

an d W illiam s

~LECTRONIC IN STRUMENTS~r@3n W00d , Holdam , and MacR ae

CAr rlloDE RAY TUBE D1 sPLAYs—&{~er ’, S ta n ’, a n d ~a lley

>flCROU7AVE REcE lvERs—~an Voorhis

THRESHOLD SIGNALs—Lawson and ~hlenbeck

TIIEORY OF SERVOME~HAXISMS-J a??WS, ,1 ’i chol s, and Ph il li ps

RADAR SC.4NXERS AND RADOMES—Ca@/, Kar~l~@ and T1~Tner

COMPUTING MECHANISMS AND LINKAGES—&JObOda

Ix DEx—Henn ev



THEORY OF

SERVOMECHANISMS

L-dited by

HUBERT M. JAMES

PROFESSOR OF PHYSICS

Pu RDUE UNIVERSITY

NATHANIEL B. NICHOLS

DIRECTOR OF RESEARCH

TAYLOR INSTRUMENT COMPANIES

RALPH S. PHILLIPS

.4SSOCIATE PIIOFESSOR OF MATHEIWATICS

UNIVERSITY OF SOUTHERN C.\ LIFOHNIA

OFF ICE OF SCIENTIF IC RESEARCH AND D13VELOP 31ENT

NATIONAL DEFENSE RESEARCH COMMITTEE

h’EW’ YORK-- TO RON TO. LONDON

.\ fcGRA W-HILL BOOK” (! OMPAAJY, INC.

1947



L, -T

THEORY OF SERVOMECHANISMS

COPYRIGHT, 1947, BY THE

hlCGRAW-HILL BOOK COMPANY, INC.

PRINTED IN THE UNITED STATES OF .<31EDICA

.411 righ ts reserved . T his hook , or

part s t hereof, may not be reproduced

in any form withou l perm ission OJ

[he publishers.

THE MAPLE PRESS COMPANY, YOl!K, P.4,



Foreword

THE t remendous resea rch and development effor t tha t went in to the

development of radar and rela t ed techniques dur ing World War II

resu lted not only in hundreds of radar set s for milit ary (and some for

possible peacet ime) use but also in a grea t body of informat ion and new

techniques in the elect ronics and h igh-fr equency fields. Because thisbasic mater ia l may be of gr ea t va lue t o science and engineer ing, it seemed

most impor tan t to publish it as soon as secur ity permit ted.

The Radia t ion Labora tory of MIT, which opera ted under the super -

vision of the Nat ional Defense Resear ch Commit tee, under took the gr ea t

t ask of pr epa rin g t hese volumes. Th e wor k descr ibed her ein , h owever , is

the collect ive resu lt of work done a t many labora tor ies, Army, Navy,

un iversity, and industr ia l, both in th is coun t ry and in England, Canada ,

and ot h er Domin ion s.

The Radia t ion Labora tory, once its proposa ls wer e approved and

fin an ces pr ovided by t he Office of Scien tific Resea rch a nd Developmen tj

chose Louis N. Ridenour as Editor -in -Chief to lead and direct t he en t ir e

project . An editor ia l staff was then selected of those best qua lified for

th is t ype of task. F ina lly the authors for the var ious volumes or chapters

or sect ions were chosen from among those exper t s who were in t imately

familia r with the var ious fields, and who were able and willing to }vr it e

the summaries of them. This ent ire sta ff agreed to remain at work at

MIT for six months or more after the work of the Radia t ion Labora tory

was complete. These volumes stand as a monument to this group.

These volumes serve as a memor ial t o the unnamed hundreds and

t housa nds of ot her scient ists, en gineer s, and ot her s wh o a ctua lly ca rr ied

on the r esearch , development , and engineer ing wor k t he results of which

a re herein descr ibed. There were so many involved in th is work and they

wor ked so closely t oget her even t hou gh oft en in widely sepa ra ted la bor a-

tor ies tha t it is impossible t o name or even t o know those who con t r ibu ted

t o a pa rt icu la r idea or developmen t. On ly cer ta in on es wh o wr ot e r epor tsor ar t icles have even been ment ioned. But to all those \ vho con t r ibu ted

in any way to th is grea t coopera t ive development en terpr ise, both in this

~. country and in England, these volumes ar e dedica ted.<:”cc,-

L. A. DUBRIDGE.G

9’4

an

2,I\’



EDITORIAL STAFF

HUBERT M. JAMES

NATHANIEL B. NICHOLS

RALPH S. PHILLIPS

CONTRIBUTING AUTHORS

C. H. DOWKER WARREN P. MANGEE

IVAN A. GETTING CARLTON W. MILLER

WITOLD HUREWICZ NATHANIEL B. NICHOLS

HUBERT M. JAMES RALPH S. PHILLIPS

EARL H, KROHN PETER R. WELSS

!.’J



Contents

FOREWORD BY L. A. DUBRIDGE . . vi

PREFACE . . . . . . . . . . . . ‘... ’.............,., ,.ix

CHAP. l SERVO SYSTEMS . . . . . . . . . . . . . . . . . . . ...1

l.l. In t roduct ion ...,.,. .,11.2. Types of Servo Systems. 2

1.3. Analysis of Simple Servo Systems 9

1.4. History of Design Techniques, 15

1.5, Per formance Specifica t ion s 17

CHAP. 2 . MATHEMATICAL BACKGROUND. .’ 23

INTRODUCTION . . . . . . . . . . ,. ..23

FILTERS . . . . . . . . . . . . ...24

21. Lumped-constant Filters 24

2.2. Normal Modes of a Lumped-constant Filter . 26

2.3. Linear F ilt ers. 28

THE WEIGHTING FUNCTION . . . . ...30

24. Normal Response of a Linea r Filter to a Unit -impulse Input 30

2.5. Normal Response of a Linear Filter to an Arbit rary Input 33

2.6. The Weight ing Funct ion 35

2.7. Normal Response to a Unit -step Input 37

28. Stable and Unstable Filters 38

THE FREQUENCY-RESPONSEUNCTION 40

29. Response of a Stable Filt er to a Sinusoidal Input 40

2 10. Frequent y-response Funct ion of a Lumped-constant Filt er 42

211. The Four ier In tegra l. 43

2 12. Response of a Stable Filt er to an Arbit rary Input 48

213. Rela t ion between the Weight ing Funct ion and the Frequency-

response Funct ion . . . . . ...48

2.14. Limita t ions of the Fourier Transform Analysis. 50

THE LAPLACE TRANSFORM.. . . . ..51

2.15, Defin it ion of the Laplace Transform 51

2.16. Proper t ies of the Laplace Transform 53

2.17. Use of t he Lapla ce Tr ansform in Solut ion of Linea r Differ ent ia l

Equat ions . . . . . . . . . . . . . . . . . . . . . ...56

‘v



CONTENTS

THE TI+ANSFEE FUNCTION . . . . . . . . . . . . . . . . . . . . . .

2.18. De6n it ion of t he Tr an sfer F un ct ion

2.19. Tr an efer F un ct ion of a Lumped-con st an t F ilt er

2.20. Th e Sta bilit y Cr it er ion in Terms of t he Tr an sfer F un ct ion

SYSTEMS WITH Feedback . . . . .

2.21, Cha ra ct er iza tion of Feedba ck Sys tems

222. F eedba ck Tr an sfer F un ct ion of Lumped-con st an t Ser vos

2.23, Th e F eedba ck Tr an sfer Locu s.

2.24. Rela t ion between the Form of the Transfer Locus and the Posi-

t ion sof th e Zer os an d P oles.

2.25. A Mapping Theorem . . .

226. Th e Nyqu ist Cr iter ion

227. Mu lt iloop Ser vo System s

CHAP. 3 . SERVO EL13MENTS

58

58

59

61

62

62

64

66

67

68

70

73

76

31.

32.

3.3.

34.

35.

36.

3,7.38.

39.

3.10,

3.11,

3.12.

313.

314.

315.

3.16.

In t roduct ion . . . . . . . . 76

Er ror -measur ing Systems 77

Synchros . . . . . . . . . . . . 78

Data System of Synchro Transmit ter and Repeat er 79

Syn ch r o Tr an sm it ter wit h Con tr ol Tr an sformer a s E rr or -mea su r-

ing System . . . . . . . ..82

Coer cion in Par a llel Synch r o Syst ems. 88

Rota t able Trans formers . 92Pot en t iomet er E r ror -mea su r ing Sys tems 95

Null Devices . . . . . . . . . . . . . . . . . . .101

Mot or s a nd Power Amplifier s 103

Modula tors . . . . . . . . . . . . . . . . ...108

Phase-sensit ive Detectors 111

Networks for Opera t ing on D-c Error Volt age. 114

Networks for Opera t ing on A-c Error Signal. 117

Opera t ion on @o—Feedback Filters. 124

Gear Tra ins . . . . . . . . . . . . . . . . . ...130

(’HAP.4. GENERAL DESIGN P1tIN(~IPLhX FOIL SERVOMECHANISMS. 134

4.1. Ba sic E qu at ion s. 134

42. Respon ses t o l~epr csen tt it ivc [n pu ts 138

43. Output Disturbances 145

4.4. Er ror (;oeficien t s ., 147

BASIC DESIGN TECHNIQUES AND APPLK:ATKJ N TO A SIMPLE SEIWO. 151

45. In t roduct ion . . . . . . .. 151

4.6. Differen t ia l-equa t ion Analysis 152

47. Tr an sfer -locu s An alysis. The Nyquist Diwgrarn 15848, At ten ua tion -ph ase An alysis 163

AmENUATION-PIiASERELATSONSHWSOR SERVO TRANSFER FUNCTIONS 169

Attenuat ion-phase Rclnt ionships . 169

(~on st ru ct io!] a nd In t e rpr rt :t tioli of At t cn un tion n nt l P ha se I )i;w

grams . . . . . . . . . ..171



4.11.

4.12.

4.13.

CONTENTS

Decibel–ph as e-a ngle Dia gr ams and F r equ en cy-r es pon se

acter ist ics . . . . . . . . .

Mult iple-loop Systems

Ot her Types of Tr an sfer Loci

12WALIZATION OF SEIWO LOOPS.

414. Genera l Dis cu ss ion of Equa lizn t iou

4.15. Lea d or Der iva twe Con tr ol

4.16. In tegra l Equa liza t ion

4.17. Equa liza t ion Using Subs idia ry I ,oop s .,.

APPLICATIONS . . . . . . . .

4.18. SCR-584 Au toma tic-t r a ck ing Loop .

4.19. Ser vo with a Two-phaw Mot or .,.

CHAP. 5. FILTERS AND SERVO SYSTEMS WITH PULSED DATA

5.1. In t roductory Remarks ,.

FILTERSWITH PULSEODATA.

5,2. The Weight ing Sequence

5.3. St abilit y of P ulsed F ilt er s.

5,4. S in usoida l Sequ en ces .

55. Filter Response to a Sinusoidrd Inpu t .5.6. The Transfer Funct ion of a Pulsed Filter

Char-

5.7. St abilit y of a P ulsed F ilt er , a nd t he Sin gu km Point s of It s Tr an sfer

Funct ion . . . . . . . . . . . . .

5.8. The Transfer Funct ion Interpreted as the Rat io of Genera t ing

Funct ions . . . . . . . . . . . .

FILTERS WITH CLAMPIN~. . . .

5 .9 . Th e Con cept of Clampin g.

5.10 . Tran sfer Fu n ct ion s of Som e Specia l Filters with Clam pin g

511 . Tran s fer Fu nct ion of a Filter with Clamping: Stab ility

5.12 . Sim plified Tran sfer Fu n ct ion s for la kT,l <<1

5.13. F ilt er s with Swit ches .

SERVOS WITH PULSED INPUT...

514 . Gen era l Th eory of Pu ls ed Servos : Feedback Tra rr s fer Fu nct ion ,

Stab ility . . . . . . . . . . . . .

515 . Servos Con trolled by Filter with Clampin g

5.16 . Clam ped Servo with Proport ion al Con trol

CHAP. 6. STATISTICAL PROPERTIES OF TIME-VARIABLE DATA.

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . .

6 .1 . Th e Need for Sta t is t ica l Con s idera t ion s . .

6 .2 . Ran dom Proces s an d Ran dom Series . .

6 .3 . Proba bility-d is tr ibu tion Fu n ct ion s

179

186

195

196

196

197

203

208

212

212

224

231

231

232

232

233

236

238240

242

244

245

245

246

249

251

253

254

254

257

259

262

262

262

266

268



CONTENTS

H~R~ONIC ANALYWS FOR S~ATIONARY ItANDO~ PROCESSES. 270

6 ,4 . Sta t ion a ry Ran dom Proces s . 27o

6 .5 . Tim e Averages an d En semb le Avera ges . 27 I

66. Cor rela t ion Fu n ct ion s 27367. Spectra lD enm ity . . . . . . . . . . . . . . . . . . . ...278

68. Th e Rela t ion between th e Correla t ion Fu nct ion s an d th e Spectra l

Den s ity . . . . . . . . . . . . . . . . . . . . . . ...283

69. Spect ra l Den s ity an d Au tocorrela t ion Fu nct ion of th e Filt ered

Sign a l . . . . . . . . . . . . . . . . . . . . . . ...288

ExAMPLEs . . . . . . . . . . . ...291

610. Radar Automat ic-t racking Example . 291

611. Purely Random Processes. 298

612. ATypical Servomechan ism Inpu t 300613. Poten t iometerN oise. . . . ,305

( ‘lIAF, 7, RMS-ERROR CRITERION IN SERVOMECHANISM DESIGN 308

7.1.

72.

73.

74.

7.5.

76.

7.7.

7.8.

79.

Prelimina ry Discussion of the Method 308

Mathemat ica l Formula t ion of the RMS Error . 312

iNatu re of tbe Transfer F unct ion . 315

Reduct ion of the Er ror Spect ra l Density to a Conven ien t Form 317

ASimple Servo Problem. .321

Integra t ion of t he Er ror Spect ra l Density. 323

Min imizing the Mean-square Er ror . 325

Radar Automat ic-t racking Example 328

Evalua t ion of the In tegra ls . 333

CHAP. 8 . APPLICATIONS OF THE NEW DJ 3SIGN METHOD 340

8,1 . In pu t Sign a l an d Nois e. .340

SERVO WTH PROPORTIONAL CONTROL 342

8,2. Best Cont rol Parameter . 342

8.3. Proper t ies of the Best Servo with Propor t iona l Cont rol. 3458,4. Ser vo wit h P ropor tion al Con tr ol, l’~ = 0...........347

TAGHOh lETER FEEDBACK CONTROL. 348

8 .5 . Mean -squ a re Error of Ou tpu t 348

8 .6 . Idea l Ca s eof In fin ite Ga in .349

8 .7 . Bes t Con trol Pa ram eters for Fin ite Am plifica t ion s 352

88. Decibel–log-frequ en cy Diagram 356

89. Nyqu ia tDiagram . . . . . ..359

MANUAL TRACKING ..,..... . . . . . . . . . . . . . . . . . 360

8.10. In t roduct ion . . . . . . . . . . . . . . . . . . . . . . ..36o

811. The Aided-t racking Unit 361

8.12. Applica tion of t he Rms-er ror Cr it er ion in Det erminin g t he Best IAided-t racking Time Constant . 363

ApPErin Lx A. TABLE OF INTEGRALS. 369

‘INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 I



CHAPTER 1

SERVO SYSTEMS

BY I. A. GETTING

1.1. In t roduct ion . -It is near ly ashardfor pract it ioners in the servo

ar t to agree on the definit ion of aservo asit is for a group of theologians

to agree on sin . It has become genera lly accepted, however , tha t a servo

system involves the cont rol of power by some means or other involving acompar ison of the outpu t of the cont rolled power and the actua t ing

device. This compar ison is somet imes refer r ed to as feedback. There

is a la rge va riet y of devices sa tisfyin g t his descr ipt ion ; befor e a tt empt in g

a more formal defin it ion of a servo, it will be helpful to consider an exam-

p le of feedback .

One of the most common feedback systems is the au tomat ic tem-

pera tu re con trol of homes. In this system, the fuel used in the furnace is

the sou rce of power . This power must be cont rolled if a reasonably

even tempera ture is to be main ta ined in the house. The simplest way

of cont rolling this sou rce of power would be to turn the furnace on, say,

for one hour each morn ing, a ft ernoon , and evening on autumn and spr ing

days and twice as long dur ing the win ter . Thk would not be a par t icu-

lar ly sa t isfactory system. A t remendous improvement can be had by

providing a thermosta t feedback tha t turns the furnace on when the

tempera ture drops below, say, 68° and turns the furnace off when the

tempera ture r ises above 72°. This improvement lies in the fact tha t

the outpu t of the power source has been compared with the input (a

standard tempera ture set in the thermosta t ), and the difference between

the two made to cont rol the sou rce of power—the furnace.

A more colloquia l name applied to such a system is “a follow-up

system.” In this example, the opera tor set s a tempera ture, and the

temper ature of the house, in due course, follows the set t ing.

The term “follow-up system” grew out of the use of servo systems for

t h e amplifica tion of mechan ica l power . Somet im es t he par t of t he system

doin g t he “followin g” was remote from the cont rolling poin t ; suchsystems wer e th en called “r em ot e con tr ol. ” Remot e con tr ol ca n in volve

t remendous amplifica t ion of power ; in cer ta in cases remote cont rol may

be required by physica l condit ions, a lthough adequate power is locally

available. Let us r esor t t o examples again . On a large naval ship it is

necessa ry to t ra in and eleva te I&in. guns. It is necessa ry to do this

1



2 SElt VO S YS1’h ’MS [SEC.12

continuously to compensa te for the pitch, the roll, and the yaw of the

ship. Such a gun and tur ret may weigh 200 tons. It is obviously

impossible to manipula te such a gun manually; power amplifica t ion is

required. ‘l’he opera tor turns a handwheel, and the gun mount is madeto rota t e so that it s posit ion agrees with the posit ion of the handwheel.

This is a follow-up system—the gun mount follows the handwheel. In

pract ice, it is possible to place the handlvhecl either direct ly on the gun

mount or a t a remote poin t , say in the gunnery plot t ing room below deck.

In the la t ter case the systcm becomes one of remote cont rol character -

ized by t r emendous amplifica tion . On the other hand, in the same ship

a ta rget may be t racked by posit ion ing a telescope at tached to the

director. There is adequate power available in the director , bu t the

posit ion of the director may need to be repea ted in the computer below

deck. It is very inconvenient to car ry rota t ing shafts over long dktances

or through water t igh t bulkheads; therefore resor t is had to a remotely

con t rolled follow-up. , The compl~tm input , shaft is made to follow the

director ; bl[t whereas the director had available many horsepolver , the

inpllt servo in the computer may bc only a few \ vat t s. Temperature

regula tors, remote-cent rol Ilnit s, find pwver dr ives a re all examples of

servo sys tems.

Dcjinit ion .—A ser vo syst em is :~ combin at ion of element s for t he con -

t rol of a source of polrer in which th t>mltput of the systcm or some func-

t ion of the outp(lt is fed back for compar ison }~ith the input and the

differ en ce between t hese quant it ies is used in con tr olling t he power .

1.2. Types of Ser vo Syst ems. —Ser vo syst ems in~rolvin g mechanical

mot ion were first used in the cont rol of underwater torpedoes and in the

automat ic steer ing of ships. In both cases a gyroscope was used to

determine a direct ion. Power was furn ished for propelling the torpedo

or the ship. A por t ion of this power ~vas a lso available for steer ing theship or torpedo through the act ion of the rudders. React ion on the

rudders required power amplifica t ion between the gyroscope element

and the rudder . Neither of these systems is simple, because in them two

sou rces of power n eed t o be con trolled: (1) power for act ua ting t he r udders

and (2) power for actually turn ing the ship. In systems as complica ted

as these, the problem of stability is very importan t . In fact , the most

common considera t ion in the design of any servo system is that of

stability.Consider a ship with rudder hard to por t (left ). Such a ship ]vill

tu rn to por t . If the rudder is kept in this posit ion t ill the ship arr ives

at the cor rect heading and is then restored to st ra ight -ahead posit ion ,

t he ship will cont inue t o t urn left because of its angular momentum about

it s ver t ica l axis. In due course the damping act ion of the water will

stop this rota t ion, but only aft er the ship has overshot the cor rect



SEC.1.2] T YPEL’3 OF S ERVO S YS TEMS 3

direction. If the remainder of the servomechanism opera tes proper ly,

the gyrocompass will immedia tely indica te an er ror t o the left . If the

power amplifier then forces the rudder ha rd to sta rboard, the restor ing

torque of the starboard rudder will limit the over shoot but under thecondit ions descr ibed will a lso produce a second overshoot , th is t ime to

starboard. It is en t irely possible tha t these oscilla t ions from left to

r igh t increase with each successive swing and the steer ing of the ship

becomes wild. It is impor tan t to note that th is instability is closely

rela ted to the t ime lags in the system. The probability of get t ing in to

an unstable situa t ion becomes mater ia lly redu ced as the react ion t ime of

the rudder to small er ror s in heading becomes ext remely shor t . The

stability can also be increased and er ror s r edu ced if the rudder displace-

ment is made propor t ional to the heading er ror (propor t iona l con t rol).The behavior of the system can be improved even fur ther by ant icipa-

t ion cont rol. Ant icipa t ion in this applica t ion implies tha t in the set t ing

of the rudder , use is made of the fact tha t the gyrocompass er ror is

decreasing or increasing; it may go as far as t o take into account the

actual ra te a t which the er ror is increasing or decreasing. Then, as

the ship is approaching the cor r ect heading, ant icipa t ion would indica te

the necessity of turning the rudder t o starboard, even though the er ror is

st ill t o por t , in order to overcome the angular momentum of the ship.This deflect ion of the rudder should be gradually reduced to zero as the

cor rect h ea din g is r ea ch ed.

The examples given above seem to imply that mechanica l servo

systems are a product of this cen tury. Act ua lly, h uman ph ysica l mot or

behavior is la rgely controlled as a servo system. A person reaches for a

sa ltcella r . He judges the distance between his hand and the sa ltcella r .

This dist ance is the “er ror ” in the posit ion of his hands. Through his

n er vou s system and subconsciou s mind this er ror is u sed t o con tr ol muscu-

lar mot ion, the power being der ived from the muscular system. As the

dist ance decreases, der iva t ive con trol (an t icipa t ion) is brought in to

play through subconscious habit , and overshoot ing of the hand is pre-

ven ted. A more illust ra t ive example is the process of dr iving a ca r . A

person who is just learning to dr ive genera lly keeps the ca r on a road by

fixing his a t t en t ion on the edge of the road and compar ing the loca t ion of

this edge of the road with some object on the ca r , such as the hood cap.

If th is distan ce is t oo small, t he lear ner r ea cts by tu rnin g t he st eer in g wh eel

to the left ; if it get s t oo large, he react s by turn ing the steer ing wheel tothe r igh t . It is cha racter ist ic of the lea rner tha t his dr iving consists of a

con tin uou s ser ies of oscilla t ions abou t t he desir ed posit ion . The more

carefu lly he dr ives, tha t is, the grea ter his concen t ra t ion , the h igher will

be the frequencies of his oscilla t ions an+ the smaller the amplitude of

his er rors. As the dr iver improves, he in t roduces ant icipa t ion , or der iva -

1



4 S ERVO S YS TEMS [SEC.1.2

t ive con t rol. Inth iscon&t ion adfiver takes in to considera t ion the ra te

at which he is approaching KIs cor rect distance from the edge of the road,

or , what is equiva len t , he not ices the angle between the direct ion of car

t ra vel a nd t he dh-ect ion of t he roa d. His con t rol on t he steer ing wheel is

then a combina t ion of disfdacement con t rol and der iva t ive cont rol. His

oscilla tions become lon g or n on exist en t, a nd his er ror s sm aller .

So long as the road is st ra ight , a dr iver of this type, act ing as a servo-

mech an ism , per forms t oler ably well. H owever , a ddit iona l fa ct or s come

in to play as he approaches a bend in the road. Chief among these is the

displacement er ror resu lt ing from the tendency of the opera tor to go

straight. The er ror due to cont inuous uniform curva ture of the road can

be t aken ou t by essent ia lly establishing a new zero posit ion for the steer -

ing wheel. A dr iver per forming in this manner exhibits ‘‘ in tegra l con-t rol. ” Actua lly, a human being is not a simple mechanism, and he has

available in th is instance information of other types. His dr i}-ing is a

complicat ed combina tion of propor t iona l, der iva ti~-e, and in t egra l con-

t rol, mixed with nonlinear elements and knowledge of the direct ion in

which the road is going to turn . This for ekn owledge is somet im es

refer red to as ant icipa t ion and is somet imes confused with der iva t ive

cont rol. Th e example ser ves never theless t o illust ra t e t he basis of ser vo-

m ech an ism s in gen er al. The power to be con t rolled in this case was

der ived from the engine of the car . The inpllt t o the system vas the

actual path of the road; the outpllt vws the posit ion of t hc cm; and the

er ror mechanism in which the output and input were compared was

t he human oper at or .

The human opera t or is a very common clement in many servo systems.

Human elements a re used in t r acking ta rgets for fire cont rol (see Chap. 8)

in con tr ollin g st eam en gin es, in con tr ollin g set tin gs on a ll sor ts of mach ili-

ery. The human opera tor is sometimes refer red to as a biomechanica l

link; much can be learned of his response by the applicat ion of servotheory.

The term servo system is not commonly used when the systcm in-

volves a human opera tor . It is sometimes rest r icted fur ther to include

cont rol only of systems that involve mechanical mot ion . For example,

the automat ic volume cont rol of a home radio reccivcr is a feedback

system, in which the output level of the receiver is compared with the

desired level (usua lly a bias volt age) and a diffcmncc, or a combinat ion of

the differences, set s the gain of the rcceivcr . This closed loop meets all

the requirements of a servo systcm, but it dot s not involve mechanica l

motion. We shall apply the t erm servo systcm to such devices but shall

r est r ict t he t erm “ser vomech an ism ” t o ser vo syst em s in volvin g mech an i-

ca l mot ion . It is, in genera l, t rue that the theory of servomechanisms is

ident ical with tha t of feedback amplifiers as developed in the com-



SEC. 1.2] TYPES OF SERVO SYSTEMS 5

munica t ions field. Ther e a re cer ta in pract ical differences which at t imes

make this similar ity not quite apparen t . Servomechanisms may involve

the con t rol of power through the use of the elect ron ic amplifier , in which

the power is furnished as pla te supply for the vacuum tubes; th is is very

similar to a feedback amplifier . On the other hand, a servomechanism

may include on ly hydraulic devices, a pump furnishing oil a t a high

pressure being the source of power . The con t rol of thk oil flow may be

accomplished by hydraulic valves. Mathemat ica lly the elect ron ic

amplifier and the hydraulic system may be very similar ; but in the

physica l aspect s and in the frequencies and power levels involved the

two may be (but are not necessar ily so) quite differen t . A hydraulic

system may be able to respond to frequencies up to 20 cps; a feedback

amplifier may be built t o opera te up to frequencies as high as thousands

of megacycles. Hydraulic systems have been made in power levels up

to 200 hp; feedback amplifiers are genera lly used in ranges of power of a

few wa tt s t o m illiwa tt s.

In pr evious examples, r efer en ce \ &asmade to the use of servo systems

as power amplifiers and as a means of remet e cont rol. Ser vo syst em s

per form t wo oth er major funct ions: (1) as t ra nsformers of informat ion or

da ta from one type of power to another and (2) as null inst ruments in

compu t ing mechanisms.It is somet imes desired to change elect r ica l voltage to mechanical

mot ion wit hou t in t r edu cin g er ror s a rising fr om va ria tion s of loa d or power

supply. Such a problem can be solved by the use of a servomechan ism.

For example, an elect r ic motor is made to rota te a shaft on which is

mounted a potent iometer . The voltage on this poten t iometer can then

be made to vary as any arbit ra ry funct ion of shaft posit ion . This

ou tpu t voltage is compared with the or igina l elect r ica l voltage, and a

dh lerence or some funct ion of it made to con t rol t he elect r ic motor .

Th is is a servomechanism .

It should have been clear tha t in all t he preceding examples a com-

par ison was made between outpu t and input and tha t the source of power

was so con t r olled as to r educe the differ ence between t he outpu t and input

to zero. In other words, a ll servo systems are null devices, somet imes

ca lled er ror -sens it ive devices. The advantages of such a system from a

sta ndpoint of compon en t design will be indica ted in t he n ext sect ion .

A null device can be made to solve mathemat ica l equa t ions such as

ar e involved in the fire-con t r ol problem. Figure 1.1 is a schemat ic for t hemech an iza tion of t he fir e-con tr ol pr oblem in on e dim en sion . Th e fu tu re

range R depends on the presen t range r , on the speed of the ta rget in

range dr/dt, and on the t ime T required for a bullet t o t r avel from the gun

to the ta rget . The t ime of flight T of the bullet is some funct ion of the

fu tu re range R—a funct ion that is genera lly not available as a simple




analyt ic r ela t ion, but on ly fr om ballist ic tables. Th e r ela tion s between

these quant it ies may be expressed thus:

dr

‘= ’-+TZ’T = j(R).

(1)

(2)

It is obvious that fu ture range cannot be obta ined without knowledge of

the t ime of fllght and that t ime of flight cannot be known without know-

ing fu tu re r an ge. It is n ecessa ry t o solve t hese equat ion s simultan eously.

Powert-i,

Ampllf!erI

I

7 CamR’=r+T$

FIG.1.1.—Servomecha nismn a comput er.

In Fig. 1.1, range is in t roduced at the lower left -hand corn er . The der iva -

t ive of range is t aken and mult iplied by an arbit rary value of t ime of

fllght T. The product is added to the observed range, to give a hypo-

thet ica l fu tu re range R’. This hypothet ica l fu tu re range actua tes the

cam giving the t ime of flight T’ cor respon din g t o t hk h ypot het ica l fu tu re

range. If T’ were equal t o T, t he init ia l assumpt ion of the t ime of fligh t

wou ld h ave been cor r ect ; t his, of cou rse, wou ld be a cciden ta l. I n genera l,

the assumed va lue T will differ fr om T’. The difference c = T – T’

ca n be fed int o an amplifier supplied fr om an in depen den t sou rce of power ,

and this amplifier used to dr ive the motor a t t ached to the T shaft . If

now T’ is gr ea ter t ha n T, t he amplifier will apply a volt age to the motor

th at will dr ive T t o smaller values, tending thus t o r edu ce the differen ce

T – T’. When this clifferen ce will have reached zero, the fu ture range

R and the t ime of fllght T will cor respon d t o t he obser ved r an ge and r an ge

ra te .

This computa t ion could be done without a servo by having a dir ect

mechanica l connect ion between the outpu t of the cam at T’ and the inputt o t he mult iplier a t T. A lit tle t hou gh t will sh ow, h owever , t ha t pr act ica l

considera t ions would limit the usefulness of th is a r rangement to simple

funct ions and to devices in which the accuracy would not be dest royed

by the loads imposed. In the above example, the servo has two impor-

tant funct ions: (1) It in t roduces a flexible link between the cam and the



SEC.1.2] TYPES OF SERVO SYSTEMS 7

mult iplier , and (2) it prevents the feeding of data in a direct ion opposite

to that shown by the ar rows.

Equat ions (1) and (2) can be writ ten in the more general form

g(l?,z’) = o,(3)h(R, T) = o. (4)

In theory, it is a lways possible to solve such a set of simultaneous equa-

t ions by elimina t ing one variable. If, however , g and h are complica ted

funct ions or implicit ly depend on another independent variable (say

t ime), the solut ion by analyt ic methods may become difficult . It is

a lways possible to have recourse to a servo computer of the type illus-

t ra ted in Fig. 1.1.

Servomechanisms can be classified in a var iety of ways. They can be

cla ssified (1) as t o use, (2) by t heir mot ive chara ct er ist ics, and (3) by t heir

cont rol character ist ics. For example, when classified according to use,

they can be divided in to the following: (1) remote cont rol, (2) power

amplifica tion , (3) indica tin g in st rum en ts, (4) con ver ter s, (5) compu ter s,

Servomechanisms can be classified by their mot ive character ist ics as

follows: (1) hydraulic servos, (2) thyrat ron servos, (3) Ward-Leonard

con tr ols, (4) amplidyn e con tr ols, (5) two-ph ase a -c ser vos, (6) mech an ica l

torque amplifiers, (7) pneumat ic servos, and so on. In general, all these

syst em s a re ma th emat ica lly simila r. Considera t ions as to choice of thet ype of m ot ive power depen d on loca l circumsta nces a nd on t he par t icu lar

character ist ics of the equipment under considerat ion . For instance,

amplidyne cont rols a re useful in a range above approximately + hp.

Below the +-hp range the equipment becomes more bulky than thyra t ron

or two-phase a-c cont rol unit s. On the otner hand, drag-cup two-phase

motors a re ext remely good in the range of a few mechanical wat t s because

of their low iner t ia but become excessively hot as the horsepower is

increased above the &hp range. Pneumatic servos are ext remely usefu l

in . a ircraft cont rols and especia lly in missile devices of shor t life where

st or age ba tt er ies a re h ea vy compa red wit h compr essed-a ir t anks. Pneu-

matic servos are also used in a la rge number of industr ia l process cont rol

applications.

For the purposes of this book, the most impor tan t classificat ion of

servomechanisms is tha t according to their cont rol character ist ics.

Hazen’ has classified servos in to (1) relay-type servomechanisms, (2)

defin it ecor rect ion ser vomech an ism s, a nd (3) con tin uou s-con tr ol ser vo-

mechanisms. The relay type of servomechanism is one in which the fu llpower of the motor is applied as soon as the er ror is la rge enough to oper -

ate a relay. The defin ite-cor rect ion servomechanism is one in which the

I H. L. Hazen, “Theory of servomechan isms,” J . Frank lin Zn.d., 218, 279(1934).



8 S ERVO S YS TEMS [SEC.12

power on th e m ot or is cont rolled in fin ite steps at defin ite t ime interva ls.

The con tin uous-cont r ol servomechanism is on e in which the power of th e

motor is cont rolled cont inuously by some funct ion of the er ror . This

book concerns it self with the cont inuous type of con t rol mechanism.

All th ree types have been used extensively. The relay type is genera lly

the most economical to const ruct and is useful in applicat ions where

cr ude follow-u p is r equ ir ed. It has, h owever , been sllcccssfu llv a pplied

with high per formance output , even for such applicat ions as inst ruments

a nd power dr ives for dir ect or s. Relays can be made to act very quickly,

tha t is, in t imes shor t compared with the t ime constants of the motor .

Under these condit ions the relay type of servo can bc made to approach

cont inuous cont rol so closely that no sharp line can be drawn. In Chap.

5, an analysis is made of the limita t ion on cont inuolls-cont rol servo-

mechanisms ar ising from the use of in termit tent data . The second

type of servomechanism, t he fin ite step cor rect ion , is used pr incipally in

instruments.

Th e con tin uou s-con tr ol system s t hem selves ca n be fu rt her classified

according to the manner in which the er ror signal is used to con t rol the

motor : propor t iona l cont rol, in tegral cont rol, der iva t ive cont rol, an t i-

hunt feedback (subsidiary loops), propor tiona l plus der ivat ive con trol,

and so on. The study of these differen t methods of con t rol is one of themajor tasks of th is book.

Befor e con tin uin g t he discu ssion on ser vomech an ism s it is wor th wh ile

to consider the terminology as it has developed over the past few years.

The defin it ion given in the fir st sect ion requires that a servo system have

the following proper t ies: (1) A source of power is cont rolled and (2)

feedback is providccf. This defin it ion applies equally well to four fields

of applied en gin eer in g, wh ich h ave developed mor e or less con cu rr en tly:

(1) feedback amplifiers, (2) automat ic cont rols and regu la tor s, (3)recording inst ruments, and (4) remote-cont rol and power servomecha-

nisms. As implied by the first sentence of this book, it is difficult to find

unique defin it ions segrega t ing these four fields. It is genera lly agreed

tha t a servomechanism involves mechanical mot ion somewhere in the

system; there is agreement that the power dr ives on a gun tu r ret con-

st itu te a servomechanism. Tempera ture regulators a re oft en excluded

fr om t he class of ser vomech an ism s an d classified as au tomat ic r egu la t or s

or con t rol inst ruments, even though such mechanical elements as relaysand motors may be used. If there is any ru le tha t seems to apply, then

perhaps it is that in a servomechanism the element of grea t est t ime lag

should be mechanica l and, in gener al, tha t the ou tput of th e system should

be mechanical. For the purposes of this book, the term servo system

will include all types of feedback devices and the t erm servomechanism

will be r eser ved for ser vo syst em s in volving mecha nica l ou tpu t.



SEC.1.3] ANAL YS IS OF S IMPLE SERVO SYSTEMS 9

1.3. Analysis of Simple Ser vo Systems.—The purpose of th is sect ion

is to presen t simple analyses showing var ious methods of approach to the

mathemat ica l descr ipt ion of servo systems; in subsequent chapter s a

formal and reasonably complete analysis is given . The mathemat ica l

tools used in th is fir st t r ea tment have been der ived from the genera l fieldof opera t ional ca lcu lus and are, th er efor e, limited t o the considera t ion of

lin ea r syst em s, t ha t is, syst em s descr ibed by lin ea r differ en tia l equ at ion s

Input ,

Load

FIG.1.2.—Open-cycleontrolsyetem. I:lcJ .1.3.—Simplelosed -cyclecontrol system,

with constan t coefficien t s.’ This limita t ion rest r ict s on ly a lit t le the

u sefuln ess of t he analysis, inasmuch as m ost pr act ica l ser vom ech anism s

either a re linear or can be approximated sufficient ly closely by a linea r

represent a t ion.

The advantages of a servo systcm in cont rast t o an open -cycle system

are illust ra ted by even the simplest type of servo system. F igure 1.2

sch emat ica lly sh ows a n open -cycle con tr ol syst em . If t he ha ndwh eel H

is t ur ned t hr ou gh an an gle 81,th e sou rce of exter na l power is so con tr olled

through the amplifier tha t the motor rota tes the load shaft L th rough an

angle Oo. In a per fect system,, % would at all t imes be equal to 0,. This

would require, of cou rse, tha t a ll the der iva t ives of 00 were instan tane-

ou sly equal t o t he der iva tives of fl~, Wer e t hese condit ion s t o be sa tisfied,

the character ist ics of the power supply, the amplifier , and the motor

would have to be held constant a t all t imes, or compensat ion devices

would have to be incorpora ted. The amplifier must be insensit ive to

power fluctuat ions; th e torque character ist ics of th e motor must be inde-

pendent of t empera ture; the system must be insensit ive to load var ia-

t ions; and so on . In genera l, these requ irements cannot . be met . The

most effect ive example of t he open -cycle system is a va cuum-tu be ampli-

fier . It is possible t o make a vacuum-tube amplifier in which the outpu tis a lways propor t iona l to the input with in limits of load and power -line

fluctuat ion. This is, however , a lmost a unique example; it is near ly

1M. F. Gardnerand J . L. Barnes,Transientsn LinearSystems,Wiley, NewYork,1942; E . A. Guillemin ,Communica t ion Networks, Vols . 1 and 2, Wiley, New York,1935;V. Bush,Operational Circui t Analysis , Wiley, New York, 1929.



10 S ERVO S YS TEMS [SEC,13

impossible to find a power-con t rol mechanism in which the cycle is not

closed mechanica lly, elect r ica lly, or th rough a human link,

In Fig. 1.3 is shown a schemat ic of a simple closed-cycle cont rol

system. It differ s from the open-cycle cont rol system in that the output

angle th is subt racted from the input angle 61 to obta in the er ror signal c.

It is th is er ror signal which is used to control the amplifier . F igure 13

represen ts inverse feedback, or , as it is somet imes called, nega t ive feed-

back. Let K, be the gain of the amplifier . Then the output of the

amplifier V is given by

V = K,~. (5)

Assume that the motor has no t ime lag and a speed at all t imes propor -

t ional to V:

de.

dt= + KmV. (6)

For simple pr opor tion al con tr ol,

~=gl—go (7)

is used direct ly as input to the amplifier . Combining Eqs. (5) and (6),

we getJ 7

z=

or

where K = K, Km. If we

0,0 sin at , we getdo.

dt

1 deo

o,–eo=+— KIK.~’ (8)

~~++eo=ol, (9)

now consider a sinusoidal input 0, equal to

+ K% = KO,O sin t it . (10)

This equat ion will be recognized as similar to the equat ion of an h!C-cir -

cu it dr iven by an alternat ing genera tor , which is, on wr it ing q for the

charge,

R#+&=VOsinuL (11)

The steady-sta te solut ion for the RC-circu it can be wr it t en

q =* sin (of – ~); (12)

where, v,go= , ,, tan $ = +. RC (13)

dR’ + $2

and ~ is the angle by which the charge lags tJ e voltage. Similar ly,



SEC.1.3] ANAL YS IS OF S IMPLE SERVO SYSTEMS

t he solut ion of Eq. (10) is

1

00 = 1900in (d – ~); (14a)

where

(14b)

If K is la rge compared with co, th is may be expanded by the binomial

t heorem as

’00=

’14-a(15)

We see immedia tely that provided only K is much la rger than u , the out -

put 60 will be essent ia lly equal to the input 0, in magnitude and phase;

the accuracy of the follow-up requires only making sufficien t ly high

t h e velocit y con st an t Km of the motor and the gain K 1 of the amplifier .

In cont rast to the open-cycle system, it is not necessa ry to use’h com-

pensated amplifier or a motor insensit ive to load in such a system. These

are th e chief and fundamenta l advantages of inverse feedback.

Equa t ion (15) implies cer ta in limita t ions on the system shown in

Fig. 1.3. In any real amplifier the ga in will be high until sa tura t ion

sets in or up t o a defin ite frequ ency. Likewise, t he mot or speed con st an t

will drop off if the speed is increased or torque exceeded. In genera l,

t herefore, ther e will be an upper va lue to a beyond which the system will

not funct ion . Actua lly, all motors have t ime constan ts, tha t is, exhibit

iner t ia l effects, and it is necessa ry to consider this t ime constan t in the

analysis.

The preceding analysis descr ibed the st eady sta te of the servo illus-

t ra ted in Fig. 1.3 when the input is a sine wave. Let us now consider the

t ran sien t beh avior of t he same pr opor tion al-con tr ol syst em for t ra nsien t

solu t ion ; instead of a sine-wave input let us assume tha t the input 19~s

zero for a ll t imes up to h and then suffers a discont inuous change to a

new constan t A for all t imes grea ter than to. In shor t , a st ep funct ion is

applied as the input to the servo. Th e differen tia l equ at ion is

de.~ + K% = Ko, (16)

and it is to be solved for

e, = o, t < to,

8,= A, t > to;1 (17)

the solut ion is& = ~[1 – ~-K(L–tO)], t > t,,, (18)

as can be shown by subst itu t ing in Eq. (16). The step funct ion is shown




in Fig. 1”4 as a dot ted line, and the response is shown as a full line. It is

clea r tha t the output approaches the input as the t ime beyond toncreases

wit h ou t lim it . The la rger the va lue of K, tha t is, the la rger the gain of

the amplifier and the la rger the velocity constan t of the motor , the morequickly will the output approach the input . The er r or a t any t ime t

is the clifference between the dot t ed line and the full line. It fa lls to l/e

of its init ia l va lue .4 in a per iod l/K.

It is evident that the t ransien t analysis and the steady-sta te analysis

display the same genera l fea tures of the system. For example, we

see immedia tely from the t ransien t solu t ion tha t if the input wer e a sine

e

I e,

‘E--C t

FIG.1.4.—ltmponseof :Lsimplesvvvos>st e,,,t o a ,tcL)fu,,ct ion

wa ve of fr equ en cy j t he ou tpu t, N-ou ld follm~ it C1OSC1Yn ly if t hr ]wr ioci

of t he sin usoid wer e gmat .r r t ha n 1/1<, t ha t, is, ~ smnllcr t ha n K.

The solu t ion of I?q. (16) for an arbit rary input ran be \ \ -r it t{,n in

t erms of a “\ vcight ing funct ion . ” If d, is any input beginning fit :~finit (

t ime, the output will be

I

.

co(t ) = K @,(/ – T)(?-K’(/ T , (If))

J rJ

as can be ver ified by subst itu t ing in to Eq. (16). Indescr ibed by Eq. (17), the outpllt ran be comput r{l

/

(–10

90(1) = .IKC-K7 (7’T

o

= .4[1 – ~–x(f -t oll,

a s fou nd bef or c. ‘1’h r i’lln ct ior r

JV(T) = xc-”~’

t he ca se of t hr in])llt ,as

(20)

(21)

is ca lled the weight ing l’unmion. l’hysicall y Eq. (19) means that a t any

t ime t the output is equxl to a sum of cont r ibu t ions from the input a t

a ll past t imes. Each clement of the input appears in the outpu t mult i-

plied by a factor W(T) dependent on the t ime in t ,erval r between the

presen t t ime t and the t ime of the input under considera t ion . W(r) thus



SEC. 1.3] ANALYS IS OF S IMPLE SERVO SYS TEMS 13

specifies t he weight wit h which t he input a t any past moment cont r ibu tes

to the present output . It will be noted that in th is example the weight -

ing fu nct ion is an expon en tia l. When the t ime in terva l between input

and ou tput is grea ter than l/K, the cont r ibu t ions to the outpu t will besmall; t he r emot e past input will have been essent ia lly forgot ten .

To summarize, the simple system including propor t iona l con t rol, a

linear amplifier , and a motor with no t ime lag connected as a servo

system with nega t ive feedback has been anal yzed (1) as a st ead y-st a te

problem, (2) as a t ransien t problem, and (3) as a problem involving the

weighing of the past h istory of the

input . All of these analyses give

essent ia lly the same resu lt that theou tput is equal to the input for fre-

qu en cies below a cr it ica l va lu e equ al

approximately to K. One should

expect that ther e would be mathe-

matica l procedu res for going from

on e t ype of analysis t o anot her , and

su ch is in deed t he case.

Another in terest ing type of

t ransien t input oft en used in servo

a na lysis is t h e discon tin uou s ch ange

in the speed of e,. Such as input is

shown in Fig. 1.5 by the dot t ed line.

t h e form

to tF1~.1S.-lbspo!lse of a si,,,pleservos>+

tem to suddencha ngein velocity,

Th e differ en tia l equ at ion no]v t ak es

+ ~+ + % = z3(f – t“) for t > f,); (22)

=0 for t < to.

The output a fter t ime t , is

[00 = B (t — t,,) — * [1 — e–~(t–~f,’1] . (23)

This is shown as a full line in Fig. 1.5. It is obviou s t ha t as t ime in cr ea ses,

the velocity of the outpu t will eventua lly equal the velocity of the input ;

t her e will, h owever , be an angular displa cem ent or velocit y er ror between

them. The ra t io of the velocity to the er ror approaches the limit K as

t ~ cc ; th is is readily seen from the equat ion

de,

z BK K—e B[l – e-~(~-k’)] = 1 _ ~-K(f-to)”

(24)

This coefficient K, a product of the gain of the amplifier KI and the

velocity constan t of the motor K m, is ca lled t he velocit y-er ror con st an t



14 S ERVO S YS TEMS [SEC.13

and will her ea fter be wr it t en K.. If the loop were opened, it would be

the ra t io of velocity to displacement a t the two open ends. It is obvious

tha t the velocity er ror in th is simple system could be reduced by increas-

ing K. In Chap. 4 we shall see tha t th is er ror can be made equal t o

zer o by in tr odu cin g in tegr al cen t r ol.

As indica ted previously, any motor and its load will exh ibit iner t ia l

effect s, and Eq. (6) must be modified by adding a term. The simplest

ph ysica l mot or ca n be descr ibed by a differ en tia l equ at ion of t he form

(25)

where J is the iner t ia of the motor (including tha t of the load refer r ed to

the speed of the motor shaft ); j- is the in terna l-damping coefficien t

result ing from viscous fr ict ion , elect r ica l loss, and back emf; and Kt is

the torque constant of the motor . If t her e is no accelera t ion , the mot or

will go at a speed such tha t the losses a re just compensated by the input

V. This value of dt?o/dt is determined by the rela t ion

jm ~ = +K,V. (26)

Subst itu t ing from Eq. (6), we see tha t the in terna l-damping coefficien t

j~ can be wr it t en as-.

(27)

Thus the in terna l-damping coefficien t of a motor can be computed by

dividing K,, the sta lled-torque constan t (say, foot -pounds per volt ),

by Kmj the velocity constan t of the motor (say, radians per second per

volt ). Equat ion (25) can be rewr it t en in the form

T. = J ‘F”;t

(29)

the motor iner t ia appea rs on ly in the t ime constan t Tm. In shor t , the

ch ar act er ist ics of t he mot or ca n be specified by st at in g it s in ter na l-damp-

ing coefficien t and its t ime constant ; these can be determined by exper i-

ment ally measu ring sta lled t or que as a fun ct ion of volta ge, run nin g speed

as a funct ion of voltage, and iner t ia .

If we now use Eq. (28) instead of Eq. (6) as the different ia l equat ionrepresen t ing the behavior of the motor , we can find the outpu t due to

an ar bit r ar y input by solving t he differen tia l equat ion

VI

“-e”= ’=%= KIKm( )T.$$+d$ ; (30)



SEC.1.4] HIS TORY OF DES IGN TECHN IQUES 15

(31)

where Kv equals K IKm. This equat ion is similar in form to the dif-

ferent ia l equ at ion of an LRC ser ies cir cu it dr iven by an ext er na l a lt er na t-

ing genera tor .

J ust as in the case of Eq. (6), we can wr it e the specific solut ion of th is

equa t ion . It will perhaps suffice t o get the genera l solu t ion of the

equa t ion that holds when 01 = O:

TdlO+d~O+Kdo=o.m dt , d~ v

Letting

pl=a+juo and ‘pt=a.

the solu t ion can be wr it ten in the form

I% = aep” + bepzf,

wh er e th e p’s must sa t isfy

Kup’+$m+~m=o.

The solu t ion of this is

J,P=– L 1 L 4;%

2T. ‘z 2’: .

(32)

jcoo,

(33)

(34)

(35)

The na ture of the solu t ion depends on whether 4K,K~T~ is less than,

equal to, or grea t er than 1. In the fir st case the radica l is rea l and the

solut ion consists of overdamped mot ion (tha t is, a < l/2T~). In the

secon d case th e outpu t is cr it ica lly damped, and in the last case th e outpu t

r ings with a Q equa l to v’-. (Q as defined in communica t ion prac-

t ice). The system is a lways stable, and the outpu t approaches the valueof a constan t inpu t as t + m.

It is ch ar act er ist ic of a secon d-or der lin ea r differ en tia l equ at ion of t his

type that the solu t ions a re a lways stable; % is a lways bounded if 0, is

bounded. It is, however , unfor tunately t rue tha t physica l systems are

seldom descr ibed by equat ions of lower than the four th order , especia lly

if the amplifier is fr equency sensit ive and the feedback involves con-

version from one form of signal to another . For example, feedback

may be by a synch ronous genera tor with a 60-cycle ca r r ier (see Chap. 4)

which will have to be rect ified before being added to the input . This

r ect ifier will be essent ia lly a filter descr ibed by a differ en tia l equa tion of

an order h igher than one. The quest ion of stability therefor e a lways

pla ys a n impor ta nt r ole in discu ssion s of t he design of ser vo syst em s.

104. History of Design Techniques. -.4utomat ic cont rol devices of

one kind or another have been used by man for hundreds of yea rs, and



16 S ERVO S YS TEMS [SE(. 11

descr ipt iousof ear ly scr volike devices can bcfounclin lit era tu re at least

as fa r back as the t ime of I,eonardo da Vinci. Th e a ccumula ted kn owl-

edge and exper ience tha t compr ise the presen t -day scicncc of servo

design , ho\ vcvcr , r cccivcd a gr ra t in it ia l ‘impulse from the ]vork and

publica t ions of Nicholas Minorsky ’ in 1922 and 11. 1,. IIazcn’ in 193-!.

Minor sky’s \ r or k on the au tomat ic st rr r ing of ships and Irazcn’s on shaft ,-

posit ionir rg types of servomechanisms both con t~ined mathmnat ical

ana lyses based on a dir ect study of t he solu t ions of dilfcrcn t ia l equa t ions

similar to those of Sec. 1.3. “1’his approach to the design problem IVUS

the only one availa lic for many year s, and it was exploit ed \ r it il signifi-

can t succes s by in t elligen t and indust r io(ls d rs igrmrs of sc’rvt )n l(~cll:~n isms .

In 1932 h’yquist ’ published a proccdurc for studying the sta l)ility

of feedback amplifiers by the usc of steady-sta te techn iques. 1[is po\ rm-ful theorcm for studying the stability of fcccfback aml~lifir rs bccamc

known as the N“yquist stability cr it er ion . In Xyquist ’s ana lysis the

beh avior of t he ser vo syst cm \ vit h t he feedba ck l(J op br oken is con si{leu ,d.

The r at io of a (complex) amp]itudc of the ser vo ou tpu t t o t he (L’L)IT)])]CY)

er ror amplitude is plot tccf in the complex plane, with fr rq(lency as LL

var iable parameter . If t he result ing cu rve r locs not encircle t lw cr it ica l

point ( —1, O), t he system is stable; in fact , the far ther the locu~ c:m he

kept away from the cr it ica l poin t the gr ea ter is t he stability of t ]w system.The theory and applica t ion of th is cr it er ion a rc discussed in (haps. 2

and 4. F rom the designer ’s viewpoin t , t he best advantage of th is mrthod

is tha t even in complica ted systems t ime can be saved in analysis and a

grea t insight can be obta ined into the deta iled physica l phcnomrna

involved in the servo loop. Some of the ear liest w-ork in th is field \ ras

done by J . Taplin a t .Nfassachuset ts Inst itu te of Technology in 1937,

and the work was ca r r ied fu r ther by H. Harr is, 3 a lso of Ma ssa cblwt ts

Inst it ut e of Tech nology, wh o in tr odu ced t he con cept of t ra nsfer fun ct ions

in t o s ervo t h eory. Th e ~va r cr ea t ed a gr ea t demand for h igh -per formance

servomechan isms and grea t ly st imula ted the whole subject of servo

design . The supposed demands of milit ary secur ity, however , confined

t he r esult s of th is st imula tion wit hin fa ir ly small a ca demic and industr ia l

circles, cer ta in ly to the over-a ll det r iment of the \ var effor t , and pre-

ven t ed, for example, t he ear ly publish ing of the fundamenta l work of

G. S. Brown and A. C. Hall.’ The rest r icted, but never theless fa ir ly

I N. Minorsky, “Direct ional Stability of Automat ical ly Steered Bodies,” J . Am.

SOS. Naual Eng., 34, 280 (1922); H. L. Hazen , “Theory of Servomechankms ,”J . Fran klin ~TMt.,18, 279 (1934).

ZH. h“yquist, “Regenera tion Theory,” Belt System Tech . J ., XI , 126 (1932).

3 H. H arr is, “The Ana lysis and Des ign of Servomechan isms ,”OSRD Repor t

454, J anua ry 1942.4G. S. Brown and A. C. Hall, “Dyn am ic Beh avior a nd Design of Ser vo-

mechanisms,” Tr an s.,ASME, 68, 503 (1946).



SEC.1.5] PERIWRMANCE S [’ECIFICATIO.VS 17

widely circula ted, publicat ion in 1943 of The Analysis and Synthesis oj

Linear Servomechanisms, by A. C. Hall, gave a comprehensive t re~tment

of one approach to the steady-st ,a te analysis of servomechanisms and

popular ized the name ‘‘ t ransfer -locus” method for this approach. Some

of the impor tant concept s in t r oduced by steady-sta te analysis a r e th ose of

“t ransmission around a loop” and the use of an over-a ll system opera tor .

In 1933 Y. W. Lce published the result s of work done by himself and

Nor ber t Wiener , descr ibing cer ta in fundamental rela t ionships bet ween

the real and imaginary par t s of the t ransfer funct ions rcprcsen ta t ivc of

a la rge class of physical systems. These basic rela t ionships have been

applied in grea t deta il and with grea t advantage by H. W. Bodel to the

design of elect r ica l networks and feedback amplifiers. Several groups,

working more or less independent ly, applied and extended Bode’s

techniques to the servomechanism design problcm, and the results have

been very fru it fu l. The result ing techniques of analysis a re so rapid,

convenient , and illuminat ing that even for very complica ted systems the

designer is just ified in making a complete analysis of his problcm. As is

shown in Chap. 4, the complete analysis of a systcm can be carr ied

through much more rapidly than the usual t ransfer -locus methods per -

mit , and the analysis of mult iple-feedback loop systems is par t icu lar ly

facilitated.

1.5. P er forma nce Specifica tion s. -In design in g a ser vomech an ism for

a specific applicat ion, th e design er necessar ily has a clear , defin ite goal

in mind; the mechanism is to per form some given task, and it must do

so with some minimum desir ed qu alit y of per forma nce. Th e design er is,

th er efor e, faced with t he pr oblem of transla t ing this essent ia lly physical

in format ion in to a math ema tical definit ion of t he desir ed per forma nce—

one that can then be used as a cr iter ion of success or fa ilure in any

a tt empt ed pen cil-a nd-pa per syn th esis of t he mech an ism .

The most impor tant character ist ic of a servo system is the accuracy

with which it can per form its normal dut ies. Ther e a r e sever al differ en t

ways in which one can specify the accuracy of per formance of a servo-

mechanism. The most useful, in many applicat ions, is a sta tement of

the manner in which the output var ies in response to some given input

signal. The input signal is chosen, of course, to be represen ta t ive of the

type of input signals encountered in the par t icu la r applicat ion. Many

servos are used in gun director s and gun data computers, for instance, to

r eproduce the mot ion of the ta rget , a ship or a plane, being followed or

t ra cked by t he dir ect or . Su ch mot ion s h ave cer ta in defin it e ch ar act er is-

t ics, because the velocit ies and accelera t ions of the t argets have fin ite

physical limita t ions. The performance of such ,servos is oft en par t ia lly

I H . W. Bode, “F eedba ck .knplifier Design ,” Bell S ystem Tech . J ., XIX, 42

(1940).



18 S ERVO S YS TEMS [SEC,15

summarized by a sta tement of the er rors tha t may exist between the input

and outpu t mot ions under cer ta in peak velocit ies and accelera t ions or

over cer t ain r anges of velocit ies and acceler at ions. Alt er na tively, on e

can specify what the er rors may be as a funct ion of t ime as the mechanismr epr odu ces some t ypica l t ar get cou rse.

The per formance of a servomechanism can also be specified in terms

of its response to a step funct ion. The pr ocedu re of exper iment ally

and theoret ica lly studying a servomechanism through its response to a

step-funct ion input is ext remely useful and is widely used for a number of

reasons. The exper imenta l techniques used in such test ing are simple

and requ ir e a minimum of inst rumenta t ion. The char acter ist ics of any

t ruly linear system are, of cou rse, complet ely summarized by its r esponse

to a step-funct ion input ; tha t is, if the step-funct ion response is known,

the response to any other arbit rary input signal can be determined. It

would be expected, therefore, and it is t rue, tha t with pr oper in terpreta -

t ion t he step-funct ion r espon se is a power ful and useful cr it er ion of over -

a ll syst em qualit y.

In some applicat ion s t he in pu t sign als a re per iodic an d ca n be a na lyzed

into a small number of pr imary harmonic component s. In such cases

the per formance of the servo system can be specified convenient ly by

sta t ing the response character ist ics of the system to sinusoidal inputs of

these par t icular ly important frequencies. With the increased use of

sinusoidal steady-sta t e techniques in the analysis and test ing of servo-

mech an ism s, it h as become fa ir ly common t o specify t he desir ed fr equ en cy

response of the system, that is, the magnitude and phase of the ra t io of

the outpu t 60 to the input Oras a funct ion of frequency—ra ther than at

severa l d iscrete frequencies . If the system is linear , it s per formance is

completely descr ibed by such a specificat ion, as it is by specifica t ion of

th e r esponse t o a st ep fu nct ion. Depen din g u pon t he pa rt icu la r a pplica -

t ion and the nature of the input signals, one or the other type of specifica-t ion may be easier to apply,

In any pract ical case a ser vo per forma nce will be r equ ir ed t o m eet con -

dit ions other than that of the accuracy with which it is to follow a given

in pu t u nder st an da rd con dit ion s. The top speed of a servomechanism,

such as will ar ise in slewing a gun or in lockin g a follow-up mechanism into

syn ch ron ism , m ay fa r exceed t he maximum speed du rin g a ctu al iollow-u p

applica t ions. It is somet imes necessary to specify the limits of speed

between which it must opera te—the maximum speed and minimum

desirable speed unaccompanied by jump. For example, a gun-director

servo system may be required to have a slewing speed of 60° per second,

a top speed dur ing actual following of 20° per second, and a minimum

speed of O.O1° per second. The ra t io of maximum to minimum foll~,,.ing

speed is h er e 2000. This speed ra tio const itu t es on e cr it er ion of goodn ess

of a servo system.



SEC.1.5] PERFORMANCE SPECIFICATIONS 19

In cer t a in applica t ions (for example, the cont rol of the cu t t ing head

of a la rge planer or bor ing mill or of the radar an tenna aboard a sh ip) the

t ransient loading on the outpu t member of the servomechanism may be

very high . Under these circumstances, a small er ror in the outpu t should

resu lt in the applica t ion of nea r ly the fu ll torque of the motor ; indeed,

considera t ions of t ransien t load may require a source of power fa r in

excess of t he dyn am ic loa d it self. It is gener a lly t r u e in h igh -per formance

servomechanisms that a lmost the en t ire init ia l load comes from the

armature or rota t ing element of the motor it self. Bet t er designs of

servomotors have tended to increase the ra t io of torque to iner t ia of the

mot or r ot or .

Three other pract ica l factors a re impor tan t in the design of good

servomechanisms and are hence often included in specifica t ions: (1)backlash , (2) sta t ic fr ict ion, and (3) locking mechanisms. Backlash

cannot be analyzed by a considera t ion of linear systems, because the

backlash dest roys the exact linea r ity of a system. Pract ica l exper ience

has sh own th at t he backlash of th e m echa nica l and elect r ica l compon en ts

limits the sta t ic per formance of a servo system. Backlash may occur

in gear t ra ins, in linkages, or in elect r ica l and magnet ic er ror -sensit ive

devices. Backlash often has the unfor tuna te effect of limit ing the gain

a rou nd t he loop of a ser vo syst em, t her eby r edu cin g its over -a ll effect ive-

ness. Increase in the gain of a servo system invar iably rmults in oscilla -

t ions of the order of the backlash ; the h igher the gain the h igher the

fr equency of t h es e oscilla t ion s. Th e in cr ea sed fr cqu cn cics of oscilla tion

a re accompanied by excessive forces that cause wear and somet imes

damage.

Sta t ic fr ict ion has the same discont inuous character as backlash ,

If the sta t ic fr ict ion is h igh compared with the coulomb fr ict ion within the

m in imum specified speed, ext rem e jumpin ess in t he ser vo per forma nce will

result . The er ror signal will have to buildup to a magnitude adequa te toover com e the sta t ic fr ict ion (somet imes called st ict ion). At this instan t

the rest ra in ing forces a re suddenly diminished and the servo tends to

over sh oot its m ar k.

Lockin g mech an ism s, su ch a s low-efficien cy .gca rs or worm dr ives, a re

t rou blesom e in ser vom ech an isms wh er e t ran sient loads a rc en cou nt er ed.

The effect is tha t of high sta t ic fr ict ion , emphasized by the resu lt ing

immobilit y of t he device.

It is impossible t o con st ru ct h igh -fldclit y ser vomech an ism s if mech an i-

ca l r igidity is n ot m ainta ined in sh aft ing and gear in g. ‘1’h c in tr odu ct ion

of m ech an ica l elem en ts wit h n at ur al fmqu cn cy compa ra ble t o fr equ en cies

encoun tered in the input is equ ivalen t t o in t roducing addit ional filter s

in to the loop. If such filt er s a re dclibcra tcly put in to produce stability,

su ch design ma y be ju st ified. Unfor tunately, it is t rue that mechanica l




elements in resonant st ructures undergo t remendous dynamic forces

which may far exceed the sta lled-torque loading on the elements. It is

gen er ally desir able t o specify, a sa por tion of design cr it er ia , t hemech an i-

ca l resonant fr equency of th e system.

Anot h er pr act ica l con sider at ion in ser vomech an ism design a rises fr om

the low power level of the input t o the amplifiers. Except under ext reme

condit ions, the er ror signal is small and the gain of the amplifier may be

higher than one million. If, for example, the feedback mechanism con-

sists of elect r ica l elements tha t may pick up st ray volt ages or genera t e

harmonics because of nonlinear elements in the circuit s, these spur ious

volt ages may exceed the er ror signal r equ ired for the minimum specified

ser vo speed and, un less supressed, may even over load the amplifier .

The applica t ion of servomechanisms to the au tomat ic t racking ofplanes by radar and the applica t ion of filter theory to the smooth ing of

observed data in genera l for gunnery purposes have brought to ligh t the

need for consider ing the effect s of noise in the system; this t oo must a t

t imes be included in the per formance specifica t ions. In the case of the

au tomat ic t racking of planes by radar (see Chaps. 4, 6, and 7), a radar

an tenna mount is made to posit ion it self in line with the ta rget . The

an tenna beam illuminates the ta rget , and the reflect ions from the ta rget

a re received by the same antenna mount tha t t ransmits the signal. The

beam is made to scan in a cone at 30 CPS,in such a manner tha t the signal

would come back a t a constan t signal st rength if the ta rget were in the

cen t er of the beam. If, on the other hand, the an tenna mount poin ts to

one side of the ta rget , the reflected signal is modula ted at 30 cps. The

phase and amplitude of th is modula t ion is the er ror signal; the phase

giving the dir ect ion and the amplitude the amount of the er ror . The

phase and amplitude a re resolved into cx-rors in eleva t ion and t raver se

and are used to actua te the servoamplifier s and servomotors on the an ten-

na mount . Were it not for the fact tha t the reflect ions from the planefade in ra ther haphazard ways, the servo problem would be of the usual

type. The presence of the fading in the er ror-t ransmission system,

however , makes necessa ry carefu l design of the system, with due regard

for the frequency dist r ibu t ion and magnitude of the fading. For exam-

ple, if the fading were character ized by a frequency of 5 cps, it would be

necessa ry to design the servo loop in such a way that a t 5 cps the response

of the system would be either zero or very small. This is, for tuna tely, a

reasonable step, since no plane being t racked will oscilla t e with such afrequency. If the fading should cover the spect rum from 5 cps to all

h igh er fr equ en cies, t hen t he fr equ en cy r espon se of t he ser vo syst em wou ld

have to be equiva len t to a low-pass filt er with cu toff somewhat below

5 cps. On the other hand, the a t t enua t ion of the h igher fr equench in

the response of the servo system is invar iably accompanied by the in t ro-



SEC.1.5] PERFORMANCE SPECIFICAT IONS 21

du ct ion of a cceler at ion er ror s; for a fixed amplit ude, a ccelera tion goes up

a s t he squ ar e of t he fr equ en cy. Su ch a system becomes sluggish and may

not follow a plane undergoing evasive tact ics. A compromise must be

made between suppression of the fading and accurate following of the

actual mot ion of the target . Methods by which this can be done are

discussed in Chaps. 6 to 8.

There somet imes ar ises the problem of designing the best possible

servo system of a given order of complexity to meet a given need. This is

the subject of the second par t of the book. The pract ice before the war

in the design of servos was to employ a mechanism adequate for t he pr ob-

lem. The difficu lt problems encoun tered in the war , par t icu la r ly in the

field of fir e con tr ol, emph asize t he n ecessit y of designin g t he best possible

servo system consisten t with a given kind of mechanism. It is not easyto give a sta tement of what is the best per formance. It has been com-

mon pract ice (though not a desirable one) to specify servo per formance

in terms of the response, say a t two frequencies, and to omit any sta t e-

ment about the stability of the system in the presence of large t ransien ts.

It is obvious that a system designed to meet these specifica t ions will not

necessa r ily be the best possible servo if the input conta ins frequencies

ot her than t he t wo specified on es. In deed, syst em s design ed t o a specifi-

ca t ion of th is type have shown such high instability a t high frequenciesas t o be a lmost u seless in t he pr esen ce of la rge t ra nsien ts.

If it is desired to design a “best possible servo, ” it is necessa ry to

defin e a cr it er ion of goodn ess. Hall’ an d Ph illips’ h ave in depen den tly

applied t he cr iter ion tha t t he rms er ror in t he following will be minimized

by the “ best” servo. For a full st a t ement of the per formance of such a

servo it is a lso necessa ry t o descr ibe the input for which the rms er ror is

minimized. In the case of the previous example of automat ic radar

t racking of an airplane, the problem was to t rack an a irplane on physi-

ca lly realizable courses of the type to be expected in the presence of ant i-

~ircraft fir e. The input , to the servo dr ives of the antenna mount , except

for fading, might be th e instan taneous coordina t es of t he plane flying any

one of a la rge number of paths, approximated as consist ing of st ra igh t

segments; the character of the fading can be observed in a number of

t r ia l runs with the radar set t o be used. The order of the differen t ia l

equa t ion descr ibing the servo system was fixed by the character ist ics of

the amplidyne, the d-c dr ive motors, and the amplifier . The problem

was t o det er r r iin e th e pr oper value of t he parameter s available for adjust -

ment in the amplifier in such a way tha t therms er ror , averaged over the

many st ra ight -line courses of th e ta rget , should be a minimum. The usc

1A. C. Hall, Th e Analysis an d S yn th rsis qf Linear Seruo?nxc?wnisms,echnologyPress,MassachlmwttsTnstituteof Technology, May 1943.

z 11,S. l%illips, ‘1St,rvomrcha~lislns,”RI, Repor t No. 372, n ay 11, 1943.




of the rms-er ror cr iter ion in this problem is just ified pr incipally by the

fact that it lends it self to mathemat ica l analysis. It is obviously not

the best cr iter ion for all types of problems; it gives too grea t an emphasis

to la rge momentary er rors. In the ant ia ircraft case, la rge momenta ry

er rors might cor respond to one or two wild shots. Obviously, it is bet ter

to have one or two wild shots, with all the rest close, than to have all

sh ot s fall in effect ively wit h a moder at e er ror . A bet t er working cr it er ion

has n ot yet been developed.

Th e rms cr it er ion of goodn ess is par ticu la rly u sefu l beca use it permit s

one to take into account the presence of noise, provided only that the

frequency character ist ic or the ‘‘ aut ocor rela t ion funct ion” of the noise

is known . The analysis given in the la t ter par t of this book, a lthough

difficult for pr act ica l designers, is important in indust r ia l applicat ionswhere t ransient loading has definite character ist ics and where best

per formance is economica lly necessa ry. Th e loa din g con st it ut es in effect

a noise and can be t rea t ed by th e meth ods t her e developed.



CHAPTER 2

MATHEMATICAL BACKGROUND

INTRODUCTION

This chapter will bc clcvoted to n discussion of the mathemat ical

concept s and techniques th:~t mm fundamental in the theory of servo-mechanisms. These ideas ~r ill, for thcl most , ~x~r t,bc dm”eloped in their

r ela t ion to filters, of which ser vomrch:misrns form a sprcizl class. Mor e

specifically, the chapter will bc concerned ~vith the }v:lys in \ ~hich the

beh avior of linear filt er s in gen er al an d sclvomc(,lla llisll~s in par ticu lar

can be descr ibed and \ vith making clear t he rela t ions bctwccn th e var ious

modes of descr ipt ion . 1

The input and ou tput of a filt er a rc often rr la tcd by a differential

equation, t he solu t ion of ~~hich gives the output for any given input .This equa t ion provides a complctc descr ipt ion of the filter , but one tha t

cannot be conven ien t ly used in dmign techniques. Other modes of

descr ipt ion of the filter a re rela t ed to the outputs produced by specia l

t ypes of inpu t:

1. The wcighiing funct ion is the filter output produced by an impulse

input and is simply rela ted to the outpu t produced by a step input .

2. The frequ en cy-w spon se f~tn clion rela tes a sinusoidal input to the

output tha t it pr oduces,

3. The transfer junction is a gen er aliza tion of t h e fr equency-r espon se

function.

Th ese modes of descr ipt ion a re simply r ela ted, a nd ea ch offer s a dva nt ages

in d iffer en t fields of applica t ion .

Discussion of these ideas requires the use of mathematica l devices

such as the Four ier t ransform and the Laplace t ransform. For complet e

dkcussions of these techniques the reader must be r efer red to standardt ext s; for h is conven ien ce, however , cer t ain ba sic idea s a r e “h er e p r esen t ed.

Although it has not been in tended that the analysis of the chapter should

be ca rr ied t hr ou gh with maximum r igor , t he r ea der w-ill obser ve th at some

pains have been taken to provide a logica l development of the ideas.

1The authors wish to acknowledgehelpful d iscussionswith W. Hurewicz in theplanningof this chapter .

23



24 MATHEMATICAL BACKGROUND [SEC.21

Th is developmen t is illu st rat ed by a pplicat ion t o lumped-con st an t filt er s,

in terms of ~vhich the rela t ions here discussed a re especially easy to

understand.

Par t icular a t t en t ion has been paid to the discussion of stability of

lilt er sj which is of specia l importance in its applica t ion to servomecha-nisms. The la t t er par t of the chapter is devoted to a discussion of the

Nyquist stability cr it er ion and its applica t ion t o single-loop- and multi-

Ioop-feedback systems. Para llel developments in the case of pulsed

filters will be found in Chap, 5.

FILTERS

2.1. Lumped-constant Filter s.—The most familiar t ype of filter is t he

elect r ica l filter consist ing of a network of a fin ite number of lumped

r esist an ces, ca pa cit an ces, a nd in du ct an ces wit h con st an t va lu es. Figure

21 illust ra tes a par t icular ly simple filt er of th is

t ype—an RC-filt er con sist in g of a sin gle r esist an ce

~a ,”,,, Rand asinglecapacitanceC.I o

The input to an elect r ica l filter is a voltage

~1~ ~.l,_alI, ~C-filte, E ,(t ) supplied by a source tha t may be taken to

have zero interna l impedance; the ou tpu t is an

open-circu it volt age Eo(t ).

equat ion . In the case of

der ived form

Input and ou tpu t ar e r ela t ed by a different ia l

the RC-filt er of Fig. 2.1 th is has the easily

~dEo ~ ~O=E,

dt ‘ ‘(1)

wher e the quant ity T = RC’ is the t ime constant of the network. In

genera l, the input and o~~tpu t are rela ted by

~ , dnEo &lEO, ~ + a,,-l —

dmE,~tn-, + “ “ . + (IOE. = h. ~

+bm_, ~+... + boE,, (2)

where the a’s ~.nd b’s are constants and m, n s 2.V, N being the number

of independen t loops in t he filter n et wor k (including on e loop t hr ough t he

.Joltage source but n on e t hr ough t he out put circuit ).

Since this formula t ion is less common than that in terms of mesh

cur rent s, it may be desirable to indica te it s der iva t ion . In a N-mesh

n etwor k of gen er al form, t he mesh cu rr en ts a re det ermin ed by in tegr o-dif-

ferent ia l equat ions which may be wr it ten’ as

1See,for instan ce,E . A. C,uillemin,Communicat ionVehoork s,Vol. I , Wiley, NewYork, 1931,p . 139.



SEC.2.1] LC’11[PED-C0.i’8 TAATT FILTERS 25

aLI~I+ a12~2+ u13i3+ “ + al.~ix = E,,

ac,il + azziz + a?sij + + a*.ViN = 0,. . . . . . .

1

(3]

a ,~lil + a.~~ij + a.\-3i3+ . . + a.v&riN= O,

~rhere i~ is the cur ren t in the kth mesh and

(4)

Thr output voltage is determined by the mesh cur rent s, through an equa-

t ion of the form

a .v+l,t i, + fh +l,: i, + + av+l,.,r iN = Eo. (5)

Ikluat ions (3) and (5) may be regarded as N + 1 equat ions in the 3N

(Iuantitics dil/dt, il, f dt i~, (k = 1, 2, . . . , N). To elimina te such

quant it ies from considera t ion and to obta in a direct rela t ion between 11~

and E,, one may form the first 2N der iva t ives ~vith respect to t ime of

thesr .V + 1 equat ions. One has then , in all, (2N + 1) (N + 1) equa-

t ions in the N(2.V + 3) quant it ies

‘1’h cw equ at ion s in ~-olve a lso Eo, E,, and their fir st 2N der iva t ives;

bet t~mn them one can allvays eliminate the unknown mesh cur ren ts and

t heir dcr iva ti~-es, obt ain in g a lin ea r r ela tion between Er, Eo, and their

fir st 2.$’ der i~-a tives, If some of t he quant it ies Ljk, R~k, and C;~ a re zer o,

it may not bc nw.-cssa ry to take so la rge a number of der iva t ives in order

t o climinat c t he unknoum cu rr en t quant it ies; m and n may then be less

than 2N, as they were found to be in Eq. (1).1JVhen t he input volt age E,(t) is specified, Eq. (2) const itu tes a non-

h omogen eou s linem differ ent ia l equa tion that can be solved t o det ermin e

Eo(t ). The genera l solu t ion of such an equa t ion can be expressed as

I If on e rxcludes n ega t ive va lu es of L, 1?,,a nd C from con sider a tion (t ha t is.dcak \ vit ll:Lpzssive filter), then nOequat ion will contain a term in di~/dt , i~,or J dt i~unlessa siulilart erm occurs iu the kth loop equat ion; this applies, in part icular,to theou tpu t equ :lt iou Eq, [5). I t follows tha t ~vhcuone d ifferen t ia t esEqs . (3) to obta in

uew rqu~t iolls wit h ~vh ich to elim ina te cu r ren t va r iables , one obt a in s a t lea st a smany new cu r ren t va r iables a s new equa tion s. On e cm in cr ea se t he n umber ofequat iousas compared with the number of current var iables only by different ia t ingEq. (5) and wltb it a sufficien tnumber of Eqs. (3) to make possibleeliminat ionof a llthe uetvvariables: this may or may not requiredifferent iat ionof the first of Eqs. (3),The number of der ivat ivesof ECIt h at must be in t roduced in order t o elim ina tea llcu r r en t va r iables is t hus equa l t o t he or igina l exces s of va r iables over equa tion srequiredfor their eliminat ion; the number of derivat ivesof E, t ha t must be in tr o-du cedmay be equal t o t his bu t n eed never be larger. Thus in the result an tEq , (’z)me will have n z m.



26 ,\ [/tT!iEMA TICAL BACKGROUND [SEC,22

thesum of a ny solu tion of t he n on homogen eou s equ at ion , plu s t he gen er al

solu t ion of the homogeneous equat ion obta ined by set t ing equal to zero

all terms in El:

dm.li’o d,,–lfio

an ““al”-+o,L -, =+. ..+aOEo=O.

In the par t icu lar case of the IiC-filter descr ibed by l;q. (1)

solu tion may be \ vr it tcn fis

E.(t ) = .4C ~ + ~/

1–,——~, : d7 E,(T)c ~ ,

(6)

t he gen er al

(7)

\ r her e th e fir st t erm is t he gcu cr al solu tion of t he homogen eou s equ at ion

,T(~+Eo=O (8)

and the sccoud is a par t icular solut ion of the nonhomogeneous Eq. (1),

as is easily ver ified by subst itu t ion in to that equat ion.

To dctm-min e th e ou tput volt age E,)(t ) it is n ecessar y t o kn ow both t he

input funct ion E~(t) and t ,h c adjusta lic con stants in th e gener al solut ion

of t he homogeneous equ :~t iou . Th rsc la tt er con st an ts a re det ermin ed by

the init ial condit ions of the problcm. One set of init ia l condit ions is

espcei:dly emphasized in what follol~s: t hc condit ion that the systemstar t from rest when the input , is first applied. The resultant output of

the filter under this condit ion will bc termed its normal wsponse t o the

specified input . In the case of Eq. (7), the condit ion that Eo(t) = o

at t = O implies .i = O; the normal response of this filter to an input .

E,(t ) beginning at t = O is thus

or , by a change in the var iable of integra t ion ,

(9(2)

(9b)

2.2. Normal Modes of a Lumped-constant Filter .—The solut ions of

t he h omogen eou s differ en tia l equ at ion [Eq. (6)] a re of con sider able in ter -

est for the discussion of the genera l behavior of the filter . The filter

outpu t dur ing any per iod in which the input is ident ica lly zero is a sohl-

t ion of this homogen eou s equat ion, since dur ing this t ime Eq. (2) r edu ces

to Eq. (6). The outpu t dur ing any per iod in which the input E, is con -

stant can be expressed as the sum of a constant response to this constant ,

input,

E. = $ E, (10)



SEC.2.2] NORMAL MODES OF A LUMPED-CONSTANT FILTER 27

(this being a solut ion of the nonhomogeneous equat ion), and a suitable

solu t ion of Eq. (6). In this case the solu t ion of the homogeneous equa-

t ion can be termed the “t ransien t r esponse” of the filt er to the ear lier

h istory of its input . Transient r esponse can, of cour se, be defined more

gen er ally, wh en ever t he in put a ft er a given t im e tOt ak es on a st ea dy-st at eform: Th e transient response of the filter is the difference between the

actual ou tpu t of the filt er for t > toand the asymptot ic form that it

approaches. This asymptot ic form is necessar ily a solut ion of the non-

homogeneous Eq. (2); the t ransient is a solu t ion of the homogeneous Eq.

(6).’

The genera l solut ion of Eq. (6) is a linear combinat ion of n special

solu tion s, ca lled t he n orma l modes of t he filt er ; t hese h ave t he form

where k is an in teger and pi is a complex constan t . The genera l form of

the solu t ion is then

E. = C,h,(t) + C,h,(t) + ~ + Cfih.(t); (12)

the values of the constant s ci depend on the init ia l condit ions of the solu-

t ion or on the past h istory of the filter .

To determine the normal-mode solu t ions, let us t ry c“’ as a solut ion of

Eq. (6). On subst itu t ion of e@for E., t his equ at ion becomes

(anp” + an-,p”-’ + + aO)@’ = O. (13a)

Thus ep’is a solu t ion of the differen t ia l equat ion if

P(p) = anpn + a.-lp”-’ + . . ~ + a. = (). (13b)

This equat ion has n roots, cor responding to the n normal modes. If all

n root s of this equat ion, PI, PZ,,P3, . . . ~P.~ are distinct, then all normal-

mode solut ions are of the form e@; if p i is an s-fold root , it can be shown

(see, for insta nce, Sec. 2“19) tha t t he s cor respon din g norma l-mode solu -

t ion s a reeP,t, ~eP,, pe,r’,i . . . , t“–lcp,t.

Let us denot e the possibly complex value of pi by

pi = CXi+ j@i, (14)

wh er e ai and u~ar e rea l. If pi is rea l, the norma l-mode solut ion is rea l:

h i(t ) = .Pe”J . (15)

If pi is complex, its complex conjugate pf will a lso be a solut ion of Eq.

1I t may be emphasizedtha t the normal responseof a filt er is it s complete responset o an in pu t, u nder t he condit ion t ha t it st ar t fr om r est ; t he n ormrd r espon semayincludea transientresponseas a part .



28 MATHEMATICAL BACKGROUND [SEC.2.3

(13b), sin ce t he coefficien ts a k a re r ea l va lu ed; t he n orma l-mode solu tion s

defined above will be complex but will occu r in the t ransien t solu t ion in

linear combina t ions that a re real:

If pi is purely imaginary t here maybe purely sinusoidal t ransien ts: sin ~it ,

Cos d.

It will be noted that the normal-mode solu t ion will approach zero

exponent ia lly with increasing t if pihas a negat ive rea l par t bu t willincrease indefinite y if the rea l par t of piis posit ive. If all the solu t ions

of Eq. (13b) have nega t ive real par t s, the t ransien t response of the filt er

will a lways die ou t exponent ia lly a ft er the input assumes a constan t

value; the filter is then stable. 1 This may not be so if any pi has a posi-

t ive real par t ; when it is possible for some input to excit e a normal mode

with posit ive ai, t hen the output of the filter may increase indefin itely

with t ime-the filter is then unstable. It may also happen that the real

par t of pi vanishes. If th is root is mult iple, there will be a normal modethat increases indefinitely with t ime and will lead to instability of the

filter if it can be excited. If the imaginary root pi is simple, the normal

mode is sinusoidal; the system may remain in undamped oscilla t ion aft er

this mode has been excit ed. It is physically obvious that in such a case

a cont inu ing input a t the frequency of the undamped oscilla t ion will

pr odu ce an ou tpu t t ha t oscilla t es wit h in definit ely in cr easin g amplitu de.

In the precise sense of the word, as defined in Sec. 2.8, such a filt er is

unstable. In summary, then , we see that a lumped-constan t filter con-

sist ing of fixed element s is cer t a in ly stable if a ll root s of Eq. ( 13b) have

nega t ive real par t s, bu t may be unstable if any root has a zero or posit ive

r ea l pa rt .

2.3. Linear F ilters.-The lumped-constan t filters discussed in th e pre-

ceding sect ions belong to the more genera l class of ‘‘ linear filters. ”

Linear filter s a re character ized by proper t ies of the normal response—

proper t ies tha t may be observed in the normal response of the RC-filt er

of Sec. 2.1:

(9b)

1 The words “stable” and “unstable” are used here in a genera l descr ipt ive sense.

We shall la ter consider the stability of filter s in more detail and with greater genera lity

and precision ; the ideas here expressed are in tended only for the or ienta t ion of the

reader.



SEC.2.3] LINEAR FILTERS 29

Th ese a re

1.

2.

3.

The normal response is a linear funct ion of the input , in the

mathemat ica l sense. If y,(t ) is the normal response of the

filt er to the input x,(t ) and y,(t ) is the normal response to the inputQ(t ), then the normal response to the input

z(t ) = Clzl(t) + C2Z2(L) (17)

(cl and CZbeing arbit ra ry constan t s) is

y(t ) = C,y,(t ) + C,y,(t ). (18)

The normal response at any t ime depends only on the past va lues

of the input .The normal response is independen t of the t ime or igin . That is.

if y(t ) is the no~mal response-to an input z(t ), then y(t + tO)is the

normal response to the input x(t + i’0). ‘1’h is r equ ir em en t is,

essen tia lly, t ha t t he cir cu it elem en ts shall h ave va lu es mdepen den t

of t ime. This const itu tes a limita t ion , though not a ser ious one,

on the types of filter s tha t we shall consider .

It should be poin ted out tha t a lthough few pract ica l filter s a re st r ict ly

linear , most filt er s have approximately th is behavior over a range of

va lues of the input . Consequent ly, the idealiza t ion of a linear system is

widely useful and does lead to valuable predict ions of the behavior of

pract ica l sys tems.

:~:

&IG.22.-A filter in FIG.2.3.—Curren tI throughawh ich the capacit y is a diode rect ifieras a functionof thefunct ionof t ime. potcnt ia l cliffcrence V betweenan-

ode and cathode.

It should be emphasized tha t the requir ement of linea r superposit ion

of responses (It em 1 above) does not suffice to define a linear filter ; the

circu it elements must a lso be constan t in t ime. .4n example of a “non-

linear” filt er can be der ived from the filt er of Fig. 2.1 by making thecapacity change in t ime—as by connect ing one of the pla tes of the con-

denser th rough a link to the shaft of a motor (F ig. 2.2). In spite of the

fact tha t the superposit ion theorem [Eqs. (17) and (18)] applies t o th is

filter , it is not linear , and its normal response cannot be wr it ten in an

in tegra l form such as Eq. (9 b).

An example of a filter tha t is nonlinea r in the convent iona l sense is

t h e familia r d iode r ect ifier . The cur ren t through the diode and the out -



30 MATHEMATICAL BACKGROUN D [SEC.~.b

pu t voltage is differen t from zero on ly when th e poten t ia l difference

between theanode andca thode isposit ive(Flg. 2“3). The superposit ion

theorem above does not hold for th is system.

Input output For example, if z,(t ) = A, a constan t , and

zz(f) = A sin t it ,. it is easy t o see that th e com -

bined ou tpu t to z,(t ) + z,(t)s not the sum ofFIG.2.4.—Amechanical61ter,

t he ou tpu ts du e t o z,(t ) and z,(t ) sepa ra tely.

An example of a mechan ical filter is sketched in Fig. 2.4. The input

and ou tpu t shafts a re connected th rough a spr ing and flywheel; the fly-

wheel is provided with damping tha t is propor t iona l to its speed of rota -

t ion . Such a filt er can be made to be linear , a t least for small angu la r

displacemen ts of the input and ou tpu t shaft s.

THE WEIGHTING FUNCTION

It will be noted tha t th e rela t ion

(9b)

expresses the ou tpu t of a par t icu la r linear filter as a weigh ted mean of all

past va lues of the input ; more precisely, the inpu t a t a t ime t – r con -.

t r ibu tes to the ou tpu t a t t ime t with a rela t ive weigh t e-~ tha t is a func-

t ion of th e elapsed t ime in terva l r . (It must be remembered tha t in th is

example EI(t ) = O if t < O.) Th is method of rela t ing the inpu t and

ou tpu t of a filt er by a weigh t ing junct ion is genera lly applicable to linear

filt er s a nd is of gr ea t impor ta nce. Th e weigh tin g fu nct ion it self is closely

rela t ed to the normal response of the filter to an impulse inpu t . We

shall begin , then , by consider ing the normal response of linear filt ers to

th is par t icu la r type of input .

2.4. Normal Response of a Linear Filter to a Un it -impu lse Input .—

The unit impulse or delta funct ion 6(f – @ is a singu la r funct ion definedto be zero everywh ere except a t t = to,nd to be in fin ite a t t= ton such

a wa y t ha t it possesses t he followin g in tegra l proper ties (tl< fz):

\

12

lit j(i) l$(t – to) = o if tl> toor t Z < to, (19a)c1

/

i,dt j(t) 6(t – to) = ZU(to) if tO = tI or tO= t% (19b)

t ,

/

i,

d j(t)(t– to)= f(h) if tl< to< h. (19C)1,

Th e fu n ct ion /i(t – f,) may be considered as the limit of a cont inuous

function 6a(t– to)hat is symmetr ica l in t abou t the poin t t= toand

depends upon a parameter a in such a way that



SEC.2.4] NORMAL RESPONSE TO A UNIT-IMPULSE

!

+.

d t a.(f – to) = 1,—.

lim C$=(t– tO) = O if t # .tO.a+Q

Examples of such funct ions a re

INPUT 31

(20a)

(20b)

(21)

as a approaches zero, these funct ions tend to take on the proper t ies

a ssign ed t o t he unit -impu lse fu nct ion .

The normal response of a linea r filter t o a unit impulse applied at the

t ime t = O is denoted by W(t ); it will be ca lled the weigh t ing junct ion , for

reasons to be made eviden t la ter . lVe have, of course,

W’(t ) = o ift <O. (22)

The weigh t ing funct ion may be discon t inuous and may even include

terms of the delt a-funct ion type for t o z O.

The normal response of a linear lumped-constant filter to an impulse

input can be determined by considera t ion of the govern ing differen t ia l

equat ion [Eq. (2)]. After the moment of the impulse, E[ will be zer o,

and the r espon se W(t) must be a solu t ion of the h omogeneous differen t ia l

equat ion [Eq. (6)]; tha t is, it must be a linear combinat ion of the normal

modes of the filt er . At the moment of the impulse W(t) may be dis-

con t inuous; it will even conta in a term of the form C6(f) when the filt er is

such that the ou tpu t con ta ins a term propor t ional to the input .

In the case of the simple RC-filter descr ibed by Eq. (1) ther e is on lyon e n orma l mode,

hl = e–$,

and the weigh t ing funct ion is of the form

W(t) = Ae-+, t>o,

w(t) = o, t<o. 1

On e can det ermin e t he con st ant A by in tegra t ing Eq. (1) from

with EO = w(t ) and El = ~(t) t h is becomes

/

t

J

t

NV(t ) + d, W(,) = dr ~(r).—. —m

(23)

(24)

m to t;

(25)



[SEC.242 MATHEMATICAL BA G’KGROUND

If t >0, we have, by Eqs. (19c) and (24),

TAe-~ – AT(e-~ – 1) = 1, (26a)

whence

A=+. (26b)

The form of W(t) can be determined by similar methods when there is

more than one normal mode. Another method of solut ion , employing

th e Laplace t ran sform, will be indica t ed in Sec. 2.19.

The normal response to a unit impulse can also be determined from

an integra l formula t ion of the response to a genera l input . In the case

of our simrde RC-filt er , one would star t with Eq. (9a). The normal

response to a unit impulse a t t ime tO > 0 becomes

/

1 t–,

E .(t ) = ; ~——

J (t – tO)e ~ d7.

then, by Eqs. (19),~ _f+

E.(t ) = ~e , t > to,

E.(t) = o, t < to. I

It follows tha t for th is filter

w(t ) = + e-i t>o,

w(t) = o, t<o. I

‘I’h is funct ion is shown as curve a in Fig. 25.

The RC-filt er shown in Fig. 2.6 has a delta -funct ion term

‘k ‘veigh’ing~~~~~~iis

m in ed by t h e differ en tia l equ at ion

o .!- where

(27)

(28)

(29)

in its

deter-

(30)

FIG. 2.5.—The normal re-sponseof th e circuitof Fig. 2.1 ~, = Rt7. (31)

with T = 1 (a) t o a unit -impu lseinpu t a t t = Oand @) The norma l-r esponse solu t ion of th is equa tionto a unit -s tepinput s ta r t ingatt=o.

for an input E, tha t begins after t ime t = O

is

/

t–,——do(t)= .Er(t) * : dr ll,(7)e “ (32)

as can be ver ified by subst itu t ion in to Eq. (30). The response of this

filt er t o t he impulse input

E,(t) = 6(1 – tO), to >0, (33)



SEC.2.5] NORMAL RESPONSE TO AN ARBITRARY INPUT 33

is t hen

1 -’+0

E.(t ) = a(t– to) –~le ,

it follows t ha t

(34)

(35)

In t he discu ssion t ha t follows we shall con sider t he weigh tin g fu nct ion

W(t) as a primary ch ara cter ist ic of a filter r at her

than as a der ived quant ity t o be obta ined, say, by -c

solu tion of a cliffer en t ial equ at ion . Experimen-

3

t ally, it is somet imes pr act ica l t o obt ain t he weigh t- E, (t) R E“(t)

ing fun ct ion by r ecor din g th e r espon se of t he filterwhen a la rge input is suddenly applied and re- FIG. 26.-An RC-moved. (It is, of course, essent ia l tha t th is input filte r with a delta-func-

t ion t erm in the weigh t -should not over load the system. ) When this is ingfunction.

don e, t he t im e du ra tion of t he in pu t pu lse At sh ou ld

be shor t compared with any of the natura l per iods of the filter ; tha t is,

on e sh ou ld h ave

AtdW’(t)~ << w(t ) (36)

for all t . Otherwise, small var ia t ions in W(t) may be obscured. An

a lternat ive method for the exper imenta l determinat ion of W(t) will be

in dica ted in Sec. 2“7.

2.6. Normal Response of a Linear F ilt er to an kbit rary Input .—The

normal response of a linear filter to an arbit rary bounded input El(t )

M

with at most a finite number of discont inui-

t ies ca n be con ven ien tly expr essed in t erms of

the response to a unit-impulse input . We

E,(t) , (t l) shall assume that Ill(t ) = O when t < 0.

This funct ion can be approximated by a sett, r+

t-of rectangles as shown in F ig. 2“7. In com-

FIG. 2.7.—Approximation put ing its effect on the filter ou tput , the por -

of a funct ion E l(t ) by a set of t ion of th e input r epr esen ted by a ver y n ar rowrectangles.

rectangle of width At1 and height El(t l), a t

m ea n t im e t l, ca n be a ppr oxim at ed by t he impu lse in pu t EI(~l) A~l ~(t — ~1);

the ability of a filter with a finite response t ime to dist inguish between a

t rue impulse input and a pulse of dura t ion At with the same t ime integra ldiminishes as At approaches zero. One is thus led to approximate the

input El(i) by a sum of impulses:

E,(t)= y E,(t.)tn b(t - tn).(37)

Ln




The norma l response of a linear filt er t o a succession of impulse inputs

will (by the defin it ion of linear ity) be the sum of the responses to each of

these inputs. We have now to sum the responses to the incrementa l

inputs of the rectangula r decomposit ion of El(t ) given by Eq. (37); t he

resu ltan t sum will approximate the normal response EO(t ) to the input

E,(t).

Let us fir st split off from t he weight ing funct ion any delta -funct ion

s ingular it ies, wr it ing

where 0<r l<7z <... , and We(t ) is a bounded but not necessar ily

con t inuous weigh t ing funct ion . To a unit -impulse input at t ime t = O

this filt er gives a bounded output WO(t), plus impulse outputs at t imes

t= O, TI, TZ, . . . , wit h r ela tlve magr ut udes CO,CI, CZ, . . . . Thesd

pa rt s of t he filt er Tespon se ca n bc con sider ed sepa ra tely.

Th e significa nce of t he delt a-fu nct ion t erms in t he weigh tin g fu nct ion

is easily apprecia ted. If t here is a term co b(t), the filt er gives in response

to an impulse input a simultaneous impulse output with magnitude

changed by a factor co; in response to an arbit ra ry input l?~(t ) it gives an

output coEI(t ). Simila r ly, cor responding to the term cl~(t — TI) in

the weight ing funct ion there is a t erm CIEJ(t – Tl) in the response tot he input E,(t ).

Now let us consider the par t of the filt er response associa ted with the

bounded funct ion W’O(t). This par t of the response to an impulse input

d(t – tJ at t ime t l is given by WO(t– t,);he response t o a differen t ia l

input EI(tl) d(t — tl) Ml is thus E{(tl) WO(t — tl) Ml, and th e cor r es pond-

ing r esponse of t he filt er t o t he a pproxima te input [Eq. (37)] is

In t he lim it as t he At ’s a ppr oa ch zer o, t his becomes

fm, E,(t,)W,(t – t,),

o

t he exact output due to the bounded par t of the weight ing funct ion .

Since W,(t) vanishes for nega t ive va lues of the argument , we may take

the upper limit of the in tegral a t t l = t , and wr it e

/

t

E .(t ) = c,E ,(t ) + c,E,(t – T,) + “ . . + dt, E,(t,)WO(f – t,). (39)o

This representa t ion will be valid even when the input conta ins impulses

occur ring between t he t ime O a nd t. A mor e compa ct and mor e genera lly



SEC.26]

u sefu l form is

THE WEIGHTING FUNCTION

/

t+E.(f) = d, Il,(t l)w (i – t ,).

o

35

(40)

Each delta-funct ion term in the weigh t ing funct ion gives rise t o one of

the terms appear ing before the in tegra l sign in Eq. (39). When the

weigh t ing funct ion cent a ins the delta funct ion C08t), it is necessa ry to

indica te the upper limit of the in tegra l as t+ (that is, t approached from

above) in order to include the whole term co.El(t ) in the response ra ther

than just half of it . The more compact nota t ion also requires that one

write

/

i,+A

d(, a (t , – t ,) J (t – t ,) = J (t – h) (41)f,– A

when t here a re delt a funct ions in both th e input and weight ing funct ions.

On in t roduct ion of th e n ew var iable of in tegra t ion

T=t —tl, (42)

we have

/

tEo(t) = d, E,(t – r)?V(r).

o–(43)

This gives the normal response to an arbit ra ry input as an in tegra l over

the past va lues of the input , each of these values being weigh ted by the

response of th e filter t o a unit -impulse funct ion.

Equa t ion (9b) illust ra tes this resu lt in a specia l case in which W(t) is

given by Eq. (29); Eq. (9a) is similar ly a specia l case of Eq. (40).

2.6. The Weight ing Funct ion .—The weight ing funct ion provides a

complet e character iza t ion of t he filter . As we have seen , the normal

response to any input can be computed by means of the weight ing func-

t ion . In addit ion , from the weight ing funct ion of a lumped-constan t

filter one can determine the normal modes of the filter , t hese being the

terms of the form Pe”,t in to which W(t) can be resolved.

The weight ing funct ion expresses quite direct ly what may be ca lled

the “memory” of the filt er , tha t is, the exten t to which the distan t past

of the input a ffect s the response a t any t ime. This is evident in the

width of the weight ing funct ion; the “memory” may be termed long or

sh or t accordin g t o wh et her t he weigh tin g fu nct ion is br oa d or n ar row.The “memory” determines the distor t ion with which the filt er output

reproduces the input to the filter ; the filt er will r eproduce well an input

that changes but lit t le with in the length of the memory of the filter , bu t it

will distor t and smooth ou t changes in the input tha t t ake place in a

per iod small compared with the memory. Another aspect of the memory

is the lag in t roduced by the filter . If the input is suddenly changed to a




new value, the output acquires the corresponding new value only after

a per iod of lag det ermined by t he widt h of t he weight ing funct ion .

Examples.—We have already examined the weight ing funct ions of

two simple l? C-filt ers, as given by Eqs. (29) and (35).

‘I%’”‘@IfW?=FIG.2.S.—(a) An LC-filter;(b) th eweight ingfun ctionof th e circuitshownin (a)

The differen t ia l equa t ion for a filt er consist ing of an inductance L

and capacity C, as shown in Fig. 2.8a, is

(44)

The weight ing funct ion is sinusoidal for t >0, with t he angular frequency

a“ :

W(t) = CO=in ant ; (45)

this is sketched in Fig. 2.8b.

A filt er wit h feecf~ack ,

w(t)

o

w(t)o

w(1)

nt+

I:lci.2.9.—\ \ -cigh t i]lgunct-ion of a simpletir!vorlmctlan-isl]l for differ en t r ela tivevaluesof the con~tants.

‘(t) = (1

such as the servo illustra ted in Fig. 1.3, may

be gover ned by t he differ ent ia l equ at ion

( )~+~+K 00= K&,(it’ dt

(46)

where K is a constan t . The weight ing func-

t ion for this filt er will be der ived by applica -

t ion of La pla ce t ra nsform met hods in Sec. 2.17.

The resu lts a re as follows. Let

(47a)

(47b)

‘l’he quant it ies w and ~ are called the un-

damped natural frequency and the damping

ra t io, respect ively. When ~ < 1, the system

is underdam ped, and t he weight ing funct ion is

% sm [(1 — ~z)W+J].*T, ,g-r.rd (48)—

When ( = 1, the system is cr it ically damped; the weight ing funct ion is

w’(t)= @;te-~”’. (49)



SEC.2.7] NORMAL RESPONSE TO A UNIT-S TEP INPUT 37

When ~ >1, the system is overdamped, and

w(t) =~–ru.t ~(p-l)%.t _ ~–(r%)%q.

2((2U: 1))5 [(50)

Th ese t hr ee forms of th e weight ing fun ct ion a re illust r ated in Fig. 2.9.

2.7. Normal Response to a Unit -step Input .—The unit -step funct ion

u (t ) is defin ed a s follows:

u(t) = o ifi <O,

u(t ) = 1 ift~O. }

(51)

The normal response of a filt er to a un it -step input is closely rela ted

to its weight ing funct ion. In par t icu lar , the normal response to a unit

step a t t ime t = O is, by Eq. (43),

/

1

u(t) = d, w (,). (52)o–

J ust as the unit -step funct ion is

the in tegra l of the unit -impulse

funct ion, so t he r espon se t o a unit -

step input is the integra l of the

response to the unit -impulse in-

3=E?=-

FIG. 2.10.—.4pproximat ionof a funrt ion.EI(t)by a set of stepfunctions.

pu~. Conversely, the ~~eigh t ing funct ion of a filt er can be cletermined

exper imenta lly as the der iva t ive of the output produced by a unit -step

input . The form of the funct ion U(t) for the RC-filt er of Fig. 2.1 is

illust r ated in Fig. 2.5.

Let us assume tha t both E,(t)nd W(t) a re well-beha ved fun ct ions,

with E~(t ) making an abrupt jump from the value O to E1(0) a t t ime

t = O. Integra t ing Eq. (43) by par ts, we obtain

t

/ (E.(t) = EI(t – 7) U(T ) –)

d . : E ,(t – 7) u(,), (53a)o– o–

/

t

E.(f) = E,(o) U(t) + dr E:(f – ~) U(r), (53b)o–

/

f+

Eo(t) = E,(O) U(t) + dt,ll; (t,)Z? ’(t– t,), (53C)o

where the pr ime is used to denote the der iva t ive ~~ith r espect to the indi-

ca ted argument . The output of the filter is here expressed as the sumof responses to the step funct ions in to which the arbit ra ry input can be

resolved (see Fig. 2.10): an init ia l st ep of magnitude El(O) a t t ime t = O

and a cont inu ous dist r ibut ion of infin itesimal steps of a ggr ega te amou nt

E~(tJAtl in the in terva l At l about the t ime t l. The cor responding forms

of the rela t ion when El has ot her discont inuit ies or U increases s tepwise

(,W con ta in s delt a fu nct ion s) will n eed no discu ssion h er e.




2.8. Stable and Unstable Filter s.-Thus far in the discussion of the

weight ing funct ion we have made no dist inct ion between stable and

unstable filters. This was possible only because a t ten t ion was rest rict ed

to input funct ions tha t differ from zero only after some fin ite t ime. Toproceed fur ther we must define stable and unstable filters. A stable

filter is one in which every bounded input produces a bounded output ;

that is, t he normal response of a stable filter never becomes infinitely

la rge unless the input does so. An unstable filt er will give an indefin it ely

increasing response to some par t icu lar bounded input , though not , in

gen er al, t o a ll such inpu ts.

The weight ing funct ion affords a means of determining whether a

given filt er is st able or unst able, t hrou gh t he followin g cr it er ion : .4 linearjilter is stable if and only if the integral of the absolute ~alue of the

weighting junction, ~~ dr 1~(,)1, is jinite. Thus, the second of the

filter s ment ioned in Sec. 2.6 is unstable, since / 1md. sin CW,l does n oto

converge.

To prove that the convergence of this in tegra l assures the stability of

the filter we need to show tha t if E1(t) is bounded, that is, if there is a

constan t M such that IllI(t)< M for all t,then Eo(t)s a lso bounded.

The filter output may be wr it ten

/

tE.(t) = d, E,(t – T)W(,),

o–(43)

since we rest r ict our a t ten t ion to Er’s tha t a re zero for nega t ive values of

the argument . As the absolute value of an integra l is cer ta inly no grea ter

than t he in tegra l of th e absolute value of the in tegrand, we have

/pib(t)l ‘ dr \E,(t – 7)l[W(~)l.

o–(54)

The inequality is st rengthened by put t ing in the upper bound for E,(t )

and ext endh g t he range of in tegra t ion:

]Eo(t)] s M\

m dr ]W(~) ]. (55)

Thus, if ~~ d, l~(T)l exists,-%(t)is~undd .The proof of the second par t of the stability cr iter ion—tha t the filter

iS unstable if ~~ d, l~(T)l doesnot converge-is somewhatlonger andwill be omit t ed here. It involves the const ruct ion of an input E,(t)

tha t will make Eo(t ) increase without limit , and is essent ia lly t he same as

t he cor respon din g pr oof given , in t he ca se of pu lsed filt er s, in Sec. 5.3.



SEC.2.8] S TABLE AND UNSTABLE FILTERS 39

The rela t ion of th is result t o the ear lier discussion (Sec. 2.2) of the

stability of linear lumped-constant filt ers is easily under stood. We have

noted (Sec. 2“4) tha t the weight ing funct ion of such a filter is a linear

combin at ion of it s n orma l-mode fu nct ion s,

w(t ) = c, a(t ) + C,h ,(i) + C,h ,(t ) + . . . + Cnh.(t ), (56)

the h’s being given by Eq. (11). Now the in tegra l ~M lh (t )[ dtwilln ot

converge for any normal-mode funct ion hi tha t has w z O, nor can one

form any linear combina t ion of these funct ions for which such an integra l

converges. Thus the in tegral ~~ IW(t )l dt will converge if, and on ly if,

the weight ing funct ion conta ins no normal-mode funct ion for which

c%2 0. Stability of the filter is thus assured if all t he roots of Eq. (13b)

have negat ive rea l par ts, in accord with the ideas of Sec. 2.2. on the

other hand, the filt er may be stable even when there exist roots with

nonnega t ive rea l par t s if the cor responding undamped normal modes do

not appear in the weight ing funct ion , tha t is, if they are not excit ed by

an impulse input . Since any input can be expressed as a sum of impulse

inputs, th is is sufficient t o assure that no undamped modes can be excit ed

by any input whatever . The convergence of ~~ lW(~)l d,asa cr it er ion

of the stability of a filt er is thus precise and complete; in effect , it offers

a method of determining what normal modes of a filt er can be excited—

not merely what modes can conceivably exist .

On ly when a filt er is stable is it possible to speak with full genera lity

of its response to an input that star t s indefin itely far in the past . We

have seen tha t for a bounded input E,(t ) which vanishes for t <0, the

n orma l r espon se isI-t

E.(t ) = ) E,(t – ,) W(,)d,.

o– (43)

If E, has nonzero va lues when the argument is less than zero, the upper

limit of in t egr at ion must be cor respon dingly ext en ded; if t he input bega n

in the indefin itely remote past , we must wr ite

[

.

E.(t) = E,(t – T)W(,) d,.o–

(57)

If the filt er is stable—and hence if ~~ IW(,)I d, < W-thenthe in tegra lin Eq. (57) will converge for any bounded input . If, however , this exten .

sion of the limit is a ttempted in the case of an unstable filter , the resu lt ing

in tegr al ma y n ot con ver ge. This cor respon ds, of cou rse, t o t he possibilit y

tha t an unstable filt er subject t o an arbit rary input in the indefin itely

r em ot e past may give, at a ny fin it e t ime, an in fin it tiy la rge out put ,



40 MATHEMATICAL BACKGROIJND [SEC,29

We shall ther efor e apply Eq. (57) only in the t rea tment of stable

filters; in dea ling with linear filt ers in genera l, and unstable ones in

par t icular , it \ vill be nmxssary to usc an equa t ion of the form of Eq. (43)

and only inputs that sta r t a t a fin ite t ime.

THE FREQUENCY-RESPONSE FUNCTION

To this poin t we have considered the response of a linear filter to two

specia l t ypes of inputs-impulse and step inputs—and t he rela ted weigh t-

ing funct ion by which the filt er may be character ized. We now turn our

a t t en t ion to another specia l type of in pu t—the pure sinusoidal input—

an d t he r ela ted fr equ en cy-r espon se fu nct ion , wh ich also ser ves t o ch ar ac-

t er ize a ny st able lin ea r filt er .

We shall see that the response of a stable filter to a pure sinusoidal

input funct ion is a lso sinusoidal, with the same frequency but genera lly

differen t amplitude and phase. The frequency-response funct ion

expresses the rela t ive amplitude and phase of input and outpu t as func-

t ions of frequency. It is defined only for stable filter s, since a pure

sinusoidal input must star t indefin itely far in the past and can thus be

considered only in con nect ion with a stable filter . [Th e in pu t

E,(t) = O, t<o,E,(t) = A sin u,t, t>o, }

(58)

which might be applied to an unstable filt er , is not a pure sinusoid but a

superposit ion of sinusoids with angula r frequencies in a band about w]

The impor tance of the frequency-response funct ion rest s on the fact

tha t any funct ion subject to cer ta in rela t ively mild rest r ict ions can be

wr it ten as the sum of sinusoidal oscilla t ions (See. 2.11). The response

of a linear filter can be expressed as a similar sum of responses to the

sinusoidal component s of th e input by means of t he frequency-responsefunct ion , which rela tes cor responding components of input and ou tput .

2.9. Response of a Stable Filter to a Sinusoida l Input .—1n dealing

with sinusoidal inputs and outputs it is conven ien t to use the complex

exponen t ia l nota t ion . The genera l sinusoidal funct ion of angular fre-

quency ~ can be represen ted by a linear combina t ion of the funct ions

sin d and cos t it or , more compact ly, by a cos (cot + o), where a is the

~mplitude and @ t he phase with respect to some refer ence t ime. An

even more compact nota t ion is obta ined by represent ing th is sinusoidby t he complex expon en tia l

A ~i~~= aei~eit it , (59)

of which a cos (a t + O) is the real par t . Here phase and amplitude are

r epresen ted together by the complex factor A = a.d~, of which a is the

magnitude and @the phase. A change of amplitude by a factor b, together



SEC.2.9] RESPONSE TO A S INUSOIDAL INPUT 41

with a change of phase by A@, is then represen ted by mult iplicat ion of

the complex exponent ia l by the complex number be’~$; th is changes the

mult iplier of eio~ to abejt++~~) and the rea l par t of the whole expression

to ab cos (tit + @ + A@).When a complex funct ion is used to denote a filter input , a complex

expr ession for t he out pu t \ vill result . Because of the linear proper ty of

the filter , the rea l par t of th is complex output is the response of the filt er

t o the rea l par t of the complex input , and similar ly for the imaginary

par ts of input and output . It is thus easy to in terpret in rea l form the

r esu lt s obt ain ed by con sider in g complex in pu ts.

Use of the complex nota t ion makes it easy to prove that a sinusoida l

input t o a stable filter gives r ise to a sinusoidal outpu t . I,et

EI(t)= Aei”’. (60)

Then , by Eq. (57), ~ve have

/

.

E.(t) = A dr ~iti(c-,) ~(T) (m l )

o–

((X)1)1

where W(7) is the weigh t ing factor of the filter . For reasons tha t ~villbe eviden t la ter we shall denote the convergen t in tegra l in Eq. (60b) by

Y(ju):

/

m

Y(ju) = dr e-i”’~~(T). (61)o–

ThenEo(t ) = A Y(ju)eiU’. (62)

Thus the filter outpu t is sinusoida l in t ime; it differs from the input by a

constant complex fact or Y (jw). For unstable filt ers the integra l in Rq.(61) will, in gen er al, n ot con ver ge.

Considered as a funct ion of the angular fr equency, Y(ju) is ca lled the

frequency-response function. This funct ion expresses the amplitude and

phase difference between a sinusoida l input a t angular fr equency u and

the response of the filt er . The input amplitude is mult iplied by the

factor]Y(ju ) I = [Y(ju ) Y*(j@)]J+ , (63)

and the phase is increased by

[

~ = tan_,~ Y(jcd) – Y*(jcO)

1Y(ju) + Y“(jid) ‘(64)

wh er e t he a ster isk den ot es t he complex con ju ga te.

Exper imenta lly, the frequency-response funct ion of a filt er can be

determined by compar ing the amplitude and phase of sinusoida l inputs




at va rious frequencies with t he amplit ude and phase of t he corr espon ding

outputs. In order to compensa te for the fact tha t any real input star t s

a t a finite t ime, it is necessary to regard as the response only tha t par t of

the output in which the amplitude and phase do not change with t ime,

t hat is, t he so-ca lled “st ea dy-st at e response. ”

2.10. Frequency-response Funct ion of a Lumped-constant Filter .—

The frequency-response funct ion of a lumped-constan t filt er is easily

determined from the differen t ia l equat ion of the filt er . In th is equa t ion

d“Eo dn–lEO

a“ F + a“-’ dta-, + “ ‘ “ + a,Eo

we may set

E, = eiut,

Eo = Y(ja)e~@t. 1

We ha ve t hen , on ca rrying ou t t he differen t ia t ions,

[a .(ja)” + am--,(ju)”-’ + “ . + adze’”’

= [b~(jw)fi + bn-,(ju)~-’ +

whence

(65)

. ~ . + bO]e’-’, (66)

Y(ju) = bM(~~)m + bm .-,(.h )m -’ + . ~ + b,a n(jo)” + an_l(ju)n–l + . . + aO”

(67)

The frequency-response funct ion of such a filt er is thus a ra t ional func-

t ion , the ra t io of two polynomials in ju with coefficien ts tha t appear

dir ect ly in t he differ en tia l equ at ion .

I

b~Ytiw)l

ow +(: .

(a) -92 [b)

Fm. 2.1l.—(a) The amplitudeamplificat ionan d (b)t he pha seshiftof the circuitof Fig.2.1.

As examples we may take the two stable filt ers considered in Sec.2.6. For the simple RC-filt er of Fig. 2“1 we have, on reading t he required

coefficien ts fr om Eq. (1),



43EC.2.11] THE FOURIER INTEGRAL

The amplitude amplifica t ion and phase shift a re

IY(j@) I = (1 + t i’T’)-fi, (69a)

@ = tan-’(–~7’); (69b)t hese funct ions a re plot ted in F ig. 2..11.

The frequency-response funct ion of the. simple servomechanism

descr ibed by Eq. (46) is

The amplitude amplifica t ion and phase shift a re

@= – tan-l“ \ w .]

()2J (71b)

1–:u .

these quant it ies a re plot t ed as func-

t ions of u in Fig. 2.12, for ~ = ~.2.11. The F ou rier In tegra l.—We 1

have n ow t o consider h ow an arbit ra ry

input can be expressed as a sum or in-lY(jw)l

tegra l of s inusoidal components.

The represen ta t ion of a per iodich

funct ion by a Four ier ser iesl will be

assumed to be familia r t o the reader .

Any funct ion g(t ) tha t is per iodic int im e wit h per iod 2’, is of bou nded va ri-

a tion in t h e in t er va l

–T~<ts;,

and is proper ly defined at point s of

discont inu ity can be expressed as an

infin ite sum of sinusoidal t erms withfrequencies tha t a re in tegra l mul-

t iples of t h e fundamen ta l fr equ en cy

(72)

-v L.----- AL-------

(b)

F1~. 2.12.—(a) The amplitudeam-plificationand (b) t h e phase sh ift of as imple servomechanism,as given byEqs. (71) wit h~ = +.

1A. Zygmund, TrigonometricalSeries,Zsubwenczi Fu nduszuKu ltur y Na rodowej,War saw-Lwow,1935; E . T. Wh it t aker and G. N , Wa t son , Modem Analysis, Mac-millan,New York, 1943.




In terms of complex exponen t ia l one can wr ite

+-

g(i) = ~ ane2”jJ@, (73a)

n-—m

wh er e t he coefficient s am ar e given by

(73b)

When t he fu nct ion g(t) is n ot per iodic bu t sa tisfies ot her con dit ion s-

t he con ver gen ce of (_+W”dt lg(t) I is sufficien t-it is possible t o r epr esen t

the funct ion , not by a sum of terms with discrete frequencies nfl, but by ~

sum of terms with all frequencies .f:

/

+-g(t) = df A (f)ez”JJt. (74a)

—.

This in tegra l will oft en converge on ly in a specia l sense. The funct ion

.4 (~), which gives th e phase and r ela t ive amplitude of t he component with

fr equency j, can be computed by means of the formula

\

+-A (j) = dt g(t)e–zrjft. (74b)

—.

Equat ions (74) provide an extension of Eqs. (73) for the limit as

T~-. The reader is refer r ed to standard text s’ for a complete dis-

cussion. It will suffice hereto show that the extension is plausible. It is

obvious tha t we can const ruct a funct ion h(t ) tha t is per iodic with the

period T and is ident ica l with g(t ) in the in terva l (– T/2 < t< T/2).

Mor eover , this can be done h owever lar ge t he (fin it e) fundamenta l per iodT is made. For each value of 2’, Eqs. (73a) and (73tJ ) hold, with h(t)

in the piace of g(t). It is plausible to assume that these equat ions hold

in the limit as T* m, If we set j = n/T, dj = I/T, and Tan = A(j),

then as T becomes infin ite h(t ) becomes g(t ), and Eqs. (73a) and (73b)

become Eqs. (74a ) a nd (74b) r espect ively.

For many purposes it is conven ien t to express the Four ier in tegra l

rela t ions in terms of the angular fr equency u = %rj. With th is change of

va ria ble, E q. (74a ) becomes

‘(’)‘i-LD”@e’”’ (75)

] E . C. Titchmarsh , In t roductwnto th e T h eory of th e Fou rier IntegralsrClarendonPr ess,Oxford, 1937.



SEC.2.11] THE FOURIER INTEGRAL 45

As an a lterna t ive form, we shall wr it e

(76a)

where

!

+-

G(jo) = di g(t.)e-~-t. (76b)—.

Considered as a funct ion of the rea l angular frequency u, G(@) will be

t ermed t he F ou rier t ra nsform of g(t ).

It is clea r that A(j) and G(ju) a r e in a sense both Four ier t ransforms

of g(t ), since they represen t the same funct ion of frequency. They are,

h owever , differ en t fu nct ion s of t heir in dicat ed a rgumen ts, wit h

A(j) = G(%jf). (77)

In this chapter we shall herea ft er dea l ordy with th e representa t ion G(ju),

in order to proceed conven ien t ly from the Four ier t ransform to the

Laplace t ransform.

It should be noted that if g(t ) is an even funct ion of t , g( –t ) = g(t ),

t,hen

/

.

G(jo) = 2 dt g(t ) Cos cu t. (78a)

oIt follows that G(u) is an even rea l-valued funct ion of u and that g(t ) can

be wr it t en as1 r-

g(t ) = ; Jo do G(jo) COSd. (78b)

If g(t ) is an odd funct ion of t , g( – t ) = –g(t ), t hen

/

G(@) = 2.i - dt g(t ) sin ut . (79a)

oSince G(&) is then an odd funct ion of u , one can wr ite

/

.m

g(t) = : du G(@) sin od. (79b)o

As an example of the Four ier in tegra l representa t ion , let us consider

the funct iong(t ) = e–”l’l, (80a)

shown in Fig. 2.13a. Then by Eq. (76b)

\

+-G(jw) = d~ e–ald-jd

—.

(81)




Thus

G(jr.o) = &,; (80b)

t his fu nct ion is’plot ted in F ig. 2“13b. Th e Fou rier in tegr al r epr esen ta tionof the given funct icm is, by Eq. (76a),

(82)

The validity of this representa t ion can be proved by evalua t ing the

in tegra l, for instance, by the method of residues. 1 As a br ief review of

C(@)

& A+

a b

FIG.2.13.—(a)Plot of th e fun ctionu(t) = e-”’ll; (b ) the Fouriertr an sformof th e fun ctiong(t).

the method of residues, th is evalua t ion will be ca r r ied th rough in some

detail. It is conven ien t for th is purpose to change the ~ar iable of

4 L

aFIG.2.14.—Pa th sof int egra tionin the”

complexp-plane. (a) Pat hof integrat ion( +@, j ~); (~), (c) pa th s of in t egr a tionfor useof methodof residues.

-.

in tegra t ion from w to j~ or , morepr ecisely, t o in t rodu ce t he complex

variable

p=cl+ju, (83)

‘of which & is the imaginary pa r t ,

and to replace the in tegra l over rea l

va lues of u by an integra l over pure

imaginary values of p. The integra l

of Eq. (82) then becomes

wit h t he path of in t egr at ion along t he

imaginary axis in the p-plane, as

shown in Fig. 2.14. By resolving the integrand in to par t ia l fract ions,

the in tegra l in Eq. (84) can be brough t in to the form

I . a/r 1

/(

j-du — eiwt= _ dp ~–~

)

ept.az + COz %J -3. p+a

(85)—. p–a

1See E . C. Titchmarsh , The Theor~ of Funct iona l Oxford, New York, 1932.



SEC.2.11] THE FOURIER INTBGRAL 47

It remains, t herefor e, toevalua te expression~of the form

/

3-

1=~.1

2Rj -j.dp — ~nt.

p–a(86)

We now apply the method of residues. If t <0, the integrand

approaches zer o as Ipl ~ m in the r igh t ha lf of the p-plane; in fact , it

can be shown that t he line integra l a long the semicircle CEof radius lt in

the r ight ha lf plane approaches zero as Rbecornes infin ite. Let us then

consider the Iinc in tegra l of th is in tcgrand around the closed con tour (b)

of Fig. 2.14. This consists of twopar t s: thcin tegra l a long the imaginary

axis from —jRto +jRandthc in tegra l around thesemicircle C~:

1I(R) = Z=j

$

1({p

“= w::+ L.)dp+’p’ ‘87”)–a”

AS R - cc, the second in tegra l t ends to zero, and the fir st approaches the

integra l 1 of Eq. (86). Thus

(87b)

t he desir ed lin e in tegr al ca n be eva lu at ed as t he limit of a con tou r int egr al.Now the integra l a round the closed con tour is equal t o 2ir j t imes the sum

of the residues of th e in tegrand at all poles enclosed by the contour , t aken

with a minus sign because the in tegra t ion is in the clockwise sense. As

R ~ m, t h e con t ou r (b) of Fig. 2.14 will come to enclose all poles in the

r igh t half plane. If a has a posit ive rea l par t the in tegrand has a pole in

the r ight ha lf plane, and

I = – &. 27r je0t= – (’”’ [( <0, Rc(a) > O].l (88a)

If a has a negat ive rea l par t , there is no such pole, and

1=0, [t <0, lie(a) < O]. (88b)

Since t he a of Eq. (85) is a posit ive r ea l qua nt ity, t he fir st t erm con tr ibu tes

noth ing to the in tegra l, and

(89)

If t >0, the integrand of Eq. (86) approaches zero as Ipl - m in

the left ha lf plane. By arguments similar t o those above, the desired

integra l is equal t o the in tegra l around the con tou r (c) of Fig. 2.14, in

the limit as R + co. This is in tu rn equal t o %j t imes the sum of the

I The symbol Rc(a) denotes t h e r ea l pa r t of a .




residues in the left half plane, taken with a plus sign because t he integra -

t ion isin the counterclockwise sense. Thus

1=0, [t> O,h!e(a) >0], (90a)

I = e“’, [t> O, Re(a) <O]. (902))

InEq. (85) thesecond term cont r ibutes noth ing tothein tegra l, and

\

+-du -~ # = e–at = e–altl

~z + @z(i > o). (91)

—.

(hcombining these resu lt s, onever ilies the tru th of Eq. (82).

2s12. Response of a Stable Filter toan Arbit ra ry Input .-Let us con-

sider the response of a stable linear filt er to an input gl(t ) with Four ier

t ransform,. Then

\

+.

g,(t ) = & _m du G@) e?w~. (92)

Since the response of the filt er t o an input e~”’is t he output Y(@)e’W’, it

follows from the linear proper t y of t he filt er that its r esponse t o t he input

g,(t ) is t h e ou tpu t

/

+.

go(t ) = & _ ~ dw G,(ju) Y(ju)e’w’. (93)

It is evident that the Four ier t ransform of the filt er output is

Go(ju) = YE,. (94)

That is, ihe Fourier transform of ihe jilter ouiput is equal to the Fourier

transform of the input multiplied by the frequency-response junction of the

jilter.

2.13. Relat ion between t he Weight ing Funct ion and the Frequency-

response Funct ion.-The rela t ion between t he frequency-response func-

t ion of a stable filt er and the weight ing funct ion has been sta ted in Sec.

2.9:

\Y(ju) = “ dr e–;ti’~(r). (61)

o–

Since W(7) vanishes for , <0, we may write

/

+-Y(jw) = dr e-jm’W(r). (95)

—.

The frequeny-response function of a stable jilter is the Fourier transform

of the weight ing funct ion. It is impor tant to note tha t this theorem isrest r ict ed to stable filtera . For unstable filt ers the in tegral in Eq. (95)

will, in genera l, not exist . The inverse of t his rela t ion is, of cour se,

(96)



t ?mc.2.13] WEIGHTIN G AN D FREQUEN C Y-RES PONS E F UN CTION S 49

The significance of this r ela t ion and th e impor t ance of t he rest rict ion

t o st able filt er s may be illu st ra t ed by a con sider at ion of lumped-con st an t

filt er s, for wh ich t he fr equ en cy-r espon se fu nct ion is

y(ja) = fAn(j@)’”+ bm-1(ju)-1 + “ “ “ +bo

G(jw)” + a ._l(ju)”–l + . . . + ao’ (m ~ n) (67)

For simplicity, let us assume that the complex constants

pi = (U + j(di, (14)

which are the root s of the equat ion

P(p) = anp” + an_lp”-l + . . . + ao, (13b)

are all dist inct . Then Y(ju) can be expressed as a sum of part ia l fract ions,

Y(j.) = co + -,U:p,+j++ ”””+,+; @7)where the constan ts Ci depend on the b’s as well as on the a’s. Th e

constan t COwill vanish unless m = n.

First , let us assume tha t all the p’s have negat ive real parts—all

normal modes of the filt er a re damped. Let us fur ther assume tha t

m < n. Then the filt er is stable, and we should be able to comrmte

W(t) asL

‘(’) ‘WW#Ih+ ~ “ “ +* )’ i ” ’ ’9 8)

To evalua te this integra l it is again convenient to in t roduce the complex

variable

p=a+jw, (99)

of which@ is the imaginary par t . Th e int egr al of Eq. (98) t hen becomes

‘(’) ‘M::’(+F1+“ “ +5%)’”“m)with the path of in tegra t ion a long the imaginary axis in the p-plane,

as shown in (a) of Fig. 2.14. This in tegra l can then be evalua ted by the

method of r esidues.

Following the procedure out lined in Sec. 2.11, if t <0 we in tegra tearound the contour (b) in Fig. 2.14. Since for the stable filter none of

the poles of the in tegrand lie in the r ight half plane, it follows tha t

w(t ) = o, t<o. (lOla)

If f >0, we in tegra te around the contour (c) in Fig. 2.14. Each term

Ci/ (p – pi) contr ibu tes t o the integra l in this case, and we obtain

W(t) = C,ep” + Czep” + . . + Cne””, f>o. (lOlb)



50 MATHEMATICAL BACKQ’ROUND [SEC.214

Thus we have found that the weight ing funct ion is a sum of normal

modes; in addit ion , we have a means of determining the constan ts C. by

r esolu tion of t he fr equ en cy r espon se Y(@) in to pa rt ia l fr act ion s.

These resu lt s for a stable filt er a re in accord with our ear lier ideas

a bou t t he weigh tin g fu nct ion . Now let us consider an unstable filt er .

We shall see t ha t for such a filt er t he fr equen cy r espon se is n ot t he F ou rier

t ransform of the weight ing funct ion; the assumpt ion tha t it is will lead

us to false resu lt s. We assume, then , tha t some of the p’s have posit ive

rea l par t and that these p’s appear in the resolu t ion of Y(jo) in to par t ia l

fract ions. Let us a t t empt to compute W(t ) by means of Eqs. (98) and

(100). For t < Owe no longer obta in W(t) = O bu t

W(t ) = – ~ Ciep”, (t< o), , (102a)

(- : o)

a sum including a term for each pi with posit ive real par t . On the other

hand, when t >0, the con tour of in tegra t ion surrounds only poles with

nega t ive rea l par t , and we obtain

(102b)

(a ,‘<o)

Both these resu lts a re er roneous: the weight ing funct ion must be zero

for t <0 and must include a normal mode with posit ive rea l par t when

t>o.

2.14. Limita t ions of the Four ier Transfoxm Analysis.-The Four ier

t ransform techniques considered above are usefu l in the discussion of

filt ers, bu t their applicability is limited by the fact tha t the Four ier

t ransform is not defined for many quant it ies with which one may need

to dea l. We have just seen that the weight ing funct ion of an unstable

filt er does not have a Four ier t ransform. The same is t rue of manyimpor t an t t ypes of filt er input : t he unit -st ep funct ion , t he pur e sinusoid,

t he “con st an t-velocit y fu nct ion ” [z(t ) = t it ], t he in cr ea sin g expon en tia l;

for none of these funct ions does the in tegra l of the absolu te magnitude

converge.

In some ca ses it is possible t o ext en d t he discu ssion by t he in tr odu ct ion

of con ver gen ce fa ct ors, which modify t he funct ions sufficien t ly t o ca use

the Four ier t r ansform to exist but not so much as to hinder the in ter -

preta t ion of the result s. This device is somet imes usefu l but may involve

ma th ema tica l difficu lt ies in t h e u se of double-lim it in g pr oces ses.

A more genera lly sa t isfactory procedure is t o make use of the Laplace

transform. This is defined for funct ions tha t differ from zer o only when

t >0 (tha t is, a fter some defin ite instan t); it is defined for all prac&al

61ter inputs, for t he normal responses t o these inputs, and for t he weight -



SEC.2.15] DEFIN ITION OF THE LAPLACE TRANSFORM 51

ing funct ions of stable and unstable filter s. The t rea tment of filt ers by

Lapla ce t ran sform met hods is closely para llel t o t he discu ssion in t erms of

Four ier t ransforms, but its rela t ive fr eedom from rest r ict ions makes it

decidedly t he m or e power fu l m et hod.

THE LAPLACE TRANSFORM

The following discussion of the Laplace t ransform and its applica-

t ions is necessar ily limited in scope and deta il. No a t tempt has been

made to sta te theorems in their maximum genera lity. For a more

extended t rea tmen t the reader must be r efer r ed elsewhere. 1

2.15. Defin it ion of the Laplace Transform.-Let g(t ) be a funct ion

defined for t z O, and let

Ig(t )l s Kc”’ (103)

for some posit ive constan t a . Then the in tegra l

\F(p) = “ d g(t)e-p’ (104)

o

is absolu tely convergen t for a ll complex va lues of p such that the rea l

par t of p is gr ea ter than a. Considered as a funct ion of the complex

variable p, l’(p) is termed t he Laplace t ransform of g(t ); it maybe den ot ed

a lso by

/S[g(t )] = mdt g(t )r@, (105)

o

the argument p being understood. If a is the grea test lower bound

of rea l constan ts for which g(t ) sa t isfies an inequality of the form of Eq.

(103), t he Lapla ce t ra nsform of g(t ) con ver ges a bsolu tely in t he h alf pla ne

to the r igh t of p = a; a is ca lled t he abscissa oj absolute conver gen ce. Th e

region of defin it ion of ~[g (f)] can usually be extended by analyt ic con-t inuat ion t o include th e en tire p-plane, except for th e poin ts a t which F(p)

is s ingu la r . In what follows, this extension of the domain of defin it ion

will be a ssumed.

The Laplace t ransformat ion can be defined for cer t a in types of func-

t ions that do not sa t isfy Eq. (103). In wha t follows we shall consider

on ly fu nct ion s t ha t con ta in a fin it e n umber of delt a-fu nct ion sin gu la rit ies,

in addit ion to a par t sa t isfying Eq. (103). When one of these delt a

funct ions occurs a t t = O, we shall define the Laplace t ransform as

/J3 [g(t )] = mg(t)e-p’.

o–(106)

1G. Doet sch , T?wor ieund A nwendung&r Lap laze Transformation , Springer,Ber lin , 1937; H. S . Cars lawand J . C. J aeger ,Opera t iona lMethod s in Applied Mathe-

matics, Oxford , New York , 1941; D. V. Widder , The Laplace Transform,PrincetonUniversityPress,Princeton, N. J ., 1941. /




This extension of the definit ion of the Laplace t ransform is necessa ry, in

view of th e two-sided ch aracter of t he delta funct ion , t o assure tha t

lim J 3[g(t – [tOl)]= @g(~)]. (107)t,+o

For pure imaginary va lues of p, p =,ju , the Laplace t ransform of a

funct ion that is ident ica lly zero for t < Obecomes its Four ier t rans-

form—if th is exist s. Thus the Laplace t ransform is, in a sense, a gen-

era liza t ion of the Four ier t ransform [Eq. (76 b)] applicable when g(f)

vanishes for t < 0.

Th e in ver se of t he Lapla ce t ra nsformat ion [Eq. (104)]is

1

/

b~j=

9(0 =~j b_jm dp F(p)e”’, (108)

where the pa th of in tegra t ion in the complex plane runs from b — j~

tob +j~, tother igh t of theabscissa ofabsolu te con\ 'ergence.

Examples.—It will be ~vor th }vhilc to give a number of examples of

the Laplzce t ransform for fu ture rcfcrcnce. They can be ver ified by

direct in tegra t ion .

EXAMPLE l. —Theun it -st t ’p funct ion u ([):

u(t ) = o, 1 <0,u(t ) = 1, tzo. I

~

m

S[u(t – to)] = d ll(t – to)c–’”

o

[

.— ~~p–it =

~–l>ta—

(0 p ‘

In par ticular , JWIn ot e t hat

C[u(t )] = ;.

lh .4MPLE 2:

(

o, (t < o),

g(t) =

sin d, (t 20);

wc(g) = -~

p + u~”

I

o, (t < o),

g(t) =

t“, (~~o);

~!C(9) = ~i”

ExAMpLE 3:

(to z 0). (10!)/

(109C)

(llOa)

(llOb)

(llla)

(lllb)



SEC.2,16] PROPERTIES OF THE LAPLACE TRANSFORM 53

In pa rt icular , we n ot e tha t for t he” unit -r amp funct ion,”

(

, ($ < o),

g(t) = (l12a)

t , (t 20);

EXAMPLE4:

J3(g) = ;. (l12b)

{

o, (t < o),

g(t) = (l13a)peal, (t> o);

s(g) =

~!

(p – a).+l- (l13b)

EXAMPLE5.—The unit -impulse funct ion b(t – to):

/

.

.s[a(t – to)] = d ~(t – .io)e-pL= e–@O, (to 2 o). (114)n–-

In pa r t icu la r ,‘C[a (t )] = 1. (115)

2.16. Proper t ies of the Laplace Transform. -We here note some

proper t ies of the Laplace t ransform that are usefu l in determining the

t ransform of a funct ion or the inverse of a given t ransform. All funct ions

con sider ed will be of t he r est rict ed cla ss defin ed in Sec. 2.15.

Lineari ty. -If gl(t) and g,(t) have the t ransforms &(g,) and ~(g,), then

J3(clgl+ c2g2) = cl&(gl) + c2c(g2), (116)

wh er e c1 a nd CSa re a rbit ra ry con st an ts.

Laplace Transform of a Derivative:

()# = fro(g).

Th e pr oof is simple:

()/

dg “

‘a=o_ /dt~e-’t= “ +P “(t)e–’” dt g(t)e–pf

(l– (l–

(117)

= g(o–) + pc(g), (118)

from which Eq. (117) follows, since for all funct ions under dkcussion

g(O– ) vanish es.

Thus we see tha t mult iplica t ion of the Laplace t ransform of a funct ion

by p corresponds t o taking the der iva t ive of the funct ion with respect to t.

An example of this rela t ion is provided by the unit -step funct ion u(t ) and

its der iva tive, t he unit -impulse funct ion ~(t).By Eq. (1 17) we must

haveJ2[a(t)]p2[?L(t)] (1191




by the resu lts of the preceding sect ion this is

l= p:.P

Lap.lute Transform ojan Integral:

=wo’g(”l%(’)]

(120)

(121)

This isessent ia lly the same as Eq. (117), with g(t ) here playing the role

of d g/ d t in t ha t equa tion .

Division of a La place t ra nsform bypt hus corresponds t o in t egrat ion

of the funct ion with respect to t,with appropr ia te choice of the lower

limit of the in tegral. Since n-fold applicat ion of the opera tor ~ dt t o\o–

t he u nit -st ep fu nct ion gives

u ),,

dt u (t) =$, (t > o), (122)

it follows t ha t

~“i-) -(-)

t“ 1 ‘~.—~! P P’

(123)

the result of Example 3 of the preceding sect ion then follows from the

lin ea rit y of t he Lapla ce t ra nsform.

Laplace Transjorm oj e-o’ g(t) .—If g(t ) has the Laplace t ransform

F(p), t henJ 3[e-’’g(t )] = F(p + a). (124)

For example, &[t nea ’u (t )] is obt a in ed by r epla cin g p by p – a in

.qtw(t)] $.

This istheresu lt sta ted inEq. (IIM).

Laplace Transjorm of the Convolu tion oj Two Functions.—Let g,(t)and gz(t ) be two funct ions of t that vanish for t < 0. The con volut ion of

t hese t wo funct ions is

\h(t ) = “

/

.

dr g,(T)g,(t – r) = d7 g,(t – T )g,(r).o– o–

(125)

If gl(t ), gz(t ), a nd h(t) ll possess La pla ce t ra nsforms, t hen

S (h) = SO. (126)

For

‘(h) = L:dte-”’l:d’g(’)g,(’-r)

/

.

\

.— dr gl(r)e–p’ d t g2(t – r)e-p(’y). (127)

o- o–



SEC.2.16] PROPERTIES OF THE LAPLACE TRANSFORM 55

Changing the var iable of in tegrat ion in the second integra l to s = t– r,

and remember ing that gz(s) = O if s < 0, we have

\

.

/

.

&(h) = dr gl(~)e–P’ ds g~(s)e–~”,o– o–

(128)

whkh is a resta tement of Eq. (126).

The mult iplicat ion of two Laplace t ransforms thus cor responds to

format ion of the convolut ion of their inverse funct ions. This is often a

convenient way to determine the inverse of a given Laplace t ransform

when it can be factored in to two Laplace t ransforms with recognizable

inverses.

Limiting Values of the Laplace Transform.—Let F(p) be the Laplace

t ran sform of g(t). When the indica ted limits exist , then the following

t heor ems a r e va lid.lim pF(p) = }~m~g(t). (129)p-o

If g(t ) conta ins no delta-funct ion term at t = O,

lim pi’(p) = g(O+). (130)p+ m

If g(t) conta ins a t erm K ~(t ), then

lim F(p) = K, (131)p+ m

andlim p[l’(p) – K] = g(O+). (132)p+ w

Th e pr oof of t hese r ela t ions can be ca rr ied ou t along t he followin g lin es

when g(t) con ta in s no delt a funct ion s. We note that

I .pF(p) = dt g(t)pe_@ = –/

m dt g(t) ~ (e-p’).o–

(133)o-

In tegrat ing by par t s, we have

wher e dg/o!t may con ta in delt a funct ion s cor r esponding t o discon tin uit ies

in g, including a term g(O+ ) ~(t) cor responding to a discont inuityy at

t = O. In the limit as p ~ @, on ly t his la st delt a funct ion will con tr ibu te

to the in tegral on the left ; one has

Iim pF(p) =

/

0+ di $ = g(o+). (135)P- m o–




Sim ila rly, r epla cin g c–’” in t he in tcgr an d by it s lim it a s p ~ O, one obta ins

lim pF(p) =\

“dtg=g(co).p+o o–

(136)

2.17. Use of t he La place Tra nsform in Solu tion of Linea r Differen tia l

Equat ions.—The Laplacc t ransform th eor y offer s a convenien t method

for t he solu tion of lin ea r differ en tia l equ at ion s, su ch as t he filt er equ at ion

&lEoa. ~ + an-l ~m_, —+”” +a~EO

= b 1~’ + bw_, ~~’ + . . . + boE,. (2)m dtrn dtm+

The formula t ion of th is method is par t icu lar ly simple when the init ial

condit ions on the solu t ion EO cor r espond to sta r t ing of the system from

rest under an input El tha t begins a t a defin ite t ime, say t = O; that is,

when it is requ ired to find the norma l respon se of the system to such an

input . Under such condit ions, both sides of Eq. (2) represen t funct ions

tha t begin to differ from zero only at t = O. Equat ing the Laplace

t ransforms of th e two sides of this equa t ion and making use of the proper -

t ies of t he t ra nsforms a s discu ssed in t he pr ecedin g sect ion , we h ave

(amp ’ + a.-lp”-’ + . . + aO)SIZlO(~)l

= (b~p’” + b~_,p”-’ + “ “ “ + bo).$[E ,(t )l. (1~~)

Writing~(p) = b~p~ + b~-,p~-l + ~ . . + b,

anp” + an_lp”–l + . . . + aO’(13!3)

as in Eq. (67), we have

mc[llo(t)] = Y(p)s[E,(t)]. (139)

Thus it is easy to obta in t ie Laplace t ransform of the filt er ou tpu t by

mult iplying the Laplace t ransform of the input by a ra t ional funct ion in p

wit h coefficien ts r ea d fr om t he differ en tia l equ at ion . The outpu t it self

can th en be det ermin ed by applica t ion of t he in ver se Lapla ce t r an sforma -

t ion [Eq. (108)] or by resolving the Laplace t ransform of the outpu t in to

pa rt s wit h r ecogn iza ble in ver ses.

As an example, let u s det ermin e t he weigh tin g fu nct ion of t he RC-filt er

of Fig. 2“6. This can be obta ined by solving Eq. (30) with E](t ) = ~(t ).

We have then , by Eq. (115),

J3[ll,(t)] = 1. (140)

In this case Eq. (139) becomes

(141)



SEC.217] SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS 57

Thus, for the delt a -funct ion input , the Laplace t ransform of the output is

11— .&[W(t)] = *1 =l–T, 1

P+~

(142)

The inver se of 1 is, of course, ~(t ); the inverse of the second term follows

from Eqs. (113), with n = 1. We thus find

W(t) = o, (t < o),

W’(t) = b(t) – +, e-;, (/ ~ f)),) (143)

in agreement with Eq. (35).

As a second example, we may der ive the weigh t ing funct ions given in

Eqs, (48) to (5o). These are solu t ions of Eq. (46), which can be rewr it t en ,

by use of Eqs. (47); as

F or a delta -funct ion input , C(%) becom es

(146)

The denom ina tor fa ct ors in t~

r esolvin g t he t erm on t he r igh t in to pa rt ia l fr act ion s, we obt ain

[

1.fJ w(o] = 2(*2 : 1)}4 p + UJ – 4(2 – 1)~~

1—p + M + W.(-P – 1)J 61

. (147)

Double applica t ion of Eqs. (113) then gives

w(t ) = o, (t < o),

W(O = 2({23 ~)k[e-m.{[email protected]~uM

1

(148)_e–u”rt -a. (r~l)W] (t>o).

This genera l resu lt takes the forms of Eqs. (48) t o (50) for ~ <1, ~ -+ 1,

and { > 1, respect ively.




THE TRANSFER FUNCTION

2.18. Defin it ion of t he Tr an sfer F un ct ion .-Th e t ransfer junct ion of a

filt er is defined to be the Laplace t ransform of its weight ing funct ion .

In th is volume, t ransfer funct ions will usually be denoted by Y(p), withdist inguish ing subscript s as required. We have, then ,

/Y(p) = J3[w(t)] = mW(t)e–”.

o–(149)

The normal response of a linear filt er (stable or unstable) to an input

E,(t ) tha t is zero for t <0 can be wr it ten as

/

.

J%(t ) = dr E,(t – ,) W(r). (57)o–

It will be noted tha t this is t he convolu t ion of the input and the weight ing

funct ion , as defined in Eq. (125). It follows, by Eq. (126), tha t when the

Lapla ce t ra nsforms exist ,

S[li!o(t)] = Y(p)s[E,(t)] : (139)

The transfer function of a jilt er is the ratio of the Laplace transforms of any

normal response and the input that produces it. The use of the symbol

Y(p) in Eq. (139) and t hr oughout t he whole of t he pr ecedin g development

is thus consisten t with the nota t ion for the t ransfer funct ion here in t ro-

duced. instead of defin ing t he t ra nsfer funct ion for t he lumped-const an t

filt er direct ly by Eq. (138), we have chosen to define it as the Laplace

t ransform of the weight ing funct ion , which has been taken t o be t he more

pr im it ive con cept in t his ch apt er .

The t ransfer funct ion may be regarded as a genera liza t ion of the fre-

quency-response funct ion . Unlike the frequency-response funct ion ,

it is defined for unstable filt ers as well as stable filt ers. It is defined forgen er al complex va lu es of t he a rgumen t p, a nd not just for pu re imaginary

values of @ (u is rea l va lued). When the frequency-response funct ion

exist s, it can be obt ained from t he t ra nsfer funct ion by r epla cing t he a rgu-

ment p by ju [compare Eqs. (61) and (149)]; the values of the frequency-

response funct ion a re the va lues of the t ransfer funct ion a long the

imagina ry a xis in t he p-pla ne.

In the preceding sect ion we have seen how, for a lumped-constan t

filter , Eq. (139) ca n be der ived fr om t he differen t ia l equa tion of t he filt er

and how it can be used, instead of t he differen t ia l equa t ion , in determin-

ing the normal response of the filt er to a given input . In solving many

problems it is possible t o dea l exclusively with the Laplace t ransforms of

input and output and with t ransfer funct ions, except perhaps in the final

in t erpreta t ion of the t ransforms in terms of funct ions of t ime. It is



SEC.2.19] TRANSFER FUNCTION OF A FILTER 59

then useful to abbrevia te the notat ion of Eq. (139) and to wr ite simply

~O(P) = y(P)~l(P), (150)

the indicat ion of the argument P giving sufficient warning that it is a

Lapla ce t ra nsform wh ich is in volved.

F ilt er s con sist ing of many pa rts can be descr ibed by differ en tia l equa-

t ions that govern the several par t s or by a single different ia l equat ion

der ived from these by eliminat ion of in termediate variables. In the

same way one can descr ibe the components of a filt er by equa t ions of the

form of Eq. (150) and can eliminate var iables between these equat ions,

by pu rely algebr aic man ipula t ion , t o obtain a similar equ at ion gover nin g

t he over -a ll ch ar act er ist ics of t he filt er . This calculus of Laplace trans-

forms provides a formally simpler descr ipt ion of the systems than that in

t erms of differ en tia l equ at ion s a nd will be much u sed in t his book .

Aver y simple example is pr ovided by a filter th at consist s of t wo filter s

in ser ies . Th e fir st filt er , wit h t ra nsfer fu nct ion Yl(p), r eceives a n in pu t

E, and yields an outpu t EM; EM then serves as input to a second filter ,

with t ransfer funct ion Y2(P), which gives t he final ou tpu t Eo. In terms

of Laplace t ra nsforms we h ave

E JZ(P) = Y 1(P)~l(P), (151a)Eo(P) = Y z(P)~~(P). (151b)

Eliminating EM(p), we obtain

Eo(P) = Y ,(P) Yz(P)~dP). (152)

We seen then, that the over -a ll t ransfer funct ion of the complete filt er is

Y(p) = Y,(p) Y,(p): (153)

The transfer j’unction oj two filters in series ia the product of ~heiT individualtransfer function.s. By Eqs. (125) and (126) one can infer that the weigh t -

ing funct ion of two filters in ser ies is the convolut ion of their separate

weighting functions:

/w(t) = “ fh w,(r) W,(t – T). (154)

()-

This result can also be der ived from the rela t ions

/

1

E~(t) = dr E,(t – 7)W,(~), (155a)o–

,!

t

Eo(t) = d, E,,(t – ,) W,(7), (155b)o–

wh ich cor respon d t o Eqs. (151a) and (15 lb), r espect ively.

2019.

function

Transfer Funct ion of a Lumped-constant Filter .-The t ransfer

is more generally useful in the discussion of filt ers than is the




fr equ en cy-r espon se fu nct ion , in pa rt beca use it is defin ed for a wider class

of filt er s. For an illust rat ion of its use weshall return tothe considera-

t ion of lumped-constant filters, for which the t ransfer funct ions a re

r at ion al fu nct ion s of p, wit h coefficien ts t hat ca n ber ea dfr om t hegovem -

ing diiTerent ia l equat ion:

Y(p) =b~pm + b~–,p--l + . . . + b.

a.p’ + an_lb”–l + . - . + aO”(138)

If m > n, the resolu t ion of Eq. (138) in to par t ia l fract ions conta ins

t erms of t he form A~–~p*–’,A”-_lp-–l,l, . . . , A ,p. The filt er ou t pu t

then conta ins terms propor t ional to the first and up to the (m – n – l)th

der iva t ive of the input . In par t icular , W(t) conta ins these der iva t ives

of the delta -funct ion input . Then ~~dt]W(i)l does not converge, as

can be seen by consider ing the funct ions approximat ing to the delt a

funct ion; the filt er is unstable. We have a lready noted tha t with a

passive lumped-constant filter one cannot have m > n.

If m s n and the root pi of

F’(p) = amp” + a._ Ipn-l + . . . + aO = O (13b)

is si-fold, the genera l resolu t ion of Eq. (138) in to par t ia l fract ions is of

t he form

cl,l’(p) = co + -—

CM cl,.,

P–P1+(P –P1)’+””’+(P– TW’~ c,, C22 C2,,2

‘(P– P2)’+”””+(p–p2)”’P– P2

-1-”””.

The inverse of th is is, by Eqs,

w(t)w(t ) = cd(t)

[+ cll+:t+ ~+.

+ . . . .

(156)

(111) a nd (113),

= o, (t < o), \

The delta -funct ion term appears in the weight ing funct ion on ly if m = n;

it represen ts a term in the genera l ou tpu t tha t is propor t ional t o the input .

The other terms in W(t) represen t the t ransient response of the filter to

t he impulse input , expr essed as a sum of norma l-mode funct ions.

It is evident tha t the weight ing funct ion will conta in an undamped

normal mode and the filt er will be unstable if and on ly if the resolu t ion of



SEC.2.20] THE STABILITY CRITERION 61

Y(p) in to par tia l fr act ion s con ta in s a pi with nonnega t ive rea l par t . This

is not the same as saying that the filt er is stable if and only if Eq. (13b)

has no such roots; it may have such root s and st ill be stable if the cor -

responding factor (p — pi)” in the denomina tor in Eq. (138) is canceledou t by a similar factor in the numera tor . When this happens, one may

say tha t the network has an undamped na tura l mode but tha t the input

terminals a re so connected to the filter that no input can excit e th is type

of response. This may be a very precar ious type of stability, since a

small ch ange in th e filter constan ts—in t he b’s—may make this cancella -

t ion in exact and t he 61ter unstable.

2.20. The Stability Cr it er ion in Terms of the Transfer Funct ion .-It

will be n ot ed t hat p’s withnonnega t ive rea l par t s will be absent from Eq.(156) and the lumped-constant filt er will be stable if and on ly if Y(p) has

no poles in the r igh t half of the p-plane or on the imaginary axis. This is

a specia l ca se of a mor e gen er al sta bilit y cr it er ion : If t he t ra nsfer fu nct ion

Y(p) is analyt ic in the r ight half plane and is well behaved on the

imaginary axis—for instance, if t he absolu te va lue squared of dY/dp is

in tegr able-t hen t he filt er is st able. If a t least one singula r poin t of Y(p)

lies in the r ight half plane, or if a t least on e pole lies on the imaginary axis,

t hen t he filt er is u nsta ble.It will be not iced tha t the preceding theorem does not cover all pos-

sible situat ions; in par t icu la r , it does not set t le the case where ther e a re

singula r it ies other than poles on the imaginary axis. It is cer ta in ly

adequate, however , for most pract ica l problems; for all filt er s with

lumped elements the t ransfer funct ion is a ra t iona l funct ion ana lyt ic

excep t for poles .

The pr oof of th is genera l stability cr iter ion will be indica ted br iefly

If the absolu te value squared of dY/dp is in tegr able a lon g t he imagin ar y

axis, then , by the Parseva l theorem, LltW(t ) 12 a nd (1 + P)\W(t)12a re

in tegrable. It follows, by Schwar tz’s inequality for in tegra ls, that

/

.

\dt lW(t)\ = Q dt (1 + t2)~~]W(t )l 1

0- o– (1 + t ’)>+

<.J ]

m dt (1 + P)[w(t )[’d]

“dt~”

o- 0- 1 + p’ (158)

/

.

dt [W(t ) I is bounded and the filt er must be stable. On the other

h&d, it is eviden t that for a stable filt er the Laplace t ransform of the

weight ing funct ion is analyt ic and uniformly bounded in the r igh t half

plane. Th e t r an sform funct ion can t her efor e have no singular it ies inside

the r igh t half plane. Fur thermore it could have no pole on the imaginary

1See,for instance,E. C. Titchma rsh,In t roduct ion to the Theory of Four ier Inkgral s,

CIar endonPr ess,Oxford , 1937,pp. 50-51,




axis and st ill remain uniformly bounded in the r igh t half plane. This

concludes the proof of the second par t of the theorem.

SYSTEMS WITH FEEDBACK

2.21. Char act er iza t ion of F eedback Systems.—A mechanica l or elec-

t r ica l system with feedback is one in which the outpu t of some par t of

the system is used as an input to the system01 at a poin t where this can affect its own

value. A servo system is a feedback system

FIG. 2.15.—Servoschemat ic.in which the actual ou tpu t k compared with

t he in pu t, wh ich is t he desired output , and the

dr ivin g elemen t is a ct iva ted by t he differ en ce of t hese qu an tit ies.

F igure 2“15 is a block diagram showing the essent ia l comect ions of a

servo system. The output I% is fed in to a mixer or compara tor (in

mechanical systems, a differen t ia l) where it is subt racted from the input

I%t o pr odu ce t he er ror signal

t(t) = Ol(t) – Oo(t ). (159)

This signal cont rols the output th rough a system of amplifiers, motors,

and other devices, here shown as a box. To complet e t he forma l descr ip-

t ion of the system it is necessary only to specify the rela t ion establishedbetween e and d. by the conten ts of this box. lf the system is linear ,

th is can be specified as a t r ansfer fu nct ion Y(p); in terms of t he Laplace

t ran sforms we can wr ite simply

(%(p) = Y(p),(p). (160)

The t r an sfer fu nct ion Y(p) will be called t he jeedba ck t r an sfer -fun ct ion.

It is the t ransfer funct ion around the en t ire feedback loop, from the

ou tpu t of t he differ en tia l (~) b ack t o t he in pu t t o t he differ en tia l (00).

The over -a ll per formance of the servo can be descr ibed by another

t r an sfer fun ct ion Yo(p), which r ela t es t he input and ou tpu t of t he system:

%(p) = Y,(p) eJ(p) (161)

This may be called the over-all transfer junction, or the transjer function oj

the system. This t r ansfer fun ct ion must be car efu lly dist ingu ished fr om

the feedback t ransfer funct ion , to which it is simply rela ted. Equat ion

(159) implies t ha t

6(P) = OI(P) — 80(P); (162)

elim in at ion of c(p) from Eqs. (160) and (162) yields the very importan t

relation

Y(p)‘O(p) = f- ~(~j’ (163)



SEC.2.21] CIIARACTERIZATION OF FEEDB.ACK S YS TEMS 63

Since Eqs. (159) and (161) a re defining equat ions for c and Y,, Eq.

(163) is valid for any feedback system for which Eq. (160) is valid. In

cer ta in types of systems Eq. (160) will not be valid if Y(p) is in t erpret ed

as the t ransfer funct ion around the en t ire loop. This occurs when the

servo out put is not combined direct ly with th e input but is fu rther filtered

in t he feedba ck loop; t he dr ivin g elemen ts of t he syst em a re t hen a ct iva ted

by the difference between the input and this funct ion of the output . For

example, a modifica t ion of the servo system of Fig. 2.15 is shown in Fig.

~

FIG.2.16,—Servowith nddcdfilterin feedbackloop.

2.16. A t ilt er has been inser t ed in the feedback loop to provide an input

t o t he differ en tia l t ha t is n ot %(p) but Yz(p)&I(p). In su ch a syst em

Yl(p)[o,(p) – y,(p)%(p)] = co(p). (164)

Th e over-a ll t ran sfer fun ct ion is t hu s

%(P) = Y,(p)YO(P) = ~

1 + Y 1(P)Y 2(P)”(165)

On the other hand, the t ransfer funct ion around the loop is

Y,(p = Y,(p) Y,(p); (166)

Yo(p) and Yz(p) are not rela t ed by 13q. (163). It will be noted that Eq.

(165) expresses the t ransfer i’unct ion of t he system as a fract ion in which

the denomina tor is 1 plus the t ransfer funct ion around the loop. This

can always be don e.

Such servo systems as tha t shown in Fig. 2.16 are not , in genera l,

sa t isfactory. If the system is to have a zero sta t ic er ror , it is clear thatthe feedback filt er must a lways give the same asymptot ic response to a

st ep funct ion ; tha t is, }~o pY,(p) (l/p) = YZ(0) = 1 [see Eq. (129)].

Since changes in the parameters of the filter may change the va lue of

Y.z(0), it is not customary to filter the output before compar ing it with

the input; on th e ot her hand, such a filter ing act ion maybe inadver t en t ly

in tr odu ced by elast icit y in t he gea r t ra ins and by ot her fa ct ors.

A servo system may conta in more than one feedback loop. F@re

2.17, for instance, show a feedback system with two loops. The

inner loop serves to modify the character ist ics of the dr iving elements;

the whole of the conten ts of the dashed box of Fig. 2.17 cor responds to

the box of Fig. 2.15. In this system we have

A = c – Y2(p) e.(p), (167a)

e.(p) = Y*(p)p. (167b)




Eliminat ing ~ fr om t hese equat ions, we obt ain

O.(P) = 1 + ;$;Y2(P) C(P) = y(P )@). (168)

Thus

Y,(p)Y(P) = 1 + y,(p) Y,(p)” (169)

The t ransfer funct ion of the system is given by Eq. (163) or , more

exp licit ly, byY,(p)

(170)Ydp) = 1 + Y ,(p) + Y ,(P) Y2(P)”

Since a servomechanism is a lumped-constan t filter with t ransfer

function Yo(p), it s stability can be discu ssed by applica t ion of t he gen er al

t heory of filter stability: A servomechan ism will be stable if and only if

YO(p) has no poles in the r ight half of the p-plane or on the imaginary

axis. From Eq. (163) it is eviden t tha t Ye(p) will have a pole on ly where

1 + Y(p) has a zero. [A pole of Y(p) is merely a poin t where Ye(p)

equals 1.] Thus a servomechanism mull be stable if and only if 1 + Y(p)

has no zeros in the right half of the

WZ’e’’z$:

sally vahd only t i Y(p) M defined

Fm. 2.17.+ervo with two feedback loops. Y(p) in th is sta tement by the loop

transfer funct ion Y,(p) only if

Ye(p) and Yl(p) a re rela ted by Eq. (163); in other cases one must r eex-

amin e t he r ela tion between t hese qu an tit ies. Su ch cases will n ot be con -

sider ed fu rt her in t his ch apt er .

The feedback t ransfer funct ion is of basic impor tance in the theor y of

servomechanisms—for the dkcussion of stability, for the evalua t ion of

er rors in servo per formance, and in genera l throughou t the design pro-

cedure. We shall therefore turn to a discussion of its proper t ies.

2.22. F eedba ck Tr an sfer F un ct ion of Lumped-con st an t Ser vos.-Th e

feedback transfer funct ion of a lumped-constant servo can be wr it t en as

(171)

where s is an in teger , K, is a constant , and Q~(p) and Pn(p) a re poly-

nomials of degr ee m and n r espect ively, t he coefficien ts of t he zer o power

of p being taken as unity. The constant K, in th is expr ession will be

ca lled t he gain; in genera l it is defined as

(172)



SEC,2.22] FEEDBACK TRANSFER FUNCTION OF SERVOS 65

t he exponen t s being so chosen as to make the limit fin ite and different

from zero.

The value of s has an impor tant bear ing on the proper t ies of the sys-

tem. Eliminat ing % between Eqs. (161) and (162), one can express the

er ror in t erms of the input ,,

C(P) = ~ + lY(P) e,(p). (173)

Given the form of 19,(p) and the limit ing form of Y(p) as p ~ O (that is,

IC,/ p’) one can determine, by applicat ion of Eq. (129), the limit ing value

approached by the servo er ror as t s cc. For example, let us consider a

st ep-fu nct ion inpu t wit h t he La pla ce t ra nsform [E q. (109c)]

e,(p) = ;. (174)

The limit ing value of the er ror as t becomes infinite is

C(CO)=lim p 11

p+o 1 + Y(p) j

= lim1

KQ. lim L. (175)

P+O 1 + —’ —“ P+01+5

p’ P. Pa

The cont inued act ion of the servo will eventua lly reduce the er ror to

zero only ifs z 1. Ifs = O, then the limit ing value of the er ror is

(176)

Now let us assume that s = 1, so tha t the sta t ic er ror of the system is

zero, and find the steady-sta te er ror ar ising when the input changes at auniform r ate. We consider then t he constant velocity input with Laplace

t ransform d{ = 1/p2 [Eq. (11 lb)]. The er ror with which th is input is

followed will approach

Thus, if the

(s = 1), the

c(~) =lim p 11

p-o 1 + Y(p) @

= lim1

P+O p + K, $“

.1. —.

K,(177)

feedback t ransfer funct ion has a simple pole at the or igin

system will follow a unit constant -velocity input with an

er ror (lag) that is the reciproca l of the gain. This gain K1 ‘is ca lled t he

vdocily-error constant,



66 MATHEMATICAL BACKGROUND [SEC. 2.23

If thesystemis to have zero er ror fora t ionstan t -velocity input , the

condit ion to be sat isfied is s Z 2; that is, the feedback t ransfer funct ion

must have a second-order pole at the or igin . If s = 2, the system will

show aner ror for a constan t accelera t ion input (OJ = 2/pa);

2=— .K,

(178)

The quant ity (Kz/2) is then called the acceleration-error constant.

2.23. Th e F eedba ck Tr an sfer Locu s.—The feedba ck t ra nsfer fu nct ionfurnishes one with a complete descr ipt ion of the servomechanism. It

is a complex-va lued analyt ic funct ion of the complex var iable p; as

such it is completely determined by it s values a long a curve. The

imagin ar y a xis h as specia l sign ifica nce in t his conn ect ion beca use Y(ju )e]”

is the steady-sta te response of the feedback loop to the pure sinusoidal

input e’”; Y(ju) can therefor e be measured direct ly by exper iment . The

plot of Y(ja) in the complex Y-plane for all rea l values of u is called the

feedback transfer locus; it is a lso refer red to ~ the Nyquist diagram of the

t ransfer funct ion. As we shall see, this locus furnishes us \ vith a very

convenien t way of determining th e stability and th e performance charac-

t er ist ics of t h e s er vomechanism .

Since the net \ \ -orks and devices \ vith which we are concerned can be

r epr esen t ed by differ en tia l equ at ion s wit h r ea l coefficien ts, we h ave

Y*(jCd) = Y(–jai). (179)

It follo~rs t hat t he rea l pa rt of Y(joJ ) is an even fu nct ion and t he ima gin ary

+’”’.=+=+”’ ‘::;:ho:::xis::

quent ly the t ransfer locus is sym-

Y-plane; in plot t ing this locus it is

(a) (b) (c) necessary to draw only the graph

FIG. 21S,-The approach of Yfjw) to for posit ive values of ~the re-in fin ity for a first-orde~ pole (b) an d a mainder of the locus is then ob-second-order pole (c) as u ~ O+ (a).

t ain ed by r eflect ion in t he r ea l a xis.

The proper t ies of the feedback t ransfer funct ion discussed in Sec.2s2’2 are readily r ecognized in the t ransfer locus. If the sta t ic er ror for

Lthe system is zero, then the feedback t ransfer funct ion must have a pole

at p = O. If the pole is of the fir st order , then , as shown in (b) of F ig.

2-18, Y(jco) becomes infinite a long t he nega t ive imaginary axis as u + O• l-

(the + indicates tha t u approaches the or igin from posit ive values as in

Fig. 2.18a). For a second-order pole at the or igin (a zero steady-sta te



SEC.2.24] FORM OF THE TRANSFER LOCUS 67

er ror for a constant -velocity input), Y(@) becomes infin ite a long the

negat ive rea l axis as shown in Fig. 218c. In genera l, if the feedback

transfer funct ion has a pole of order s at the or igin , Y(j.0) approaches

infin ity a long the direct ion tha t makes an angle of (–7r/2)s with theposit ive r ea l axis.

The limit ing form of the t ransfer locus as u - m can also be easily

obta ined. Let b~ and an respect ively be the coefficient s of pm in Q~(p)

and p“ in P.(p) [Eq. (171)]; then as a ~ + ~,

(180)

In genera l, s + n –m > O; and Y(ju) approaches zero from a direct ion

tha t makes an angle of ( –7r /2) (s + n – m) with the posit ive rea l axis.

For example, if Q-/P . is a constant divided by a polynomial of fir st

degree in p, Y(ju) approaches zero a long the negat ive rea l axis for s = 1

and along the posit ive imaginary axis for s = 2.

2.24. Rela t ion between the Form of the Transfer Locus and the

Posit ions of the Zeros and Poles.—In the preceding sect ion the feedback

transfer locus was defined as a mapping in the Y-plane of the imaginary

axis in the p-plane. It is now necessary to define this locus more care-

fully. Usually the feedback transfer funct ion will have a pole at the

or igin , and it may have other poles a long the imaginary axis. As p

pa sses t hr ou gh t hese poles, t her e a re discon tin uit ies in t he t ra nsfer locu s;

thk locus then falls in to segments cor responding to the par t of the

imaginary axis between + ~ and the fir st such pole reached with decreas-

ing jq the par t between the first and second poles, and so on . In or der

to define the connect ion between these segments we shall now modify the

pr eviou s defin it ion of t he t ra nsfer locu s.

Let us consider a path in the p-plane tha t

+

A p-plane

lies a long the imaginary axis, except that it

shall include a small semicircular detour in

t he r igh t half plane a bout t he singula r point s

on t he ima gina ry axis, and a la rge semicir cu- C

lar pa th in the r ight ha lf plane, from very

la rge n ega tive imagin ar y va lu es t o ver y la rge

posit ive imaginary values (see Fig. 2.19).

For su fficien tly small det ou rs a bou t t he poles Bon the imaginary axis and for a sufficient ly FIG.2,19.—Closedpath in th e

large semicircular connect ing path , th isp-plane.

closed pa th will enclose all of t he poles and zer os of a ny r at ional funct ion

Y(p) inside the r ight ha lf plane. We shall call the closed curve that

resu lts fr om mapping a ny such con tou r on to t he Y-plane t he t r an sfer locus

of Y(p).




These-defined transfer locus can be determined exper imenta lly. It

is evident tha t the locus for nonsingular frequencies can be obta ined as

the steady-sta te response to sinusoida l inputs a t these frequencies. The

mapping of the semicircles in the p-plane on to the Y-plane can be det er -

mined as follows. Near the singular frequencies the system will, in

genera l, over load . One can, however , determine from such test s the

or der of the pole, which as we shall see is sufficient to define the t ransfer

locu s a bou t t he det ou rs. The same is t ru e for t he lar ge semicir cle (about

t he poin t at in fin it y).

If the pole a t pl is of order s, then in t raversing the semicircle from

PI + ~~ to p, – jP (for su fficien tly small bu t posit ive p), t he cu rve t ra ced

in the Y-plane by Y(p) will be essent ially an arc of a circle t raversed in

the coun terclockwise direct ion through an angle of about .m. This

follows immedia tely from the fact tha t in the neighborhood of p, the

t ra nsfer fun ct ion is appr oxim ately of t he form

AY(P) = (p _ ~,). (181a)

or

Y(PI + Pej+) = AP-se-is~, (181b)

wh er e .4 is a complex-va lu ed con st an t.Similar ly, for very la rge values of Ip[ (the ou ter semicircle in the

p-pla ne) we h ave

(180)

As t he la rge sem icir cle is t ra ver sed by t he poin t p = Re@, from o = –T/2

to @ = +7r/2, Y(p) t ra ver ses a cir cu la r a rc of small r adiu s (a ppr oa ch in g

zer o with l/ZP+m–n) th rough an angle —(.s + n — rn)~.

We have now seen how the feedback transfer funct ion maps the closed

curve of Fig. 2-19 into a closed curve I’ in the Y-plane. The r ight half

of the p-plane is thus mapped by Y(p) in to the in ter ior of r . Conse-

quent ly if Y.(p) is equal to – 1, for instance, for some point in the r ight

ha lf plane, then the con tour 1? will encircle the point ( – 1,0) in the

Y-plane. Since the servomechanism will be stable if and only if Y(p)

does not equal – 1 for any point in the r igh t ha lf of the p-plane, or on

the imaginary axis, it is clear tha t the t ransfer locus furnishes us with

another means of determin ing the stability of the servo. How this can

be don e will be discu ssed in t he followin g sect ion s.

2.26. A Mapping Theorem.—We shall now prove a theorem of

ana lysis t ha t is u sefu l in t h e discu ssion of t h e st abilit y of ser vomechan isms.

Let G(p) be a ra t ional funct ion of p. If the point p in t ,h e p-pla ne



SEC.2.25] A MAPPING THEOREM 69

descr ibes a closed con tour C in the posit ive’ sense, the poin t G(p) in the

Gplanedescr ibes aclosed con tou r r (Fig.2”20). ljt hecont ou r Cent ircles

in a positive senke Z zeros and P poles (this takes into account the multi-

plicity of zeros and poles), the corresponding contour r in the G-plane enm”rcles

the origin

N=Z– P (182)

timesin a positive sense.

We here assume tha t none of the zeros or poles lies on the contour in

the p-plane; a sligh t modifica t ion of the con tou r will a lways allow it to

sa tisfy t his con dit ion , sin ce t he zer os

and poles of G(p) are isola ted. The

t erm “en cir cle” is defin ed a s follows:

Consider a radia l line drawn fromthe poin t pl t o a r epr esen ta tive poin t

P on the closed con tou r C. As the

point P descr ibes the con tou r C in

some sen se, t he r adia l lin e sweeps ou t +’+’IG. 2.20.—Mapping of a closedcoutour in the p-planeonto the G-plane .

an angle tha t will be some mult iple of 27r . If th is Angle is 2rn, then the

con tou r en cir cles t he poin t pl, n times. The sign of n will depend upon

th e sense in which P descr ibes C.

We shall prove the theorem in stages. Suppose first tha t the funct ionhas a single root a t p = p,:

G(p) = A(p – p,), (183)

where A is a constan t . In polar form this becomes

G(p) = APe~*. (184)

It is clea r that as the poin t p dcscr ibcs a con tou r C in the posit ive sense,

t he cor respondin g poin t in th e G-plane descr ibes a con tou r r in a posit ive

sense. The number of t imes that r encircles the or igin is precisely the

tota l change in @/27r which occurs when p t races C once. Thus r

en cir cles t he or igin on ce if C con ta in s PI in its in t er ior but ot herwise does

n ot en cir cle t he or igin .

If the funct ion has two dist inct root s pl and pz, then

G(P) = A(P – P,)(P – P2). (185)

When wr it t en in pola r form, th is is

G(p) = ~plpze~[$’+~’), (186)

where p — p, = plei~l and p — pz = p~el~’. If the contou r C encircles

no root s, the tota l change in (o1 + OJ is zero and r does not encircle the

or igin . If the contou r C encircles one or both the root s, the tota l change

1The interiorof the contour is alwayson the left as the point describesthe contour.




in (@l -!- 42) is% or4u and r encircles the or igin ina posit ive sense once

or twice, respect ively.

If the two roots are equal, any con tour enclosing one will, of course,

enclose the other , and the cor responding con tour in the Gplane will

encircle the or igin twice. In genera l, if the con tour C encircles a root

of mth order , or m dist inct zeros, the con tour r encircles the or igin m

times in a posit ive sense.

Now let us consider a funct ion with a pole of first order a t p = pl,

(187)

If the con tour C encircles the pole p = p, in a positive sense, the con tour r

in the G-plane encircles the or igin once in a negative sense. This can be

genera lized in exact ly the same fashion as was done above with respect

t o the zeros; if the con tour C encircles n poles in a posit ive sense, then the

cent our r in the Gplane encircles the or igin n t imes in a negat ive sense.

We can combine these types of funct ion to form one tha t has both

zeros and poles. Suppose G(p) is a ra t iona l fract ion

(188)

where Q-(p) and Pn(p) are polynomials of degr ee m and n respect ively.

This can be wr it t en in factored form as

( ‘P1)(P– P2)”””(P– P*)G(P) = ~ ~~ _P,)(p –-P,) . . “

(p – pm )’(189)

where some of the root s p 1, pz, . . . , pm and some of the poles PI,

P2, . . . , Pm may be repeated. Asp descr ibes in a posit ive sense a closed

con tour tha t encircles Z zeros and P poles, the phase angle of G changes

by +% for each of the enclosed zeros and by – 27rfor each of the enclosed

poles. The tota l number of t imes that the cor responding con tour in the@plane encircles the or igin is exact ly Z – P. Th is est ablish es t h e va lid-

it y of the mapping theorem.

2.26. The Nyquist Cr iter ion .-Now let us apply the theorem of the

pr ecedin g sect ion t o t he fu nct ion

G(p) = 1 + Y(P) (190)

and to the closed con tour in the p-plane illust ra ted in Fig. 219. The

map of a contour in the p-plane on to the Y-plane can be obta ined by

shift ing the cor responding map on the G-plane to the Ieft by one unit .It follows tha t the con tour C in the p-plane, descr ibed in a posit ive sense,

will map into a con tour r in the Y-plane tha t encircles the point ( – 1,0)

in a posit ive sense N = Z — P times. The transfer locus—a curve of

th is type for which C encloses all zeros and poles of 1 + Y(p) tha t lie



SEC.2.26] THE N YQUIS T CRITERION 71

inside the r igh t half of the p-plane—will thus encircle the poin t (– 1,0)

a number of t imes that is the difference between the tota l number of

zeros and the tota l number of poles of 1 + Y(p) inside the r igh t half of

the p-plane. This resu lt , t ogether with the fact that the system will bestable on ly if the number of zeros is zero, can be used in the discussion of

the s tabilit y of servomechan isms .

There is a well-known theorem due to N’yquist ’ that applies to feed-

back systems in which the feedback t ransfer funct ion is that of a passive

network. The feedback t ransfer funct ion of even a single-loop servo is

not usually of th is form; the very presence of a motor in t roduces a pole

at the or igin . It remains t rue, however , tha t single-loop servos of a

la rge and impor tan t class have feedback t ransfer funct ions with n o poles

inw”de the r igh t half of the p-plane. To these sin gle-loop servos t he

following theor em applies: The servo will be stable if and only if th e locus

of the feedback transfer function does not pass through or encircle the poin t

(– 1,0) in the Y-plane.

This can be proved as follows. First of all, let us assume tha t the

over -a ll t r ansfer funct ion

(163)

has a pole on the imaginary axis. Then 1 + Y(p) has a zero for some

poin t p on the imaginary axis, and Y(p) = – 1; t he feedba ck t ra nsfer

locus passes through the poin t (– 1,0). We know that in this case the

servo is unstable (Sec. 220), in agreement with the sta tement of the

Nyqu is t cr it er ion .

Now let us assume tha t there is no pole of Yo(p) on the imaginary axis.

Then the servo will be stable if and only if 1 + Y(p) has no zeros in the

r igh t half of the p-plane. Now single-loop servo systems of the class tha t

we are consider ing have no poles of Y(p) in the r igh t half of the p-plane;

that is, P = O. The con tour C does not pass through any poles or zeros

“of 1 + Y(p), since the poles a re bypassed by detours and we are now con-

sider ing only the case where there a re no poles of YO(p) [i.e. no zeros of

1 + Y(p)] on the imaginary axis. It follows tha t the feedback t ransfer

locus will encircle the poin t (– 1,0) a number of t imes tha t is exact ly

equa l to the number of zeros of 1 + Y(p) in the r igh t half plane. The

servo will thus be stable if and only if the number of encirclement s is

zer o. This con clu des t he pr oof of t he cr it er ion .Examples.—As an applica t ion of the Nyquist cr it er ion we shall now

consider the type of servo descr ibed by Eq. (46). The feedback t ransfer

funct ion for th is syst em is

K

‘(p) = p(Tmp + 1)’(191)

1H. Nyquist, 1’Regener at ionTheory,” Bel l .’S@em Tech . Jour., 11, 126 (1932).



72 MA THEM A TICAL BAG’KGR07J ND [SEC.226

where K is the gain and T- the motor t ime constant . A rough sketch of

the locus of the feedback t ransfer funct ion is shoum in Fig. 2“21. The

full line shows the por t ion of the locus obta ined for posit ive values of a;

the dot ted line tha t obta ined for negat ive values of w The semicircle

T-b.-=

; (J=O--,1 \

\~~ Y-plane j,1 \‘\ \”---

\_ +<0

(ti -+-

a l. o+

FIG.2.21.—Locusof th efeedback tra nsfer functionl’(p ) = K/[p (TmP + 1)1.

abou t the or igin cor responds to the inden ta-t ion made to exclude the or igin . The a r rows

indica te the direct ion in which the locus is

t raced as w goes from + m to – ~. Actua lly

what is drawn is the locus of Y(p)/lY; the cr it ica l

poin t is then (– l/K,O) instead of (– 1,0).

(The reason for drawing the locus in th is way is

that it is much easier to change the posit ion of

the cr it ica l point than to redraw the locus for

differen t va lues of the gain .) According to the

Nyquist cr iter ion this servo system is stable,

since t he feedba ck t ran sfer fu nct ion locus does

not encircle the poin t [ – (l/K), O].

Th eor et ica lly, t he syst em \ Villbe st able h o~vever la rge t he gain . Th is

is not actua lly the case, because the feedback transfer funct ion in Eq.

(191) only approximates tha t of the physica l servomechanism. A closer

approxima tion includes t he t im e con -

stant T. of the amplifier ; then

Y(P) = 1—.K p(Tmp + l)(~op +—~” (192)

The locus of th is feedback t ransfer

funct ion is shown in Fig. 2“22. For

sm all va lu es of t he gain K,,t he cr it ica l

poin t [– (l/K), O] is not enclosed by

the locus and the system is stable; for

la rge values of the gain , the point[– (l/K), O] is encircled twice by the

locus, and the system is unstable. It

is easy to obta in the limit ing value

of K for wh ich t he system becom es un-

stable. The value of u for which the

,/ “=o-

‘\Y-pla ne 1

~/‘.)

\

/-\ \

?f#. +c.a I

(-+?0)

‘y )

,W=-.m

/.-4/

L/

@=o+

FIG. 2.22.—Locusof the feedbacktransferfunction

[Y (P)/ Kl = I/ [p(l”mp + l)(T.p + l)].

locus crosses the negat ive rea l axis is easily found to be uo = l/~m~

Since Tn is much la rger t ha n T~, the magnitude of Y at th is point is

(193)For st abilit y, t hen , KTa must be less than unity; tha t is,

(194)



SEC.2.27] M(J LT ILOOP S J YRVO S YS TEMS 73

2.27. Mult iloop Servo Systems.—In single-loop servo systems the

feedba ck t ra nsfer fu nct ion h as n o poles in side t he r igh t h alf of t he p-plan e.

As a consequence the stability cr iter ion has the simple form given in the

previous sect ion . For mult iloop servo systems, however , the feedback

t ransfer funct ion may conta in poles in the in ter ior of the r ight half plane;th is can occur when one of the inner loops is unstable. When this is the

case, the more genera l form of the theorem in Sec. 2.24 must be applied.

It remains t rue tha t the system is stable if and only if there are no zeros

of 1 + Y(p) in the r ight half plane, but the number of t imes tha t the

feedback t ransfer locus encircles the poin t ( – 1,0) is equa l to the number

of zeros minus the number oj poles. An in dependen t determin at ion of th e

number of poles inside the r igh t half of the p-plane must be made, a fter

which the number of zeros can be obta ined by refer ence to the t ransfer

locus and use of Eq. (182).

In genera l, because of the form of the feedback t ransfer funct ion for a

mult iloop system, it is not very difficu lt t o determine the number of

these poles . As sta ted in Sec. 2.21, the feedback t ransfer funct ion for

mult iloop systems will often be of the form

(169)

If there are th ree independen t loops in the system, Yl(p) may itself be

of th e formY,(p)

Y1(P) = 1 + y,(p) Y4(P)” (195)

A procedure for determin ing the stability of the mult iloop servo system

will be made clear by consider ing such a thr ee-loop system. The poles

of Ys(p) Yl(p) can be obta ined by inspect ion , since this, in all likeli-

hood, will be a rela t ively simple funct ion . The locus of Y3(P) Yl(p)

is then sketched to determine the number of t imes tha t it encircles the

poin t (– 1,0). F rom this number N and the number of poles of

Y3(P) Y4(P)

in the r igh t half of the p-plane, the number of zeros of 1 + Ys(p) Y,(p) in

thk region can be obta ined; th is is the number of poles of Y1(p) inside

the r igh t ha lf of the p-plane. The number of poles of Yz,(p) in the r ight

ha lf of the p-plane can be determined by inspect ion; the number of poles

of Yl(p) Y2(P) is the sum of these numbers. The locus of Yl(p) Y2(p) is

then drawn, and the number of poles of Y(p) in this region is determined

as before. Fina lly the locus of Y(p) is drawn, and from this locus is

deduced the number of zeros of 1 + Y(p) in the r ight half of the pplane.

Thus by use of a succession of Nyquist diagrams it can be determined

whether or not the system is stable.




Examples of Multiloop Syst em s.—We shall first discuss a t wo-loop

system of the type sketched in Fig. 2“17. Let us suppose tha t the inner

loop consists of a tachometer plus a simple RC-filt er of the type shown in

Fig. 2.6. The tachometer outpu t is a voltage propor t ional to the

der iva t ive of the servo ou tput . The t ransfer funct ion for the combina-t ion is t h er efor e

Y,(p) = K,pTop

Top + 1’(196)

where K1 is the factor of propor t iona lity y, We shall suppose tha t the

amplifier -plu s-mot or h as t he t ra nsfer fu nct ion

K,‘1(p) = p(Tm p + l)(Z ’.P + 1)-

The feedback t ransfer funct ion isY,(p)

Y(P) = 1 + Yl(p)y2(P)’

(197)

(169)

To determine whether or not Y(p) has any poles in the r igh t ha lf of the

p-plan e it is n ecessar y t o exam in e

K, K,TOpY1(P)Y2(p) = (Tm p + l)(T .p + l)(Z’OP + 1)”

Th e t ran sfer locu s for Y,(p) Y,(p) is shown in Fig. 2.23. Since

Y,(p) Y,(p)

(198)

has no poles in the r ight ha lf of the p-plane, the N-yquist cr it er ion can be

+

Y,Y2-planeapplied to determine whether 1 + Y1Yj has

any zeros in the r ight ha lf of the p-plane. It

is eviden t from Fig. 2.23 tha t the locus does

@o+ not encircle the poin t (— 1,0) for KIKZ ~ O;

Kl=+ca consequent ly 1 + Y1Yz has no zeros in the

(

r igh t h alf pla ne, a nd Y(p) has no poles in th is

region. It follows tha t the simple Nyquistcr it er ion can be applied in th is case.

FIG.2.23.—Transferocusof As a second example we shall consider aY,(p)Y,(p) = (K11c2T0F)/[(~mP+ 1)(~.P + l)(~OP + 1)1 .

ser vo for wh ich t he simple Nyquist cr it er ion

is not va lid. We shall aga in consider the sys-

tem shown in Fig. 2“17 and shall suppose tha t the tachometer and filter

combin at ion h as t he t ra nsfer fu nct ion

‘2(P)K1p(&)and tha t the amplifier plus motor can be descr ibed by

(199)

(200)



SEC.2.27] M ULTILOOP SERVO SYSTEMS 75

We shall fu r ther assume that To > T~. As a bove, t he feedba ck t ra nsfer

fu nct ion is

Y1(P)(169)(P) = 1 + Y,(p) Y2(P)’

wher e n ow

‘1(P)Y2(P)&%(TD%) (201)

The locus of Y,Y2 is sketched in Fig. 224 for posit ive values of w only.

This locus in ter sect s the nega t ive rea l axis a t about the frequency

The magnitude of YIYJ a t th is frequency is

IY,Y,(UO)I = *KIK,.

Thus the inner loop is stable if ~KX, <1.

loop is unstable and, indeed, an applica t ion of

(202)

(203)

If ~KXZ >1, the inner

the theorem of Sec. 2.25

shows tha t in th is case there a re two zeros of the funct ion 1 + Yl Yz in

the r igh t half of the p-plane. l@=o -

Y-plane \ ~- =,\

@ A$+iiijkIG.224.-Tran sfer 10CUBf FIO.2.25.—Feedbackrans-

Y,(p) Y2(P) = fer locus .[K,K,/(z’mP + 1)1

[TOP/(T’OP 1)1’.

We shall now assume that the gain is set so that the inner loop is

unstable. A sketch of the feedback t ransfer locus is shown in Fig. 2“25;

the or igin of the p-plane has been excluded from the r igh t half of the

pplane by the usual detour . If the ga in KI is such as to make the poin t

(– 1,0) be at A, then N = O. Since Y(p) has two poles in the r ight half

of the p-plane, O = Z – 2, or Z = 2; the system is unstable. If the

simple Nyquist cr it er ion were applied to th is system, it would lead to the

fa lse conclusion tha t the system is stable. If the gain KI is such as to

make the poin t ( – 1,0) fa ll a t B, t hen N = – 2. In th is case – 2 = Z – 2

and Z = O; the system is stable. In appl ying the simple N yquist cr it er ionone would have obta ined a double encirclement of the cr it ica l poin t in a

nega t ive direct ion ; the resu lt would then have been ambiguous.



CHAPTER 3

SERVO ELEMENTS

BY C. W. MILLER

3.1. In t roduct ion . -In th is chapter there will be presented some

examples of the physica l devices that a re common components in elec-

t ron ic servo loops. Its purpose is to descr ibe a few actual circu it s and

mecha nism s th at ma y a ssist in t he ph ysica l u nder sta ndin g of t he pr oblem s

discu ssed in followin g ch apt er s—t he design a nd mat hemat ica l con sider a-t ion of the en t ir e servo loop. This chapter can , however , serve only as a

cu rsor y in tr odu ct ion t o t he field of ser vo compon en ts.

FIG.3.1.—Simpleservoloop.

Since the types of devices commonly ut ilized in the more .eomplex

ser vo loops a re t he same a s in r ela tively simple s er vo loops, it is su fficien tly

informat ive to consider the possible elements in such a simple loop as

tha t presen ted in Fig. 3.1. In th is figure the common mechanica l dif-

ferent ia l symbol has been employed to indica te any device that has anou tput propor t ion al to the difference of t wo inputs. Thus,

6=01—00. (1)

The var iou s “boxes” employed in th e loop have t ran sfer character ist ics

defin ed by

(2)

In geneta l, a t t en t ion will be focused on the frequency dependence of

these t ransfer character ist ics ra ther than on amplificat ion or ga in.

Because of th is, no space will be devoted, for example, t o the quite

involved problem of the design of vacuum-tube amplifier s. Specia l

76



S =c. 3 .2] ERRORMEASURING SYS TEMS 77

a t ten t ion will be paid, however , t o er ror -measur ing systems and their

inaccuracies. This emphasis is based on the fact tha t such elements a re

not included in the servo loop and therefore no amount of ca re in the

design of t h e loop will decr ea se t he er r or r esu lt in g fr om t heir in adequ acies.

The equat ion rela t ing the input quant ity 8, and the outpu t quant ity

OOn Fig. 3.1 is

L&= Y]*(P) ‘23(p) yin(P)

el 1 + Y12(P )Y23(P )Y,l(p) + Y23(P )Y,,(p) ‘

The rela t ion between er ror e and the output quant ity is

00 = Yu(P) Y 23(P) Y 31(P).—

e 1 + Y23(P) Y32(P)

(3)

(4)

It is t o be not ed tha t the loop in the example involves two devices of

the differen t ia l type. Differen t ia l 1 of Fig. 3.1 is an er ror -measur ing

system; such systems will r eceive extended discussion in this chapter .

Amplifiers may be employed in either or both of the boxes Y,Z(p) and

Y23(p). In one of these boxes there may be also a device for changing

the na ture of the signal ca r r ier ; such elements will be ment ioned in this

ch apt er . Also, specia l t ra nsfer ch ar act er ist ics ma y be desir ed for oper at -

ing on the er r or c; suitable networks for this will be presen ted. Equaliz-ing or stabilizing feedback is employed by the path th rough Yaz(p), and

a sect ion of this chapter will presen t examples of the devices commonly

used to obtain a desir ed t ransfer character ist ic Ysz(p). The remaining

box, the t ransfer character ist ic of which is given as Yil(p), is often a gea r

t ra in and as such has (if it is proper ly designed) no interest ing frequency

depen dence. Som e discussion, however , will be included on the problem

of gea r t ra in s.

Th er e is, of cour se, no clear -cu t ru le for separa t ing the components of aservo loop. Indeed, even with the same servo loop it is somet imes helpful

t o make differen t sepa ra t ions tha t depend upon the par t icu lar in terest a t

the t ime. In this chapter , the a t t empt will be made to separa te t r ansfer

funct ions according to physica l pieces of equ ipment . To maintain con-

sistency with Fig. 3“1, pr imes will be used if addit iona l equipment is

involved.

3.2. E rr or -mea su rin g Syst em s.—Necessa ry compon en ts of a ny closed-

loop system are those devices which measure the devia t ion between theactua l ou tpu t and the desired outpu t .

It is imbor tan t tha t th is difference, or er ror , be presen ted in the f orm

most suitable for the other components in the cont rol system. Thus

the loca t ion of equ ipment , as well as the choice of the physica l type of

er ror signal and its t ransmission, is impor tant . For example, a mechani-

cal different ia l is ra rely used as a device to obtain the difference between



78 S ERVO ELEMEN TS [SEC.3.:

the actual ou tpu t and the desir ed ou tpu t , because it is oft en impract ica l

to rea lize these two quant it ies as ph ysically a djacen t sha ft r ota tion s.

Also, one may or may not desire the er ror as a shaft rota t ion , the choice

depending upon the amplifying system to be used and its r equ ired

placement.

Cer t a in ly on e must a lso choose an error -measu r ing system tha t has an

inheren t accu racy grea ter than that r equ ired of the over-a ll loop. Often

both sta t ic accu racy and dynamic accu racy of the system must be

examined. Many error -measu r ing devices produce “noise” of such a

na tu re tha t the componen t frequencies a re propor t iona l to th e ra t e of

change of the input and ou tpu t quant it ies. The response of the loop to

t his n oise r ou st be con sider ed.

In addit ion t o su ch n oise, er r or -m ea su rin g system s ut ilbing a -c ca rr ier

voltages commonly have in the final ou tpu t not on ly a voltage propor -

t ional to the er ror , with a fixed phase sh ift from the excit a t ion volt age,

bu t a lso a volt age tha t is not a funct ion of er ror and is 90° ou t of phase

with the so-ca lled er ror voltage. Harmon ics of both volt ages a re usually

also presen t . The phase sh ift of the er ror voltage, if reasonably constan t

for th e class of device, is not a ser ious problem bu t must be considered in

th e design of the other componen ts. The quadra tu re voltage and the

harmon ic voltages a re oft en very t roublesome because they tend to over -

load amplifiers and to increase the hea t ing of motors. Specia l design ,n ever th eless, ca n elim in at e t his pr oblem (see Sec. 3-12).

F&dly, the componen ts of the er ror -measu r ing system must be

mechanically and elect r ica lly su itable for the applicat ion ; that is, t he

er ror -measu r ing system must fu lfill it s funct ion for a sufficien t length of

t im e u nder velocit ies a nd a cceler at ion s of t he va ria bles bein g t ra nsm it ted

and with the exposu re, impacts, tempera tu res, etc., encoun tered in the

an ticipa ted use of th e equ ipment .

Beca use of t he ea se of t ra nsm ission of sign als a nd t he r esu lt in g fr eedom

in pla cement of equ ipment , elect r ica l devices h ave had wide applicat ion sas er ror -measu r ing devices. A few types will be discussed in the follow-

ing sect ions .

3.3. Synchros.-Th is discussion , elemen tary in nature, will be

rest r icted to a type of synch ro that is in common use in Army, Navy,

a nd some n onmilit ary equ ipmen t.

If the axis of a coil ca r rying an a lternat ing cu rren t makes an angle 9

with th e axis of a second concen t r ic coil, th e induced emf in the second

coil will be K cos e, where K is a constan t dependen t upon the frequency

and t he magn itude of the cu rren t in the pr imary coil, t he st ructu re of thecoils, and the character ist ics of the magnet ic cir cu it . If t wo a ddit ion al

secondary coils a re added with their axes 120° and 240° from the axis of

the fir st secondary coil (see Fig. 3.2), t he emf’s will be



8Ec. 3.4] S YNCHRO TRANSMITTER AND REPEATER SYSTEM 79

Eo,, = Kcosf3,

Em, = KcoS(8– 1200),

I

(5)

Em, = Kcos(O –2400),

where the subscr ipt s indica te the poin ts between which the voltage is

measured and their order gives the sense of the measurement . The

t erm ina l volt ages will be

E,,.s, = K W COS(8 + 300),

Es,s, = K V% COS (d + 1500),

1

(6)

E~,,, = K @ COS (0 + 2700).

In Fig. 3.2thelabeling is tha t for Navy synchros, and the circu it asdrawn is physica lly equiva len t to a Navy synchro viewed at the end of

the synchro from which the rotor shaft extends. It is standard pract ice

in the Navy, however , t o consider counter - S2clockwise rot a t ion posit ive ra ther than clock-

A

~._.-/“

-.

wise, as in Fig. 3.2. / ‘\ \e

Va riou s modificat ions of t he con st ru ct ion of

3

\\\ \

a syn ch ro exist t o suit differ en t fun ct ion al u ses. R2~‘.

1

In this sect ion , on ly synchro t ransmit ter s, ~ -” i o I

synchro repea tem, and synchro cont rol t rans- }, /’

former s will be men t ioned. In these units, t he \ /

t h r ee st a tor coils a re loca ted on a lamina ted ‘.. /’---- S1

magnet ic frame that sur rounds and suppor t s th e S3rotor . Synchro t ransmit ter s and repea ter s Fm, 3.2.—Schemat icdi.-

have a salient pole or “dumbbell” type rotor ,gramof a synchro.

but a con t rol t ransformer has a cylindr ica l rotor . A mechanica l damperis bu ilt in to synchro repea ter s to decrease oscilla tory response. For a

m or e deta iled discussion of t hese t ypes of synchros (and for discussion of

other types), one or more of the var ious references should be consu lted. 1

Table 3.1 gives a br ief summary of some of the character ist ics of a few

of the Army and Navy synchros.

3.4. Data System of Synchro Transmit ter and Repea ter .—Such a

device as tha t descr ibed in Sec. 3.3, so const ructed that the angle o can

be changed a t will by turn ing the rotor (the pr imary), is known as a

I oSynchroa and Their App lica t ion )” Bell Telephone Labora tor ies , Sys temsDevelopmentDepar tm ent ,Report No. X-63646, Iesue2, New York, Mar . 19, 1945;“Specificat ionsfor Synchr oTransmissionUnits and Systems,”0.S. No. 671, Rev. D,Bureauof Ordnance,Navy Depar tment,Washington,F eb. 12, 1944; “Specificat ionsfor Units , A. C. Synchronousfor Data Transmission,”FXS-348 (Rev. 7) Tentat iveSpecifica t ion ,F rankford Ar sena l, Ma r . 9, 1943; “Unit ed St a t es Na~ Synch r cm ,Deec~ipt ion and Oper at ion ,” Ordn ance Pamph let No. 1303, A Join t Bu reau of

Ordnanceand Bureauof ShipsPublication,Na vy Departm ent.Washington,Dec. 15,1.944.



80 S ERVO ELEMEN TS [SEC.3.4

synchro t ransmit ter (or genera tor ) when it is used in transmit t ing elec -

t r ica lly the value of the angle t?. By use of a syn chr o r epea t er (or motor),

the angle .9may be reproduced at a remote posit ion as a shaft rota t ion .

A synchro repea ter is similar in st ructure to a syn ch ro t ra nsm it ter ,

except for the addit ion of the damper previously ment ioned. Becauseof the difference in use, the rotor of the repea ter is fr ee to rota te’ with

lit t le fr ict ion and low-iner t ia loads, whereas the t ransmit ter rotor is

mechanically const ra ir ied to the va lue of d that is to be t ransmit ted. A

r epea ter is con nect ed t o a t ran sm it ter as in dicat ed in Fig. 3.3.

TABLE3,1.—SYNCHRONITSFOR115-VOLT.60-CYCLEOPERATION

SynchroWeightlb

l—5G Navy genera tor . 56G hTavygenera t or . 87G Navy genera t or . 18IF Navy motor . 25F Navy motor . 5lCT Navy con t rolt ransformer . 2

3CT Navy con t rolt ransformer . 3

5CT Navy con t rolt ransformer . 5

IV Army transmit ter 4.8I -4 Armyt ransmit t er 11.8X Army repea t er . 1.3

V Army repea ter . 4.8II-6 Army r epea ter . 10.3XXV Army trans-former. . . . . . . . . . 1.3

XV Army trans-former. . . . . . . . . . 4.8

katoI

9090909090

90

90

90105105105

105105

104

104

btol

volts

115115115115115

55

55

55115115115

115115

60

58

Exci-

,a t iorcul--rent ,amp

0.61.33,00,30.6

0.04!

0.04:0.551.00.12

0.550.55

.,.

Jn i t torqufgradient,z-in./degre

0.41.23.40.060.4

.,.

. . . . . . . .0.481.0

.020-0.02

0.480.53

Unit s tat i(

accuracy,degrees

Avg

0.20.20.20.50.2

0.2

0.1

0.10.2!0.2!1.0

0.2!0.2!

Ma>

0,60.60.61.50.6

0.6

0.3

0.30.50.52.0

0.50.5

0.9

0.9—

RotorIertia,b in .z

0.310.942.40.0260.31

0.026

0.31.,

If the rotor of the repea ter has the same angular rela t ionship to it s

sta tor coils as the rotor of the t ransmit ter has to its sta tor coils, no cur -

rent will flow in the sta tor leads. This is a minimal-energy posit ion.

If the angles a re different , cu rrents flow in the sta tor leads and equal

torques are exer ted upon both the t ransmit ter and the repea ter rotor s in

such a sense as to reduce the misalignment and thus to approach a mini-

ma l-energy pos it ion . Since the t ransmit ter rotor is const ra ined, the

repea ter rotor turns unt il it assumes an angular posit ion 0, to within the

er ror s ca used by con st ru ct ion difficu lt ies a nd fr ict ion . It will be sho~vn



SEC.34] SYN .CHRO TRANSMITTER AND REPEATER SYSTEM 81

(Sec. 3.6) tha t the restor ing torque when a synchro unit is connected to

a similar unit act ing as a t ransmit ter is very near ly 57.3 Tu sin (d’ – O),

where T. is the unit torque given in Table 3.1. There is a posit ion of

zero torque at 180° misalignment , bu t it is u nst able.

FIG.

1-.—. S2 I /-.. S2

RI ,’ e “, ~1 ,/% e’ ‘“\ \

\ 9

~~r!i

\ .I \

.“ II R21R2f’ 1 \

\’ : I 1’ I\\

;\\ /\ \~.> 0“

\\ /’

s: - “ —-- c,

II +ieJ*1’

1 1

A-c SUDDIV

Transmitter - Repeater -

3.3.—Synchror an smitt erconnectedto synchr orepea ter .

In some cases where it is necessary to reproduce a var iable quant ity

remotely as a rota t ion , the pr imary system can easily stand any torque

tha t might be r eflect ed upon it . It might seem tha t a simple t ransmit ter -

r epea ter system would have wide use in such cases. As is seen , however ,

from Table 3.1, small loading would cause very appreciable er r or s even

for the la rger synchros. If the er ror ever exceeds 180°, the synchro

repea ter will seek a null 360° from the proper angle. In fact , the syn-

chro repea ter may lock to a zero torque posit ion after any number of

mult iples of 360° measured at it s own shaft . This makes it difficu lt t o

obta in gr ea ter tor qu es by gear ing down t o th e load, th ough some schemes

have been devised.

As a fu rt her disa dva nt age, t he t ra nsfer ch ar act er ist ic of a t ra nsm it ter -

repea ter system exhibits a high resonan t peak (an amplifica t ion of 2 to 8,

depending on the manufacturer) a t a frequency between 4 and 8 cps.

Any increase in the iner t ia a t the synchro shaft result s in a st ill poorer

t ransfer characteris t ic.

For these reasons, simple t ransmit ter -repea ter data systems are

genera lly used to dr ive only light dia ls. In such a use, a coarse dial

(so-called “low-speed” dial) is dr iven by one t ransmit ter -r epea ter com-

binat ion in which shaft values are such tha t the 360° ambiguity is of no

consequence or such that the ambiguity will be avoided by limits on the

var iable. A fine dia l (so-called “high-speed” dial) for accura te in ter -

pola t ion between gr aduat ions on th e low-speed dial is dr iven by a secon d

t ransmit t er -repea ter combina t ion . Siice it is used on ly for in terpola-

t ion , th e 360° amb;guity causes no con cer n.



82 S ERVO ELEMEN TS [SEC.35

By adding torque-amplifying equipment tha t is act iva ted by the rota-

t ion of the synchro-r epea ter shaft , many types of servos can be devised.

They range from simple types in which the rotor act iva tes a switch tha t

direct ly cont rols an elect r ic motor , t o more complex systems in which ,

for in st an ce, t he syn ch ro r epea ter con tr ols a va lve in a h ydr au lic amplifier .

3.6. Synchro Transmit ter with Con t rol Transformer as Er ror -meas-

ur ing System.— Wit hou t torque amplifica t ion, a synchro-t ransmit ter -

52 S2,--

‘S?/

R1 Z IX,zt’

3

A-c \supply

R2\ I \

\ \

\\ /

53‘\

CpntrolTransmltier transformer

I?m. 3.4.—Synd)rotr an smitt erconn ectedto a synr hr ocontr oltra nsform er.

r epea ter combina t ion is not , in genera l, s~t isf act my as a follow-up

because of the low torque and poor response cha racter ist ic. As a result

of t he ext ensive developmen t of the science of elect ron ics and of design

a clva nccs in elect ric m ot or s a nd gen er at or s, t or qu e amplifica tion is oft en

obta ined by elect r ica l means. I?or th is, the na tura l input is a voltage

ra ther than a shaft rota t ion . To meet th is need, synchro con t rol t rans-

former s h ave been developed. Figu re 3-I shows how a con t rol t rans-

formers connected to a t ransmit ter .

With neglect of source impedance, the voltages given inEq. (t l)will

be impr essed a cr oss cor respon din g st at or lea ds of t he con tr ol t ra nsformer .

Thes e volt ages will p roduce p ropor t iona l fluxes , wh ich ~vill add vect or ia lly

to give a resultan t flux having the same angular posit ion in r espect t o the

cont rol-t r ansformer st a tor coils as the t ransmit ter rotor has to its sta tor

coils. If therotorof thecon t rol t ransformer isseta t r igh t angles to th is

flux, novoltage isobserved across termina ls R1-R2. Iftheangle of rota -

t ion 0’ of t he con tr ol-t ra nsformer r ot or is less t ha n O,a n a -c volt age a ppea rs

across R 1-R2 with a slight phase lead (about 10°) with respect t o the line

volt age. If t ?’is grea t er than 0, a n a-c voltage is aga in observed across Rl -

R2, but with an addit iona l 180° phase shift . In fact , t he volta ge appear ing

across R1-R2 maybe expressed as E~l~z = Em sin (O — o’). Since Em is

usua lly about 57 volts, t her e is, for small er rors, about a volt per degree of

er ror measured a t the synchro shaft . In add~t ion , a quadra ture volta ge(with h armonics), usua lly less than 0.2 volt in magnitude, is presen t even

at O’ = O. Note tha t ther e is a “fa lse zero” 180° from the proper .mo.

By use of a phase-sensit ive combina t ion of amplifier and motor , a



%c. 3.5] TRANS MITTER--GONTROL- TRANSFORMER S YSTEM 83

t orque can be obta ined tha t can be used to make O’just equal t o (?. This

is accomplished by connect ing the input of the amplifier to terminals

R1-R2 of the con t rol t ransformer and cont rolling a motor , gea red to the

con t rol-t r ansformer rotor , with the output of the amplifier . If the equiP-m en t t o be posit ion ed is a lso gea red t o t he mot or , a simple ser vomech an ism

is t hu s obt ain ed. Examinat ion of the phase of the volt age for angles

very near the “fa lse zero” ment ioned above shows that it is an unstable

posit ion . The servo will qu ickly synchronize to the proper zero.

It is to be noted that here, as for the synchro-t ransmit t er -repea ter

data system, the er ror in following must not exceed 180° at the synchro

shaft , or th is servo system will seek a null 360° or a mult iple of 360°

(measured at the synchro shaft ) from the proper angle. Therefor e, thesame requiremen ts on shaft va lues must be imposed here as were men-

t ion ed for t h e sin gle syn ch ro-t ra n sm it t er -r epea ter combin at ion .

Difficu lt ies in fa br ica tion in tr odu ce in accu ra cies in syn ch ros a s sh own

in Table 3.1. These inaccuracies a rea ser ious limita t ion on the use of the

previously descr ibed system using a single t ransmit ter and con trol t rans-

for mer . Such an er ror -measur ing system is oft en called a “ single-speed”

syn ch ro da ta syst em .

Th e effect of in accu ra cies of syn ch ros on t he pr ecision of t ran smission

of a quant ity can be grea t ly decreased by opera t ing the synchro a t a

smaller range of the var iable per revolu t ion of the synchro. For example,

if the servo problem is to reproduce accura tely the t ra in angle of a direc-

tor , a ‘‘ l-speed” synchro system will be needed. For th is there a re

needed a synchro t ransmit t er gea red one-t~one to the director and, of

course, a synchro cont rol t ransformer geared one-to-one to the r emote

equ ipmen t t ha t is t o follow t he dir ect or . Sin ce 360° r ot at ion of t he dir ec-

tor rota t es the synchro through 300°, a 0.5° inaccuracy of the synchro

system would give 0.5° inaccuracy in r eproduct ion of the angle. This

l-speed synchro has no stable ambiguous zeros. It can now be paral-

leled with a “high-speed” synchro system, so geared, for example, that

10° of dir ector rota t ion turns th is second t ransmit ter 360°. A similar ly

gear ed con t rol t ransformer is added to the r emote equipment . The

syn ch ros a re n ow a lign ed or elect rica lly zer oed (see pr eviou s r efer en ces in

Sec. 3.3 for technique) by so clamping their frames that when the er ror

between director posit ion and the posit ion of the remote equipment is

zero, both synchro systems produce zero-er ror volt age. With such apair of h igh-speed synchros the inheren t inaccuracy in reproducing the

t ra in angle has been reduced by a factor of 36; for example, it is 0.013°

if an inaccuracy of 0.5° exist s a t the synchro shaft . Unfor tuna tely, the

high-speed (wh ich can in this case be called 36-speed) system has 35 false

zeros, or lock -in poin ts.

The l-speed system gives approximate informat ion of the tota l t ra in




angle; the 36-speed system gives very accura te indica t ion of the angle,

but on ly if it is synchron ized a t the proper zero.

A switching system is needed to pu t the l-speed system in con t rol

unt il the reproduced t ra in angle is approximately cor rect and then th row

the con t rol to the 36-speed synchro system. Such a circu it is oft en calleda “synchr on izin g” cir cu it . An example of a type of such a circu it is

shown in Fig. 3.5.

The act ion of the circu it is qu it e simple. The fir st tube VI acts as an

amplifier . The second tube is biased below cutoff. As the l-speed er ror

,=-,Tn R? nf Wspeed synchro / \

I \I

speed synchroo

\T

\ \‘Tl

/J DC relayhigh impedance

+ 300V lowwattage

R,lM

o. - -. --

TOR2 of 1-!

T:’lpf

v,

+6sN7

Rz

I

51K

R3

1,o/uf

5.6K* o

To R1 of 36-speed synchroand R1 of l-speed synchro

FIG.3.5.—Synchr oni.zingircuit for dua l-speedsynchrosyst em.

, To servo-amplifier

volt age increases, there is no effect on the pla te cur ren t of V2 unt il the

er ror has reached a cr it ica l va lue rela t ed to the bias on V2, a t which poin t

the average pla te cur ren t of V2 increases rapidly. This cr it ica l va lue of

er ror is ch osen eit her by ch an ging th e gain of t he first sta ge or by ch an ging

the bias on V2 so that the l-speed system assumes con t rol befor e there is

a ny danger of lock ing-in on on e of t he a dja cen t fa lse zer os of t he 36-speed

system.

Since the er r or should be low while the servo follows mot ions of the

dir ect or , th is cir cu it will n ot n eed t o fun ct ion except wh en t he follow-u p

system is first tu rned on or when there is a severe t r ansient in the mot ion

of t he dir ect or .

There a re some considera t ions tha t t end to limit the exten t to which

t he sh aft va lu e of t he h igh -speed syn ch ro con tr ol t ra nsformer is decr ea sed

in an effor t t o improve the accuracy of a servo and, incidenta lly, a lso to

increase the gain of the t ransmission system in volts per unit of C..xsr .

On e limita t ion is that the high-speed synchros should not be dr iven at too

I

I

I

I

I

1



fjnc, 35] TRAN&lfZTTER-CONTROL-TRANSFORMER SYSTEM 85

high a ra te a t the maximum velocity of the t ransmit ted var iable. For

example, Navy units listed in Table 3“1 should not be dr iven faster than

300 rpm, though there are specia l un its, similar in other r espect s, tha t

can be dr iven up to 1200 rpm. Iner t ia considera t ions should not beover looked. Often the synchros, even if at 36-speed, reflect to the motor

shaft iner t ia tha t is not negligible, especia lly for small motors. Syn-

chronizing difficu lt ies may ar ise a t low shaft va lues of the high-speed

synchro. Because the region of high-speed con trol is ext remely nar row,

a compara t ively shor t t ime is required for the servo to pass completely

through th is region . A large number of overshoot s or even susta ined

oscilla t ion may resu lt . Finally, lit t le is ga ined by lower ing the shaft

va lue of the high-speed synchro if backlash in the synchro gear ing iscausing more er ror than that inheren t in the synchros or , in genera l, if

the other components in the loop do not mer it increased accuracy from

th e da ta t ra nsm ission .

The above dir ect or follow-up system will ser ve t o illust rate a problem

encoun ter ed in du al-speed syn ch ro syst ems. If t h e follow-up mechan ism

is turned off and the dir ector is rota ted through an angle tha t is close to

an in tegra l mult iple of 180°, the 36-speed system will be a t or near a

stable zero (see Fig. 3.6a). Fur thermore, the system is at or nea r thefa lse zero of the l-speed synchros, and there will be insufficien t signal to

act iva te the relay of the synchronizing circu it . Thus the follow-up,

if tu rned on, would remain 180° from the proper angle until some condi-

t ion arose tha t caused sufficient er ror to throw the relay. Such an

ambiguous zero exists in any dual-speed data system at those values of

the t ransmit ted var iable which require an even number of rota t ions of

the high-speed synchro and an integra l number of half revolu t ions of the

low-speed synchro.The problem of this ambiguous zero can be solved by put t ing in ser ies

with the er ror voltage from the 1-speed synchro cent rol t r ansformer a

voltage ESO tha t has a magnitude in the range

(7)

where cc is the cr it ica l er ror angle tha t will th row the synchronizing

relay, ~HSis t he shaft va lue of th e high-speed synchr o syst em, and E-, isthe maximum er ror voltage tha t the synchro can deliver . The voltage

Eso is oft en ca lled a n “ a nt ist ick-off” volt age.

The purpose of th is volt age E.. is to displace the false zero of the

l-speed system so that there is enough voltage from the l-speed system

to throw the synchronizing relay at wha t was the ambiguous zero. Of

course, the l-speed fa lse zero must not be displaced so much that it

approaches the next stable null of the high-speed system.



86 S ERVO ELEMEN TS [~EC.35

The displacement of the fa lse zero can bet t er be understood if one

follows through the procedure of adding the volt age step by step. F igure

3.6a is a plot of the magnitude of the volt ages observed from both the

1- and the 36-speed synchro systems as the dh-ector is rota t ed through

360° of t ra in angle with the follow-up tu rned off. In this figure, when the

1 Ill3;0” & 1’o”

I90” @+ 170”180”190” 270°

(Distortedangular scale)(a)

Ill I Ill

350”0° 10° 90° ~+ 170”180°190” 2;0”

(Distorted angular scale )

(b)

FIG. 3.6.—Addingst ick-offvolt aget o du ai-speedsynchrosyst em.

plot of the magnitude of the a-c er ror volt age passes below the baseline,

it indica tes a phase sh ift of 180°. For clar ity, on ly the volt ages in the

region of 0° (the angle at which the r emote equipment has been left )

and in the region of 180° (the region of the ambiguous zero that we wish

to eliminate) a re plot ted. This is merely a par t ia l plot of

E, = E.,,, sin 0, (&’a)

andEii = Em.. s in (360), (8b)

where ,?l~ is the er ror volt age from the l-speed system, E~ is t he er ror

voltage from the 36-speed system, and @is the angle through which the

dir ector has been turned.

If a volt age I& from the same a-c source tha t is used to excit e the



SEC.3.5] TRANS MITTER—CONTROL- TRANSFORMER S YS TEM 87

synchro genera tor s is added to the l-speed er r or voltage, the voltages

become

E; = E= sin O + ESO, (9a)

E~ = E~., s in (360). (9b)

This will displace upward the cu rve in Fig. 3.6a labeled “ l-speed er ror

voltage. ”

Now it is st ill necessa ry tha t when the db-ector is a t the same angle as

the r em ot e equipment , the er ror volta ge from the 1- and 36-speed systems

be zero and of the same phase for small er ror s. Therefore, we rea lign

the low-speed con t rol t r ansformer to obta in these condit ions, and the

r esu lt is sh own in F ig. 3.6b. The equat ions for th e voltages ar e now

E~ = E.= sin (I9— 1#1)+ Eso, (lOa)

E. = E- s in (36I9). (lOb)

Since at 0 = O, E; = O, we have

E,o = E-. sin ~. (11)

The new posit ion of the fa lse zero Ofof the l-speed system is given by

t he solut ion of

o = Em. sin (0, – +) + Em,. sin O,

or

1

(12)

0, = 180 + 2@.

Thus the fa lse zer o has been sh ift ed through an angle 2q$. It is desired

tha t t he fa lse zero be sh ift ed by an amount grea t er than ~c, the cr it ica l

er ror angle associa ted with the throwing of the synchron izing relay.

Then the follow-up will be unable to set t le in the stable zero a t 180°

of t he 36-speed syst em beca use t he syn ch ron izin g r ela y will be en er gized.Also, the fa lse zero must be sh ift ed by an amount tha t differs by EC

fr om a n in tegr al mult iple of t he sha ft va lue S=s of t he h igh-speed syn ch ro.

For pract ica l r easons, it is well to limit the shift to an amount less than

the shaft va lue of the h igh-speed synchro. Therefore,

SHS– CC>! M>6C. (13)

Siice #1is fa ir ly small, we may wr ite

s.s–ec>2; Em > cc (va lu es in degr ees). (14).

For a 1- and 36-speed system, such as ou r example, S=. = 10° and

w is usually about 2°. Thus a reasonable choice of ESO is about 3 volt s,

because E- = 57 volt s .

We may genera lize the above formula to include systems in which the



88 mwo ELEbfENT.5f [SEC.343

Iow-speed synchro is not at l-speed and obta in

(15)

in which SW is the shaft va lue of the low-speed synchro and S~s, SB, and

CCa re a ll expr essed in t he same unit s of t he va ria ble.

3.6. Coer cion in Par allel Synchr o Systems.—So fa r in t his discussion

there has been no ment ion of the effect s of the in ternal impedance of

syn chr o units. As m or e follow-up systems composed of syn chr o r epea ter s

or combina t ions of synchro repea ters and t ransformers are added in

~, t ti. .“. r ,

Transmitter I%eaysize “a” !. ,,

4

E It 1 ~

FIG.3.7.-—Simplifiedynchrocircuit for torqu ecalcula t ions.

parallel to a single synchro t ransmit ter , impedance effect s become mor e

pronounced.

Simple relat ionships, predict ing these effect s with sa t isfactory

accuracy for small angular er ror s between t ransmit ter and follow-up,

can be obta ined from elementary reasoning. These rela t ionships a re

suppor ted both by exper imen t and by precise analysis. I

We are in terested in a synchro system that is t ransmit t ing a fixed

angle 0. For genera lity, a t ransmit ter of size a and a repea ter of size b

(such as, for instance, a Navy 7G dr iving a Navy 5F) are chosen. The

Y-connected sta tor system is replaced by a sta tor system composed of

1?. M. Linville and J . S. Woodward, “ Sslsyn Inst rumentsfor Posit ion Systems,”Illec. E’rw.,New York, 53,953 (1934).



SEC.3.6] COERCION IN PARALLEL S YNCHRO SYSTEMS 89

one coil with its axis at an angle 8 (thus parallel tothe t ransmit t er rotor )

andasecond coil with itsaxisat an angled + 90°. This is an equiva lent

circuit for comput ing torque, provided appropr ia te circu it values a re

used. By applying a small torque to the rotor of the repea ter , a small

angular er ror A is in t roduced into the system. The configura t ion is

drawn in Fig. 3.7.

Examinat ion of Fig. 3“7 shows tha t the cur ren t il will be, to lowest -

order terms, propor t iona l to 1 — cos A = Az/2. Thus both il and its

der iva t ive in respect to A are negligibly small, and the torque will a r ise

predominate y from the curren t s ig and is. If one ignores loop 1, the

equat ions for L and ia arejuME sin A

‘2 = Z,b(Z,. + Z.b) + w’M~ sinz A’ (16a)

Eis = (16b)

Z,b +w2M2 s in2 A’

Z,* + Z,b

where M is the mutual inductance, a t A = 90°, between the repea ter

rotor and the repea t er sta tor coil car rying the cur ren t iz.

The energy of the simplified circuit is

U = ~(La + Lb)ijiz + M Sin Ai.& + ~L,t,isis, (17)

where La and Lb are the induct ive coefficien t s of the sta tor windings of

the synchros size a and b r espect ively and L,b is t he in du ct ive coefficien t of

the rotor of the size b synchro.

The torque is propor t iona l t o the ra te of change of thk energy with

the angle A. If it is assumed tha t M = <&f+b and tha t the resistance

of t he windings is small compa red wit h t he r ea ct an ce, on e obt ains

T=K

E; sin A

Oz(La + Lb)’ (18)

in wh ich E’ is the secondary voltage for A = 90° with no loadlng and K

is cons tan t .

We note that the torque obta ined when a synchro unit is used with a

similar unit is, for units size a and b,

Ts=KE~= (T.)- sin A,

E: sin ”A =Tb = K ~2Lb (T”)b SiXI A.I

(19)

The symbol T. is unit torque in ounce-inch per radian and is 180/u t imes

the value listed in Table 3.1. With subst itu t ion of these expressions, Eq.

(18) becomes

(20a)



90 S ERVO ELEMEN TS [$Ec. 36

where

~ ~ (~u)b~“

(20b)

If, now, we dr ive n repea ter s of size b, each following with the sameer ror (as they well may, because of similar fr ict ion loading), the term Lb

in Eq. (18) is reduced by a factor I/n . Th e t or qu e is equ ally dist ribu ted

at then shafts, so that a t each repea ter shaft the torque is

j“=z( Tt i)b sin An+R

(21)

With similar reasoning, the torque at each repea ter shaft can be com-

pu ted wh en differen t sizes of r epea ter s a re con nect ed t o a sin gle t ransmit -

t er . The decrease in the torque gradien t as more repeater s a re connected

to a t ransmit ter will result in grea ter er ror , because the fr ict ion load

at each repea ter remains constan t . For conserva t ive est imates, one may

assume that three-quar ter s of the er ror for a given repea ter ar ises from

fr ict ion and proceed on that assumpt ion to compute the increased er ror

as loading decr ea ses t he t or qu e gra dien t.

21= _~1 Zlb

E COS (9-240”)~ ~0~~0-120.~EC05(e-A-240”)

Z3= , z*~

FIG.3.8.—Simplifiedynchr ocircuit for terminal-volta gealculations.

dlwazZ,. .Z. +T C092O—240”),

,.

It is often desired to connect

that is a lready car rying a load of

z,, = Z, + ‘~ COS20 - A),r

Z?b=zb+~2J fb2~0~20 _ A _ 120°)

Z.1,“2Jfb2

‘* = ‘b + z,*COS20 —A – 2400).

a con t rol t ransformer to a t ransmit ter

one or more repea ter s. Therefore it is

wor t h invest igat ing the accuracy of such as ystem. This involves obta in-

ing expressions for the sta tor volt age of a t ransmit ter under load. We

shall a ga in u se on ly simple approximat ion s. F igure 38 is the circu it of a

t ransmit ter (size a) connected to a motor (size b) from which , by Theve-

nin’s theorem, the rotor circuit s have been removed. The mesh equa-



SEC,3.6] COERCION IN PARALLEL S YNCHRO SYSTEMS 91

t ion s a r e

E w’%[COS(8 + 30°) – COS(8 – A + 300)]

= il(zl. + Zlb) — i2(z2a + 22*),

1

,

E fi[COS (6 + 150°) – COS(d – A + 1500)] (22)= il(Z1. + Zlb) — ~3(Z3. + z3b),

E @i[COS (O + 270°) – COS(I9 – A + 2700)]

= i2(Z2a + Zzb) – i3(zt i + z3b).

The meaning of the impedance terms is indicated in Fig. 3.8.

In manipulat ion of Eqs. (22), we will need to examine the genera l

factor

z 1 1

Z.. ;Z.b =

[ ( )1

=—=C..zb 1+

U2~f; COS2 (O — A) Z,bZ:b

1+;z a

1 + 2.[1 + ((.#M: COS2@/z:a ) (2,./2.)]

(23)

The factor C is a constant for a given configura t ion of synchros because

.%/.% = Z,a/Za, and because (u’.kf~ COS2 ~) /Z~o = – E~+/E~, ~vhere

E,$ is a ppr opr ia te st at or volt age for t he gen er al a ngle 1#1n d E. is t he r ot or

voltage. Mult iplying each side of Eqs. (22) by Eq. (23) and reca lling

tha t A is a small angle, we obta in

GE @ A cos (I9+ 120°) = ilzl~ – izZ’.,

C’E M A cos (I9+ 60°) = ilZ1. – i3Z3~,

)

(24)

G’E & A cos 0 = i,Z2a – i3Za0.

These expressions a re the interna l voltage drops in the t ransmit ter .

Ther efore, the apparen t sta tor voltages a re

E @i[COS (o + 30°) – AC COS(e + 1200)]

= E & COS (d + 30° – CA),

E @COS (8 + 150°) – AC COS (6 + 600)]

– E W COS (8 + 150° – CA),(25)—

E ~[COS (0 + 270°) – AC COS8]

= E W COS (0 + 270° – CA).

Compar ing these apparent voltages with Eq. (6), we note tha t because

of loading, the sta tor voltages for the system that should be transmit t ing

the angle 0 are voltages which would be associated with the angle 6 – CA.

Ther efore, any ser vo employing contr ol t ransformers in this system will

have an addit ional er ror of – CA”. With reason ing ident ica l with tha t

presen ted in obta ining Eq. (20), it can be shown that



92 S ERVO ELEMEN TS [SEC. 37

It should be emphasized that in a complex system of synchros fed

from a single genera tor , there a re both “steady-sta te” er rors due to

loading and in teract ion er ror s dur ing any transitory response. Since

synchro repea ter s a re fa ir ly unstable devices, the t ransmit ter tha t is

chosen must be so la rge that th e r epea ter oscilla t ions ar e not excessively

r eflect ed in t he con tr ol-t ra nsformer sign al.

3.7. Rota table Transformers.-A rota table t ransformer resembles a

synchro in that it has a wound armature that can be rota ted rela t ive to a

sta tor system. The sta tor system, however , is composed of a single

winding ra ther than th ree dist r ibuted in space, The device is, in fact , a

simple two-winding t ransformer so const ructed tha t one winding can be

rota ted in respect to the other winding. As an element for simple

t ransmission of data , it has ra ther rest r ict ed use. It possesses on ly onemajor advantage over a three-sta tor winding synchro-it requires one

less lead. Occasionally, for instance in t ransmit t ing data from a gyro

suppor ted in a gimbal system, the reduct ion of leads is impor tan t . If,

for some such reason , it is decided to use a follow-up system employing

rota table t ransformers, one t ransformer is appropr ia tely mounted on ,

say, the gimbal system of the gyro so that the angular rota t ion of the

rotor with respect to the sta tor is the desired datum. This datum could

well be the t ru e eleva t ion angle.

For thermal reasons, it is preferable to excit e the st ,a tor winding of

such a t ra nsfor rqer r at her than its r ot or . Th e volt age .??,fr om t he semmd-

a ry (r ot or ) win din g a t n o loa d \ villbe, t o a close a ppr oxim at ion ,

(27)~P

wher e t he su bscr ipt p r efer s t o the pr imary, s to the secondary, and e

is the angle between the axis of the tuw coils. In terms of the angular

var iable f? to be t ransmit t ed, the expression for the secondary volt age is

(28)

where K is an a lignmen t pa ramet er .

An iden t ica l rota table t ransformer is mounted on the remote equip-

ment tha t is to r eproduce the angle ~. It is so aligned that its volt age is

iden t ica l with that expressed in 13q. (28). The units ar e then connected

as suggested by the equivalen t circu it presen ted in F ig. 3.9. The ser ies

connect ion of th e pr imar ies will be ment ioned la ter .The er ror volt age is given by

~ = (~,)m~[cos (~ + K) – COS (~ + K + A)]

= (Es)~,x A sin (@ + K),

with the assumpt ion of a small va lue of A.

(29)



SEC.3.7] ROTA TABLE TRAN SFORMERS 93

It is eviden t that the gain (volt s per degree er ror ) is not constant for

th is er~or -measur ing system but var ies with the t ransmit ted var iable as

sin (~ + K). Therefore, it can be used on ly in t ransmit t ing a var iable

tha t is limited to some range such that P has a var ia t ion of less than 180°.

FIG,3.9.—Er rorsystemusingrota ta bletr an sform ers,

Fur thermore, it is impor tan t to adjust K to minimize the gain var ia t ion . I

This establishes the appropr ia te va lue of K as

K = 90°- ~+, (30)

where & and fl~ are the upper and lower limits of D. The a ssocia t ed

per cen tage chan ge of ga in is

( )%L ~oo.

AC= ]–COS-

2

(31)

The permissible change of gain for a ser vo \ vill not be discussed in th is

sect ion . The analysis for invest iga t ion of the effect s of change of gain is

usually n ot difficu lt ; a gr ea ter pr oblcm is often tha t of loca t ing and det er -

mining all of the possible ga in var ia t ions. A gain change of more than

20 per cen t in the er ror -measur ing system is often undesirable. This

wou ld limit th e r an ge of P t o about 75° in t he above equa t ion .

A pract ica l difficu lty is indica ted by Eq. (29). If the a-c voltages

from the secondar ies differ in t ime phase by a dcgmcs, the subt ract ion is

imper fect ; a qu adr at ur e volt age, \ vith a small phase difference a and a

(E.]~,,

(

i3r. – / 3”small amplitude a —57 -- sin @— ~

)volt s, is pr esen t in a ddit ion

to the t rue er ror voltage. For cxamplcj if (36).,., is 20 volt s and a is

1In some easesit m:k.yx, preferredto ch<mwz v:~lucrr fK influcmrxxly an impor-l:mt regionof O:ts well :1sth e tot:t! r:ingr of t , ltrvlll~le(,f p.




2°, the quadra ture volt age is about 0.4 volt a t the ext reme of d suggested

above; th is is a volt age of grea t er magnitude than the er ror volt age for an

er ror of 10. This quadra ture volt age can be diminished by excit ing the

pr imar ies in ser ies, as has been done in F ig. 39. Often th is is a good

solution. The fact tha t frequent ly the units a re necessa r ily small in size

because of mechanica l limita t ions of the applica t ion , tends to set a

requ irement of lower pr imary volt age; a lso, the ser ies circu it helps t o

assure tha t the magnitude of E, will be iden tica l for bot h u nit s.

O-J e(t.

&(line) ~

~.

FIG.3,10.—Compu ta tionalrr or-tr an smissionystem .

Oft en a r ota ta ble t ra nsformer is pr esen t in an er ror -m ea su rin g syst em

as, a computa t iona l device. As a simple example, suppose tha t it is

desir ed t o solve con tin uou sly with t he aid of a ser vo t he following explicit

equ at ion for x(t ):

sin ~(t )z(t ) = -y—

Cos e(t)’(32a)

wh ere 7 is a constan t , d(t ) is without limit , and O(f) has limits tha t rest r ict

z(t ) t o finite values. For ease of solu t ion, the equat ion can be rea r ranged

as

cos 6’(t )z(t ) – ~ sin @(t) = O. (32b)

The sine and cosine terms can be obta ined from rota table t ransfor -

mers. The circu it is tha t of F ig. 3.10.

The er ror volt age across RA is

V. = K cos d(t ) Az, (33)

where K is a dimensional design parameter and AZ is the er ror in x. Th e

gain is K cos 8(t ). The limits of d(t ) determine whether or not it may be

n ecessa ry t o r em ove this gain va ria t ion . One simple way is indica ted in

F ig. 3“10, in which the resistor R, is a poten t iometer or a t tenuator in



SEC.38] POTENTIOMETER ERROR-MEASURING SYSTEMS 95

ser ies with a fixed resistor . The poten t iometer is so wound that the

resistan ce measured from t he arm t o th e lower data line is

sec{Na (t ) + [e(~)lcn i.}

“sec {N[a (t )]mx +[O(t )]m ,n )”

In this expression [a (t )]- is t he t ota l usefu l rot at ion of t he pot en tiomet er

and N indica tes the proper gear ing, defined by

[8(t)]_ – [o(t)]m ,. = N [a(t)]m = (34)

The resistor from the lower terminal of the poten t iometer to the data

3.8. P ot en tiom et er E rr or -m ea su rin g Syst em s.—In some specia l a ppli-

ca t ions it is desirable to use a d-c signal ca r r ier . The inputs may exist

“natura lly” as direct cur ren t , as in the problem of cont rolling or record-

ing the outpu t of a d-c genera tor , or an input may have been brough t

14II

f Q*I1

1I iL----- -— ---- ------- --- -----

FIQ.3.1I.—Data system for mu ltipleadditiveinput s.

delibera te y to a d-c level, so that smooth ing of the data can be accom-

p lished with grea ter facilit y. In some such cases, it may be a dvan ta geou s

to ut ilize poten t iometer s as elements in the da ta system; since accu racy

and life a re of pr ime impor tance, t hese are invar iably of t he wire-woun d

type. If a slide-wire type is used, the data a re cont inuous, but it is

difficu lt to obta in reasonable values of over-a ll r esistance. For th is

reason the more common type of poten t iometer in which a contact arm

makes turn-to-turn contact is in wide use.

An er ror -measur ing system utilizing poten t iometer s might be of the

form of Fig. 3.11, which has been genera lized t o indica te the possibility

of more than one input quantity. It will be assumed that the inputs a re

to have equal weight ing. Some or all the input voltages might well be

obta ined from poten t iometer s .




The output of the net , with the servo off, is

The following rela t ions may be obta ined by design:

The output volt age of the net E. is, of course, the er ror voltage and is

expressible as Gc, where G is the voltage gain of t he net . Ii’

(37)

then Eq. (35) reduces to

E. = G,, (38a)

where

C=e,, +e,z+. ..+el. —00. (38b)

The gain G of the net is

R.

The net shown in Fig. 3.11 is only a simple example of what may be

used. Mult iplica t ion and division of funct ions of var iables can also be

achieved; in fact , net s can be designed that , by the aid of a servo, obta in

con tin uou s solu tion s t o qu it e complex equ at ion s.

Addin g n etwor ks of t he form pr esen ted above is oft en u sefu l in accom-

plishing the funct ion of Differen t ia l 2 of Fig. 3.1. In such a case, the

feedback quant ity from the output of Box 32 is one input quant ity to the

network, the output of Box 12 is the second input quantity, and the out -put of the network feeds in to Box 23.

In the type of poten t iometer in which the contact arm makes turn-to-

turn contact , t he data a re discont inuous. The na ture and posit ion of

t hese discont inuit ies a re of in terest bot h in accu ra cy con sider at ions and in

understanding the “ noise” that such a poten t iometer system adds to



SEC.3.8] POTENTIOMETER ERRORMEAS UR ING SYSTEMS 97

the t rue data . Such discon t inu it ies a re of the order of Es/T, where ES

is the voltage a cross the poten t iom eter and T is the tota l number of tu rns

on the poten t iometer ca rd. Thus they a re small, bu t the developmen t of

a ccu ra te ser vo-con tr olled pot en tiomet er win din g mach in es ha s a dva nced

to the sta ge wh er e it is of va lue t o examine even such small er r or s.

We shall examine these discont inu it ies for a poten t iometer tha t has

been wound per fect ly with T ident ica l turns. A physica lly equiva len t

cir cuit is pr esen ted in F ig. 3.12. The effect ive width of the contact arm

is Kd — b + d/c, where d is the dktance between cen ter s of the wires

a long the con tact sur face, b is the width of the effect ive contact sur face

P

b-d -l

2=0

. . .

123

E&EEI Z-(z)mx

Q9..a ‘“ R.Rs ““”d-2 n-l n n+K n+K+l n+K+2 N-2 N- I N

& Es K~ 1-Zc>l

FIG. 3,12.—Equivalentcircuitfor a potent iometer .

of a single wire, K is an in teger equal t o or grea t er than 1, and c is a

number grea ter than 1. For a contact -a rm mot ion of d/c, K turns will

he shor t -circu ited; for a contact -a rm mot ion of [1 – (1/c)ld, K – 1 turns

will be shor t -circu ited. In each of these regions, the voltage observed at

t he con ta ct a rm will r em ain const an t. Actua lly, since this cycle may be

used to determine K, d, and 1/c, the following der iva t ion is valid for any

shape of con ta ct arm or wir e sur face. If we consider such a cycle in the

region of the n th contact sur face, we have

(40)

where xl and X2are the ext r eme values of x for which the K t u rn s between

the n th and then + K contact sur faces are shor t -cir cu ited and m and z,

a re the ext r eme values of z for which the K – 1 turns between the n + 1

and the n + K contact sur faces are shor t -circuited. & seen from Fig.




3.12, the assumpt ion has been made tha t the poten t iometer is

[ ()1(K–1)+; d

shor ter than its actual physica l length , and half of th is cor rect ion is sub-t racted from each end. This cor r ect ion is needed to minimize the er ror s.

In tu it ively, one rea lizes tha t it a r ises from the shor t -circuit ing act ion of

the con tact arm.

The inherent poten t iometer er ror due to resolut ion will be defined as

(41a)

Obviously, the ext reme er ror s occur a t x,, x2, x,, and/or x,. At theext r emes of the r egion of shor t -cir cu it ing K t ur ns, E q. (41a ) becomes

n—1 X(1,2)

‘( ’’2)= T–K– -T–K+l–:

(41b)

At t he ext rem es of t he r egion s of sh or t-cir cu it in g K – 1 tu rns, Eq. (41a)

becomes

(41C)

If we ignore terms tha t are a lways of the order of T–2 and assume

T>> K and l/c, we obta in from Eqs. (41b) and (41c):

1

“=m

1

“=m

‘@T-:)+,_,],c

( )]n2— C

T– 1,

(2

)

__l--~&

“=w c CT

Examinat ion of Eqs. (42) shows tha t

(dN--n = – (62).,

(~2)N -m= – (dn,

(e3)w+ = – (c,).,

(,,)N _n = – (,3).. I

(42)

(43)



SEC.3+3] POTENTIOMETER ERRORMEASURING SYSTEMS 99

If we define A a s equal to (l/c – ~), it is a lso t rue that

(6,)+* = – (+A

I(44)

(C2)+A= – (63)-ABecause of Eqs. (43) and (44), it&of in terest to examine the er r or s for

only O < n < N/2 and 1 < c ~ 2. Figure 3.13 shows the er rors given

by Eqs. (42) for some values of n and l/c. It is to be noted tha t the

poten t iometer , if per fect ly aligned, has apparent resolu t ion of at least

J/’

/

,//’

J’

/“/

FIG.3.13.—Potentionmterwolutionanderrors.

l/2T and, for c = 2, consistent resolu t ion at the cen ter of the poten-

t iomet er of l/4T.

Often it is impract ica l to avoid using the poten t iometer in the region

Ofx=o. This is especia lly t rue when a funct ion capable of posit ive

an d n ega tive va lu es is bein g con ver ted in to an elect rica l volt age by mean s

of a tapped potent iometer . There is a region of zero voltage % the tap

tha t is [Kd + (d/c)] unit s wide. This represen ts a separat ion of

()K–l)d+ :

between t he in ter cept on th e z-axis of th e best voltage slope for nega t ive z




and the in tercept cm the x-axis of the best voltage slope for posit ive z.

This, if the er ror is split , makes the voltage always too low on either side

of the tap, and an addit ional constan t er ror of magnitude

[

K – 1 + (l/c)

2T 1

is present . Obviously, in such tapped poten t iometers K must be kept

small.

The broad zero-voltage region may be avoided by “floa t ing” the

potent iometer , that is, not grounding the zero poten t ia l point . This,

‘-’’->-. --—-. ---1--

‘--’’:S-’--IG. 3.14.—Diff erent ia t ion and integrat ion by error-measuring elements .

however , permits the coordina te of the zero poten t ia l to dr ift if, becauseof loading, +Es is not ident ica l with – E. or if th ermal differ en ces cau se

n on un iform ch an ges of t he r esist an ce in t he pot en tiom et er .

It can be seen that the noise arising from the fin ite resolu t ion of the

poten t iometer will have a fundamenta l frequency dependen t upon spac-

ing of the wires, the por t ion of the poten t iometer in use, the contact arm

parameter c, and the velocity of the con tact arm. The amplitude

of the noise will be dependen t on the first t h ree of these parameters.

Because of this noise, it is genera lly unwise to a t tempt to ut ilize er ror -der ivat ive ‘n et s in h igh-gain ser vos using su ch er ror -m ea su rin g syst em s,

for the afnplifier or motor may over load or overhea t as the result of the

high gain for such noise.

Since only discrete voltages may be obta ined from such poten-

t iometer s, h igh-ga in ser vos may display a t en den cy t o cha t ter with a ver y

small amplitude in seeking a voltage va lue tha t is not obta inable. With

a higher degree of stability in the loop, the tendency usually disappears.



Ssc. 3.9] NULL DEVICES 101

Computa t ional er ror -measur ing elements are not rest r ict ed to those

which per form t he algebra ic manipulat ions of mult iplicat ion, division,

addit ion, and subtract ion. There are a lso elements tha t can be used to

per form t he oper at ion s of in tegr at ion a nd differ en tia tion . To a ccomplish

t hese opera t ions, any device can be used that has an out put propor t ional

to the t ime ra te of change of the input . Such a device will be called

her e a “t a chometer .” If the input exist s as a shaft rota t ion , a simple

tachometer is a small d-c genera tor . Usually such a tachometer has a

permanent -magnet field, bu t there is no reason why the field cannot be

supplied by an external elect r ica l source; the output volt age will then

be the product of some funct ion and the speed of the shaft of the tachom-

eter . Indeed, one might thus obtain the second der iva t ive of the mot ionof a sha ft , a lt hough a ccu ra cy gr ea ter t han a few per cen t wou ld be difficu lt

to obta in .

Simple examples of circu its using elect r ica l t achometers and poten-

t iomet er s a re pr esen ted in F ig. 3.14. In Fig. 3. 14u, the er ror voltage is

00v, = K,pel – ES ~m) (45a)

and, for the steady sta te,

‘0=[% O’mab=c@(45b)

In Fig. 3“14b, the er ror volta ge is

e,

“ = ‘“ (O,)m=r– “peo’ (46a)

whence

(46b)

In t he above equat ions, K, is, of cou rse, t he volt age-speed con st an t

of the tachometer . For a cer ta in permanent-magnet tachometer , 1 which

has been used in many equipments, K. = 35.50/1800 volt s per rpm

+ 10 per cent , with 10.5 volt depending on direct ion of rota t ion. The

linear ity of this t achometer is 0.5 per cent . The tachometer has a la rge

number of commutator segments (18) to reduce noise.

3.9. Null Devices. -There exist er ror -measur ing systems that ,a lt hough oft en n ot , employin g elemen ts ma ter ially cliffer en t fr om t hose

descr ibed above, a re unique in their applica t ion and do deser ve separa t e

classificat ion. The y will be called null devices, because they cont ribu te

no er ror tha t is a funct ion of the magnitude of the t ransmit t ed var iable.

An example will serve as a definit ion . Here the problem is to posi-

t ion a shaft with high torque to the same angle as a shaft on which only

I Type B4.1,E lect ric In dica tor Co,, St am for d, Corm. /’-~

,.. ,



102 S ERVO ELEMEN TS [SEC,3.9

a slight load maybe imposed. If the shafts can be so placed as to have a

common axis, the low-torque shaft can car ry the rotor of a rota table

t ransformer and the high-torque shaft the sta tor . If the unit is proper ly

a ligned, t her e will be zer o out put volta ge at zer o er ror ; and if er ror exists,

the phase of the output volt age will indica te the sense of the er ror . Since

the null posit ion is used, no er ror will be contr ibu ted by the rota table

t ransformer . It is impor tan t tha t the er ror -volt age output be fa ir ly

linear with er ror , bu t this is a problem of servo-loop gain and not of

posit ional accuracy.

It is apparen t tha t many types of null device a re possible. One

shaft may, for instance, ca r ry one set of the pla tes of a condenser , and

the other shaft the other set of pla tes for the condenser . If, then , this

condenser is one arm of a br idge circu it excit ed by alterna t ing cur ren t ,

proper er ror signals can be obta ined from the output of the br idge. On

the other hand, coupling between the two shafts may be by a nar row light

beam. The low-torque shaft may car ry a ligh t sou rce or a mir ror to

reflect a beam of light from a source on the follow-up shaft . The fol-

low-up shaft then car r ies a pr ism \ vhich split s the received light beam

and sends it t o two photoelect r ic tubes. Zero er ror , of course, should

exist when the outputs of the two photoelect r ic t ,ubes are equal. Both

of th e examples in this paragraph, howe~er , are subject t o possible er ror s

ar ising from var ia t ions in severa l elements; for example, var ia t ions in

the other arms of the br idge in the first case or differences in the charac-

ter ist ics in photoelect r ic tubes in the second. On the other hand, such

systems impose t ru ly negligible loads on the low-torque shaft . The

design er must weigh su ch consider at ion s for ea ch pa rt icu la r a pplicat ion.

In some problems, the displacement to be followed exists as a linear

motion. Many of the devices used in following rota t ional displace-

men t a re t hen a pplica ble wit h obviou s modifica tion s.

One device, a so-ca lled “E-t ransformer ,” has been widely ut ilized,

especia lly in following linea r mot ions. A sketch is presen ted in Fig.

3.15a. The E sect ion is ca r r ied by the follow-up, while the bar sect ion is

ca r r ied by the displacement to be followed. Mot ions are in the plane of

the paper and across the figure as indica ted by zo and xl. Terminals

1 to 2 are excit ed from the appropr ia te a -c source. l’here is one posit ion

of the bar for which zero voltage (ideally) is observed across terminals

3 to 4. This is the posit ion in \ vhich the bar gives such coupling that

equal and opposit e emf’s are induced in the secondary. Pract ica lly,

there will be harmonic and quadra ture volt ages left a t the null posit ion.

For a displacement of the bar to the left of tha t zero posit ion , a volt age

of phase o appears across the secondary terminals 3 to 4; and for a dk-

placement to the r igh t , a voltage of phase @ + 180° appears across the

secondary. This voltage, then, may be used as an er ror volt age, and it is



SEC.3.10] MOTORS AND POWER AMPLIFIERS 103

n ot difficu lt t o obta in a t least approximat e pr opor tion alit y between er ror

a nd volt age for small er ror s. One must be cer ta in , however , that var ia-

t ions in coupling ar ising from causes other than the difference between

z, and m, such as mechanica l looseness, do not occu r , because suchvar ia t ions impair t he accu racy of t he device.

There is an impor tan t extension of the idea of an E-t ransformer tha t

is useful in follo}ving an object tha t moves in two coordina tes. Such

a problem is encountered when a radar director is t o be stabilized by

reference to a gyro axis tha t is kept poin ted a t the ta rget . Th e device

consists of two single-coordina te E-t ransformers a t r ight angles to each

ot her , ~vit h a common ccn tcr , or pr imary pole. This assembly is moved

N’!’

(A) (b)

l:IG.3 15.—E-tr an sform rr b,

by the follo\ \ -up. The input mot ion, from a gyro in our example, movesa dome, shown dot t ed in the plan vie~r of Fig. 3. 15b. The separa te er ror

voltages for the two servo systems involved, t ra in and eleva t ion , a re

ava ilable fr om t he two secon da ry syst em s.

3.10. Motors and Power Amplifiers. -Most d-c servo motors, used

wit h con tr olled a rma tu re volt age an d fixed-field excit at ion , h ave t or qu e-

speed ch aracter ist ics t ha t , t o a ver y good approximat ion, may be defin ed

by either of the set s of convent iona l coefficien ts. If one sta r ts with the

assumption tha t the torque M is linear with the armature cu rren t and thatthe cur ren t is a funct ion of motor speed Q and applied voltage E-, one

has




or one may direct ly assume M = f( il~, E~), and then

‘=--~~)..[~)Em+(~)”nL=fEmEfmf48

In the above equat ions Em and i~ are the voltage applied to the motor

a rmature and the cur ren t flowing through the armature. The coefficient s

defined by Eqs. (47) and (48) have descr ipt ive names: K, is the torque-

cur ren t constan t ; K, is the back-emf constant ; Km is t h e speed-volt age

constan t ; and f- is the in ternal-damping coefficien t . It is seen tha t

Km = K;l. Str ict ly, Eq. (48) assumes tha t there is no t ime lag between

FIG.3.16.—D-cmotorcharacteristics.

i~ and Em, Figure 3.16 illust ra tes the constants for a hypothet ica l d-c

servo motor . Except a t h igh va lues of i~ where sa tu ra t ion effect s may

appear , exist ing d-c servo motors have character ist ics that a re, for prac-

t ica l purposes , simila r ly linea r .

Two-phase low-iner t ia motor s have been developed that a re very

convenien t for low-wa tt age con tr ol applica t ions. For many of these

a-c motors the convent iona l coefficient s a re far from constan t over the

whole range of excit a t ion and speed. More accura te analysis of the

motors, however , does not yield a simple rela t ion for M = f (Q~,E~).

For this reason it is common pract ice to use the coefficien ts defined in

Eqs. (47) and (48). For tuna tely, they are oft en fa ir ly constant for low

speeds and voltages. F igure 3.17 shows how the coefficien ts vary for one

two-phase motor . For th is same motor K,/Z~ is reasonably constan t

over a wide range of E and ~.. It is seen tha t for speeds less than one-

four th of maximum speed, f~ and Km are fa ir ly constan t . Thus, for

I



SEC.3,10] MOTORS AND POWER A MPLZFIERS 105

applicat ions where accura te per formance is not required at the higher

speeds, one may ana lyze the loop using low-speed evalua t ions of the

constants. It is advisable, however , to invest igate the stability of the

loop a t in termedia te a nd h igh speeds, beca use differ en t va lu es of t he mot or

t ime constan t and th e loop gain will be en cou nter ed.

In many cases it is possible to employ gain devices or amplifiers that

a re near ly independen t of the frequency over the region of frequencies

~n ,lf= 4 in.oz

\Jm,Em=Iio VOItS

\

/

400~

/’4,‘f_, Em= ‘Ovolts

7fm,Em=40volts

~o)0 2000 3000 4000

IIm in rpm

FIG.3.17.—Character is tics of two-phase a-c motor , Diehl FPE49-2 . Sixty cycle , 110 volt son fixed ph ase, n -wa tt maximum ou tpu t r at in g.

that is of in t erest in the closed servo loop. For example, most vacuum-

tube amplifiers can be represen ted as having a t ransfer charact er ist ic

G,,

Y12(P) = ~p + ~ (49)

to a sat is factory degree of exactness. Sin ce t he a tt ainmen t of an a ppr opr i-

a te va lu e of Glz is t he pr im ar y con cer n in amplifier design , t he equ ivalen t

t ime constant 2’ is oft en somewhat accidenta l. For tuna tely, it is usually

smaU. The wise designer , however , will exper imenta lly determine

Yl,(p) t o reassure himself concern ing the small magnitude of 1’ and to

obta in it s value. Simila r sta tements hold for any vacuum-tube ampli-

fica t ion that might exist in Box 23 or Box 32.

The final, or power , stage of amplifying equipment may employ any




of severa l element s depending upon the magnitude of power required and

t he designer ’s pr efer en ces. F or applica t ions ut ilizing ver y small mot or s

(less than +T hp) sa turable r ea ct or s or elect ron ic tubes, eit her t he gaseous

or vacuum type, a re in common use. Power vibra tors, thyra t rons, or

rota ry magnet ic amplifiers a re oft en employed for cont rolling la rger

motors.

Since the rota ry magnet ic amplifier (amplidyne) has received wide

applica t ion as a power stage dr iving a d-c mt itoi-, it may be of value to

=X’Rotary magneticamplifler D-c motor

FIG. 3.1S .—l{otary magnet ic amplifier and d-c motor ,

obta in an expression for the t ransfer character ist ic of such a combina-

t ion . The equivalent circuit is shovm in Fig, 3. 1S. The axis of the

r ota r y amplifier established by the direct ion of the film from the excita -

t ion cur r en t ij is commonly ca lled the direct axis. Quant it ies associa ted

\ vith th is axis will be char acter ized by the subscr ipt d, a nd t hose a ssoci-

a t ed with the axis a t r ight angles to this dir ect ion (the quadra ture axis)

will h ave a subscr ipt q. In this pr imary direct -axis field, ~, is a wound

rotor dr iven by an auxiliary motor a t a speed S. It is apparent that

(50)

in wh ich KI is a constan t descr ibing the magnet ic circuit . Beca use of

the rota t ion of the rotor , an emf is developed between the brushes b

with a magnitude

Th e con st an t Kz is determined

machine. The quadra ture-axis

field +, that may be augmented

by the mater ia l and st ructure of the

cur ren t iq crea tes a quadra ture-axis

by the coil. Q of impedance R, + Lqp.

I



SEC.3.10] MOTORS AND POWER AMPLIFIERS 107

The quadratu re field causes an emf to be developed between brushes a .

An expression for th is emf can be der ived from Eqs. (50) and (51),

G.ef

ed = SK8iq = (1 + Tfp)(l + !l’,p)”

S2K,K,KS

I

(52)

G. ER~(Ra + R,]

The load cur r en t in the armature circu it would produce a direct -axis

flux tha t would oppose the excit ing flux. Thus the machine would have

very poor regula t ion . To eliminate th is difficu lty, compensat ing coils C

a re presen t t o cancel ou t the flux in the direct axis due t o load cur ren t .

Since the emf ea is applied to the motor -a rmature circu it , t he rela t ionma y be wr it ten

ed = i~R(Tap + 1) + K.L, (53a)

in which

R = R. + R. + R- = t ot al a rmat ur e cir cu it r esist an ce, (53b)

(53C)

The torque result ing from the motor -a rmature cur r en t i~ will be con-

sumed in accelera t ing the iner t ia J m of the motor rotor , t he iner t ia J O of

the gear t ra in which is reflected to the motor , and the iner t ia of the load

.J Lreduced to motor speed; it a lso will be dissipated in overcoming any

viscous fr ict ion fg Q in t he gear ing and bear ings. Thus the torque

equa tion is

in which N is the gear ing rat io t o the load. With .J ’ as the tota l effect ive

iner t ia a t the motor shaft , Eq. (54) maybe rewr it ten as

K,~ (ed

‘KeQ’”) ‘wp+l)(T@+l)Q~

Since J’/f@ >> T., th is equa t ion is usually approximated as

(55a)

(55b)

Usually j, << (K, Kt)/R; t hen , using Eq. (51), the final t r ansfer funct ion



108 S ERVO ELEMEN TS

for the power stage and motor is, since L = pe~,

00 KmG.

– = p(Tm.p + l)(!f’,p + l)(!rfp -t 1)’f

[SEC.341

(56)

Suppose that a simple ser vo is const ructed using an elect ron ic ampli-

fier (with gain G, and lag 2’) in Box 12, a motor -amplidyne combinat ion

in Box 23, and no equalizing-feedback path ., If an er ror -measur ings ys-

tem is employed with a voltage gain of G, volt s per unit of er ror and Box

31 is a simple gear tra in of r educt ion N, Eq. (2) becomes

$= K.

e p(z’mp + 1)(1’,p + l)(T,p + l)(Tp + 1’

~ ~ G,G,G.K.

1

(57)

.N“

3.11. Modula tors.—It is somet imes desirable to conver t a d-c signal

to an a-c signal tha t has a determinate phase rela t ive to an exist ing a-c

voltage. For example, this is desired when a t we-phase a-c servomot or

is used in a loop that has an er ror -mea su rin g system ut ilizing d-c voltages,

Beca use of t he difficu lt ies of d-c amplifica tion , it is oft en pr efer r ed t o mak ethis t ransit ion before any d-c amplifica t ion and to postpone amplifica-

t ion until the signal is in a-c form. Thus it is common that lit t le or no

ext raneous noise or fa lse er ror signal can be tolera ted. This oft en ru les

ou t var ious schemes of modula t ion employing vacuum tubes in favor of

m ech anica l modu la tion, su ch as is obt ain ed wit h a vibr at or .

One form of a synchronous vibra tor employs a fla t r eed, or leaf, fixed

at one end and with the other end free and car rying an iron slug. Encircl-

ing this fr ee end is a fixed coil, excited from the alternat ing cur ren t t o

which the vibra tor is to be synchronized. The coil cr ea tes an oscilla t ing

magnet ic field with its vector lying along the length of the reed; as a

result the face of the slug at the t ip of the reed has an alternat ing magnet ic

polar ity. J ust above this end of the reed arc the poles of a permanent

magnet , so placed that the face of the slug is subjected to a fixed field

perpendicular , to the fla t of the reed. Thus the reed is subjected to an

alternat ing force and will oscilla te with a fundamenta l fr equency equal

to the fundamenta l frequency of the cur ren t flo\ r ing in the excit ing coil.

If the reed is mechanically resonan t a t a frequency near the excit ing fr e-

quency, large amplitude is obta ined with lit t le expenditure of excit ing

power. This resonance frequency, ho\ vever , must not be too near the

excit ing frequency or a small change of excit ing frequency will pr oduce a

la rge change in the phase of the oscilla t ion of the reed. One commercia l

designl has made t he compr omise that a vibr ator for 60-cycle use employs

1Synchr onous converter No. 75829-1, Brown Instrument (~o., Philadelphia,P a.



SEC. 3.11] MODULATORS 109

a reecl resonan t a t about 80 cps. Thk gives about a 12° lag between the

r eed mot ion and the excit ing voltage; the la t ter may, of cou rse, be shifted

in phase by a fixed amount if so desired. In servos using two-phase

motors, it is often conven ien t to obta in the requ ired pha~e shift for thecon tr olled ph ase in t his fa sh ion . The same vibra tor , t o be excit ed from

6.3 volt s, has a coil impedan ce of abou t 120 ohms, a lmost pu rely r esist ive.

Beca use of th e h armon ic con ten t in th e excit in g volta ge, it is pr efer able t o

,,, .+

T

~z Eo(t)

“’iiill:?~

b

,+ ~b

(3J a

-g

k

Eoif] ~

“= g

a

Fm. 3. 19.—Vibrators as modulators, (a) Three possible circuits; (b) generalized output.

obtain t he phase shift by a ser ies inductance ra ther than by using a

series capacitan ce.The ~eed car r ies elect r ica l con tact s and acts as the pole of a double-

th row single-pole switch with a noise level less than 2 ~v. For a cer ta in

fr act ion of t he cycle, t he pole may shor t-cir cu it bot h fixed con ta ct s.

F igu re 3.19a pr esen ts t hr ee possible cir cu it s employin g syn ch ron ou s

vibra tor s as modula tor s. A genera lized outpu t , for which it is assumed

tha t the input voltages do not vary sign ificant ly dur ing a modula t ion

per iod, is given in F ig. 3. 19b. It may be expressed as

‘(’)‘WY “)+SE2.

+ : (E1 + E,)

2 [ 1~+ cos s ~ (2WL– 1) sin [t io(2m – l)t ]

m-l

+ : (2E, – E, + II,)

z

‘1~n sin (smn) cos (2wont ), (58)

n=l




where the dr iving voltage has been assumed to be V sin (d + L?), ~ is

equal t o the lag observed in the vibra tor reed, and s is t he fract ion

of the cycle for which the pole and the two contact s are shunted to each

other.For Circuit 1 in Fig. 3.19a, Ez = ES = O, and the outpu t can be

expressed, from Eq. (58), as

j(t)=~R,y~2(1–s)+E~R1yR,

{[1-(91 sin~~++[’-rwsin’u”’+ ““” }

–E:R, &2 (s7r Cos Zwt + . . ~ ) (59)

if s is considered to be a small quant ity. The harmonic terms will cause

some heat ing and over loading; they are small, however , and are usually

so a tt en ua ted by t he amplifier th at t hese effect s a re n ot ser iou s.

If the input t o the vibra tor is E sin mt , then, when only the fir st two

terms of Eq. (59) are considered, the ou tpu t is

j(t ) =

:+2 [+)l~lysyn~’

+ Cos (u, – ?n)t - Cos (Qlo+ ?n)t]. (60)

In such a case the sole object ive is usually to obta in the sideband terms of

frequency CM– m and a, + m. In many cases the presence of the

addit ional (sin mt)-t erm in the outpu t causes no apparen t difficulty. In

feedback loops, however , it has been observed tha t a circuit similar to

Circu it 2 in Fig. 3“19 permit s h igher gain . Such a circu it has only the

(sin rwd)-terms in its ou tpu t , so tha t no term similar to the fir st in Eq.

(60) is p roduced.

Circu it 3 is of in terest because it combines the opera t ions of obtain -

ing the difference of two voltages and of modula t ing tha t difference.

Often 2* is so chosen tha t it is very large compared with RI and RS. If

22 is complex, ca re must be taken to maintain the gain a t the car r ier

frequency.

The outpu t of Circuit 3 may be expressed in terms of servo er ror .

If RI = Ra, Z~ = w, E. = K@,, and – E~ = K%, it s ou tpu t r edu ces t o

j(’)K[’@+o+e4s+)l+’%l’-(a lsinuo’ “ “ “ ~ ’61)

When modula t ion is per formed at a low signal level, ca re has been



i?.Ec.3.12] PHAS E-S ENS ITIVE DETECTORS 111

taken toavoid coupling by st ray capacitance tea-c voltage leads. For

example, in Circuit 1 there is st r ay capacit ance Cm between a volt age

source Em sin (cd + +) and the leaf con tact and ungrounded con tact .

Since th is st r ay capacit ance will be quite small, for usual cir c~it s thecon tr ibu tion t o Eo(t) from this coupling is

+$%]CnE. COS(d + 4)

for the half of the modula t ion cycle tha t Eo is not zero. For simplicity,

assume thats is zero and that the volt age dr iving the reed is sin (wd + ~),

where, as before, @is the phase lag of the reed. Then the effect on Eo oft he st ray cou pling is, in fundamenta l fr equ en cy compon en ts,

It is seen that , in genera l, both an in-phase componen t and a quadra-

tu re componen t have been added by the st ray coupling. If the volt age

dr iving the reed has been sh ifted approximately 90° to obta in phase shift

for a two-phase servomotor , the st ray coupling causes lit t le or no servo

er r or from “standoff” in the volt age E. A quadra tu re compon en t, h ow-

ever , will exist . If the volt age dr iving the reed has not been mater ia lly

sh ifted in phase, the quadra tu re componen t is small but th ere is a‘ (stand-

off” er r or equal to tha t which would cause a change in E of amount

AE = ; uOR,CSE%. (63)

For example, if C. = 2 X 10-8 ~f, R, = 1 megohm, wj = 377,

“En= 6.3 volts, then AE is abou t 6 mv.

3.12. Phase-sensit ive Detectors.-It is oft en necessa ry to change

from an a-c er ror voltage to a d-c er ror voltage. For instance, if a syr l-

ch ro er ror -mea sur in g system is u sed with a ser vo loop in which th e power -

ou tput st age is an amplidyne, it is clea r ly desirable to have a circuit tha t

will have a d-c ou tpu t with the sign depending on the phase of the a -c

er ror voltage and with the magnitude propor t iona l to the magnitude of

the a-c er ror volt age. Such a device is ca lled a phase-sensit ive detector .

There a re many var ia t ions in design; one possible circu it is presen ted in

Fig. 3“20.

In Fig. 3.20, it is seen tha t the volt age applied t o the full-wave rect if y-

ing d iode D1 is V1 = V, + V, — Vol and the voltage applied to the otherdiode Dxis V,= V,-V. – VO,. The voltage V, is oft en called the

“r efer ence volt a ge. ” If it is assumed that




v, = v, Cos(d + e), (64a)

v, = v, Cosd, (64b)then

VI = v’(V, + V, cos 8)2 + (Vt sin 0)2 cos (d + 1#11) VOI, (65a)

V, = V(V. – V, cos 0)2 + (V, sin 0)’ cos (cot – &) – VO,, (65b)

If @is small, Eqs. (65) reduce to

VI = (Vr + v, Cos 19)Cos cd – v“,, (66a)

V2 = (Vr – v, Cos e) cm t it – v“,. (66bj

It will be noted that the ga in is not very sensit ive to the phase angle 0.For frequencies of the order of u , the load impedances across which

VO, and VOZare developed are kept la rge compared with the in ternal

FIG. 3.20.—Ph ase-sensicive detector..

impedance of the diodes and t ransformers. This avoids the effect s of

nonsimila r ity in these in terna l impedances and in the loads themselves

and helps to rea lize the full voltage gain GI of t ransformer 2’1. A high

load impedance requires a high va lue of R, and a low value of C,. To

decrea se t ime delay, the product R, L’, is kept small, and poor filt er ing

r esult s (10 per cen t r ipple is usually t oler ated). For t un ately, if o is small,

t he r ipple due to the referen ce volt age tends to cancel in the circu it given

here; rela t ively lit t le r ipple is presen t in tha t ou tpu t when the er ror is

small. In connect ion with r ipple it is wor th not ing tha t noise similar to

r ipple ca n be ca used by small differ en ces in t he volt age gain of t he secon d-ar y sect ions of T,. Again th is will be pr opor t ion al t o er ror volt age.

In the steady sta te, V, and V2 ~ O, and the outpu t voltages are

a ppr oxim at ely d-c, wit h valu es

Vol = v, + v,,

V,2 = v, — v., (v, s v,);

therefore

I(67)

v“ = 2VC if V, s V,.



SEC. 3.12] PHAS E-S ENS ITIVE DETECTORS 113

Thus V~ is linear with the magnitude of the er ror . If an er ror of the

opposite sense is assumed, the phase of V, shifts 180°, and the sign of the

ou tpu t volt age is n ega tive.

If V, > V,, VO, = V, – V, and V, = 2V,. This limit ing act ion issomet imes desir able if cir cu it s of in tegr al t ype follow t he r ect ifier , beca use

it establishes the maximum outpu t tha t will be “in tegra ted.”

If @approaches 7/2, Eqs. (65) can be writ t en

v, = V“ yTm Cos (d + 0’) – v,,, (68a)

v, = V“v: + w Cos (d – 4’) – V02, (68b)

(68c)

It is seen tha t no net direct cur ren t result s from the circu it , provided

tha t either the voltage gains of the secondary sect ions of TI are ident ica l

or that the differences average in such a way as to produ ce zer o net effect .

In the ou tpu t , however , t here is increased r ipple due to the difference in

phase bet ,ween v, + V, and V, – V,, becau se th e r ipple does n ot can cel.

This insensit iveness to a quadra tu re component in the er ror signal is

very advan tageous, not on ly in elimina t ing a quadra ture component that ,

has ar isen from some sor t of unfor tuna te character ist ic of the er ror -

measur ing circuit but a lso in separa t ing for a par t icu la r servo the appro-

pr ia te er ror signal from a tota l er ror signa l that conta ins da ta for two

servos. For instance, in radar t racking the tota l er ror voltage may be

wher e CEand CTare the er ror s in eleva t ion and t raverse respect ively. BY

applying this er ror voltage }0 two phase-sensit ive-detector systems in

para llel, on e of which is supphed with a r efer en ce voltage V, = V, sin t it

and the other with a refer ence voltage V; = V: cos d, the er ror data

for the t raverse and eleva t ion servos a re separa ted into two isola ted

voltages. When it is desirable to eliminate a quadra ture component but

t o obta in t he final er ror sign al as an a -c volt age, a ph ase-sen sit ive det ect or

may be used and followed by a modula tor , such as the vibra tor discussed

in Sec. 3.11.

In some var ia t ions of the phase-sensit ive-detector circuit , the refer -

en ce volt age is in tr odu ced sepa ra tely in to t he two diode-r ect ifier sect ion s

in such a way tha t a nonuniformity of t ransformer secondar ies may

produce a small d-c ou tpu t of the circu it with zero er ror . If a zero-er ror

d-c ou tpu t exist s, r egardless of cause, the servo will “stand off” from its

proper zero. Indeed, such” standoff” may exist in a servo because of any

ext raneous er ror data tha t appear in the forward channel, such as the

pickup ment ioned in Sec. 3“11 in connect ion with the synchronous

vibra tor . Its magnitude can be easily established if limits can be set onthe magnitude of the fa lse data . If a t the poin t of or igin -of the ext rane-




ous data (of magnitude A) one unit of er ror produces G units of datum

quant ity (volts in the above case), the standoff is A/G units of er ror .

Thus the la ter in the loop the ext raneous data appear the less is the resu lt -

ing inaccuracy. Amplidynes, for instance, oft en have an appreciable

ou tput a t zero input due to hysteresis; bu t because they are used in the

fin al power st age, lit tle in accu ra cy r esu lt s.

(a ) (b) (c)

FIG. 3.21 .—Equa lizing networks for opera t ing on d-c e rror volt age.

(a ) Eo/ E , = (Tap + 1)/(Go!i”.P+ 1); (b) Eo/EI = Go[(Z’ap+ l)/(AoTOp + l)];(C).%/ ’E I = Go[(T IP + 1)(~24P+ 1)1/[(?’aP + l)(!fb~ + l)].

3.13. Networks for Opera t ing on D-c Er ror Voltage .-A simple servo

t ra nsfer ch ar act er ist ic, su ch as wa s pr esen ted in Sec. 3.10, is of t he form

‘0 – ; (T ip + l)(z’zp +Kk3P + 1)— —c . . . (70)

As the gain is increased in an effor t to reduce the er r or in following, the

servo often becomes unstable before the gain is h igh enough for sa t is-

factory per formance. By inser t ion of circu it s having proper t ransfer

cha ract er ist ics in Box 12 (F ig. 3”1), it is oft en possible t o incr ease t he gainto the desir ed value and yet maintain sa t isfactory stability. Examples of

such networks, useful if the er ror is a t a d-c volt age level, a r e shown in

Fig. 3.21. The t ransfer character ist ics a re plot t ed asymptot ica lly.

This t ype of plot and its depa rt ures fr om t he actua l cu rve will be discussed

in Sec. 4.10.

Th e n etwor k given in F ig. 3,21a h as t he t ra nsfer ch ar act er ist ic

(71)

Such a network is often called a “ der iva t ive-plus-propor t iona l net ,”

because it haa the same character ist ics as an “idea l” der iva t ive-p lus -

propor t iona l device [Y = G(Tp + 1)] for fr equ en cies less t ha n l/(GoTa ).

It is not difficu lt to obta in any reasonable values for the frequt ...vies

l/Ta and l/(G,T.).



SEC.3.13] lXC ERRORVOLTAGE N ETWORKS 115”

Then et wor k given in Fig. 3.21 bhasthe t ra nsfer charact er ist ic

~,=~o=GOT.p+l

E, ml’(72a)

whereT. = RICl, (72b)

Tb= [R, +R,(l –@o)]C,, (72C)

R,GoE—

R, + R,”(72d)

The res is t ance R3 is usua lly det ermin ed by r equ ir em ent s of t he following

stage. Because th is net has the same character ist ic as an ideal in tegra l-

plus-propor t iona l device {Y = G[( l/Tp) + 1]] for frequencies grea ter

t han l/Tt ,, it is som et im es ca lled an int egr al-plu s-P ~oPor tiona l n etwor k.

Again, T. a nd Tb ca n be est ablish ed in depen den tly.

In servo design, cases often ar ise in which it is desired to have a

t ransfer chara cter ist ic combining t he pr oper t ies of bot h of t he a bove net -

works. In order to reduce the number of d-c stages, the networks are

usually combined direct ly. Such a composit e network is shown in Fig.

3.21c. The t ransfer character ist ic is

E.%

(Tip + l)(T2@ + 1)—

( )[

7R,

~1~,, + ~a TI,Z’2 p’ +1

T,+~z4+&(T13+T2) P+l+:a

(73a)

where T1 = Rlcl; T24 s (R2 + R4)C2; TM ~ (RI + Rs)CI; T2 ~ RzC2.

We may wr it e Eq. (73a) in the form

E. (T,P + l)(T24p + Q,

E = ‘0 (Tap + l)(TbP + 1)Go G ~.

R, + R,(73b)

The design of the servo loop involves adjust ing the magnitudes of

T,, T*4, T ., Tb, and GO. The gain at very high frequencies G= is

(74a)

and is also of in terest because of the presence of noise in the er r or signal.

Since it is obviously defined if T,, T24, T., Tb, a nd GOa re fixed, ch oice of

t hese qua nt it ies shou ld be in fluenced by con sider at ions of an a ccept able

Gm.

If R8 is determin ed by gr id-circu it requirements or ot her impedance

requirements of the following stage, there are precisely the requisite

number of parameters to establish the four t ime constant s and the low-



116 S ERVO ELEMENTS [SEC.313

frequency gain GO. It is to be noted that the inequality

GoT~Tz, < ~G.=T. (74b)

a

cannot be viola ted; the upper limit occurs at Rz = O.

Compar ison of Eqs. (73a) and (73b) yields, in t erms of Cl and C*, and

the quant it ies tha t it is desired to fix,

TeT~ = G, T, T,, + (1-G”’’Tl+R~cl)[T,4-(%)~3c,l ‘“a )

‘a+Tb= ‘1+ T24+R8(1 -G”’[C’-(%9CJ ‘75’)Solut ion of Eqs. (75a) and (75b) for C* gives

G,

c’ = 2R,(1 – G,)’ {T,G, + T,,(2 – Go) – T. – T,

J [ (%%lT@ -4)A (T. + T, – GoT, – GoT,,)’ – 4G0 l’eT,

(76a)

Commonly the fir st t erm under the square-root sign is much grea ter than

the second, and so we may obtain the approximat ion

c, =GO

[T2~ –

TaTb – TITZ,GO

R,(I – G,) 1. + Tb – Go(T1 + T2,) “(76b)

With this va lue of C,, Cl is obta ined from Eq. (75b):

cl= T.+ Tt, -T1-T’J+l– Gocz

R,(I – Go) Go “

(77)

Then

RI = ::

1–G,

“’R’r I(78)

“=%-’4+”))A common difficu lty is that the tota l capacity Cl + Cz becomes very

large, especia lly if a very la rge t ime constant in the denomina tor is

desired. The expression for the tota l capacity becomes, with use of the

appr oximation of Eq. (76b),

1

[

TaT, – !i’’I!f’z4Go1, + CZ = R,(1 – Go) Ta+T,–Tl–

T. + T, – Go(T, + 7’,4) ,“I

(79a) I

I



SEC.3.14] A-C ERRORSZGNAL NET WORKS 117

Apparently TI should be chosen grea ter than T,,, a lthough the economy

is not usually impressive. It is seen that the tota l capacity requir ed

decre~s if Ra can be increased. For usual design requ irements on the

t ime constants, th e tota l capacity also decr eases as lower values of GOa re

a ccepted and appr oa ches t he lower limit

(Cl + Cz)]im= ~, T.+ T, – T, –TaTb

)—.T. + T~

(79b)

8ince the components of the network are oft en determined empir i-

cally, an analyt ical examinat ion of the result ing loop character ist ic is

desirable. The values of Go, TI, and T,, a re easily obta ined, but the

va lues of T. and Tb are not so obvious. If Ta is assumed to be the larger

of the two t ime constants, then

Jp+~ 1–;

T. =2’

(80a)

and

@–P&$

T~ =2’

(80b)

where

[ 1=G0 Tl+Tz, +&( T13+T2)~ (80c)

—

[

R,

1(80d)~ GO Tiff’24 + ~3 ‘13T2 “

Frequently 4a c @2,and t hen , approximat ely,

T,T2, + ~ T13T2

Tb=~=

2’, + 7’,4 + ~ (T13 + T2)’

(80e)

T@-;[

= G, T, + T ,, + ~ (T13 + Tz)1– l’b. (80f)

3.14. Networks for Opera t ing on A-c E rr or Signal.—It can be demon-st ra t ed that if a voltage V = M(t) cos ud, in which m is the fixed car r ier

fr equ en cy, is impr essed on a n etwor k t he t ra nsfer ch ar act er ist ic of wh ich is

Y(j(J ) = G[l + jTd(u - 00)1,

the output is

[

dM(t)VO=G M(#)+Td~

1Cos CL@.

(81)

(82)

Thus such a network has precisely the character ist ics associa ted with a



118 A ’ERVO ELEMENTS [SEC, 3.14

t ru e pr opor tion al-der iva tive device a nd as such would be usefu l in equa l-

izat ion of ser vo loops employing a -c er ror data .

Many networks have been ut ilized to approximate the t ransfer

character ist ic expressed in Eq. (81). The network that will be discussedfir st is the parallel “T” net ,l as given in Fig. 3.22a.

*. 222°%0a) (b) (c)

FIG.3.22,—Pa ra llel-Tan d bridge-Tnetwork s.

It is to be noted tha t a load resistor and a sou rce impedance have not

been included in Fig. 3.22. If Y(p) is the t ransfer cha racter ist ic of one

of the networks of Fig. 322, Zr the input impedance with the outpu t open-

circu ited, Z{ the impedance looking back in to the outpu t with the input

shor t -cir cu it ed, t h en

‘=(’+a(’+%’y(””(p)’8where Z is the source impedance to the voltage Eo, ZL is the load imped-

ance, and 222 = 21 + 2,[ Y(p)]z. The devia t ions of Y’(p) from Y(p)

have been invest iga ted, both by mathemat ical analysis and by actua l use

of the net s in servos, and have been found, in genera l, to be small when the

values of Z and ZL employed are those associa ted, for example, with a

synchro or a low-impedance phase-shift ing net as a source and a vacuum

tube as a load. In the following dkcussion the source impedance and load

impedance are neglected, with the approximat ion that they are zero and

infini te, respectively.

The genera l expression for the t ransfer character ist ic of the para llel

T of Fig. 3.22a is

E. [TIZ’2T,P’ + T](S2 + T3)P’ + (T, + Sl)p + 1]

E = \ T,TzT@ + T,(S* + Z’,)p’ + 7“2(2’, + s, + !r,)p’+ (T, + S, + T, + s, + T3)P + 1] (84)

where T1 = R, C,, Tz = RZC9, T3 = R&a, A’1 = RI C’, and S2 = RZCa.

This expression may be made to assume the form

1A. Sobczyk, “ Paral le l ‘T‘ Stabilizing h’et works for A-c Servos, ” RI, Repot i

No. 811, Mar. 7, 1946,



Snc.3.14] A-C ERROR-S IGN AL N ETWORKS 119

Eo=(:’+’)(i’’+i:fi’+l)l+’Td”*([email protected]

“ (:’+)($2+$T=E1+’’W(”-85)if t he following r ela tion s a r e sa tisfied:

~, = ~(d7du’+2Z L+LLloz’d)

T~

()

)

2wl 1 – ;d

u’Tz = ~J

I

T, = &,

\

(86)

&=;T:+g – T,,

2S’=; –—– -&u – T,.

u~Td

If the modula t ion of the input volt age does not crea te sidebands lying

fa r from m, Eq. (85) approximates the ideal character ist ics of Eq. (81),

except for the presence of the t ime lag 1.

The physical requirement tha t SI and S, be posit ive imposes on u the

simultaneous restr ict ions

12—–;odl–l’u ; <u<&o–

~ – T@o*O + ;0 dl – l%: (87)

2& – MT~ (1 + v’1 – t ’u~) < ~ < %0 – t iol’d (1 – /1 – t ’t i$)

~ _ ~ _ ~Tdu2

Tdo 4 – ;d – i?’du:

(88)

It is to be noted that , for realizability, 1 s l/wo.

A design procedure is apparent . It is assumed that theoret ica l or

exper imenta l considera t ions have determined acceptable values of Td

and 1. The car r ier fr equency wo is, of course, known. The res t rict ionson u are established by Eqs. (87) and (88). The t ime constants Tl,

T’, 2’3, S1, and St a re computed, with a va lue of u not t oo near the end

poin ts for a reason discussed la ter . Then if one component , C~, for

example, is specified, t he ot her compon ent s a re det ermined, becau se

The choice of u and of the one component should be influenced byimpedance considera t ions. The input impedance, with the ou tput open-




circuited, ZI, and the output impedance with

Z., can be expressed as functions of u and Ca:

[S EC. 3.14

the inpu t shor t-circu ited ,

()I (78,E =Wo

(:o’+’)(+’+a’+’)%+++)+:15’3

, (Wa)

+(&-&) (;+*) P,+(*+:+l)P

(). Cs,~ =

‘,($+:)”+[(+d+9(’+s’w0’T’l’+(’+s’wO). (,,,,

@4:’+1)(+?2+a’+1)It is evident that as u approaches an end point of one of the critical

regions, so that SI or S2 approaches zero, ZI approaches zero. Also, to

avoid low input impedance, the gain G = l/ 1’d should not be chosen too

near the upper bound of l/( Two).

Shce odd sizes of condensers are d ifficu lt to obtain , it is usefu l to

present a design procedure that may be applied when values of Cl, C~,

and Ca are known and values for & R2, and &are desired for a given Td.The relations are

1Rs=—

U4mca’

(91a)

(91b)

(91C)

in which u must be a solu tion of

2 c,(c,+cJu*-g*u+ 2 Cal + Ca“,c,

— — = o. (9M)uOT.i C*C, c1 u#d C:

For Cl= C;= Ct,the positive solu tions for u are u = l/@ and

u = 2/( TJwo). The gain G = l/Td can be obtained from the expression



122 SERVO ELEMENTS [SEC. 3.14

If th is resonant T is then connected as shown in Fig. 3.23 to a poten-

t iometer of impedance low compared with the input impedance of the T,

the ou tput can be wr it t en as

E . R ,-0 (. – .0)+ j T ,i Z u

R , + R ,=—(93)

“*” (. – .,) ‘+j l Z w

in which T.i = [1 + (R l/Rz)zU and can tkus be adjusted with in the

limit s 1< T.i < [1 + (R,/R,]L

E*O

FIG . 3.23.—Circu it for variab le 2’,+

As in ter est in g simplifica t ion of t he para llel T follows if, in F ig. 3.22a,

either RZ = O or Cl = cc. Then circu it s of the form shown in (b) and(c) of F ig. 3.22 are obta ined. Such networks are commonly ca lled

br idge-T networks. In an effor t t o discuss the circuit s simultaneously,

the par t s have been renumbered.

Th e t ran sfer ch ar acter ist ic is

E. T,T,P’ + (T, + S,)p + 1—.E,

(94)T,TsP’ + (T, + & + TS)P + 1’

where T1 = R,C1l Tg = R&a, and S1 = RK~ in Circu it b and S1 = RL’~in Circu it c. To obta in the approximate der iva t ive character ist ic of th e

form

E. 1“~” (U – .0)+ jT.i Zu

—.—

E ,-0 (. – t oo) ‘d l+jl Zu

t he t im e con st an ts mu st be defin ed by t he followin g r ela tion s:

()3=$0 ;–;d,

T, =

01

17

2;– J -Td

(95)

(96)

“2”1S1 = TN:

—

( ))

11”

2 i–~d



sEc.3.14] A-C ERROR-S IGNAL N ETWORKS 123

For posit ive S,, the gain G = l/T~ must sa tisfy t he in equ alit y,

(97)

Since for the parallel T the upper bound of gain is l/( T,w,), it is observed

tha t for a given Td, less gain can be obta ined from a br idge T. Because

of noise in the signal, the smaller t ime lag 1 for a given Td in a br idge T

is a disadvantage when compared with a para llel T.

TARLE3.3.—CONSTANTSOR SYMMETRICALRIDGET. CIRCUIT b

c, = c, = 1.000 /.If

NotchT ,m interval,

Cps

— — —

2.5 &24.O

5.0 *12. O7.5 * 9.0

10.0 k 6.015,0 i 4.020.0 & 3.030.0 * 2.040.0 * 1.550,0 * 1.2

60.0 * 1.0100.0 + 0,6

R, X 103,megohms

1.061000.530500.35360

0.265300.176800.132600.088410.066310.053050.044200.02653

Input outputR, X 10J ,

Gainimpedance impedance

megohms z, x 103, z, x 10’,megohms megohms

6.63 0.2424200 1.70790-2, 135j 1.60750-2, 009j13.26 0.0740700 0, 9878G2 470j O.9824C+2,456j19,89 0.0343300 0.6838132.564j O.6S30&2 ,561j

26,53 0.0196100 0.52030-2. 602j O.5201&2. 601j39.79 0.0088110 0.350502. 629j 0.35060-2. 629j53.05 0.0049750 0.26390-2. 639j O.2639&2 .639j79.58 0.0022170 0. 1764W2.647j 0.17640-2. 647j106.10 0,0012480 0.13250-2. 649j 0,13250-2, 649j132,60 0.0007994 0.10600-2. 650j 0.10600-2. 650j159.20 0.0005552 0, 0883&2, 651j o 0883&2. 651j265,30 0.0002000 0. 05304–2.652j O.0530~2. 652j

For Circuit b, in terms of arbit ra ry condenser s,

R, =2

T,u;(c, + c,)’

01

T,ll

‘3=7 Z+ T,’

‘=T+4)+*)

(’38)

Similar formulas for Circuit c, in terms of a rbit ra ry resistors, a re

obta ined for Cl, Cs, and G if in 13q. (98) RI is in terchanged with Cl and

RS is in terch an ged with C’s.

For Circuit b

Z,= R,+T,p + 1

C,T ,p’ + (Cl + C3)P’

R3[(S1 + TI)P + 11

‘0 = !l’,!l’,p z + (T, + A’, + T3)P + 1’ I

(99)




and for Circu it c

1 R,(!f,p + 1)zr’~p+

(T , + Sl)p + 1’

R ,T ,p + R, + Rz

‘0 = !l’,!l’,p’ + (S , + T , + T3)P + 1’ I

[SEC.31.5

(loo)

in which 21 and ZOh ave t he meaning established above for th e para llel T.

Table 3“3 gives some circu it constants for the br idge T for Circu it b;

for Cir cu it c, RI is merely in t er changed ~vit h C’, and R~ wit h Ci.

Tolerance requ irements have been full y invest iga ted’ for the T net -

wor ks given in th is sect ion . The expressions, however , a re quite lengthy

and will not be presen ted. In genera l, it may be said tha t to hold theM of the T to with in 5 per cen t , t he notch width to with in 2 per cen t of

CW,and the phase shift t o zero + 10°, for a parallel T with a !i’’dt i, of 15

each componen t must be held to about 0.5 per cen t .

3.16. Opera t ion on 00—Feedback Filter s.—In the design of servos

tha t a re to employ an interna l feedback loop, it is necessary to obta in a

signal, usually a voltage, from a poin t in the loop as near the outpu t as

possible. This is commonly obtained from a tachometer dr iven direct ly

from the motor shaft . Thus there result s a volt age tha t is propor t iona lto the speed of th is shaft and that may be modified by one of the nets,

discu ssed in t his sect ion , befor e bein g su bt ra ct ed fr om t he er r or .

If it is impr act ical t o employ a ta chometer , a simple br idge cir cuit with

d-c motors may be used to obtain a voltage propor t iona l to speed. The

motor cur r en t may be expressed to a reasonable approximat ion as

ed — 1<<Q,n

““ = R., •1- R. + R.’(101)

in which the terms have the same defin it ion as those used in Sec. 3.10.

With th is rela t ionship and the torque equa t ion

it can be shown that the outpu t volt figc Eo from the br idge circuit as

shown in Fig. 3.24, is

(102)

if it is assumed that f~ >> jO.

I Sobczyk, op. cit.



SEC.3.15] OPERA TION ON 90-FEEDBACK FILTERA’ 125

Often R. includes the resistance o the compensa t ing coil in the motor

r equ ir ed for p roper commuta tion . The posit ive limit of l’~, (at R, = O)

may be increased, if desired, by shunt ing this coil with a resist ive divider

and obta ining EO between a point on the divider and Terminal 2 of the

motor . When the br idge is balanced, T~,, = O, and a voltage direct ly

pr opor tion al t o speed is obt ain ed. hTega tive va lu es of T~, a re obt ain ed

when Rz/Rl > R~RC. The possi-

ble infinite negat ive value of T~, ~R.

su ggest ed by Eq. (102) is a fa llacy

result ing from the fact that the e~ R.

viscous damping of the gea r t ra in

was considered negligible com- 0pared with internal damping of

the motor . Actua lly, the nega-0 EO 0

t ive values of T-O are of lit t leFIG.3.24.—Bridgecircu it for obta in ing feed-

back voltage.

in terest ; usually the br idge is

eit her balanced t o obtain t he speed volta ge or design ed t o obta in a posit ive

!l’~O. The presen ce of t he posit ive T~Oin trodu ces a der iva t ive t erm in t he

feedback loop and thus can be used to increase phase margin, permit t ingh igher feedba ck ga in .

In a simple servo loop, if a voltage propor t iona l dO~/di is subtracted

from the er ror voltage, the effect is ident ical with tha t which would be

obta ined by in creasing t he viscous damping, say, in th e gear t ra in . This,

of cou rse, has stabilizing act ion but also results in gr ea t er er ror in follow-

in g con st an t-velocit y in pu ts. Thus, in servos using such an interna l

feedback loop at d-c level for equalizat ion, it is desirable to use a high-

paas network between the source of d-c voltage and the mixing stage inwhich the modified d-c voltage is subtracted from the er ror signal.

Single-, double-, and t r iple-sect ion RC-filter s are commonly used. Net -

works employing inductors have also been used, but the usefulness of

such networks is limited, owing to the nonlinear proper t ies of the la rge

in du ct an ces r equ ir ed (a bou t 1000 t o 10,000 h en rys).

The single-sect ion high-pass RC-network shown in Fig. 325a is

ch aracter ized by a single t ime const an t T = RC, and the t ransfer func-

t ion may be wr it t en as

Eo = TP

z l’p + 1’

T = RC, (103)

which, asymptot ica lly, is represen ted by a r ising 6-db/octave sect ion t o

o = l/T and a constant O-db gain for high frequencies. Asymptotic

plot s and the depar ture from the actual curve are discussed in Sec. 4.10.

The phase angle is +90° for , low frequencies, decreasing toward zero a thigh frequencies. At u = 1/7’ the phase shift is ~ 45°.




The two-sect ion high-pass RC-network is shown in Fig. 3.25b. Its

t ra nsfer ch ar act er ist ic is

E. T1Tzp2

—.E

( )

(lo4a)Z’,T,p’+ 1+~+~ T,p+l’

where

T, G R, C,, (104b)

Tz = R,C,, (I04C)

Equat ion (104a) can also be wr it t en in two other ways:

Eo = l’.!f’@2

% (T.p + l)(T,P + 1)’

E. (TP)2—.E (Tp)’ + 2{Tp + 1’

(lo4d)

(104e)

where T. = longer effect ive t ime con st an t,

Tb = Smaller effeCt iVe t ime COnSt an t,

f = damping rat io (~ > 1),

T = ~TaTb = flT* = mean t im e const an t.

E_o . T. TbP2

E (T@+l)(Tbp+l)

.~r .,F--..:Z...,. ,

J -T Ta ~

logWlog62

a bF IG. 3.25.—High -pa ss filt er s for d-c feedba ck . (a ) Sin gle-sect ion h igh -pa ss filt er ,

Eo/ E = To/ (TP + 1); (b) two-section high-pass l?C-network,

Eo/ E = (TaT,pz)[(Z ’.p + 1)(’hp + l)].

When the asymptot ic t ransfer funct ion is wr it ten in the form of Eq.

(104G?), it becomes apparent tha t it r ises at 12 db/octave to u = 1/7’=,

then at 6 db/octave to w = 1,/Ti. and is then constant a t Odb for higher



SEC.3.15] OPERATION ON $0—F’EEDBACK FILTERS 127

frequencies (see Sec. 4.10). The12-db/octave por t ion ifextended to the

O-db a xis wou ld cr oss it a t

(105)

It is a lso apparen t from Eq. (104d) tha t t here a re only two pr imary

pa rameters, !l’~and !!’~; th e form inwhich th ese pr ima ry parameters will

beusedhere isl’~/!l’, and @i’~1’, = T’.

55

50/

45 /

40 c,/

~=0.25

352

/

0 .50 ~

30%

c

<25 /

20

15,

10

5 \+

o-12 -9 -6 –3 +3 -I-6 +9

L;n &+12

T2

FIG.3.26.—Graph for designing two-section high-pass RC-fi lter .

Th e followin g r ela tion s h old between t he va ria bles:

(106a)=,’+(:)(1+:)2 T

)

F,

T.

()

T C,_=7=2p-1+2~ <{*-l =.f~2J ~’T,

(106b)

The rela t ionship indica ted in Eq. (106b) is plot t ed for severa l va lues of

C1/C~ in F ig. 3.26.



128 SERVO ELE i14EN lqS [SEC. 315

In the usual circuit s, t he ou tpu t resistor Ra is requ ired to be less than

some cr it ica l va lue due t o gr id-r etu r n requ irements or ot her impedance-

level considera t ions. With this fact inmind, apossible design pr ocedu re

becomes apparent . Exper imenta l or theoret ica l work has determined

T., Tb, and R,. Thus Y, T, and R, are fixed. By the use of

as given by Fig. 3“26, T/T2 can be determined for a par t icu lar choice of

C,/C,. The value of 1’, and thus the value of C, can be then established.

By the par t icular va lue of C,/C, employed in determin ing T/ T2, c, k

established. Since l“, = T’/T,, T, is known , R, is esta blish ed, a nd t he

compon en ts of t he filter a re complet ely det ermin ed.

Such a pr ocedu re, h owever , ignor es the fact tha t ther e a re fou r circu it

elements, but it is desired to establish on ly th ree pa rameters. It seems

desirable to use the other parameter t o minimize the tota l capacitance

CO ~ C, + C, used in the circu it .

For tunately, y increases monoton ica lly with -t . Thus establishing

y, T, and Rt is equiva len t to establishing ~, T, and R2. Equat ion (106a)

may be rewr it t en as

(106c)

Solvin g for C, a nd minim izing C, for fixed ~, T, and Rt pr odu ce a n ew r ela -

t ion sh ip bet ween ~ a nd t he cir cuit compon en ts:

Compar ison of th is with Eq. (106a) gives

(), 2

(-), c,

(-)

T*C= C,’_l

c,

(106d)

(106e)

a s t he n ecessa ry con dit ion for t he m in im iza tion , wh ich makes it possible t owrite

[~c=; ;+’;+ J ($y - 1]; (106j9

use of th is in Eq. (106b) yields 7C = f(T/Tz), a plot of wkch has been

in clu ded in F ig. 3“26.



SEC.315] OPERATION ON $0—FEEDBACK FILTERS 129

Thus, for economy in tota l capacitance, the design procedure sug-

gested above is modified in tha t the choice of the value of Cl/C, should be

det erm ined by t he in ter sect ion of t he desir ed value of ~ wit h t he ye-cu rve.

Genera l equat ions rela t ing the tolerance on the componen ts of the

fiker with the var ia t ion of T., 2’,, 2’, and -Ymay be der ived in the usual

manner .

%++’”)%+(:-’”)%+(~+BD):?+(~-BD)07”)

TABLE3.4.—TOLERANCEOEFFICIENTSORMINIMUMCAPACITANCEASES

c, T ,

C, E.4L) ED ;+AD ;–AD ~+BD +–i3L)

2 1,333 0.522 –0.174 1.022 –0.022 0.326 0.674

4 1.067 0.307 –0,184 0.807 –0.193 0.316 0.6846 1.029 0.235 –0.168 0.735 –0.265 0,332 0.668

8 1.016 0.196 –0.153 0.696 –0.304 0.347 0.653

Table 3.4 gives the tolerance coefficien t s for two of the minimum

capacitance cases. It will be observed that the main cont r ibu t ion to

t he er ror s com es from t he squa re-root rela t ion bet ween T and TIT, and

that the change in the ra t io of the effect ive t ime constan ts is of a smaller




order . This means tha t a given percen tage er r or in one of the corn -

ponent s will produce about one-ha lf of tha t percentage er r or in the two

effect ive t ime const ant s.

3.16. Gea Trahs.-For useincon trol, agea r t ra in must meet cer t a inrequ irements of efficiency, reversibility, r igidity, backlash, st r ength ,

and iner t ia . Unfor tuna tely, theoret ica l work has not progressed to the

stage where all of these standards can be expressed mathemat ica lly.

In r ega rd t o efficien cy, usu al en gin eer in g r ea son in g a pplies, sin ce t his

will direct ly a ffect the size of the motor tha t must be employed. Fur ther -

more, it is t o be n ot ed that h igh sta rt ing fr ict ion produ ces r ough following

of the input a t low speeds. High viscous fr ict ion decreases the effect ive

velocity-er ror constan t . Also, since it oannot be t r ea ted as a constan tfrom gear t ra in t o gea r t ra in or as a funct ion of t ime, it s stabilizing act ion

can be a source of ser ious design er ror .

At presen t t her e is n o sa t isfact or ily complet e analysis of t he problem s

encoun tered when servo cont rol is employed with an ir r ever sible gea r

t ra in , such as cer ta in worm reduct ions. Bit t er exper ience, indica tes

however , tha t except possibly where the load iner t ia and load torques a re

small, ir reversible gea r t ra ins should be avoided. It is felt tha t the dif-

ficu lty is caused by the locking that t akes place in many ir reversible gea rt ra ins when su bject t o loa d t orques.

The r igidity should be such that with the expect ed load iner t ia , there

will be no natura l frequencies tha t a re less than, as a rough est imate, five

to ten t imes the highest resonant fr equency of the rest of the cont rol

loop. Natura lly the r igidity of any por t ion of the gear t ra in not enclosed

in the loop must be such that er ror s ar ising from deflect ions under load

are tolerable .

Ba ckla sh in gea r t ra in s, if en closed in t he loop, oft en pr odu ces in st abil-it ies of either very high or very low frequency. Usually the amplitude

is of the order of magnitude of the backlash . A rule based onlY on

exper ience is tha t the backlash must be less than one-half of the accept -

able following er ror . If the backlash is not enclosed in the loop, the

resu lt ing er ror can, of course, be equal to the backlash.

Obviously, the gea r t ra in should be sturdy enough to withstand the

maximum torque load that it will exper ience. An est ima te of the maxi-

mum load can be obta ined. The torque exer ted on the motor rotorMm and the torque exer t ed on the load Ml a re r ela ted by

Ml =J,N

J, + J&’ ‘“’(108)

where Jr is the iner t ia of the load and Jm is the iner t ia of the motor

a rmature and fir st gea r and the iner t ia of the rest of the gea r t ra in , of

r edu ct ion N, is ign or ed.



SEC, 316] GEAR TRAINS 131

L’sually it is on ly in cont rol problems employing fa ir-sized motors

that concern exist s about t ihe st rength of the gear t ra in . Thus the motor

is usually direct curren t , and the rela t ion M = KTin holds over a wide

range of i~. Under severe condit ions, however , sa tura t ion effects mayoccur.

Th e grea t est t or qu e load will be exper ien ced in on e oft wo cases:

CASE I: If the load is blocked and a maximum voltage E.,, is applied

to the motor , then from Eq. (108)

Ml = N(Mm)i”,, (109)

since in such a case .l~ = cc . The quant ity (M~),m is the torque

associa ted with a cur ren t i~ = E~/Rn and may be less than Kti~

du e t o sa tu ra tion .

CASE II: If the maximum voltage Em is suddenly reversed after the

motor has a t ta ined full speed, then

M, =J,N

JI + JmN’(Mm),,,” (110)

if the motor is assumed to run with a back emf equal to the applied

emf for constant speeds. This is approximately cor r ect with high-

efficiency gea r t ra ins and no torque load on the output of the gear

t ra in . The quant ity (Mm) Z,mis the torque produced by a cur ren t

im = 2Em/Rm,

.k compar ison of Eqs. (109) and (1 10) shows that if

(Mm),. < _ 1

(Mm),,. J; “

l+ZN’

Case 11 gives the grea ter torque but , if

(111)

(112)

Case I gives the grea t er torque.

In applicat ions involving ser~omotors of a few mechanica l wat ts ou t-

put , the iner t ia of the gea r t ra in , especia lly the first mesh or two, is fa r

from negligible compared \ vith the iner t ia of the motor armature. If

the iner t ia l of the gear ing past the four th gea r (Fig. 3.27) is considered

negligible, the following equat ion may be wr it t en for the total iner t ia .TL

observed at the motor shaft , N’ being the tota l gear reduct ion to the

iner t ia load Jl:

(l13a)

1The fcdlowingtrea tment is from an unpublishedpaper by N. B, Nichols ,



132 SERVO ELEMENTS {S EC. S -16

If it is assumed that the gear s (of diameter D, width W) are all solid and

of a mater ia l with a density p,

In Eq. (llOb), the small iner t ia of the idler shafts has been ignored

or , if desir ed, absorbed by change of pin ion width W. With the use of

N12 s D2/D1 and N = (D2/DL) (Dd/DJ , Eq. (113b) may be writ t en as

where

as the

JO k the moment of iner t ia of a motor pin ion of the

second gea r .

same widt h

h Ill

eF% -J1, J

J .

FIG.3.27.—Spurgea r t r a in .

The in terest now lies in the value of NIZ that will min imize J, for

fixed values of W,/W2, W@Z, Wd/?V*, N, and D,/D,. Such a value

will be a solu t ion of the equat ion

() ()6 _~3 &’Nz _2N2~4 g34 ~

12 W2 D, 12 W, Dl=’(l14a)

which can be wr it t en asZ3+2=3KZ (l14b)



GEAR TRAINS 133EC. 3.16]

if

‘=~2N-%(w’(2r’‘K=(mw%)”N-”

In genera l, Kwill be small. Toazero-order approximat ion

~. = —-2)47 (115)

a nd t o a fir st -or der a ppr oximat ion

Z1 = —2~~(1 — ~Kzo))$ = —2%(1 + >% K). (116)

Usually the zero-order approximat ion is sufficient . For instance, if

W3/Wz z Wh/W2 = D3/Dl = 1, N = 8, the fir st -order cor rect ion con-

t r ibu tes only about 1 per cent to the value of xO. When th is cor rect ion is

neglected, IVlz = 2.25 and JO = 8.6.J o. It has often been the pract ice

to make the fir st reduct ion as large as po~sible in a mistaken effor t to

minimize iner t ia . It is wor th not ing in the above case tha t if the reduc-

t ion of 8 had been taken in a single mesh , the iner t ia of the gear ing at the

motor shaft .l~ would have been 65~0.

The following table compares the iner t ia of the gea r t ra in for the

minimal case with tha t iner t ia resu lt ing from an equivalent single reduc-

t ion . Again , the assumpt ion has been made that

K3=E4=Q3 =1,W2 W, D,

TABLE35.-INERTIA OF VARIOUSGEARTRAINS

NM

8.00

2,25

12.00

2.5718.00

2.94

27.003.36

Iv,,

3:56

4.67

,.. .

6,12

8:03

N

8

8

12

12

18

18

2727

65.0

8.6

145.0

11.1

325.0

14,0

730.018.1

(a), in.z 02

0.00740.00100.0165

0.00130.03690,0016

0.08280.0020

(b), in.z 02

0.3720.0490.832

0.0641,870

0.080

4.1800.103

Table 3“5 gives in Column (a) the result ing iner t ia if J O is from a

9T, 48P, &in. -wide brass pinion and Column (b) is from a 24T, 48P,

&in. -wide brass pin ion. These iner t ias maybe compared with a motor -

armature iner t ia of 0.077 in. 2 oz for a Diehl two-phase 2-wa t t motor

(FPE25) and 0.66 in .’ oz for Diehl 11- and 22-wat t motor s (FPE49-2

and FPF49-7).



CHAPTER 4

GENEIUIL DESIGN PRINCIPLES FOR SERVOMECHANISMS

BY N. B. NTICHOLSj W. P. MANGER, AND E. H. KROHN]

The systemat ic t rea tment of mult iple-loop servomechanisms is quite

complex and will not be at tempted in this chapter . Systems with only

w

one independen t in pu t va ria ble will fir st be t r ea t ed:

the discussion can be readily extended to systems

with more than one independent input var iable by

FIG. 4 1.—Single-differ- the usual superposit ion theorem which applies toentialservosystem.

all linear systems. ,

4.1, Ba sic Equ at ion s.—The simplest sin gle-in pu t ser vomech an ism h as

only on e er ror -m ea su rin g elem en t (t he different ia l in t he usua l symbolic

diagram) and on e t ransfer element . A simple example of this type has

been discussed in Chap. 1; it s symbolic diagram is shown in Fig. 4.1.

It s equat ions may be writ ten~=el —go, (1)

Y,,eo –K 1 + Y1l’

c 1

x= 1 + Y1l’

(2)

(3)

(4)

where Yll, 190,01, and c are funct ions of p or @, depending on the type of

solu t ion desired. The funct ion YI I is ca lled the loop transfer funct ion or

the feedback funct ion and is usually composed of a number of product s

tha t a re the t ransfer funct ions of the individual elements of the servo

loop. The hTyquist stability test discussed in Chap. 2 can now be

appfied to the loop t ransfer funct ion Y1l, to determine \ vhether or not

the expressions in Eqs. (3) or (4) cor respond to a stable systcm. The

in terpret at ion of t he test is simplified if, as is usually the case, YI I(p) has

no zeros or poles in the r ight half plane when p is rega rded as a complex

variable.

For the simple servo considered in Chap. 1 \ ve have

(5)

~N. B. Nich ols is t he a ut hor of Sees. 41 a nd 4.14 t o 4.19 in clu sive; W’, P . Ma nger

of Sees. 42 to 4.9, and 4.12 to 4.13; E. H. lirohn of Sees. 4.10 and 4.11.

134



SEC.41] BAS IC EQUAT ION S

wh er e K, = velocit y-er ror constant ,

Z’~ = motor t ime constant .

Adding an equalizing lead net wor k, we obta in

K.(l”lp + 1)Y1l(P) = ~(~mp + 1)(T2P + 1)’

135

(6)

wher e 7’1 = lea d-n etwor k t ime con st an t, som et imes ca lled t he der iva tive

t ime const ant ,

Z’Z = lea d-n etwor k lag (less than T’,).

Proceeding in th is manner , we can build up more complex funct ions for a

s ingle-loop or s ingle-d ifferent ia l sys tcm,

In a two-different ia l single-input system ther e ar e two possible con-figura t ions, as illust ra ted in Fig. 4.2. These systems include another

‘i!ailigm

e~

(a) (b)

FIG. 4.2a and b.—Two-differential servo systems.

t ype of junct ion , which may be called a branch poin t. A branch poin t

has one incoming funct ion and two outgoing funct ions which are ident i-

ca lly equ al t o t he in com in g on c, Th e differ en tia l ju nct ion h as two in com -

ing funct ions and one outgoing funct ion which is equal to the algebra ic

dffer en ce of t he two in comin g on es. Th e symbolic dia gr am or t he a ssoci-a t ed equat ions must , of cour se, indica te which of t he incoming funct ions

reta ins its sign on t raversing the junct ion and which changes its sign.

The equat ions for Fig. 4.2a may be wr it ten

c+eo=e+Y31e3=er, (7)

~2+ 1902— 012 = C2+ Y3283— Y12E= O, (8)

03 – Y2362= o. (9)

Using Eqs. (8) and (9) and eliminat ing c,, we find

(lo)

where the expression on the r ight may be looked upon as the effect ive

t ransfer funct ion between the output of Ylt and the input to Yt l. The

expression on the r ight may be rewr it ten to obta in

03 1 Y22

Y12E= Y3, 1 + Y,;

(11)



136 GENERAL DESIGN PRINCIPLES [SEC. 4.1

wh er e Yzz is the loop t ransfer funct ion for th e subsidiary loop associa ted

with different ia12 in Fig. 4.2a. Using Eq. (7), one obtains

y12ya1 Y220

‘= YS2 1+Y22(12)

or

:0 = Y12Y2aYs1 = Yh

e 1 + Y23YS2 1 + Y22’(13)

where Y~l is the loop t ransfer funct ion associa ted with different ia l 1

when Y,z = O. We also have

00 y~l

Ii = 1 + Y;l + Y22’

‘f 1 + Y22

ii= 1 + Y~l + Y22”

(14)

(15)

In spect ion of Eq.(13)-shows. tha t if Y?J (l + Yzz) has no zeros or

poles in the r ight ha lf plane, then th; nor rna l..Nyquist stability test may

be used to determine the stabihty of the over -a ll system character ized by

Eqs. (14) and (15). Sin ce Y~l by it self u su ally sa tisfies t his r equ ir emen t,

it is neceaaary to inquire if 1/(1 + YzJ separa tely does. It follows tha tapplica t ion of the simple Nyquist t est t o the over -a lJ system requires

tha t the subsidiary loop represen ted by Eq. (11) be stable. The two-loop

t ransfer funct ions en t er Eqs. (13) to (15) in a ra ther simple manner tha t

permit s an easy der iva t ion of the system equat ions. Combhing Eqs.

(12) and (7), one obtains

YlzYal Y22

00 = Ya2 1 + Y22 Y1l

ii ~ + Y12Yal Y22 = 1 + Y1l-(16a)

Ya2 1 + Y22

Equat ion (16a) emphasizes the reason for calling this a twdoop

system: YZZappea rs in Y1l in the same manner tha t

Y,2Y3, Y22Y1l = Y32 ——

1 + Y22(16b)

appears in t he complete expression for oJ o1.

In Fig. 42b there are two single-element pa ths for da ta to leave junc-

t ion 3 and enter junct ion 2. The equat ions associa ted with Fig. 4%

are easily shown to be

00— = Yla(y . + y b)y 21 , (17)e

g= Yla(ya + Yb) Y21

8, 1. + yla(y~ + Yb) y21’(18)

1I

I

I



SEC. 41] BAS IC EQUAT ION S 137

where differen t ia l 2 has been assumed to have the equat ion

22 = Ya03 + Y#&

One observes tha t Y. and Yb en ter Eqs. (17) and (18) in a differen t man-

ner from that in which YW and Y23 en ter the previous equat ions. The

t ransfer funct ion between junc-

t ions 2 and 3 is rea lly only the

sum Y= + yb; there is no loop

equ at ion a ssocia ted wit h differ en -

t ial 2. In other words, Fig. 4.2h

is in r ea lit y on ly a sin gle-loop sys-

tem with a loop transfer funct ion

y13(Ya + Yb) Y21 = Ym

There are many three-differ -

ent ial syst em s, an d n o exh au st ive

t rea tment is contempla ted here.

The schemat ic shown in Fig. 4“3

has been used in amplidyne ser -

vos employing quadra ture ser ies

I 1FIG, 43,-A thr ee-cliffrentialservosystem.

and armature voltage or tachometer

feedback for equaliza t ion, together with an er ror -signai equalizer . The

equat ions for Fig. 4“3 may be writ ten as

~+eo=e+yblob=el, (19)

Cz— Y126 + Y3223 = o, (20)

23 – Y4384 – Ys3es = o, (21)

e4 – Yz4~2 = O, (22)

e, – Y,se4 = o, (23)

where Ylz = er ror -equa lizer t r ansfer funct ion ,

YZA= pOWer-amplifier and amplidyne t r anSfer fUnCt iOn (Open

circuit),

Y4S = quadr at ur e-ser ies field t r an sfer fu nct ion ,

Y4S = mot or -amplidyn e t ra nsfer fu nct ion ,

YSS= t ach omet er t ra nsfer fu nct ion ,

YSZ= feedback -equa lizer t r ans fer funct ion ,

YE ., = gea r-t ra in t r an sfer fu nct ion .

Solving t he five simultaneous equat ions for a number of t he var iables, we

obtain

~= Y12Y24Y45Y61

0, 1 + Y12Y24Y46Y61 + Y24Y43Y32 + Y24Y46Y68Y82’

(24)

e 1 + Y24Y43Y32+ Yz4Y45Y53Y~*

– = 1 + Y12Y24Y46Y41 + Y24Y43Y32 + Y24Y46Y68YS2’,(25)



138 CE.VERA L l)ti,~IGiY I’R I.VCIPLEL5 LSJW.42

Y,21’241”45Y5,~~= _. ---–- ._ __,e 1 + Y24Y43Y32 + Y2; Y45Y53Y3Z

(26)

Y241’43Y3, + Y24Y451’53Y32e“~= _____ ____________7; 1 + Y24Y43Y32 + Y24Yi5Y53Y32’ (27)

0.,—. YZ4Y1SY3Z + YzlY~~Y~3Y32.~z

(28)

Other rela t ions between the var iables may be obta ined by simple com-

binat ion of the above, as, for example,

Combinat ion of Eq. (29) with Eq. (24) gives

04. ___ Y ,, Y 2,

e; 1 + Y121724Y43Y5, +–Y24y43Y32 + Y24Y45Y53Y32”

(30)

The previous equa t ions a re in proper form for a A’yquist t est of suc-

cessive loops in order to determine the stability of the system. Inspec-

t ion of Eq. (24) shows tha t the equat ion for this circu it is the same as

tha t of a two-loop system [see Eq. (14)] with an inner -loop t ransfer func-

t ion of Y24Y43Y32+ YZ4Y45Y53Y32 an d an ou ter -loop t ra nsfer fu nct ion

given by Eq. (26). This means tha t one can fir st study the loop

character ized by Eq. (28). The difference between the number of zeros

and poles in the r igh t half plane of the funct ion

1 + Yz4}’43Y32 + Y24Y45Y53Y32

can be found by means of the N-yquist t est . In th e usual case t her e will be

n o poles or zer os (t he inner -loop equaliza t ion n et works being designed t o

give no zeros), and the system will be stable when the ou ter loop is opened

by making Y~~zer o (reducin g preamplifier ga in t o zero). Having applied

the IYyquist t est to the inner 100P, one may draw the A’yquist diagram

for the complete system by the use of Eq. (26) and determine the dif-

ference between the number of zeros and poles of 1 + Oo/c. In order

tha t the system be stable, 1 + O./c can have no zeros in the r igh t half

plane. If the inner loop is stable and, as is usual, Y12YZ4YMY,l has no

poles in the r igh t half plane, then 1 + O./c will have no poles and the

simple Nyqu ist cr it er ion will a pply.

4.2. Responses to Represen ta t ive Inputs.-A genera l discussion of

t he na tur e of typica l servomechanism r esponses t o r epresen t at ive inputs

requires a genera l defin it ion of the funct ions tha t cha racter ize servo-

mechanisms. All the mathemat ica l knowledge available from an inspec-

t ion of the st ructu re of a servomechanism can be given by a sta tement of

the loop t ransfer funct ions for each closed loop in the system. The



SEC. 42] RESPONSES TO REPRESENTATIVE INPUTS 139

following discussion will apply toa two-loop system. The analYsis can

readily be extended to include mult iple-loop systems if desired. Th e

t ransfer funct ion of the pr incipal, or er ror , loop, when the other loop is

open , will be den ot ed by Y~l(p); it will be of t he form

p (,)= K,,., IA&1

The loop gain of the inner loop may be writ t en as

Yj,(p) = K,,% *f,

(31)

(32)

where f and g are polynomia ls in p of the genera l form

and the a’s are in tegers.

The t ransfer funct ion rela t ing the ou tput 0. to the input O, can be

written

: ‘p)= 1-++;;! Y$,(p) = 1 ;’}:\ p)” (34)

We now proceed to establish cer ta in limita t ions imposed on the loop

t ransfer funct ions Y(p). Suppose that a unit -step funct ion input is

applied to th e ser vomechanism \ vith t r ansfer fun ct ion given by Eq. (34).

‘Then

Y;,(p) 1t%(p) = ~ —.—.

1 + v, (P) + Y!Z(P)(35)

The asymptot ic behavior of 00(t ) for la rge values of t is given by (see

(’hap. 2)

lim t?.(t ) = ~lo p%(p)t+ cc

or

lim %(t) = lim(

K,p”,

t+ m )+o 1 + K,p% + K@a~

(36)

If the servomechanism is to have zero sta t ic er ror , tk,(t ) must approach

1 for la rge va lues of t . Evident ly czl must be nega t ive for th is to be t rue.

Equat ion (36) may be wr it t en

lim %(t) = lim

(

K,

t+ m )-O P–a’ + K1 + K’p”l–”z(37)

It is seen tha t a , must be less than O and a’ must be grea t er than al in

or der tha t th e servomech an ism h ave zer o sta t ic er ror .



140 GEN ERAL DES IGN PRIN CIPLES [SEC. 4.2

The condit ion on al can easily be der ived from physica l reasoning.

We consider Y1l(p) as the loop gain and imagine the loop to be opened as

shown in Fig. 4.4. If a steady signal c is applied as shown, then the out -

put Ooshould move cont inuously in an at tempt to balance ou t this er rorsignal. This will be the case only if the exponen t m is equal to or less

than – 1. We call any feedback loop having a t ransfer funct ion of the

form of Eq. (31) or (32) a” zero-sta t ic~r ror loop” if al ~ – 1.

The rest r ict ions on m are somewhat more complica ted. When

al = —1, we have the condit ion a2 ~ O. Thus the subsidiary loop can-

F[c. 4.4.—Simple servo with feedbackloop opened.

not be of the zero-sta t ic-er ror type if

the over -a ll loop is to be of the zero-

sta t ic-er ror type. In a system withal ~ —z (we will see la ter tha t such a

servomechanism is character ized by

zer o fina l er ror when following a con -

stan t -velocity input) it is necessa ry

tha t a , z – 1. Such a servomech~

nism may then have a subsidiary loop of the zero-sta t ic-er ror type. It

will be seen subsequent ly, however , tha t in the presence of such a sub-

sidiary loop the over -a ll loop will noi be character ized by zero velocityerror.

A fur ther rest r ict ion is placed on the loop t ransfer funct ions of Eqs.

(31) and (32), in tha t they must approach zero at infinite frequency.

In pract ice this is assured by the presence of par~sit ic elements in the

loops. In terms of Eq. (33), we must have r + a < s. It is somet imes

convenien t to ignore th is rest r ict ion when invest iga t ing only the low-

frequency character ist ics of a system. One must , however , a lways be

ca refu l not to draw unwarran ted conclusions when this condit ion isneglected.

It can be shown that the over -a ll t r ansfer funct ion O.(p) /O,(p) has the

same gener al form as t he t ransfer funct ion of a low-pass filt er ; such servo-

mechanisms can be considered as a specia l class of low-pass filt er . The

frequency response of the two-loop system is given by Eq. (34), with

ju subst itu ted for p. Making use of the least severe rest r ict ion on az,

cm = al + 1, Eq. (34) becomes

: (j.) = K, #

(j.)-. + K, Af + I@=” ’38)

For zero frequency thk reduces (for a , nega t ive) to

(39)



SEC. 4.2] RESPONSES TO REPRESENTATIVE INPUTS 141

Forvery la rge frequencies OO/Olapproaches zero a t leaa t as fast as I/u ,

and we see tha t a servomechan ism does indeed behave like a low-pass

t it er . An ideal servomechan ism, able to reproduce an inpu t signal

exact ly, would of course have a t ransfer funct ion of unity, tha t is, wouldbe an idea l low-pass filt er with a very high cu toff frequency. (This

descr ipt ion of an idea l ser vomech an ism is r eason able on ly in th e absen ce

of noise, since on ly then is it desirable to reproduce the input signal

t

(b)

FIG.4.5.—(a) Cha ract er is tics of idea l low-pa ss Slt er ; (b) cha ract er is tics of r ea l low-pawlilter.

exact ly.) Such a system cannot , however , be physica lly rea lized, and

our problem is tha t of syn thesizing a low-pass st ructu re tha t will r epro-

duce inpu t signals with sufficient fidelity for the purpose being con-

sider ed in a ny given a pplica tion .

Communica t ions engineers have defined an idea l low-pass filt er to be

one having a t ransfer funct ion with magnitude and phase such as ar eshown in Fig. 4.5a. This filt er has a gain of un ity in a fin ite pass band

from a = O to u equal to some frequency m. The gain ou tside th is band

is iden t ica lly zero. The phase of the t ransfer funct ion is zero for u = O

and var ies linear ly with u in the pass band. The phase can be left unde-

fined outside the pass band. It shou ld be poin ted out tha t it is not

physica lly possible to const ruct a filt er with such a rapid cu toff, and we

might thus expect some nonphysica l behavior of the filt er . This par t icu-



142 GENERAL DESIGN PRINCIPLES [SEC,42

lar t ransfer funct ion is chosen , however , since it enables one to obta in

some ra ther simple rela t ionships between the frequency and t ransient

behavior of filter s. If we suppose that a unit st ep funct ion is applied to

thk filt er , we can compute the ou tput by means of the inverse Four iert ransform. The result is’

where Si(z) is the sine in tegra l of x, given by

(40)

(41)

Equa t ion (40) is plot ted in Fig. 46a. Examinat ion of Eq. (40) shows

that the response is small but not zero for negat ive t ime—one of

the nonphysica l character ist ics in t r oduced by the arb~t ra ry choice of the

Bu!ldwime ‘+ ~ ---Ak7b~

1 .0=

-005

_/ _

o 1-~— go .-J

Delaytime (a)

Pericdof

k

BuJdupime

.~;qt

~rb~

1.0=

;005

Oy o+ ‘—

Delaytime (b)

F IG-,46.-(a ) Tr an sien t r espon se of idea l low-pa ss filt er ; (b) t ra nsien t r espon se of r edlow-pass filter.

t ransfer funct ion . Aside from this behavior , one observes that the

response is a damped oscilla tory one which reaches a value of 0.5 at a

time 7d = oJ u ,, tha t may be callecl the delay t ime. ‘I’he oscilla tory

per iod is 27r /wo, and the fir st overshoot is 9 per cent . We can define the

bu ildu p t ime Tb as the t ime tha t would be required for & to increase from

O to 1 at the maximum rate. To eva luate the buildup t ime we calcula te

the slope of the response curve at t = 1$0/uo:

Thus ,bwo = m, or , with UO= 27r~,,

Tbfo = ;. (43)

Th is expr ession , t oget her \ ~it ht he expr ession s for t he dela y t ime a nd oscil-

1E, A. Guillemiu, C’mnmunicat icm etworks, VO1.II, Wiley, New York , 1931,

p , 477.



SEC. 4.2] RESPONSES TO REPRESENTATIVE INPUTS 143

la tory per iod, gives the sa lient rela t ionships between the fr equency-

response behavior and the t ransien t response of an “ideal” low-pass

filter.

The forms of the at tenuat ion and phase character ist ics of a physica l

low-pass filt er a re shown in Fig. 4.5b. This rea l filter differs from the

ideal one in tha t there is some transmission outside the nominal pass

band and there are some frequencies in the pass band that a re t ransmit t ed

with gain grea ter than unity. The phase character ist ic is also not a

1.8

1.6

1.4

1.2

1.0000.8

0.6

0.4

0.2

00 1 2 34 56 78 910

;/.

FIQ.4.7.—Transientesponsefort het ra ns fer fun cticm(T’P’ +2(TP +l)-l.

lin ea r fu nct ion of fr equ en cy. We shall define uo for such a filter as the

value of u at which It90/ozl is —3 db. In the case of filter s having a

peaked or resonance response it is somet imes convenien t to define m as

that va lue of u at which the response becomes unity a fter the peak. The

step-funct ion response of such a filt er is sketched in Fig. 4.6b; it is ofthe same form as the response of the ideal low-pass filter . The response

is, of course, zer o for negat ive t ime.

A low-pass filt er chara cter ist ic tha t occurs frequent ly is given by

(44)

The responses of thk filter t o unit -step funct ions, for differen t va lues of

(, a re plot t ed in Fig. 47, and the fr equency response character ist ics for

similar va lues of ~ are plot ted in Fig. 4“8. The cor rela t ion exhibited

bet ween t he pea k height [maximum value of 100(u)/&(0) I in t he complet e

fr equency range] and the magnitude of the overshoot of the t ransien t

response is r epresen ta t ive of most servomechanism performances. The

magnitude of the fir st over shoot of the step-funct ion response, the fre-

quency-response pea k height , and t he frequen cy at which t he peak occu rs

(or the frequency at which the gain has dropped to unity) are often used



144 GEN ERAL DES IGN PRIN CIPLES [SEC. 42

as figu r es of mer it of per formance. Th e bu ildu p t ime is oft en con sider ed,

a lso, and the genera l rela t ionships between these quant it ies which have

been illust ra ted are very useful in the synthesizing of a system to meet

given specificat ions.

+15

+10

+5

o

:—~

~

-lC

-15

-2C

-25

-3C

o

-10°

-20”

-30”

–40°

–50°

–60°

-70°

-80” Q

_go. :

-1oo”

-110”

-120”

-130°

-140”

-150°

-160°

-170°

1 0,2 0.3 0.4 0.6 0.8 1-180”

23468

COT

l~lG.4.S.—F requ en cy r espon se for t he t ra nsfer fu nct ion (Tzpz + 2~Tp + l)-l. (a ) Ga incurves ; (b) phase-shift curves .

A constan t -velocity input 61 = W is a good approximat ion to many

typica l servomechanism inputs, and the response to such an input is of

interest . Making use of Eq. (34), we can wr it e the asymptot ic form of

e(t ), for la rge t , as

(Qlime(t ) =lim ––sK1pUI

t+ m ~o P )1 + K,puI+ K,P”’ “(45)



SEC.4.3] OUTPUT DIS TURBANCES 145

If al = – 1 and az = O, t his reduces to

1+K2lim f(t ) = fl —KY = #.t+ m v

(46)

The output thus follows the input with a constant posit ion er ror equal to

the input velocity divided by a constant ca lled for obvious reasons the

“velocit y-er r or con st an t. ” This is a character ist ic of all zero-sta t ic-

er ror servomechanisms. If al = – 2 and az 2 0, the servomechanism is

said t o h ave zer o velocit y er ror , sin ce a ca lcu la tion sim ilar t o t he pr ecedin g

one shows tha t for a constant velocity input 81 = Qt,

lim c(t ) = 0.t+ m

If a constant accelera t ion input 01 = ~At* is applied to this servo-

mechanism, and if a l = —2 and at = O, then

1+ K2 =:.lim ,(t)= A ~t+ m a

(47)

Th is equat ion defin es t he” a cceler at ion -er r or con st an t” for t his pa rt icu la r

system. Th e defin it ion and inter pr eta t ion of velocity and accelera t ion-

er ror constants and other er ror constants will be discussed more thor -ou gh ly in Sec. 4.4.

4.3. Output Disturbances.-The performance of a servomechanism

may be in flu en ced by man y ext ra neou s’ cin pu ts” t o t he syst em . Changes

t .

~“(”Fm . 4.9.—TwIAooPservomechanism.

in the gain of vacuum tubes, changes in the values of resistors and con-

densers, r ipple on the pla te-supply voltage of vacuum tubes, changes in

fr ict ion of bear ings, and t ransient torque loads are all examples of such

externa l influences. It can be shown with grea t genera lity that the

presence of feedback in a system result s in a reduct ion of the effect s of

such influences. Ra ther than a t tempt such a general t r eatment , we shall

discuss only the impor tant case of t ransient tor que loading on t he outputof a servomechanism.

Suppose that we have a twdoop servomechanism, as shown in Fig.

4.9, where the ou tput of Element 2 is the servomechanism output t%.

Element 2 is assumed to be a motor character ized by iner t ia J and a

viscous damping coefficient j.. An externa l torque Z’(t ) acts on the out .

put shaft . The input signal to the motor is denoted by .E. The differen-



146 GEN ERAL DES IGN PRIN CIPLES [SEC.43

tia l equa t ion for Oocan then be wr it t en

J p’eo + fmpeo = Kmjm&+ z’(p) (48a)

or

~. = ~2t + ~, y~),Kmfm

where

Y, = ““p(l”mp + 1)”

From the basic equat ions for a two-loop system

T(p)~el– Y:l —

<=Kmfm

l+ Y;, +Y~2”

If O, is set equa l to zero, then

(48b)

we have

(49)

h’ote t ha t t he r espon se of t he syst em w ith ou t je e~b ack t o T(t) is simply

co(p) .

K& l ;

(50)

(51)

t he presen ce of feedba ck modifies t he response by t he funct ion (or opera -

tor ) in the parentheses in Eq. (50). For a single-loop servo Eq. (5o)

r edu ces t o

(),T(p) 1Oo(p) = n

l+ Y,,”(52)

This r esu lt is similar t o t he familiar t heor em of feedba ck-amplifier t heor y

which sta t es tha t the effect s of changes in parameter s and externa l influ-ences a re reduced by feedback in the ra t io 1/(1 + L@), where P@ is

the gain around the feedback loop.

As an example, let us use Eq. (50) t o compute the asymptot ic response

of a zer o-st at ic-er ror ser vomech an ism t o a su st ain ed t or qu e TO suddenly

applied to the outpu t shaft . We have as before

Y,T,lim %(t) = lim —

(

1 + Y$2

t+ m P+o Km jm)

+ Yq, +Y:2”

Usin g t he ch ar act er ist ics of Y!, a nd Y$, discu ssed in Sec. 4.2, we fin d

()’0 1+K2lim do(t ) = ~ ~ — _— .,!-) . . = f.:. – ::

(53)

Th e qu an tit y K? is ca lled the “torque-er ror constan t” and is direct ly



SEC. 4.4] ERROR COEFFICIENTS 147

propor t iona l to the velocity-er ror constan t . It follows tha t a zero-

velocity-er ror system will h ave an in fin ite t or qu e-er ror con stan t ;tha t is,

a con st an t load t or qu e will n ot cau se an er ror in ou tpu t posit ion .

Equat ion (50) can be used to compute the response of a system to anyarbit r a ry torque loading if such a ca lcula t ion is necessa ry in a given

applica t ion . Usua lly, however , a computa t ion of KT gives enough

informat ion . Since we are dealing with linear systems, the er r or s resu lt -

ing from torque loadlng or other externa l influences add direct ly to the

er ror s tha t a re caused by the dynamics of the system when following any

inpu t s igna l.

It should be noted tha t the response to a torque distu rbance T(t ) is

iden t ica l with the response to an equiva len t input signal

(),T(p) 1 + Y:,

‘“’(p) ]e’’”iva’e’”= Kmjm Y~l(54)

Frequen t ly, par t icu la r ly for simple systems, th is equ iva len t input is

easily ca lcu la ted and is of such a na ture tha t the response is readily

est im at ed fr om a k nowledge of t he st ep-fu nct ion r espon se of t hes yst em .

4.4. E rr or Coefficien ts.-The discussion in the preceding sect ions led

na tura lly to the defin it ion of cer ta in system parameter s ca lled “er ror

coefficient s, ” which charact er ized the per formance of a given system to

some given inpu t . Specifica lly, we have defined the velocity-, accelera -

t ion-, and torque-er ror constan ts for severa l simple systems. The con -

cept of er ror coefficien t can be considerably genera lized, and such a

genera liza t ion provides a very useful and simple way of consider ing the

na ture of th e response of a system to a lmost any arbit r a ry inpu t .

We consider the Laplace t ransform of the quant ity c/t?, for a genera l

servomechan ism and assume tha t it can be expanded as a power ser ies

in p, valid at least for small p. Calling th is funct ion M(p), we have

(55)

Proceeding in a formal manner , we have

6(P) = ~(P)eJ (P) = CoOr(P) + ClP@I(P) + : P2&(P) + “ “ “ . (56)

The region of convergence of the power ser ies for M(p) and c(p) is the

neighborhood of p = O. These ser ies can therefor e be used to obtain an

expr ession for c(t j h ha t is valid for la rge va lu es of t , th at is, for t he steady-

s ta t e response. This result is

~(t)coor(t)+cl~;+:y + “ “ “ .t+ .

(57)




The er ror is seen to consist of terms propor t iona l t o the input , the

in pu t velocit y, in pu t a cceler at ion , a nd, in gen er al, st ill h igh er der iva tives

of the input signal. The constan ts C., defined by Eq. (55), a re clear ly

a ser ies of er ror coefficien t s which can be used to ca lculate the steadyst at e er r or of t he ser vomechan ism . Assuming tha t the Cn’s a re known,

let us examin e a few t ypical calcu lat ion s. If O,(t ) is a u nit -st ep fu nct ion

of posit ion, then the steady-sta te er ror is simply Co. If O,(t ) is a step

fu nct ion of velocit y, t hen COmust be zer o for a fin it e er ror ; t his fin it e er ror

is then simply CIQ, where fl is the input velocity. If 81(t ) = ~AP,

for a finite er ror , COand C, must be zero. Compar ing these results with

Eqs. (46) and (47), Sec. 4.2, we see tha t the C’s are rela ted simply to

t he er ror coefficien ts pr eviou sly defin ed. F or in st an ce,

c, = +, c, = #“ a

(59)

Let the input be a single-frequency sinusoid: OI(t ) = o sin uoi. Equa-t ion (57) formally gives the result

The er ror is sinusoidal, and its amplitude is given by o t imes the square

root of the sum of the squares of the two quant it ies in parentheses in

Eq. (60). The rest r ict ion to small va lues of p, considered ear lier as a

rest r ict ion to la rge values of t , cor responds in this case to a rest r ict ion to

small values of w,. This is perhaps more easily seen if \ vego back to Eq.

(55) and consider ~ (jo) ra ther than ~ (p). We would then have

~(j@)=CO+CJjU–~–#jti3+~@’ +. . . .

(61)

Tak~n g t he absolute vah ~e of both sides, \ \ ’efind

[( )co-g+ %-... 2t l = 10,1 ,

( )1

2 }$

+ c,. – c;;+”0. .(62)

Equat ion (61) will converge if the C’s are bounded. In this event



150 GEN ERAL DES IGN PRIN CIPLES [sMC.44

able in indica t in g desir ed gain levels in th e subsidia ry loops for opt imum

.~alues of the C’s, tha t is, for minimum error .

In a la ter sect ion th ere will be given a very simple way of determin ing

approximate values of the fir st few C.’s, a t least , direct ly from the fre-

qu en cy-r espon se cu rves of t he system .

The er r or coefficien ts ca n be given a st ill differ en t physica l sign ifica nce

which is somet imes useful and of more than academic interest . Let us

suppose tha t a uni~step funct ion of posit ion is applied to a genera l

serve. Then

(67)

and

‘[1’’’’”1 = %)(68)

Furthermore,

[- dt c(t) = lim

/

t-If (P)

dt c(t ) = lim — (69)o Z’+m o p+o P

We are familia r with the fact that a zero-sta t ic-er ror servo is charac-

t er ized by an M(p) that has a zero of order one at p = O; tha t is,

M(p) = pN (p).

Subst itu t ing in Eq. 65, we get

Cn=$ ’N’P)IP=O

This gives immedia t ely

andC’o=o (as assumed)

[c1 = ho N(p) + p ~T:

1= lim N(p),

N (p) = ~, ‘0P

(70)

(71)

(72)

(73)

which is precisely the expression in Eq. (69) for the integra l of the er ror

for a step-funct ion input . It follows that Cl or h’:’ is a good measur e of

the speed of response of a servo tha t has an aper iodic or near ly aper iodic

response but may easily be a very poor figure of mer it for a servo having

a damped oscilla tory response, because a very poor system of this type

could conceivably be adjusted to give a very low C, (very high K.).

In a similar manner one can show tha t CZ or K~l, for a zero-s t a tic-er ror

a nd zer o-velocit y-er r or ser vo, isr .

(74)



SEC. 4.5] ZNTRODUCTION 151

tha t is, the accelera t ion-er ror constan t is the reciproca l of the t ime

in tegra l of the step-funct ion er ror weigh ted by the t ime. Similar ly c ~ is

propor t iona l to the integra l of the step-funct ion er ror weighted by the

square of the t ime and so on for the h igher er ror coefficien ts. Th ein terpreta t ion of the er ror coefficients in this manner is often inst ruc-

t ive. Similar correla t ions may be der ived if the input is considered as

a st ep fu nct ion in velocit y.

Most of the resu lts of th is sect ion have been obta ined in a purely

formal way, proceeding from Eq. (55), and the conscient ious reader will

obser ve man y st eps in volvin g oper at ion s th at a re open t o qu est ion . Th e

inversion of Eq. (56) in order to pass to Eq. (57), for instance, result s not

on ly in the terms given in the la t ter but a lso in a whole ser ies of h igher-order impulse funct ions (delta funct ions) tha t have been discarded.

The just ifica t ion for th is procedure, as well as for the validity of the

term-by-term inversion of the t ransform in Eq. (55), is t oo involved to

be presented here. The reader in t erested in these quest ions, as well as

in the genera l subject of the asymptot ic behavior of funct ions and their

Laplace t ransforms, is r efer red to Par t III of G. Doetsch’s excellen t

bookl TheoTie und Anwendung der Laplace Transform ation .

BASICDESIGN TECHNIQUESAND APPLICATION TO A SIMPLE SERVO

4.6. Int roduct ion. -In the following sect ions we shall consider @

typica l servo design problem and examine the var ious procedures $~

techniques available for its solu t ion, a t tempt ing to emphasize t~e

a dva nt ages a nd lim ita tion s of t he differ en t modes of a ppr oa ch . 1.

e

fj

3FIG.4.10.—Simpleervo loop.

L.-J d- J an

Figure 4.10 is a block diagram of the system to be analyzed. The

er ror -m ea su rin g device is assumed t o give a voltage propor t iona l to the

er r or . The t ransfer funct ion of the equalizing network is taken to be

(75)

which is the t ransfer funct ion of the network shown in Fig. 4.11, where

2’ = R,C1 and a = (R, + R,)/R,. The reasons for using this so-called

“integral” type of equaliza t ion will be eviden t from the result s of the

an alyses in t he followin g sect ion s; a discu ssion of in tegr al equ aliza tion

1G. Doetsch, Them”e und AriwenduW da Laptace Tra rw forrnu .t it rn ,Dover Publi-cations,New York , 1943.



i.,

152 GEN ERAL DES IGN PRIN CIPLELY [SEC,443

is given in Sec. 4,16. Theamplifier can becharacter ized by a constan t

gain G.. The motor is cha racter ized by an iner t ia J~ and a viscous

~

RI

c,

0

Fm . 4,11.—Equalizingne twork.

damping coefficien t f~; it s t ra nsfer fu nc-

t ion is

Km(76)#m (~) = p(’l’mp + 1)’

where T. = Jm/j~ and K. has t he dim en -

sions of angular velocity per volt . The

study of this system is impor tan t because

simt ie servos of this kind are fr euuent l~

used and because many mor e complica ted systems can be a pproxima ted b~this simple system for t he purpose of st udying t he effect of adding in t egra l

equ~lization.

The importan t physica l parameters of the complete system can now

be lis ted:

T~ = the motor t ime constan t ,

Km = the motor gain,

G. = the amplifier gain,

T = t he in tegr al t ime con st an t,

a = the at t enuat ion factor of the equalizer .

In a given design problem the motor character ist ics a re usually consid-

ered as being given , while a , T, and G. a re parameters tha t we can vary

in any way we choose in order t o improve the per formance of the system.

In gen er al a ny “solu tion ” to the design problem should tell us how the

per formance of the system is influenced by changes in any of the param-

et er s and should pr ovide a ra t ional basis for select ion of t hose parameters

whose values can be adjusted at will.4.6. Differ en tia l-equ at ion Analysis. —The method of direct solu t ion

of differen t ia l equat ions is commonly employed in the study of most

physical systems and immedia tely suggests it self in connect ion with the

servo problem. Following the basic out line given in the fir st sect ion of

this chapter , we can writ e for our chosen example

(77)

where

: (P) = “’(p) J1 +

Y1l(P)

‘,l(P) ‘(flU%P)(Gc)[P(T~”’+-l)] ‘p(Ti;:~%+lj(,8)

It is impor tan t t o note tha t the product GsK~ a ppea rs in t hese equ at ion s

in such a way that it can immedia tely be ident ified with Kv, the velocity-

er ror constan t . The different ia l equat ion for this system is clear ly



SEC. 46] DIFFERENTIA>EQUA TION ANALYS IS 153

aTz’~”+ (az’+z’)d~”+ (l+ K*z’)@+Keo~ d~a m dp dt v

This different ia l equa t ion is t o be solved subject to the condit ion tha t the

system is init ia lly a t rest . The Laplace t ransform method of solu t ion is

the most convenien t approach , since Eq. (79) is a simple ordinary dif-

ferent ia l equat ion with constant coefficien t s. The t ransform of the ou t -

put is

th(p). (80)X%(t)l = ~(~mp + 1)(:;) ++17; K,(1 + ~P)

We now assume some represen ta t ive or a t least in terest ing form of

inpu’t funct ion , the most commonly used one being a suddenly applied

displa cement of velocity, as discussed in Sec. 4.2. F or a suddenly applied

velocity %, Eq. (80) can be writ t en as

K.%(1 + Tp)(81)~[e”(f)l = pqp(Tm p + l)(aTp + 1) + K.(1 + ~P)l’

or

e.(p) = %(1 + Z’p)

P2(::+1)($+2’2+1) ’82)

where the denomina tor has been factored and the parameters a , (, and

u. in t rodur .d. The inversion of Eq. (82) to find (?o(t ) can be per formed

in gen era l by means of th e inversion in tegra l and the ca lculus of residues,

or it may be expanded in par t ia l fract ions, and the separa te inversions

looked up in a standard table of Laplace t ransform pairs (see Chap. 2).’Th e result is

18ss, for in st ance,M. F . Gardner and J . L. Barnes ,Transien ts in L irwar System-s,

WiIey , New York, 1942,Table C.




Limit ing forms of this expression must be used for the cases ~ = 1 and

a=l. The er ror s result ing from this suddenly applied velocity are

plot t ed in Figs. 4.12 and 4,13 for differen t va lues of f and a . The motor

t ime constan t , a fixed system cha racter ist ic, is used as t he normaliza t ionconstan t . It is necessa ry to select a value of the parameter a before

drawing these curves ;thefigures aredrawnforu = 10. It@illbeappar-

en t la ter tha t changes in the value of a have a direct effect on the magni-

tude of the steady-sta te following er ror but pract ica lly no effect on the

t ransien t na ture of t he response.

3.0

2.5

2.0

1.5

1-

E~oc1.0

0.5

0

0.50 1 2 34 .6 7

:Tm

8 9 10

FIG.4.12.—Transientesponsesor differen t va lues of ~. All curves are for a = 1.

We have not as yet defined the pa rameters a , c, and Q. in terms of the

pa rameter s of the actua l system, so tha t ou r results a re of no pract ica l

va lue as yet . For any given single set of data , of course, the system

parameters cor responding to a set of values of a , ~, and ~fi can be ca lcu-

la ted, but it r equ ires a considerable number of such calcula t ions to

determine the effect s of genera l var ia t ions in system constan ts. This

is an inheren t difficu lty of the differen t ia l equat ion approach--even inthis simple example it shows up quite st rongly.

Since the character ist ic equat ion is here of the th ird order , we can

expect to find no simple rela t ionsh ip between the system parameters and

t he coefficien ts of t he fa ct or ed cu bic in 13q. (82); we a re t hus led na tur ally

to some sor t of graphic presen ta t ion of th is relat ionsh ip. Char ts rela t ing

t he coefficien ts of t he gen er al cu bic

(84)@+a2z2+a lz+l=0



SEC.4.6] DIFFEREN TIAl=EQUA T ION AN ALYS IS 155

to the values of a , {, and U“ in the factored cubic

5+:’+’)(:’+1)=0(85)

h ave been pr epa red in con nect ion wit h this sam e pr oblem , 10Z’3but t heir

use involves a fair amount of calcula t ion , since the coefficients al, az,

and as are st ill not rela ted in any simple way to th e actual system parame-

3.0

2,5

2.0

~ h: 1,5 ) i

c

1.0

0.5

o— — .

%

0.50 24 6 8 10 12 14 16 18 20

t/2’m

FIG.4.13.—Tran sient esponsesfor differ en t va lues of a . Ml curves are for .? = 0.25.

ters. It is apparent that the only sat isfactory answer is a char t thatdirect ly rela tes the system parameter s to a , {, and a.T~. Such a char t

can be prepared using the equat ions that result when coefficients of

powers of p in Eq. (81) a re equated to coefficients of like powers in Eq.

(82) :

1Y. J . LIU, Servomechaniwn.s: Charts for V erifyin g Th eir S tability an d for Fin din g

tk eRoots of Thxir Th iTd and Fourth Degi-ee Characteristic Equations, privatelyprintedby Massachuset tsInst itute of Technology, Department of Electr ical Engineering,1941.ZAlso,L. W. Evans, Solutionof th eCubicEquation and th e C’Ubic Charts, privately

prints&by MassachusettsInstitute of Technology, Department of Electrical Engi-neering,1943.

aAlso, E. J ahnke and F. Erode, Funkt ionentafeln(Tabfes of Funct ion-s), Dover,N ew York, 1943, pp. 21–30 of the Addenda.

Ii



156 GEN ERAL DES IGN PRIN CIPLES [%c, 46

The three dimensionless parameters a, T/T~, and KmT~ r epr esen t t he

actua l constants of the physical system; and in order to presen t cor re-

sponding values of a , f, and u.T~ it is con ven ien t t o assign a fixed valu e t o

one of a, T/ T~, and K,T~ and plot the remaining t wo against each other .F igure 4.14 is such a cha r t drawn for a = 10, in which T~/T is plotted

“’l=%bkk%

0.2\

1A

0,

FIQ.4.14.—Designcha rt for single-time-lagervowith integralequa lizat ion. Hea vysolid linesa re for consta ntva lu es of (; ligh t solid lin es a re for m nst an t va lu es of w_T~;dashed lin es a r e for con st a nt va lu es of K. T~z. The en t ir e family of cu rves is dr awn fora = 10.

against K. T~ and the curves a re for constant va lues of ( and ~nTm.

Thk is, then essent ia lly a plot of the reciproca l of the integra l t ime

constan t against the loop gain or velocity-er ror constant . F igure 4.15

is a similar figure drawn for a = 5.We know from the response curves of Figs. 4.12 and 4.13 that va lues

of r in the range fr om 0.25 t o 0.75 r esult in r ea sona ble t ra nsien t r espon se

and tha t it is desirable to have as high a value of K. T~ (velocity-error

constant) as is consisten t with sat isfactory t ransient response. The

design ch ar ts t hen sh ow tha t t he int egr al t im e con st ant should be bet ween

four and eigh t t imes as large as the motor t ime constant and tha t loop



~EC.4.6] DIFFEREN TIA&EQUA TION ANALYS IS 157

gains of the order of K~T~ = a can be used. If the in tegra l equa lizer

were not used, a maximum usable va lue of K, T~ would be 1 or 2; thus

t he equ alizer en ables us t o in cr ea se t he velocit y-er ror con st an t by r ou gh ly

the at tenuat ion factor a of the network or possibly by as much as twice

th is factor . Values of KG!&~, a dimen sion less pa ramet er pr opor tion al t o

\ l \ m !, I

\,

“\

\

0.4 \ \\ \ \ \ \

“\ ‘\ ““- ‘,&glh0.3 \ \ ’.\ I

I \ “\ [ M ~i I I

-L1-l

I \ i 11“ iii II I . I I

I 1 1-- -–r”–––

I I I J-0,1

L0.2 0.3 0.4 0.6 0.8 1 2 34

KvT~

aFIG.4,15.—Des ign cha r t for s ingle-t ime-lag servo with in t egra l equa liza t ion . Heavy

solid lin es a re for con st an t va lu es of ~; ligh t solid lin es a re for con st an t va lu es of u nT”;d ashed lin es a r e for con st ant va lu es of K. T~z. Th e en tir e family of cu rves is dr awn fora=5.

t he a cceler at ion -er ror const an t (see Sees. 4.2 and 4.4), ha ve been plot ted

on the design char t s; they indica te clear ly an opt imum value of in tegra l

t ime constan t ; opt imum, tha t is, if accelera t ion er ror s a re considered

impor tan t . It can be shown tha t such opt imum adjustment cor responds

to making the parameter a equal t o roughly unity.

Examinat ion of the t ransien t response cu rves show that if the system

is adjusted to ~ = 0.25 and a = 1 (KvT~ = 2a), then the r ise t ime or

buildup t ime of the step-funct ion response will be about 1.5T~ sec and

the per iod of the oscilla tory par t of the response will be about 5 T~ sec.

According t o Eq. (43), the cutoff frequency of th is servo, considered as a

low-pass filt er , is about 2/7’~ radians per second. We can conclude that ,



158 GEN ERAL DES IGN PRIN CIPLES [SEC. &~

(as long as a is chosen grea ter than about 5) the maximum usable gain

for thk system is between K“T~ = a and K~T* = 2a and tha t a va lue of

T equal to approximate] y 4T~ is required for opt imum opera t ion a t

these gain levels. The buildup t ime is not mater ia lly affected by changesin the value of a.

The different ia l equat ion analysis’ ca rr ied out in deta il yields a gr ea t

dea l of informat ion about the system character ist ics; the amount of wor k

involved, however , is rela t ively large, even in the simple example tha t we

have selected. In a more complex case, char ts displaying the effect s of

va rying the numerous system parameters cou ld not be prepa red withou t

rm unreasonable amount of effor t . If a deta iled t ransient analysis of

such a system is requ ired, it is common pract ice to set up the problem ona different ia l ana lyzer and determine a la rge number of solut ions by

va rying the system constant s one at a t ime or (hopefully!) in appropr ia te

combinat ions. More often than not such a procedu re leaves the designer

with a la rge amount of da ta tha t a re ext r emely difficu lt to in terpret in

terms of opt imum performance from the system.

For tunately, more powerfu l and convenien t design procedures not

based on an explicit solut ion of the different ia l equat ions ar e available;

t heir a dva nt ages a re so ma nifold t ha t t he differ en tia l-equa tion a ppr oa chis seldom used by servo engineers a t the presen t t ime.

Th e st udy of simple syst em s fr om t he differ en tia l-equ at ion viewpoin t,

however , is inst ruct ive, par t icular ly t o the neophyte, in demonst ra t ing

t he u se of va riou s ba sic m et hods of equa liza tion a nd t he effect s of va ryin g

system parametem. In this connect ion , char ts similar to those presen ted

in this sect ion can be prepared for simple servos with other kinds of

equalizat ion . An int imate knowledge of the behavior of such systems is

useful to the designer , since so many complex systems can be approxi-

mated by these simpler systems.

4.7. Transfer -locus Analysis. The Nyquist Diagram.-The transfer -

10CUSanalysis is a study of the steady-sta te response of the servo system

to sinusoidal input signals. . I t is common pract ice, for the sake of

simplicity and convenience of interpreta t ion, to study the loop t ransfer

funct ion ra ther than the over -a ll t ransfer funct ion, tha t is, to study the

t ransmission of signals around the servo loop. The essen t ia l advantage

of this method arises from the familiar fact tha t the sinusoida l steady-

sta te solu t ion of the differen t ia l equat ion can be wr it t en down immedi-

a tely if the t ransfer funct ion of the system is known.

I G. S. Brown and A. C. Hall, “Dynamic Behavior and Design of Servomechwnisms,“ Trans. ASME, 68, 503 (1946). S . T$r.Herwald, “ Considerat ionsin Servo.mechan ismDesign,” Trans.AIEE, 63, 871 (1944). A. Callender ,D. R. Hart ree,andA. Por t er , “Time Lag in a Con t r ol Syst em ,” Trans. R oy. S ot. (L on don ), 235A, 415

(1936),



SEC. 4.7] T RAN SFER -LOCUS ANALYS IS 159

We now t ake up t he t ransfer -locus analysis of t he simple syst em shown

in Fig. 410. The t ransfer funct ion defin ing the t ransmission of signals

around the servo loop is evident ly given by

: (~) = P(l +K&;lT:) .~p)”

To fin d t he sinusoida l st ea dy-st at e solut ion we simply

Eq. (87), obt a in ing

K,(I + jwT)

% ‘~u) = ju(l + ju!!’~)(l + jauZ’) “

(87)

replace p by jo in

(88)

This is, in genera l, a complex number tha t can be expressed in it s polar

form, tha t is, in terms of it s magnitude and phase. A pola r plot of th is

90”

‘=8.“Tm=*&J ’m = 10

Tm

4.16.—Nyqu is t d iagram for s ingle-t ime-lag servo. (a ) No equa liza t ion , @),with integral equal izat ion.

funct ion with the dr iving frequency u as a parameter is ca lled a t r ansfer -

10CUSlot or , m or e commonly, a Nyquist diagram.

Let us fir st consider the Nyquist diagram corresponding to the seNo

with no equaliza t ion, which is simply a plot of

(89)




The rea l factor K,T~ affect s only the magnitude and not the phase of

thk quant ity and hence is simply a radia l sca le factor on the plot . We

will t a ke Ku T~, which is pr opor tiona l t o t he loop gain (or velocit y-er ror

constan t ), equal t o unity for the purpose of plot t ing Eq. (89). Curve Ain F ig. 4.16 is the Nyquist plot of Eq. (89). The values of u l”~ cor re-

sponding t o va rious point s of t he cu rve a re labeled in t he figure.

The loop phase angle a t any frequency (taken as posit ive) dimin ished

by 180° is defined as the “phase margin’~ at tha t frequency, and the

frequency at which the cu rve crosses the 10./el = 1 cir cle is t ermed the

“feedback cu toff fr equency.”

In th is ext rem ely simple syst em we ha ve just on e a djustable par amet er

a t ou r dkposal, namely, the loop gain , and we must determine wha t va luesof ga in will r esu lt in t oler able per forma nce. This is commonly done by

rel’st ing t he Nyquist diagram t o t he over -a ll frequ ency-response cur ves

of t he syst em, a pr ocess t ha t is ra ther easily car ried out .

It was poin ted ou t in Sec. 42 tha t the over -a ll fr equency-response

curves defined by the funct ion ~ (ju) a re similar t o the frequency-

response curves of a low-pass filter and tha t the peak height and

cor respon din g fr equ en cy a re u sefu l cr it er ia of per forma nce. Cu rves ofconstan t va lue uf 10o/(l,I can be drawn on the N yquist plot in order to

determine the genera l na ture of the over -a ll frequency-response curve.

These cu rves a re drawn in Fig. 4.16 in dot ted lines for var ious values of

00

‘=%”It is ea sily sh own t ha t t his family of cir cles is defin ed by t he equ at ions

M, MCen ter = – M, _ ~, radius = M, _ ~. (90)

From the figure we see clear ly tha t for K,T~ = 1 the peak height will

be approximately 1.2 and will occu r a t a fr equency given by wT~ equals

about 0.8. Increasing the gain is equivalen t to changing the radia l

sca le factor on the plot . One can easily determine, for instance, tha t

increasing K. Z’~ from 1 to 2 will increase the peak height from 1.2 to

about 1.5.Now let us examine wha t happens when the in tegra l equa lizer is added

to the system. The quant ity ~ (j~) is then mult iplied by the factorI

1 +j$~wTm

G(jti!!’.) = — —.

l+ja~tiT.in



SEC.4.7] T RAN SFER -LOCUS ANALYS IS 161

This t ransfer locus of the in tegra l network is plot ted in Fig. 4.17; it is

a lways a semicircle, the zero frequency poin t being a t r = 1, 0 = O and

the infin ite frequency poin t a t r = a–l, 0 = O. Th e pa ramet er T/T~

det ermin es t he dist r ibut ion of fr equ en cies a lon g t he a rc of t he sem icir cle.To combine this cu rve with the t ransfer locus .4 of Fig. 4.16, we must

fir st select va lu es of a a nd T/T~, then “mult iply” the two loci in the way

in which complex numbers a re mult iplied, tha t is, by mult iplying the

90°

120”~ ..0

270°Fm . 4.17.—Nyquistdiagramfor int egra lequa lizer,

magnitudes of the radius vector s a t a given fr equency and adding their

phase angles. The curves in Fig. 4.16 show the resu lts of such a process

when a is taken to be 10 and the curves a re for var ious va lues of T/T~.

These curves a re drawn with KoT_ = 10, as compared with KVT. = 1

in the case of the curve for the unequalized servo. If T/Tn = 1, we

sse tha t the t ransfer locus passes very near the cr it ica l poin t —1 + jO;

the seNo is decidedly unstable when KVT~ = 10. If, however , T/T~ is

increased to 8, a ga in of K$T~ = 10 can be used, giving a frequency-

response peak height of less than 1:6 a t oT~ equal t o approximately 0.8.

The gain cou ld be increased to 20 without causing the peak height to

grow t o more than 1.8. If T/T~ is increased to 20, this same high gain

can be used, and the per formance will not differ essent ia lly from the

per forma nce obt ain ed wit h T/Tin = 8. Fur ther analysis would show



162 GENERAL DES IGN Prin ciples [SEC.47

tha t T/T~ could be decreased to about 4 and tha t a gain of K,T~ = 20

could st ill be used.

It is quite clea r from these curves tha t the phase margin at feedback

cutoff is a good cr iter ion of stability when the t ransfer locus is more or

less para llel or tangent to one of the circles of constan t M in the vicin ity

of feedback cutoff. This condit ion obta ins in many, if not most , actual

systems. More than 30” phase margin at feedback cu toff is usually

desirable; more than 60° will usually result in a system that is grea t er

t ha n cr it ica lly damped.

Assumin g t ha t K,T~ = 20 with T/T~ = 8, feedba ck cu toff occu rs a t

about uT~ = 1.3, giving a buildup t ime of 2.4T_ (see Sec. 4.2)—some-

what la rger than the va lue found from t~e different ia l-equat ion analysis

in the preceding sect ion ; if T/ T~ were r educed to about 4 (we have

a lready ment ioned that this is possible), t he fr equ en t y of feedba ck cu toff

would be increased, resu lt ing in a somewhat shor ter buildup t ime.

It is inst ruct ive t o in t erpret the curves of Fig. 416 in terms of the

Nyquist stability cr iter ion developed in Chap. 2. The t ransfer loci have

been drawn only for the range of frequencies from O t o LTI and since the

stability cr iter ion requ ires the curves for the complete range of all r ea l

fr equ en cies, – @ to + ~, we must imagine the curves of Fig. 4.16 to be

complet ed by dr awing in t heir complex conju ga tes, i.e., th eir r eflect ions

about the rea l axis. We can see that no mat t er how high the ga in is

ra ised, the cr it ica l poin t —1 + jO will never be enclosed, so tha t in the

mathemat ica l sense the system never becomes unstable. This apparent

paradox and it s explana t ion have a lready been discussed in Chap. 2,

a nd we h ave a lr ea dy seen in th is sect ion h ow mor e det ailed con sider at ion s

of stability set an upper limit to the usable ga in .

The Nyquist diagram is most commonly used in con junct ion with

curves of constan t magn itude of 100/@~lto determine pa rameter va lues

tha t resu lt in a sa t isfactory over -a ll frequency response ra ther than for

determin ing stability in the absolu te mathematical sense. For more

complicat ed systems, addit ional factors will appear in t he expression for

00/., and these can be combined one a t a t ime in the same way tha t the

two simple factor s were combined in our example; thus the effects of

addit ional equalizat ion or more complex mot ive element s a re easily

studied. The economy of though t and t ime inherent in th is approach ,

as compared with the direct solu t ion of the different ia l equa t ions, is

much more st r iking in more complex examples where the differen t ia l-

equa t ion meth od is all but unfeasible; even in our simple example, how-

ever , t he t ransfer -locus method is much more conven ien t and less t ime-

consuming, as the reader can ver ify by going th rough the deta iled

calcula t ions involved in solving the same or a similar problem by the two

cliffer en t t ech niqu es. Th e pr in cipal adva nt age of t he t ra nsfer -locu s t ech -

I

I



SEC.48] ATTENUATION -PHAS E ANALYS IS 163

nique is the manner in which the resu lt s of varying system parameters

ca n be det erm in ed.

This type of analysis has been exhaust ively t rea ted by A. C. Hall,’

who gives a deta iled account of the basic philosophy of the method andt r ea t s many differen t pract ica l examples. The t r ea tmen t of mult iple-

100p systems by th is method is somewha t involved, since the loop trans-

fer funct ion is then no longer a product of simple factors. Severa l

wr it er s have poin ted ou t the advantages of using a diagram in which

C/OOs plot ted ra ther than 00/c for such mult iple-loop systems. Since

the genera l approach to the design proklem to be discussed in the follow-

ing sect ion is ideally su ited for mult iple-loop systems, these inverse

Nyquist diagrams are not descr ibed here in any deta il.

4.8. At ten ua tion -ph ase An alysis. —The’ (a tt en ua tion -ph ase” or ” deci-

bel–log-frequency” type of analysis to be in t roduced in this sect ion has

been found to be the most sa t isfactory approach to the servo design

problem and is the method tha t will be used’ in the la ter sect ions which

dea l with the genera l servo design problem in deta il.

The genera l theoret ica l founda t ions of this type of analysis a re dis-

cussed in Sec. 49, and the presen t sect ion is in tended to give no morethan a br ief in t roduct ion to the method, which , like the t ransfer -locus

m et hod, is ba sically a st udy of t he st ea dy-st at e t ra nsmission of sinu soidal

signa ls a round the servo loop. The rea l and imaginary par t s of the

logar ithm of the loop t ransfer funct ion a re plot t ed as funct ions of the

frequen cy, on a logar ithmic fr equen cy sca le. Writ ing

(92)

we see that the rea l par t is the logar ithm of the magn itude of $ (jco)

and the imaginary par t is simply the phase of the same funct ion . The

quan tit y 20 log,O ~ ~ propor t iona l to the rea l par t of the above expres-6,.

sion, is usua lly plot ted ra ther than just in ‘~ and is then ca lled the loop.,a t tenua t ion in decibels. It should be remarked here tha t the terms

“attenuation” and ‘‘ ga in” are used in terchangeably in this t ext for the

same quant ity, even though the ga in in decibels is the nega t ive of the

LA. C. Hall, The Analysi s and Syn thesis of L inear Servomechanisms,echnologyPress ,Massachuset tsInst itu te of Technology, May 1943. A. C. Hall, “ Applicat ion

of Circu it Theory to the Des ign of Servomechan isms,” J . F rank lin Inst ., 242, 279

(1946).Seealso:H. Lauer,R. Lesnick,and L. E. Matson, Servomechan ismundamentals ,

McGraw-Hill,New York , 1947.



164 GENElL4L DES IGN PRINCIPLES [SEC.4.8

a t tenua t ion in decibels. The con text in any par t icu la r instance will

p reven t con fu sion . The sign ifica nce of t his seem ingly t rivia l modifica tion

of the usual Nyquist diagram stems from cer ta in asymptot ic proper t ies

of the result ing curves and from cer ta in rela t ionships between the twodiagrams. It will be shown la ter tha t for a la rge class of t ransfer func-

t ion s, t he so-ca lled m inimum-ph ase class, t he a tt en uat ion ch ar act er ist ic

is complet ely det ermin ed wh en t he ph ase ch ar act er ist ic is pr escr ibed, an d

vice versa. A knowledge of the na tu re of th is funct ional rela t ionship

between t he a tt en uat ion an d ph ase ch ar act er ist ics oft en makes it possible

to car ry through a la rge por t ion of the design procedure using only the

a t tenuat ion curve, which is ext remely simple to const ruct , even for ver y

complex sys tems.Let us now consider the a t tenuat ion and phase diagrams for the t rans-

fer funct ion of Eq. (88). First we not ice tha t the rea l par t of the loga-

r ithm of this expression can be wr it ten

A = 20 log,, KoT~ + 20 log,o 1(1 + jtiT)l – 20 Iog,o ljd’~[

–20 log,, 1(1 + jmT~)l – 20 Iog,o 1(1 + jacoT)l; (93)

the at tenua t ion character ist ic is the sum of the character ist ics of the

individu al fa ct or s of t h e complet e expr ession . The first term is simply anaddit ive constant ; we ther efore take K, T~ = 1 and elimina te th is t erm

for the t ime being. The at tenua t ion character ist ic cor responding to a

typica l t erm, say A, = 20 log,, \(1 + jcoT) 1, is easily const ructed. For

ver y low frequen cies th is t erm appr oaches ‘Xl loglo 1, or zer o. For ver y

la rge fr equ en cies we h ave

20 log,, ](1 + ju!f’)1 -20 Ioglo UT. (94)

ThusAIwO, UT < 1,

A, =20 log QT, UT > 1. (95)

In th is a sympt ot ic r ela tion A 1 is obviously a linear funct ion of the loga-

r ithm of the frequency and becomes zero a t a frequency u = I/T. To

determine the slope of the linear plot of A 1 as a funct ion of log10 tiT~, we

not ice that if a given va lue of u is doubled, that is, if the frequency is

r aised on e oct ave, t hen A 1is increased b y 20 log10 2. Thus the slope may

be expressed as 6 db per octave. These high- and low-frequency asymp-

tot es a re shown in Fig. 4“18, where z = QT. The exact funct ion 20

log,, \(1 + juT) ] is also plot ted, in dashed lines, and is seen to differ

from the asymptot ic curve by at most 3 db, a t the corner frequency

w = l/!f’. The term 20 log,o ljtiT~} in Eq. (93) is easily seen to have an

at tenua t ion character ist ic tha t is simply a st ra ight line with a slope of

6 db per octave passing through zero at the frequency u = l/1’~.

I

I

I



SEC.48] ATTENUATION-PHAS E ANALYS IS 165

If we now assume that T/T~ > 1, we can draw the asymptot ic curves

for each of the terms in Eq . (93) all on one drawing and finally take the

sum of these curves to find the asymptot ic a t tenua t ion character ist ic

+30

+25

T+ +20=9+15m: +10N

+5o

-50.06 0.1 0.2 0.4 0.6 1.0 2 461020

xFIG.418.-At tenuat ion of 1 + jz.

+50

+40

+30

+20

+10%

:0-

p... l_+- ! I I

I

~20

-30

-40

-50

-600.0040.0060.01 0,02 0.040,06 0,1 0.2 0.4 0.6 1 24u T=

F]o. 4.19.—Asympt ot ic a tt enua tion ch ar act er ist ic. Cu rves a re dr awn for Ya = 10, and K,,T~ = 1.

A - -20 log,o ju l’n l,B w -2o log], (1 +juT,n)l,c -20 10!310(1 +M31,D N –20 log,, ](1 +j@T)l,E - sum of th e above.

. 8,

for the en t ir e loop. This is shown in Fig. 4.19. In this and the following

figures the symbol v has been used to denote the quant ity T/T~. Actu-

ally, of course, the individual curves A, B, C, D need not be drawn in



166 GENERAL DES IGN PRIN CIPLES [SEC.48

order to draw the complete curve E once a lit t le insight in to the nature

of t he pr ocess is gained.

Severa l impor tant fea tures of the over -a ll cu rve E should be noted.

(1) The final break from 6 to 12 db per octave will a lways occur a t

uT. = 1, independent of the par t icular values of a > 1 and T/T~ tha t

a re selected, as long as T/ T~ > 1. (2) It is ext remely simple to observe

th e changes that r esult when t he pa rameter s T/T_ and a are var ied. The

rat io T/ T~ determines the length of the 6-db per octave st retch between

uT. = T~/T and uT~ = 1, and a determines the length of the 12-db per

octave st retch between u T~ = (a T/ T~)–l and ~ Tm = T*/ T. Finally,

a value of loop gain K~Z’~ cliffer en t from unity result s simply in a shift of

the A = O line up or down on the plot , according to whether K,T~ ismade less than or grea ter than unity.

We now take up the const ruct ion of the phase character ist ics. It is

eviden t that the complete phase angle for the loop t ransfer funct ion is

equal to the sum of the phases of each of the separa te factors of the

t ransfer funct ion . The factor K,T~, being real, cont r ibu tes noth ing t o

the phase, while the factor (jaT_)–l cont r ibutes –7r/2 radians or – 90°,

independent of the frequency. The phase of a typica l factor (1 + jz) is

given by\ q5 = tan-’ x. (96)

This is plot ted in Fig. 420. If z = UT and o = l/ T , then z = 1; thus

this cu rve is the phase associa ted with the a t tenuat ion character ist ic of

Fig. 4.18, the poin t x = 1 cor responding to the frequency tiT~ = T~/T.

We now have all the data that we need to const ruct the over -a ll loop

90

80

703&60%.5 50*

; 40

%z 30CL

20

10

00.02 0.040.06 0.1 0.2 0.4 0,6 1 246102040

xF IG. 4.20.—Pha se of 1 + jz,



SEC.443] ATTENUATION-PHAS E ANALYS IS 167

phase character ist ic. It should be not iced tha t rapid changes in phase

can occur only in the vicinity of those frequencies a t which the slope of

the a t tenuat ion character ist ic changes and tha t if the at tenuat ion

char act er ist ic has a constan t slope over an y appreciable fr equ en cy r an ge,t hen t he associa t ed phase ~villbe essent ia lly con stan t. This is illust ra t ed

in Fig. 4.21, ~vhere the phases of the individua l factor s of the t ransfer

funct ion a r e sket ched (A, II, C, ,!)) a long with their sum E. Compar ing

---i

*1OO

+80

+60

+40

+20

i??!0-F

,= -.20*; -40

%~ -60

j-so

-100

-120

-140

-160

-1800.0040006001 0.02 0.040.06 0.1 0.2 0,4 0.6 1 24

u T~FIG.4.21.—Phasecharacteristics.Curves are drawn for u = S, a = 10, K“!f’~ = 1.

SeeFig. 4.19 for ident ificat ionof curves.

the over-a ll phase character ist ic with the over-a ll a t tenuat ion cu rve of

Fig. 4.19, we see that for very low frequencies, where the slope of the

at tenuat ion cu rve is constan t a t – 6 db per octave, the phase is essen t ia llyconstant at – 90°. As we approach the frequency of the first break

uZ’~ = (a T/ T~) ‘1, the phase begins to change rapidly and tends toward

a new constan t value of —180°, associa ted with the long —12-db per

octave st retch of the a t tenuat ion curve. As the a t tenuat ion cu rve

breaks back to – 6 db per octave, the phase once again changes rapidly

back toward the – 90° value ahvays associated with a – 6-db per octave

slope. However , since the frequencies aT~ = T~/T and uT~ = 1 are

rela t ively close t ogether (the 6-db per octave por t ion of th e curve is rela -



168 GEN ERAL DES IGN PRIN CIPLES [SEC.4.8

t ively shor t ), t he phase does not reach –90° again but ra ther decreases

asymptot ically to —180° under the in fluence of the break from —6 to

–12 db per octave at the frequency o!l’- = 1. It is eviden t from Fig.

4.21 tha t as long as the frequencies at which the breaks occur a re fa ir ly

well separat ed, the phase changes associa ted with the breaks are more or

less in depen den t of on e a not her .

The discussion in the previous sect ion showed tha t the amount of

phase margin at the frequency of feedback cu toff was a good pract ical

cr it er ion of system stability, a t least 30° and preferably 45° or more

phase margin being requ ired. F igure 4.21 indica tes tha t a phase-margin

maximum occurs about midway between the two cr it ica l frequencies

uT . = T _/ T and uT* = 1, in the cen ter of the – 6-db per octave slope.

We now change the loop gain KvTm in such a way as to make feedback

cu toff occur at a frequency with the desired phase margin , tha t is, we

slide the A = O line down in Fig. 4.19 unt il it in t er sects the a t tenua t ion

curve at the desired cu toff frequency. The curve has been drawn for

T/T~ = 8, a = 10, and in th is case we find that a loop gain of Kt,T~ = 26

db will give sufficient ly stable per form an ce (30° phase margin). A va lue

of Kt,Tm = 12 db will give 45” phase margin a t cutoff and a more stable

system. It will be shown later tha t a 6-db per octave st retch of th is type

must be at least 2* octaves long in order to develop sufficient phase

mar gin ; it follows t ha t Z’/T~ must be grea t er than or equal to rough ly 4.

We also observe from Fig. 4“19 that the maximum usable va lue of KWT.

will a lways be very near ly equal to 2a, twice the a t tenuat ion factor of the

in tegr al n etwork. To ar r ive at an est imate of the r ise t ime of the

tr an sien t we con sider t he equ at ion

(97)

This tells us that when 100/cl is la rge, then 10o/0~[ is very near ly 1 and

tha t when 10~/cl is small, I%/0,1 is approximately equal to Itb/c 1. Thus

we can form an asymptot ic curve for 100/011by taking 100/0,1 equal to 1

from zero frequency ou t to the frequency of feedback cutoff and equal to

I@o/c] at a ll h igher frequencies. We then have once again a typica l low-

pass filt er character ist ic that cuts off a t 12 db per octave. According toEq. (43), the r ise t ime of the step funct ion response is T/uo, or 2.25T~ SSC,

when T/T~ = 8, a = 10, K.T. = 26 db.

The

curve.

a cceler at ion -er ror con st an t is ea sily fou nd fr om t he a tt en ua tion

Equat ion (66) gives us

I

I

1 aT+T. —T 1—.Ka K. – ~~

(98)



SEC. 4.9] ATT ENUATION -PHAS E RELATION SHIPS 169

a nd we ca n wr it e, a ppr oximat ely,

(99)

If we extend the 12-db per octave slope in the low-frequency region unt il

it in tersects the line of selected gain level, the frequency of in t ersect ion

will give t he a cceler at ion -er ror const an t t hr ou gh t he formula

(loo)

wh er e w, is t he fr equ en cy of in ter sect ion .

The reader should observe tha t given a cer ta in amount of pr ior knowl-

edge of at tenuat ion-phase rela t ionships, it would not have been neces-

sa ry to compute the phase character ist ic in deta il and that most of the

design pr ocedur e is based on t he a t tenua t ion character ist ic, which can be

drawn in asymptot ic form with no computa t ion and negligible effor t .

The mor e complex examples consider ed la ter in t he cha pt er will illust ra t e

even mor e st rik in gly t he su per ior it y of t he a tt en ua tion -ph ase con cept s.

ATTENUATION-PHASE RELATIONSHIPSFOR SERVO TRANSFER FUNCTIONS

4.9. At tenuat ion-phase Rela t ionships.-A complete mathematical

t rea tment of a t tenua t ion-phase rela t ionships will not be presen ted here

because the deta iled result s of the theory are not actually used in this

book, and the theory has been exhaust ively t r ea ted elsewhere.’ ‘A brief

survey of the theory is given, followed by a deta iled exposit ion of the

pra ct ica l pr ocedu res in volved in t he analysis of ser vo problems.

The t ran sfer funct ion s consider ed may r epr esen t t he physical char ac-ter ist ics of many different kinds of devices. For example, they may be

over -a ll loop t ransfer funct ions, subsidiary loop t ransfer funct ions, or

perhaps the t ransfer funct ions of simple passive equalizing networks.

In genera l, we consider the logar ithms of these funct ions, the real par t

A(w) being the at tenuat ion, or gain, and the imaginary par t I$(w) being

the phase. The general symbol Y(p) is used here to represent the

t ransfer funct ion, considered as a funct ion of the complex frequency

P = a + ju. We can divide the class of t ransfer funct ions considered

into two subclasses, depending on the locat ion of the zeros and poles of

Y(p). If Y(p) has no poles or zeros in the r igh t half of the pplane, then

I See, in par t icular ,H. W. Bode, Network Ann lysis and Feedback Amp li$er Design ,

VanNoetrrmd,NewYork, 1945;a briefreadabletreatmentis given by F. E. Terrna n,Rodti Ew”neera’ Ha&k, McGraw-Hill, New York, 1943;L. A. MacColl, Funda-

mental Theory of S emomechan isms , Van Nost r and, New York, 1945; R. E . Graham,“bear SCrVOne.ory,” Bell S ystem Techn til J ournaz, XXV, 616 (1946). E. B.

Ferrell,“The &rvo Problem as a TransmimionPr oblem,)>roc, ~RE, M, 763 (19.45).



170 GENERAL DES IGN PRINCIPLES [SEC,49

Y(p) is ca lled a “minimum-phase” funct ion. The grea t major ity of the

fu nct ion s en cou nt er ed in ser vo t heor y belon g t o t his cla ss, wh ich possesses

the following impor tant proper ty: If the a t tenua t ion A(u) is known over

the ent ir e r ange of frequencies, then the phase O(U) is uniquely deter -m in ed; a nd sim ila rly, if o(m ) is kn own over t he en tir e r an ge of fr equ en cies,

then A(u) is uniquely determined. This proper ty is not possessed by

funct ions Y(p) having poles or zeros in the r igh t half of the p-plane, and

this case, which ar ises occasionally in connect ion with systems having

more than one feedback loop, must be t r ea ted differen t ly.

4-

3

w

/ ,

52 / \8

s /

1 , //

/L

00.1 0.2 0,4 0.6 1 2 4610

W/ u.

FIG. 4.22.—Weighting function.

The formula expressing the minimum phase associa ted with a given

at tenuat ion cha ract er ist ic can be gi~-en in a va riet y of forms, and numer-

ous other a t tenuat ion-phase rela t ionsh ips can be der ived by funct ion-

theory considera t ions. One form of the rela t ion is

l+(q) =

where P = in co/uO. This formula places in dir ect evidence th e impor ta nt

character ist ics of the a t tenuat ion-phase rela t ionship. The phase inradians a t any frequency t iO,@(mO),is expressed in terms of the slope of

the a t tenua t ion diagram and a weight ing funct ion, where

()

dA

G=slope of a t t enuat ion cur ve in decibels per oct ave.

The weight ing funct ion in coth 1~1/2 is plot t ed in Fig. 4.22; it has a tota l

weight , with r espect t o in t egra t ion over p, of 1. Severa l impor t an t resu lt s,



SEC,4.10] CONS TRUCTION OF PHASE DIAGRAMS 171

obtained in a more exper imenta l manner in the last sect ion, can be

der ived from Eq. (101). It is clear , for instance, that the phase associ-

a ted with a constan t a t tenuat ion slope of 6n db per octave is just r im/2

radians. It is a lso clear from the form of the weight ing funct ion that thephase must always change most rapidly in the vicinity of changes in

slope of the at tenua t ion character ist ic and that the phase at any given

frequency is influenced appreciably only by the changes in at tenua t ion

slope n ear t hat given frequ ency. The actual use of Eq. (101) for com-

puta t iona l purposes would be ext remely involved; and when a phase

character ist ic must be computed from a given at tenua t ion character -

ist ic, it is mor e convenien t t o approximate the given curve with st ra ight-

line asymptotes and to compute the cor responding phase by means of the

procedures and char t s developed by Bodel for this purpose. In servo

problems it is near ly always possible to approximate the at tenua t ion

character ist ic with sufficient accuracy, using only st ra ight lines with

slopes that a re integra l mult iples of 6 db p~r octave; consequent ly the

phase can be computed using the simplified techniques and char ts pre-

sen ted in t he following sect ion .

Occasionally a nonminimum-phase n et wor k may be used for a passiveequalizer . The t ransfer funct ion for such a network will have one or

more zeros in t he r ight half of the p-phmc and will thus have, for instance,

a factor of the form (—1 + Tp), \ rhich has the same at tenua t ion charac-

ter ist ic as (1 + i“p) but a rcwrscd pht isc character ist ic. Another non-

minimurn-phase situat ion ar ises in the case of an unstable subsidiary

loop, for \ vhich the t r t insfcr funct ion has a factor (T’p2 – 2(Tp + 1)-’;

th is agoin has thr same at tcnuat if)n character ist ic as (l’”p’ + 2~Tp + 1)-’

but a r eversed phase character ist ic. The occur rence of these specia l

ca srs is n ot t rmlbkwomc in ser vo pI’(JbIC’IIISf t he t echniqu es of t he follo\ v-

in g sect ion s a re CIUP1OYC(1,sin ce Ne h :ivc a llr ays sufficien t kno~r ledge of

the or igin of the a t tcn~mt ion character ist ic to kno~v ]rhethcr or not such

nonmin imum-pha se st ru ct ur es a rc pr esen t. C’lca rlyj h o~vever , if on e is

s imply given an attmutitkm rhamctcrkticj it cannot he assumed auto-

mat ically that , it is the character ist ic of a minimum-phase st ructure, and

the phase cannot be cornputcd ~vith cer ta in ty. Obviously the at tenua-t ion and phase may be computed from the t ransfer funct ion .

401O. Const ruct ion and Interpreta t ion of At tenua t ion and Phase

Diagrams.—l?or the const ruct ion of a t tenuat ion and phase diagrams t he

feedback t ransfer funct ion 00/t is exprcssml, as far as possible, as the

product or quot ien t of factors of the form (7’p + 1); its decibel magni-

tude 10~/c\ ,,~and phase angle Arg (oJ E) arc then plot t ed on semilog

coordina t es a s a funct ion of fr equcmc,y, In this t ype of plot an asympt ot ic

method can be used to approximate the cllrves,1~orh?,Op.Cd., Chap. XV.



172 GENERAL DES IGN PRINCIPLES [SEC,4.10

As an illu st ra tion of t he con st ru ct ion of decibel-log-fr equ en cy gr aph s

let us take the t ransfer funct ion

(102)

As @~ O, l@O/.Sldb ~ log,o K, – 20 loglo w The argument then

proceeds as in Sec. 4.8. Since doubling the frequency diminishes the

value of Ioo/cl by a factor of 2, the asymptote to the actual curve for small

values of a has a slope of – 6 db/octave [the log of 2 is 0.30103;

20(0.30103) = 6.0206 db,

which is usually approximated by 6 db]. When u >> I/ T , tha t is,

u T >> 1, 100/cI va ries inver sely as the square of the fr equency. Ther efor e

the asymptote has a slope of – 12 db/octave. If ]@O/Clis represen ted by

these two asymptotes in tersect ing at o - 1/T, the maximum depar ture

from the actual curve is only 3 db and occur s a t the in tersect ion . For one

octave above or below this poin t the depar ture is 1 db.

When more t ime constants a re present , the approximate plot is made

similar ly, with a change of 6 db in it s slope at each va lue of a for whichu t imes one of the t ime constants equals unity. The slope is decreased

for t ime constants in the denomina tor but increased for those in the

numerator. The depa rtu re fr om th e actual cur ve nea r ea ch slope-change

poin t is the same as tha t for Eq. (102) if the t ime constants a re not too

close to each other .

Th e plot of Ar g (0~/c) for E q. (lo2) is a ppr oxim at ed fr om t he f ollowin g:

()st i-O, Arg $ + —900,

()m-+~, Arg ~ + —lsOO.

()Atw=&Arg ~ = –135°,

1

()

%0u= —,Arg ~

2TM —26.5° fr om t he —900-pha se a sympt ot e,

2

()

eo ,~= —,Arg —T

M +26. 5° fr om t he —1800-ph ase a sympt ot e.e

For convenience, the “ decibel–log-frequency” and ‘‘ Arg–log-fre-

quency” plots a re usually made on the same sheet .

In using th is method of plot t ing t ransfer funct ions for servo design ,

numerous shor t -cut methods can be devised.

The at tenuat ion plot may be const ructed ent irely by project ing the

line at one slope to the next break poin t and then project ing the new



SEC.4.10] CONS TRUCTION OF PHAS E DIAGRAMS 173

slope to the next break poin t and so on. However , since the er ror s in

this process a re cumulat ive, bet t er accuracy is obta ined for the asymp-

tot ic-a t tenua t ion plot if the va lue of the at tenuat ion at each break poin t

in the asymptot ic curve is computed direct ly. The following procedurehas been found usefu l: The decibel value is computed at the fir st break

poin t (a = 1/7’,). If lI?o/ Kcl (K being the gain term) is being plot t ed

and the low-frequency asymptote has a slope of 6 db/octave, the required

value is – 20 logl~ (l/T,). If the asymptote preceding the next break

poin t (u = 1/ T2) has a slope of 6m db/octave, the decibel value at this

br ea k poin t is compu ted by su bt ra ct in g 20 loglo (Z’,/’i”,)~ fr om t he decibel

a t the previous break poin t . The decibel values at the other break poin ts

a re sim ila r ly compu t ed .A log-log du plex vect or slide r ule is u sefu l for ca lcu la tion of t he decibel .

magnitude or phase cont r ibuted by any of the (Tp + 1)-terms and of

the depar ture of the asymptot ic plot from the actual cu rve.

In calculat ing the magnitude -v’-, use is made of the rela t ion

l/tanh p = ~1 + l/sinh’ p, with ~u2T2 + 1 = l/tanh ~ and

1UT . ~—.

smh ~

In det ail, t he pr ocedu re in volves

1. Computa t ion of l/coT.

2. Determinat ion of g = sinh-’ (l/uT).

3. Determinat ion of tanh p.

4. Computa t ion of l/tanh p.

5. Determinat ion of log (1/tanh p).

By leaving the slide exact ly in the mid-posit ion these opera t ions are

completed by only two set t ings of the cur sor . To illust ra te the method,

let co!!’= 0.5.

1. Set the cur sor to 0.5 on the C1-scale and read the value ~ = 1.442

on the Sh2-scale. (The Shl-sca le is used if COT> 1.)

2. Set the cursor to 1.442 on the Th-scale. The figure 1.12 on the

Cl-sca le is the magnitude t i~l, but 0.05, the log of the

magnitude, may be read direct ly on the L-sca le by reading thk

linear scale as if the zero gradat ion were at its r ight -hand end andthe one gradat ion at the left end. Menta l mult iplica t ion by 20

gives 1 db for the cont r ibut ion of thk t ime constant .

Since the asymptot ic plot uses O db/octave for a (Tp + I)-t erm wh en

u l’ < 1, the magnitude of 1 db obta ined in the example is a lso that of the

depar ture of the asymptot ic plot from the actual curve for UT = 0.5;

and since the depar tu res a re symmetr ica l about UT = 1, this is a lso the

depa rt ur e a t OT = 2.



174 GENERAL DEL5’1GNPRINCIPLES [SEC.4.10

The phase angle is obta ined for UT < 1 by set t ing the cur sor to UT on

the C-sca le and readkg the angle on the T-sca le; for UT > 1 it is obta ined

by set t ing the cur sor to UT on the CI-sca le and reading the angle on the

T-sca le, but using the complement , which is usua lly given in red num-bers on this sca le. The ST-sca le is used for aT > 10 or <0.1.

When two t ime constants of the t ransfer funct ion are close together

and one is in the numera tor and the other in the denomina tor , t he maxi-

mum depa r tu re of the approximate plot from the actua l curve is less

than the 3 db resu lt ing from one t ime constan t a lone. F igure 4.23 shows

both plot s for TI = 2T,.

I?IG.4.23.–

At the

+2

+1

o

-1

-2

-3 — —I

-41, I

-5I I 2.0d b

-6

0 d b

I 0.7db 02db

-7I

I

-8L L _!_8T1 4TI 2T, $k,~ * + ;

u,. ic tud and a pp r ox ima t e plots of 1[2’m + 1)1(2’LP + 1)1 wit h T, = 2Tz.

geomet r ic mean (w~ = l/~ Z’IZ’Z) the depa r tu re is a lways

zero. If 2’, is the la rger t ime constan t , a maximum depar tu re of

‘“’o+([’+(wodboccurs both a t u = 1/1’1 and at u = l/T2. The phase angle is zero

for small CO,ncreasing to a maximum of –ir ,’2 + 2 tan-’ V“T ,/ T , a t the

geomet r ic mean ( TI being the la rger t ime constant ) and then decreasing

t o zero for h igh er fr equencies.

When two t ime constan ts a re close together and both are either in thenumera tor or in the denomina tor of the t ransfer funct ion , the maximum

depar tu re of the approximate plot from the actua l curve is grea t er than

the 3 db resu lt ing from one t ime constan t a lone. If the two t ime con-

stant s a re equal, the maximum depa r tu re is 6 db at u = 1/T. If th ere

a re n eq”ual t ime constan ts, t he maximum depar tu re is 3n db at u = I/T.

For two t ime constants, with l’, = 2T2, the maximum depa r tu re is 4 db.

This is shown in Fig. 4.24.



SEC.4.10] COA ’A’TR IJCT ION (IF I’IIASE I>IAGRAMS

If !l’’, ist hela rger t ime const an t , amaximum depa rt ureof

occur s both at u = 1/7’1 and at o = Ii’!l”j. .lt the geomet r ic mean

(u* = 1/ @,l”J , the pht ise angle is – 90°; t he depa rt ur e of t he a ppr oxi-

mate cu rve rca chcs a loca l minimum and is 20 log [1 + (TLJ ~J l d b .

The phase-angle asymptotcw arc 0° for low fmqucncics and t 180° at

high frequencies for t he t ime constan t s in t he n umera tor or denomina tor

‘z~– 1.-.. l.. .! I T~

-6 I \ l A3.5dbl

-8 III

-lo - \ \ ~ -

40db

-1 21

-1 4\ \

1I

-1 6,1 \\

Ii

1 \ \ I

%-18 1,~–zo 12d~ ,

c 1&

1-2 4

\

1

-2 6I

-2 8\

I

-3 0I 0 .3 d b

-32I

-34 \1

-36 !1

-38 1 \ .1

-401

.I

-42}

II I I 1 I I I 0,1db

(/ J

FIG, 4.24.—.kct ual a nd a ppr oxim at e plots of 11/[(T,P + l)(T~P + 1)11wit h T, = 2Tz.

respect ively. The dot ted project ion of the O- a nd the 12-db asymptotes

in Fig. 4.24 indica tes tha t the region above u = 1/ Tz may be approxi-

mated by a 12-db break at u = l/@z.

When a quadra t ic factor having a pair of con juga te complex root s

appears in the t ransfer funct ion , the shape of the actual cu rve is tha t

shown in Fig. 4.8. This plot is for th e dimensionless quadra t ic fa ct or

1

T’p’ + 2~Tp + 1“



176 GENERAL DES IGN PRIN CIPLES [SEC. 4.10

Only the cases where ~ <1 need be considered, since when ( = 1 the

denomina tor = ( Tp + 1)‘ and, when ~ > 1 it may be factored in to

(T IP + l)(T zP + 1).

+1 4

+1 3 o

+1 2 5

+11 10

+10 15

+9 20

+8 25

+7 30

+6 35%c

401~+s

.i !W4 455

~ :

2+3g

30$

:+2 55f

+1 601

0 65

-1 70

-2 75

-3 so

-4 85

-5 90

-60.1 0.2 0.4 0.6 0.8 1 2 4 6

alt8 1;5

FIQ.4 ,25.—Depa r tu r es of t h e a sympt ot es fr om the act ual cu r ves , for t h e quadr a tic fa ct orl/(z’2p~ + z~r ’p + 1).

The asymptot ic plot is constmcted as for ~ = 1, using a 12-db change

I

in slope a t w = 1/T, The phase-angle asymptotes a re 0° for low fr e-



SEC. 4.10] CONS TRUCTION OF PHASE DIAGRAMS 177

quencies and – lSOOfor h igh frequencies with – 90° at u = l/T. The

depar tures of the asymptotes from the actua l cu rve both for a t tenua t ion

and phase a re given on Fig. 4.25. When the quadra t ic factor occu rs in

the numera tor of the t ransfer funct ion, the signs of the depar tures on

thk plot a re rever sed. The depar ture curves a re symmetr ica l about

u = l/T. For the fr equencies not covered by the plot , below UT = 0.1

a nd a bove UT = 10, the phase-angle depar ture may be computed very

closely by using a change of a factor of 2 for each change of one octave in

frequency.

When nonminimum-phase terms such as (Tp – 1) or

(T ’p’ – 2fTp + 1)

(resu lt ing, for example, from an unstable in terna l loop) appear in the

t r ansfer funct ion , the approximate a t tenuat ion plot is const ructed in

the same way as for (TP + 1) or (Tzpz + 2(TP + 1), and the depar ture

from the actua l cu rve is also the same. The associa ted phase angle is,

however , not the same as for the minimum-phase terms. (The actua l

cu rves can , of course, always be computed direct ly.) Since these factors

a re seldom encoun tered, t hey a re n eglect ed in this discussion .

For const ruct ion of the actua l cu rve from the asymptote lines, it isconven ien t to use the following rela t ions tha t hold t rue for all of the

fa ct or s con sider ed in t his sect ion .

1. For both phase angle and at t enuat ion the depar tu re contr ibu ted by

a single t ime constan t or by a quadra t ic factor is symmet r ica l

about u = 1/T and that from two t ime constan ts close together is

symmet rkal about ~ = 1/ V’TITZ but may be t rea ted as two sepa-

ra te t ime constan t s if desired.2. The phase-angle depar tu re decreases by a factor of approximately

2 for each octave along the &sca le in the direct ion away from the

maximum depar ture poin t . From the data given in the previous

discussions it is seen that th is approximat ion does not hold in the

region close to the maximum depar ture point .

3. The at t enuat ion depar tu re decreases by a factor of approximately

4 for each octave along the u-sca le in the direct ion away from the

maximum depar ture poin t . Since the at t enuat ion depar tu redecreases a t a more rapid ra te, this approximat ion may be used

closer t o the maximum depar ture poin t than that for phase but it is

not sa t isfactory less than 1 octave from this poin t or 2 octaves if

mor e a ccu ra cy is r equ ir ed.

To illust ra te an approximate method for comput ing tota l phase

angles, let the equa t ion for the phase angle be



178 GENERAL DE S[G.V PR INCIPLE S [SEC.410

This may be approximated by

Arg~= ~n~+u~

Tl+k; –:z

1

Z,>(104)

uh

where2 z

1’1 includes only the t ime constants for which t il” < 1, ~T,

1 h

t hose wit h WT >1, and k is the number of those in ~1/Tk. For UT = 0.5 or

2, the er ror is about 2°; and for aT = 0.25 or4j the er ror isabout0.4°. The

maximum er ror is 12.4” for UT = 1. The terms for which 0.5 < COT< 2would ordinar ily be handled separ ately by mor e accu ra te methods.

A pair of divider s maybe used to facilita te th e ca lcula t ion of th e phase-

angle and at tenua t ion depar tu re for any fr equ en cy, using th e asymptot ic

a t tenuat ion plot . Suppose that the a t tenuat ion plot consist s of 6-db

a t tenuat ion to w = l/T1, 12 db to o = l/TZ, and 2+ db for the h igher fre-

quencies and that the phase angle is desired at a cer ta in frequency u,,

lying between a = l/ T l and u = 1/ T,. The dividers a re set to the

distance a long the u-axis between 1/T, and a=. Then one point of thedivider is placed on 1 on the co-scale, and the value of wZT1 is read wher e

t he ot her poin t on t he divider lies on t he o-scale, WT being grea t er than 1

when 1/T is less than ~. and less than 1 when l/T is grea ter than OJ Z.

From the va lue of CJ Z7’he phase angle or at tenua t ion depar tu re may then

be computed by the methods previously suggested. The same procedur e

may be used with all the t ime constants but would not be necessary with

t hose for which l/T differ s by severa l oct aves from a=, since their cor l-

t r ibut ion is close to the asymptot ic value. If the double t ime constantT2 in the example is due to a quadrat ic factor , t he angle is obta ined by use

of Fig. 4.25. The tota l phase angle is the sum of the angles for the terms

in the numera tor minus the sum for the denominator terms, with !30n0

for p“.

The at tenua t ion diagram for a t ransfer funct ion may be const ruct ed

before determina t ion of the va lue of the gain t erm K associa ted with it .

Frequen t ly I&J Ke I is plot ted. At very low frequencies the expression

reduces to lpml where n is a posit ive or nega t ive in teger . Zero db is

loca ted on the sca le at the poin t where the low-frequency asymptote

crosses u = 1 on the frequency scale.

The case where n = – 1 is frequen t ly encoun tered. In th is case K

is t he velocity-er ror coefficien t Ku, which may be read off the plot in two

ways. On the at tenua t ion diagram the line represen t ing a unit va lue of

10o/cl is usually loca ted for opt imum phase-angle condit ions. It s ;mi-

t ion on the decibel scale of the I@O/KCI plot gives the decibel va lue of



SEC.4.11] DECIBEL-PHAS E-ANGLE DIAGRAMS 179

Ko for t he ga in set tin g ch osen . The value of u at which the project ion of

t he 6-db/oct ave low-fr equ en cy a sympt ot e cr osses t he u nit lin e is t he va lu e

of Kv.

When the low-frequency asymptote has a slope of – 12 db/octave,the in ter sect ion of it s project ion with the unit line in the plot of 180/Kcl

occur s a t u. = ~=, where K. is t he a cceler at ion -er ror coefficien t. If

a 6-db/octave low-frequency asymptote is followed by a long 12-db/

octave sect ion due to a (Z’IP + I)-t erm in t he den om in at or (K.Z’I >> 1),

the project ion of the 12-db sect ion maybe used in the same way to obta in

K. = u:. Under the same condit ions K. = K,/Z’l.

The er ror c(t ) for an input O,(t ) can be obta ined with sufficien t accu-

r acy fr om t he equat ion

(105)

when the frequency componen ts of the input a re low enough so tha t the

h igher -order t erms a re negligible. This is usually the case, for example,

in a servo loop used for automat ic t racking of an airplane.

4.11. Decibel–phase-angle Diagr ams and F requen cy-r espon se Cha r-

a cter ist ics.-This sect ion cen t sins a discussion of meth ods tha t facilita te

+30

+25

+20

+15

+10

%+5~c-%~3-5

0 ,, , , “ I I , ! . \ 1

w/1 I Ill

-lo

-15

-20

-25

-30–360” -270° -180° -90” 0° +90° +180° + 270° +360”

Phase angle

-180° –90” 0° + 90° ~ 180° -90” 0° +90” + 18C”Phase margin

FICA4.26.—Constant-ampli ficat ion and phase-angle contours on the loop-gain phase-angledia gr am and illu st ra tive plot s of Y’,,, 1/ Y., a nd l/Yb.




the determina t ion of cont rol-syst em constan ts compat ible with good

frequency-response character is t ics . By means of a study of the t ransfer

funct ion rela t ing the outpu t 00 t o the input 0,, we shall see how to adjust

t he servo paramet er s so tha t t he ra t io ikf = 1.90/0,[has a limited depar t ure

from the ideal Ioo/01I = 1.0 over a su itable bandwidth of opera t ion ,To do thk it is convenien t to plot both M-contours and the loop t ransfer

funct ion /30/e on the same diagram. The con tours a re analogous to

t he M-circles in t he complex plane of t he t ransfer-locus plots.

If 00/8, = Y, and 0~/c = Y,,, t hen

Y1ll“, = —

1 + Y1l”(106)

Contours of constan t M = IYll in decibels and contou rs of constan t# = Arg (Y1) in degrees a re plot t ed on Fig. 426,’ against IYlll,b on the

ver t ica l axis (loop gain in decibels), and Arg ( 1“1,) on t he hor izonta l axis

(angle in degrees). Since these plot s repea t for each successive 360°

sect ion of Arg (Yll) and are symmetr ica l about the middle of each sec-

t ion it is possible to use a la rger and more accura te plot showing only a

180° region of Arg (Y,,), as in Fig. 4.27.

For convenience the angle is a lso indica ted on both figures in terms of

the degrees depar tu re from – 180° and labeled “phase margin, ” Theequa t ions for the contou rs a re

and

IY,lI,,. = 20 log,,[

sin (@ – *)

1in ~ ‘(107/))

where @ = Arg Yil, ~ = Arg Y1.On Fig. 427 the M-con tou rs a re given from +12 to – 24 db. Ilelo\ v

–24 db, since IY,,] << I, Eq. (106) yields IY,l = IYlll. At the same t ime

Arg (Y,) = Arg (Y,,) a lso; the Arg (Y,)-contours asymptot ica lly

approach th e Arg ( Y,,) -lines and are not separa tely labeled, except t hose

for –5° and –2”. As Y,l increasw, YI + 1, IYII ~ O db, and Arg

(Y,) + OO.

For each con tou r for M > 1 there a re two values,

lY,,ldb = –20 log,, (1 + M-’),

where the phase margin is zero. When IY,,l,, = – 10 log,, (1 – M-2),

the phase margin reaches a maximum and is COS–l~1 – 1}1-2. (In—.—

the above formulas the numer ical value of M is used ra ther than the

1The examplesY*, Ya, and Y’,, arc discuswxila ter in this sect ion.



SEC.4.11] DECIBEL-PHASE-ANGLE DIAGRAMS 181

decibel value.) For the +contours the highest va lue of IY,,l is reached

where the phase margin is equa l to 90° + + f rnr a nd in decibels equa ls

20 loglo n/sin $1.

N I II

M I N

I 1 1 II 1 II II II II II

Phase margin in degrees

32° 20 40 60 80100 120 140 160 180

28

24–0.5

20

-116

12 -2

8 -3

+4-4

.E -5

‘~ o -6

E4-4-9

-8-12

-12

-16 -18

-20

-. -24

-28

F

-180 -160 -140 -120 -loo -80 -60 – 40 -20 -50

Anglen degrees‘IO. 4.27.—Constan t–phase-angle and constan t-amplification contours on the decibel–ph

angle loop diagram.nse-

It is obviously possible t o t ran sfer t he M-con tour s and th e #-contour s

from this decibel–phase-angle diagram to the type of decibel–log-fre-

quency graph discussed in Sec. 410. Then, a fter a study of the manner

in which the a t tenuat ion curve crosses the M-contours, it is possible to

a lter the gain and, if necessary, the shape of the a t ten lla t ion plot , t o



182 GENERAL DE ti’IGN PR INCIPLES [SEC. 4.11

obt ain a sa tisfa ct or y fr equ en cy-r espon se ch ar act er ist ic. Since a la rge

number of M-curves must be plot ted, it is easier (if the designer has

available enla rged copies of F ig. 4.27) to plot the t ransfer locus of Yll

on th is decibel–phase-angle diagram, by use of data from t he at t enua t ion

and phase diagram. This Y,, plot should, in genera l, remain away from

the O-db and O-phase-margin poin t and should not cross over the M-con-

tou r tha t cor responds to the qua lity of per formance that can be tolera t ed.

A change of ga in moves the Y,l plot ver t ica lly on th is diagram. If

th is t ransfer funct ion has a phase margin tha t is very la rge a t low fre-

qu en cies and decr ea ses cont in ua lly wit h fr equ en cy in cr ea se, t he gain ma y

be increased to the va lue where the plot on th is diagram is t angen t to

t he M-con tou r r epr esen tin g t he t oler able per forma nce.

Frequent ly the t ransfer funct ion has a maximum phase margin a t afrequency other than zero. It is customary to design so tha t the gain

that is used places th is in the region where the plot crosses the la rger

M-contou rs. The lowest peak amplitude in IY,l is obta ined by loca t ing

the maximum phase-margin poin t on the Yl 1 plot tangent to an M-con-

tou r near the maximum phase-margin poin t on the M-contou r . If

the Yll plot on Fig. 4.27, when adjusted in this way, passes between two

of the M-con tours that a re plot t ed, in t erpola t ion may be used with due

considera t ion of the fact tha t the maximum phase-angle poin t on anM-curve is h igher in decibels than that for the next lower va lue M-curve.

With the best gain adjustment for a given maximum phase-margin ,

the peak height of If3,/011is obviously the va lue of the .If-contour that is

t angen t to the maximum phase-margin line, provided that the cu rva ture

of the YII plot does not exceed the cu rva ture of the M-con tour . The

numer ica l va lue of the peak amplifica t ion is then

M = (1 – Cos’ f$)-~~, (108)

The proper gain adjustment may be determined by set t ing the Y,l gain

a t the maximum phase-margin poin t equa l to

\ Y,,\ db= –lo log,, (1 – ,1P) = –lo log,, ((!0s’ 0), (109)

If the Arg ( Yl,) decreases in phase margin much more rapidly on one

side of the maximum than on the other and if la rge ga in changes in the

loop are expected, a ga in should be used tha t differ s from the adjustment

men t ioned above in tha t it provides equal per formance (tha t is, the samemaximum M is rea ched) at bot h ext remes of t he gain var ia t ion .

In order t o illust ra te the deta ils of these methods and in terpret a t ion

of t he r esu lt s, two examples \ \ Tille given , u sin g t he loop t ra nsfer fu nct ion s

(1 Oa )



SEC.4.11] DECIBELPHAS E-AN GLE DIAGRAMS 183

and~, = K(z’lp + 1)11 p’(z’,p + l)”

(nob)

with T I = i$j sec and T? = ~]v sec.

The at tenuat ion and phase diagrams for these t ransfer funct ions a re

given in Fig. 4.28. The asymptotes from which the actual a t t enuat ion

curves were plot t ed by the depar ture method (see Sec. 4.10) a re shown

dot ted. The curve for M = +3 db for Y,, [tha t is, plot t ed against Arg

(Y,,)] illust ra tes the use of these curves on this type of diagram. Ithas

+50 -1oo

+40 -120

+313 -140

+20 -160$2

+10-180~

.—

% :,E

0,:-200 m

u:mf

–lo -220

-20 -240

-30 -260

-401 2

-2804 6810 20 40 6080100 200 400 6008000M

w

FIG.4.28.—~ttenua tionandPh asediagramfor Y,, an d Y’~,.been found conven ien t to use the same linear distance for a degree of

angle on th is a t tenuat ion and phase diagram as on the decibel–phase-

angle diagram of Fig. 4.27. A pair of dividers can then be used in t rans-

fer rin g eit her M-cu rves t o t he decibel–log-fr equ en cy dia gr am or a t ra nsfer

locus to the decibel–phase-angle diagram. In either ease the divider s

are set to the distance corresponding to the phase margin a t u for a given

value of decibels on the a t t enuat ion diagram and then used to mark off

th at dista nce a lon g t he cor respon din g decibel lin e on t he decibel–ph ase-

angle diagram.



184 GENERAL I~iL $IGN PR INCIPLES [SEC,4.11

Figure 4.29 shows plots of Y,l and Y~l on the decibel–phase-angle

diagram. The~-parameter values aremarked along these plots.

The decibel scale on the a t tenua t ion plot cor responds to a cor rect

gain set t ing for Yllbut is not cor rect for }’~l. Athirdcurve shows thata

Phase margin in degrees

+32 o

+30

+28

+26

+24

+22

+20

+18+16

+14

+ ]2

+4-%KF?E

-12~

-22

-24

-26

-28

—u \ l \ l \ l \l \ k l\ l\ I\

-8-lo

-14

-16-18

-20

-30-1

Angle in”deg&s ‘- ‘-

FIG. 4.29.—Illustrative plots of Ylland Y’],.

gain lowered by slight ly over 8 db is the best adjustment and gives a

peak of slight ly over 6 db.

The plot for Y{l actually cr osses the – 180° phase-angle line shown on

Fig. 4.26. On Fig. 4.29 it is shown by reflect ing the curve at

Arg (Yj,) = – 180°



SEC.411] DECIBEL-PHAS E-ANGLE DIAGRAMS 185

and plot t ing back across the same set of con tours, with – 170° for – 190°,

– 1500 for – 210°, and so on . The phase margin is then the nega t ive of

t ha t r ea d on t he diagr am , bu t t he decibel sca le applies u nch an ged.

Th e plot s on t he decibel–ph ase-a ngle diagr am a ppr oa ch an a sympt ot eof an in tegra l mult iple of 90° for both low and high decibel values. The

plot of Y~, approaches – 180° at h igh decibel va lues and – 270° at low

decibel va lues. In const ruct ing the plot , use may be made of the

approximate ra te of approach , which is such tha t the depar tu re from a

90n0 asymptote changes by a factor of 2 for a change of 6n db.

It is a genera l ru le tha t a fa ir ly long 6-db/octave sect ion between two

long 12-db/octave sect ions on the a t tenuat ion plot will provide a region

of posit ive phase margin that var ies in exten t direct ly with the length ofthe &db/octave sect ion . In the case of Y,, th is sect ion is about 15 db

long; best servo per formance result s when the un ity gain line crosses the

6-db/octave sect ion about a th ird of its length from its h igh-frequency

end. Sin ce th is t ype of plot is frequent ly obta ined fr om lead or der iva t ive

equaliza t ion , a fur ther discussion of it is given in Sec. 415, where Table

41 gives the maximum phase-angle cont r ibu t ion toward posit ive phase

margin for var ious length s of th e 6-db/octave sect ion .

When th e 6-db/octave sect ion is followed by an 18-db/octave sect ion ,

the phase margin is less than with the 12-clb/octave sect ion , and the gain

must be lower ed, as is illust ra ted in th e case of Y~l. If desir ed, t h e va lu es

of M from the diagram can be plot t ed against o, as is done in Sec. 415,

but usually the shape of th is plot can be seen with sufficient deta il

dir ect ly fr om th e decibel–phase-angle diagram. With a kn owledge fr om

past exper ience of the a t tenuat ion and form of phase diagram requ ired

for sa tisfa ct or y per forma nce, it is possible t o om it a lso t he decibel–ph ase-

angle plot in the ear ly design stages. This plot is genera lly made only

aft er the gain has been determined roughly by inspect ion of the a t tenu~

t lon and phase diagram.

The decibel–phase-angle diagram may be used for funct ions other

than those of the type of Eq. (106) by plot t ing and/or reading reciproca l

values. As examples, we may take

(111)

or

(112)



186 GEN ERAL DES IGN PRIN CIPLES [SEC,412

The reciprocal funct ions a re handled by changing the signs of both the

decibel and phase angle of the funct ion or changing the signs (on the

decibel-phase-angle diagram) of the sca les tha t apply. On Fig. 426

the plot of 1/Y. represen t s this quant ity with the signs cm the loop-gainand phase-angle sca les as labeled. If the signs of these two scales a re

reversed, the plot is tha t for Y.. To read 1 + Yo from the~- and #-con-

tours, the signs on the sca les must be rever sed.

It is of in terest to study the case where the plot on the decibel–phase-

margin diagram passes close to or through a O-db and O-phase-margin

poin t . The rela t ion of Eq. (112) is used as an example, with the plots of

– 270°

-180°0

-90°

FIG.430.—Nyquist diagramfor Y. and Yb.

l/Ya and 1/ yb on Fig. 4.26 rep-

resen t ing a t ransfer locus for twodifferen t ga in adjustments.

With IYal increasing, Ar g (1 + Ya)

changes rapidly from —270° to

– 90° and, if the gain of Ya is r~

duced by 1 db, will make the

cha nge inst an ta neou sly, sin ce t he

n ew plot passes t hrough t he – 180°

phase angle and O-db poin t . Thisis seen to be the actua l case by in-

spect ion of Fig. 4.30. Fur ther re-

duct ion of the gain by 1 db leads

to the plot of l/Yb in ~lg. 426.

Here Arg (1 + Yb) appears t o

jump from – 360° to 0° at the O-phase-margin poin t . This does not

actua lly happen , as may be seen from Fig. 4.30.

4.12. Mult iple-loop Syst ems.—The design of mult iple-loop systemsis a very impor tant topic, and a la ter sect ion will ca r ry through a deta iled

design of an actua l system, showing the advantages of equaliza t ion by

means of subsidiary loops. In this sect ion we shall consider a simple

double-loop system in order to in t roduce the techniques tha t will be

needed in a la ter analysis and in order t o illust ra te fu r ther the ideas

developed in t he pr ecedin g sect ions. The equat ions for a genera l two-

loop system are given in Sec. 41, Eqs. (13) to (15), and the schemat ic

diagram is given in Fig. 4“2. As a simple example let us takeKVT

‘$’ = Tp(z’p +T)

and

‘2’= (Tp + l;~;o!;;p + 1)”

(113)

(114)

The transfer funct ion of the principa l loop when the subsidiary loop is



SEC.4.121 MULT IPLE-LOOP S YS TEMS 187

330

10 270

0 D K,T=K22=0b210

% .<- !,~-lo -- -..c -.. 150g.- E.+ . ~,,, ~,> Bj

\ \

K22=IIb ~& 7 g

s / ’ #d -20 90 :

Gc #,-. E

-30‘,

,..,\ 30

>.......-----/

#

. . 0‘.\

-40B

-30

-5001

-901.0 10 ,!,T 100 1002

FIG.4.31a .—Phase and a t t enua t ion cha ract er is tics for a doub le-loop sys tem with a s tablesubsidiary looP.

+20

+10

o

%-,0.s.-;]-20

-30

-40 -

-5 00.1 1.0 10 1

UT

-

330

270

-2’0

a+90:::

-..E---- --- +30i) K2Z=1db

o

-30B

c -90) moo

FIG.4.31b.—Phase a nd a tt en ua tion ch ara ct er ist ics for a ~- ,>h le-loop syst em wit h a nunstable subsidiar} loop.




open , Y~l, represen ts asimple amplifier and motor combinat ion , and the

t ra nsfer fu nct ion of t he subsidiary loop r epr esen ts a t ach omet er feedback

circu it with a ra ther complex equalizing network. The physical rea liza-

t ion of this network (w-hich , incidenta lly, would require act ive imped-

ances) is not discussed here, since it would take us t oo fa r afield from

the purposes of the present discussion. The curves labeled .4 in Fig.

4.31a and b are the asymptot ic and actual a t tenuat ion character ist ics

cor respondin g t o Y~l, drawn with KOT = 1. This simple character ist ic

falls a t – 6 db/octave from low frequencies up to UT = 1 and then cuts

off asymptot ica lly to —12 db/octave. The phase of this funct ion is

not plot t ed but is clear ly asymptot ic to —90° at low frequencies and

changes rapidly in the vicin ity of WT = 1 to become asymptot ic t o

– 180° at high frequ en cies.

The a t tenuat ion character ist ic for t he subsidiary-loop t ransfer func-

t ion is propor t iona l to T3p3 at low frequencies and hence r ises a t 18

db/oct ave t owa rd wT = 1. At high frequencies Y,z cuts off a t a ra t e

of —6 db/octave, the asymptote star t ing at UT = 4. This cu rve and

also the actual a t tenua t ion curve, easily plot t ed from the asymptot ic

character ist ics by the methods given in Sec. 4.10, a re labeled B in Fig.

4.31a and b. The phase of Y,,, which again is easily computed using

the methods already refer r ed to, is labeled C in the same figure. Thesu bsidia ry-loop t ra nsmission ch ar act er ist ics, we see, a re t hose of a t ypica l

ba ndpa ss filt er , wh ich , for KZZ = 1, has asymptot ica lly unity gain in the

pass band tha t extends from WT = 1 to UT = 4.

We must now determine what values of subsidiary-loop gain K,, will

r esu lt in a complete system that is suitably stable. The equa tion

(115)

tells us that we are in terested in the funct ion 1 + YZ1. The behavior of

this funct ion is most easily invest igated by replot t ing th e at tenua t ion and

phase of I’,, on a gain–phase-angle diagram as descr ibed in Sec. 411.

It was sho\ vn that by plot t ing a funct ion l/ F on t he specia l coor din at e

system shown in Fig. 4.26, the at tenuat ion and phase of 1/(1 + F) are

immediately determined. In Fig. 4.32 the reciproca l of Y,, has been

plot ted on a decibel–phase-angle diagram by reading off cor responding

va lues of a t tenua t ion and phase from the YZZcurves in Figs. 4.31a and b

and reversing their signs. The gr id in Fig. 4.32 has been reflected about

both the 0° and – 180° phase-angle lines in order to make the single-

sect ion gr id serve in place of an extended diagram of the kind shown in

Fig. 4.26. The upper curve in Fig. 432 is for KZZ = 11 db, while the

lower curve is for K22 = 31 db, a 20-db increase in loop gain as compared

with the upper cu rve. The two curves are, of course, ident ica l in shape.



SEC.4.12] MULTIPLE-LOOP SYS TEMS 189

As men tion ed in Sec. 4.11, caut ion must be exer cised wh en r eadin g va lues

of the magnitude and phase of (1 + Y22)–I from the labeled curves in

any such plot in which the ext r eme phase change in Y22 is grea t er than

Phase angle in degrees

–180 –200 –220 –240 –260 80 60 40 20 (1

-0,5

-1

-2

-4

-5

-6

–9

-12

-1 0-18

–20

-24 -24

-28

I I I I u II II II 1[ II II II II II II II II Ill]-180 -160 -140 -120 -100 -80 -60 -40 -20 -50

FIG.4.32,—Gain—phase-angle diagram for subsidiary looP.

180°. ‘rhe observa t ion tha t the phase of 1 + YZZmust be con t inuous

as long as 1 + YZZis cont inuous is usua lly sufficient to resolve any ques-

t ion as to what phase should be assigned any given va lue of 1 + YZZ. A

crude sketch of the N’yquist diagram for YZZcan always be made with no

difficulty, eit her lJ Y inspect ion of Y12 itself or from the }’ZZ curves of




F ig. 4“31a a nd b; such a diagram will a lways show clear ly the behavior of

the phase of 1 + Y2z. The Nyquist diagram for Y*Z, drawn for KZZ = 11

db, is given in Fig. 4.33. Let us now consider the asymptot ic behavior

of 1 + Y22 when the gain K22 is set a t a level of 11 db. When Yjz is

small compared with 1, then 1 + YZZ is essent ia lly unity; when YZZis

la rge, 1 + Y2Z is essent ia lly just Yjz. These considera t ions lead us to

the asymptot ic curve for 1 + Yz2 which is labeled D in Fig. 4.31a and b.

The actual a t tenuat ion cha racter ist ic can be r ead direct ly fr om th e upper

FIG. 4:33.-Nyqu is t p lot for subs id ia ry loop , K,, = 11 db.

curve of Fig. 4.32 and is plot t ed as Curve E in Fig. 4.3 la . The phase

cor responding to the asymptot ic a t t enuat ion curve D is the dashed

curve F, while the actua l phase, cor responding to E (taken from Fig.

432), is the solid cu rve G. We not ice, fir st , tha t the asymptot ic and

actual a t tenuat ion curves differ quite considerably from each other ;

the phase character ist ics a lso differ , but not as markedly as the a t tenu-

a t ion character ist ics. It \ vill be seen la ter tha t use of the asymptot ic

a t t enuat ion curve for 1 + Y2Z and the cor responding phase in place of

t he actua l cha racter ist ic would not appreciably affect th e fina l resu lts of

the ana lys is .

Inspect ion of the form of Yza shows that it has no poles in the r igh t

half of the p-plane, and, therefore, neither does 1 + Yzt . Applica t ion of

the usual N-yquist t est to the t ransfer locus of Fig. 4.33 then shows that

1 + Y,l has no zeros or poles in the r igh t half plane; we conclude that

t he subsidiary loop is stable and, fur t hermore, tha t t he usllal simple form



SEC. 412] MULT IPLE-LOOP S Y SZ’EMS

of the Nyquist cr it er ion will su ffice to t est t he

system.

191

stability of th e over -a ll

F rom Eq. (1 15) we see tha t the a t tenua t ion cha racter ist ic for the over -

all loop t ransfer funct ion Y1l is found simply by subt ract ing the a t tenu-a t ion (in decibels) and phase of 1 + Y!Z from the a t tenua t ion and phase

of Y~l, The result s of th is procedure a re shown in Fig. 4.34, where the

Curves A are the asymptot ic and actual cha racter ist ics for the case

K,, = 11 db, KtZ’ = O db. The actua l character ist ic was computed

using the Curve E from Fig. 4.3 la , not direct ly from the asymptote,

10 -40-520

-10. -80-480 ;u~

D

/:0 I \ \ .l Kcy ~.w..

II - 120 - 440 .5

\----- D I \

-..-160-400j

D . . . .

‘-50 k

-. \

-70E

w -200-3603

-90-. \

/-240-320

–11001

-2801 10 100 1000

UT

FIG.4.34.—Over-allpha seand at tenu at ioncha ra cteristicsor casesof stableand un stablesubsidiar yloops .

a lthough the la t ter procedu re would be sa t isfactory in this par t icu la r

case. The phase of Yll is given as Curve B in Fig. 4.34, on ce again com-

puted from the curves of F ig. 4.31cc, a lthough a direct computa t ion

from the a t tenua t ion asymptote would be permissible. If such a ca lcu-

la t ion were ca r r ied ou t , we would find tha t the phase-angle maximum

tha t occurs nea r UT = 10 on Curve B would be shifted to u!l’ = 8and would be about – 132° instead of – 145°. It will be seen tha t th is

change would have only a sligh t effect on the va lue of loop gain Km tha t

is to be select ed. Examina t ion of the over-a ll character ist ics shows tha t

if we select K,T equa l to 44 db, then feedback cu toff (the poin t a t which

the a t tenua t ion cha racter ist ic crosses the O-db line) will occu r approxi-

mately in the cen ter of a —6-db/octave slope and near the phase-angle

maximum of – 145°) cor responding to a phase margin of +45°. Accord-

ing to the discussion given in the preceding sect ion th is represen ts a



192 GENERAL DE SIGA’ PR INCIPLES [SEC.412

satisfactory adjustment of gain, and we see that we can obta in a dimen-

sionless velocit y-er ror con st an t (K~Z’) of 44 db, or 158, a gr ea t impr ove-

ment over the value of 1 or 2 obtainable with the simple unequalized

single-loop system. A good approximat ion to the speed of response is

easily obta ined by forming

%/$,. F rom the equa t ion

the asymptot ic curve for the magnitude of

00—

(116)

it is seen that when @o/e is la rge, the magnitude of %/(?r is simply 1, or

O db, and when L%/c is small, the magnitude of 00/81 is very near ly just

the magnitude of 00/c; thus the desired asymptote for the magnitude of

0~/0~ is flat (zero slope) from zero frequency out to the frequency of feed-

back cutoff ancl is ident ica l with the asymptote of RO/C at all h igher fre-

quencies. Having set ~u!i” = 4+ clbj we find that the – 12-db/octave

cutoff asymptote of the OCI/@lcharacter ist ic in ter sects t he O-db line at

tJ CT = 13. According to 12q. (43), Sec. 42, the buildup t ime is given

closely by

1

‘*=m.=:=iw(117)

In the same manner the buildup t ime of the unequalized single-loop

system is found to be ,b = rZ’. Thus we have mater ia lly improved the

speed of r espon se of t he syst em by a ddit ion of t he su bsidia ry loop.

.4 rough sketch of the Nyquist diagram of the over-a ll system is

easily drawn from Fig. 431 and is shown in Fig. 4.35a. If this cu rve

is imagined to be completed for the complete range of rea l frequencies,— co to + co, by adding the complex conjugate of the curve shown,

the usual NTquist test will show that 1 + YII has no zeros in the r ight

half plane; in view of ou r ear lier discussion, t his est ablish es t he stabilit y

of the over -a ll system. This drawing is given here pr incipally for use

in a la ter discussion. If a deta iled picture of the frequency response

of the system were desired, we could plot the data of Fig. 4.34 on a

gain–phase-angle diagram. This is not actually necessary, however ,

since we have a lready obta ined the asymptot ic 00/Or curves, and we

know that with the system adjusted to give a 45° phase margin in the

vicinity of feedback cutoff, the resonance curve will have a peak of

cer ta in ly not more than 6 db.

We n ow consider the effects on the over-a ll system of changes in va lue

of the subsidiary loop gain KZZ. Refer r ing to Fig. 4.32, we see that

increasing KZZ over 11 db simply slides the upper cu rve down on the

diagram. In par t icu lar , if KZZ is increased to 21 db, the cu rve passes



SEC. 412] h f 7J LTIPLE-LOOP SYSTEMS 193

th rough the singular poin t of the diagram, cor responding to an infin ite

a t tenuat ion value of I + Y,,. This ~itua t ion cor responds to increasing

the radial sca le factor in Fig. 4.33 until the t ransfer locus passes exact ly

th rough the cr it ica l poin t – 1 + jO. If XZZ is increased st ill more, sayto 31 db, we ar r ive at the lower curve in Fig. 432, and we find the

behavior of 1 + Y~~for th is case in the same way that we determined the

behavior of 1 + Y,, with K,, = 11 db. The solid curve D in Fig.

431b is the asymptot ic behavior of the at tenua t ion of 1 + Y,Z with

K,, = 31 db, and the dashed cur \ -e E is the actual a t t enua t ion ehar -

I Y],- plane

~IG. ‘i.35a.-Ovrr-all Nyqu ist d ia gr am for FIG.4.35h.—Over-al l Nyqui.s t diagram forca se of a s table subs id ia ry loop . ca se of an unst able subs id ia ry loop .

act er ist ic. The actual phase character ist ic is shown as Curve F; it

was plot ted direct ly from Fig. 4.32. We not ice immedia tely tha t the

increase of KZZfr om 11 to 31 db has radica lly changed the behavior of the

pha se char acter ist ic while t he beha vior of t he a ttenuat ion char acter ist ic

is rela t ively unchanged. The phase character ist ic is quite eviden t ly no

longer the minimum phase shift a ssocia ted with the given at tenuat ion

characteristic. This situa t ion is fu rt her clar ified by con sider at ion of th e

Nyquist diagram of Y22 (Fig. 4.33). The diagram as drawn is forK2* = 11 db; by allowing the t ip of the vector 1 + Yaz, drawn from the

poin t – 1 + jO to the curve, to t raverse the curve as o var ies from O to

co, we easily ver ify the genera l behavior of the phase character ist ic G

in Fig. 4.31a. Increasing KZZ to 31 db changes the scale factor in Fig.

4.33 so that the cr it ica l poin t – 1 + jO is enclosed by the t ransfer locus,

we see tha t the behavior of the phase of the vector 1 + Yza is indeed

changed and is as given by Curve F in Fig. 4.31b. We now apply the

Nyquist stability cr iter ion to this subsidiary loop. We easily see tha t



194 GEN ERAL DES IGN PRIN CIPLES [SEC, 412

the funct ion 1 + Y2Zhas no poles in the r ight half plane. The curve of

Fig. 433 (with the changed radia l sca le factor ) is com pleted by adding the

complex conjuga te curve, cor responding to frequencies from —~ to O,

and the Nyquist t est is applied. The vector 1 + Y2Z undergoes two

complete revolu t ions in the clockwise sense as the t ip of the vect or t r aces

the curve from a = – cc to ~ = + cc ; we conclude that 1 + YZZhas two

zeros in the r igh t ha lf of the p-plane, or , in other words, tha t the sub-

sidiary loop is unstable. This br ings ou t clear ly the nonminimum-phase

cha racter of 1 + YZZwhen KZZis large and shows why its phase chara cter -

ist ic is so r adica lly a lt er ed wh en KZZ is in cr ea sed.

ATOWet us invest iga te what happens to the over -a ll system when the

gain of the subsidiary loop is increased enough to make it unstable. Atfir st sight , a t least , it appears in tuit ively obvious tha t the complete

system will become unstable, but closer analysis will show that th is is

another of many situa t ions in which intu it ion fails one. Th e at tenua -

t ion and phase funct ions for the over -a ll systems are obta ined from the

curves of Fig. 4.31 b in the same way as in the ear lier case and are shown in

Fig. 4.34, C designat ing the a t tenuat ion curves and D the phase curve.

The phase-shift cu rve has been reflected about the – 280” line, since the

phase shift exceeds – 450° a t low frequencies and is asymptot ic to –4500a t ext r emely low frequencies. Select ing a value of Km equal to 86 db,

we see tha t the phase shift a t feedback cutoff will be —130°, or the phase

margin will be 50°. So far , we apparent ly have a sa t isfactory system.

Now let us examine the h-yquist diagram, a rough sketch of which is

given in Fig. 135b. Again, ~ve must imagine this diagram to be com-

pleted by adding the complex conjugate curve and a large semicircle in

the r ight half plane join ing the two zero frequency por t ions of the curve.

Applica t ion of the Nyquist test shows tha t the vector 1 + Y,, undergoes

two complete revolu t ions in the counterclockwise sense as the en t ire

cu rve is t ra ver sed. Thus, the number of poles of 1 + Yll in the r igh t

half plane exceeds the number of zeros in tha t region by two. Since

1+ Y,l =1+*2 (118)

and an ear lier Nyquist test of the subsidiary loop has disclosed two

zeros of 1 + Y~Zin the r igh t ha lf plane, we see tha t 1 + Yl: has two poles

in this region ; it follows tha t 1 + Yll has no zeros in the cr it ical r egion

and tha t the over -a ll system is stable. The selected gain level of 86 db

gives a velocit y-er ror con st an t of a ppr oximat e y 20,000 T-’ and a buildup

t ime of 7~ = (7r/140) T , r epr esen tin g su bst an tia l impr ovemen ts in syst em

performance.

The reader should apprecia te tha t a designer with a cer ta in back-

ground of exper ience with these methods could ca r ry through the above



SEC. 4.13] OTHER TYPES OF TRANSFER LOCI 195

design procedure a lmost completely, using only the asymptot ic a t tenu-

a t ion character ist ics tha t can be const ru t t ed in a mat ter of minutes.

The deta iled curves have been presen ted in an effor t t o supply the reader

with some of the requisite insight in to the design procedure.4.13. Ot her Types of Transfer Loci. —0ccasionally t her e ar ise specia l

problems in servo design that a re best t r ea ted by procedures other than

those already presen ted. The vast major ity of problems, however , a re

readily handled using the “standard” techniques. The ingenious

designer will cont inue t o develop new procedures ad infin itum to suit his

own out -of-the-ordina ry problems, and it would be fu t ile t o a t tempt to

give any comprehensive discussion here of all the specia l methods tha t

have been devised. A few, however , a re perhaps wor th ment ion ing.Severa l wr it ers’ have proposed the use of the reciprocal of the usual

Nyquist diagram for the t rea tmen t of mult iple-loop systems and systems

in which there are elements in the feedback path or paths such that a

t rue er ror signal does not actua lly exist in the systcm. The advantage

of this recipr oca l diagram in discussing mult iple-loop systems is readily

apprecia ted by writ ing the equat ion for c/@o for a double-loop system.

Using the nota t ion of Sec. 41, we have

(119)

Thus the c/00 diagram can be const ructed by a simple vector addit ion of

two preliminary diagrams ra ther than by a process of ‘( mult iplying”

two diagrams together , as with the usual Xyquist diagram. The in ter -

ested reader will find ample discussion of these ideas in the references

a lr eady cit ed .

The drawing of Nyquist diagrams is often complica ted by the ext r eme

range of values of the radial coordina te tha t must be plot t ed. The

example in the previous sect ion serves as a good illust ra t ion of this dif-

ficu lty. Refer r ing to Fig. 435b, a simple calcula t ion will show that if

the sketch wer e actua lly drawn to scale, the following values of radius

would have t o be plot t ed:

At the point A, r = approximately 100.

At the point B, r = approxima t ely 5000.

At the poin t C, r = approximately 115,000.

At the point D, r = approximately 150,000.

At feedback cu toff, r = 1.

Th e obviou s difficu lt y is usua lly su rmou nt ed by plot tin g va rious por tion s

of t he cu rve wit h cliffer en t sca le fa ct or s. This is a sa tisfa ct or y solu tion ,

1 H. T. Marcy, “ Para llel Cir cu it s in Servomechan isms ,” Trans. A IEE, 66, 521

(1946), H. Harr is, J r ., “Th e F requ en cy Respon se of Au toma tic Con tr ol Systems,”

T ran s. AZEE, 65, 539 (1946).




especia lly since on e is usually in ter ested in t he det ailed sha pe of t he locus

on ly inthe vicinity of the feedback cu toff frequency. A deta iled drawing

can be made of this por t ion of the 10CUS,while crude sketches of the

r emain in g por tion will su ffice. Some workers, however , prefer to usea

logarithmic radial scale, effect ively giving a polar form of the decibel-

ph ase-a ngle diagr am a lr eady discussed in this ch apter .

Diagrams similar t o Nyquist diagrams can also be used in the t r ea t -

ment of ser vo problems involving pulsed or discon t inuous data . These

problems are discussed in Chap. 5.

EQUALIZATION OF SERVO LOOPS

4.14. Gen er al Discussion of Equaliza tion. -Equa liza tion cir cuit s a nd

networks are employed in servo circu its in order to obta in a desiredbeh avior for t he complet e system . In the USUaldesign some of the par ts

of the system are selected with an eye to availability, t est , ease of mainte-

nance, or other reasons. For example, a synchro data t ransmission is

quite often specified because the completed system may have to t ie in to

a shipboard free-con t rol system !vhere the synchro system has a lready

been st anda rdized. The po~ver-supply fr equency is often specified m,

for example, 60 cps. If in addit ion the use of 1- and 36-speed ordnance

syn ch ros is specified, t he er r or sign al will con sist of a 60-cycle volt age ~vit han er ror gain for small er rors of 1 volt per degree on the 36-speed shaft

or 36 volts per degree on the l-speed shaft . Ot her con sider at ion s may

have dicta ted the choice of an amplidyne and d-c motor as the power

dr ive elemen t. Th is la st ch oice l~ill t hen r equ ir e a scr voa rn plificr ca pa ble

of accept ing a 60-cycle er ror signal and deli~er ing :~ d -c cur ren t to the

con tr ol field of t he amplidyne; th is deman ds t he usc uI’a pha se-scnsit ire

det ect or in t h e servoa rnplifier .

The per t inent constant s prcs~nt~d to the servo dwixncr }Vill then be

app roxima tely the following:

1. Combined motor -amplidyne t imc ctmst a nt , 0.25 scc = T~.

2. Effect ive t ime constant of the phase-sensit ive detector , 0.02

sec = T ,.

3. Amplidyne quadra ture-field t ime constan t , 0.02 sw = 1’1.

Th e loop t ra nsfer fu nct ion ~Vit h a fl~t fr c(l\ lcllcy-r cspor lse amplifier

may be writ t en

!!?= .—-––—- --- --- -—---.6 p(Tnp + I)(T ,P + I)(TWP +- ])’

(120)

where K.= “KY” = wlocit y-er ror con st an t, dct g/sec pcr deg,

K, = er r or -measu r ing elemen t sen sit ivit y, volt s, ideg,

K..i = amplifier ga in , ma /volt ,



197EC. 4.15] LEA D OR DERIVATIVE CONTROL

Kc = amplidyne open -cir cu it ga in , volt s/ma,

Km = motor speed-voltage ra t io, deg/sec per volt ,

l/N = motor -load gea r rat io, deg/deg.

Thk t ransfer funct ion has a slope of – 6 db/octave from zero to w = I/T_

and – 12 db/octave to a = l/ T , = 1/ Tq and then decreases a t 24 db/

octave. It has a 180° lag and thus zero phase margin at

2.5u = — = l,6cps.

T .

A value of Kv = 6.25/ T in = 25 see-’ will make the system unstable; a

va lue of Kt i = 0.75/T~ = 3 see–l is requ ired in order tha t the phase

margin be 45” a t the feedback-cutoff frequency u, = 0.75/ T., = 0.48 cps.

The buildup t ime ~~ill then be approximately 7; = 1/ (2j.) = 1 sec. This

system would have a maximum er ror of 10° when the input is a 30 °-ampli-

t ude 6-see sin e ~~ave and wou ld ordin ar ily not be a ccept able.

Hysteresis in the amplidyne magnet ic circuit may also cause con-

siderable er ror in a system \ vith as small a velocity-er ror constant as

the above. An amplidyne may commonly have a hysteresis loop as

wide as + E~h = t 20 volt s. The result ing hysteresis standoff er ror ,

.s~= K., Eah /NK., with K“, = 85° per sec/’volt and N = 300, k then

q = 2°.

Three more or less genera l methods may he used to modify the

above system in order to improve its per formance a t either high or 10J V

frequencies.

1.

2.

3.

The propor t iona l in tegra l method is applicable ~vhen a buildup

t ime and cutoff frequency of the same order of m~gnitude as tha t<

of t he simple syst em a re a ccept :ll)lc or desir able. In th is m et hod,the loop t ransfer funct ion is lr ft substant ia lly unchanged for fre-

quencies above one-four th the frcq(lency of the – 6 to – 12-db/

oct ave transit ion (tha t is, abovr a poin t 2 oct javcs below l/Tn,) and

its magnitude is in cr eased in th e loirer freqllcn cy r ange.

Lead or der iv~t ive equa liza t ion is used to improve the system per -

forma nce at all fr equ en cies; in par ticu lt ir , t he fccdba .ck-cu toff fr e-

qu en cy is increased tvith a cor respon din g r edu ct ion in th e buildup

time.One or more subsidiary 100PS may be int roduced in order to.improve t h e ser vo per formance. ‘l’h is con st it ut es a ver y poiver fu l

method of equa lizat ion and decreases the effect s of var ia t ions in

some of t he elemen ts.

4.15. Lead or Der iva t ive Con tr ol. —Lead or der iva t ive equ aliza t ion is

u sed t o r aise t he feedba ck -cu toff fr equ en cy. It will, in gen er al, in cr ea se

the velocity-er ror constant while keeping a sa t isfactory stability or




phase margin in the region of cutoff. Th e feedback t r ansfer funct ion of

the simple servo ment ioned in Chap. 1 may be writ t en

eo=~ I

E p ‘fm$l + 1-(121)

This t r an sfer fun ct ion has an asymptot ic character ist ic of – 6 db/octave

for u < 1/T- and – 12 db/octave for 1/ Tm <0. The phase angle is

– 135” at a = 1/ Tm , decreasing to – 153.5° a t u = 2/T~. For KUTn = 1

the heigh t of the resonance peak will be + 1.25 db (1.15 ra t io) and the

feedback-cutoff fr equ en cy will be w = 0.78/ Tm . For &7’~ = 2(+6 db)

the resonance peak will be 3.6 db with w = 1.2.5/ T~, and for

K,l’m = 4 (+12 db)

the resonance peak will be +6.3 db with o. = 1.75/ Tm . Thus the usual

stability requ irement of a resonance peak between +3 and +6 db would

a llow a K. between 2/ 1“~ and 4/ Tm . Assuming T- = 0.25 see, t h is wou ld

cor respond t o Ku between 8 and 16 see–’.

.4ddlng either an a-c or d-c propor t iona l-der iva t ive equalizer of the

type discussed in Chap. 3 makes the feedback t ransfer funct ion

OO Kc, 1 G,(T ,p + 1)— .— (122a)e p Z’wp+l GOT,p+l’

eo=~ 1 Tdp + 1—e

(122b)P T-P + 1 GoT@ + 1’

where Td = der iva t ive t ime constant ,

G, = d-c or ca rr ier ga in (G, < 1),

Ku = KU,GO = velocity-er ror const an t .

The complete study of this t r ansfer funct ion involves the two parame-t er s T,j/ Tm and Go as well as the velocity-er ror constant Ko. It will be

most conven ient to study first the case where Td/T~ <<1. With this

assumpt ion, Eq. (122) can be approximated as

.9. KuTm T~p + 1— -—-—c (T~p)2 GOTdp + 1.

(123)

Th e fir st fa ct or in Eq. (123) pr odu ces a ph ase la g of 180” at all fr equ en cies;

thus the propor t iona l-der iva t ive equa lizer must supply all of the leadrequ ired to give the desired phase margin near feedback cutoff. Inspec-

t ion of the decibel–phase-margin con tour diagram (Fig. 4.27) shows

that the over-a ll fr equency response is 16’o/6,~= +3 db and the loop

phase margin is +45° where the loop gain is +3 db, whereas ]00/0,] = +6

db and the loop phase margin is +30° when the loop gain is + 1.3 db.

The propor t ional-der iva tive equalizer h as a n a sympt ot ic cha ra ct er ist ic

of (Go).~ up to u = l/T~, increases at 6 db/octave to u = 1/GOT~, and is



SEC. 415] LEAD OR DERIVATIVE CONTROL 199

then constan t a t O db for h igher frequencies. It s phase angle is zero for ,

small O, increasing to a maximum at the geomet r ic mean of the two.——above frequencies, w~ = l/(T~ v’Go), and then decreasing to zero for

h igher frequencies. The maximum phase angle is easily shown to be

& = ~ — 2tan–1 &;2

(124)

solvin g for Go, we fin d

()GO=tana~–~- (125)

The gain a t the geomet r ic mean frequency is /G,. Table 4. I gives a

few cor responding values of G, and ~~. .\ n equa lizer with G, = – 15.31

TABLE$i.1.-}f.aIMuM PHASEANGLEFORPROIWRTIONAkDERIVATIVEkjIJ;IJZER

75° 60° I 45° 30° I 15. I 7,5.

(:0;,, –35.22 –22.88 –15.310 –9 .5400 –4.600 –2.2801

G,57.70 13.93 5.827 3.000 1.698 1,300

db will ther efor e have a resonance peak of +3 db when K“ !l’m is ad jus ted

to put the maximum phase margin a t a loop gain of +3 db. When this

is th e case,K. Z’. 1

()

l’.— = 1.414 = (+3 db),

—’&o(126)

T, &or

(128)

F igu re 4.36 is a decibel-log-fr equ en cy plot of th is syst em for TJ T , = 10.

Curve a is the a t tenua t ion—log-frequency plot . The lower phase-angle

cu rve c is drawn from th e approximate Eq. (123); t h ere is rela t ively lit t le

difference between curve c and the exact phase-angle curve b in regions

nea r the phase-margin maximum for a va lue of TJ Td = 10. The main

differ en ce is a slight ly lower maximum phase margin in the approximate

curve. To obtain the desired phase margin of 45° it would suffice to use

a lead equalizer with GOso ch osen tha t th e appr oximately compu ted phase

margin is 45° — tan-’ Td @o/T~ instead of 45”. To eva lua te th is

expression the approximate va lue GO = —15.31 db may be used; one

finds tha t the requ ired approximate maximum lead is 42.5°. The cor -

~ct ion of Go for th is small change in maximum phase margin is ordinar ily

not necessa ry.



200 GE.\ ’ERAL I) ILS’IGN l’RINCfF’LES [SEC.415

l-sing ~q. (127), onecomp~ltes aval~~[:ofh”.1’~ = +50.7 db for the

velflcitY-er r or con st an t. If 1“~ = ~sec, t hiscor respondst o

and I’d = 0,025sec. ‘~henotch lr iflth (see Sec. 31-1) of th is leudcqual-

izer for a -c use is then 40/27r = +6.4 cps. It should be poin ted ou t ,

h owe~r er , tha t apar allel-T leadequ alizcr use{i~vith a60-cps ca rr ier \ vill

yield a lower~alue of G, than is requiredin th is example and will thus

FIG.

+5 –90

o –loo

-5 –110

q

-lo –120

-153

–130 ~

–140 :T ‘“20W .-

~Z–25 -150~—

S–30 –160 ~m.

.E~ -35 -170 :

~ -40 –180 ~

-45j

-50

–55

-60

-650.50.6 1 2 4 6 10 20 40 60 100 200u T~

4.36.—Lead equa liza t ion applied to a s inglet ir ne-lag servomotor . (a ) At t enua t ion ;(b) ph a se angle; (c) a ppr oxima te phweangleby Eq. (123).

provide ala rger phase margin than isnecessary for the +3-db resonance

peak.

F igure 437 is a decibel–phase-margin diagram for the system with

K. and Td/!f~ selected as above. F requency parameter values havebeen marked on the curve. The frequency response 10./0,1 can then be

plot ted fr om this cu rve by obser ving t he fr equen cies a t which t he decibel–

phase-margin curve crosses the respect ive resonance con tour s. The

result ing curve is shown in Fig. 438. The asymptot ic frequency r esponse

is, of cour se, constan t for frequencies less than feedback cu toff and

follows t he 10o/c[ a sympt ot e for fr equ en cies a bove feedba ck cu toff.

Use of the lead equalizer has thus result ed in an increase of KaT~ by

50.7 – 3.0 = 47.7 db (increase of K, from 5.7 see-’ to 340 see-’). The



SEC. 415] LEAD OR DERIVATIVE CONTROL 201

feedback-cutoff frequency has been raised from u, = l/T~ to a. = 32/T~

(w = 128 radians/see = 20 cps).

It would appear that a h igher value of Ku could be obta ined through

the use of a smaller value of Td, with a cor responding fu r ther increase in

the cutoff frequency. In the usual case, however , Eq. (121) will require

modifica t ion because of the presence of other t ime constants which will

begin to make their influence felt as the cutoff frequency is ra ised. In

the use of a 60-cps car r ier frequency there appear to be limita t ions

FIG.

o 10 20 30 40 50 60 70 ~ 90

Lwp phase margm in degrees

4.37.—Lead equa liza tion a pp lied t o a Singlet ime.la g s er vomotor c K, = 34o See-L;T~/ T m = 0.1.

pla ced on t he u se of cu toff fr equ en cies t ha t a ppr oa ch t he ca rr ier fr equ en cy

or , in some instances, even one-half of the car r ier frequency. Another

sou rce of difficu lty ar ises when a phase-sensit ive detector is used. A

ripple filter is then required t o decrease th e r ipple voltage, in or der not t oover load the output stages of the power amplifier . This r ipple filter

must have appreciable at tenuat ior i a t twice the car r ier frequency in a

fu ll-wave rect ifier and will thus cont r ibu te appreciable phase shift a t

fr equ en cies a bove on e-h alf of t he ca rr ier fr equ en cy.

Inspect ion of Fig. 4.37 shows tha t an increase in loop gain of approxi-

mately 12 db will increase the resonance peak to +6 db, and a reduct ion



202 GEN ERAL DES IGN PRIN CIPLES [sm . 415

in loop ga in of approximately 27 db will increase the resonance peak to

5.4 db—the la rgest resonance peak that can be obta ined by reducing the

loop ga in . Th ese two qua nt it ies give t he m agn it ude of t he amplifica t ion

tolerances tha t should be placed on the system in order to mainta insat isf a ctor y stability if the other quant it ies a re constan t . The velocitY-

er ror constan t would, of course, change by the same factor , and over -a ll

syst em specifica t ions may not permit a reduct ion in K. of more than 12

+55

+50

+45

+40

+35

+30

~ +25

.=+20

.:0+15

+10

+5

o

-5

-10

–150.612461020 40 60 100 200

UJTm

FIG.4.38.—Frequency response for a s ingle -t ime-lag servomotor with lead equa liza t ion ,Ku = 340 see-l; TJT~ = 0.1.

db. A check can be made on the effect of a change in T~ on t he syst em

stability by use of Figs. 4.36 and 4-37. Change in T~ is pract ica lly equiva-

len t to a change in ga in , since the T~–phase-margin cont r ibu t ion near

feedback cutoff is small. An increase of Tm by a factor of 2 wou ld require

a factor of 4 (+12 db) increase in K. t o k eep t he ph ase-ma rgin maximum

in t he gain r egion for maximum st abilit y sin ce it wou ld in cr ea se t he len gt h

of the —12-db/octave sect ion by 1 octave. Allowing an increase of T.by a factor of 2 without changing the velocity-er ror constan t would

cor respond approximately to a reduct ion in ga in of 12 db and wouJ d

increase the resonance peak to +3.9 db.

No a t t empt will be made a t th is t ime to make a complete toler ance

discussion of the above circu it . The following equat ions for the lead

equa lizer will be usefu l in a fu r ther study:



SEC. 416]

where

4.16. In t egr al

I.\ rT EGRAL EQUALIZ A IJ ’IO,V

E.– Go

T~p + 1

E; – G,T,p + 1’

h = ~ – 2 ta n-’ v@,

G, = -Q%,,

T, = R , C,,

6 Ta~’ + %1

T~ = R,(131)

6G0 _ R, 6RI R, bRZ

Go – R , + R; RI RI+ R2E’(132)

1 6G0

6+” = – 2 Cos 4“ G“(133)

Equaliza t ion . —Integra l equaliza tion is used in a ser vo

203

(129)

(130)

loop in order t o increase the loop

may be used in conjunct ion \ r ith

lea d equ aliza tion in a loop.

For the moment ~re shall as-

sume tha t a given t r ansfer func-t ion has been selected and tha t it

is th en desir ed t o in crea se t he loop

gain at lo\ v frequencies, Onc ex-

ample of this type ar ises in con-

n ect ion ~t -it h a s er vomotor h avin g

a qua dra tic lag fact or ,

(134)

This basic loop equat ion cou ld

a r ise from a tachometer -equa lized

servomotor of the t ype discussed

in the next sect ion . An equat ion

of approximately this form also

ar ises in the design of a gyro-

gain a t rela t ively low frequencies; it

+55

+50

+45

+40

+35g

+30% %.5+25 ~c *~+20 m

.:

:+15 wJ z+ 10 %

%+5 3

0

–5

-lo

–15o

0

1:,<,.4 :;9,– ~l,artt ,teristic.of a ser~on,~t”~withUu:d,:iticlug. ~ = 0.25.

stabilized fir e-cont rol direct or , In t he la tter ca se Eq. (134) r epresen ts t he

t r ansfer funct ion rela t ing the precession torque applied to the gyro and

the angle through which the dir ector turns in response to er ror signals

bet ween t he gyr o and t he dir ect or ,

The feedba ck tra nsfer funct ion of Eq. ( 134) has an a symptot ic cha rac-

ter ist ic of – 6 dl]?octave for frequencies bclo}v u = 1/T and – 18 db/

oct ave for fr equ encies above 1/T. The usual design for the subsidiary




loop would yield a value of S?of the or der of 0.25, resu lt ing in a r esonance

peak in ]pOO/e] of +6 db. Figure 4“39 is a decibel–log-frequency and

+25

+20 -+1

— _

+ 15 +2 I

4.2 ;~+ 10

$3.5 +5,:

mso~

-5

-lo

-15

–20o 10 20 30 40 50 60 70 80 90

LooPphasemarginindegrees

FxG.4.40.—Characteristicsof a servom otor with quadra t ic lag. { = 0.25,

pha se-a ngle–log-fr equencyplot of t het ra nsfer fun ct ion of Eq. (134), for

~ = 0.25. Since theloop gain isrelat i~elyc onstant i n theregion~ ~-here

t 20

+15

+10+5

%?0.5_5.S5–10

-15

-20

-25

-300.02 0,04 0.1 0.2 0.40.6 1 2WT

FIQ. 4.41.—Frequency response of ser vO-motor with quadra t ic lag. r = 0.25.

flectecl in the zero-phase-margin line.

the phase angle is going through

–180° (O phase margin), it will

be necessary to choose avalueof

KUZ’ t ha t ~vill give a loop gain of

– 5 db or less when the phase

mar gin goes t hr ou gh zer o, in or der

to have a resonance peak of +3 db

or less in t he fr equ en cy-r espon se

curve. A value of Km!f’= – 11

db will sa t isfy this requ irement .

F igure 440 is a decibel–phase-

margin plot of th is system with

over -a ll fr equ en cy-r espon se con -

tours drawn in for K, T = – 11

db; a par t of the curve has been re-

The fr equency-r espon se maximum

increases to +6 db with a fur ther + 1.5-db in~rease- in KVT. Th e ser vo

becomes unstable for an increase in KVT of only +5 db and would be



SEC. 4.16] INTEGRAL EQ UALIZA TIOA’ 205

ra ther unsat isfactory. F igure 4“41 is the cor responding over -a ll fre-

quency response 100/0,1.

Feedback cutoff occurs a t U. = 0.3/!!’, a lthough the resonance peak

occurs a t c+ = 1/2’. The frequency response exhibit s a fa ir ly deep

minimum at a fr equency between feedback cutoff and the resonance

peak. If one assumes T = ~ see, then ~, = 5.3 see–l, ac = 0,7 cps,

and w = 2.4 cps.

This system can be equalized by the use of a propor t ional-in tegra l

n etwor k h avin g t he t ra nsfer ch ar act er ist ic

Eo = T,p + 1

x, A,T,p + 1’(A, > 1). (135)

This has an asymptot ic character ist ic of O db for u < l/(A oTJ, – 6

db/oct ave for l/ ( AoT ,) < u < l/T, and is constant a t l/Ao for higher

fr equ en cies. The at tenuat ion A 0will be chosen quite large, and the t ime

constant T1 will be la rger than the t ime constant T appear ing in the

quadra t ic factor . The phase angle of the propor t iona l-in tegra l equal-

izer will be zero for low and high frequencies and will approach a maxi-

mum nega t ive }-a lue at the geometr ic mean of the frequencies l/(A ~Z’l)

a nd l/Tl.Addit ion of the equalizer phase angle to tha t of the or igina l loop will

give an over -a ll phase angle having a maximum nega t ive value of about

– 180° at u = (<A O/ TJ1 and a minimum negat ive value between the

frequencies u = l/Tl and ~ = I/T; it then decreases toward – 270° at

h igher frequencies. It will then be desirable to place feedback cu toff

near the minimum nega t ive phase angle between u = 1/ T1 and o = 1/T.

The phase angle near cu toff should be in the neighborhood of – 135°.

Th e ph ase-a ngle con tr ibu tion of t he pr opor tion al-in tegr al equ alizer is

Arg ~ . – tan–l wAOTI + tan–l UT1. (136)

For large A 0 (A 0 = 40 = +32 db in the present example) we have

a ppr oxim at e y

(137)

From Fig. 4.40 it appears that the phase margin of the or iginal loop can

be decreased by approximately 30° at u = 0.3/T. This would mean tha t

we can set

tan–l ().3 !# = 60° (138)

or~1 =57

T. . (139)



~06 GENERAL DES IGNP R INCIPLES [SEC. 4.16

‘l’he feedback transfer funct ion can t hen be wr it t en

KVT T,p + 1eo – 1—e Tp AoT,p + 1 (Tp)2 + 2{Tp + 1“

(140)

F igu re 4.42 is a decibel–log-fr equ en cy and phase-angle-log-fr equ en cy plot

for AO = +32 db, T1/ T = 5,62 = +15 db, f = 0.25. The factor

.4O/ KtiT has been taken out in order to facilita te compar ison with the

previous system. The loop gain curve is st ill fla t in the region of zero

+100 -110

+go -120

+80 -130

+70 – 140

+60 -150 g~

g +50 -160 =.5

:; +40 -170&

%~ +30 –lM:

s 2~ +20

-190 A+10 -200?

2

0 -210

-lo -220

-20 -230

-30 -2400.002 0.0040.0060,010,02 0.040.06 0.1 0.2 0.4 8.6 1 2

0.)2’

F]G. 442.-Character is tics of in tegra l-equa lizedservomotor with quadra t ic lag.A, = 32 db; TI/T = 15 db; ~ = 0.25.

phase margin ; it will again be desirable to set KUT at such a value tha t

the loop gain will be —5 db when the phase margin is zero. This will

cor respon d t o K. T / A o = — 12 db (K, T = +20 db) for this example.

F igu re 4.43 is t hen t he decibel–pha se-ma rgin diagr am , wit h t he a ppr opr i-

a te 100/6,1 cent ours. Figure 4.44 shows the frequency-response cu rve

and t he a sympt ot ic 100/cl-curve. The [0~/~\ -curve is useful for est i-

mat ing the magnitude of the er ror a t low frequencies where 100/c\ and

10,/,] a re nea r ly equal. The frequency response is seen to have two

maxima of +3 db, one below and one above feedback cutoff. Inspect ion

of the decibel–phase-margin diagram shows that any smaller va lue of

T1 would make the phase-margin maximum so nar row that either one or

both of the frequency-response maxima would be la rger than +3 db.

The choice of T ,/ T = +15 db has a lso made th is system quite sensit ive



SEC. 416] IN TEGRAL EQUAL IZAT ION 207

to a ch an ge in Kol’. .4 change in K.T of +1.5 or –12.5 db will ra ise

on e of th e frequen cy-response maxima t o +6 db.

+25

+20

+15

+10

4c“;+5.=m

so2

-5

-lo

-15

–20o 10 20 30 40 50 60 70 80 90

LooPphasemarginindegrees

FIG. 4.43,—Charact ,e r is t ics of integra l -equal ized servomotor with quadra t ic lag.

Compar ing this with t he or iginal system, we see that t he addit ion of an

integra l equa lizer with an at t enua t ion A o of +32 db has increased KtiT

from – 11 to +20 db, an increase of +31 db. The feedback cutoff fre-

quency is now u. = 0.34/T, as +eo

compared with the previous UC= +~

0.3/ 2’. In other words, the low-

frequency loop gain has been in- ~ ‘a

creased without an appreciable :: +20,

change in cu toff frequency. This ~ ,0

is the usual resu lt obta ined with

integra l equaliza t ion. -20

It should be poin ted out , how- -w

ever , that the use of ~ = 0.25 and (1.002.0U40.010.020.04 0.1 0.2 0.40.61.02.0UT

TJ 2’ = +15 db has made this FIG.4.44.—Fr equea cYesponseof ser vesystem more sensit ive t o changes with integral-equalizeduervomotor with

in KVT than would ordinar ily bequadraticlag.

desired. The most st ra igh t forward method of improvement would call

for a la rger value of ~ (perhaps 0.5) and a slight increase in the ra t io

T ,/ T .




4.17. Equa liza t ion Using Su bsidia ry Loops. —F eedba ck h as lon g been

used to change and improve the per formance of elect ron ic amplifiers.

Feedback may be used t o linear ize the power -outpu t stages, to hold more

near lY constan t gain , t o obta in a specia l frequency response, and for

m an y ot her pu rposes.

Su bsidia ry loops a re u sed in ser vomech an ism s for t hese same pu rposes,

with the added complica t ion tha t more kinds of element s a r e ava ilable,

since h ere on e admits mechanical and elect romechanica l devices. The

‘wI~.4.45.—Subsidiary loop.t achometer genera tor , for ex-

ample, delivers a volt age propor -

t ional to the velocity, or the

der iva t ive of t he rot at ion angle of

a shaft . A motor acts like an in-t egra tor in tha t it s velocity is

propor t iona l to its applied volt -

age. A poten t iometer delivers a

volt age propor t iona l to it s rota t ion angle. A synchro t ransmit t er and

con tr ol t ra nsformer deliver a volt age pr opor tion al t o t he differ en ce in t heir

sha ft a ngles.

The MIT differen t ia l ana lyzer ’ is an example of a complex mult iple-

100P servomechanism which is used in the solu t ion of differen t ia l equa-t ions. Each of it s major unit s has two or more servomechanisms incor -

pora ted in it . The in tegra tor , for example has two servo follow-ups

which r eceive elect r ica l da ta and dr ive t he lead screw and in tegra t or disk.

The in tegra t ing wheel merely turns an elect r ica l t ransmit ter which is

followed up by another servo at the poin t where the in tegra tor wheel

angle is used. The var iou s unit s a re in terconn ect ed t hrough an elect r ica l

swit chboard. The whole mechanism may be looked upon as one or more

major loops r epr esen tin g t he cliffer en tia l equ at ion , wit h m any su bsidia ry

loops in volved in t he ser vo follow-u ps.

Consider the subsidiary loop represented symbolica lly in Fig. 4.45.

Its t ransfer funct ion may be wr it t en

or

Y,, Y3, Y,3Y3,y,,=;==

1 + Y23Y32(141)

(142)

E qu at ion (142) is con ven ien t t o u se in t he fr equ en cy r egion wh er e

YZSYSZ= Y2Z

is small compared with 1; it indica tes that t he subsidia ry loop has pract ic-

1V. Bush and S. H. Caldwell, “A New Type of Differen t ia l Ana lyzer ,” J .

Fran k lin In st., 240, 255 (1945).



SEC.417] 17QlJ ALIZAT10N US ING SUBS IDIARY LOOPS 209

ally no effect on the direct t ransmission through it s element YZ9. In

the frequency region where YZ, is la rge compared with 1, the direct -

t ransmission element YZ3 is replaced by the reciproca l of the rever se-

t ransmission element YW as can be seen frOm Eq. (141). The abovewill be recognized as the usual result : one can obta in the reciproca l of a

network over a given frequency range by placing it in the &por t ion

of a feedba ck amplifier .

It should be poin ted ou t tha t a lthough it is usually convenien t , it is

not necessary tha t a t ransfer funct ion be stable when it is used as an

elem en t in a lar ger feedba ck loop. Th e funct ion YM, for example, cou ld

have a nega t ively damped quadrat ic factor in its denomina tor , a r ising

from t he funct ion (1 + Y2z)-’. The use of su ch an elem en t will, h owever ,requ ir e careful use of the genera l Nyquist cr iter ion as discussed in

Chap. 2.

Proceeding to an example, let us consider an amplidyne coupled to a

d-c motor with a d-c t achometer dr iven by the motor . F igure 4’2a of

Sec. 41 is t he cor respon d~n g symbolic d~agr am , wit h

em KAKaK.—. (143)C2 ’23 = p(T~op + l)(Tqp + l)(T~P + 1)’

002= YSZ= K.p,

x(144)

00= Y,, = ;,

E012 K,– = Y,, = ml;e

(145)

(146)

Omhas been used in place of the 83 of Fig. 4.2a . The over -a ll feedback-

t ra nsfer fu nct ion a ssocia t ed wit h differ ent ia l 1 is t hen

do =y;,

— — = Y,,,e 1 + Yn

(147)

where

, (148)YY, = Y12Yz3Y31 = T ~(Tmap+ l)(T ,~’~T fi(T ,p+ l)(TJ P+ 1)m .

‘“ = ‘ 2 3 Y3 2~Tmap+ l)(T$’\ l)(T jp + 1)’(149)

Equa t ion (148) has been par t ia l] y nondimensionalized through the

u se of T~a. This is usually a rela t ively fixed parameter and cannot be



210 GEN ERAL DES IGN PRIN CIPLES [SEC. 4-17

var ied in the solu t ion of a par t icu la r design problem. The following

t im e-con st an t va lu es will be a ssumed:

Tma = motor -amplidyne t ime constan t = 0.25 see,

T , = amplidyne-quadra ture t ime constan t = 0.02 see,T , = amplidyn e-con trol-field t ime con st an t = 0.002 SCC,

T , = r ipple-filter t ime const an t = 0.04 sec.

Th e con st an t K] 1has t he dimen sion s of see-’ a nd is r ela ted t o t he velocit y-

er ror constan t . The constan t KM is dimensionless; it has been ca lled the

+40

+30

+20

+ 10

0

~-lo

:-20

:.30

-4 0

-5 0

-6 4

-7 0

-s o0 .0 2 0 .0 4O .LK0 . 1 0 .2 0 .4 0 ,6 I ? 46102040

wT..

FIG.4.46.—Asym@otic cha ra cter ist icsof amplidynewith direct ta chomet erequalizat ion

ant ihunt ga in . Equa t ion (147) may be rewr it t en as follows:

(152)

It will be usefu l to make fir st a decibel-log-frequency plot of both Y~l

and YZZ on the same piece of paper , as shown in Fig. 4“46. Curve

A, t he asymptot ic plot of YM, has a break from – 6 to – 12 db/octave a t

oT_ = 12.5; it s cor responding phase margin a t this break would be

approximately 45°, decreasing to approximately 30° a t uTW = 25.

This wou ld mean tha t the largest usefu l va lue of K2.zwou ld be between

+22 and +34 db since it is most conven ien t in th is case t o make the

subsidiary loop stable. The va lue of KZZ = +30 db gives the asymp-

t ot ic Cu rve B for Yza/(1 + Yaa);’ th is coin cides wit h YZZfor frequ encies

a bove a cz = 19.5/T~a . The Odb axis for Curve B has been set a t –30db. From Eq. (152), the asymptot ic Curve D for YII is Iyql!db (Cu rve

C) minus Iy221ab (Cu rve A) PIUS [y22/ (1 + y22)]db (Cu rve B); th is CO~-

cides with Y~l (Curve C) at frequencies above CM The resu lt ing

Y,, = oo/c asymptote has a break from – 6 to – 12 db/octave at u = l/T,

and a break from —12 to – 18 db/octave a t u = u.,. The subsidiary

loop has sh ifted the break in Y~l a t l/T~a up to it s cu toff frequency u,2.



SEC.4.17] EQUALIZATION US ING SUBS IDIARY LOOPS 211

The phase margin associa ted with me Y,, asymptote would then be

approximately 45° at co = 1/ T,; the complete system would probably be

opera ted with feedback cutoff occur r ing at a slight ly h igher frequency of

perhaps WI = 8/T_a. This would cor respond to a value ofK,,T~a = +50 db

Curve E is the asymptote for 0./0,, with its O db at – 50 db. Curve D,

when refer red t o this axis, is then the loop gain O/c; it gives an indica t ion

of the er ror for lo~v-frequency sinusoidal inputs. The low-fr~quency

– 6 db/octave, if extended to higher frequencies, Ivill in tersect the

– 50-db ordinate at a frequency equal to the velocity-er ror constant :

KuT_ = 10 or K. = 40 see-’. The single-loop system vith Y,l = Y~,

wou ld h ave been oper at ed u-ith feedba ck cu toff a t a ppr oxim at ely

and with a velocity-er ror constan t of K.Tma = 2 or K, = 8 see–l. Th e

addit ion of the subsidiary loop has thus made an appreciable impi-ove-

ment in the system.

It may be remarked that a reduct ion in T , that would permit a larger

ant ihunt gain and higher feedback-cu toff frequency in the tachometer

loop would not make an appreciable improvement in t he over -a ll system;the – 6 to – 12-db/octave break will st ill occu r at o = 1/ T,, and this

set s an upper limit on the feedback-cu toff fr equency for the combined

system. The feedback transfer funct ion Y,, could, of cour se, be modified

by a lead equalizer in Y,,, and in this way a higher cu toff frequency could

be obta ined for the complet e system.

A more accura te computa t ion of the over -a ll feedback transfer func-

t ion Yll can be obta ined by plot t ing the tachometer t ransfer funct ion

YZZon t he decibel–ph ase-mar gin dia gr am w-it h t he a ppr opr ia te con tou rsfor IY22;i(l + Y2.z)I and Arg [Yz.z/(1 + Y,,)]. These curves can then

be used together with Y~,/ Y,, to give accura te values of Y,l. This

pr ocedu re ~vill be illust rat ed in t he t he n ext sect ion .

The feedback transfer funct ion Yll can be increased for low fre-

quencies by including in YS, a high-pass filter that will reduce Y22 to

zero at low frequencies. The cutoff frequency of this h igh-pass filter

should be made so low that the —6-db/octave asymptote in Yll is a t

least 2.5 octa ves in length (15 db), .4 single-sect ion high-pass RC-filtergives

“P .–K,p,,, = —T~p + 1

which changes YZZin to

(153)

Kt,T,py?? = ‘ZS1’S? = ~T~ + l)(T~~p + l)(Tqp + l)(T,p + ]j” ‘154)

Th e t ra nsfer fun ct ion Y~, is left un ch anged,



212 GENERAL DES IGN PRIN CIPLES [SEC,4.18

Equat ion (152) can also be used to const ruct the asymptot ic decibel-

log-fr equ en cy cu rves for Yn. Figure 4“47 is const ructed with the same

upper cutoff frequency for the tachometer loop and with

Z’h = 0.28 sec = l.l!f’n..

The var ious curves have been let tered in the same manner as Fig. 4.46.

+40

+30 -q \ ~

+20

+10I ~Dl \ I

J 1 ! I

5-30

-40

-54

-60

-70

-so

aoz 0.040.%0.1 0.2 0,40.6 _] 2 46103040

--i

UJy.,

FIG.4.47.—Asympt oticcha racter isticsof amplidynewith high-passtacllomet erequa li-zation.

Curve E is t he I%/0, I a sympt ot e. Feedback cutoff has been reduced to

u. 1 = 4/T_, and the systcm would be opera ted \ vith

(KvTm),,, = (KnTJ ij = +39 db,

as compared with the previous value of +20 db. A compar ison of the

loop-gain Curve D with the previous result shows that the two systems

have equal gains, ‘+28 db, a t u = 4/T~.; the high-pass systcm has

the higher loop gain at lower frequencies and the lower loop gain at

higher frequencies.

APPLICATIONS

4.18. SCR-584 Automat ic-t racking Loop.—This system serves as a

good illu st ra tion of t he decibel–log-fr equ en cy design t ech niqu es beca use

of the availability of the considerable amount of exper imen ta l data con-

ta ined in Radia t ion Labora tory Repor t 370.1 The &Tquist 10CUSand

different ia l-equat ion met hods wer e used in th e design of this equipment .

The system’ compr ises a radar t ransmit ter and a receiver which

delivers a modula ted video signal to a diode detector . A so-ca lled

“slow” automat ic gain cont rol opera tes on the radar r eceiver gain in

LG. J , Plain and S. Godet , “ Data on SCR-584 ControlEquipment ,” Dec. 17,1942.1 “The SCR-584 Radar,” Electronic&,8, 104 (1945).



SEC. 4.18] SCR-584 A UTOMATIC-TRAf7KIN G LOOP 213

such a way that the amplitude of the video modula t ion envelope is pro-

por t ional t o t he angle bet ween the parabolic-reflector axis and t he t arge$ ‘,$

tha t is, to the angular er ror in t racking the ta rget . The signal is mod% ~~~

Iated at 30 cps, since recept ion takes place through a conica l-scanning ]antenna placed at the parabola focus and rota ted at this speed by an J

1800-r pm in du ct ion mot or ; t his mot or a lso dr ives a two-ph ase permanen t-

magnet reference genera tor . The 30-cps reference genera tor is used it ii

the commutat ing or phase-sensit ive detector to enable separa t ion of the )

eleva tion a nd t ra ver se (or a zimut h) er ror signals. 1

e,Gear

I , -+--,_]

rainMotor

6Speed

Radar feedback

transmitted bridgeand receiver

tEVideo error Antihu;t I

signalII Em

feedback

Thirdfilter

detector and

30 cps filterEJ

I(3 Commutating

6L630Cps

circuit andripple filter .%

error stgnal r

-A0 cps reference ~.’-’. d

FIG.44S.-SCR-584 t rm lcil!g-wn,~block cliagmn l.

In what follonx it \ villbe assumed that the commutat ing detector has

per formed it s fun ct ion and t ha t t he analysis ca n be ct ir ried ou t sepa ra tely

for the elevat ion and azimuth channels. ‘Nw circuits of the tJ vo channels

are ident ica l and can therefore IN considered separa tely. Figure 4.48

is a block diagram of the servo systcm. The torque limit circu it s \ vill

not be discussed in the follo\ ving and ha ve not been included in Fig, 4,48.

Neglect ing noise modula t ion effect s, the modulat ion envelope is a t rue

represen ta t ion of the antenna misa lignment and will be assumed to be

in phase with the actua l er ror c Radia t ion Labora tory Repor t 370

gives an over -a ll frequency response for the third detector and 30-cps

filter . The data are fa ir ly well fit ted by a single-lag t ransfer funct ion

with 7’3 = 0.013 see:

(155)

where CSis the amplitude of the 30-cps er ror signal. The commuta ting

circui$ can be considered as producing a slo\ vly varying voltage \ vith

super imposed ripple. hlost of the r ipple power is 60 cps and higher ,

since it is fllll \ rave. The a\ cr :~ge value of the commutator outpllt is

equal to the amplit ll(k of th(’, 30-cps er ror signal associa ted wit}~ the




azimuth or eleva t ion er ror . The commuta tor ou tput is then fed th rough

a r ipple filter of the type illust ra ted in Fig. 4“49. The nominal va lues

indica ted would give a quadra t ic lag with ~ = 0.35 and a cu toff frequency

of 71 radians/see or 11.3 cps. Actua l measurement indica tes that thefrequ en cy response depends on th e amplitude of t he input voltage E, bu t

is fa ir ly a ccu ra tely r epr esen ted by t he t ra nsfer ch ar act er ist ic

i?,—.E, (T,pl+ i~’

(156)

with T, = 0.01 sec.

The 6L6 outpu t stage can be looked upon as a combined mixer and

amdifier . The ou t~ut of the r ipple filter is applied to the gr ids of a

2000h lOOk

T

o

E.

o 1 0

FIG.4.49.—SC R-5S4 commu tat or-ripplefilter.

pair of 6L6 tubes, and the amplified

ant ihunt feedback voltage is differ -

ent ia lly applied to the screen gr ids.

The pla t e cur ren t s of the two 6L6’s

flow through two amplidyne cont rol

fields; t he differ en ce cu rr en t is effec-

t ive in producing the amplidyne ou t -

pu t volt a ge. The induct ance of each

cont rol field is approximately 30

henrys, which, with t he assumpt ion of

100 per cent coupling between the two cont rol fields and a 30,000-ohm

plate resistance for th e 6L6, gives

(157)

with t he cont rol-field t ime constan t T . = 0.002 sec. The quadra ture-

field t ime con st an t T* was measured and found to be 0.02 sec. Using

the armature resistance of 6.9 ohms, rotor iner t ia of 7 lb inz, and the

name-pla te data of 0.5 hp, 3450 rpm, 250 volt s, 1.9 amp, one obta ins

Tfi = 0.041 sec. Combining the motor and amplidyne and using the

armature circu it resistance of 11 ohms together with the motor resist -

an ce of 6.9 ohms, on e finds t he combined mot or-amplid yn e t ime con stan t

T- = 0.10 sec. This compares with a measured value of 0.11 sec for

the combined t ime constan t in the system. Using Eq. (157) and the

above t ime constants, one then obta ins

(T-p + 1)(1’,p + l)(l”cp + l)ptL = K.K,E, – KmKJEJ . (1Z8)

The speed-feedback br idge is of the type discussed in Sec. 3.15. It is

redrawn in Fig. 4.50, a long with the connect ions between the motor and

amplidyne. Refer r ing to Fig. 324, we see that



SEC.4.18] SCR-584 A UTOMA TIC- TRACKI.VG LOOP

R,=l?, +R , =3.1 ohms,

R.= R,+ R, =5.8 ohms,

R. = R, + R, = 10.1 ohms.

Th e speed-feedba ck volt age is t hen r ela ted t o d- by”-- ----- ”--,lleequabluu

215

(159a)

is t he effect ive t ach om et er con st an t, an d

~ _&Rm

R,’TT .~=Ra+Rm+Rc ‘;

t he loading effect of C, in Fig. 4.50 has been neglected. On subst itut ing

in the numerical va lues, on e obta ins !l’~o = 0.22, Tm~ = 0.024 sec.

R,

5.ik

%C4 Al

F1R,

RI Cl

1~~~:

2 4.7

A2C1 10k0,5pfF2 R,

R , E,F3

c2il1.1

F4;:

R6 R5

2 1,1cl C4

450

I 1

.—Amplidyne-motor conn ections and speed-feedback bridge.

The capacit or Cl in Fig. 4“50 is a capacity load on t he divider resistors

R, and R,. It looks in to a very high impedance at the input to the feed-

back filter . Equat ion (159a) then

becomes

~=x T .,p+l* ‘ T=p + ~ pfk (159b)

where T= = (R IRz/ R ~Rz)cI = 0.0017 E*

sec. This capacitor is in t roduced to

pr ovide a Klgh -fr equ en cy cu toff a nd:pr even t br ush noise a nd gen er al pick -

up in the system from act iva t ing the o

feedback-amplifier s tages.FIG.4.51.—SCR-5S4ant ihunt filte r .

F igure 4.51 is a schemat ic diagram of the high-pass equalizer or an t i-

hunt filt er used in this system. The voltage E. is supplied by the low-



216 GENERAL DES IGN PRIN CIPLES [SEC. 4.18

im peda nce br idge and may be consider ed t o ha ve a zer o sou rce impeda nce.

Th e volt age EY is applied to a vacuum-tube con t rol gr id R, can then be

t aken as the on ly load. The t ransfer funct ion can be shown to be

E, ‘,c2R@’(&+’)—.E,

‘1C1P(R,C2P+’+%) (+ P+1)+R2C2P+’- ‘16”)

This t ransfer funct ion can be factored and rewr it ten as

E, T, ()

~ ‘ (TLp + 1)

u.—.—E,

[()

2 (161)

“ (T,p + 1) ~ +2r:+l1

where TL = L/Rl and T l, u ., ~ are obta ined by fact or ing t he denomina-

t or . Put t ing in the constants given in Fig. 4.50, one finds

T. = 0.253 see,

TI = 0.558 see,

us = 3.835 radians/see,f = 0.608,

T1/TL = 2.206 = +6.9 db,

u.T I = 2.140 = +6.6 db,

W%l’L = 0.970 = –0.3 db.

The asymptot ic character ist ic r ises at 12 db/octave to u = l/ T,, a t

6 db/octave for l/Tl < u < w-, a t –6 db/octave for u . <0 < l/TL;

it is then constant a t O db for h igher frequencies. Using the previously

defin ed qu an tit ies, on e obt ain s

0. =

KmK,K, KmK~K,(TI/T.) (P/un)’(T.P + 1) (T~oP + 1) ~

(T,p+l)(T,p+l)2’– ~T1p+ ~, ~ ‘+2*E+ 1 (Tp+ I) m

[() 1z

p(Twp + l)(T,p :l)(Tcp +“~)(162)

Solu tion for 0~/6 gives

O*—.c

[

KmK,Ka

P(Tm.P + l)(T,P + l)(~,P “+ 1)(T3P + l)(T,p + 1)21()mK,KO FL ~~1 ~ 2 (z’~p + l)(TmoP + 1)

l+(T=P+l)(TmaP+ l)(T,P+l)(TCp+l) (T,P+l) [(P/U. )2+211P/t i.)+1](163)



SEC. 4.18] SCR-584 A UTOJ 1 A T IC-TRACKING LOOP 217

In t erms of the gear ra t io N between 0. and t l~ and the loo~-t ransfer

funct ions Y!, and- Y,,, one can wr it e

0.—= Y,, = *2,f

where

y :, =K.

p(7’m@ + l)(T,?J + 1)(7’.P + 1)(7’3?-JY,, =

()K,, > ~

2

7’,, Un(~’~ + 1)(7’mLO~

(T’=p + l)(Tm@ + 1)(7’<,P + 1)(7’.p + 1)(7’lp +

(164)

+ l)(Trp + 1)” (’(;5)

+ 1)

l)[(l?\ 2+2re. +1]’

L \ cI)./ w’

(1(i).K. = KmK,Ks/ N = velocity-er ror const an t ,

KZZ = K~KJK, = an tih un t gain .

The ant ihunt gain KZ, has been so defined tha t it is the zero-frequency

loop ga in aroun d th e speed-feedback loop wh en t he capacitor s in th e ant i-

hunt filt er a re shor t -circuited ou t (the filt er then has unity gain a t all

frequencies). The ant ihunt gain is a dimensionless number ; the ant i-

hunt gain con t rol on the servoamplifier can be adjusted to a maximum

value of K,j = 70. The forward gain con t rol on the servoamplifier

cou ld be adjusted to make K. a maximum of 540 see–l.

The following is a summary of the constants for the system:

1

“ = 0“558 ‘ec = ~8’ ‘“

w. = 3.8 radians/see,

1

“ = 0“253 = 3:9’

T1

ma= 0“11 ‘ec = 9:0’

T~, = 0.022 = ~, &

Tq = 0.02 = &,

TS = 0.013 = ~,

T,= O.01=~, -., .

T. = 0.002 = &i, I ,>,.;.1 .

T, = 0.0017 = +,f\ ). . .

f = 0.608.

F igure 4.52 is an asymptot ic decibel-log-frequency plot of Y~l/KO

and Yt2/K2P. A value of K22 = +26 db or an ant ihunt ga in of 20 was

chosen in drawing the combined curve for Y~I/( 1 + YZJ .

As long as the ant ihunt loop is reasonaby stable (perhaps 15° phase

mar gin a t cu toff), it is possible t o con st ru ct a ph ase-a ngle–log-fr equ en cy



218 GEN ERAL DES IGN PR ~N CIPLES [SEC. 4.18?-.,

+10-.. “-;

o

-lo

–20

–30

–40%

“s-50.s

‘-60

-70

-80

–90

–loo

–1100.40.612461020 40 60 100 200 400

Frequency in radians/see

FIG. 4.52.—Asymptot ic loop gains .

+175

+150

+125

+100

+75+50g

;i-25=

iOc.

-25 ~e

–54

02 0.4 0.6 1 2 46

\ ,\

-7 5

\-1oo

-125

-150

; 10 20 40601W20U 4ooaolomfreuuemcyn radiany%c

FIG.4.53.—SCR-5S4antih untloop gain an d phase angle.



SEC. 4.18] S CR-584 A UTOMA TIC-TRACKING LOOP 219

cur ve from the asymptot ic curve for Y11 and to use this phase-angle cur ve

along with the asymptot ic decibel–log-frequency curve in select ing the

value of K. tha t will give sa t isfactory system stability. In the presen t

example, however , all of the steps will be car r ied through in deta il, inor der t o illust ra te the gener al meth od.

-281. I I I 01 I

-24

-20

-16

-12

-8

3-4

.E

“: o

;44

~8

+12

q 16

-’-+Z(J

+24

+28-180 -160 –140 –120 -1oo -80 –60 –40 -20 0

LooDDhaseamzle in dearees

FIG. 4.54.—SCR-584 ant ihunt -loop decibel–phase-angle diagram . Kzz = +26 db.

Figure 4.53 is a plot uf the exact decibel–log-frequency and phase-

angle–log-frequency curves for ant ihunt -loop t ransfer funct ion Y22.

Examinat ion of this curve indica tes that KZZ = +26 db will give a

stable system. The loop gain and loop phase angle can now be plot t ed

on the decibel–phase-angle diagram (Fig. 4“54). The con tour s on this

diagram permit (the sign having been changed as discussed in Sec. 411)

a determinat ion of the t ransfer gain and phase angle for the funct ion

Y, = 1 + Y22. (167)




The loop gain IY221..,is plot t ed against Arg ( Yz,), with the radian fre-

quency u as a parameter . The contours a re those of IYzl and Arg ( Yz)

and are cor rect ly marked in decibels and degrees, for

–180° < Arg (Y,,) <00.

The contour diagram could be extended to posit ive values of Arg (Y,);

but since the form of the contour s is obta ined by a reflect ion in the loop-

gain axis, it is sufficient t o r eflect the cu rve for Yz2 in t he line Arg ( Y,,) = O

and plot back across the same set of contours. For this par t of the curve

the IY~l contours then reta in the same values, but the values to be

+60 +150

+50 +i25

+40 +Ic!o

+30 +75 ~‘0

~+20 +50 “:-~

“:+1O +25 -

30%

0:

-10 – 25

-20 –50

-30 –75

-4002

–loo0.4 0.6 1 2 461020 40 60 100 200 400 600

Frquencym radians/sec

FIQ. 4,55. —SCR-5S4 ant ihunt -t r .ans fcr gain and phmc ang]c.

associa ted with the Arg ( Yz) contours must be regarded as the negat ive

of those marked on the diagram. Thus star t ing at high frequencies the

cu rve crosses the —180° axis. The loop gain increases in value to

approximately +24 db at u = 6 and Arg (Y~.J = 0°; the cu rve is then

reflected and becomes asymptot ic to the —180° axis as co-+ O.

The value of KW = +26 db used in const ruct ing Fig. 4.54 gives a

stable ant ihunt loop with approximate y 12° phase margin at low-

frequency feedback cutoff and a peak gain of approximately – 9 db.

Increasing K,Z by 16 db would make the ant ihunt loop unstable at the

h igh -fr equency end.

Figure 4.55 is const ructed by reading off the values of the t ransfer

gain ~YZld. and the t ransfer phase angle Arg ( Yz) from the contours for

Y2 = 1 + Y*2.

Figure 4.56 gives the loop phase angle for the loop transfer funct ion

Y~l tha t remains when the ant ihunt loop is inact ive (Y,, = O). The



SEC.4.18] SCR584 A UTOMA TIC-TRACKING WOP 221

-75

-1oo

-125

-150

~-175

‘-200.E

%.225g

#.250

f

-275

-300

-325

-3500.2 0.40.612461020 40 m 100

Frequency in radiansfiec

FI~. 4.56.—Primary-loop phase angles.

+20

+10

o

-10

-20

%-30

“,~-40

.E

‘-50

- IJ j

-70

-80

-90

-1000.2 0.40.6 1 2 4 6 10 20 40 60 100 200

Frequency in radians/sac

l;lO. 4.57.—Pr imnry-loop glbir,~.




cu rve for Ar g (Yu ), a lso sh own in th is figu re is con st ru ct ed by su bt ract in g

Arg ( Y,) (Fig. 4.55) from Arg ( Y~l):

Arg (Y,,) = Arg (YY,) – Arg (Y,). (168)

Figure 457 shows th e cor responding curves for th e pr imary-loop gain

and for

Iyllldb = IY!,l,, – Iy,l~.. (169)

Refer ence to Fig. 4“56 shows tha t there is a maximum phase margin of

+40° (180° – 140°) in the frequency region near a = 8 radians/see.

+25 ——<

+20 -+1 \~ \ +o,5 +0.25 o

+152,~y ~ \

)

=0 J / /

8a. +6

–3–6

-lo ~ — 20_ — . — — —

—

-18

-20-180 -170 –160 -150 -140 -130 –120 -110 -102 -90Loopphase angle in degrees

FIG. 4.5S.—SCR-5S4 decibel-phase-angle diagram.

It follows tha t opt imum stability will resu lt when K. is so adjusted tha t

th is maximum phase margin occurs a t a loop gain of approximately +2

db. Select ion of KO = +46 db (200 see-’) accomplishes this; Fig. 4.58

is t h e r esu lt in g decibel–ph ase-a ngle dia gr am for t h e syst em wit h

K. = 200 see–l.

Th e amplifica tion for Ioo/0,1 ca n be r ea d fr om t he amplifica tion con tou rs;

it is seen to have a resonance peak of +3.8 db at u = 8 radians/see

(1.3 Cps).

The over -a ll frequent y response for the system with K. = 200 see-l

and KZZ = 20 is given in Fig. 4.59. It will be seen that the extension of



SEC.418] SCR-584 A UTOMAT IC-T RACKIA’G LOOP 223

t he low-fr equ en cy – 6-db/oct ave slope for 1190/~in ter sect s t he O-db axis a t

u = 200 = K,.

It follows from Fig. 4.58 tha t an 1l-db decrease or an 8-db increase in

K, will ra ise the resonance peak to -i-6 db. This means tha t the over -a ll

syst em does n ot h ave much sen sit ivit y t o ch an ges in t he er ror -sign al ga in—one of the requ irements for an automat ic-t racking system of the type of

the SCR-584. ... ffl, ..! ‘.,n’, ~,

+60.. ‘r a ~? ‘t “~

+50

+40

+30

+20

+10a.E f)

$_lo

–20

-30

-40

–50

–600.2 0.40.612461020 406O1OO2CHI

Frequency in radim@ac

FIG. 4.59.—SCR6S4 frequency response .

One observes from Fig. 4.58 tha t the stability of the system is deter -

mined by the phase angle and gain of YI I in the frequency range

3<u <20.

Changes in KZZ and Km will obviously have no effect on Ar g (Y$,). Refere-

nce to Fig. 4.54 shows tha t a change in KZZwill have lit t le effect on Arg

(Yz), since the contours of Ar g (Yz) a re almost parallel t o the contour s of

Ar g (Y.z2). The cont ou rs of [yzld~ ar e a lso a lmost pa ra llel t o t he con tou rs

of ]y22\ db in th is fr equency region . Thus an increase in KZZwill m er ely

move the complete curve of Fig. 4.58 downward by the same number of

decibels, a ssuming t ha t Kv remains constant . Also, increasing KV and

KZ* by the same factor wiU leave the stability and cu toff frequency

unchanged.

A study of the power spect rum of the angular er ror in automat ic




t racking with the SCR-584, by the method of Chap. 6, indica tes that the

locat ion of the peak var ied dur ing normal opera t ion from 0.4 to 0.9 cps

(2.5 to 5.6 radians/see) and tha t the peak height was approximately

+8 db when the resonant fr equen cy was 0.9 CPS,with gr ea ter peak heights

occur r ing as the resonant peak shifted to lower frequencies. This is

precisely the behavior that would be expected from Fig. 4.58 as either

K“ is decr eased or K,, is incr eased fr om t he value used th er e.

Referr ing to Fig. 4.52, one can invest igate the behavior of the system

as some of the parameter s are var ied. Among these are the combined

mot or -amplidyne t ime con st an t T - and the t ime constant 7’”0. These

t wo t ime constants can change appreciably as th e r egula t ion resistance of

the amplidyne’ is changed through its manufacturing tolerance of 25

to 42.5 ohms (compared with 11 ohms as used in the foregoing analysis).For R. = 25 ohms,

Tma = 0.11 ff j ~:~ = 0.19 see,

5.8 – 1.6 = 0,025

‘“0 = 0“19 25 + 6.9sec.

For R. = 42.5 ohms,

T.= = 0.1, 4:~~~~ = 0.3 mc,

5.8 – 1.6

‘m” = 0“3 42.5 + 6.9 = 0“026sec.

The change in R. is thus seen to cor respond pr imar ily to a change in TmO

with lit t le change in Tmo. The change in Tma over the above range can

be almost complet ely r emoved fr om t he over -a ll syst em by a r eadju stm en t

of KZZtha t leaves K, and the frequency response unchanged. This would

indicate that , as has been shown by exper ience, the manufactur ing

t oler an ce on Ra is sufficient ly precise. Var ia t ion of amplifica t ion in t he

var ious elements can, of course, be cor rect ed by adjustment of the two

gain con tr ols th at clian ge Km and Kzz.

The t ime parameters tha t could give t rouble are T,, u*, and T .;

t hese determine the length of the – 6-db/octave slope in the YII asymp-

tot e. The manufactur ing tolerances on these parameter s a re not known

to the authors, but it would seem reasonable to require Tq to be less than

0.03 see, T~ t o be grea ter than 0.35 see, and U. to be less than 5.6 radians/

sec. These tolerances were probably held in product ion , since they

correspond to a 40 per cen t change from the nominal value.

4.19. Servo with a Two-phase Motor .—The two-phase motors men-

t ioned in Chap. 3 are quite useful in low-power servo applicat ions. They

1GE amplidyne model 5.4M65FB2A, 500-watt , 250-volt , 115-volt , thr ee-phasedrive motor.

I



SEC. 4.19] SERVO WITH A TWO-PHASE MOTOR 225

have been widely used in comput ing mechanisms and remot e posit ioning

applicat ions. As an example, we may consider a servo system designed

origina lly t o dr ive a computer shaft in synchronism with t ra in angle data

provided by an ant ia ircraft dir ector . This uses a Diehll two-phase

140

130

120

110

100

90

g 80

;c 70~

~60&

50

40

30

20

10

0.

3600

3400 T\

3200

3000

2800

- Rpm2600 /

2400

\ ,2200

2000 /

s 1800:

‘1600: Total power Inputd:1400 0.7“:0~

\~

a 1200 ~ 0.6 ~2 xxlcrco

Amperes per phase

\

0.5E2

800 / +/

/ ~ ~

600/

.Poweroutput

400 ,/

200 /

o0 5 10 15 20 25

Torque in oz-in.

FIG. 4,60.—Brake t est a t 115 volt s, 60 ~P~OfD ie~ mot or FPF49.7.

servomotor. First a br ief out line will be given of the exper imenta l

pr ocedu re in volved in obt ain in g t he mot or ch ar act er ist ics

The motor ra t ing is 22 ‘mechanica l wat ts output a t 2200 rpm; Fig.

4.60 is a copy of the manufa@r&’s data. The rotor iner t ia is 0.66

oz-in . 2, and the impedance looking in to either of the phases var ies from

350 ohms at not load and maximum speed to 180 ohms when sta lled.

Each winding t akes 70 wat ts or 75 volt -amperes when t he mot or is sta lled,

with 115 volts on each winding.

1S.S.No. FPF49-7, 115 volts , 60 cps, Diehl Manufactur ingCo., Sornet ille , N. J .




Figure 4“61 shows the free-runn ing speed and the sta lled torque as a

funct ion of the cont rol phase voltage with 115 volts on the fixed phase.

In spect ion of t hese cu rves shows t ha t Km = 840° per second per volt and

K, = 0.25 oz-in . /volt . The measured iner t ia of the motor and its

3500

3000 /

Free running speed —

2500 /10

– Torque into stalled motor

2000 8?/ 0

E .s

E :

1500 b~f

1000

y /

- 4oz-in.

_ 0.25 ~

500 - 2

00 10 2Q 30 40 50 60 70°

volts

FIG. 4.61.—Character is t ics of Dieh l FPF49-7 two-phase motor .

associa ted gear t ra in gave an iner t ia of 1 oz-in . 2 Conversion of these

values to a common system of units gives

f“ = g = 0.018 oz-in . radian per see,.

(170)

T. = ~ = 0.15 sec.m

The speed-torque cu rve of Fig. 4-60 gives f~ = 0.022 oz-in ./radian per sec

for t he in ter na l damping coefficien t, th at is, t he slope of t he speed-t or qu e

cu rve a t zero speed point . Assuming tha t the sta lled torque is linear

wit h volt age, on e obt ain s fr om F ig. 4.60 Kc = 0.26 oz-in./volt.

The value of Km can then be obta ined from Eq. (170). All of t he

quant it ies ar e seen to be in fa ir a gr eement , an indica t ion tha t t he common

brake test da ta can be used to obta in the motor t ime constan t T~ and

t he speed volt age con st an t Km .



SEC.4.19] SERVO WITH A TWO-PHASE MOTOR 227

The er ror signal in the system we shalf consider is obta ined from a

synchro cont rol t ransformer tha t is gea red down from the motor by a

ra t io of 10. The er ror signal is then a 60-CPS voltage with a gradient of

O.l volt perdegreej refer red tothe motor shaft . The fixed phase of the

204.4k + 300

9.lk 4.6k 19.9k

CTerrorsignal

o

FIG.4.62.—Ph ase-lag etworkalld bridged-Tequa lizer.

motor is put across the 115-volt a -c line, and theer ror ismadeto lag the

line by 90° through the use of a two-sect ion RC phase-lag network. A

convent iona l a-c amplifier is used with a pair of push-pull 807’s in the

output ; the amplifier has a volt age gain of 22,000 (+86.5 db). It

160

14 \ /0

12 \ (

10 r%

,; s- <> ~o

) 0

86 /A

4 /0 I /( >“000’

2

030 40 50 60 7(I 80 90

Frequency in CPSFIG. 4 .63.—Frequency response of equalizer and phase-lag network ,

delivers approximately 45 wat t s in to the sta lled motor—enough to

dr ive the motor to its maximum power output a t 22oO rpm. Inverse

feedback is used in the amplifier in order t o keep its output impedanceat a low value (approximately 350 ohms). The phase-shifted er ror

signal is passed through a br idged-T equalizer with a notch width

of 5.5 cps (T’~coo= 11). F igure 4.62 is a schemat ic diagram of the fil-

t er and phase-lag network. Figure 4.63 gives a plot of the frequency

response of this filter and phase-lag network, together with a plot of

1 + i (U – coo)~d, which is the exact response for a lead equalizer in a

ca rr ier -fr equ en cy syst em wit h ca r rier fr equ en t y jo = COO/2mnd der iva t ive



228 GENERAL DES IGN PRINCIPLES [SEC.4.19

t ime constan t Td. In each curve the gain has been plot t ed with it s

6@cps va lue as a r efer ence level. The input a t t enua tor and outpu t

ca thode follower a re included in F ig. 4“62 as an example of one method of

coupling the equa lizer in to the circu it ; in th is example the over -a llvolt age a t tenua t ion a t 60 cps in 69 db. The input a t tenua tor has a

loss of 9 db; the phase-lag network a loss of 20 db; and the br idged-T

at tenuat ion is 40 db. The amplifier used in this par t icu lar applica t ion

was also used in 11 other servo channels with differen t equa lizing net -

works. This a r rangement requ ired cabling between the equa lizing

80

0

60

% 50,E

‘~ 40

~ 30

20

10

00

Radian frequency

l?IG. 4.64.—Experimental decibel-log-frequency plot of the loop transfer function.

network and the servoamplifier . Since a low-impedance line is less

likely to pickup signa ls from adj scen t circu its, it was advan tageous to use

the ca thode follower as an impedance t ransformer . The line fr om the

servoamplifier to equa lizer thus opera t es with approxima tely 500 ohms

impedance to ground.

F igure 4.64 is a plot of the loop gain for this system. The exper i-

menta l setup made use of a synch ro-con t rol t r ansformer as a genera tor of

a modu la t ed 60-cps signa l wh ich was connect ed to the inpu t termina ls of

the network shown in Fig. 4.62. A constant 60-cycle volt age was

impressed on its sta tor ; and as the rotor was tu rned, its volt age was

E~ = E COSu~t COSu~t, (171)

where E = maximum value of th e 60-cycle ca r r ier ,fo=; =c a rr ier fr equ en cy, cycles per secon d,

.f~ = ~ = rota t ion speed, revolu t ions per second.

The volt a ge E. is of the same form as that which would be produced by

a small-am plitude sin usoidal m ot ion of a syn ch ro-con tr ol t r an sformer in a

syn ch ro da ta -t ra nsm ission syst em . The magnitude of E was changed



SEC. 419] S ERVO WITH A TWO-PHASE MOTOR 229

th roughou t the cour se of the exper iment in order to mainta in the maxi.

mum volt age applied to the cont rol phase of the motor a t 50 volt s rms.

The amplitude of oscilla t ion of the motor shaft or a geared-down shaft

was observed opt ica lly at the low frequencies and elect r ica lly a t h igh

frequencies. The ra t io of the motor-shaft amplitude in degrees to the

voltage E of Eq. (171) is plot t ed in Fig. 4“64.

Th e system was oper at ed with a velocit y-er ror con st an t K* = 500 see–’

which placed feedback cut off a t 96 radians/see (15.3 CPS). Th e in ter cept

of the —12-db/oct ave sect ion on the O-db axis of the loop gain cu rve

occur r ed at 57.7 radians/see and gave an accelera t ion -er ror constan t of

(57.7) 2, or 3330 see-z. E xper imen ta l over -a ll fr equ en t y-r espon se cu rves

show a rapid fa lling off a t frequencies above 15 cps and thus ver ify the

above ana lys is .

.1-2

4 ]-250

FIG.4 .65.—Ins tan t aneous angu la r velocit y and acceler a t ion of a t a rget on a s tr a igh t -linecourse.

This system was designed to dr ive a computer shaft in synchronism

with the t ra in angle of an ant ia ircraft dir ector tha t was t racking an a ir -plane ta rget . The computer shaft was gea red down by a factor of 360

from the motor shaft . For a hor izonta l deck and a st ra igh t -line constan t -

velocit y t ar get , on e fin ds

01= tan-’ “(~- ‘~), (172),*

where 81 = input t ra in angle,

V~ = hor izon ta l velocit y,

Rm = minimum hor izon ta l range,t~ = t ime a t crossover .

The angular velocity and angula r accelera t ion are

(173)

(174)



230 GENERAL DESIGN PRINCIPLES [SEC.419

Theangular velocity reaches amaximum of V~/R~at crossover , ordI = O.

The angular accelera t ion reaches a maximum posit ive va lue of

at 30” befor e cr ossover and a corresponding maximum nega t ive value a t

30° after crossover . F igure 4.65 is a plot of the angula r velocity and

acceler a t ion when V~/R~ is~ radian/see (500 angular roils per secon d).

This cor responds to a ta rget with Rm = 500 yd and V~ = 250 yd/sec

(450 knots). Using the values of K, and Kc pr eviously determined, on e

computes for the servo following of such a ta rget a maximum velocity

er ror of 1 mil and a maximum acceler at ion er ror of 0.05 r oils.



CHAPTER 5

FILTERS AND SERVO SYSTEMS WITH PULSED DATA

BY W. HUREWICZ

5.1. In t roductory Remarks. —Servos considered so far in this book

opera te on the basis of er ror data supplied continuously, in an uninter -

rupted flolv, The presen t chapter is concerned with servos that am

actuated by er ror data supplied infermit fen t lg, a t discrete moments

equally spaced in t ime. In other terms, the er ror data a re supplied in

the form of pulses, and the servo receives no informat ion wha tsoever

about the er ror dur ing the per iod between two consecut ive pulses.

As an ext remely simple example of a servo with pulsed data , let us

consider a system for which the input and the output are shaft rota t ions.

We denote as usual by o, and % the angles determin ing the posit ion of

the shafts and suppose tha t the act ion of the servo consists in (1) measur-

ing the er ror E = Or— @oat specified, equ ally spa ced moment s, sa y on ce

every second, and (2) instan taneously rota t ing the output shaft immedi-

a tely a fter each measurement by the angle K~, where K is a fixed pos it ive

constant. In order to study the per formance of the servo, we may sup-

pose tha t the system begins opera t ing with an init ia l er r or COand that the

angle Of is kept fixed t herea fter . Immedia tely following t he fir st er ror

measurement , t he er ror will acquire t he value

Cl = (1 — K)t o. (1)

Immedia tely a fter the next measurement the er ror will have the value

~z = (1 — 101 = (1 — K)%o. (2)

Denot ing by c. the value of the er ror immedia tely after the nth measure-

ment , we have

c. = (1 — K)”@. (3)

When O < K <2, the er ror approaches zero with increasing n; the servo

is therefore stable. On the other hand, for K >2 the er ror increases

indefinit ely an d t he ser vo is u nst able. It is t o be noted that instability

is caused in this case by overcorrection. This kind of instability is quite

typica l for ser vos with pulsed data .

A less t rivia l example will be obt ain ed by t he following modifica tion s

of the preceding example: Instead of assuming that the er ror s are cor -

231



232 FILTERS AND SERVO SYS TEMS WITH PULSED DATA [SEC.52

r ect ed by instantaneous rota t ions of the outpu t shaft , let us now assume

tha t the cor rect ive act ion of the servo consists in cont inually exer t ing a

torque on the outpu t shaft in such a fashion tha t the torque is always

propor t iona l to the er ror found at the immediately preceding measure-ment . The torque then remains constan t dur ing the interva l between

two measurements, changing stepwise a t each measurement . Again it is

clear tha t overcor r ect ion and instability will occur when the cor rect ive

torque per unit er ror is too large. In this case a quant ita t ive analysis of

stability condit ions is not simple, nor can it be obta ined by methods used

in the theory of servos with cont inuous er ror da ta . Th e complet e

analysis will be given at the end of this chapter . Another example of a

servo with pulsed data is the au tomat ic gain cont rol system for radart r ack ing syst ems.

In the following we shall denote by the rep et it ion p eriod of a servo the

t ime in terva l T, between two consecu t ive moments at which er ror data

a re received. The quant ity I/T, will be ca lled the “repet it ion frequency

~,” (in cycles per second). It is a lmost needless to ment ion tha t when l“,

is ver y small compared with other t ime constan ts involved in the system,

the servo can be t rea ted to within a sufficient ly good approximat ion as a

servo with cont inuous data . The need for a differen t t r ea tment ar ises

when the length of the repet it ion per iod cannot be neglected in compar i-

son with the remaining t ime constan ts of the servo.

In analogy with the procedure adopted in Chap. 2, we shall base the

theory of servos with pu lsed er ror data on the theory of jilters w ith pu lsed

inpu t data; t his will be t he su bject of t he f ollowin g sect ion s.

F ILTERS WITH PULSED DATA

6.2. The Weigh t ing Sequence.—By a linear filter with pu lsed data

(abbrevia ted in the following discussion t o pulsed filt er ) we shall mean a

transmission device tha t is supplied with input da ta a t specified equally

spaced moments and, in response, furnishes outpu t da ta at the same

moments in such a way tha t (1) the ou tput da ta depend linear ly on the

input da ta r eceived previously and (2) the per formance of the device does

not change with t ime. An example of such a device is a four -terminal

passive networ k tha t receives its input volta ge in the form of pulses; if we

consider the values of the voltage a t the ou tpu t terminals on ly a t the

instants when a pulse is applied to the input terminals, th is network can

be regarded as a filt er with pulsed data .

In order to formula te mathemat ica lly the condit ions 1 and 2 of the

preceding paragraph, let us assume tha t input da ta are received at the

moments t = n!f’, (n = O, i 1, *2, +3, . . ). Let x. be the value Of

the input and V. the value of the output a t the t ime t = nT,. Condition

1 sta tes that



SEC.5.31 S TABILITY OF P lJ LSED FILTERS 233

Y . = xG, kx~–k, TL =0, *1,+2,.-. ? (4)

k =’1

wher e t h e cn ,k ar e r ea l con st an ts. Con dit ion 2 st at es t ha t t he coefficien tsc.,~ depend only on k, since x.–k must en ter yn in the same way as xm_~

en ter s the expression for y~. Set t ing

Wk = C.,k l (5]

we wr it e Eq. (4) as.

yn =z

Wkxm—k. (6)

k=l

In order to avoid difficu lt ies caused by the fact that the infinite ser ies

in the r igh t -hand term of Eq. (6) may not converge, let us assume for the

t ime being tha t the input up to a cer t a in moment is zero; that is, x. = O

for sufficien t ly la rge nega t ive values of n . Then there a re only a fin ite

number of terms in the sum which a re differen t from zero, and the

qu est ion of con ver gen ce does n ot ar ise.

The sequence (w.) is quite ana logous to the weight ing funct ion W(t)

in tr odu ced in Ch ap. 2; it will be ca lled t he weigh tin g sequ en ce of t he filt er .

It may be noted in passing tha t a pulsed filter can be rega rded as a con-

t in uou s filt er wit h t he weigh tin g fu nct ion

w(t) =z

Wna(t – ?LTr), (7)

“=’1

wh er e 6(t ) den ot es t he Dir ac delt a fu nct ion .

The meaning of the numbers W. will be made clea rer by the following

remark. I.et , the input to the filt er be a single unit pu lse applied at t = O:

Zo = 1, 2%=0 for n # O. (8)

Then by Eq. (6)

y. = w. (n=l,2,3, ...). (9)

In words, the number W. represen t s the response to the unit pu lse n

r epet it ion per iods a ft er t he pu lse has been r eceived.

If for a sufficien t ly la rge n, say for n > N, all the w“ a re zero, then the

outpu t a t a given moment depends on ly on the N input da ta receivedimmediately preceding th is moment . One can say in th is case that the

filt er h as a ‘ ‘fin it e memor y” lim it ed t o N pieces of in forma tion .

5.3. Stability of Pulsed Filters.-In complete analogy with the con-

cept of stability developed in Chap. 2, a pulsed filter will be ca lled stable

if to a bounded input t her e always cor r esponds a bounded ou tput . We

shall see that a neazssary and wq?kient condition for stability is t he absolu te



234 FILTERS

convergence: of

of t he filt er .

AND SERVO SYSTEMS J f”[TH PIILSED DATA [SEC.5.3

?the ser ies ~ w., where (w.) is the weight ing sequence~=1

.

We first prove the sufficiency. Suppose tha t the sum ~11 w.n=l

is fin ite, and let A denote it s va lue. Let the input sequence (z.) be

bounded; tha t is, for a cer ta in posit ive number M,

Ixn[ < M (lo)

for every in teger n. From Eq. (6) we obtain

.

IY.I < ~ J flwkl = MA. (11)

k=l

The output is therefore bounded by the number MA.

In order to prove the necessity of the condit ion let us assume that the

filt er is stable. It was remarked in the preceding sect ion that the weight -

ing sequence (w.) can be regarded as the output sequence cor responding

to the unit -pu lse input . Hence (by the definit ion of stability) the

sequence (UAJis bounded. Let M be a posit ive number such that

[Wnl < M (n=l,2,3, ..). (12)

Suppose now that the ser ies ~ ,w., diverges. Th er e cer ta in ly exist sn=l

a posit ive in teger N1 such that

N,

zIwnl >1. (13)

n=l

We fur ther select an in teger Nz > NI such that

N,– N,

zIwnl > MN, + 2. (14)

~=1

The exist ence of such an in t eger follows from t he divergence of the ser ies.

We con tin ue t he pr ocess a d in fin it um . Aft er h avin g select ed t he in teger s

N,, N,, . . . , Ni_I, we choose an in teger Ni > iV~-1 sa t isfying

. .

1A seriesz

w. is said to be a bsolu tely convergen t ifI

Iwnl < m .

n-l ~=1



SEC. 5.3] S TABILITY OF PULSED FILTERS 235 I

(15)

We now define a bounded input sequence as follows:

(16)

x. =0 for n s O,

x. = sgn w~,–n1 forl~n<N1,

xt i = sgn wN,_rI for NI s n < Nt ,. . . . . . . . . .

x. = sgn w~~_. for N<, s n < N,, I

Let (v*) be the corresponding outpu t sequence. By Eq. (6)

N.–l

yN, =2

wkz~i–k. (17)

k=l

For 1 s k s Ni – N~_,, the kth t erm in this ‘sum has (in accordance with

the defin it ion of the numbers G) the va lue Iw,l. Then by Eq. (15)

N,– N,.,

2wkx.v,_~> MN,-I + i. (18)

k=l

On the other hand, for any n we have lw~l < M, and hence

N,–1

2w,x.v,_k > – lfN ,_,, (19)

~=N,–N,-,~1

Addin g t he la st t wo inequa lit ies, \ ve obt ain

y.v, > i. (20)

Thus (yn) is an unbounded sequence, con t r a ry to the assumption tha t thefilt er is stable. This concludes the proof.

An impor tant proper ty of stable filt ers is the existence of a steady-

sta te response to a step-funct ion input . Let a unit -step funct ion be

applied to the filter a t the t ime t = O. The input z. has then the va lue

zero for n < 0 and the va lue one for n a O F or t he cor responding ou t-

put data we have, by Eq. (6),n

zym = wk.

(21)k=l

.

Since t he a bsolut e con ver gen ce of t he ser ies2

W. implies or din ar y con -n=l

‘ The symbol “sgn b“ denot es t h e number t ha t is O i f b = O, is 1 if b >0, andis–lifb <O.



236 FILTERS AND SERVO SYS TEMS WITH PULSED DATA fS ~C. 5.4

vergence, the outpu t y. in Eq. (21) approaches with increasing n the

fin it e va lue.

z~ = wk. (22)k-l

The constant .S represen t s the steady-sta te ou tpu t cor responding to a

unit -step inpu t . It should be remarked tha t the steady-sta te response S

may exist even for an unstable filter , since the ser ies2w. may conver ge

withou t converging absolu t ely.

So far we have always assumed tha t the input z. is zero up to a cer ta in

moment . In the case of a stable filt er the ser ies in the r igh t ihand termof Eq. (6) converges for any bounded infin ite sequ en ce (z.). This follows

from the fact that if M is an upper bound for the absolute values lzm1,the

convergen t ser ies ~ 1-1”w M a majoran t ser ies for the ser ies in Eq. (6);

n=l

the ou tpu t sequence is well defined even if G is differen t from zero for

arbit rar ily la rge negat ive values of n. In the case of stable filters one

may therefore speak of an outpu t for a bounded input which has been

“goin g on for ever .” In par t icular , the response to the constan t un it

input z. = 1 (n = O, ~ 1, +2, “ . . ) is the constant outpu t

(23)

which is, of cour se, equal to the steady-sta te response to the unit -step

input.We shall ca ll a filt er normalized if

.

zw.=.

n=l

(24)

In this case a constan t input is faithfully reproduced by the filter , with-

ou t either a t t enuat ion or amplifica t ion. In the case of a normalized

filt er , each of the outpu t da ta Y- can be regarded as a weigh ted average ofpreviously applied input da ta x*1, x-a , x-a , . . . .

5.4. Sinusoida l Sequences.—Let (z.) (n = O, +1, +2, “ “ . ) be a

two-sided discr ete sequence of data obser ved at equally spaced moments

nT, where T, is fixed once for all. (In this sect ion it is not necessa ry to

assume that the members of the sequence are inpu t da ta of a filter .)

We shall ca ll (x.) a sinusoida l sequence with a frequency of w ra&ans per



SEC.5.4] S IN US OIDAL S EQUENCES 237

second or j = W/2Wcps if

z. = A sin (uT, + O) = A sin (2irnfT, + o), (25)

where A and @ (’‘amplitude” and ‘‘ phase, ” r espect ively) a re constan ts

and A > 0.

We observe fir st of all tha t un like the cont inuous funct ion , the se-

quence of Eq. (25) is not periodic unless the frequencies j and j. = I/T,

are commensurable .’

We fur ther note tha t the amplitude A does not necessar ily represen t

the maximum value a t ta ined by the members of the sequence. 2 For

instance, if f/j, = %and @ = ~/4~ach number in the sequence of Eq. (25)

has one of the two values + (~2/2)A = ~ 0.7A. If, however , f and j,a re incommensurable, the amplitude A is a lways the upper bound

(a lthough not necessar ily the maximum) of the sequence; tha t is, there

a re m embers of the sequence arbit ra r ily nea r t o bu t less than A.

The following remark is of the grea test impor tance: The sequence of

Eq. (25) remains uncha nged if the frequency j is repla ced by the f requency

.f + k.f,, where k is an in teger . No d istinction can be m ade between two

frequencies that d ifler by an in tegral m ultiple oj the repetit ion jrequency.

For insta nce, a sinusoidal sequ en ce of fr equ en cy j, obviou sly consists of asingle number repea ted infin it ely many t imes and is hence the same as a

sequence of the frequency zer o. It is clear from the foregoing tha t by

varying j between O and j, the en t ire range of frequencies is covered.

As a mat ter of fact , every sinusoidal sequence can be writ t en as a

sequence with the jr equency not exceeding one-halj of the repetition jre-

quency. In order to show th is, let us consider two frequencies j and j’,

such tha t f + j’ = f,; such frequencies will be ca lled complementary to

each other . Clea r ly one of these fr equencies, say f’, is S j,/2. Henceit is sufficient t o demonst ra te tha t every sinusoida l sequence with the

frequency j can also be writ t en as a sinusoida l sequence with the fre-

quency j’. NOW

A sin (2@T, + O) = A sin (– 2rnj’T, + O)

= A sin (2rnj’T, + T – o). (26)

Thus a sinusoidal sequence with the frequency .f can also be represen ted

as a sequence with the frequency j’, with the phase changed from b to(lr – @).

As in the discussion of con t inuous sinusoida l da ta , it is conven ien t t o

1By a per iodic sequ en ce is meant a sequ en ce such tha t for a cer ta in fixed m,

zn+~ = Zmfor all in t egers n . A sin usoida l sequ en ce a lwa ys belon gs t o t he class of

sequences ca lled ‘(a lmos t per iod ic” r ega rdless of wh et her or n ot it is per iodic.

2 It is, of cour se, clea r tha t n o member of the sequence can exceed A.



238 FILTERS AND SERVO SYSTEMS WITH PULSED DATA [SEC.5.5

su bst it ut e for t hesequ en ce of Eq. (25) a sequ en ce of complex n umbers,

where c is a complex constan t . With suitably selected c the rea l par t s of

the numbers U- a re the members of the sequence of Eq. (25). The

sequence (27) can be writ t en in an even simpler way if we make the

substitution

~ = @T,. (28)

it n ow becomes

0“ = Cz”. (29)

In thk represen ta t ion z can be an arbit rary complex number of

absolu te value 1. Each jrequency w is represen~ed by a dejinite point z oj

the unit circle; moreover , equivalen t frequencies (tha t is, frequencies

differ ing by mult iples of the repet it ion frequency) a re represen ted by

the same poin t on the unit circle. (Hence the advantage of using a circle

instead of a st ra ight line for represen t ing t he cont inuum of frequencies.)

The frequency co = O cor responds to the number z = 1, and the frequency

equal to one-half of the repet it ion frequency corresponds to the poin tz = —1, whith is one-half of the cir cumference away from the poin t

z = 1. The frequency equal to one-fou r th of the repet it ion frequency

is r epresen ted by the poin t z = j, which is one-fou r th of the tota l cir -

cumference away from the reference poin t z = 1, and so on . When u

is var ied from O to 27rf,, the poin t z makes a complete turn a round the

unit circle in t he cou nt er clockwise dir ect ion . It should be noted that

two point s on the unit cir cle symmetr ic with respect t IOthe rea l axis (or ,

what amounts to the same th ing, two con juga te complex numbers ofabsolu te value 1) represen t complementary frequencies, which , as was

poin ted out before, a re not essen t ia lly differen t . Hence the poin ts of

the upper (or the lower) semicircle, including + 1 and – 1, suffice to

r epr esen t t he en tir e r an ge of fr equ en cies.

5.5. F ilter Response to a Sinusoidal Input .—Since a sinusoidal

sequence is necessar ily bounded, it follows tha t a sinusoidal sequence of

data applied as the input t o a stalie filt er will pr odu ce a well-defin ed

output . In comput ing the output it is conven ien t to replace the sequencein Eq. (25) by an imaginary input sequence of the type in Eq. (27) or ,

equivalen t ly, by a sequence of the type in Eq. (29). The “ response” to

such a complex input sequence is a sequence of complex outpu t data with

rea l par t s represen t ing the ou tput data cor responding to the rea l par ts of

the input data .

From the input sequence

~n = czn = Cemwi”r (n=o, *l, i-2,.”.) (29)



SEC.55] / tES POATSE TO A S INUSOIDAL INPUT 239

one obta ins the ou t put sequence

. .

i,, =

zc%>z“-k = Cz’$

2U)kz-k = Cyz”, (30)

k=l k=,

wh er e t he complex number

y= t“’’z-k = 2W’’-’”’”(31)

k=l k=l

depends only on the fr equency u. Th e sequ en ce (fn )det ermined byEq.

(30)isof exact ly thesame form asthesequence in Eq. (29). Transla ted

in to “rea l” t erminology, th is yields the fundamenta l resu lt tha t (in

complete ana logy }vith t he t heor y of con t inuous filter s) the res pons e ofa

stable filter 10 a sinusoidal input isa%”nusoidal output ojthe sam e frequency.

The change in the amplitude and phase is obta ined in a familiar way

from the complex number Y: The ra t io of the amplitude of the ou tpu t to

tha t of the input is the absolu te value of Y; the differ ence between thephase of the ou tpu t and the phase of the inpu t isthe angular coordina te

ofw. Oneseestha t forz = l,that is, for theinpu t fr equency zero, the

response factor is Y = WI + WZ+ WS+ . “ . ; whereas for z = —1,

tha t is, for theinput frequenc~' ~j,, t heresponse factor assumes the value

Y=–w1+w2–~3 +.... Since the la t ter va lue is a lways a real

number , we are led to the following impor tant conclusion : The phase shift

at the input frequency oj +j, is either 0° or 180°.

Exam ple .—Let W. = K“, where K is a r ea l constan t and ]K] < 1.For the frequency-response ra t io Y we obtain from Eq. (31)

.

y= 2 ~n z-n – ~K K

Z—K= eWTr _ K = cos uT, — K + j sin ~T,~ (32)

n=”l

The ra t io of ou tpu t amplitude to input amplitude is

The outpu t lags behind the input in phase by an angle

~ = tan-lsin uT,,

cos uT , — K.(34)

Regarded as a funct ion of the frequency a, Y is, of course, per iodic

with the per iod%/ T, = %j,,

and according to Eq. (31) the weight ingnumbers W. are coefficien ts in the Four ier expansion of Y as a funct ion of

W. This implies incidenta lly that the numbers w. (and consequen t ly the



240 FILTERS AND SERVO SYSTEMS WITH PULSED DATA [SEC, 5.6

per formance of the filter ) a re completely determined by its frequency

response.

Before concluding this sect ion , let us consider a bounded input that

becomes sinusoidal only from a cer ta in moment on and is completely

a rbit rary up t o this moment (subject merely t o the condit ion of bounded-

ness). Suppose, for instance, tha t Eq. (25) is sa t isfied only for n z O

whereas, if n < 0, the numbers G form an arbit rary bounded sequence.

Let ymbe th e output in th e present case, and let (ysn) d en ote the sinusoidal

response to the sequence that sa t isfies Eq. (25) for all posit ive and

nega t ive values of n . By using the absolute convergence of the ser ies

2W. it is easily shown that

lim (y. – y,.) = O. (35)n- M

In ot her terms, the sequence (y,.) represent s t he steady-sta te ozdput cor -

responding to an input tha t is sinusoidal only after a cer ta in moment ,

without having been sinusoidal “ forever . ” The dt ierence Y. — g..

represent s th e t ransien t par t of the out put , which gradually disappears.

5.6. The Transfer Funct ion of a Pulsed Filter .—In the preceding

sect ion t he qu an tit y

was considered

formal way we

.y= zWkz+ (31)

k=l

only for values z on the unit circle Izl = 1. In a purely

can consider Y as being defined for any complex number.

z for which the power ser iesI

~k~~ converges; for any such number

k=l

the sequence Yz” (n = O, t 1, ~ 2, +3, “ “ ) can be regarded (againfrom a purely formal poin t of view) as the output sequence that cor res-

ponds by Eq. (6) to the input sequencel z“ (n = 0, f 1, t 2, +3, “ ~ ).

Th ese forma l con sider at ion s a re va lid even if t h e filt er is u nst able.

It is well known from the elements of funct ion theoryz tha t the ser ies.

zwkz–kconver ges for ]ZI > R and diverges for Iz I < R, wh ere t he con-

k=l

ver gen ce r adiu s of t h e ser ies R is t he u pper limit of t he sequ en ce

lwm]’/n (n = 1,2, . . . ).

1We sha ll speak in this case of the “output sequence” despite the fact tha t the

sequence z is unbounded unless IzI = 1 (previously we agreed to consider output

sequ en ces on ly in t he ca se of bounded in pu t sequ en ces).

2 See , for instance, E. C, Titchmarsh, Tb Z’heory of Frmdions, Oxford, New York,

1932.



SEC.56] TRAN SFER FUNCT ION 241

In ext r eme cases R may have the va lue O, which means that the ser ies

converges everywhere except a t the poin t z = O, or R may have the value

cc, which means that the ser ies diverges everywhere except a t the poin t

Z=m. In the following we shall a lways assume that R < m or , equiv-a lent ly, tha t a fin ite posit ive number M can be determined in such a way

that

Iwn l s Mn (36)

for every n. Observe that this condit ion is automat ica lly sa t isfied if

the weight ing sequence is bounded and, in par t icu lar , if the filter is

stable. 1

For Izl > R the quant ity Y is an analytic funct ion of z with the valuezero at z= m. By using the process of analyt ic cont inuat ion one

may be able t o assign values of Y even to points with Izl < R (for such

poin ts w will not be represen ted by the power ser ies2

t o~z-~), Th e

.k=l

complet e analyt ic funct ion obta ined by t he pr ocess of analyt ic cont inua-

t ion from the power ser ies may very well turn ou t to be a mult iva lued

funct ion; for example, set t ing w. = ( – 1)”/n , we obta in

Y(z) = – in (1 + z-’),

with infin itely many values assigned to each z.

The t ransfer funct ion of a filter is defined as follows: Given a jilter

(stable or unstable) with the weighting sequence (w,,), the junct ion oj the

complex variable z de$ned by.

y(z) =

z-vkz-k (37)

k=l

and ezten.ded by the process Oj adyt ica l cont inuat ion is called the iransjer

junction oj the jilter.

Example 1.—Let W%= K“, where K is an arbit rary rea l constant .

Then (see t he example at th e end of t he pr ecedin g sect ion)

y(z) = +. (38)

We observe that Y(z) is defined over the ent ir e complex plane

[if the value m is included among the values of Y(z)], despit e the

I It has been sh own in Sec. 5.3 that stability implies t he a bsolute con ver gen ce of.

th e sum2

W. a nd, con sequ en tly, t he bou ndedn em of t he sequ en ce (w-).

n-ls Tit chma rsh , op . cit.



242 FILTERS AND SERVO SYSTEMS WITH PULSED DATA [S .(-, 57

fact tha t the power ser ies by which Y is defined converges only

for Izl > IKI. The poin t z = K is a pole of the funct ion Y(z);

at all t he r ema ining point s of t he plan e Y(z) is r egula r.

Example 2.—Let w. = l/n!. Then

Y(z) = e“ – 1. (39)

In this case Y(z) has an essent ial singular ity at z = O.

Exampte 3.—Consider a filter such that w. = O for sufficien t ly large

n, say for n > nO. Then

y (z) = P (2)2-””, (40)

where P(z) is a polynomial in z; conver sely, when Y(z) is of this

form, then all sufficient ly high terms in the weight ing sequence

vanish and the filter has a ‘(finite memory. ”

It is wor th remarking that the subst itu t ion

z = eTr P (41)

t ransforms the funct ion Y(z) in to the funct ion of p defined by the ser ies.

.2w~e–~’r r . This ser ies is quite analogous to the I,aplace integra l

k=l

1‘Mdt W(t)e–@, which served to define the t ransfer funct ion of a “ con-o–

tinuous” filt er in terms of it s weight ing funct ion W((). In the case of

pulsed servos it is much more convenien t to use the var iable z than p,

since in most impor tan t cases the funct ion Y(z) turns out to be a rational

junction of Z, whereas w, rega rded as a funct ion of p, can never be rat ional

or even a lgebr aic. This last r emark follolvs from the fact tha t z and

consequent y Y are per iodic in p with the imaginary per iod 2~j/ T ,.5.7. Stability of a Pulsed Filter , and the Singular Points of It s Trans-

fer Funct ion.—This sect ion rela tes only to filter s with single-valued

t r an sfer funct ion s. This rest r ict ion is made to at ioid terminologica l

complica t ions; the following discussion could easily be extended t o the

case of mult iva lued t ransfer funct ions by subst itut ing for the complex

plane t he Riemann sur facel det ermined by t he t ransfer funct ion.

With the assumpt ion that Y is single valued, it is clear tha t the weight -

ing sequence is completely determined by the t ransfer funct ion Y (z),since the numbers w. a re coefficien ts in the Taylor expansion of Y(z)

at the point z = cc. It follows that all the proper t ies of the filter are

determined by the funct ion Y(z). The most impor tant proper ty of a

filter is its stability or lack of stability. We shall now prove that the

stability proper t ies of a pulsed filter , like those of cont inuous filters,

depend on the locat ion of the sin gu lar poin ts of it s t r ansfer funct ion .

1E , C. Titchmarsh, The Theory of Funct ion s, Oxforal, hTewYork, 1932.



SEC,5.7] S TABILITY OF A PULSED FILTER 243

We consider fist a stable filter . According to Sec. 5.3, the ser ies. .

I wh converges absolutely. In other worda , the ser iesz

wkz–k

k-1 k=l

converges for z = 1. Hence the convergence radius R must be ~ 1.The funct ion Y(z) is regular for Iz] > R and afor t ior i for Izl > 1 (includ-

ing z = ~ ); all the singular it ies of Y(z) are therefore conta ined inside

the unit cir cle or on its boundary..

On the other hand, if the filter is unstable, then the ser iesz

wkz–k

k=l

doesnot converge absolutely for z = 1, and hence R z 1. We now recall

that by a fundamenta l theorem of funct ion theory, there is a t least one

singular poin t (which may be either a pole or an essent ia l singular ity of

Y) on the boundary Iz[ = R of t he conver gen ce circle. Since R z 1, t his

poin t is loca t ed either in t he ext er ior or on the boundary of t he unit circle.

Combin ing these remarks, we obtain the following fundamental

theorem: If all the singu lar poin ts of the transfer junction are located inside

the un it circle, the jilter is stable. Zj at least one s ingu lar poin t lies ou tk -icfe

the un it circle, the jilter is uns table.

It should be observed that there is one ambiguous case which is not

covered by the preceding theorem, the case of a t ransfer funct ion with all

singular it ies inside or on the boundary of the unit circle and at least one

sin gu la rit y loca ted exa ct ly on t he bou nda ry. In this case the filter may

be stable or unstable. From the physica l point of view such a filt er

should be regarded as unstable, since a very small change in the physica l

constan ts of the filter may throw the cr it ica l singular ity from the bound-

ary in to the exter ior of the unit circle, causing actual instability. In

this connect ion we may fur t her r ema rk that for a stable filter t he posit ion

of t he singular it ies of the t ransfer funct ion indica tes the degr ee of stabil-

it y; the fur ther the singular it ies a re from the boundary of the unit circle

the more stable is the filter .

The above theorem can be illustra ted by the example w. = K“.

We have found for this case

y(z) = &K. (38)

The only singular ity of Y(z) is the poin t z = K, a nd t he t heor em in dica tesstability for IK I < 1 and instabilityy for IK I > 1. The same result fol-

10WSimmediately from the definit ion of stability or from the result of

Sec. 5.3.

In the following sect ions we shall dea l for the most pa r t with filter s

with t ransfer funct ions tha t a re rat ional funct ions of z:

I Since Y(z) is n ot a constant , th ere must be at lea6t on e singu lar poin t ,




(42)

where P(z) and Q(z) a re polynomials’ tha t may be assumed to be without

common factors. The singular it ies of Y(z) a re the roots of the a lgebra ic

equation

Q(z) = 0. (43)

The filter is stable if a ll t he root s of th is equa t ion are conta ined inside the

unit circle. For example, the filt er with the t ransfer funct ion

y(z) = --+22 — 7J Z

is stable, whereas the filter with the t ransfer funct ion

1Y (z) ‘ z~

– 22

(44)

(45)

is unst able.

Compar ing the result of th is sect ion with the theory developed in

Chap. 2, we recogn ized that in our presen t considera t ions the unit circle

plays the role that the ha lf plane to the left of the imaginary axis played

in the theory of cont inuous filter s. This is in accordance with the fact

t ha t t he su bst it ut ion~ = ~T,v (41)

ment ioned at the end of the preceding sect ion t ransforms the unit circle

of t he z-plan e in to t he left half pla ne of t he p-pla ne.

5.8. The Transfer Funct ion In terpret ed as the Rat io of Genera t ing

Funct ions.—It was shown in Cha~. 2 tha t the t ransfer funct ion of a

cont inuous filter can be in terpret ed as the ra t io of the Laplace t ransform

of the output t o the Laplace t ransform of the input . In order t o gain ananalogous in terpreta t ion for t ransfer funct ions of pulsed filters, let us

con sider an in pu t sequ en ce (x”) wit h X* = O for su fficien tly la rge n ega tive

values of n . Suppose, fu r thermore, that ther e exist s a constant M >0

such th at

X. < M“ (46)

for ever y n (th is condit ion is cer ta in ly sa t isfied if the input is bounded).

Th en t he fu nct ion

.91(2) = z Xkz–k

k=–.

(47)

of the complex var iable z is defined when IzI is sufficient ly la rge. We

10bserve that Q(z) must be of higher degr ee t ha n P (z), sin ce by its defin it ion t he

t ransfer funct ion has the va lue zero at z = m.



SEC.5.9] CONCEPT OF CLAMPING 245

shall call th is fun ct ion t he gener at ing junction of the input . In the same

way we associa te with the ou tput (Y.) the funct ion

.

go(z) =

2ykz (48)

k=–-

ca lled the genera t ing funct ion of the outpu t . We now obta in , using

E q. (6),

9+)Y(Z) = 2‘+w’z-k2‘-’’2W’X=’=—. k=l ,,=_m k=l

.

—“z ynz–” = go(z) 1. (49)

,,=—m

It follows tha t t he t ra nsfer fu nct ion Y(z) ca n be expr essed as t he r at io

y (z) = g+).

g,(z)(50)

This resu lt will be useful in the theory of pulsed servos. It cm also

be applied to the t r ea tmen t of two filt ers in ser ies, where the ou tput of thefir st filt er is the input of the second filt er . In such cases the transjer

junction oj the tota l jilter is the product oj the trans jer junctions oj its com -

ponents . Let g,(z), go(z), and g’(z) be the gener at ing funct ions, respec-

t ively, of the input to the tota l filt er , t he ou tput of the tota l filt er , and

the ou tput of the fir st componen t ; then we can wr ite

go(z) = 9&) 9’(Z).

9{(Z) g’(z) g,(z)(51)

This proves our asser t ion , since by the above theorem the two factors in

the r ight-hand term are the t ransfer funct ions of the componen t filt ers.

F ILTERS WITH CLAMPING

5.9. The Concept of Clamping.—Let us suppose that pulsed da ta

measured a t the moments,t = O, 2’,, 22’,, . . . , a re fed in to a device tha t

yields an ou tput with va lue at any t ime equal to the va lues of the input

a t t he immedia tely pr ecedin g pu lse. E lect rica l cir cu it s t ha t per form t hisfunct ion are often refer r ed to as ‘‘ clamping circuits”; we shall speak of

t h eir a ct ion a s “clamping.”

Denot ing by Z. the va lue of the pulsed input to a clamping circuit a t

1This shows, incirfmrtnll.v, that the series defin ing go(z) rmnvcrgcs in the r egion in

wh ich t he ser ies defin in g gl(z) a nd Y(z) m ! bot h m nwwr gc]lt .



246 FILTERS AND SERVO SYSTEMS WITH PULSED DATA ~~Ec.5.10

the t ime t= nT, and by x(t)he va lue of the output a t the t ime t, we

havez (t ) = x. for nTr < t s (n + I) T,. (52)

The funct ion z(t) changes discont inuously at the moments t = nT,, and

its graph is of sta ircase form. It will be noted tha t the va lue of z(t)t

t h e discon tin uit y y poin t nT, has been set equal t o z(nT, – O) = X-L

[and not t o z(nT, + O) = z.]. Th is is, of cou r se, a n a rbit ra ry conven t ion .

Suppose now that the ou tput z(t ) of the clamping device is fed in

tu rn in to a linear filter of the cont inuous type dea lt with in ear lier chap-

t er s. Let y(t ) be the outpu t of th is filter , and let y- = y(rzT,). We can

regard the combinat ion of filter and clamping device as a single puked

jilter t r ansforming the input data Z. in to the ou tput data y.. An example

of such a filt er with clamping is the so-ca lled ‘(boxcar” det ect or used in

radar devices. F ilter s with clamping occur in many applica t ions.

In the following discussion we shall let Y(p) be the t ransfer funct ion

of a cont inuous filter , t o dist inguish it from the t ransfer funct ion y(z)

of a pulsed filter . The quest ion ar ises: How may one compute the

t ransfer funct ion Y(z) of the pulsed filter obta ined by combining a con-

t inuous filter with a storage device as descr ibed above? Since the

beh avior of th e pulsed filt er is complet ely det ermin ed by t he na tu re of t he

cont inuous filter and the t ime constan t T, of the clamping device, itfollows that the t ransfer funct ion Y(z) is completely determined by the

funct ion Y(p) and the t ime constan t T,. In s tu ciyin g th e problem of

comput ing th e fun ct ion Y(z) we shall confin e ou rselves t o cases in which

Y(p) is a m tion alju rwt ion , t ha t is, t o t he ca se of con tin uou s filt er s descr ibed

by a fin ite system of linear differen t ia l equat ions with constan t coef-

ficien ts. Such a filter can always be visualized as an RLC-network.

6.10. Transier Funct ions of Some Specia l F ilter s with Clamphg.—

Befor e passing t o th e genera l t rea t ment of filter s wit h clamping, we shallcon sider some impor ta nt specia l ca ses.

Example 1.—Let Y(p) be con st an t;

Y(p) = K. (53)

The act ion of the filt er withou t the storage device consist s merely in

mult iplying the input by the constan t K. For pulsed input and ou tput

we have

Y. = Kz(nT,) = Kz._,. (54)

Hen ce t he pulsed filter has th e weigh tin g sequence

WI = K, Wz = o, ‘W3 =0,..., (55)

and the t ransfer funct ion is

y(z) = Kz-’. (56)



SEC.5.10] TRANSFER FUNCTIONS OF CLAMPED FILTERS 247

Example 2.—Let

Y(p) = +aJ (57)

where a and B are constan ts. If the filter is stable, then a <0. Aconcrete rea liza t ion of a filt er with t ransfer funct ions of this t ype is the

simple RC-network of Fig. 51. The weight ing funct ion cor responding

to the t ransfer funct ions Y(p), tha t is, the funct ion with Laplace t rans-

form equal t o Y(p), is R

W’(t ) = /3e”’. (58) ,nPUto

output

Let the pulsed input be the unit pu lse ‘o]iagec voltage

o A o

Zo = 1, Zn=o for n # O. (59) FIG,5.1.—SimpleItC.ne twork .

Aft er having been passed through the stor age device, th is input becom es

x(t ) = 1 for O<t~Z’,, z(t ) = o for t s O or t > T ,. (60)

The cont inuous outpu t of the filt er is’

/

.

\

t

y (t) = d , W(T)z (t – T) = d . W(T). (61)o– 1–T,

The values of y(t ) for t= T ,, 2T ,, 3T ,, . . . , const itu te the pulsed out -

pu t sequence yl, y2, y3, - “ “ . As has been shown (see Sec. 5.2), the

ou tpu t sequence of the pulsed filter for a un it -pulse in pu t is ident ica l

with the weigh t ing sequence (w.). Hence

\

n Tr

/

n T,~m = W(T) d, = @ear d, = ~ (en”’ –

(n–l)T,l)e’w’’n~. (62)

(n- I)?’r

F rom this resu lt we obta in for the t ransfer funct ion Y(z) of the pu lsed

filterm .

y(z) = z :(k~k c – ea~~ — l)z–l

2

(emrz-l)k

k=l k=O

=P; (e”’, – l)z-’(l – earvr ’)-’, (63)

or in more symmet r ic form

y(z) = – : g. (64)

This formula is cor rect even for a = O, provided the factor (1 – e-”’)/a

in Eq. (64) is replaced by its limit ing value as a -0, namely, —T,;

h en ce cor respon din g t o Y(P) = B/ P, one has Y (z) = OT~/ (Z – 1). Let

us remark fur ther tha t Example 1 discussed above can be t r ea ted as the

lFor7<0 weeet W(7) =0.




limit ing case of Example 2 for a and B increasing “indefin itely, with con-

stant r at io ~la .

Suppose now tha t a <0 and consequent ly emT’<1. The filt er is

then stable both with and withou t the storage device. Suppose, in

addit ion , that the filt er is normalized; tha t is, - a = B, and consequen t lyY(1) = Y(0) = 1. It is quite inst ruct ive to consider the jr equenc~

r espon se of t he pu lsed filt er , a s t hu s r est rict ed, for fixed filt er in g con st an t

a a nd for va riou s va lu es of t he clamping t im e const ant Z’,.

According to Sees. 5.5 and 5.6, the frequency response a t fr equency

a is obta ined from the values of Y(z) for z = e~”‘“. Now it can easily be

shown tha t as z moves on the unit circle, the point Y(z) given by Eq.

(64) moves on a circle C which has its cen ter on the rea l axis and cuts that

axis at the point sy(1) = 1,

y(. ~) . _ !S

1 + ea~r ()= tanh $ (65)

(see Fig. 5.2). When the point z moves clockwise, the point y(z) moves

counterclockwise. A complete 360° rot a t ion of z corresponds to a com-

plete 360° rota t ion of Y(z); a

rota t ion of 180° of z from z = 1 t o

z = —1 corresponds to a 180’

()aTr

tan h ~rotation of Y(z) from Y(l) = 1 to

04 ~ y(–1). If we denote the point1

w(e~”’r) by P, then the length of

t h e segmen t OP gives t he a tt en ua -

t ion factor and the angle XOP

FXQ. 5.2.—Transfer function 10CUEforgives t he pha se shift cor respon d-

Example2. ing to the frequency W. It is

clear from Fi~. 5.2 that thea t tenua t ion factor has its smallest va lue, tanh (a 2’,/2), for uT, = m, tha t

is, for u equal to-one half of the repet it ion frequency. The phase shift

a t th is frequency is 180°.

Suppose now tha t T, is very small compared with the t ime constan t

– (l/a ) of the filt er . Then the cir cle C is pract ica lly tangen t to the

imaginary axis at the or igin . In th is limit ing posit ion C coincides with

the 10CUSof the point s [– a /(ju – a)] with ( – co < ~ < co), which reP-

resen t the frequency response of the filt er withou t the storage device. Itis, of cou rse, t o be expect ed t ha t”for ver y small T, t he in t roduct ion of the

st or age device ca nn ot ma ke any a ppr ecia ble differ en ce.

If the t ime constant T, is increased, the cen t er of the circle C moves

nea rer and nea rer to the or igin ; there is less and less at tenua t ion . In

the limit case aT , = — w t he cir cle C coincides with the unit circles



SEC. 5.11] TRANSFER FUNCTIONS OF CLAMPED FILTERS 249

around the origin . By Eq. (64) the t ransfer funct ion in this case becomes

y(z) = z_’, which is the same as in Example 1 with K = 1. This is

again evident a pr ior i, since when 2’, is la rge, the filter remembers only

t he last piece of informat ion supplied and consequent ly y. is det ermined

by zfi_, a lone.

Example 3,—Let

(66)

where n is an arbit rary posit ive in teger . The cor responding weight ing

is obta ined by applying the opera tor [l/(n — 1) !](dn–l/da”–l) to the

weight ing funct ion of Example 2. By going through all the steps of the

comput at ion of E xample 2, on e sees ver y ea sily t hat t he r esult ing t ra nsfer

funct ion Y(z) is obta ined by applying the same opera tor to the t ransfer

funct ion computed in Example 2. Hence, by Eq. (64),

(68)

For example, if v = 2,

5.11. Transfer Funct ion of a Filter with Clamping; Stability. -We

are now prepared to compute the t ransfer funct ion when Y(p) is an

a rbit r ar v r a tiona l funct ion .

Y(p) = ~, (70)

where N(p) and D(i) are polynomials in p without common factors and

the degree of the numerator does not exceed the degree of the denomina-

tor . Let al, aa , . , G, be the different zeros of the denominator ,

tha t is, the differen t poles of Y(p). We can then decompose Y(p) into

a fin it e n umber of simple par tial fr act ion s

Y(p) = K +~

~ks

,, (p – ak)”-(71)

Using the same methods as above, we find that the t ransfer funct ion

w(z) is obta ined by adding up the cor responding expressions as given by

Eq. (68). That is,



250 FILTERS AND SERVO SYS TEMS WITH PULSED DATA [SEC.5.11

y(z) = : + 21—c

(

_ ~~. 1 – e“’”

(s – 1)! l%;-’ ffk Z — eak Tr)

(72)

k,,

(The fact tha t the numbers ak and pk. a re not necessar ily rea l has, of

cour se, n o effect on the formal computa t ions. )

It should be noted tha t Y(z) is a ra t iona l funct ion of z, the poles of

which , aside from a possible pole a t z = 0,1 are a t t he point sz Zk = e“~~r .

The degree of the denominator of Y(z) cannot exceed the degr ee of the

denominator of Y(p) by more than 1.

When Y(p) has no mult iple poles, the decomposit ion of Y(p) into

par t ia l fract ions assumes the simpler form

z~kY(p)=K+ —

~p–ak’(73)

and Eq. (72) simplifies to

y(z) = : +2

_ & 1 – euk’r

ffk z — e“h rr(74)

kExample 1.—Let

Y(P) =*l=l– L.

One obt a in s

p+l

y(z) = : – ::::::.

(75)

(76)

Example 2.—Consider next a resonan t filt er . Denot ing the resonan t

frequency of the filt er by w., we can wr ite

(77)

In order to compute Y(z) we fir st decompose Y(p) in to part ia l q

fractions:

( )(p)=J$ *n –-L_. (78)

Applyin g E q. (74), on e obta ins

= (1 – Cos %2’,)1+2

1 + 22 – 22 Cos @.T,”

1This pole occurs if and on ly if N(p) has the me degree as D(p).

z Obseme that two differen t poles al, al of Y yield the ea rne pole of Y if

aI — az = %kj/ T. , wherek is sn in teger.



SEC.5.12] S IMPLIFIED TRANS FER FUNCTIONS 251

Suppose fir st t ha t u.T, is n ot an in tegra l mult iple of m; t he reson ant

frequency is then not an in tegra l mult iple of half of the repet it ion

frequency. The formula above shows Y(e)”.r r ) = cc and Y( – 1) = ().

This means tha t the filt er with the stor age device st ill r esona tes a t

the frequency W. and in addit ion gives comp lete a tte nu ation a t one-

half of the repet it ion frequency. The a t tenuat ion factor of the

filter opera t ing without the stor age device is, of course, differen t

fr om zero for a ll fr equencies . In case unT, = kr, where k is an odd

in teger , on e has

y(z) = + (80)

The pulsed filter resona tes in this case at half of the repet it ion

fr equ en cy. All fr equ en cies except t he zer o fr equ en cy a re amplified,

since 12/(2 + 1)1 > 1 when Izl = 1 and z # 1. Suppose finally

tha t unT , = lm r,where k is an even number . In th is case

y(z) = 1 for z = 1, y(z) = o for z # 1. (81)

This means tha t the filter reject s all the frequencies except the zero

frequency. In an actua l physical syst em (which can only approxi-

mate this idea l case) such a pulsed filt er will act as a low-pass

filter with an ext remely sharp cutoff a t a low frequency.

We conclude th is sect ion with the following impor tan t remark. We

have seen tha t to every pole a of the funct ion Y(p) cor respon ds a pole

euT~of the funct ion Y(z). If the rea l par t of a is nega t ive, the poin tea~, is inside the ufit circle, and conversely. Reca lling the stability

cr it er ia for con t inuous and discont inuous filt er s, we conclude that if a

filter is stable without the storage device, it will r ema in stable with the

storage device, and conversely.

6.12. Simplified Transfer Funct ions for Icw2’,1 >> 1.—The expressions

occur r ing on the r igh t in Ilqs. (72) and (74) can be rewrit t en as follows:

_ pk, 1 – eak’, @k,— —~k z_e a k Tr—z._l akT,

(82)

T, eakT, – 1 – ‘k

SettingakT,

‘Y k = ;=~lJ (83)

and in troducing t he n ew var iable

z—1p=~,

-,one can wr it e Eq. (72) as

(84)

K—– +

~

—e1 LL.

Y(z) = Y(I + ‘*P) = 1 + TrP ~,(s – 1)! aa j-’ ‘y,p – a,

~ (85)



SEC.5.13] FILTERS WITH SWITCHES 253

to the poin t jw on the imaginary axis and can be ident ified with it .

This is in accord with the expecta t ion that for frequencies tha t a re low

compared with the repet it ion frequency there should be pract ica lly no

difference between the responses of the filter with or without storage

device. It would, however , be ent irely wrong to ident ify the two

responses for high frequencies. For instance, the response of the con-

t inuous filt er to the frequency u = mf,, which is one-half of the repet it ion

fr equ en cy, is det ermin ed by t he va lu e Y(m~f,), wh er ea s t he r espon se of t he

pulsed filter to the same frequency is (approximately) determined by

the value Y( – 2j,). This la t t er va lue is a lways rea l, cor responding to

a phase shift of either 0° or 180° (see Sec. 5.5).

The formula of Eq. (87) can be used only when T . is small compa r ed

with all the t ime constan ts of the filter . One might object tha t this con-

dit ion can never be sa t isfied in a concrete physical device, since there

alwa ys rem ains t he possibilit y of small u npredict able t im e lags wh ich ma y

be comparable to the t ime per iod T, or even smaller . In order to meet

this object ion , let us consider the simplest filt er with a single t ime con-

stant T,, h avin g t he t ra nsfer fu nct ion

(88)

Suppose now that this filter has an addit ional t ime lag T , which is small

compared with !7’1. The exact t ransfer funct ion is then

1 1

(

T, T ,Y (P) = AI T~I = T , – T , T IP + 1

)– — (89)

T ,p + 1

By Eq. (85) we have, exact ly,

where 1~1 – 1I is a small number if the ra t io T,/ 7’l is small. The num-

ber Y* is not necessar ily close to 1. The second term in the expression

for Y(z) is, however , very small compared with the fir st t erm because of

t he small coefficien t Tz/ (T l – Tz); it therefore does not make any

appreciable differ en ce whet her this term is left unchanged, simplified by

replacing ~Z by 1, or left ou t a ltogether . The same reasoning can be

applied in the genera l case, where it leads to the conclusion that the

simplified Eq. (87) can be applied without r ega rd for ‘~parasit ic” t ime

lags, provided the “essent ia l” t ime constant s a re la rge compared with

T,.

5.13. F ilters with Switches. -Instead of assuming a storage device

tha t holds the pulsed-input value for the ent ire repet it ion per iod T,, we

could consider a storage device tha t , reta ins the pulse dur ing only a frac-



254 FILTERS AND SERVO SYS TEMS WITH PULSED DATA [SEC.5.14

t ion of the repet it ion per iod. It is not hard to der ive in this more gen-

era l case formulas for the t ransfer funct ion Y(z) that are analogous to

those der ived in Sees. 5“10 through 5.12. We shall t rea t in some detail

on ly the ext reme case in which the pulse is held for a per iod T, small

compared with the repet it ion per iod T ,. Such a system can best be

visua lized as a filt er fed by a cont inuous input and supplied with a switch

tha t is closed only dur ing very shor t per iods of length T, following each

of the moments t = O, T ,, 2T ,, . . . .

During the shor t per iod T , we may rega rd as constan t the fin ite

weigh t ing funct ion W(t ) of the filt er without the switch. When the

switch is in opera t ion , the input r eceived at the t ime t = O ha s t he weigh t

7’,W(nTr) a t the t ime t = nT,. In other terms, the filt er with the switch ,

r egar ded as a pulsed filter , has the weight ing sequence

W. = T , IV(nT ,) (91)

a nd t he t ra nsfer fun ct ion.

y(z) = l“,s

W’(nT ,)@. (92)

~=1

For example, if

Y(p) = *a! W(t) = Bed,

we havem

(93)

In the more genera l case when Y(p) is a r a tiona l t r an sfer funct ion sa t isfy-

ing Y( cc ) = O, we have, in the nota t ion of Eq. (72),

z T . r3’-1 /3k8em’Try(z) = — —

(s – 1)! ikl~-’ z – e.mp”k ,s

(95)

If all the numbers akT, ar e small, we can repla ce the exponent ia l ec’~r in

this formula by the approximat ion (1 + akT,); with the subst itu t ion of

Eq. (84) this yields, a fter obvious simplifica t ions, the approximate

formula

y(z) = y(l + T,p) = # Y(p). (96)1’

Except for the fa ctor T ,/ T , th is is the same as Eq. (87).

SERVOS WITH PULSED INPUT

5.14. Genera l Th eor y of Pulsed Ser vos: Feedback Transfer Funct ion ,

Stability.-In a pu lsed servo the input and the outpu t are considered



SEC.5.14] PULS ED S ERVOS 255

only at discret e t lmest = n l’’,(n = 0, ~1, t2, o “ A ). Weshall denote

by & the input and by f30.t he ou tpu t a t the t ime t= nT,- The servo

outpu t is act iva ted by the pulsed er ror

E. = 81. — eon. (97)

As in the theory of cont inuous servos, we shall assume that the ou tpu t

sequence (oOm )s r ela ted t o th e er ror sequ en ce (cm)as outpu t and input of a

linear filt er . In th is case, however , the filt er will be a pulsed filter . The

t ransfer funct ion Y(z) of thk filter will be ca lled the jeedbaclc t ransfe~

junction of the servo. In order to obta in the rela t ion between the ou tpu t

and the input , we shall assume tha t up to a cer ta in moment both the

input and the output a re zero:

e,. = o, 0.. = o, c. = o, (98)

for sufficient ly la r ge n ega t ive valu es of n.

Following the ideas of Sec. 5.8, we set

.

gl(z) =xO,&–”, (99a)

-m

.

go(z) = zOo.r”, (99b)

—.

9,(2) = zqnz—’ = g,(z) – go(z). (99C)

-.

By the in terpreta t ion of the t ransfer funct ion discussed in Sec. 58 we

havego(z) = y(z)g.(z) = y(z)[g,(z) – go(z)] (100a)

ory(z)

g“(z)= 1 + y(z) g’(z)(100b)

‘The rela t ion between the sequence IO,.) and the sequence (O..) is thus

the same as that between the input and ou tpu t of a pulsed filter with the

t ransfer funct ion

y(z)Y“(Z) 1 + y(z);(101)

y.(z) is the over-all transfer junction of t he ser vo.

Since the pulsed servo is, in effect , a pulsed filt er with the t ransfer

funct ion Yo(z), it is clea r tha t the theory of stability of pulsed servos is

con ta ined in the theory of stability of pulsed filter s developed in the

preceding sect ions of this chapter . Hence the servo will be stable if all



256 FILTERS AND SERVO SYSTEMS WITH PULSED DATA [SEC. 5.14

t he sin gu la r poin ts of t he over -a ll t ra nsfer fu nct ion Yo(z) a re loca ted in side

the unit circle. If a t least one singular poin t lies outside the unit circle,

then th e ser vo is unstable.

We shall su ppose, in wh at follows, th at t he feedba ck t ran sfer fun ct ionY(z) is a ra t ional funct ion of z. Th e sin gu la rit ies (poles) of t he over -a ll

t ransfer funct ion Yo(z) a re then the roots of the algebraic equat ion

1 + y(z) = o. (102)

The stability cr it er ion becomes this: If all the roots of Eq. (102) are

inside the unit circle, the servo is stable. If at least one of the roots of Eq.

(102) lies in the exterior oj the unit circle, then the servo is unstable.

The stability cr iter ion for pulsed servos is, of course, very similar to

t he stabilit y cr it er ion for con tin uou s ser vos; in fa ct on e n eed on ly r epla ce

the ‘unit circle by the left half plane in order to obtain the cr iter ion

(see Chap. 2) for the cont inuous case. As in the cont inuous case, the

stability cr iter ion for the pulsed servo can be brought in to geomet r ic

form. To this end we define the t ransfer locus of the pulsed servo to

be the closed curve descr ibed by the point Y(z), as the point z descr ibes

the boundary of the unit circle in a counterclockwise direct ion . If any

points exter ior to the unit circle a re mapped onto the point – 1 by Y(z),

then at least one of the roots of Eq. (102) is in the exter ior of the unit

circle and the servo is unstable. In such a case, the boundary of the

exter ior of the unit circle will enclose the point —1; this boundary is, of

course, the t ransfer locus. One can, in fact , show that the number of

roots minus the number of poles of Y(z) + 1 is precisely equal to the

number of t imes that the t ransfer locus encircles the point —1 in a clock-

wise direct ion . This is, of course, the analogue of the cont inuous case,

except that the circumference of the unit circle is used instead of the

imaginary axis. If, in par t icula r , the funct ion Y(z) has no poles outsidethe unit circle (which means tha t the servo is stable when the feedback

is cut off), on e can use, a modified form of t he Nyquist cr iter ion: !f’he servo

is stable ij and only ij 1 the transjer locu s d oes not surround the point — 1.

The procedure to be used when Y(z) has poles on the boundary of the

unit circle is analogous to that developed in Chap. 2.

If Y(z) is a ra t ional funct ion of degree 2, Eq. (102) can be writ ten in

t he form

P(z) =z2+Az+B =0, (103)

where A and B are rea l. In this case (he stability cr iter ion , that the root s

of Eq. (103) lie in side t he u nit cir cle, r edu ces t o t he followir ig simple form:

The servo is stable if and only if

1In this and in the rem aiuder of the cha pt er if a root of Eq. (102) lies oa or outside

t he cir cle of con ver gen ce, t hen t he ser vo ie con sider ed t o be u nst able.



SEC. 5.15] SERVOS CONTROLLED BY FILTER WITH CLAMPING 257

P(l)= l+ A+ B>O,

ZJ (-1)=1-.4+B>O,

)

(104)

P(o) = B <1.

The last condit ion excludes imaginary roots of absolute va lue ~ 1

and, together with the first two condit ions, excludes rea l root s ou tside

the in terva l – 1 < z < +1.

Similar but more complica ted cr it er ia can be der ived for equat ions

of h igh er or der .

6.16. Servos Cont rolled by Filt er with Clamping.-The discussion of

the preceding sect ion applies in par t icular t o a servo cont rolled by a

jilter with clamping. Such a servo can be regarded as a cont inuous servoc~nta ining a clamping device tha t holds the er ror signal at a constan t

va lue dur ing the per iod nT, < t < (n + 1) T ,; the rest of the servo is

then act iva ted by the er ror s measured at the instants T,, 2T,, 3T ,, . . . ,

in st ea d of. by t he er ror s mea su red con tin uou sly.

Let Y(p) be the feedback t ransfer funct ion of the servo withou t

the clamping device. If Y(p) is a rational function of p, t h e feedback

t ransfer funct ion Y(z) of the pu lsed servo can be computed from Y(p)

by the method developed in Sees. 59 through 5.12. The stability

equation

l+y(z)=o (102)

ca n t hen be h an dled eit her a lgebr aica lly or geomet rica lly.

Especia lly simple is the case when the t ime in terva l T, is small

compared with all the numbers la~[–1, the CW’Sbeing the poles of the

funct ion Y(p). As was poin ted out in Sec. 5.12, in this case one obta ins

t he appr oxima te formu la

Y(l + T ,p) = Y (p). (87)

The variable p is rela ted to z by

z—1p=~, (84)

and the unit circle in the z-plane goes over in to the circle C (see Fig. 5.3),

11P+~= T ,. (105)

The servo will be stable if and ordy if all root s of the equat ion

l+ Y(p)=o (106)

lie within the circle C defined by Eq. (105). It is eviden t tha t the

graphica l procedu re for determining the stability of the servo consist s

in t racing the locus of the point s Y(p) as the point p descr ibes t h e cir cle




C in a cou nt er clockwise dir ect ion . The number of t imes tha t th is locus

sur rounds the poin t – 1 in a clockwise dir ect ion gives the difference

between the number of root s and the number of poles of Eq, (106) tha t

lie out side the cir cle C. This pr ocedu re differs fr om the ordinary Nyquistpr ocedur e only in the fact tha t the circle C is used instead of th e imaginary

axis.

As an illust ra t ion of var ious methods for determin ing the stability

of a servo, we may consider the simple example in which

Y(p) = KT,p + 1“

(107)

Aside from the clamping device, the cont roller of t he servo is an expo-

nen t ia l-smoothing filter with the t ime constant Z’1 and the gain K.

Subst itu t ing in Eq. (64), we obta inT.——

y(z) =K(l – e ‘:)

——z–e:

(108)

First we apply the stability cr it er ion of Sec. 5.14. The root of the

s tabilit y equa t ionl’,

——

l+y(z)=l+K(l–e~l)=O

z – e-g

(109)

is numer ica lly less than 1 if and only if

l+y(–1) >0, (110)or

m1,

K < coth ~; (111)

the servo is stable on ly when the gain is thus limited.

The same condit ion can be obta ined geom etr ica lly by consider ing the

locus of the point s Y(z), when z descr ibes the unit cir cle. As has been

shown in Sec. 5.10, th is locus is the circle which has its cen t er on the

rea l axis and cu ts the rea l axis a t t he point s

w(l) = K, and Y(–1) = –K tanh ~,. (112)

According to the geometr ica l-stability cr it er ion , the ser vo will be stable

if the poin t —1 is not con ta ined in this circle, 1 tha t is, if

–Ktanh & > –1. (113)

Th is, of cou r se, coin cides wit h t he st abilit y condit ion der ived ana lyt ica lly.

IThe only pole of Y(z) does not lie ext er ior to t he un it circle.



SEC.5.16] CLAMPED SERVO WITH PROPORTIONAL CONTROL 259

Suppose now that 2’, is small compared with T ,. Then we can use

t he simplified geomet rica l cr it er ion , wh ich involves dr awin g t he locu s of

for

(107)

(105)

This locus is a circle in the p-plane, symmetr ic in respect to the real axis

and in tersect ing the rea l axis a t the points

Y(0) = K and()

Y–; =~.2TI

(114)?

–~+1,

The point – 1 lies in the exter ior of this cir~le when

(115)

This is an appr oxima te st ability con dit ion , wh ich , for t he case T ,/ Z ’, <<1,is pract ically ident ical with t he exact condit ion der ived above, since it is

permissible in this case t o r eplace coth (Z’J 2T,) by 2T,/T,.

6.16. Clamped Servo with Propor t ional Cont rol.—We now consider

in some detail the servo ment ioned in Sec. 5“1, in which the er ror (that

is, the angular displacement between the input shaft and the output

shaft ) is measured at the moments T ,, 2T ,, 3T ,, . . ., and in which the

cor rect ive torque is always propor t iona l to the er ror obta ined at the

immedia t ely p receding mea su remen t .

If the er ror were measured cont inuously, we should have an ordinary

ser vo with pr opor tion al gain cont r ol. The feedback t r ansfer funct ion

of such a servo has the form

(116)

where Tm is t he motor t ime constant and K is the velocity-er ror con-

stant . The equalizer of the pulsed servo can be regarded as a filter

with t he t ran sfer fu nct ion Y(p) combined with a clamping device. In

order to compute the t ransfer funct ion Y(Z) of the equalizer (see Sec.

5.11) we fir st decompose Y(p) in to par tia l fr act ions:

Y(p) = $–+ .P+~m

(117)



260 FILTERS AND SERVO SYSTEMS WITH PULSED

It now follows from Eq. (74) tha t

or ,

——

y(z) = ~ –KZ’~(1 – e ‘~) .

z – e-$The stability equa t ion is

1 + y(z) = o,

a ft er clea r ing fract ions,

DATA [SEC.5.16

(118)

(102)

7’.——P(z) = (z – l)(z – e ‘ M)1 + y(z)] = O. (119)

This is a quadrat ic equat ion . According to the remark a t the end of

Sec. 514, the necessary and sufficient condit ions for stability a reP(l) >0, P(–1) >0, P(o) <1. (104)

The fir st of these condit ions is sat isfied ident ica lly. The second

an d t he t hir d, a ft er simple a lgebr aic m anipu la t ions, yield, r espect ively,

& > $ + ~_ :T,,Tm.r 1’

In Fig. 5.4 the values of T,/T~ ar e mar ked

(120)

(121)

on the hor izonta l, t he

values of l/Kl’. on the ver t ica l axis. The two curves represen t the

funct ions on the r ight -hand sides of the above inequalit ies: the regicm of

0 .5

0 .4

0 .3

K% 0,2

0.1

00.1 02 0 .40 .61 2 4 6 10 20 4060 100

~

TaI

FZQ.5.4.—Region of instability.

instability is shaded. The graph

shows that when T , << Tm or

Tm << T,, t h e r ela t ion

KT, <2 (122)

is the approximate condit ion for

stability. For TJ Tm = 3.7, one

has the opt imum stability condi-

tion KT, < 4.2. F or ot her r at ios

T ./ Tm , KTr must be smaller than

a number C tha t var ies between 2

and 4.2. Let us consider a t r iple of va lues T ,, Tm , K, cor r esponding t o

a poin t on the boundary of the region of stability in Fig. 5.4. We shall

dist in gu ish between two ca ses.

1. T,/T~ Z 3.7. In th is case the point (l/K T,, T,/ T*) is on the

boundary of the curve represen ted by the r ight -hand side of the

inequality (120); that is, t he secon d inequality in (104) is r eplaced

by the equa t ion P( – 1) = O. In other words, the equat ion

y(z) + 1 = O has the root z = – 1, and the over-a ll t ransfer func-



SEC. 5.16] CLAMPED SERVO WITH PROPORTIONAL CONTROL 261

t ion Yo(2) is infinite at z = –1. This indicates that the servo

r esonu tis a t t he fr equ en cy if,.

2. T,/Tn <3.7. In this case the th ird of the inequalit ies (104)

is replaced by the equat ion F’(0) = 1. It follows that the equa-

t ion Y(z) + 1 = O has a pair of conjuga te complex root s on the

boundary of the unit circle. The servo then resonates at the fre-

quency cor responding to these roots, that is a t a frequency differ -

ent fr om *f,.



CHAPTER 6

PROPERTIES OF TIME-VARIABLE DATA

BY R, S. PHILLIPS

INTRODUCTION

6.1. The Need for Watist ica l Considera t ions. -Up t o th is poin t we

ha ve limited our design cr iter ia for servomechan isms t o consider at ions

of stability, of suitable damping, and of the na ture of the er ror for a step,constan t -velocity, or constant -accelera t ion input . We have not con-

sidered the er ror tha t will resu lt from the, actua l input tha t the servo-

mechanism will be ca lled upon to follow. Likewise noth ing has been

said about the effect on the servomechanism of uncon t rolled load dis-

turbances or of the effect of random noise sources, which a re oft en found

in the er ror -measur ing device and the servoamplifier . Clear ly the

fundamental ent it y by which a servomechan ism shou ld be judged is

the actua l er r or tha t result s from the actua l input and these random

disturbances. It is t rue tha t the design cr iter ia a lr eady developed,

together with ingenuity and common sense, will in most cases lead to a

sa t isfactory solu t ion of the design problem. It is, however , essent ia l

t o come to gr ips with the basic problem, not on ly in order to obta in a

good solut ion for the except iona l servo system but a lso to build up a

r at ion al a nd syst emat ic scien ce of ser vomech an ism s.’

The actual inpu t to a servomechanism, the uncon t rolled load dis-

turbances, the noise in t er ference, and the actual servo ou tpu t can, in

gener a l, be descr ibed on ly st a tist ica lly. Befor e developin g t h e n ecessa r y

machinery for such a descr ipt ion , we shall discuss some examples of

these quan tit ies .

Usua lly a servomechan ism is r equ ired t o follow many differ en t input

signals. If, on the con t ra ry, the input wer e per iodic, it would in genera l

be simpler to dr ive the ou tpu t by a cam than to use a servomechanism;

the la t t er is most useful when its input is va r ied and to some exten t

unpredictable. The set of input signa ls for a single servomechan ism is

similar to the set of all messages t r ansmit ted by a single telephone init s lifet ime. Like the telephone messages, the input signals w-ill be

confined to a limited frequency band but will he somewha t var ied in

deta il. Fur thermore, in neither case is it possible to predict the fu tu re

with cer ta in ty on the basis of the past . On the other hand, the sounds

in the telephone message are not complet ely unrela ted. ‘I’he remainder

262



SEC.6.1] NEED FOR S TATIS TICAL CONS IDERATIONS 263

of an uncompleted sen tence could in many cases be guessed; likewise,

the possible fu tu re values of a servo input cou ld be predicted if the ext r a -

pola t ion were not ca r r ied too far in to the fu ture. Hence, despit e the fact

tha t the lifet ime input t o a given servomechanism cannot be descr ibedin a st ra igh t forward way as a funct ion of t ime, it is clear tha t there is a

grea t deal tha t can be said about it .

As an example, consider an automat ic-t racking radar system that is

requir ed to t r ack all a ircra ft t raveling through a hemisphere of radius

20,000 yd about the system. Because of the limited accelera t ion to

which aircra ft and pilot can be subjected, a ll the a ircraft t r a jector ies

have about the same degree of smoothness. The loca t ion of cer ta in

aircraft object ives near the t r acking system also induces a degree of

u niformit y in t he t ra ject or ies. In or der to assess precisely the demands

on the system, one would need to know the probability of occur rence

and the st r a t egic impor tance of the differen t possible paths. It is well

to remark that a lthough an exact h istory of the system would furnish

the probability of occur rence of the pa ths, th is in format ion could be

much mor e easily deduced from other sources.

There a re some servo systemsj designed to follow a simple input , t o

which the above considera t ions apply in on ly a t r ivia l sense. This is

the case for a thermosta t designed to mainta in a constan t t empera tu re

in a building; the input signal is simply a constant tempera tu re. It is,

however , necessa ry that the thermosta t be a closed-cycle system in

order tha t it may cor rect the random var ia t ions in the building tem-

pera tu re caused by fluctua t ions in the tempera tu re of the atmosphere.

This is an example of an uncon t rolled load distu rbance. Another

example is the effect of a gusty wind on a. heavy gun-mount servo-mechanism. It is clea r tha t one could not hope to design an ideal thermo-

sta t -cont rolled hea t ing system withou t knowing how the tempera tu re of

the a tmosphere fluctuates; the system must be built to respond to the

dominant frequency band of the atmospher ic t empera tu re fluctuat ions.

In order to descr ibe these fluctuat ions it is again necessa ry to use the

language of sta t ist ics, since it is on ly cer t a in probability funct ions of

t hese flu ct ua tion s wh ich a r e pr edict able.

There a re many other uncont rolled disturbances oper at ing on a servo-mechanism besides the load di&urbance. The most t roublesome are

those which occur where the er ror signal is a t a low power level, as it may

be in t he er ror -mea su rin g device or in t he fir st sta ge of t he ser voamplifier .

For example, in the first stage of the servoamplifier it somet imes happens

that small volt age fluctuat ions caused by thermal agita t ion of elect rons

in a meta l or by the er r a t ic passage of elect rons through a vacuum tube

(shot effect ) a re of the same order of magnitude as the er r or signal. Here

a ga in st at ist ica l con sider at ion s a re ca lled for .



SEC.6.1] NEED FOR STATIS TICAL CONS IDERATIONS 265

h example of a noise source occur r ing in the er ror -measur ing device

is found in ra dar t ra cking systems. H er e t he r ada r beam sca ns con ica lly

about the axis of the t r acking system. The result ing modula t ion of the

rada r signal reflect ed from a target provides informat ion about the er rorwith which tha t ta rget is being t r acked. The received signal is a lso

modu la ted by fluctu at ions in t he over -a ll r eflect ion coefficien t of t he a ir -

craft , caused by propeller rota t ion , engine vibra t ion , and change in the

a irplane’s aspect result ing from yaw, roll, and pitch . This type of dis-

tu rbance is known as fading. An actua l r ecord of the fading in the

received rada r signal r eflect ed from an aircraft in fligh t is shown in

Fig. 6.1. The fading is given in terms of fract iona l modula t ion and the

roils t racking er ror tha t would produce such a modula t ion in the absenceof fading.

The per formance of a servomechanism will depend both on its input

a nd on t hese uncon tr olled dist ur bances. Since the input can, in genera l,

be expr essed on ly in st at ist ica l t erms, a nd sin ce t he dist ur ba nces cer ta in ly

can be only thus expressed, it is clear tha t the ou tpu t of the mechanism

can be assessed only on a sta t ist ical basis. Thus what is of in terest is

not the exact per formance of the mechanism but ra ther the average

per form ance and the hkely spread in per form ance.The uncont rolled factors are not necessar ily uncontrollable. In

most ca ses on e ca n, by pr oper design , con tr ol a distu rba nce so complet ely

tha t it s effect is negligible. In some instances, however , it is impossible

to do this without badly impair ing the mechanism’s usefulness. For

instance, if a filter in a telephone system is designed to t ransmit a mes-

sage, it must of necessity t ransmit some of the ever -presen t noise; if the

noise and the message are in about the same frequency band, one cannot

eliminate the noise without a t the same t im e preven t ing the t ransmission

of the message. This same situa t ion holds for servo systems. Here

it is desired to follow a signal and at the same t ime to ignore the dis-

turbances; as both a re not simultaneously possible, a compromise must

be made. It will be the purpose of the remainder of this book to presen t

and discuss a method for making this compr omise expedien t ly.

This chapter will be devot ed to developing the sta t ist ica l tools

of the theory. Sect ions 6.2 th rough 6.5 furnish background mater ial—

a discu ssion of st at ion ar y r an dom pr ocesses. Although the concept of a

random process is basic in what follows, it is actua lly used in the ca lcula-

t ions on ly t o obta in cer ta in inpu t examples; these sect ions can be omit ted

on a fir st reading. On the other hand, it is impera t ive tha t the reader

understand t he meaning of th e au tocor rela tion funct ion and the spect r al

density if he is t o apprecia te the developments in la ter chapters. The

au tocor rela t ion funct ion is dea lt with in Sec. 6.6; the spect ra l density in

Sec. 6.7; and the rela t ion bet ween the two in Sec. 6.8. The spectr a l



266 S TATIS TICAL PROPERTIES OF TIME-VARIABLE DATA [SEC.6.2

density and au tocor rela t ion funct ion of the filt ered signal a re der ived

in terms of the filt er input in Sec. 69, In Sec. 6.10 t he a ut ocor rela tion

funct ion for the er r or of a radar automat ic-t racking system is der ived;

the resu lt s compare very favorably with exper iment . The remainderof the chapter is devot ed to examples.

6.2. Random Process and Random Ser ies.-A ran dom processl con-

sists of an ensemble of funct ions of t ime having cer ta in sta t ist ica l

properties,

The not ion of a function of tim e y(t) is familiar enough , z An ensemble

of such funct ions is simply a given set of funct ions of t ime. This concept

is most usefu l when these funct ions are typica l r ecords of some physica l

quan t ity taken from a set of essent ia lly similar systems conta ining some

uncon tr olled elemen ts. The member funct ions of a random-process

ensemble need not be completely random, and, in fact , we do not exclude

ca ses wh er e t he fu nct ions exhibit n o r andomness wh at ever .

In genera l, it will not be possible to predict the fu ture values of a

funct ion of an ensemble from its past va lues; nor will the similar ity

between the physica l systems, which genera t e the ensemble, imply

tha t one can predict the values of one funct ion by observing another

funct ion of the ensemble. It is not at all obvious tha t one can formula te

a theory for such an ensemble of funct ions. In order to do so it is, in

fact , necessary to place rest r ict ions on the ensemble. Only those

en sembles wh ich meet t he r equ ir emen t t ha t t her e exist cer ta in pr oba bilit y

dist r ibut ions for the funct ion values will be ca lled random pr ocesses; th e

precise na ture of these dist r ibut ion funct ions will be discussed in the

next sect ion . Random processes a re then subject t o sta t ist ica l discus-

sion; on e can ma ke sta t ist ica l predict ions concer nin g the funct ions of th e

ensemble a nd t he cor respon din g physica l system s.

Examples of random processes a re plent ifu l in na ture. For instance,

we can obta in a random process by recording the fluctua t ing voltages

due to thermal “noise” between two poin t s on a set of iden t ica lly cu t

pieces of similar meta l. The funct ions of another random process might

descr ibe the possible mot ions of the molecu les of gas in a box. In th is

case we assume tha t we have a sufficient ly la rge number of similar boxes

of ga s so t ha t a ll possible init ia l con dit ion s of t he molecu les a re r epr esen ted

with equal likelihood. If we then record the posit ion and velocity

1Severalaspects and applicat ions of the general theory of random processesarereviewedby Ming Chen Wang and G. E. UMenbeck,Rev.Mod Phys. 17, 323 (1945);

by S. O. Rice, l?el~Systim Tech. J our . 23, 282 (1944) and 25, 45 (1945); and by

S. Chandrasekhar , Reu . Mod. Phys. 15, 1 (1943). These papers include ra ther

complet e r efer en ces t o t he lit er at ur e. A mathemat ical t rea tment of the subject

can be found in a paper by N. Wiener , A eta Math . 65, 118 (1930).

2 For each va lue of t , y may consist of a set of nombers. In this case y(t ) tan be

con sider ed m a vect or fu nct ion of t ime



SEC. 62] RANDOM PROCESS AND RANDOM SERIES 267

of each molecule in every box for all t ime, we shall have an ensemble

of funct ion s comprisin g a random process. The fading record shown in

Fig. 61 is a sample of a funct ion belonging t o a random process, genera ted

by t he r eflect ed r adar sign als fr om an en semble of a ir pla nes in all possiblest at es of mot ion .

It is not difficult to devise funct ion ensembles with the sta t ist ica l

pr oper ties of r an dom pr ocesses. I?or example, con sider funct ion s t hat .

assume only the values O and 1 and are constan t th roughout successive

unit in terva ls. We can define an ensemble conta in ing all such funct ions

by sta t ing the probability of occur ren ce of every subclass of funct ions in

the ensemble. For inst ante, we may sta te tha t—

1. All funct ions differ ing only by a translat ion in t ime are equally

likely.

2. The funct ion values O a nd 1 are equally likely in each in terval.

3. The probability of a funct ion taking on the value O or 1 in any

in terva l does not depend on its values elsewhere.

These condit ions can be sta ted more concisely as follows. Let

{

for n+sst <n+l+s

y.(f) = a. }=o, +1 +2,.. ‘—,— (1)

where the a.’s are independent random variables assuming the values O

and I ~vith equal likelihood. For each set of values of the a’s ( ,

a—l, aO, a l, ,), let the probability tha t s lies In any region within the

interva l (O,1) be equal to the length of that region . Then the set of

fu nct ion s obt ain ed by all possible ch oices ofs a nd of t he a ’s will con st it ut e

the random process. It is well to note tha t this defin it ion does not

explicit ly descr ibe any member funct ion of the ensemble as a funct ionof t ime. It is evident tha t t here are funct ions in the ensemble, such as

y(t ) = O or y(t ) = 1, that hold no interest for us, since we are concerned

on ly wit h a ver age pr oper ties. Such funct ions are very rare; in fact the

probability of choosing one at random is zero.

A r an dom series consists of an ensemble of funct ions defined over all

posit ive and negat ive in tegral va lues of an index; often the in tegers

r epr esen t equ ally spa ced in st an ts of t im e. Such an ensemble is a r an dom

pr ocess on ly if it meet s st at ist ica l specifica tion s en tir ely pa rallel t o t hoseplaced on random processes; the proper t ies of random ser ies are exact ly

a nalogou s t o t hose of r an dom pr ocesses.

An example of a random ser ies can be genera ted by a very large group

of men, each busily engaged in flipping his own coin . If the record of

each man’s flips is r ecor ded (heads as 1 and tails as —1), t hen t he r esu lt ing

set of records will form a random ser ies. Th e discr et e r andom -wa lk

problem in one dimension also involves a random ser ies. Here each

member of a la rge group of men takes a unit step either forward or back-



268 STATISTICAL PROPERTIES OF TIME-VARIABLE DATA [SEC.6.3

ward, with equal likelihood, a t successive unit in tervals of t ime; the

record of their posit ions as a funct ion of the number of steps taken is a

random ser ies. The ser ies obta ined by taking the first differences of the

member funct ions of this random ser ies is precisely the random ser ies

gen er at ed by t he coin flippers.

6.3. Probability-distr ibut ion Funct ions.—Before the concept of a

random process can be fully understood it is necessary to discuss proba-

bility-dist r ibut ion funct ions. Let us consider a fin ite ensemble of func-

t ions. At a defin ite t ime t , we can determine the fract ion 6, of the tota l

number of funct ions y(~) that have a value in the in terval between yl

and YI + Ay,. This will depend on the specified YI and t and will be

roughly propor t iona l to Ayl for small Ayl; that is,

al = ~,(v,, ~) Ay,. (2)

The funct ion P,(y,t ) is ca lled the “first probability distr ibut ion. ” Next

we can determine the fract ion & of the member funct ions for which Y(L]

lies in the range (vI, yl + Ay,) a t a given t ime t l and also lies in the range

(Y2, YZ + AY2) at a given t ime IZ. This fract ion is

82 = l’2(y1, L1;Y2, i2) AY1Ay?; (3)

P2 is called the “secon d pr obability dist ribu tion . ” J Ve can cont inuein this fashion, defining the th ird probability dist r ibut ion in terms of

the fract ion of funct ions that lie in three given ranges at three respec-

t ively given t imes, and so on. 1

The probability-dist r ibut ion funct ions so defined must fulfill the

following obvious condit ions .

1.

2.

3.

P. 20.

Pn(yl, tl; y2 , t,; “ ‘ “ ; ylt) ‘n) is a symmetric f~lnct ion in t he set of

var iables y,, t i; y2, tz; . . ; y ., t,,. This is clear , since P. is ajoint probability.

Pk(yl, tl; ; yk , fk )

=/““ b+,“ ~ “ ‘Ynp’’(y]’];~ “’’)tn); ‘4a),-1=

\dy P ,(y ,t ). (4b)

Since each funct ion pk can be der ived from any P%with n > k, t h e func-

tions Pm descr ibe the random process in more and more deta il as n

increases.

Although we have defined probability-dist r ibut ion funct ions for

I If y takes on on ly a discrete set of va lu es, t hen P. will be defined a s a p robabilit y

it self a nd n ot as a pr obabilit y den sit y, Th us P ,(y,,t ,) will be t he pr oba bilit y of ~(t )

t aking on the value VI at the t ime t,, and so on .



SEC.6.3] PROBABILITY-DIS TRIBUTION FUNCTIONS 269

fin ite ensembles, it is clear tha t t he funct ions themselves have stat ist ica l

meaning when applied to infin ite ensembles. Thusi

bdy P,(y,t ) can

a

be thought of as the probability tha t an arbit ra ry member funct iony(t ) of the ensemble lies in the in terval a < y < b at the t ime t . With

this in mind it is now possible to define a random process precisely:

A random process consists oj an ensem ble of junctions oj tim e that can

be characterized by a complete set oj probability-distribution junctions.

It is easy to see that an exper imenta l determinat ion of a probability-

d is t r ibu t ion funct ion Pm is a tedious task. Frequent ly one can determine

the funct ion s P. fr om st a tist ica l con sider a tion s. For example, this is

the case for the random process descr ibed by Eq. (1). It is evident tha tat any t ime there is equal likelihood that y is either O or 1:

P,(y,t) = + fory=Oorl,

P,(y ,t) = o otherwise. 1(5)

Since a. is independent of a,. for n # m, it follows that

P,(y,, t,; y,, t,) = P1(Y1,tl)1’1(Y2, t2) (6)

~vhenever It l — fz[ > 1. On the other hand, if Itl— h S 1,then the

probability tha t both t l and t2 lie in the same unit in terva l is just

(1 – It , – t,l);he common value is then either O or 1 with equal likeli-

hood. The probability tha t t,and t,do not lie in the same unit in terva l

is clear ly lt l – t,(;n this case, as in the case descr ibed by 13q. (6), any

of t he four possible combinat ions of (y I ,y2) occur with equal frequency.

Hence, for IL, – h S 1,

P,(y,, f,; y,, t,) = +(1 – It,– q) + * Ill– f,lforyl=y2=Oorl,

I

(7)

= +Itl— t,l for (Y ,,YZ) = (0,1) or (1,0).

Thus Pz depends on the difference between LIand tq. Th e h igh er pr ot ~a -

bi]ity dist r ibut ions can be discussed in the same way.

To t ake a nother example, consider th e previously ment ioned discrete

random-walk problem in one dimension. Let us suppose tha t in all

exper iments the walker remains sta t ionary at the or igin for n s O and

therea fter t akes a unit step either forward or backward with equal

likelihood at successive unit interva ls of t ime. Because of this init ia l

condit ion we need to consider only n > 0. For this case it can be shownl

1 See, for example, Kfing Chen Wang and G, E, Uhlenbeck, Rev. Mod. Phys. 17,

327 (1945), Th e con dit iona l probability is usually der ived in the litcra tum. This is

the probability tha t an individua l will be at y at t = n if it is known that a t t = O

h e was at the or igin . Beca use of ou r in it ia l con dit ion , this condit iona l probability

is precisely our P, (y,n ) for n z O.




tha t

“(”n’ = r+;~+)’(;)”

(8)

if n and y a re even or odd in teger s together ; otherwise, P1(y,n) = O. It

can likewise be shown for th is problem tha t pk can be wr it t en in terms of

PI; for ins tance,

Z’2(V,, n,; Y2, n2) = P1(VI,nI)P I[(y2 – VI), (n2 – nl)l, (9)

where n~ > nl. In words, the probability tha t an individua l is a t yl

a fter n l steps and then a t YZ after nz steps is equal to the probability

tha t he fir st walks t o y, in nl steps, t imes the probability tha t he t her ea fter

walks a distance (yz — Y1) in (nz — nl) steps.

HARMONIC ANALYSIS FOR STATIONARY RANDOM PROCESSES

6.4. Sta t ionary Random Process.—In most applica t ions the under-

lying mechanism tha t genera tes the random process does not change in

t ime. In addit ion , one is usua lly in terest ed on ly in the steady-sta te

ou tpu t tha t occur s a ft er t he in it ia l t ransients have died down. When

thk is the case, the basic probability dist r ibut ions are invarian t undershift s in t ime.

A r a ndom process ch ar a ct er ized by pr obabilit y-dist ribu tion fu nct ion s

tha t a re invar iant under a change in the or igin in t ime is said to be a

stationary random process. Such a process is descr ibed in increasing

det ail by t he dist ribu tion fu nct ion s:

Pl(y,) dyl = pr oba bilit y of finding a va lu e of a m ember of t he en sem ble

between yl and y, + dyl.

PzIY1, (~1+ T); Y,, (L2+ 7)] dvl dy2 = join t probability of finding a

pa ir of va lues of a member of the ensemble in the ranges (y,,

Y1 + dyl) and (YZ, YZ+ dyz) a t r espect ive t imes t l + ~ and t2 + T.

This funct ion will be independen t of ,; it will be conven ien t to

abbrevia te it as Pz(yl, yz, t),where ~ = h — tl.

P,[y,, (tl + T); Y*, (t! + 7); y3, (t3+ T)]dy, dyz dy, = join t probability

of finding values of a member of the ensemble in the ranges

(Y1, y, + dyl), (YZ,YZ+ dy2), (Y3, Y3 + dy3), a t the respect ive t imes

t l + T, tz+ T,ts+ T;and so on .

This and all similar P’s will be independent of T.

The thermal motion of fr ee elect rons in a meta l a t constan t t empera -

tu re and the Brownian mot ion of molecules of gas in a box at constant

t empera tu re each genera tes a sta t iona ry random process. The random

process descr ibed by Eq. (1) is a lso sta t ionar y.

I



SEC,65] T IME A VERAGEi3 AND ENSEMBLE AVERAGES 271

The discrete random-walk problem, formula ted at the end of the

previous sect ion , does not genera te a sta t ionary random ser ies, since the

set of possible posit ions cont inually increases with n ; the dependence of

PI upon n is shown explicit ly in Eq. (8). On the other hand, the ser ies

produced by the coin flippers is st a t ionary; in this case

P~[Y ,, (W + m ); “ “ “ ; Y k , (W + ?n)] = (*P for

~i=+l(~=l, ”””, k),

1

(lo)

Pk=o otherwise.

As has a lready been noted, the fir st difference of the ser ies obta ined by

the discrete random-walk problem in one dimension is essen t ia lly the

ser ies produced by the coin flippers. Some of the servomechanism

inputs to be considered la ter are similar t o the random-walk ser ies in

tha t they are not themselves sta t iona ry random processes whereas their

t im e der iva tives a re.

6.5. Time Averages and Ensemble Averages.—In dealing with sta-

t ionary random processes it is usually assumed tha t t ime averages a re

equ iva len t t o en semble a ver ages. This is the so-ca lled “ ergodic hypo-

thesis” of stat ist ica l mechanics. It is usually applicable only to sta-

t ionary ra ndom pr ocesses tha t a re (or might be) gen er at ed by an ensemble

of systems for which the uncont rolled elements of any one system

approa ch arbit rar ily nea r t o ever y possible configura t ion in t he cour se of

t ime. In such cases it is expected that any one system can be taken as

represen ta t ive of a proper ly defined ensemble, not only as rega rds the

nature of the possible configura t ions but a lso as rega rds the probability

tha t any given set of configura t ions will be observed. In other words,

since the na ture of the under lying mechanism does not change with

t ime, it is expected that a large number of observat ions made on a

single system at randomly chosen t imes will have the same sta t ist ica l

proper t ies as the same number of observat ions made on randomly chosen

systems at the same t ime. A r igorous mathemat ica l proof of the ergodic

hypothesis has been found for very few systems.

A simple example will serve t o clar ify t he meaning of this assumption .

Consider an idealized billia rd table with per fect ly reflect ing cushions

and a mass poin t as a billia rd ball. Once star t ed, the idealized ball will

maint ain it s speed for ever an d, except for cer ta in specia l init ia l posit ion s

and direct ions, will eventually approach arbit rar ily near to any given

poin t on the table. Now let us define an ensemble of t ra jector ies

star t ing from all possible posit ions and dir ect ions with the probability

of star t ing in any given region and in any given angular range being

propor t iona l to the a rea of tha t region t imes the magnitude of the angular

range. The er godic hypothesis will t hen st ate t ha t t he ensemble a vera ge

of any physica l quant ity defined by posit ion and dir ect ion of a t ra ject ory



272 STATIS TICAL PROPERTIES OF TIMh’- VARIABLE DATA [8 EC. 6.5

will be equal to the t ime average for any one of the nonper iodic t ra jec-

tor ies, For instance, the average t ime spent by any nonper iodic t ra jec-

tory in a given region of the table is propor t iona l to the area of that

region ; for th is is cer ta in ly t rue of an average over the ensemble at thet ime when the ensemble is set up and likewise at any other t ime. In

making an ensemble average in this case it is actually not necessary to

exclu de per iodic t ra ject or ies fr om t he en sem t)lr , since t hey occur , in any

case, with zer o probability,

The ergodic hypothesis can be given a more formal sta tement as

follows: Let ~(yl, vi, . . . , v.) be an arbit ra ry funct ion of the var iables

y,, v2, . , y~, and let VI = Y(L + 71), YZ = Y(L • 1-72),

We ~h a~l ‘d~~o~e(t + 7.), where y(t ) belongs to a random_ process. ~

the t ime average of F for yi = y(~ + ,i) by F, and the ensemble average

by ~. That is, by defin it ion

/F=lim 1 T dtF[y(t +71), . . . ,

T -iw~T .Ty(t + 7.)] (11)

and

‘=/”” \ dy1”””dy$(y1y2 J ””yn)x Pn[yl, (t + T1); “ ; y“, (t + 7fl)]. (12)

The average ~ is clea r ly independent of t ; ~ will be independent of t

if t he process is sta t ionary. The ergodic hypothesis sta tes:

If the random process is stationary, then

F=F (13)

with a probability oj 1.

A few examples will serve to illust ra te the significance of th is hypo-

thesis. We can determine for any random process the ensemble average

or mean of y, a t th e t ime t , from the first probabilityy dist r ibut ion I’l(y,t):

~=/

dy yP,(y,t). (14)

From the way in which P, is determined, it is clea r that J is the mean of

all the y(t)’s of the ensemble. For a sta t ionary random process, P,(y,t)

and consequent ly ~ do not depend on t. On the other hand, the time

aver~e, which is defined as

\

jj=lim-!- T d t y(t),T -MO 2T -T

(15)

will, in general, differ for the var ious funct ions of an ensemble. The

ergodic hypothesis sta tes that for a sta t ionary random process these two

methods of averaging give the same result , no matter a t what t ime the



SEC. 66] COIiIWLA T ION FIJNC1’IONS 273

ensemble average is made or with what funct ion (except for a choice

of zero probability) the t ime average is made. That is

5=V (16)

The same thing can be said for the higher moments of the dist r ibut ion

PI ) which can be determined by set t ing F(y) = y’, The nth moment is

defin ed a s

7=/

dy g“p,(y,t). (17)

For a sta t ionary random process this is equal to the t ime average

(18)

The second moment is ca lled the “mean-square” value of y, and its

square root @ is called the “ root -mean-square” (abbrevia ted rms)

value of ~. From the fir st and second moments one can der ive the

variance

(Y – Y )z = 7 – (v)’

—/

dy (y – j) ’~,(y,t), (19)

which is a measure of the width of the dist r ibut ion Pl(y,t) about it s a ver -

age value ~.

6.6. Cor rela t ion Funct ion s.—The aut ocor rela t ion function of a func-

t ion y(~) is defined as the t ime average of y(t)y(t + 7). It is a func-

t ion of the t ime interval 7 and of the funct ion y. It will be denoted by

h?V(~)or , where th is is not ambiguous, by R(7). By defin it ion , t her efore,

R(7) = y(t)y(t + r) = lim ~/

Tdt y(t)y(t + 7).

T+m 2P _~(20)

As we have seen in the previous sect ion, in the case of a sta t ionary

r andom process R(~) will not (except for a choice of zero probability)

depend on the member of the ensemble on which the t ime average is

per formed. Fur thermore, the t ime average of ~(t )y(t + r ) will be equal

t o t he en semble a ver age

y(t)y(t + ,) =— H dy, dyz YIYJ’2(Y1,Y2,7) (21)

with probability 1, Thus one can define an autocor rela t ion funct ion

for a sta t ionary random process as well as for a single funct ion . This

funct ion gives a measure of the cor rela t ion between y(t l) and y(h), where

t t – t l = r . In case y(t l) and y(h) are independent of each other ,

P,(y,, t,;y,, t,) = P1(Y1,W1(IM2) (22)



274 S TATISTICAL PROPERTIES OF TIME-VARIABLE DATA [SEC. 66

and .—I?(t, –h) = y(k)y(t,) = y(tl) v(k). (23)

For noise this situa t ion is approximated when the t ime in terva l r is

sufficiently large.

Some of the proper t ies of R(r ) a re fair ly evident .

1.

R(0) = ~. (24)

2. R(r) is an even funct ion of r , since

R(T) =y(t)y(t +T)=Y(t– T)U(t) ‘Y(t) Y(t– T) ‘R(–T). (25)

3. [R (~)l S R(0). This r esult s fr om t he in equality

O ~[y(t ) ~y(t+T)]’ =y2(t )+y2(t +7) +2v(~)Y(~ +7). (26)

Hence

+2y(t )y(t +7) ~y’(t ) +y’(t +7). (27)

Averaging both sides of this equat ion gives

+2R(,) ~y~+y’(t +,) = 2R(0). (28)

4. Given any set of r ’s (T1, 72, . . . , T.), the determinant

r

(7,–71) R(Tl~Tz ) . . . . . R(T1-Tn)

(T ,–T I) R(T , –72) . . . . . . R (T2– Tn)

. . . . . . . . . . (29)

. . . . . . . . . . . . . . .

R (~m ‘T ,) R (rn–7J . . . . . “ R(T . –7.)

is symmet r ic and nonnegat ive in value. It can be shown tha t con-dit ion (4) is a n ecessa ry a nd su fficien t condit ion t ha t R(7) be an au to-

cor rela tion funct ion . 1

It is somet imes convenien t to work with the funct ion

p(7) =[Y (t) – iil[v(~ + r ) – ?71,

(30)j5_g2

wh ich will be called t he normalized autocorrelation function. It is eviden t

tha t p(0) = 1 and that for noise P(7) ~ O as T ~ @.

The autocor rela t ion funct ion for a random ser ies is defined as

N1

R(m) = lim —z

Y.Y.*.N+. 2N+ 1

n--N

It is clea r tha t R(m) has proper t ies ana logous to those of R(r).

I A. Khiotchine, “Korr elat ionet heorieder Sta tion&en Stocha stischenProzesse,”

M@. Ann. 108, 608 (1934).



i?.Ec. 6.6] CORRELATION FUNCTIONAS 275

Examples.—It will be inst ruct ive to consider a few examples. In the

case of the purely incoheren t sta t ionary random ser ies genera ted by the

coin flippers one would expect R(m) t o decrease very rapidly with

in cr ea sin g m . As was seen in Sec. 6.4, P1(yl) = 4 and PZ(yl,yz,m) = ~,

where y 1 and gz take on the values + 1. It follows from Eq. (21) that

R(0) = 7 ,Wl(?h) = 12 x *+ (–1)’ x + = 1, (32)4

and that for m

R(m ) =

——

#o

zYcyipz(YiJYi,m).

(1)(1)+ + (1)(–1)3 + (–1)(1)+ + (–1)(–1)+

I(33)

o.

The other ext r eme is represented by the’ funct ion y(t ) = 1, for which

R(T ) = lim ~/

Tdtlxl=l. (34)

T-. 2T –T

If y(t ) is per iodic, t hen t he per iodicit y per sist s in R(T). For example,

le t

y(t ) = A sin (d + @). (35)

Then

R(T) = y(f)~(~ + 7) = lim ~/

Tdt ~ sin (cd+ ~)~ sin (t it+ cOT+ +)

T-. ZT –T

[

= ~im & Cos ~, _ ~sin (2wT + Q. + 24)

T -)cc 2 2T 2U1 sin (2t iZ’ – U, – 2@)——2T 2U 1

A,. — Cos UT.

2(36)

Although R(7) has the per iod of y(t ), it is an even funct ion independent

of the phase @ of y(f). For the funct ion

y(t ) = A, +2

A, sin (co,t + Ok), (37)

wher e G # con for n # m, the autocorrela t ion funct ion is

z

Al~(r ) = A~ + ~ COSu,,. (38)



276 STATIS TICAL PROPERTIES OF TIME-VARIABLE DATA [SEC.6.6

Here again the per iods of y a re presen t in R(r), but the phase rela t ions

have been lost . If an apparent ly random funct ion conta ins hidden

periodicities, R(r) will a ppr oa ch asympt ot ically an oscilla tin g fu nct ion

like tha t in Eq. (38).In the case of the sta t ionary random process descr ibed by Eq. (l),

we can obtain the autocorrela t ion as an ensemble average by means of

Eq. (21). The second probability dist r ibut ion P, is given in Eqs. (6)

and (7). If ~ = t l — k is grea t er in absolu te va l!le than 1, then , applying

Eq. (6), we obtain

R(,) =z

yl@2(Y1, Ll;Y2~2)

=+(OXO+OX1+l XO+ 1X1)=+. (39)

On the other hand, if 171s 1, we make use of Eq. (7):

R(r) = [*(1 – Ir \ ) + ~171](0 X O + 1 X 1) + ~1~1(0X 1 + 1 X O)

= * – +ITI. (40)

It follows from Eq. (5) that ~ = + = ~. Hence the normalized auto-

cor rela tion fu nct ion is

,0(,) = o for ITI> 1,

= 1 – 171 for 171S 1. )(41)

To obta in an au tocor rela t ion funct ion from exper imen ta l da ta , one

of necessity sta r t s with a r ecord y(t ) of fin ite length : O s t s T . If the

data are not dkcrete, it is possible to consider them as such by using on ly

the va lues at t imes t= nA (n = 1, 2, . . ~ , N = T/A). The t ime

interva l A should be chosen so small tha t the funct ion y(f) does not vary

sign ifican tly in a ny in ter val A. If y(t ) is to be used as an input to somemechanism, it is sufficien t that A be small with r espect to the system

t ime constants. Set t ing Y; = y(nA), then

N–m

R(m) = -&-z

Y.Y.*, for m ~ O. (42)

n=l

Equat ion (42) loses its reliability for very la rge m. For , as can be seen

from Eq. (36), in working with a fin ite in terva l T a r ela t ive fr a ct ion aler ror of about l/u T is in t roduced for each per iodicity present . The

er ror in determining th e con tr ibut ion of a per iod P t o the au tocor rela t ion

funct ion will be less than 2 per cent if 27r [(N – m)A/P] >50. Hence

for t his pu rpose T = NA should be about 10P, and m should not exceed

iT/A.

The normalized autocor rela t ion funct ion of the fading data shown

in Fig. 6.1 was obta ined by use of Eq. (42). The per iod of observa t ion



SEC. 6.6] CORRELATION FUNCTIONS 277

was T = 20sec. Thevalue of Awaschosen to be firsec. The resu lt ing

funct ion p(~), ca lcu lated over the range O S r s 1.28 see, is shown in

Fig. 6.2. As can be seen , there is very lit t le cor rela t ion between data

more than 0.1 sec apar t . It follows from this tha t a very conserva t ive

choice of T was made; 2 or 3 sec would have been adequate.

+ 1.0

+ 0,8

+ 0.6 -

c +0.4

<

+ 0.2

0 n \ ~ / \ / \/

-0.20 0.16 0.32 0.48 0,64 0.80 0.96 1.12 1.28

7 in sec

FIG.6,2.—Normalized au t ocor r ela t ion funct ion for fading r ecord. The rms va lu e is 10.3mik.

Cor rela tion Ma tr ix.—If y is a two-dimen sion al vect or (u ,v), on e defin es

a cor rela t ion mat r ix instead of a correla t ion funct ion. For a sta t ionary

r an dom pr ocess, t he er godic h ypot hesis gives

1 a=l= ‘1(t )u(t + T) U(t )v(t + T) U(t)u (t + T ) U(t)o(t + T )

?J (t)u(t + T )(43)

O(t )u (t + T) V(t)u(tT)“

The funct ion of r , u(t)z’(t + T ), is ca lled the cross-correla tion funct ion

and will be designa ted as Rt i~(7).

The cross-cor rela t ion funct ion is not symmetr ic, nor can one in ter -

change the order of u and v withou t changing its va lue. There does,

however , exist t h e r ela tion sh ip

R..(T) = IL.( –T). (44)

By usin g t he in equ ality

[ 1(t)J (t+ T ) 2

Os —fl~ - @

(45)

and averaging over t ime, one finds tha t

(46)

Ru,(.) is a measure of the coherence between u(t ) and o(t+ T).



278 STATISTICAL PROPERTIES OF TIME-VARIABLE DATA [SEC. 6.7

6.7. Spect ra f Density—We shall have occasion to consider the effect

of a filter on the funct ions of a random process. It is n at ura l, t her efor e,

t o at tempt to resolve the funct ions of a sta t ionary random process in to

their Four ier component . Such an at tempt will, in turn , lead us tothe concept of the spectral density G(j) of a funct ion ~(t ) and of a

stat ionary random process to which y(t ) belongs. If y(t ) is the voltage

across a unit resistance, then G(j) df is the average power dissipated

in the resistance in the frequency interva l (f, f + df). If y(t ) is the input

t o a linear filt er with t ran sfer fu nct ion Y(2rjf ), then, as will be shown in

Sec. 6.9, the output has the spect ra l density lY(2mjf)1’G(j).The spectr al den sity of a fu nct ion y(t) is defin ed in t he following way.

Let

!/2’[0 = Y(t) for– Tsfs T,

o elsewhere. )(47)

.

The Four ier t ransform of y.(t)lways exists ~nd is by definit ion

If A * denotes the complex conjugate of A, then , since y. is rea l va lued,

A,(f) = A$(–f). (49)

The average power density for y,(f) a t the frequency f is[ 1A,(f) 12]/2T;

both posit ive and negat ive f must be taken in to account . Since

lA~(f)I = IA=(–f) 1, one can limit a t t ent ion to posit ive va lues of j

and take the average power density to be [IAdf )I*l/T. The wec~~aldensity of the funct ion y(t ) is defined as the limit of this quant ity as Tgoes to infinity;

G(f) = lii~; lA,(f) l’. (50)

As should be expected, one can obtain the average power in y by

in tegra t in g t he average power densit y G(f) over all posit ive frequencies.

In symbols

\

j?= ~“ dfG(f). (51)

This can be proved as follows. From Eqs. (48) and (49)

\ mf1AT(f)12J:.df[A~(-f)/ -”md52.In terch an gin g t he or der of in tegra t ion gives

/ - ‘f1AT(f)12= l-mm (’’(y’’):mfmfA~( ~f-e’”-’”’’fll” ’53).It follows from the Four ier integral theorem (see Chap. 2) that the



SEC, 67] S PECTRAL DENS ITY 279

quant ity inside the brackets is just y~(t ). Hence

/

m

/

T

dL7J ;(Q = dty’(t) =

/

m djl~df)l’

—m –T —. r.

Dividing through by 27’ and passing tothelirnit give

Ifwe noninterchange the order of limits, inser t ing G(j) for it s equiva-

lent , t he pr oof of I@. (51) incomplet e.The hbove discussion of the spect ra l resolut ion of the funct ion y(t )

has dealt with the power ; as a consequence, all informat ion concern ing

the phase rela t ionships of y has been lost . It is vdl t o remark on the

difficu lt ies encountered in at tempt ing to deal more direct ly with the

F our ier t ransform of~. The Four ier t ransform itself willexisto n lyify (t )

a ppr oa ch es zer o a st becomes in fin it e. F or funct ions such as t hose found

in sta t iona ry random processes, A ~(j) will, in gen er al, eit her oscilla te

or grow without bound as 2’ becomes infinite. Even a mean Four iertransform such as lim (1/2T)A ~(j) or ~~m~ [1/(22’) ~f]A.(j) will oscilla te

T+ ~

ra ther than approach a limit if y is, for example, a “per iodic funct ion”

with a suitably varying phase.

Cross-spect ra l Densi ty.—I t is possible t o defin e a cr oss-spect ra l den sit y

for two funct ions (u,v) as follow-s. Let

and

Th e cr-oss-sp ectTa l d en sit y is then defined as

(56)

(57)

F rom t he r ela tion s A ~(.f) = A ~(’–j) and B.(f) = B;( –j) one obta ins

G..(j) = G:.( –f) = Gw( –j). (58)

Hidden Per iodiczt ie.s—The spect r al den sit y G(j) may con ta in sin gu la r

peaks of the type associa ted with the Dirac delta funct ion . 1 This will

1Th e Dir ac delt a fu nct ion h as t he followin g pr oper ties:

I~’dy a(y) = /0 i y6(y) = ; for a ll c >0—,

and

b(y) = o for y # o.



280 S TATIS TICAL PROPERTIES OF TIME-VARIABLE DATA [SEC. 6.7

be the case if the mean value of y is not zero or if y conta ins hidden

per iodicit ies. The peaks occur a t frequencies a t which (1/2’) li4,(~)12

becomes in fin it e wit h T; the coefficien t of the delta funct ion at such a

frequency is given by lim (1 /2T2) 1A ~(j) 12. We shou ld ther ef or e r edefin eT+ m

G(j) a s follows:

[ 1(f) = ;lm~ & lA,(f,)12 ~(f – fl)except wh er e this

is zer o,

I

(59)

G(f) = lim ~ lAT(f)12 otherwise.T+.T

If the mean of y(t ) is not zero, then G(j) will have a singular ity

a t t he or igin :

G(j) = 2(ii)’L5(j) + G,(j). (60)

For pure noise the peak at j = O, cor responding to the d-c term, will

usually be the only peak, and Gl(j), defined by Eq. (60), will be a

regular funct ion represen t ing the cont inuous spect rum. In this case

it is somet imes convenien t t o in t roduce t he normalized spectral density:

s(j) =

GU)

/o-djG,(j)” (61)

This quant ity has the dimension of t ime, since j has the dimension

(t ime)–’. The denominator in Eq. (61) is simply the var iance of y:

(62)

If y(t ) is a t r igonomet ric polynom ia l

y (t) = Ao +z

A, sin (2~jk t + @, jk # O, (63)

then the spect ra l density is

G(j) = 2A~6(j) +z

$ b(~ – [j,]). (64)

In th e genera l case of noise with h idden per iodicit ies t he spect ra l density

will consist of a cont inuous par t and a number of peaks a t discrete

frequencies.

Spectral Density oj a Stationary Random P rocess.—The member fu nc-

t ions of a sta t ionary random process will (except for a choice of zero

probability) have the same spect ra l density. This, then , may be called

the “spect ra l density of the sta t ionary random process. ” In comput ing

it , one can deal with any typica l funct ion of the ensemble or , as desired,

can ca r ry out averages over the ensemble.



SEC. 67] S PECTRAL DENS ITY 281

As a fur ther example, let us obta in the spect ra l density of the sta -

t ionary random process descr ibed by Eq. (1). Since the funct ions a re

given only in sta t ist ica l t erm s, it will be necessa ry t o obta in the ensemble

average of the spect ra l density. Since, however , the process is sta -t ion ar y, t his will be t he spect ra l den sit y for t he in dividu al fu nct ion s except

for a set of probability zero. Since s serves on ly to shift the t ime axis

and since the spect ra l density is independent of phase, it is clear tha t we

can suppose s to be zero. To simplify the ca lcula t ions let us subt ract

ou t the mean at the very star t . We have then

y(t ) = a. – + forn~t<n+l, (65)

where the an’s are independent random variables and t ake on the values

O and 1 with equal likelihood. By Eq. (48),

N–1

A.(f) ==z (“- ~) f+’’’’-’m ’”

.=— NN–1

= y ~ (.. - ;)e-,.f(w.

1

(66)

.=— N

Squaring the absolute va lue of A~(j) and dividing by i’i give

We now take the ensemble average of 13q. (67). Since

(~:; ifn #m,

ifn=m, )(68)

x

it follows that

()

F yf ’;.— . (69)

Taking the limit as N becomes infin ite, we obta in the average spectr a l

density:

(70)

Spectral Density of a Random Ser ies.—The spect ra l density for a

sta t ionary random ser ies is defined in an analogous way. Given the t ime

ser ies v., consist in g of da ta r ecor ded a t un iform int er va ls of len gt h T.,’ we

define, in analogy to Eq. (48),u.

A , ( f ) = T.z

y~e–z”’l “r’. (71)

..—.V

1The quan t it y T . is t he same as the repet it ion per iod of Chap. 5.



282 S TATIS TICAL PROPERTIES OF TIME-VARIABLE DATA [SEC. 67

Th is fu nct ion is per iodic in f wit h per iod 1/2’,; it is complet ely det ermin ed

by its va lues in the range (–1/2T,, l/2T,). Since the Y. a re rea l,

A .J f) = A:( –j). We then define the spect ra l dcnsit y of the t ime

ser ies, in ana logy to Eqs. (59), as

[

1W) = ,;~mm 2(N + i)2T :

‘1

l~vul)l’ ~(f – jl),

except where the quant ity in the bracket is zero, and as

1G(j) = ~l:mmN + ii)~, lA,df)l’ elsewhere.

)

It follows from the per iodicity of .4 ~(j) and the or thogonality

funct ions e-’”jfm’r tha t

$’= ~llf’A,(~”2 = ~1’’2T’d~’AN(f)’2–N

Hence

(72)

of the

(73)

(74)

The spectra l density of a sta t ionary random ser ies is, of course, defined

as the spect ra l density of any typica l ser ies from that ensemble.

0.10

0.08

~ 0.06mc-

<; 0.04

0.02

00 2 4 6 8 10 12 14 16 18 20

Frequency in CPS

F IG. 63.-Norma lized spect r al den sit y of fading r ecord.

~pectral Density jor Experimental Da ta .—Ther e a r e sever a l pr ocedur es

whereby one can obtain the spect ra l density from exper imenta l data .

The best method is to calcu late the spect ra l density as the Four ier tmns-



SEC. 68] CORRELATION F UNCTIONS AND SPECTRAL DENS ITY 283

form of the autocorrela t ion funct ion . This will be expla ined in deta il

in the following sect ion . F igure 6.3 shows the spect ra l density obta ined

in this way for the radar fading record shown in Fig. 6.1.

It is, of course, possible to obta in the Four ier coefficien ts for a fin ite

length T of data and then to compute G(j) by means of Eq. (50),

with the limit process omit ted. Numerica l-in tegra t ion methods require

tha t the data be discrete. From N pieces of data it is possible to obta in

N/2 harmonics. This is, however , a very tedious task even when one

takes advantage of shor t cu ts. 1 Var ious machines have been devised

to obta in the Four ier coefficien ts; one such is the Coradi harmonic

a na lyzer , wh ich per forms t h e r equ ir ed in tegr at ion , h armon ic by h armon ic,

as the appara tus is dr iven so as to follow the curve represent ing the data .

It is a lso possible to have a voltage follow the data and send the voltagethrough a wave analyzer .

6.8. Th e Rela t ion between the Correla t ion Funct ions and th e Spect ra l

Density—Both the autocorrela t ion funct ion and the spect ra l density

depend on the product of the funct ion y(t ) by it self. Both funct ions

likewise depend on t he per iodicit ies in t he message but a re independent

of the rela t ive phases of the Four ier components. It is therefore not

surpr ising to find that they are Four ier t ransforms of each other .2 In

fact , as will be proved la ter

and

!(T ) = “ dj G(j) COS 27rfro

/G(j)=4 “ dr R (,) COS 27rj7.

o

(75)

(76a)

Similar rela t ions hold for the normalized funct ions p(~) and S(f); for

instance

/s(j) = 4 “ d , P(7) COS 27rj7.

o(76b)

This in t imate rela t ionship bet ween th e spect ra l density and t he auto-

correla t ion funct ion sheds more light on the in terpret a t ion of each . For

example, a delta -funct ion singular ity in the spect ra l density at the

frequency jl cor responds to a cos 27r j17 t erm in the autocor rela t ion

funct ion . An exponent ia l-decay autocor rela t ion funct ion, such as is

I There exist shor t -cu t methods for the cases N = 12 and N = 24. See, for

instance, E. T. Whit taker and G. Robinson , The Ca fcdu .r of Observa tion s, Blackie,

Gla sgow, 1929, pp. 26W264.

ZThis rela t ion is conta ined in a paper by N, Wiener, Act s Math. 55, (1930). Th e

reader will find in this r eference a r igorous trea tment of the subject mat ter of the

presen t chapter , which , incidenta lly, avoids use of the delta frmct ion by working

wit h t he in defin it e in tegr al of t he spect rum G(f).



284 STATISTICAL PROPERTIES OF TIME-VARIABLE DATA [SEC. 6.8

somet imes a ssocia ted wit h noies,

R(7) = e–~1’1, (77)

has th e spect ra l density

G(j) = Qj3’ + (%f)’”

(78)

This fu nct ion is bell-shaped, decr easin g t o half its zer o-fr equ en cy ampli-

tude at f = fI/2u.

Aside from their theoret ical impor tance, Eqs. (76) a re ext remely

useful as an aid in the calcula t ion of the spect ra l density. It is oft en

simpler to compute the, autocor rela t ion funct ion first and then the

spectra l density by means of Eqs. (76) than to compute the spect ra l

density direct ly. This is, for instance, the case with the sta t ionary

random process descr ibed in Eq. (1). The autocorrela t ion funct ion

[Eqs. (3~) and

subt ract ion of

becomes

(40)] is easily computed as an ensemble average. On

the cont r ibut ion due to the square of the mean, this

R,(7) = +(1 – 171) for 1~1S 1,. 0 elsewhere. 1

(79)

Applying Eqs. (76), one obtains very simply the spect ra l density that

was der ived in Sec. 6.7 by a ra ther involved argument:

(70)

A second example is provided by the normalized spect ra l density

(Fig. 6.3) of the radar fading record shown in Fig. 61. This was obta ined

by means of Eqs. (76) from an approximat ion to the normalized auto-

cor rela tion fu nct ion of F ig. 6.2:

p(~) = e–z~l’l cos 40T. (80)

This funct ion is plot ted, a long with the exper imenta lly obtained auto-

correla t ion funct ion , in Fig. 6“4; it has all of the proper t ies of an

autocor rela t ion funct ion . On compar ing the spect ra l density and auto-

cor rela t ion fu nct ion, on e sees th at t he p,ea k in t he spect ra l den sity occu rs

a t t he fr equ en cy of t he damped oscilla tion of t he a ut ocor rela tion fu nct ion .

If the Four ier coefficients wer e t o be obta ined dir ect ly from Eqs. (76)by some method of numerical in tegrat ion , use could be made of the fact

that the autocorrela t ion funct ion vanishes for all pract ical purposes for

r ~ 0.2 sec. On the other hand, in order to obtain the spect ra l density

direct ly one would be forced to work with 20 sec of data . Thk is no

easy mat ter when the frequencies of in terest a re as high as 30 cps. One

could, however , a rgue that it is no easy mat ter to obtain the au tocor rela -



SEC. 6.8] CORRELATION FUNCTIONS AND SPECTRAL DENS ITY 285

t ion funct ion . It is not difficu lt t o calcula te the number of computa-

t ions involved in both methods. To obtain N/2 harmonics (sine and

cosine) from N pieces of data , N2 mult iplica t ions and N2 addit ions are

necessary. To compute N/4 values of the au tocorrela t ion funct ion from

N pieces of data&N’ mult iplicat ions and & N2 addit ions are necessary.

The wor k needed t o obtain t he spect r al den sity fr om t he au tocor rela t ion

depends upon the au tocorrela t ion funct ion . In genera l, for a random

+1.0

+0.8 \

\

\

+0.6 \

\ \

\c- +0.4 \

\

+0.2

o / //

/p ~

I -- ‘‘Original

-0.20 0.02 0.04 0.06 0.08 0.10 012 0.14 0.16 0.18 0.20

? in sec

FIQ. 6.4.—Normal ized autocorrelat ion funct ion for fading record.

ser ies the autocor rela t ion funct ion is effect ively zero after the (aN)th

lag, where a <<1. In the case of the radar fading record a was ~.

Computa t ion of th e Four ier cosine t r an sform then in volves only (wN)i/2

addit iona l mult iplica t ions and (aN) 2/2 addit ions; the ra t io of the

number of opera t ions in the two methods of calcula t ion is 1 to

(*+ a2/2).

Even when the au tocor rela t ion funct ion does not vanish for the

la r ger values of,, it is st ill of advantage t o compu te it first . Its usefu lness

lies in the fact tha t one can compute the indefin ite in tegral of the nor-

malized spect ra l density [see Eq. (61)] direct ly from the normalized

a ut ocor rela tion fu nct ion [see Eq. (3o)]:

/

IO

I(f”) =\

p(T)dj S(j) = : “ ,17 –— s in 2~j07.

o n(81)

[Equat ion (81 ) is ol)t,i~i~(,(l on in tcgra t in~ l;q. (76b) with respect to j,]



286 S TATIS TICAL PROPERTIES OF TIME-VARIABLE DATA [SEC. 6.8

The integra l equals the fract ional power in the message below the fre-

quency fo, except for the d-c component . If 11(j~) – I(fl) I is much

less than 1 [note tha t 1( co) = 1], then the values of S(j) in the in terval

~,,~,) a re negligibly small and need not be computed. By comput ingI(j) for severa l va lues of j one can therefore make a quick survey of the

r egion s in wh ich S(f) is sign ifica nt ly differ en t fr om zer o. Th e advant age

of knowing the impor tant frequency regions of the spect rum needs no

fu r ther emphasis .

We now return to the proof of Eqs. (75) and (76). It is conven ien t

t o wor k with t he auxiliary funct ion

(82)

.where y ~ is defined by Eq. (47). It is clear that the autocor rela t ion

funct ion is the limit of C,(,) as 2’ becomes infinite:

The Four ier t ransform of c. can be rewr it t en as follows:

In terchange of the order of in tegra t ion and in t roduct ion of s = t + T

as one of the var iables of integra t ion give

where A ~(j) is defined in Eq. (48). Finally, making use of the fact tha t

CT(T) is an even funct ion and passing to the limit , we obta in Eq. (76a):

Since the funct ion lim (1/2’) 1A ~(f) [2 is even , Eq. (75) can be obta inedT’+ m

direct ly from Eq. (76a) by means of the Four ier integra l theorem (see

Chap. 2).



SEC. 68~ CORRELATION FUNCTIONS AND SPECTRAL DENS ITY 287

When y(t) has a nonzero mean and conta ins per iodic terms, its auto-

correla t ion funct ion has cor responding constant and cosine terms. In

evaluat ing Eqs. (76) use must then be made of the following rela t ions: 1

(87)

where j~ # O. It is very easy to show that Eqs. (75) and (76) proper ly

rela te the autocor rela t ion funct ion [Eq. (38)] and the spect ra l density

[Eq. (64)] for a t rigon omet ric polyn om ia l.

Rela t ions similar to Eqs. (75) and (76) exist between the cross-

correla t ion funct ion and the cross-spect ra l density. If g is a two-

dim en sion al fu nct ion (u ,o), t hen t he followin g t heor em holds:

/

G..(j) = 2 m d~ Ru%(~)e-’”if’, (89)—.

where G.v (j) is defined as in Eq. (57).

In the case of discrete da ta taken a t successive instants of t ime 1“,

sec apar t , t he ana logues of Eqs. (75) and (76a) a r e

/

l/2T,

R(m) = dj G( j) cos %jT,m (90)o

and

‘(n‘4Tr[~+--l” “1)11=1

The proof of these results is similar to that for the cont inuous case.

In evalua t ing Eq. (91) when g has a nonzero mean or conta ins per iodic

terms, use must be made of the following rela t ions:

] If in the spect ra l density the ent ir e frequency range were used instead of the

(O, co) in ter va l, E q. (87) cou ld be r ewr it ten a s

in wh ich ca se t he e.econ d equ at ion cou ld be der ived fr om t he fist .



288 S TATIS T ICAL PROPERTIES OF TIME-VARIABLE DATA [SEC. 6.9

.

(z2’, ; +)

cos 2rf T ,m = 28(f),

m -l. )

[2T, ~ +(-’’”cos%fT,ml ‘2’(’-+)

m-lm

(z47’, ; +

)cos %fOT ,m cos %fT,m = J (f – Ifol).

m-l 1

(92)

6.9. Spect ra l Density and Autocorrela t ion Funct ion of the Filt ered

Signal.-The discussion up to th is poin t has dealt with cer t a in usefulsta t ist ica l proper t ies of t ime-var iable data in genera l. These ideas

apply equally well to the input and the ou tput of a servomechanism.

It is t he pu rpose of this sect ion t o study t he response of a servomechanism

to an input of which only cer ta in sta t ist ica l proper t ies are known. We

shall suppose that only the spect ra l density of the input (or its t ime

equiva len t , the autocor rela t ion funct ion) is kn own . If th e mechanism

is linear and does not change with t ime, it is then easy to determine the

spect ra l density of the output . Therein lies the pr incipal usefu lness of

t he spect ra l den sit y.

The theorem is simply sta ted: If Y(%rjf ) is the t ransfer funct ion of a

linear t ime-invar ian t mechanism and G,(f) is the spect ra l density of

the input , then the output spect ra l density Go(f) is

Go(j) = IY(27r J )12G@. (93)

One would cer ta in ly expect a theorem of this type to be t rue; but

because of the way in which we have been forced to define the spect ra l

density, the proof ite@f is not st ra igh t forward. The spect ra l density

has been defined as a limit involving the Four ier t ransform A,(f) of y,,

which vanishes outside the in terval (– T, T). Although Y(~fiA r(f)

is the Four ier t ransform of the output under these condit ions, it is not

the Four ier t ransform of a‘ funct ion tha t vanishes outa i~e any fin ite

interval. For th is reason the proof of Eq. (93) which follows has its

star t ing poin t in the t ime representa t ion of the linear system.

It was shown in Chap. 2 that any stable linea r t ime-invar ian t mecha-

nism which acts only in the papt 1 tan be represented by a weight ing

funct ion on the past of the input y(t). Let the weight i~g funct ion be

wr it ten as W(t). Then

1For the purposesof th is d iscussionit is not neceaaarytha t the opera toract onlyin t he pa st .



SEC. 69] THE FILTERED S IGNAL 2S 9

w(t) = o for t <0,

and

/

.

dt l W(t)l < m.o– I

(94)

The weight ing funct ion may contain Dirac delta funct ions. The output

z(t ) is then

\

m

/

.

z(t) = ds y(t – 8)~(S ) = ds y(t – s) W(S ). (95)o– —.

The autocor rela t ion funct ion of the output can be writ t en in terms of

t he input as follows:

““)=::m*/:Td’[/-”mdsy(’s)w’s’. 1

Int ercha nging t he or der of in tegr at ion gives

[JT

‘D-T 1dt y(t – s)y(t + r – r) X W(r). (97)

The limit , as 2’ becomes infin ite, of the quantity inside the brackets is

A!.(r+ s-r )= lim ~/

Tdt y(t)y(t + , + S – r). (98)

y+. 2T –T

Pa ssing t o t he limit , t her efor e, gives

HR,(T) = ds “ dr W(S )RU(T + s – r) W(r). (99)—. —.

This is the t ime equivalen t of Eq. (93). The mean-square value of

the output ~, obta ined from Eq. (99) by set t ing T = O, is

//

. .~= ds dr W(S)RV(S – r) W(r).

—. —.(loo)

Equat ion (93) can now be proved by taking the Four ier t ransform

of Eq. (99). Thus

\Go(fi = 2 -

—. “R(’)”-2”’J ’=2J ” d’~” ‘s/-”. dre-’r if+a+)-)“2miJ,e:2gf.~y(,-~– T)W(S)W(r). (101)

On change of the var iable of integra t ion ~ to (~ + s – r ), the volume



290 S TA TIL!+’T ICAL PROPERTIES OF TIME-VARIABLE DATA [SEC. 6.9

in tegra l breaks up in to the product of three independent in tegra ls:

Go(f) = I Y(!-k j.f) 12GJ U), (93)

where

/Y(27r jj) = “ dt W(t)e–2*~r’, (102)

—.as in Chap. 2.

We shill be in terested in a case sligh tly more complica ted than the

problem just considered. Suppose tha t the input consists of two par ts

—signal and noise. To use the terminology of Sec. 62, the random

process y, which descr ibes the input , has two components (u ,v), where

u is the signal r ecord and v the noise record. It fr equent ly happens tha t

t he signal and noise ent er t he mechanism at differen t poin ts or in differen t

forms. In such a case, the mechanism will opera te on these two compo-

nent funct ions differen t ly. Let the weight ing funct ion opera t ing on u

be W, and that opera t ing on v be W2, where both W1 and W2 sat isfy

the condit ion of Eq. (94). The output is then

\“ ds u(t – s) W,(S) +

/

.

z(t) = ds ti(t – S)W,(S). (103)o- o–

Applying the same reasoning as before, we obta in

//R.(T) = - ds “ dr [w,(s)Ru(, + S – ~)W,(r)

—. —m

+ W2(S )R .(T + s – T)W2(7-) wl(s)a.(~ + s – ~)wz(~)+ W1(T)RVU(Ts – 7) W2(S)], (104)

where R..(,) = u(t)v(t + T ). The Four ier t ransform of this equat ion is

Go(j) = lY1(27rjj) I’G.(j) + ty2(%.f)I’Q.(i)+ Y~(27r jj)Gu J j) Y,(27r jj) + Y,(27r jj)GoJ j) Y;(27r jj), (lo5)

wh er e Gum(f) is t he cross-spect ra l density defined as in Eq. (57).

We shall conclude this sect ion with a resum6 of the analogous results

for the case of a random ser ies. If the filter is stable, linear , and t ime

invar iant , it can be represen ted as a weight ing funct ion w-, where

w. =0 for m <0,

.

zIW*I < co.

m-o 1

For an input ser ies [y(m)], the output ser ies is then

.

z(m) = 2 y(m – n)w..

D=o

(106)

(107)



SEC. 6.10] RADAR AUTOMATIC-TRACKING EXAMPLE 291

The t ransfer funct ion for w is

Y(27r jj) =

x

wine-Zzjf T,in. (108)

m=(l

The output spect rum is again rela t ed to the input spect rum by Eq. (93),

and the output autocor rela t ion formula is

(109)

EXAMPLES

6.10. Radar Automat ic-t r acking Example.—It ~vill be inst r uct ive t o

der ive t he er ror spect r al density and a ut ocor rela t ion funct ion for a gyr o-

stabilized automat ic-t racking radar mechanism on which a great deal

of exper imenta l data are available and to study the effect of fading in the

reflected radar signal in causing t racking er r ors. In order to simplify

the discussion it will be assumed that the a ircra ft being t racked is flying

a radia l course direct ly away from the t racking system. In th is case

the problem of following the maneuvers of the plane is tr ivia l, and the

only sour ce of er ror will be the fading.

An abbrevia ted descr ipt ion of the t racking system ~rill now be given ,

A measure of the difference between the direct ion to a ta rget and the radar

reflector axis is obta ined by conically scanning a pulsed r -f beam. The

received signal is fir st amplified and rect ified, The result ing envelope

is a modula ted signal of the form

u[l + c(I) sin (27r j,t + ~)], (110)

wh er e u is pr opor tion al t o r eflect ion coefficien t of t he pla ne, ~(t ) is pr opor -

t ional t o the magnitude of the angular er ror in t racking, and j, is the

scan frequency (30 cps). The phase angle @ determines the direct ion

of the er ror rela t ive to the reflector axis; @ = O corresponds to an er ror

in t raverse only.1 Random var ia t ions of the reflect ion coefficien t of the

plane cause the signal envelope to be of the form

[1 + g(t )][l + t (f) sin (2Tj,t + 1$)] (111)

A typical record of g(t ) is tha t shown in Fig. 6.1.

The signal is then sent through a high-pass RC-coupling t ransformer

which serves to take out the d-c term. The result ing signal is next

] Th e t ra ver se a ngle is mea sur ed from the line of sight in a plane conta ining the

line of sigh t and a hor izon ta l l ine pmpmdicl llar to the line of sight . The error in

azimuth is rough ly equal to the er ror in t raver se mult iplied by the secant of the

elevat ion angle .



292 S TATIS TICAL PROPERTIES OF TIME-VARIABLE DATA [%C. 6.10

commuta ted in order to determine the phase @of the er ror proper . The

commuta tor produces two signals, one equal to the input mult iplied by

2 sin %rj,t,he second equal to the input mult iplied by 2 cos %rj,t.[The

normalizing fact or 2 is in tr odu ced t o make th e peak value of th e incom ingsignal qt ) sin (%-j,t ~) equa l the resultant of the mean va lues of the

output signals c(t ) cos 1#1nd ~(t ) sin o.] The fir st of these signa ls, the

t raverse er ror signal, is then used as input to a servoamplifier tha t

con t rols the azimuth of the antenna axis, whereas the second is used as

input to another servoamplifier tha t cont rols the eleva t ion of tha t axis.

The t raverse and eleva t ion signals a re thereaft er handled similar ly.

The commuta ted signal is sen t th rough a low-pass filt er in order to

eliminate the 60-cycle r ipple. The resultant signal goes in to a servo-amplifier with equalizing cir cuit having a t ran sfer funct ion (see Chap. 2).

Y(p) = K“ ~1, (112)

where p is the complex var iable of the Laplace t ransform and

T I = 0.36 see,

7’* = 16 see,)

(113)K, = 80 see–’.

The equalizer was chosen to have a la rge velocity-er ror constant K, and

to cu t off rapidly above frequencies common in t racking. In Chap. 4

t hk wa s ca lled ‘‘pr opor tion al-plu s-in tegr al con tr ol. ”

A cur ren t propor t iona l to the ou tput volt ages is used to excit e the

t raverse and eleva t ion torque motors, which , in turn , precess a free-

floa t ing line-of-sight gyro. The ra te of precession is propor t iona l to

the cur ren t s th rough the torque motors. F in ally, t he “r eflect or is sla ve

to the gyro. The servo tha t per forms this task is so much fast er than

the equa lizing circuit that for purposes of th is ca lcula t ion it can be

assumed per fect . The net effect is to dr ive the ou tpu t 00, so that in

t erms of Laplace t ransforms (see Chap. 2)

z (eo) = KV(T,p + 1)

s (commu ta ted er ror signal) P(TZP + 1) ‘(114)

wh ere all propor t iona lit y y constants have been absorbed in K,. When

the system is in proper adjustment , t here is no in teract ion between the

eleva t ion and t raverse con t rol systems. This permits us to limit the

discussion t o a single compon en t of t he er ror , t he t raver se componen t.

Let us begin our analysis with the r eceived signal [Eq. (11 l)]. It is

found in pract ice that Ie(t )I r emains less than 0.05 and tha t the rms va lue

of g is approximately 0.25, To a first approximat ion it is ther efor e pos-

I

I

I

I

I

I



SEC. 6.10] RADAR AUTOMATIC-TRACKING EXAMPLE 293

sible t o n eglect t he cr oss-pr odu ct t erm; t he r eceived signal ca n be wr it ten

as

Sl(t ) = 1 + g(t ) + t(t ) sin (2m~,t+ ~). (115)

In discussing the behavior of the azimuth servomechanism we can set@ = O, provided we allow c(t ) t o assume posit ive and nega t ive values;

c is then , st r ict ly speaking, the difference between the t r aver se input O,

and the t r averse ou tpu t f?o. That is,

The pr incipa l effect of th e high-pass RC-coupling tr ansformer on th e

signal sl(t ) of Eq. (115) is t o r em ove t he d-c term ; its ou tpu t is essent ia lly

sz(t ) = g(t ) + (01 — 00) sin 2r j,t . (117)

It is convenien t a t th is poin t to conver t g, O,, and 6. to angular roils.

The conver sion factor is determined by and var ies inversely with the

fract iona l modula t ion in signal in tensity caused by moving the r eflector

axis a given angle away from the ta rget .

The signal s,(t ) n ow goes thr ough t he commuta tor , which conver ts it t o

S3(f) = [g(t ) + (0, — O.) sin 2Tf, t] 2 sin %r~d

= 2g(t ) sin 27r~.t+ (O, – 0.) – (O, – 0.) cos 47tf,t . (118)

This commuta ted signal is then sen t th rough a low-pass filt er which

eliminates the 60-cycle r ipple term (01 — 190)cos h j,t . The low-pass

filt er has no effect on (O, – 6.), since most of the spect rum of (O, – 0.)

lies below 2 cps. In the case of 2g(t ) sin 27r j,t it serves to a t t enuate tha t

pa r t of the spect ra l density which lies above 10 cps. Let us designate

by f?~(t ) tha t par t of the filt er output due to the input 2g(Q sin 27r j~t . The

net effect of the low-pass filt er is then to change the commuta ted signalSJ( t ) in to

s4(t ) = ON+ (1%– 00). (119)

This signal is the inpu t t o the network with t ransfer funct ion given

by Ilq. (114). Therefore

Kv(!r,P + 1) S (ON + 0[c(%) = – e.).

p(T,p + 1)(120)

Rea rra ngem ent of Eq. (120) givesK“(!r ,p + 1)

s(t ) =~(eI) p(!r ,p + ,) ‘(’N)

~ + Kv(T,p + 1) – ~ + Kw(z’lp + 1) “(121)

p(z’,p + 1) p(T2p + 1)

We are now in a posit ion to apply the theory developed in Sec. 6“9.



294 STATIS TICAL PROPERTIES OF TIME-VARIABLE DATA [SEC. 61CI

In terms of the quant it ies in Eq. (105), we have

u = 01,

v = 8N,

I(122)

output = 6,

(T,27rjj +“l)27rjjYIUW = ~,(zmjf)z + (1 + KmTl)%jf + ~v’

Ku(z’127r jf + 1)

I

(123)

Yz(2~jJ ) = ~z(2Tj~)z + (I + K. T1)2r jj + L“

Spect ral Den sit ies of the Input and Noise.—In gen er al, t he t ra ject or y

of the airplane has zero cor rela t ion with the fading, which is caused by

fluctuat ions in the a irplane reflect ion coefficien t due to the er ra t ic par tof the plane’s mot ion . For th is reason the funct ion Gt i.(~) = G*U(–f)

vanishes ident ical ly.

When considera t ion is limited, as it is here, t o a plane flying a radia l

course, the on ly source of er r or is the fadifig it self. The input angle Of

is a constant ; we may take it t o be unity, and

Gt i(j) = 2L$(f). (124)

AS Yl(2@) con ta in s j a s a fa ct or , t he qu an tit y [Yl(21r jj) 12Gt i(~)va nish esident ica lly. The er r or spect rum G,(j) consequen t ly reduces to

G.(f) = IY,(2Tjj9j2G,(j). (125)

There remains the determinat ion of Gv(j). Since v = d~(t ) was

obta ined by sending the signal 2g(f) sin 2~fd thr ough the low-pass filter ,

we shall fir st obta in the au tocor rela t ion funct ion and then the spect ra l

density for the funct ion 2g(t ) sin 27rJ ,t . As we have seen

R(T) = 4g(t )g(t + 7) sin 2r f,t sin 2r~.(t + 7). (126)

Since the fading is independen t of the posit ion of the beam in a scan ,

a pa rt icu lar t ime ser ies g(f) is a ssocia ted in th e en semble wit h all possible

phases of the commuta tor . Hence all t r aver se er ror signals 2g(t ) sin 27r j,t ,

differ ing only by a t ransla t ion in t ime of g(t ) or a t ransla t ion in phase @

of the commuta tor , a re equally likely. We therefore replace Eq. (126) by

Averaging the sinusoidal factor over all possible commuta tor phases

gives

It follows that

R(T) = 2 Cos 27r f,Tg(t )g(t + T~= 2(COS27r j$T)Ro(T) (129j

The normalized au tocor rela tion funct ion P,(7) = [R ,(T) l/[11,(0)1, cOr n -



SEC. 6.10] RADAR A UTOMA TIC-TRACKING EXAMPLE 295

puted from the fading record of Fig, 6.1, is shown in Fig. 62. Figure

63 gilts the normalized spect ra l density S,(j) of g(t ), computed as the

Four ier t ransform of 2P.(7). It is a simple mat ter to go from the Four ier

10-2X5

4

10.003(

I

\

\

~-: 35 40 45 50

Frequency In cpsFIG. 65.-Normalized spect ra l density of the commutated fading recor d,

t ransform of P,(7) to that of p,(,) cos 27r j.r , which is. of course, the nor -

malized spect ra l density of 2g(t ) sin 27r j,t ; t his is easily shown t o be

s(j) = $[~.(j – f.) + S,(j + js)l. (130)

The normalized spect ra l density of the commutated fading record is

plot ted in Fig. 6,5; it is quite fla t ou t to about 10 cps, after which it r ises

to a double peak.In order to obta in Gu(j) we must mult iply S(f) by

R(o) = Zl?g(o) = 212 mi12 (131)

and by the square of the absolute value of the t ransfer funct ion of the

low-pass filter . The la t ter ~vill not affect the low frequencies but will

a t tenuate the higher frequencies, fla t ten ing out the peaks in Fig. 6.5.

It will therefore be sa t isfactory to approximate Go(j) as a constan t

equal to G,(O) for frequencies up to 5 cps; beyond this it does not mat ter

since the servoamplifier is insensit ive to frequencies above about 5 cps.

We t her efore set

Go(j) = (0.0030) (212) = 0.636 milz sec. (132)

The final equat ion for G,(j) is

KJT,2rjj + 1)2Go(o). (133)

“($) = T2(27r j’)’ + (1 + K, T,)2Ki$ + ~0



296 S TATIS T ICAL PROPERTIES OF TZME-VARZABLE DATA [SEC. 6.10

The normalized er ror spect ra l density h’,(f) = G.(fl/~ mG,(j) dj is

plot ted in Fig. 6“6 for specia l choice of constan ts given in Eq. (113).

1.2

l.o~

0.8

3.-:0.6\G’

0.4

0.2

I r [ I 1 I I f

FIG. 66.-Theor et ica l norma lized spect r al densit y of t he t r a ck ing er ror .

By means of the in tegra t ion tables in the appendix (see Sec. 7“6), it is

easy to show for these constan t va lues tha t

/m d fG.(j) = 0.703 Inilz. (134)o

The normalized autocorrela t ion funct ion can be computed from

S,(j) by means of Eq. (75). The computed normalized funct ion is,

for the choice of constan ts given in Eq. (113),

~p (~)= e–0.93i”1COS2T + 0.106 sin 2171). (135)

Equat ion (135), which is th e final result of this theoret ica l a rgument ,

is plot t ed in Fig. 6“7 along with an exper imenta lly obta ined normalized

autocor rela t ion funct ion of the t raverse t racking er ror for a receding

plane. The exper imenta l data from which the la t ter was obta ined are

shown in Fig. 6.8. The theoret ica l rms er ror was 0.84 roil, and the exper i-

menta l value was 1.04 roil. The close cor respondence between theory

and exper iment gives a good indicat ion of the reliability of the method



SEC. 610] RADAR AUTOMATIC-TRACKING EXAMPLE 297 I

+1,0

+0,8

+0.6

+0.4

c

K

+ 0.2

c

-0.:

-Of

\

\

\

\

\\

\ . Experimental funtior

\-— —_

\\

(

\

\_ ~

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Tin sec

Fm. &7,—Theoretic&l a nd exper iment al norma lized a ut ocor rela tion fu nct ion s of t ra ver set ra ck in g er r or .

+4

+2

o

-2

~5

+2

o

-2

-4tinsec

FIG. 6.8.—Traver~ tracking data, outgoing radial course.




in predict ing the behavior of a servomechanism with given design

constant s. It will be the purpose of the remain ing chapters in this book

to exploit these ideas fur ther in developing design cr it er ia for

servomechanisms.6-11. Purely Random P rocesses.—A sta t ionary random process hav-

ing a constant spect rum is ca lled a purely random process. In fin din g

the outpu t of a servomechanism with a noise inpu t it is often convenien t

t o t rea t the noise as though it were a purely random process. This can

be done because the noise usually has a fla t spect rum that extends far

beyond the cu toff frequency of the servomechanism; the change in

amplitude a t the high frequencies does not affect the outpu t of the device.

This was the case in the example t r ea ted in the last sect ion . It is eviden t ,however , that no such noise can ever be found in nature, since the

Javerage power of a purely random process ~” d f G (f) is infinite. It will

be inst ruct ive to determine som e of the ot her pr oper ties of purely random

processes.

A constant spect rum can be obta ined as a limit from a var iety of

processes. A case in poin t is a modifica t ion of the sta t ionary random

pr ocess descr ibed by Eq. (1).

Let

(~+s)Ll st<(7t+l+s)A,

u,(t) = a., n=o, +1, +z, ...,

}

(136)

A>O,

where (1) an = + - with equal likelihood; (2) am is independent

of am for m # n; and (3) for each set of a .’s, the probability tha t s lies

in any region with in the interva l (O,A) is propor t iona l to the length of

tha t region . Following the argument byon e ca n ea sily der ive

R*(,) = &()

@

. 0

whi~h Eq. (41) was obt~ined,

for jr ] S A,

)

(137)

elsewhere.

As before, the spect ra l density can be obta ined by means of Eqs. (76):

(138)

Passing to the limit as A -O, one has

G,(j) = f:. G.(j) ==N. (139)

The limit ing process is thus a purely random process.

As A a ppr oa ch es zer o, t he a ut ocor rekd,ion fun ct ion s RL(,) approach



SEC. 6.11] PURELY RANDOM PROCES SES

a delta funct ion . Since the a rea under each curve

N / 2, th is must be t rue of the limit curve. Thus

R(r ) = lilio R.(,) = : 6(T).

299

Rd(r) is equal to

(140)

If the rela t ion in Eqs. (76) is to reina in valid, this must cer t a in ly be the

case.

Fina lly, the member funct ions of the process of Eq. (136) become

increasingly wild as A approaches zero. In the limit the funct ions a re

made up of an infinitely dense sequence of independent delta funct ions;

the values of the funct ions a t any two differen t t imes a re completely

uncorrelated. Hence in the limit

P2(yl,y2,T) = P,(y1)P1(y2) for 7 # O. (141) “

If purely random noise of constan t spect ra l amplitude N is sen t

through a linear mechanism with t ransfer funct ion Y(2Tj~), then the

output spect rum will be, by Eq. (93),

Go(j) = N Y(2ir jf) 12. (142)

If W(t) is the weight ing funct ion cor responding t o Y(27r3”) [Eq. (102)],then by Eq. (99) the ou tput au tocor rela t ion funct ion is

//R.(T I = ; _“ ds “ dr ?V(S )6(, + s – r)W(~)

m —.

[

N“— ~ _ ds ~V(S )~(, + S ). (143).

Con sider at ion of t his expr ession su ggest s a not her example of a pu rely

random process. Since W(t) is the response of the mechanism to a single

impulse, one can easily show by direct ca lcu la t ion that Eq. (143) is the

au tocor rela t ion funct ion of the output when the input consists of any

sequence of independen t delta junctions having a zero average value, a

mean-square value of N/(26), and a random dist r ibut ion in t ime with

density ~. Fur thermore, any sta t ionary random-process input tha t gives

an output with autocor rela t ion funct ion that of Eq. (143), wha tever

the form of the weight ing funct ions, must have a constan t spect rum.

Hence another example of a purely random process can be defined as

follows :

y(t) =z

a~~(t — t.), (144)

where the an’s are independent random var iables all having the same

dist r ibu t ion with zero mean and the in tervals (L – tn_J a re likewise

independent random var iables all having the same dist r ibut ion with



300 S TATIS TICAL PROPERTIES OF TIME-VARIABLE DATA [SEC.612

mean value I/@. If UY= N/(2P), then the spect ra l density will have the

constant value N. The cor responding output of a linear mechanism

will be

z (t) =

z

a ,, w (t — LJ . (1 45)

A purely random ser ies is defined analogously as a sta t ionary random

ser ies having a constant spect rum. Any such ser ies can be represent ed by

y. = an> n=o, fl, f2, ”.., (146)

where the n’s are independent random var iables all having the same

dist r ibu t ion , wit h

Na=O,

2 = 2P,’(147)

T, bein g t he t im e bet lveen su ccessive va l~les of n (t he r epet it ion in ter va l).

It is clear that , in this case

(148)

Tbe spect rum obtained I)y means of Eq, (!) 1) is simply

(;(f) = .V, (1 N)

The ser ies produced I)y t lw coin fli!}pcm (Sec. (;2) is :m example of :~

pu rely r an dom ser ies.

6.12. A Typica l Ser vomech an ism Inpu t. —’llis sect i<m will Iw dcv (J ted

to an example of a servomechanism input that is :lppropr i~ltc for an

au tom tit ic-t r ack ing md:Lr syst cm , ‘1’h is exwn ple IMS Iwr n (Lsed in t he

following chapters. An :~iit{Jlll:Lti(-tr:l(kiIlgWI:W syst em is rcllllir edt o t ra ck a ll a ir cr tift t ra velin g t hr ough a llcm isph rr e, s:~y u f r :~(ii~ls20,000

yd, about the system. ,t s a first approxim:~t ion, (mc m:ly t rea t the

angular ra te uf the :Lirplanc as c~mstant t l~ro~@ cxtrn(led illtervak,

with abrupt ch :mges from O]lC V:LIUC!to mnot]lcr’ vallle indepcn(~cnt of

t he first , et c. ‘Mc tingll]ar displxcrn cn t a lml]t t he t rackin g syst em )~uu ld

then vary as indica ted in Fig. G!la . The changes in angul:lr velocit y

might cor rcsptm(l to the maneuvers of the aircraft . This type uf input

should not be confused }~ith an input having a constan t angular ra t e

through extended intervt ils, with the chanqvs in nz fc random and inde-

pendent ; in our example it is the ra tes themselves tha t arc random and

independent.

A bet t er approximat ion than that , of Fig. 69a to the t r :~jer tory of a

maneuver ing aircra ft could be obta ined by rounding off the corners

so that within each interval the angular ra te of the aircra ft approaches a



SEC. 6.12] A TYPICAL SERVOMECHANIL7M INPUT 301

con stan t va lu e asymptot ically, with expon en tia l decay of t he differ en ce.

Such a t ra jectory is shown in Fig. 6“9b. If y(t ) is the fir st -ment ioned

t ra jector y, then the smoothed t ra jectory will be

\

.z(t) = v ds y(t – s)e–vs, (150)

o

where v is the reciproca l of the t ime constan t for the exponen t ia l decay.

The weigh ting funct ion

W(t ) = ve–’~ (1,51)

h as for its t ra nsfer fu nct ion

Y(2Tjj) = , +“2Tjf. (152)

If G,(j) is t he spect ra l density of Y, the spect ra l density of z will then be

[Eq, (93)]v’

Gz(fl = v, + (Zmf)z G.(j). (153)

Our problem is therefor e to determine the spect rum of y. Unfor -

tuna tely, the funct ion y wanders 1 1 II

without bound in the course of it s I Ih istory and cannot be considered ~ ~

{I

as a member of a sta t ionary ran- I {

dom proces s. On the other hand, ‘.,, ~I 1I

in the applicat ions to be made we

~

(0Jcurve#--

‘.shall be in terested not so much in

.‘.

y as in its der iva t ive. The de-‘.‘. I

r iva t ive of y, which will bc des- ‘. I. (~j curve

igna ted by z, does belong to a-.~~

w---

sta t ionary process; it is plot ted f14

in Fig. 6.9c. The funct ion z(t ) is b Q3 ~3~

of con stan t value over successive~’j

1 (c) curve

interva ls, and its values over any t in sec ‘

two in ter vals a re in depen den t bu t ~,o. G.’J—(a)Typical servoinput ;(~)expo-

have the same dist r ibut ion . Wencnt iwllysl1100t1lednpu t; (c) irlputvclocit~

shall proceed to det ermine the autocorr (:lt it ion f~lnct it )n and then the

spect ra l den sity for z.A pr ecise defin it ion of th e sta t ion ar y ran dom prwcss follows:

z(t ) = a., for t . s t < ~.+,,

n=o, *l, +2,.’

where the an’s are independen t random var iables

t r ibut ion and the in terva l lengths

(t .+, – t .) = L

J (154)

h a~’ing th e same dis-

(155)



302 STATISTICAL PROPERTIES OF TIJ IE- t-A RIABLE DATA [Sm . 612

ar e likewise in dependent random var iables having th e same probability

d is tr ibu t ion H(1).

The t imes . . . , t_2, t_l, to, tl, t?, . . , may, for example, have the

Poisson dist r ibut ion of shots on a line. In Poisson’s pr oblem each sh otis independent ly placed on the line unt il t he resu ltant set has a mean

density of 6 shots per unit length t. The probability that any given

interval of length A conta ins a shot Cl the set is approximately PA; the

probabilityy tha t th is in terva l does not contain a shot is (1 – PA). These

expressions become exact as A -+ O. To determine 11(1) in this case we

sta r t a t an arbit rary shot and determine the probability that the closest

shot lies within an interval A at a distance 1 t o the r ight . Subdividing

the interva l of length 1 into subintcr~als of lengths A, it follo~vs that th~pr oba bilit y of n o shot lyin g wit hin a ny of t he fir st [(l,~A) – 1]-su bin ter va ls

is (I — @A)f1i3J –1,wh er eas t he proba bilit y of a sh ot Iyt in g ~rit hin t he last

su bin ter va l is PA. Hence the probability that both of these condit ions

a r e fu lfilled isII(l)A = (1 – ~A)<l’3j–l~A. (156)

Passing to the limit as A = dl approaches zero, one obtains for the Poisson

distribution

H(1) d l = De-fl’ d l, (157)

In order to determine the au tocorrela t ion funct ion it is necessary to

know the probability Q(7) that z(t ) and z(t + T) lie in the same defin ing

interva l. The probability Q(7) is equal to the sum over intervals of all

lengths of the probability that t is in an inter}’a l of length bet ]veen 1

and 1 + dl mult iplied by the probability that th is in terva l includes both

t and t + T. The probability that an ar t ibra ry point t lies in an interval

of length between 1 and 1 + dt can be found as follows. Consider a long

sect ion of t he t im e axis con tain in g, say, K intervals. The tota l number of

in tervals in this sect ion having lengths between 1 a nd 2 + d l is KH(l) d l,

and their aggrega te length is lZ{H(l) dl. The tota l length of the sect ion

is K/

M d l lH (Q. The desired probabilityy is equal to that fract ion of

the se~t ion filled with interva ls of length between 1 and 1 + dl:

; H(1) dl,

i=/

m d l lH (l)o

(158)

(159)

is the average interval length . The probability that an arbit ra ry interva l

of length 1contains both t and t + r is (1 — ~~1), if 7 < 1. It follo~vs that

(160)



SEC. 6.12] A TYPICAL SERVOMECHANISM INPUT 303

F or t he ca se of t he Poisson distr ibut ion

i=!P

and Q(7) = S-@l’l. (161)

It is now an easy mat ter to determine the autocor rela t ion funct ion

for z. For a given funct ion of the ensemble

~(t )~(~ + 7) = a: or anan+k , (162)

according as t and t+ T lie in the same in terval or in two differen t inter -

vals. For a given dist r ibut ion of L’s we fir st average over the an’s,

Because amis in dependen t of an+k, we get ~ or ii’ r espect ively for t he a bove

cases. Since the probability tha t t and t + 7 lie in the same interval is

just Q(r ), avera ging over all dist r ibut ions of t he tn’s gives

R(T) = Z(t)z(t + T) (163)

= ~2Q(7) + az[l – Q(T)].

The spect rum can now be computed by means of Eq. (76a):

/

.

Gz(j) = 4(7 – ii’) dr Q(T) Cos hf. + 2ii’a(j). (164)o

If H ra ther than Q is given , the in tegra l on the r igh t of Eq. (164) can be

expr ess ed a s

H z dr (1 – ‘) H(z) cOS %jr , (165). dl —o i

/

1° - 1 – Cos 27rjl)

‘7 o

dl H(l) ( ~%j), .

Setting

/Q(j) = “ dl H (l) COS %@, (166)

o

this becomes

I.d, Q(,) COS%rf, =

1 – $J (fl ,.

0 (%j)’i

“-.

In the case of the Poisson dist r ibut ion of poin ts

obtain

Subst itu t ion of this in to Eqs. (164) and (167) gives

(lot)

(L), with h = O, we

(168)

(169)

in agreement with Eq. (78).



304 S TATIS TICAL I’ROI’ERTIES OF TIME-VA liIA [il,l{ 1),177A [S W, 6.12

The sta t ionary random pr ocess descr ibed by Eq. (1) is a nother speciaI

case of the process descr ibed by Eq. (154). In this case

H(l) = 3(1 – 1), ;=+, ~v=lz. (170)

The reader can readily ver ify tha t in this case the cont inuous par t of

Eq. (164) ch ecks ~vith Eq, (70).

In order to apply Eq, ( 169) in an aircraft example it will be necessa ry

to have values for @and ~. Under presen t condit ions &I, the average

0.06

0.05

0.04

“: 0.03

0.02

001

L-~loo -75 – 50 –25 o F25 +50 +75 +1 10

Angularvelocity in mils/sec

~lG. 6 10,—P robabilit y d is tr ibu t ion of a ngu la r velocit y.

dura t ion of a st ra ight run by an a t tacking plane may range from 10 to

30 sec. An est im ate of a%has been made by consider ing t he dist r ibut ion

of angular velocit ies for all st ra ight -line paths in a plane t r aver sed at 150

yd/sec. Only those paths ~vere considered for ~rh ich the minimum dis-

tance to the or igin was less than 1500 yd; for these paths on ly tha t pa r t

from 9000 yd before crossover to 9000 yd after crossover was used. Any

par t of a path in which the angular velocity exceeded ~ radian /see was

om it ted. Aside from these rest r ict ions, por t ions of pa ths included in like

areas in the plane were weigh ted equally. The result ing probability

dist r ibut ion M(d) is plot t ed in Fig. 610. For this dist r ibut ion the mean-

square angular velocity ~ is 2.62 X 103 (mils/see) 2; the rms angular

velocit y is 51 mils/sec.

1Crossover is that poin t on a st ra igh t -line pa th for which the distance to the

or igin is a m in imum ,



SEC. 6.13] POTENTIOMETER NOIS E 305

6.13. Poten t iometer Noise.—It somet imes happens that the input tu

a servo is a volt age determined by a poten t iometer set t ing; for instance,

the automat ic-t racking radar system output may dr ive a poten t iomet er ,

the output volt age of which is used as the input to a computer servo-

mechanism. The t ransmit ted voltage \ vould be a fa ithful reproduct ion

of t he t racking-servo out put if it ~veren ot for t he fin ite granula t ion of th e

poten t iometer , which in t roduces in to a signal the type of noise that will

be con sider ed in t his sect ion .

We shall suppose tha t the input to the t racking servomechanism is

the input discussed in the previous sect ion and that the er rors in follow-

ing by the t racking servomechanism are negligible compared with the

poten t iometer resolu t ion ; the input to the poten t iometer will then be

pr ecisely t h e or igin al t ra ck in g-ser vomech an ism inpu t.It will be convenien t to make a few idealiza t ions \ vhich will not

appreciably affect t he resu lt s. In the fir st place, since the input is

unbounded, let us suppose that the poten t iometer is an infin ite helica l

poten t iometer . Let us suppose fu r ther tha t the \ vinding steps a re uni-

formly A roils apar t and that when the velocity of the input is a mils/see,

the poten t iometer output differ s from the input by a sinusoid of fre-

quency a/A cps and amplitude A/2 roils.

The poten t iometer input has already been discussed in the previoussect ion . It s velocity, given by Eq. (154), assumes independent constant

values over a sequ en ce of in terva ls. Let t he probabilit y dist ribut ion for

t hese an gular velocit ies be den ot ed by -itZ(a) an d let

;=O. (171)

The poten t iometer output will be the or iginal input plus sect ions of

sin usoids, all of amplit ude A/2. In fact , if 0, is the or iginal input , then

the noise t erm in the output will be precisely

(172)

The funct ions z(t ) const itu te a sta t ionary random process, It is evident

tha t the phases of adjacent segments of sinusoids a re rela ted, since z is

continuous. For in tervals long compared with the sinusoid per iod

A/a , t his cor rela tion of ph ases will h ave lit tle effect ; t h er efor e in obt ain in g

t he au tocor rela tion fu nct ion we shall assume t hat t he ph ases of adjoin ing

sinusoida l segmen t s a r e independen t .

The autocor rela t ion funct ion of-th e noise is th e ensemble average of

z(t )z(t + 7) = ~z sin (!kr .flt + IA) sin [%f2(~ + 7), + 1#121. (173)

Here +1 = 1#12nd f, = .fZ = a/A if t and t + r a re in the same segment ,

and fl is in depen den t of fz and 41 is independent of & if t and t -I- T lie in




differen t segments. Cases for which t and t + 7 lie in dhleren t segments

do n ot cont r ibu te t o th e au tocor rela t ion funct ion as on e sees by averaging

~1 and dz independent ly. When tand t+ T lie in the same segment , one

can average over all phases ~ for a given frequency j = .fI = jZ t~ get(A’/8) cos 2irfr. Aver agin g over all frequ en cies t hen gives

(A3/8) f_”~ df M(Af) COS2rfr.

To obtain the autocorrela t ion funct ion th is must now be mult iplied by

the probability tha t t and t+ T lie in th e same in terva l:

(174)

where Q is defined as in 13q. (160).

Wc can no~v obta in the spect ra l density by applying Eqs. (76):

This can be rewr it t en as

Th e fu nct ion G,(f) is t he F ou rier t ra nsform of t he pr odu ct of t wo fu nct ion s

and is th erefor e equal t o the convolu t ion of their t ransforms. Th e pr oof

of thk sta tement is similar t o that of the conv oiut i on theorem for the

Laplace t ransform given in Chap. 2. If we now assume J lf(A~) to be an

even fu nct ion, th en

/

.

.l!f(Af) =/

d~ ~ -2T,J , m d f ,11(Af ) (’0S 2rfT . (177)—. —.

We thus find

\

,mG,(f) = $ _ ~ d s K(S ) M[A(f – s)], (178)

where.

K(f) = ~ d, Q(,) COS 2~f7. (179)o

F or pu rposes of illu st ra t ion , let u s SUPPOWtha t (1) t he pot en tiom et er

is wound in steps of A = 1 roil, (2) the probability dist r ibu t ion of angular

velocit ies M(d) is the dist r ibut ion plot ted in Fig. 6.10, and (3) the set of

poin t s where a change in velocity of the input occurs is the Poisson dis-

t r ibu t ion of shots on a line. As we have seen in the previous sect ion ,

Q(7) = e-~1’1,where @is the mean density of shots. It follows tha t

(180)



SEC. 613] POTEN7’IOMETER NOISE 307

Applying Eq. (178), weseetha t G.(j) isaweighted average of M(Aj).

The total weight for th is averaging process is

(181)

If B = 0.1 see–l, K($) a t tenuates very rapidly ;in fact half of its a rea is

in a frequency band of 0.032 cps about the or igin . Hence its act ion as a

weight ing funct ion on M(Af) is much like tha t of a delta funct ion when A

is less than 1 roil. For such A,

G.(j) = : M(Aj). (182)

For A = 1 roil, the dist r ibu t ion funct ion M(Aj) is sufficient ly fla t , forfrequencies less than 5 cps, for G,(f) to be t rea t ed as a spect ra l density of

a pu rely r andom process in servomechan ism problems (see Sec. 6.11).

In order to complete th is study of poten t iometer noise it will be

necessary to compute the cross-correla t ion funct ion. In the usual

servomechanism applicat ion the er ror depends on the noise and the

der iva t ive of the input signal. (See, for instance, the example of Sec.

6.10.) It will t her efor e be su fficien t t o fin d t he cr oss-cor rela tion fu nct ion

between the der iva t ive of the input and the noise. This has the advan-tage of enabling us to wor k with sta t ionary random processes.

Th e der iva tive of t he in pu t, 8,, has been ca refu lly defin ed in 13q. (154).

For purposes of the presen t ca lculat ion we must be equa lly precise about ,

& it self. In order tha t z(t ) be a sta t ionary random process it is necessary

that all possible funct ions z(t ) bc represen ted in the ensemble. This

will be so if to a defin ite 8,, defined by a given set of constants ,

a–l, ao, al, az, . . . and a given sequence of t imes’. . . , t–,, t~, /,,

h ,. ... t here cor responds a set of inputs 0, defined by

/

t

e,(t) = [/ s e,(s) + C#l, (183)o

where the ‘( phase angle” @ takes on all of the values in the in terva l

(O, A) wit h equa l likelihood .

The cross-cor rela t ion funct ion is the ensemble average of the quan-

tities

‘(’’’l(’+’)Xsinz++’) (184).

If one averages over all inputs 0, tha t cor respond to m defin it e 0,, tha t is.

over a ll phase angles O, it is clear tha t t he average will van ish . Th e

ensemble average is then the average of this zero average over a ll possi-

ble 81; the cross-cor rela t ion funct ion therefore ~anishes iden t ica lly.




different segments. Cases for which t and t + , lie in different segments

do n ot con tr ibu te t o th e autocor rela t ion funct ion as on e sees by aver aging

I#J l nd +2 independen t ly. When t and t + r lie in the same segment , one

can average over all phases @ for a given frequency j = jl = ~Z tG get(A’/8) cos %j.. Averaging over all frequencies then gives

(A’/8)/_m~d.. M(Aj) COS %j,.

To obta in the autocor rela t ion funct ion th is must now be mult iplied by

the probability that t and t + 7 lie in the same interva l:

R(T) = ; Q(,)

/

.

df M(Aj) COS %TfT, (174)

—.

where Q is defined as in Eq. (160).

We can now obtain the spect ra l density by applying Eqs. (76):

/G.(j) = 4 “

o “cOs%f’[:Q(’’Jf(A~)1This can be rewr it t en as

The fun ct ion G.(j) is th e Four ier t ransf orm of t he pr oduct of t wo funct ions

and is therefor e equal to the convolut ion of their t r ansforms. The proof

of th is sta tement is similar to that of the convolu t ion theorem for the

Laplace t ransform given in Chap. 2. If ~ve nwv assume flf(Aj) to be an

even funct ion , then

/

M(Aj) = m

/

d~ ~–2@7 m(ifJ lf(Aj) COS 27rj,. (177)

—. —.We thus find

G.(j) = >3 /_” ds K(S) itf[A(j – s)], (178).

where

/K(j) = - d. Q(T) COS %rjT. (179)

o

F or pu rposes of illust ra t ion , let us suppose tha t (1) t he pot en tiom et er

is wound in steps of A = 1 roil, (2) the probability dist r ibut ion of angula r

velocit ies M(d) is the dist r ibut ion plot ted in Fig. 6“10, and (3) the set of

poin ts where a change in velocity of the input occurs is the Poisson dis-

t r ibut ion of shots on a line. As we have seen in the previous sect ion ,

Q(7) = e-~l’l, where@ is the mean density of shots. It follows tha t

(180)



SEC. 7.1] PRELIMINARY Y DISCUSS ION OF THE METHOD 309

We now come to the quest ion of wha t kind of ou tput is desirable for

t h e ser vo. If it were not for the presence of uncon t rolled disturbances

in the input , ou r goa l would be to make the ou tput follow the input

per fect ly. In the presence of these disturbances, however , per fect

followin g of t he in pu t in volves at least good, if n ot per fect , followin g of t he

noise. It is apparent tha t a compromise must be made between faithfully

,follom”ng the input signal and ignoring the noise.

Since a compromise must be made, it is necessa ry that we have a

cr it er ion of goodness or figure of mer it for any given design . In order

to be of use, such a figure of mer it must be in reasonable accord with

pract ica l requirement s, it must be of genera l applicability, and it must

not be difficu lt to apply. The wide var iety of servo problems precludes

a cr it er ion of goodness tha t is universa lly applicable. Even when acr it er ion is not st r ict ly applicable, design methods based upon it can

oft en fu rn ish u sefu l in forma tion . A fur ther requiremen t for the figure

of mer it is that it be unaffected by unlikely shor t -lived aberra t ions from

the mean or sh ift s in the t ime axis; instead, it should be a measure of the

a ver age beh avior of t he ser vo. This is in accordan ce with t he sta t ist ica l

na ture of the actua l input and of the noise. We sh all h er e lim it ou rselves

to a single figure of mer it , the rms er ror in following. If Oris the input

signal to be followed, if & is the output , and if e = & — 00 is the er ror infollowing, t hen th e rms error is %@, wh er e

/

lT~=lim _ dt e’(t).l’+. 2T –T

(1)

We shall consider that servo best which m in im izes the rms error.

The rms figure of mer it has been used in many types of problem.

It s use in th is chapter was inspired by N. Wiener ’s work on the ext rapola -

t ion , in terpola t ion , and smooth ing of sta t ionary t ime ser ies. 1 The ideaof applying t he in t egra t ed-square-er ror cr it er ion t o servo design has also

been considered by A. C. Hall.z One of the reasons for the wide usage

of the rms cr it er ion stems from its mathemat ica l conven ience; there is a

highly developed body of mathemat ica l knowledge that has been built

around the not ion of a mean-square va lue-the harmonic analysis

descr ibed in Chap. 6.

Th e rms-er ror cr it er ion weigh ts t he u ndesirabilit y of an er ror a ccor d-

ing to the square of it s magnitude, as indica ted in Fig. 7“la, and thisindependen t ly of the t ime at which the er ror occurs. In genera l such a

weight ing is adequa te whenever the undesirability of an er ror grows

I N. Wien er , The Extrapolat ion , In terpolat ion , and Smoothing of Stationwy TimeSeries,NDRC Report, Cambridge,Mass., 1942.

2 A. C. Hall, The Ana lysis and Synthesis of Linear Servomechanisms, Massa-

ch uset ts In st it ute of Tech nology P ress, Cambr idge, Mass,, 1943, pp. 19-25.



310 RMS -ERROR CRITERION IN S ERVOMECHA iVIS ,W DES IGN [SEC.7.1

with its magnitude. There are, however , cases where th is weigh t ing

would not be suitable. If, for example, all su fficient ly la rge er ror s wer e

equally bad, it wou ld be necessary to have a weight ing similar to tha t

shown in Fig. 7. lb, Alor is it a lways t rue tha t the undesirability of an

er ror is independen t of the t ime at which tha t er ror occurs. For ins tance,

an er ror dur ing a t r an sit ion fr om

/

one mode of opera t ion to anotherWeight Weight

Mk

may be mor e or less desirable than

an er r or dur ing a given mode of

operational

When applicable, a figure of

mer it such as the rms er ror is ex-

ccechngly useful. By means of it

Error Error one can determine the “ best” sys-(U) (b) t ern possible under suitably re-

1’IG. 7.1.—(IZ) ,s (JU<lrc-error weig}lting; st r ict ive condit ions. Thus if the(h ) nonw]uarc.e r ror wcigh tmg.

only limita t ion is tha t the system

bc a linear filt er , one can find the best such filt r r by a method developed

by Wicmer .’ 13ven if such a filt er cannot be r ea lized in the form of a ser vo,

it \ t ill st ill bc of grea t in terest to know ho}v WC1lthe rms er ror of the

r ea lized ser vo a ppr oxima tes t he rms er ror of t he idea lly })est filter .

As J VChave seen, the servo must compromise bct lvecu following the

signal and smooth in g ou t t he noise. Ir ur th ermor e, t he out pu t is a l~va ys

po!vcrcd by a source externa l t o the input . 1t is reasonable to ask why

these t t vo opera t ions, smoothing and follolving ]r ith increased power ,

cou ld not bc done in ser ies, ‘~h ~t is, \ vl]y not fir st sen d t he in pu t t ~~r ou gh

‘~ If the only source of1 filt er tha t sep~r :it c,s t he signal fr om ihc nuisc.

noise is in t he input , signal, t his is in deed feasible. If, however , the dis-

t urbancrs ar ise in the loading or at an in t cr ier point of the servomecha-

nism, then t lm smm)thing must Ijc dfmc in t}lc sr rvo it self. There is

s t ill an{~th[,r d ifhcolty ij-ith filtcr ir l~ the bigna] fir st . In pract ice there is a

limit to the :wcuracy J \ith l \ ”hi(’11 a filt ,(r can lx: made. In par t icula r , if

t he in pu t r a nge is cxccssivr , t llc u ut l)llt of t he filt er will n ot }W su fficien tly

accurate. Ncvcr thclcss, I rhcrcvcr p(JS,Sibk!,t is \ \ -cllt o pur ify the input

by firstsen din g it t hr ollg}l a filt er , Since the servo will in any case act

as a filter , it s character ist ics ]r ill have to h taken into account in the

prefiltcring

The tiesign procwlurc to lx descr ibed in th is chapter is basicallysimple a d st ,r :~igh tfor lrwd. In pmct icc it is difficult t o rea lize a given

I ‘r r al,sicr ,t s ca ll I II : trcatrd i}) a mwln [~ r sim ila r t o the pr occdurc used for the

mea n-squ ar e er r or . In th is C:M, t l]c i]itcgr :~tcd-s(l~larc er ror in the t ransient is

computed (SW:frmtmotc 011p . 3 14).

‘ Scc Wicucr , op . ci~.



SEC. 71] PRELI,!41NARY DISCUSS ION OF THE METHOD 311

weight ing funct ion as an elect r ica l network, to say noth ing of rea lizing

it as a servomechanism. It is much more pract ica l t o star t with a servo

of a given type having cer ta in adjustable parameters. The mean-square

er ror can then be computed as the integra l of the er ror spect ra l density

dir ect ly fr om integra l tables. This mean-square er ror will, of course,

depend on the adjustable parameters of the servo; the best servo of the

given type is then determined by finding the parameter values tha t

minimize t he mea n-squ are er ror .

It is wor th remarking that the method proposed jor jind ing the rm s

error does not involve solving for the roots of the characteristic equation of t he

differen t ia l equat ion descr ibing the servo. As a resu lt , th is method has a

grea t advantage over the familia r t ransien t analysis approach , which

requires t he use of a differen t ia l analyzer t o handle effect ively problems

that have character ist ic equat ions of degree five or higher . IWost of the

discussion is concer ned with reducing t he er ror spect ra l densit y t o a form

suitable for the use of the integra l tables. The er ror spect ra l density is

fir st expressed as a funct ion of the er ror -signal t ransfer funct ion , the

er r or -n oise t ra nsfer fu nct ion , a nd t h e elemen ts of t h e sign al-n oise spect r al-

density matr ix. Since the servo is a stable lumped-constant system, the

t ransfer funct ions a re ra t ional funct ions of the frequency. N’ow use of

the in tegra l tables requires tha t the er ror spect ra l density be expressed

as the sum of squares of absolute values of rat ional funct ions. Thisj in

t ur n, r equ ir es t ha t t he elem en ts of t he spect ra l-den sit y m at rix be a ppr oxi-

mated by ra t ional funct ions. It is shown in Sec. 7“4 that the spect ra l-

densit y matr ix element s can always be so approximat ed.

The use of the in tegra t ion tables is st ra igh t forward, and there a re

st anda rd pr ocedu res for minimizing t he r esult in g expr ession wit h r espect

to the servo parameters. The method is illust ra ted by two servo-

mechanisms, on e of which was independent ly designed by exper iment al

methods. The resu lts of the theoret ica l and exper imenta l procedures

are in good agreement . A fina l sect ion of the chapter is devoted to the

method used in der iving the table of in tegra ls.

Although the presen t discussion deals ordy with servomechanisms

with a cont inuous flow of data, it will be evident to the reader tha t servo-

mechanisms with pulsed data can be t r ea ted similar ly. The necessa ry

ma ch iner y for t he discussion of pu lsed ser vos has a lr ea dy been developed

in Chaps. 5 and 6.1

It is well t o add a word about the dkxuivantages of the method pro-

posed in this chapter . In the fir st place, it is assumed that the mathe-

mat ica l represen ta t ions of the differen t par ts of the servo system are

1In applying th e rm s-er ror cr it er ion to th e pu lsed servo it is con ven ien t to wor k

with the complex var iable z = ezr jf~~ on the unit circle ra ther than with the rea l

va r iable f k the in terva l ( – 1/2 T,, l/2T,),



312 RMS -ERROR CRIT’ERIOX I.V S EE VOMECHANISM DES IGA- [S EC. 72

known. In pract ice it may be difficu lt t o determine the constan ts

r equ ir ed for su ch a r epr esen t at ion . The steady-sta t e analysis of Chap. 4

does not require a knowledge of these values, requ ir ing instead only tha t

measurements be made on the stead y-st a te response of the syst cm.Another difficu lty lies in the fact tha t it is necessa ry to sta r t with a proper

type of equa lizer . The steady-sta t e design pr inciples of Chap. 4 can be

employed in t he ch oice of a good t ype of equa lizer ; t he rms-er ror analysis

can then be used to make the final adjustments of the constan ts. Lastly,

one might complain that this procedure works on ly with the rms er ror

and furn ishes no insight in to what goes on in the mechanism when

the equalizer and input parameter s a re var ied. On the other hand ther e

is n ot hin g to prevent on e from st udying t he decibel–log-frequ en cy dia-

grams as a funct ion of these parameters; this, in fact , has been done for

two common servo types in chap. 8.

7.2. Mathemat ica l Formula t ion of the RMS Error .—In this sect ion

we shall obta in an expression for the rms er ror in terms of the t ransfer

funct ions charact er izin g t he servomechanism and t he spect ra l densit ies

character izing the input signal and noise. We shall sta r t with the

assumption tha t the servomechan ism is a linear filter and that the er ror

ca n be r epr esen ted as t he sum of a linear opera tor act ing on the input

signal plus a linear opera tor act ing on the noise. In symbols, th is

assumption t akes t he form

\

m

/

.

6(1)== (is 0,(( – s) u-,(s) + ds 6,v(1 – s) W,(S ),o–

(2)o–

!vhere d~(t ) is the noise record and, M in Ghap. 2, the W,(~) (i = 1. 2) a re

st able weigh t in g fun ct ion s sa tisfyin g t h e condit ion s,

JVi(t)= O for t < (),

/

.

dt lW,(t)[ < m ,I

[i = 1, 2).

o–

(3)

This is the usual linear assumpt ion; it will be valid provided that such

nonlinear effects as sa tura t ion , backlash , and st ict ion ar e negligible, and

provided tha t t her e is no in teract ion between input and noise.

The two weight ing funct ions W, and Wz will gen er ally be differ en t

and will depend on the way in which the input and noise en ter the system.If the input and noise en ter the system at the same poin t , then the output

depends in the same way on both . In th is case if W(L) is the weight ing

funct ion for t he over-a ll system, th en

/

.co(t) ds [~,(t – .s) + e~(t – s)]~(s). (4)

o–



SEC. 7.2] MATHEMATICAL FORMULA TION OF THE RM8 ERROR 313

It follows tha t

/

.

f(t ) = o,(t ) – ds th(t – S )W(S ) –/

m (is e.x(t – S )w(.s). (5)n– o–

Hence, in this case,w,(t ) = a(t ) – w(t),

w,(t ) = – w(t ). }(6)

If the servo is subject to a random load disturbance, then still a differen t

rela t ion will exist between WI and Wz.

We shall now make use of the harmonic analysis developed in chap. 6.

we set

/

T

A,(j) = dt C(t)e–~riJ’ (7)–T

and similar ly define 11~($) and Cr( j) to be the Four ier t ransforms of

o,(t ) and ~~(t ), respect ively, over the limited range ( – ?’, 2’). Then, as

in Sec. 6.7, the spect ra l density of c is defined as

G.(.f)= $+mm IATU)12. (8)

Similardefinitions

hold for G,(f) and G&f), the spect ra l densit ies of81 and o.N,r espect ively. The cr oss-spect ra l den sity is defin ed as

Gw(f) = ;&m ~ WWTU)) (9)

from which it follows tha t

G,)/(j) = G~,(f) = G~~(–f). (lo)

F inally t he input -n oise spect ra l-densit y ma tr ix is defin ed as

[

GO GIN(f)1(f) = GN1(f) G.M) “

It is eviden t t ha t*

G(f) = ~“(~) = G*(–~).

(11)

(12)

It was shown in Sec. 6“7 tha t the mean-square value of the er ror can

be computed as the in tegra l of it s spect ra l density over all nonnegat ive

frequencies. Since the spect ra l density is an even funct ion of fr equency,

we may write r .

(13)

Thus our problem is reduced to tha t of obta ining a suitable expression

for t he er ror spect ra l density.

IThe symbol ~ denotes the t ranspose of the mat r ix G.



Taking the Four ier t ransform of both sides of liq. (2) ~vith funct ions

19~and 6N, t~hicb vanish ou tside the in terva l ( – T, 7’), v’e obtain, ap-

proximately,

.4,(j) = l“, (2~jf)Bl (j) + F ,(27r jf)C’1(~). (14)

Here the t ransfer funct ions }-t a rc, of course, the F{nlr icr t r anslorms ci’

t h e weigh t in g f~mct ,ion s TV,:

‘1’o obtain tkw crr~lr spect ra ] density ire ncwd only tala ’ 117’ t imes the

absolu t e va lu e s(lllarcd of .l~(, f), given in ]{kl. (1-4):

1 ‘} ’L(2T.joBr ’(,/) + 1’,(27r jJ ’’)C’7,ol’.~<(.1) = ~!:yx ~Tl (16)

‘his equation can be re\\”rittenin tcrrnh ()[IIlt’ck,m(,nts of tllrspe[”tral-

dcnsity mxtr ix as t ’ollolr s :

Asshownin Sec. 69, this result can bccler ivedby amore r igorous argu-

ment . It is ~r or th r ema rkin g tha t for sufficicn tly la rge T t he expr ession

on the r igh t-hand side of 13q. (l(i) approxim~tes closely tof l,(j). Hence,

lk~. (16) ~vithout thelimit s ymbolcan be use d to rompllte G,(f) I~hen d,

and OXarc obt a in ed exper imen t ally.

Combining ICqs. (13) and (17), we nolr have an cxprcssiou for the

reran-square er ror in terms of the t ransfer funct ions, character izing the

ser vo sl,st cm , a nd t he elemen ts of the input -noise spect ra l-dent iit y matr ix,

character izing the inputs. 1 The t ransfer funct ions ~vill usua lly be

rela ted. For instance, YI = 1 + 1“?, in accordance J vith I;q. (0), ~~henthe signa l and the noise en ter the systcm at the same poin t . Our next

problem is to find the t r ansfer funct ions that minimize th r mean+ quare

er ror , a llowing, of cou rse, for t he in tcr dcpcndence of t he 1“s.

It is in terest ing t o see the physica l significance of th is minimiza t ion

problem. The quant ity on the r igh t of Eq. (17), the in tegra l of which

wc wish to minimize, weights the frequency-t ra nsfer funct ions a t a given

I The in tegra ted-square er ror can be given in a for m similar to the rncan-square

er ror . Suppose, for instance, that the input and noise vanish for ~ <0 and tha tt heir I,a pla cc t ra nsforms a r e (31(p) a nd e.,,(p). r espect ively. Th en

\

.

/

.d [C(t)]z= d f IY ,(2~jf)e, (2~j’) + Y ,(2qf)H \ -(2~jf) l’.

0 —.

The iutegra ted-square er ror is a measure of the t ransient r esponsr if the inpu t is,

for example, a step fun ct ion a nd n oise is not pr esent .



SEC, 7.3] NATURE OF THE TRANSFER FUNCTION 315

frequency according tothe rela t ive impor tance of tha t frequency in the

signal and noise. This is precisely the kind of rela t ion which is needed

for a quant ita t ive ext en sion of th e st eady-st at e design methods of Chap. 4.

Fu rt her mor e we have an exper iment al ch eck on t he reliability of Eq. (17),for in t he example of Sec. 6.10 we wer e able t o compa re an exper iment ally

obta ined au tocor rela t ion funct ion with on e det ermin ed by Eq. (Ii’).

7.3. Na ture of the Transfer Funct ion . -In minimizing the rms er ror

it is, of course, possible to seek the ideal t ransfer funct ions. This

involves a ra ther long computa t ion ; even when one has determined the

ideal t ra nsfer fu nct ion s, t her e rema in s t he difficu lt t ask of r ealizin g t hem.

In pract ice it is more conven ien t to choose a su itable type of filter with

cer t a in adjustable element s and then to determine the best possibleadjustmen t of these elements. We shall therefor e sta r t with a given

family of t ransfer funct ions and find tha t t ransfer funct ion of this set

wh ich min imizes t he m ean -squ ar e er ror .

This filt er will consist of a n et work of lumped elements, some of which

are adjustable. Such a filter can always be represen ted by a differen t ia l

equat ion of the form

A,,(’) + A, J ”-1) + . “ . + A., = Bee,@” + B,e,(”-” + . . + I?m o,

+ C,e.$ + C,O\ -” + . + cltJ N ,

(18)

where m, 1 s n; y(k) denotes the kth der iva t ive of y with respect to t ime t.

The numerica l coefficien ts A, B, c a r e all rea l and depend on the adjust -

able parameter s. For any given servomechanism, each of the com-

pon en ts can be r epr esen ted by a differ en tia l equ at ion wit h cer ta in dr ivin g

funct ion s. By combining t hese equat ions on e can elimina te all funct ions

except C, &, and ON. The resu lt ing differen t ia l equat ion will be of the

form of Eq. (18). As shown in Chap. 2, the t ransfer funct ions for the

weight ing funct ions WI and WZ, descr ibed in the previous sect ion , a re

.

1 131(27rjj)m-’

ico?

n

2Ai(27rjj)”-i

ieo I (19)

i=O /

If the per formance of a filt er is to be at a ll sa t isfactory, it must be

stable. This, of course, is assured if the filt er conta ins only passive ele-



316 RMS -ERROR CRITERION IN SER VOMECHANISM DES IGN [SEC. 73

ments. The servo, on the other hand, is a specia l kind of filt er , conta in-

ing act ive elements and a feedback; it is therefore possible for a seNo

system to becom e unstable and oscilla te. Whether or not the system is

unstable will usually depend on the adjustment of cer ta in quant it ies inthe equalizer , such as the amplifier gain , and the magnitudes of cer ta in

of t he r esist a nces and capacit a nces. F or some valu es of t hese par amet er s

the system will oscilla te, but for others it will not oscilla te. The set of

all values of the parameter s for which the system does not oscilla te will

be called the region of s tability .

As was shown in Chap. 2, the system will be stable if, and only if, the

r oot s of t he polyn om ia l.

H(f) = z4, (27r jf)”-’ (20)

i=o

lie in the upper half plane. NTOWhe root s a re cont inuous funct ions of

the coefficients A~ of If(j). Hence, as the coefficien ts a re cont inuous] y

var ied, the system can go from a stable to an unstable sta te if, and only

if, a t least one root assumes a value on the rea l axis. Since the coefficients

a re cont inuous funct ions of the parameters, t he above sta tement is like-

wise t ru e of t hese pa ramet er s. It follows that in t he space of par ameter s

the region of stability will be bounded by those values of the parameter s

for which the root s of the polynomia l H(j) lie on the rwd axis.

In the polynomial H(f)j A,, t he coefficient of (2TjJ ) ‘-k, is rea l, ~on-

sequent ly the roots of H(j) = O are symmetr ically situat rd about the

imaginary axis. Tha t is, t hey are either pure imaginar ies or occur in

con juga te pairs of the type (jv f u), where o and u are real. If a root i-k

lies on the rea l axis, there are two possibilit ies: 13ithrr rk = O, or there is

a second rea l root r ; such tha t r~ = — rk. If r~ = O, then the product ofall r oots will vanish and we will have .4,, = O. If rk is r eal an d n on zer o,

then the product of sums of all pairs of roots, ~ (r i + r~), ~vill vanish.

i<k

It follows that the boundary of the region of stability is con ta ined in the

su rfa ce defin ed by

.4. n (r, + rk) = O. (21)

i<k

An explicit expr ession for this surface in terms of t he coeilicients of H(j)

can be found by the methods of Sec. 7.9 [see Eqs. (lo4) and (105)].

Not all poin ts on th is sur face are boundary points of the region of

stability, since (T; + r~) can vanish without r i being rea l. It remains to

be determined which of the bounded regions are actually regions of

stability. It is clea r tha t either all poin ts in each such bounded domain



f+Ec. 74] REDUCTION OF THE ERROR SPECTRAL DENS ITY 317

cor respond to stable sta tes of the system or all poin ts cor respond to

unstable sta tes. The stability of each domain is therefore determined

hythe stability of asingle poin t in tha t domain. Each domain can thus

be tested a t a single poin t by means of the Routh cr it er ion or the Nyquistcr iter ion (see Cha p. 2).

7.4. Reduct ion of the Er ror Speclra l Density to a Convenien t Form .—

As we have seen , the mean-square er ror can be expressed as the integra l

of the er ror spect ra l density. Our next step is to put G,(f) in to a form

amenable to computa t ion . We shall make use of Eq. (17) ~ in which

G,(f) is given in terms of the t ransfer funct ions and the elements of the

in pu t-n oise spect ra l-den sit y mat rix. Since we have limited ourselves

to filt ers of lumped elements, the t ransfer funct ions will be of the form

shown in Eq. (19); each Y is a ra t iona l funct ion with poles in the upper

half plane, symmetr ically placed about the imaginary axis. 1 It is there-

fore convenien t and natura l t o a t t empt to express G,(j) as the square

of the absolute va lue of such a ra t ional funct ion in j or as the sum of such

expressions. It will be shown in Sec. 7”6 tha t th is is, in fact , a convenient

form for t he in tegr at ion .

In many problems the elements of the spect ra l-density mat r ix ~vill

be known rat iona l funct ions. In such cases it is rela t ively easy to br ing

G, in to the desired form. When, however , the matr ix elements are not

ra t iona l funct ions or are known only exper imenta lly, approximat ions

will be requ ired in the t r ea tment . We shall now discuss each of these

ca ses in det ail.

Reduction when the Elements of G .4 re Rational Functions .—Suppose

fir st tha t the elements of the spect ra l-density matr ix are ra t iona l func-

tions. We shall show how to express G, as a sum of terms, each of which

is the absolute va lue squared of a ra t iona l funct ion with all poles in the

upper half plane, syrpmet r ica lly placed about the imaginary axis.

Since G,(f) is ra t ional, it can be expressed as the product of two

ra t iona l factor s XII (f) and Zl 1(f ) such tha t Xl I has for it s zeros and

poles those of G, in the upper ha lf p!ane and Zll has for it s zeros and poles

those of Gr in the lower half plane.

Gdj) = XH(j)z,,(fl. (22)

Since Gl(f) is rea l va lued for rea l j, both its zeros and poles must be sym-met r ic in pairs with respect t o the rea l axis; since it is an even funct ion of

j, both its zeros and poles must be symmetr ica lly placed about the

imaginary axis. The zeros and poles of X,, will a lso be symmet r ica lly

placed about the imaginary axis. Fur thermore, by proper ly choosing

I The poles of Y will be symmetr ically placed about the imaginary axis if, mnd

on ly if, t h e coeffir ir n t ,s A ~ of If(f) [s ee Eq. (20)] are realwdlled.



318 RMS -ERROR CRITERION IN SERVOMECHANISM DES IGN [SEC. 7.4

constant factors one can wr it e

x,,(j) = z?,(j) = X,( –f) (23)

for rea l va lues off. Then

Gdj) = Ix,l(j)l’. (24)

Thus t he first t erm of G,, namely, IY1(21r jj) 12G~(j), ha s t he desir ed form.

Similar ly, one can writ e

G.v(f) = lX,,(f) l’, (25)

with the zeros and poles of XZZ in the upper half plane, symmetr ica lly

placed about the imaginary axis.

It remains to show that the last two terms of G,, namely,

Y~(fijf)Gr.\ r(~) Y2(2~j”) + Y~(%jj)G.v,(.f) l“~(2Tjj),

can be brought in to the desir ed form. lVe sh all t r ea t t h es e cr oss-spect ra l

terms together . In the fir st place, it is evident from Eq. (6.89) tha t

G1~(j) is the Four ier t ransform of a rea l-va lued funct ion, the cross-

cor rela t ion funct ion RIN(7). It follows that both the zeros and the poles

of Gr&f) ar e symmetr ica lly placed about t he imaginary axis. Let us now

factor G,i@ into two ra t ional factors X,,(f) and Z,,(f) such that X,2 has

for its zeros and poles those of G,~ in the upper half plane, and 21, has

for it s zeros and poles those of G,.v in the lower half plane:

G,.}(f) = Xr,Z(f)Z,Z(.f). (26)

For each factor , both the zeros and the poles will likewise be s-ymmetr i-

ca lly placed about the

ra tiona l fa ct or s X21 an d

From Ilq. (644)

it follows t ha t

imaginary axis. We can similar ly define the

Zzl of G.,,:

G.vr(f) = X21(.f)Z21(.f). (27)

R,N(T) = I?.V,(-7) ; (28)

/

.G,>,(u + j,) = 2 h R~.v(7)e–2r’cu+j*J

—.

/

.

. 2 dr RX,(r) e~r,(U+,U)r (29)—m

= G;,(u – jL).

Hence the zeros and the poles of G,~(j) a re the complex conjugates of

the zeros and the poles of G.~-~(f). Fur thermore we can therefore con-

clude from Eq. (29) tha t

X,2(U + jv) = Z;, (?J – jv) (30a)

znd



SEC. 74] REDUCTION OF THE ERROR ISPECTRAL DENS ITY 319

Z,,(u+)) = X;,(u–j)). (30b)

Hence, for rea l va lues of j,

Gin(j) = X12(j) Xi, (j) = G},(j). (31)

FinallY, we note that the poles for each expression Y,(2hr jj)X,z(j)

and Yj(2t r jf)XZl(j) lie in the upper half plane and a re symmetr ica lly

placed about the imaginary axis. We can ther efor e express G,(j) in the

desired form as

G,(f) = IY,X,,I’ + IY2X221’

+ +IY1X12 + Y2X211‘ – ~[Y,X,, – Y,X,,12. (32)

Reduction when G 1.s Given by Experimental Da ta .—A r ea son ably

simple method is available for reducing the er ror spect ra l density to thedesir ed form wh en t he spect ra l den sit ies a re obt ain ed fr om exper imen ta l

data . As is ment ioned in Sec. 7.2, in this case the er ror spect ra l density

can be wr it t en as

Gt (~) = + IY1(Z7VY)BTU) + Y2(%rj~)c,(j)12. (3)

To suit the needs of the following method of approximat ion , let us suppose

that o,(t ) and d~(t)ave been determined for the t ime range (0,27’) and

that the limits of the in tegra ls defin ing B, and C. [Eq. (7)] a re O and 2T.It will be our purpose to obta in ra t iona l-funct ion approximat ions for B,

and CT, with poles all in the upper half plane and symmetr ica lly placed

about the imaginary axis. It is clea r that by subst itut ing these approxi-

mat ions in to Eq. (33) we will br ing G, in to the desired form.

The usual technique in making such an approximat ion is t o approxi-

mate the funct ion by a fin ite par t ia l expansion in terms of a complete

or th on orm al set of fun ct ion s. The Four ier t ransforms of the Laguer re

funct ional is a su itable set of or thonormal funct ions. The kth Laguer refunct ion can be wr it t en as

vk+>~~kLk(t) = e-’ ~ – ~(~:;;’ + “ “ + [;.2:-;;:;; (– 1)’ . . ~

The Four ier t ransform of Lk is simply+ 2“(-1 )’). (34)

(1 – 2Tjj)’

‘k(f) = w (1 + 27Tjf)’+’”(35)

This funct ion has it s only pole in the upper half plane and on the imagi-

nary axis; it is thus sa t isfactory for ou r purposes. F ur t hermor e, sin ce

the Laguer re funct ions form a complete or thonormal set for funct ions

that vanish for t <0, it follows that their t r ansforms can approximate

1N. Wien er , op. cit .



.320 RMS -ERROR CRITERION IN SER VOMEC’HA N IS M DES IGN ~8EC.7.4

t o any desired accuracy the t ransforms of such funct ions. Hence we can

approximate B,(j) by

N

a,(f) =2

Cklk(j-), (36)

k=O

where

ck =

/m ~f ~T(m (f). (37)—m

A similar approximat ion may be used for CT(j). Another , and perhaps

a more conven ien t , way of obta in ing the coefficien ts ck follows from the

fact tha t the bilinear form of Eq. (37) does not change its value if thefunct ions are replaced by their Four ier t ransforms. Hence

/

2T~k = dt &(t)L,(t), (38)

o

It is eviden t tha t the ser ies of Eq. (36) will not converge rapidly unless

the t ime sca le is so adjusted that 2’ is of the order of 1; even then the

con ver gen ce may be slow.

General Reduction of G.—If the element s of the spect ra l-density

matr ix are not ra t iona l funct ions, one t r ies to obta in a suitable ra t iona l-

funct ion approximat ion for these element s and thus reduce the genera l

case to the ra t iona l-funct ion case a lready considered. For instance,

when the element s a re analyt ic except for poles, one can frequent ly get

such an approximat ion by taking the sum of the pr incipa l par t s of each

matr ix element a t a fin ite set of poles symmetr ica lly placed about the

imagin ar y a xis.

Another method for accomplish ing th is end involves factor ing the

specta l-density mat r ix G in to two factor s X and Z:

G = XZ. (39)

The elements of X are ana lyt ic and bounded in the lower half plane,

whereas those of Z are analyt ic and bounded in the upper half plane. In

addit ion , for rea l values of ~

x(f) = X’(–f) (40)

and

X(J -) = z’(j).

By Eq. (41)

(41)

G= Xx’ (42)

for rea l values off. It is sufficient , t herefor e, to obta in suitable ra t iona l-

funct ion approximat ions for the elements of X. Because the element s

of X are analyt ic and bounded in the lower ha lf plane and becauw



SEC. 75] A S IMPLE SERVO PROBLEM 321

of the condit ion expressed by Eq. (40), it is possible to make these

approximations.

F act or ing t he spect ra l-densit y matr ix is unfor t unat ely a ver y t edious

and, for most engineer ing applica t ions, an impract ica l task. For deta ils

of th is factor ing process the reader is r efer r ed to the previously cit ed

memoir by Wien er .

7.6. A Simple Servo Problem.—It will be inst ruct ive to apply these

ideas t o a simple servomechanism to be used in dr iving a heavy gun mount

in t ra in . We shall suppose tha t the target s a re essent ia lly sta t iona ry

but tha t the gun is requ ired to slew rapidly from one ta rget to the next

closest . The input will be a ser ies of step displacements through random

angles, occur r ing in some random fashion in t ime. The input can then

be expr essed as

e,(t ) =z

Cku(t — h), (43)

k

wher e u (t ) “is a u nit -st ep fu nct ion :

u(t) = o for t <0,

=1 for t ~ O. }

(44)

The intervals (t i – L,) are independent random variables all having the

same dist r ibut ion , and t he c’s a re likewise independent ra ndom variables

all h aving t he same dist ribu tion.

It is clea r tha t 0, defines a random process. It s der iva t ive is, in fa ct ,

one of the purely random sta t ionary processes discussed in Sec. 6.11 and

was shown there to have the spectral density

G,(f) = 2&, (45)

where P is the mean density of the t ’s and u is the mean-square value of

t he C’S. (We here assume that the mean va lue of the c’s is zero.) We

shall see in the presen t problem that we need t o know the spect ra l density

for the input der iva t ive [Eq. (45)], r a ther than the spect ra l density of

the input it self. The subscr ipt I denotes the input der iva t ive in this

section.

Let us now suppose tha t there is a noise source within, the er ror -meas-

ur ing device, such that the output of the different ia l is (@I – L90+ ON),

where ON is the noise funct ion. For purposes of simplicity we shall

assume that ONis a purely random sta t ionary process having a spect ra l

densityGx(j) = N. (46)

Finally, we shall assume that the noise and the der iva t ive of the signal

are uncor re la ted:Gw(fl = O. (47)



322 RMS -ERROR CRITERION IN SER VOMECHANISM DES IGN [Sm . 75

In the design of this servomechan ism we shall limit our selves to an

equalizer with a fixed t ime delay Z’1 and an adjustable gain III. The

ou t pu t volt age V of the equalizer is t hen determined by

,2’,: + v = K,(e + e.). (48)

As was shown in Chap. 3, the equat ion of mot ion of the motor is

J ~fu+~,@. K3Vdt ’ dt ‘

(49)

wh er e J is t he rot or -plus-ou tput iner t ia , hrz is t he ba ck-em f and viscous-

damping factor , and K3 is the ou tpu t torque pe~ volt input . If we now

set 2’2 = J /K2 and combine Eqs. (48) and (49), we obta in

KIK,T1T2d; +( T1+T2)$+$+re

K,K,.T ,T2~#+(T1+T2)y+3 –~ eff, (50)

This, then , is the differen t ia l equa t ion of the servo. The on ly adjustable

par ameter is K,.

It will be noted tha t the servo differen t ia l equa t ion depends only

upon the noise and the derivative of t he in pu t. H en ce t he spect ra l den sit y

of the inpu t der iva t ive and the associa ted t r ansfer funct ion t ake the

place in our analysis of the usual inpu t spect r a l density and its t ransfer

funct ion . Equat ion (17) can then be wr it t en as

G,(j) = IY,(2mjf) I’G,(f) + I1’2(Wf)1’G.Vf), (51)

where

Y,(27rjj) =T17’2(27r jf)’ + (T, + T2)(2m.ij) + 1.

T, T,(27rj~)3 + (T , + T ,)(%rjj)2 + (%r~j) + ‘~)

and

K,K,

1

(52)—

Y,(27rjf) =Kz

K,K,T, T,(27rjj)3 + (T, + T,) (%jj)’ + (27rjf) + ~.

Thus G,(j) is a lready in the desired form, being the sum of two expres-

sions, each of which is the absolute va lue squared of a ra t iona l funct ion

with all poles in the upper half plane, symmetr ica lly placed about the

imagin ar y a xis.

The t r ansfer funct ions can be put in to a more conven ien t form by a

change of var iable. Let



SEC. 7.6] INTEGRATION OF THE ERROR SPECTRAL DENS ITY 323

Z’=T,+ Z’*,

z = %fT, )

(53)

The va lue of A is fixed and less than or equa l to ~ for all posit ive 2’,, T ,;

a r epr esen ts t he a dju st able ga in . It then follows from 13q. (13) tha t the

m ean -squ ar e er ror is

—2

/

1-

‘=% -.

“x’ + ‘1 .– 2A)!’ + 1 PUT

‘x Ill(jz)’ + (jZ)’ +J z + al’

/

lm

‘% -. ‘Z lA(jz)3 + (j$’ + jX + alz % ’54)

7.6. In tegra t ion of the Er ror Spect ra l De’nsity-We have shown tha t

the mean-square er ror can be expressed as the in tegral of the er ror

spect ra l density and tha t G, can be expressed as the square of the absolu te

value of a rat ional funct ion , or the sum of such terms, with poles all in the

upper half plane, symmetr ica lly placed about the imaginary axis. We

shall now see how to obtain a numerical va lue for the mean-square er ror

by means of the table of in tegra ls given in the appendix.

Any rat ional funct ion can be wr it t en as the quot ient of two poly-

nomials: N(j) /D(f). The above condit ion on the poles of the rat ional

funct ion is equivalen t to the condit ion tha t the roots of D(f) all lie in the

upper half plane and be symmetr ically placed about the imaginary axis.

We can th er efore express D(j) in factored form as fo~lows:

D(j) = a. rI (j – uk – j~k ) ~ (j”+ u, – jh ) ~ (j” – j@, (55)

k k 1

where CLO,t he coefficient of the highest power in f, can be assumed to be a

real number , The u’s and the u’s a re likewise real numbers. If D(j) is

of degree n , then for rea l values of j

= (–l)”D*(;).

We can ther efore wr ite

fu nct ion as

for any rea l f.

k

t he square of the absolu te

y f J 2 = (–l)nl~(j)l’D(j) D(j) D(–f) ‘

1

(56)

value of our ra t ional

(57)



324 RMS-ERROR CRITERION IN SERVOMECHANISM DESIGN [SEC.7.6

We have in this fashion reduced the problem of comput ing the mean-

square er ror to tha t of obtaining the in tegral of Eq. (57) over all fr e-

quencies. The numera tor (– I)”lN(j) I‘ is, of course, a polynomia l in f.

The contr ibut ion of any odd-power t erm in the numera tor to this in tegral

is zero, since the denomina tor is an even funct ion off. We need therefor econsider only the even powers in the numera tor . Finally, if t he mean-

square er ror is fin ite, the degree of the numera tor N(fl must be less than

the degree of the denomina tor D(j). It follows tha t our problem is

solved if we can eva lua te in tegra ls of the form

1. = ~./ - ‘f h.(f~.~– j)’lrJ . .

(58)

wherehn(~) = aofl + a lj’-’ + “ - . + a.,

g.(f) = bor -z + blf’”–’ + “ “ “ + b._,,

and the roots of h.(j) all lie in the upper half plane. Explicit formulas

for all in tegra ls of th is type for which hn is of degree seven or less are

given in the appendix. The method by which these integra ls were

eva luated is presented in Sec. 7.9.

For purposes of illust ra t ion, let us now evalua te the mean-square

er ror for the servo used as an example in the previous sect ion. As can

be seen in Eq. (54), th is involves two integrals of the type shown in Eq.

(58). The denomina tor polynomia l is in each case

h(z) = –A’jd – x’ + j.z+ a, (59)

th e den omin ator of th e t r ansfer funct ions.

Let us first determine under what condit ions the roots of h(x) lie in

the upper half plane. The root s are clear ly of the form

TI = J ’s,

b=ju+u, I (60)

r’=jv —u,

where s, V, a nd u are real numbers. Since

:= n~k = –J%(V2 + U2), (61)

it follows tha t if r l is t o lie in the upper half plane (tha t is, if s > O), then

cc/A >0 and hence a >0. On the other hand, by Eq. (104)

–2jlJ [(s + v)’ + U2] = r l (r , + n) =–j(l – afi)

A, “ (62)

k <1

consequen t ly if r , and I-3 are to lie in the upper ha lf plane (that is, if



SEC. 7.7] MIN IMIZING THE MEAN-SQUARE ERROR 325

v > O), it is necessa ry tha t a < l/A. The region of stability for th is

s er vo is t h er efor e

0<.<;. (63)

Only when a is with in this region of stability can one make use of the

table in the appendix to determine the mean-square er ror ; when a is out -

side t he region of stability, th e mean-square er ror is infin ite;

The in tegra l table is very easy to use. For instance, in evalua t ing

the first of the in tegra ls in Eq. (54), one subst itu tes in the formula for

13, setting

aO=— Aj, b, = A’, \

al= —1, b,=l– 2A,

az = j, b, = 1,

a~ = a. /

One finds tha t

~= BaTl+a(l– .4)

2a(l – Aa) “

(64)

(65)

This would be the mean-square er ror in following the input signal in the

absence of noise. It is eviden t tha t ~ becomes infin it e on the boundar ies

of the region of stability. In a simila r fash ion the second in tegra l canbe evalua ted; the cont r ibu t ion of the noise to the mean-square er ror is

The mean-square er ror it self is the sum of these t wo componen ts:

l+a(l– A)+La2~~ = POT

2a(l – Aa) ‘

where

L=>2&r T2”

(.66)

(67)

(68)

In genera l the polynomial h .(j) will be the product of two poly-

nomials, one from the signal or noise spect ra and the other from the

t ransfer funct ion . The signal or noise polynomia l is fixed once for a ll;

it s r oot s lie in the upper half plane. On the other hand, the t ransfer -

funct ion polynomial var ies as we vary the equa lizer parameter s. It s

root s will lie in the upper half plane if, and on ly if, these parameter s lie in

t he r egion of st abilit y.

7.7. Minimizing the Mean-square Er ror . -We have now obtained an

explicit formula for the mean-square er ror as a ra t ional funct ion of the

equa lizer parameter s for va lues of these parameter s inside the region of

stability. The next step is, of course, to obta in the va lues of the equalizer



326 RMS -ERROR CRITERION IN SERVOMECHANISM DES IGN [SEC. 77

parameters that minimize the mean-square er ror . These “best” va lues

can be determined by set t ing equal to zero the par t ia l der iva t ives of

the mean-square er ror with respect to the parameter s and solving for

the values of the parameters. Somet imes this process beiomes very

involved; in such cases it is simpler to loca te the minimum by directexplor at or y ca lcu la tion s of mean-squ ar e-er ror va lu es. We are also in a

posit ion to study var ious other proper t ies of the servo. We can , for

instance, determine how sensit ive the mean-square er ror is to small

devia t ions of the equa lizer parameters from their best values. Since it

is oft en possible t o elimina te some of t he n oise at its sou rce by su fficien tly

elabora te filters, it is of in t erest to see how the minimal mean-square

er ror var ies with the noise level. We can also draw any desired decibel–

log-frequ ency diagram; this will be of help t o th e exper imenta list whosepr incipal method of adjustment makes use of the servo steady-sta te

response.

It will be inst ruct ive to apply some of these ideas to the example dis-

cussed in Sees. 7.5 and 7.6. In the previous sect ion we obta ined an

expression [Eq. (67)] for t he mean -squ are er ror :

l+a(l– A)+Laz~~ = DOT

2.(1 – A.) ‘-(67)

In order to find the minimum mean-square er ror , we differen t ia te ~ with

mspcct , t o t he gain parameter a and set t he der iva t ive equal t o zer o. Th e

resu lt ing equa t ion is quadrat ic in a . Only one of its two roots,

1

a“=A+~A+L’(69)

lies in t he r egion of st abilit y. Su bst it ut in g am in to Eq. (67), on e obt ain s

t he minimal mean -squar e er ror ,

(70)

Figure 72 sholvs plots of am and e~/@u~’, as funct ions of L, for

A = ~ (tha t is, for 1’, = 7’,). As was to bc expected, the “ best” gain

value decreases as the noise level increases. For no noise what ever (tha t

is , L = O) and A = ~ the best value of the gain is $. F or t his minimal

condit ion, th e root s of the character ist ic equat ion are

(–3.276 –0.362 k 1.224j

T’ T )(71)

The logar ithmic decremen t for the complex roots is & = 1.85. A graph

of t he er ror response t o a unit -step funct ion is shown in Fig. 7.3. Accor d-

ing to the usual standards, one would say that the system is a bit under -



SEC. 77] MIN IMIZING THE MEA.N -SQUARE ERROR 327

damped. It must be remembered, however , tha t in th is case L = O.

As L in crea ses (.4 rema in in g equal t o ~), t he r oot s of t heopt ima l-con clj-

FIc..72.-Optim

1.5

67 ~“BuT

1.0 \

n iz- —

0.50 0.2 0.4 0.6 0.8 1.0

L[a l v alu e s of m ea n-squ ar e er ror , ga in, a nd loga rit hm ic decn

+1,0

+ 0.8

+0,6

+ 0.4E+0,2

o

-0.2

‘0’40123456789 10._b_T

FIG. 7.3.—Error response to unit -step input . A = +,

.ement.

L=O.

A=i.

t ion character ist ic equat ion vary in the following way: The rea l root .

decr ea ses slight ly in magnit ude u ntil for L = 1 it is – (2.99/7’); the rea l

par t of the complex root s increases slight ly while the imaginary par t



328 RMS -ERROR CRITERION IN SERVOMECHANISM DES IGN l~EC.7.8

decr ea ses un til for L = 1 t he complex root s a re ( – 0.50 + 0.85j)/ 2’. Thus

as L increases, the system becomes more damped. A graph of the

logar ithmic decremen t as a funct ion of L for the opt imal condit ions is

shown in Fig. 7.2.

7.8. Radar Automat ic-t racking Example.—It will bc the purpose of

the presen t sect ion to apply the rms-er ror cr it er ion to the pract ica l prob-

lem of radar aut omat ic t rackin g. We shall determine the “ best” va lues

of the equalizer parameter s for the gyrostabilized t racking mechanism

descr ibed in Sec. 610. This system has actually been built and pu t in to

opera t ion . The techniques used in the design of the system were those

descr ibed in Chap. 4; the final adjustment of the parameter s was made

by t r ia l-and-error methods. As we shall see, the resu lt ing va lues a re

very close to the best values as determined by the rms cr iter ion if cer ta inrest r ict ions imposed by the mount power dr ive are taken in to accoun t .

This, in effect , furn ishes us with an example of the validity of the design

met hod pr oposed in th is chapt er .

In this example, as in tha t of Sec. 7.5, we may focus a t ten t ion on the

der iva t ive of the input ra ther than on the input it self. In Eqs. (6.122)

and (6. 123), the input is t l~, and the factor 2mjf appears in Y1. The

presence of th is factor calls ou r a t t en t ion to the fact tha t the servo differ -

en t ia l equat ion does not involve 01it self but only der iva t ives of 01. 1“Ow,a t ransfer funct ion Y opera t ing on the der iva t ive of t l~gives the same

r esu lt as t he t ra nsfer fu nct ion 2rjfY act ing on 01. It follows tha t in the

er ror spect ra l den sit y

G,(j) = \Y,(2mjf)12GI(j) + [[email protected])12GNf)

+ l’T(2rjf)G,,~(f) Y ,(2rjf) + Y ,(2rjf)GN ,(j) Y~(2~jf), (17)

we can take G,(f) as the spect ra l density of the der iva t ive of O,and G,~(j)

as t he cr oss-spect ra l den sit y between t he der iva tive of 81an d t he n oise if a tthe same t ime we drop the factor 2mjf from Y, as previously defined,

writ ing

(Z’22=jj + 1)yI(%j) = ~2(2=jj)2 (1 + K.T,)2r jj + K.’ (72a)

K.(T ,2rjf + 1)(72b),(2~jf) = T2(2m jj)Z + (1 + K.1’’,)%jf + K.”

This is a necessa ry change in the poin t of view because the O, tha t we

propose to use does not have a well-defined spect ra l density whereas it sd er iva t ive does .

Let us fir st determine the region of stability for the t ransfer funct ion

of this servomechanism. This is the set of all pa rameter values (Z’l,

T ,, K,) for which the root s of

H(j) = T2(27r j.f)2+ (1 + KvT,)(2r jf) + K, (73)



SEC, 78] RADAR A UTOMA TIC-TRACKING EXAMPLE 329

lie in the upper half plane. It will be found that the region of stability is

defined by the inequalit ies

K* >0,

1 + KoT , >0,

1

(74)

T , >0.

Alt hou gh t he r egion defin ed by r ever sin g all t hese inequa lit ies would a lso

lead to roots of H(f) in the upper ha lf plane, such a region is ruled out of

considera t ion by the physical na ture of the parameter s. We see then tha t

the system will be stable for all of the inheren t ly posit ive values of the

pa rameter s. Fu r thermore, t he system does not tend to become unstable

as the gain K. is increased. (In any actual system, however , one finds

tha t as the gain is increased, th is second-order idea liza t ion of the servobreaks down because of nonlinear effect s and t ime lags tha t a re no longer

negligible. )

The pr incipa l source of noise in radar t r acking is fading. It was

shown in Sec. 6.10 tha t one can approximate the noise spect ra l density

by a constant ,G.(j) = N, (75)

where, as in Eq. (6.132),

N = 0.636 milz sec. (76)

The t r ajector y of the a irplane will, in genera l, ha ve zer o cor rela t ion with

the fading. Hence we shall assume tha t G,~(f) and G~,(f) vanish

identically.

Finally, for the servo input we shall use an input of the type studied

in Sec. 6.12 and pictu red in Fig. 6.9b. We shall suppose tha t the t ime

axis is divided in to interva ls with end poin ts . , t _.2,t–l, to, t l, t~, . . . ,

tha t sa t isfy the condit ions for a Poisson dist r ibu t ion with mean density B.

Within each in terva l the angular velocit y approaches a new value

exponent ia lly, wit h t ime consta nt 1/V. The values approached do not

depend on any of the previous or subsequen t in terva l va lues and have a

zero mean and a mean-square value equal to ~. Combining Eqs.

(6.153) and (6.169), we see tha t

We shall t ake

(77)

~ = 0.04 see-’

v = 0.10 see–l

)

(78)

~’ = 2.62 X 103 (mils/sec)2

for reasons discussed at the end of Sec. 6.12.

In this case the spect ra l-density matr ix consist s on ly of diagonal

terms. The spect ra l density GN, being a constan t , is a lready in the form



330 RMS -ERROR CRITERION IN SERVOMECHANISM DES IGN [SEC.78

requ ir ed by Sec. 7.4. It is a lso an easy mat ter to factor (l], for , as can

be seen by inspect ion ,

G](j) = lxIi(f) l’, (79)where

2V %@(80)ll(~) = (, + %jf)(fj + %“f)”

The poles of A’11 lie in the upper half plane on the imaginary axis.

We are now in a posit ion to compute the mean-square er ror . Since

t he cr oss-spect ra l density vanishes ident ica lly, t he m ea n-squ ar e-er ror

in tegra l becomes the sum of two integra ls—one par t due to the signal,

?, and another par t due to the noise, ~V. Th e compon ent in tegr als ca n be

\ vr it t en as follo\ vs :

The integra l in Eq. (81) can be brough t in to the form of Eq. (58) by

taking

h(z) = [– T2x2 + (1 + K. T,)jz + Kvl[(jx + ~)(j~ + p)]== l’jx~ – j(a + T jS )Za – (K. + atl + T ,P)z2

+ j(KoS + @)z + KuP, (83)where

a = 1 + KmTl,

S= D+P,

1

(84)

P = p..

The evalua t ion of the in tegra ls [Eqs. (81) and (82)] by means of the

appendix table is then st ra igh t forward and leads to the resu lt

a-2PvaT ; + TW + *P (aKV + CY2S+ aTJ Y + Z%S ’P)

T l. —as [(K. – T&)’ + (a + T2A’)(KJ ’ + c@)] ‘

I

(g~)

~_ N(. –1)’+T*K.,N – F2

2a ‘



SEC. 7.8] RADAR A UTOMA T IC-T RACKING

As before

EXAMPLE 331

(86)

There remains the formidable problem of finding posit ive ra lues of

(a, T ,, K,) tha t minimize the mean-square er ror . Since a direct analyt icsolu t ion of the equa t ions obta ined by set t ing the der iva t ives of ~ equal

to zero is impract icable in this case, one is forced to invent another

approach.

We can gain some insight in to the nature of ~ by studying the

a sympt ot ic beh avior as t he pa ram et ers becom e in finit e. If we set

a = C21z,

K. = K,x,

1

(87)

T , = .r,

and t ake t he limit a> z becomes infin ite, we obta in a m~lch simplm expres-

sion for ~:

It is not difficu lt to obta in the minimum of ~ by scmiernpir ica l mrtho[ls,

Table 7.1 gives the opt imal values of (a l, A’1), together with devia t ions

from these values ~vhich cause a 10 per cent incr rase in ~. .1s can bC

Optimalvalues. 2.54 6.10 0.199 I 0786 I o \)&j

[

1.50 6.10 0 218 ~ 0.875 I ] 093

Nonoptimal values. .,.! 20 6.10

405 I :.::0 896 1.090

2.54 0,652

9.75 I 0.080 I 1.012 I 1.092

1 ()~)o

2.54

seen , the values of ~ are not very sensit ive to var ia t ions in (~1, Kl)

about t heir opt imal values,

We shall now determine how ~ var ies with z, for a , = 2.54 and

KI = 6.10. This is easy, as ~ is of second degree in (l/x); in fact

[

-3+

a-v a,Kl + a;S + alS 2 + SP–+:;

10

~?

CYIKJ3 (K1 — P)z + (al + S ’)(K,S + alP) x. (89)

A graph of ~ as a funct ion of x is shown in Fig. 7.4. There is a T.erYsha]-

low minimum which occurs a t

G = 314.5 (a = 800, K. = 1918, T2 = 314.5),



332 RMS -ERROR CRITERION IN SER VOMECHANISM DES IGN [SXC. 78

At this poin t

~ = 0.1991 + 0.7846 = 0.9837 milz. (90)

The graph r ises rapidly as z decreases past 50.

For z = z~, the (1/z) 2-term in Eq. (89) is only 0.05 per cen t of ~.

It is eviden t tha t th is term does not vary rapidly with al or K,; we must

t herefore expect the absolute minimum of = to be close to its minimum

along the line (a , = 2.54x, K1 = 6.1OZ, Tz = z). This, in fact , is the case,

and for all pract ica l purposes the absolute minimum can be consider ed t o

beat the above determined point . To obta in an est ima te of t he va ria tion

100

6040

20m

‘F 10

,; c ; ‘\

4 -/

2 \

1\ _

0 50 100 150 200 250 300 350z

Pto. 74.-L’ar ia t io,] of e; along the line (a = 2,541, K, = G,l.r , T% = r ).

of C2with a and K,,, one need only subst itu te a = 314.5 al for a ,, and

K“ = 314.5 K, for K, in Table 7.1.

TABLE7.2.—> FORA’. = 80 six-l

I K.

1-Optimal values. I 80Exper imenta lva lues. I 80

{

8080

Nonoptimalvalues, So

80

l’,

13

16

13

13

17

9

a

33.029.849.022.033.0

33.0

;

0.4860.5930.4770.4950.629

0.382

yC’v

0.7650.7030.8350.8230,676

0.934

.-,’

1,251

1. Zwi

1.312

I ,318

1.305

1.316

In the actua l t racking system it was found tha t if K, was increased

beyond the value 80 see:’, ‘one could no longer ignore the effect of the

mount power servo and the system performance rapidly deter iora ted.

With this limita t ion in mind, the best parameter va lues for a and 2’,

wer e det ermin ed for K, = 80 see–’. Listed in Table 7“2 a re the mean-



SEC. 7.9] EVAL.UA T ION OF THE INTEGRALS 333

square er r or s for t hese optimal values, for t he actua l exper iment al va lues

[see Eq. (6.113)], and fina lly for values tha t give a 6 per cent increase in

the mean-square er ror . We see from Table 7.2 tha t limit ing K, to 80

see–’ increa ses t he mean-square er ror only 25 per cent a bove its absolu teminimum. This results in a 12 per cent increase in the rms er ror , which

is quite acceptable. The exper imenta lly determined parameter values

give a mean-square er ror ext remely close t o t he rest rict ed minimum value

and, in a sense, confirm the arguments of this chapter .

7.9. Evaluat ion of the In tegra ls.—It remains to show how the

integra ls in the appendix have been evaluated. These in tegra ls a re of

t h e form

a nd t he r oot s oj h“(z) all lic i71heupper h alf pla ne. No greater(mrrality

J ~ould be achieved by allo]~ing g to

conta in odd poiver s of z, since t he con-*m

tr ibut ion of such terms to the integra l

&

wiwould be zero.

gZ-plane

We now apply the method of resi-

dues’ t o the integra l in 13q. (91). In

the presen t applica t ion this method

requires tha t the value of t ile integra l –R R ReaI

t aken along a semicircle CR of radius

R, which has its een tcr a t the or igin

and lies in the upper half plane (see

I1~1,>,7.5.– P;,l I1of i!,t,xl L,t,,,l.

Fig. 75), approach zero as R becomes

infinite. This condit ion is clear ly sa t isfied, since for sufficient ly large lt

the in tegrand is less in absolute value than 111/1<:, for some posit ive

constant it f. It follo~vs tha t

The integra l about the closed path [(– R, R) + C’,] is independent ofR for sufficien t ly large R’s and is, in fact , equal to the sum of the residues

a t t he poles of t he in tegr an d con ta in ed }~-it hin t his closed pa th .

In the fur ther developments, the n may be omit t ed as a subscr i~t

wh er e this cannot lead t o confusion .

Since t he root s of h (z) all lie in the upper half plane, the root s of h( – x)

lie in t he lower half plane. Consequent ly for sufficien t ly large R the poles

1See E . C. Titchmarsh, The Theory o.fFunct ions , Oxfor d, N ew Yor k, 1932.



334 RMS -ERROR CRITERION IN SERVOMECHANISM DES IGN [S 13C.7.9

of t he in tegr an d con ta in ed in t he closed pa th of in tegr at ion [(– R, R) + C*]

will be pr ecisely t he root s of h(x). Therefore 1 is equal to the sum of the

residues taken at the roots of h(z). It will be assumed for the moment

that these root s a re all simple; it will be shown la ter that the sta ted resu lt

holds even when some of the roots a re mult iple roots.Since each residue is a ra t ional funct ion of a root of h(x) and since all

root s a re t r ea ted alike, the in tegral is a symmetr ic ra t iona l funct ion of

the roots of h(z). It follows t ha t I can be expressed rationally in t erms of

the coefficien ts oj h and g. 1 It will be the purpose of th is seet ion to der ive

su ch an expr ession .

Let x,, x2, . , , Z. be the roots of h(z), assumed dist inct . Writing

1 as the sum of residues in the upper half plane, one has

n

zg(xj)

I= —h’(zJh(-xJ’

wh er e h ’(z) is t he der iva tive

By the factor theorem,

h(x)

k=”l

of h(z) with respect to x.

(93)

.

= aOrI (f’ – Xi). (94)

i=l

Hence

h(–~~) = 2a0(– I)nZkn

(X!f + Xi). (S!5)

i#k

The least common mult iple of the factors (~k + xi) is t he product of all

sums of pairs of root s, ~ (x~ + xl). Equat ion (93) can now be writ ten

1<m

as

The evaluat ion of Eq. (96) for n = 2 is simple enough, but a systema-

t ized approach is required when n >2. The following pr ocedur e con-

sists of two par ts. First an expression is obta ined for the product of all

sums of pa irs of root s of a polynomial in t er ms of its coefficien t s. This is

clea r ly necessary for the term ~ (G + z~) and will a lso be usefu l in

1<mi,i #k

eva lua t ing the expression ~ (z, + z!), ~~hich is the product of all pa irs

j <i

1See L. E . Dick son , F ir st cou r se in & T bory of E qu ation s, W’iley, New Yor k,

1922, Chap. 9,\



SEC. 7.9] EVALUATIO.V OF THE I.h’TEGRALS

of r oot s of t he polyn omia l

335

f,(z) . 2 Q = aOz”–l + (alm+a,)z”-’X—xk

+ (ao.z~+ alzk+a,)x”-’ . + (aoz~-l+ ~ ~ . +an_l). (97)

The r esu lt in g expr ession is, in t his ca se, a polynom ia l in x, wit h coefficien t s

tha t a realgebra ic funct ions of thecoeficien ts of h(z). The problem will

then bereduced toeva luat ing symrnet r i cfunct ions of the form

n

z–;h’(xk)’(1=–1,0,1,2,..). (98)

k=l

This will be accomplished in the second par t .Let us now obtain an expression for the product of all sums of pa irs of

r oot s of a n a rbit ra ry polyn om ia l,

N

Q(z) = z ,zN–i. (99)

j=(.l

Suppose Q(z) has the root s z,, Zzj . . . , z~, all dkt inct . Then, given the

tWO rOOtSZk a nd 21, let

(loo)

It

in

will be noted tha t Q(a — y) and Q(a + ~), considered as polynomials

the var iable y, have a common root y = (a – z~)/2, since

()I—ka———

2 ‘Zk a+t ’=)=z /

and(101)

Q(zk) = o = (J (z,). )Addin g and su bt ract ing t he equ at ions

Q(a – y) = O,

Q(a + y) = O, (102)

one obta ins two equa t ions in yz which have the common root (z1 — z~)/2.

The resu ltan t ’ found by eliminat ing Y2 from these two equa t ions must

therefore vanish for a given by Eq. (100). On set t ing the resu ltan t equal

to zero, one gets an equat ion of degree [N(N – I )]/2 in a. It is clea rthat a ll sums of the type (22 + zk)/2 will sa t isfy this equat ion . Since

ther e a re precisely [N(N – 1)]/2 such sums, all r oot s a re of th is type;

that is, t here is a one-to-one cor respondence between the root s of this

equat iqn and the terms (ZZ+ z~)/2 for a ll possible choices of 1 and k . It

follows tha t t he constan t t erm of th is equat ion , divided by t he coefficien t

of h ighest power in a, is precisely

1Seefor instanceL, E. Dickson, op. cit ., Chap. 10.



336 RMS -ERROR CRITERION IN SERVOMECHANISM DES IGN [SEC, 79

1~:-~ji,i(i--Tl12 rI (2, + 2,). (103)

1<k

The actua l computa t ion of this ra t io is tedious but none the less st raight-forward. The resu lt is tha t for

N odd, AO AZ A4 — A,.–] O — O 1

n

N–1O .40 .4, — A,,_, A,,_, — O -- ~-—

(z, + 2,)— — — — —

l<krows

= (–. 1)1(,~-1)(~-vm ——— —- —— __ J

A]-1[

(104)

A, .4, As — .4?J O —O_. N–1

O .41 A3 — .4,v_, .4~ —2

— — — rows— — –- 1

N even,

I----E

AO AZ A4 — A.v_2 A,v O — O T

n

.40 Az — A.,-4 A~_z A,N — O ; – 1(z, + z,)

— — — — rows1<k= (-1)(’’’+2”1/ ’ – – – – – – _ J

A ~-l ‘--(105)

A, A3 .4, — A,v_l O 0 — O j

Al .43 — AN-S AN–I O — O ~

— ——— —— —— rows— — — — -1

As one might expect , these determinan ts appear in Routh’s cr it er ion.

As has a lready been ment ioned, th is resu lt is used in two ways in the

evalua t ion of the expression in Eq. (96). In order to obtain

~ (Zm + 24,

1<m

one merely replaces Q(z) by h(z); .4, is then replaced by a,, and N isi,) #k

r eplaced by n. On the other hand, in order to obta in ~ (xi + ZY), onej <i

r epla ces Q(z) by t he pOlynOmial~k(~) of Eq. (97); A, is replaced by

(arc; + ah-’ + “ “ - + a,),

and N by (n - 1). In the la t ter case it is eviden t tha t the determinan ts

becom e polynomials in x~, say A(xk), with coefficien t s tha t are ra t iona l

fun ct ions of t he coefficient s of h. Equat ion (96) therefore assumes the

form



SEC. 7.9] EVALUATION OF THE INTEGRALS 337

.2a0

(–l)”

~ (Xm + xl)

9(@A(~k).

xkh’(~k)

(106)

1<m k=iThe pr odu ct g(Zk)@k) is again a polynomial in xk. Since each term of

th is polynomia l can be eva lua ted sepa ra tely, it remains on ly t o ca lcu la te

expr ession s of t he t ype

n

z–;h’(m )’(1=–1,0,1,2,”””). (98)

k=l

This is the sum of all residues in the plane’ of the funct ion z’/ [h (x)].

When the degree of the numera tor is a t least two less than the degree of

the “denomina tor , the sum of the residues in the plane van ishes. It

follows tha tn

z 1 1

‘=–h~’kh’ (~k)k=l

(107a)

.

2

d—=0~=~h’(z,)

for O~lsn–2. (lo7b)

In order to obtain the result for la rger va lues of 1, it is conven ien t to wr ite

.

z-r— for 1x1 > maxlx~l, (108)x’

,=0

wher e r , is the sum of all symmetr ic funct ions of weigh t r :

r . = 1, \

.

(109)

r4 = etc. I1For a more complete developmen t of th is a rgument we refer the reader to

W. S. Burnside and A. W. Panton , Theor~ of Equations , 9th cd., Vol. I, Longmans,

London , 1928, pp. 171-179.



338 RMS -ERROR CRITERION IN SERVOMECHANISM DES [G.V [Sm . 7.9

Returning to Eq. (108), ~~emult iply th rough by xi-n and obta in for the

sum of the residues in the plane

.

forlzn–1. (107C)

It is possible to der ive a recur rence formula for the I“s by making use

of Eq. (108) in the following way:

(24._ Mx) m,x“ x“

,=(1.

(1 1

a0+aI~+a2;;+”.”+ an:)(2 )

r ,— – (110)x’

,=0

Compa rin g like power s of z on bot h sides of Ilq. (110) gives

aOrm+ alrm_l + + amro = O. (111)

Here am = O for m > n. One can avoid the use of the r ’s by succes-

sively reducing the degree of the numera tor of Eq. (106) to n – 1, by

means of the formula

2-1 + ~,~~–z + + an&’aoxj = alzk for (1 > n). (112)

Since Eq. (112) is merely the sta tement tha t h(z~) = O, this subst itu t ion

can be made at any stage of the ca lcula t ion .

When the computat ion is car r ied th rough as out lined above, one

obta ins In as a ra t ional funct ion of the coefficien t s of gin(z) and hn(z),

This expression has been shown to hold for the root s of h=(z) dist inct and

in the upper half plane. Fur thermore, since the coefficient s of h .(z)a re con t inuous funct ions of the roots, both the computed expression for

In and its in tegrand are cont inuous funct ions of the roots. The expres-

sion for In equals the in tegra l as the roots approach a mult iple sta te; it

follows tha t it is equal to the in tegra l in the limit .

It will be inst ruct ive to car ry through the above process for Is. In

t his ca se

h(z) = a0x3 + alzz + a2z + as,

}

(113)

g(z) =bDX4+ b1X2 + bz,

and the r oot s of h(z) a re assumed t o lie in the upper half plane. It follows

from Eq. (104) tha t

11

1 a. az(Xm + z,) = ~ a, ~, = aoa’ ; a1u2”

1<m

(114)I

I



SEC. 79] EVALUATION OF THE INTEGRALS 339

In this case Eq. (97) becomes

then , by Eq. (105),

i,j+k

II (Zi + z,) = ~ (aOZk+ al).

1 <z

Equation (106) can no\ \ ’be wr it ten as

1—

2(aOa3 – a,a2) ,

a1b2

~ombin in g t hcw r ct iu lt s \ vit h 1’11. (11 7 ), (me obt ain s

This can bc ar rxmged so as to give

(115)

(116)

(117)

(118)

(119)

(120)

which checks \ vith Is as gi~,cn in t he appen(iix



CHAPTER 8

APPLICATIONS OF THE NEW DESIGN METHOD

BY C. H. DOWKER AND R. S. PHILLIPSI

In the preceding chapter a method was developed by which one maY

choose the best design for a servomechanism of a given type. In or der

to make this choice, the type of servo must be decided upon in advance

and the stat ist ica l proper t ies of the input signa l and the noise disturb-

ances must be known. The best servomechanism per formance is t aken

t o be tha t which minimizes therms er ror in the output .

The usefulness of this method of servo design would be grea t ly

enhanced if the opt imal designs for genera l types of input signals and

noise wer e det ermined on ce a nd for all. In that case one would not have

t o go thr ough the deta iled calcula t ions for each par t icula r applica t ion;

instea d, a ft er comput in g t he cor rela tion funct ion s for t he input signal and

noise, on e could t hen look Up t he specificat ions for t he best ser vomech a-

nism. One would, in addit ion , be able to rela te these results t o those of

the steady-sta te analysis by transla t ing the character ist ics of the best

ser vos in to t he la ngu age of t he decibel–log-fr equ en t y dia gr am .

It is the purpose of the fir st par t of the presen t chapter to make a

modest beginning on such a program. The opt imal ser vo parameters

for two simple servomechanisms of standard type a re found for a var ietY

of inputs. These result s a re present ed by means of graphs, Nyquist

diagr ams, and decibel–log-fr equency diagr ams.

In the second par t of th is chapter the rms-er ror cr it er ion is applied tomanual t r acking of a type that has, for instance, impor tant military

applica t ions. The t racking apparatus plus the human opera tor forms a

ser vo syst em; t he novel fea tur e of such a syst em lies in t he biomecha nica l

link. The best t ime constant for the given t racking unit is determined.

8.1. Input Signa l and Noise.—We shall assume in this discussion

spect ra l densit ies of input signal and noise tha t depend on th ree param-

eters. This dependence is so flexible tha t many actua l input spect ra l

densit ies can be approximated by a suitable choice of these parameters;

yet the dependence is simple enough to a llow at least approximate solu-

t ion for t he best ser vo pa ramet er s in t erms of t he input pa ramet er s.

We shall assume that the input to the servo is 61 + o.., where & is

1Sect ions 8.1 th rough 8,9 by C. H. Dowker ; Sees. 8.10 th rough 812 by R, S.

Phillips.

340



SEC. 8 .1] INPUT S IGNAL AND NOISE 341

the t rue input signal and ONis the noise. We shall t ake as our typica l

servo input the one descr ibed in Sec. 6“12. For this input the velocity is

constant th roughout extended in tervals of t ime, changing abrupt ly a t

the beginning of each interva l to a new value independent of any other ;the end point s of the t ime interva ls have a Poisson dist r ibut ion. The

mean length of the t ime interva ls is denoted by I/@. If ~z is the mean-

square value of the velocity, then the spect ra l density of the der iva t ive

of the input is [see Eq. (6169)]

4@G,(j) = —

~z + p’(1)

where o = 27r j.

Th e spect ra l den sit y given in Eq. (1) r epr esen ts a t lea st a ppr oximat ely

the spect ra l density for many other servomechanism inputs. As long

as t he power of t he input der iva tive is con cen tr at ed a t t he low fr equ encies,

Eq. (1) is an adequate represen ta tion ; d/27r can be thought of as the cutoff

fr equ en cy of th e input der iva tive.

Th e spect ra l densit y of 0.,-is assumed t o be of t he form

G,,(f) = N, (2)

where N is a constan t . In the t erminology of Sec. 6.11, the noise inpu t

is assumed to be a purely random process. Such a spect ra l density can

be considered to approximate a spect ra l density tha t is essent ia lly con-

stant for all frequencies low enough to pass through the servo without

ser ious a t t enuat ion . In Sec. 6.10 this approximat ion was found to give

a sa tisfa ct or y r epr esen ta tion for r ada r fa din g in t he a ut omat ic-t ra ckin g

servo sys tem.

We shall fur ther assume tha t the cross-spect ra l density vanishes

identically:

G,,v(~) ~ O. (3)

This will, in genera l, be the case whenever the sources for the input signal

and t he n oise a re in depen den t.

In the par t icular examples of servo inputs tha t we shall consider the

rms input speed W&z is about 50 mils/sec and 1/6 is about 10 sec (see

Sec. 6.12). Thus G,(O) = 4~2/13 = 10’ mils2/sec. In addit ion,

G.(f) = N = 1milz sec

as in Eq. (6.132). These compara t ive magnitudes will form the basis for

a pproxima tions m ade in t he cou rse of t he following discussion.

When the cross-spect ra l density vanishes, the spect ra l density for

the er ror (Sec. 7.2) is given by

G,(f) = IY,(%jf) I’G,(j) + IY,(%jf) l’Gs(f), (4\

where Y1 is the t r ansfer funct ion for the input-signal der iva t ive and Y,



342 APPLICATIONS OF THE NEW DES IGN METHOD [SEC.8.2

is the t ransfer funct ion for the noise input . The mean-square er ror can

then be found as in Chap. 7 from the expression

/

.~=

df G,(f ).o (5)

SERVO WITH PROPORTIONAL CONTROL

8.2. Best Cont rol Parameter .—We begin with the simplest and best -

known type of servomechanism—a servo with propor t ional cont rol.

This type of servo is of considerable in terest in it self; it is, in addit ion ,

suitable for exhibit ing the complete analysis of a simple problem. The

minimum mean-square er ror for th e propor t ion al-cont rol servo will a lso

furnish a standard with which to compare the per formance of morecomplica t ed types of ser vomechanisms.

It will be convenien t in wha t follows to omit the symbol C( ),

denot ing the Laplace t ransform of the funct ion lvithin the parentheses.

Th e r ea der sh ould h ave n o difficult y in differ en tia ting bet \ \ ’een t h e fu nc-

t ion of t ime and its I,aplace t ransform by the con text .

The output 00 of the servo with propor t ional cont rol is rela ted to the

input by the equa t ion [see Eq. (4.5)]

(Tmp + l)pflo = K.(8, + 0., – 00), (6)

wh er e K. is t he velocit y-er ror const an t and T~ is t he mot or t ime con st an t.

Here Ko is a t rue parameter , var iable o~’er a wide range, whereas T~ can

be ch an ged on ly by repla cin g t he m ot or or ch an ging t he loa d iner t ia .

We shall show that if the motor t ime constan t T~ is too la rge for t he

par t icular servo applica t ion , the minimum mean-square er ror depends

st rongly on T~. If Tn is less than a cer ta in cr it ica l value, the minimum

mean-square er ror and the opt imal va lue of K, are pract ica lly inde-pendent of !/’~. If T~ has this cr it ica l value, t he peak amplifica t ions a re

la rger for the opt imal servos than might be expected on the basis of the

st eady-st at e analysis of chap. 4; bu t if l’~ is ~vellbelow t his cr it ica l va lu e,

the usual peak amplifica t ions a rc obta ined. The decisive factor in the

design of the propor t iona l-cont rol servo is the noise-to-signal ra t io

N/~. }Vhcn this ra t io is small, t he minimllm mean-square er ror is

propor t ional t o (~N’) “I and th e opt irn t il K. is prupor t iomd t o (~/N) ~~.

Ilquat ion (6) of t he ser l’o can be solved for k as follo~~s:

eO=e, –T .p + 1 Km

— ON . (7).- peI + ~w + (T -P + 11PKo + (’/ ’~p + l)p

Thus 00 differ s from O,by an er r or c consist ing of t wo par ts—the er ror due

t o th e fa ilure of t he servo t o follo!v t he input signal and t he er ror resu lt ing

from the noise. Since the spect ra l density of pfl, is 4P%/(u’ + p’), the



SEC. 82] BES T CONTROL PA RAhfE1’ER 343

mean-square er ror in follo\ ving the input is.

7

/

1 ‘du _ T;t i’ + 1 -lp;~c,=% “

~K, +jkl – 7’WU1]202+&”(8)

on in tr gr at in gt h isc,xpr cssi,~n ll yt l~( m et hodof Sec. 7.6 a nd simplifying

t he r esult , on e obta ins

(9)

Simila r ly, since G,,(j) = i~j the mean-s(luarc er ror result ing from the

n oise is

(10)

wh ich , wh en in tegr at ed, becomes

Y NK.C:v=

“4 “(11)

The mean-square er ror in following is then

.Lt th is poin t it is convrn icn t to make cer ta in approximat ions which

will normally be just ified. The number O, tha t is, the cu toff frequency

of the input . signa l, is usually small; for instance, ~–l is betv-cen 10 and

30 sec in the input , discussed in Sec. 612. ‘he motor t ime constant

T~ is u su ally bct ~~-een 0.5 a nd 0,05 sec. Hence ~T ,. is likely t o be small

compared with unity, If a lso @/h-, is small compared with 1, we may

appr oximate t o ~z by the formula

(13)

Th e compu ta tions ~~+ille simplified if we subst it ut e for Ku in t erms of

t h e dimen sion less pa r amet er

()

KNbh~=y= .

az(14)

The form of the mean-square er ror tha t we obta in is

‘= @4+,+G)%+o(15)

We are now in a posit ion to find the best va lue of K,—the value of K.

and hence y tha t minimizes the mean-square er ror . To this end, we



344 .4PPL1CATIONS OF THE NEW DES IGN METHOD [SEC. 8.2

differen t ia~e ~ with respect to y and equate the resu lt to zero:

(17)

Subst itu t ing the opt imal va lue Of y into 13q. (15), we find the min imum

mean-sauare er ror

(18)

For each value of v > 1’ Eqs. (17), (18), and (14) give respect ively a

value of T~ and, cor respon din g t o this T~, t he minimum ~z and the opt i-mal value of K.. Figure 8.1 shows how the minimum ~ and the opt imal

2.0 r

1.5

A

/ k’

1.0

& ,/ ’”/ ‘.

‘y.

.,1” I0.5

,/ ’/

,/ ”o~o 0.5 1.0 1.5 2.0

‘m (%)+

FIG. 8.1 .—Best velocit l --e r ror cons tant and mean-square er ror of servo with propor -

t ional con trol as a funct ion of motor t ime constant . Curve A, y = (K ./ 2) (,V / ~Z)M;

Curve B, ~ / (~Nz)’A .

Kv vary with T*. One sees that for la rge Tn, ~ is pr opor tion al t o T~.

On the other hand, if T~ is small, ~ is largely independent of T-; t he

best value of K“ is t hen appr oxima tely 2 [(~’/N)~].

If—-

()2 M

K.=2R ,t hen y = 1 and, by 13q. (15),

;2

()

-3 -

(~N2)$5

_:+; f$>5T~.

I ~ z 1 cor respon ds t o t he r ea l r an ge Of T.,.

(19)

(20)



hc. 83] BEST SERVO WITH PROPORTIONAL CONTROL 345

Consequen tly, if

()

N%Tm<0,4= ,

~2/33 (21)

then ~z is less than 11 per cen t above its absolu te minimum value, which

is approached as 1“~ ~ O. If we now separa te the er ror in fol!owing the

signal from the er ror due to noise, we see that

As T~ -O, the mean-square er ror in following the signal becomes half of

the mean-square er ror due to noise. For Tm <0.4 (N/~2@s)~~, th e com-

ponent mean-square er ror s are rela ted by

0.503 < ~ <0.663. . (23)

These computat ions and the approximation [Eq. (13)] on which they

ar e based are valid only if for Kz n ea r it s opt imal va lu e, @/ K. is small com-

pared with unity. But ~ is normally very la rge compared with N,

and thus K. = 2(~/N)~5 is large. In the example below [Eq. (33fJ )]

K, > 30 see–’. Hence, since ~ is usually small compared with 1 see–’,

B/KV is likely to be very small.

8s3. Proper t ies of the Best Servo with Propor t iona l Cont rol.-Let us

now examine the proper t ies of a servo with propor t iona l cen t rol, when

t he best choice is made of the parameter K,,

From Eq. (6) one sees that

K.

‘0 = (T~p + l)p 6’

where ~ = 8J + 8,V— 80 is the er ror signal (not

lowing the in tended input). Hence

00 = K,—e (T~ju + l)ju “

Thus we have the limit ing rela t ions

e. Ku 1— = —, u <<—Jc u T .

00 K. 1— .—e T~u2’

u >>—.T.

the er ror O,

(24)

00 in fol-

(25)

(26a)

(26b)

If T~ is small enough, tha t is, if T~ <0.4 [N/(~2P3)]~~, then , by F ig.

S.1, K, can be chosen equal t o 2(~’/N)~ independent ly of T~. For



346 .4PPL ICA T IO,VS f)F T IIE iVE 1[’ Ill?,T IG.V ,WETJ IOD [S m . 83

small values of w the loop gain Ivill t hen he equal to

(27)

Thus t wo decibel–log-fr equency dizgrarns, for differen t small va lues of

T~, will h ave t he same a sympt ot e at IOJVfr equ en cies (F ig. 8.2).

If th r fcxxfback cu toff fr equ ency (th e frcqu rn cy f. a t ~}-hich t he loop

gain is unity) is much greater than 1 ‘(27r7’n ,), J VChave

K. = l’mu:,

.f =; O=+($)”] (~>> +)

(28a)

(28b)

It follolvs fr om Eq. (6), wit ,h n eglect of (?~r ela t ive t o 8,, tha t t he over -

+4 0

+30

* ’20.s

‘;+1O

gCJ

o

– 10

-20

(34

FIG. 82.-Decibel-log-frequency dia gramfor ser vo wit h pr opor tion al con tr ol.

:dl simplificat ions of the servo is

The peak over-a ll amplifica t ion is

a ppr oxim at ely t he amplificat ion at

the cutoff frequmcy if KwTm is su f-

ficien t ly l:~rge; for in st ance, if KOTm

> 2.5, then J 5 per cen t er ror is

m ade by t he a ppr oxima tion

If the approximat ions of the three

pr ecedin g par agr aph s a re all va lid,

t he cu toff fr equ en cy is, by Eqs. (19),

(21), an d (28),

Simila rly, t he pea k amplifica tion is

(31)

(32)

It should be noted that a lthough ~/N@3 is usually a very la rge number ,

its twelfth root is not large.

In the case of the radar automat ic-t racking example in Sees, 6.10 and6.12 we have ;2 = 2620 milsZ/sec2, N = 0.636 milz sec., ~ = 0.1 See–l.



SEC. 8.4] SERVO WITH PROPORTIONAL CONTROL, T in = O 347

.If

()>g

T. <0.4 –X = 0.316 seca2f13

and

()2 $’J

Kv=2N = 32.1 SeC–l,

then , by Eq. (20),

~ < 0.83(~2N2)~ = 8.46 mil 2,

~ <2.91 mil. }

The cu toff frequency is

()

>~

j. >0.356 q = 1.60 Cps,

and the peak amplifica t ion is

(33a)

(33L5)

(33C)

(33d)

(33e)

It maybe poin ted ou t tha t in pr act ice, T~ is fr equ ent ly ch osen sma ller

than is required by Eq. (21). This gives a smaller er ror and a smaller

peak amplifica t ion . If in the previous example we choose

Tm ==0.1 see, (34)

we find tha t

V’= = 2.75 roils,

f. = 2.85 Cps,

IPI9 1

(35)

e, ~ax= 1,79.

8.4. Servo with Propor t iona l Cont rol, T~ = O.—We have thus farbeen using the approximate formula of Eq. (13) for ~. If T-is negligible,

however , a bet t er approximat ion to Eq. (12) is

(36)

Assuming !!’~ to be negligible, ~re shall now study the effect of noise on

the pw-formance of the servomechan ism. The rela t ive magnitude of

the noise can be expressed in terms of the noise-to-signa l r a t io N/~.This ra t io (in sec3) is usually a ver y small number, and its dimensionless

pr oduct by ~3 is a st ill smaller number . H ence it is convenien t t o in t ro-

du ce in st ea d of t he n oise-t o-sign al r at io, t he n umber [~f. Eq. (32)]

(37)



348 APPLICATIONS OF THE NEW DES IGN METHOD

Equ at ion (36) t hen becomes

~ 1

- = 2~(2-g + T’)(a’N’)~+ ;“

Now if K, and hence y are chosen so as t o minimize ~, then

p2~—

4Y + T4

d y (~N2)% = – y’(2y + r ’)’+1=0.

Ther efor e, for opt imal per formance

Y’(Y + +-4)2

Solving for y and subst itut ing back

F1~, 8.3.—Best K. and ~ of servo withproportional control and small motor t ime

constant, plot ted against noise-to-signal

parameter r , Curve A, (K./ 2) (N / ay)%;

Curve B , ~ / [(a~N~ ) %].

=Y +*4.

[SEC. 8.5

(38)

(39)

(40)

in Eqs. (14) and (38), one obta ins

the best K, and its cor responding

~for each value of r .

If r is small, we can solve Eq.

(40) as a power ser ies in r ,

y=l–~~4+~@

— *?-16 +.... (41)

Subst itu t ing back in 13q. (38), we

get

~ = ~(~Nl)~(l — &-4 + +@m— &~Is+ “ . ). (42)

The opt imal (K,/2) (N/~) ~ and

T—c~/[(a 2N2)~] a re plot ted a ga inst r

in Fig. 8“3. It is seen tha t for

small va lues of r , C3 is propor t iona l to (~Nz) ~ and the opt imal K. is pro-

por t ional to (~2/N)~. e,

TACHOMETER FEEDBACK t eo

CONTROL Tachometer

8.6. Mean-square Er ror of ~, , t

Ou t p u t .—By adding a t achom-

eter -feedback loop (see Fig. 8.4)

to the servo with propor t iona l con-

t rol, one can reduce st ill fur ther

the dependence of the mean- F IG. S4.-Ser vo wit h t achomet er -feedba ck

square er ror on the motor t imeloc>p.

constant T~. One thereby makes possible an increase in Km and a con-sequ en t decr ea se in t he er r or for a con st an t-velocit y in pu t.

The equat ion of the servo with tachometer-feedback loop is

(T .p + l)pL90 = K.(o, + 0. – %) – A T ~$ ~ pf?o. (43)a



$EC. 8.6] IDEAL CASE OF INFIN ITE GAIN 349

There a re now three servo parameters—the velocity +wror constan t Km,

the tachometer -loop gain A, and the tachometer -filt er t ime constan t

Ta—in addit ion t o t he limited par am eter Tn.

Solving Eq. (43) for 8. gives

eo=ol–(Tap + l)(T~p + 1) + Al”.p

K.(Tap + 1) + (Tap + l)(~np + l)P + AZ’.p2 ‘o’

Kti(Tap + 1)

+ IG(T .P + 1) + (T .P + l)(T .p + l)P + A1’.p’ ‘N”(44)

Sin ce t he spect ra l den sit y of pd, is (41@)/(u2 + ~’) [see Eq. (l)], the mean-

square er ror in following the input is

wh ich on in tegr at ion becomes

;2

? = ~ [(KT. + 1)(T . + T. + AT.) – K,TCTJ 1

x [K. + (KvTa + 1)13 + (T . + T in + ATJ i32 + T . Z’.B3]-1x {(KuTa + 1)(TC + T . + AT .) – KvT .Tm + (T . + T . + AT .)*B

+ (T . + T . + ATJTaZ’m@2 + KJ (Ta + T . + AT .)’ – 2T .3TJ

x [(TC + T . + ATJ 13 + TaTm@2]

+ K. TgT’’[K.d + (K.T . + 1)/32]].46)

The spect ra l density of the noise is G~(f) = N [see Eq. (2)]; the mean-

square er ror due t o noise is th erefor e

?_

!-

‘“’–z 0 du

K;(I + T~ti’)

IK. + (K.Ta + l)ju – (T . + Trn + ATJ u2 – TaT~ju312 “ ’47)

wh ich , wh en in tegr at ed a nd simplified, becomes

The mean-square er ror of the output is, of course,

z=~+q. (49)

8.6. Ideal Case of Int in ite Gain.—We shall see that if the tachometer -

lu op ga in A is la rge enough , a servo with tachometer feedback can be



350 APPLICATIONS OF THE NEW DESIGN METHOD [SEC.86

made to have a smaller mean-square er ror than a propor t ional-con t rol

ser vo even if t he former h as a ’la rge T~ and the la t ter a small Tn.

With this end in view, let us take the limit ing case where K. = CA

and A -+ m. In this limit , Eqs. (46) and (48) become

and

a nd h en ce

(50)

(51)

(52)

It will be noted that in this limit ing case ~ is independent of T~.

We now choose the parameters C and T . so’ a s t o minimize ~. Then

aP -2

8T. = (C + GTt@B+ T,#2)2 – 4!: = 0“—. (54)

Equat ions (53) and (54) may be solved for C and T . as follows. Elimi-

nat ion of ~z between Eqs. (53) and (54) yields

C -I- CTafl + Tab’ =(C2 + (3’)T=

2’

Subst itu t ion of this expression in to F.q. (54) now gives

or

where r is the parameter defined in Rq, (37). Solut ion for C yields

—.c = D 44r-’ – 1.

Equat ion (55) can now be solved for T .,

The above procedure is valid only if it I(wIs to

(55)

(56)

(57)

(58)

(59)

(60)

r ea l posit ive fin it e



SEC. 86] IDEAL CASE OF INFI.VITE GAIN 351

values of the parameters. In par t icular , ~is a minimum for finite Ta

on ly if th e den omin ator in Eq. (60) is posit ive, th at is, only if

r < (2 – <z)~f = 0.915. (61 )

If r sa tisfies t his condit ion , bot h C and Z’. \ villbe r ea l a nd posit ive.

Now, having found the opt imal values of C and T. in terms of the

parameter r , we can subst itu te back in Eq. (52) to get the minimum value

of ~. Combining Eqs. (52) and (54), we obtain

()=:& (c+cTaP+T.D’)+x14 E+c

—~& (2C + 2CTad + T.@’ – CT.D + C2TJ , (62)a

~~h ich , t oget her wit h Eq. (55), gives

-T=%7 & (C’Ta – CTa~ + C’T .J = : (2C – /9). (63)

.

If we now express C in terms of T, u sing Eq. (58), we get

From the definit ion of r it follows that

(64)

(65)

Our final expression for ~ is then

As we have already remarked, ~ and hence ~ are independent of l’~

when t her e is high-gain t ach omet er feedback in addit ion t o propor t iona l

con t rol. In the case of a servo with propor t iona l con t rol only, the

minimum mean-square er ror is dependent on Tm; its smallest va lue,

when Tm ~ O, is given by Eq. (42). The ra t io of the minimum mean-

square er ror of a servo with tachometer loop and arbit rary Tm (but

infinite A and Ku) to the minimum mean-square er ror of a servo with

pr opor t ional con t rol a nd negligible T- is

Tcmwith tachometer 4 <~ - +,3—

~r l–~r4+Z~r8–&&+ . . . +. (67)

~ without tachometer

This ra t io is shown plot t ed against r in Fig. 8.5. It is seen that for all

reasonable values of r the tachometer -feedback servo performs bet t er

t han t he pr opor tion al-con tr ol ser vo even if in t he la tt er ca se T~ = O,



352 APPLICATIONS OF THE NEW DES rGN METHOD [SEC.8.7

1,0

0.8

0.6

0.4

0.2

00 0.2 0.4 0.6 0.8 1.0

r

FIG. 8 5.—Ratio of nlin imum mean-square er rors of servos with tachometer looPt o minimum mean-squar e er ror of ser vo without tachometer looP plot ted against

r = (N /3z/~z)XZ.

8.7. Best Cont rol Parameters for F inite Amplifica t ions. -In actual

pract ice, A and Kv cannot be made arbit rar ily la rge; the preceding t rea t -

ment is therefor e an analysis of a somewhat idealized situa t ion. We

shall now modify this analysis and seek an approximat ion to the opt imal

va lues of the parameters KU and Ta for the case of a fin ite A.

We first make some approximat ions to Eqs. (46) and (48). We

assume that Km is large compared with each of the following: I/T=,

l/ T in , TJ T ;, 8, P2Tm , 6A. The last assumpt ion is just ified because (1)

if A is la rge and ~ is less than ~, then by Eq. (58), K./ ~A = 2/ 7’, which is

la r ge i“ela tive t o unit y, wher ea s (2) if A is small it is obvious tha t K, >>13A.

Using t hese a ppr oximat ions, Eqs. (46) and (48) becom e

)We now make the fur ther assumpt ion that L?is so small tha t L?T~-4<1

and flT * << 1. (In the examples below’, these rela t ions and approxima-

t ions a re valid; if o is not small, the formulas below must be used with

cau t ion. ) The formula for the mean-square er ror then becomes



SEC. 87] FIN ITE AA4PLIFICA TIONS 353

Given A and l’~, we n ow ch oose K. and T. so as to minimize ~. Then

N+ 4(1 + A) = 0’ ‘70a)

a? a&4(2 + z’i) _ NA

r3T a = K; 4(1 + A)T~ = 0“(70b)

We solve Eqs. (70) for K. and T= as follows. Solu t ion of Eq. (70~) for

K: gives

K: = ~&P4(l + A)(2+A)T~. (71)

Eliminat ion of;’ by means of Eq. (37) yields KVin terms of T.,

K = 2 <(1 +A)(2+A)&Ta“ ~6 (72)

Also, subst itut ion of Eq. (706) in to Eq. (70a) gives

1 + 3BAT . + iIA(2 + A)Ta + ~~ = :Ko6(2 + A) T:. (73)

If we no~v make use of Eq. (72), we obta in

On subst it ut ion for T~ of t h e dimen sion less pa ramet er z, defin ed by

~z = 1 fi2T:A<2+A<1+A ~6’

(75)

th is becomes

1 + 3@ATm + @A(2 + A)T .(1 + ~’) = ‘2 + ‘)$* ~P3T~. (76)

In order t o simplify this equa t ion we in t roduce another parameter ,

[ 1(1 + A)(2 + A) ‘i @TO

x=A, T’(1 + zz))~;

(77)

eliminating To, we obtain

[ 1

AC(2 + A)’ ‘ir3(l + z’)$$z1 + 3@ATm +

l+A

[ 1

A’(2 + A)’ ‘4~3(1+ z’)%’,.l+A

(78a)



354 APPLICATIONS OF THE NEW DES IGN METHOD [SEr.87

which may also be writ ten as

[

l+A

1

)4 1(78b)’ – Z = (1 + 3!3A!fm) A,(Z + A), 73(1+ #)?,’

This equa t ion can now be solved for z. Finally, solving

(77) for K, and Z’. in terms of z, we have

~ = 2D[A2(1 + A)(2 + A)]}~(l + 22)EX“

r ’,

[

A,

1‘6?J (l + 22)%

‘“ = (1 +.4)(2 +A) B“

llqs. (72) and

(79)

(80)

Equa t ions (78 b), (79), and (80) give us the opt imal va lues of Km and z’.

for ea ch set of values of A, Tmj a nd t he in pu t pa ramet er s,

We now may subst itu te in to Eq. (69) to find the minimum value of~. First , use of Eq. (70b) gives

Use of Eq. (73) then gives

But , by Eq. (72),

K.T~ T: 2(1 + A)~4(2 + A)~*02

2A(1+A)(2+A)T: = 2A(1+A)(2+A) ~6 , (82)

and hence, by Eq. (75),

K. T;

2A(1 + A)(2 + .4)T. = ‘z” (83)

Therefore

~=[

N3m

1K. ++(1+ z’).

4(1+A) 2 .

Subst it u t ing for K. and Z’a fr om Eqs. (79) and (80), we ha ve

Fina lly, by Eq. (65),



SEC. 8.7] FIN ITE AMPL IFICAT IONS 355

In Fig. 8.6, ~ is plot t ed against A/(10 + A) for T~ = z = O a nd for

th ree different va lues of r : r = 0.215, 0.368, and 0.585, or r ’ = 0.01,

0.05, and 0.20. It can be seen that for th is range of values of r , if

A>:, (86)

then the mean-square er ror is less than 5 per cen t above its limit ing value

when A * m .4 .0 4 .0

I

3.5 3.5

3.0

\ !

3.0

I

\ k - 22.5

\

2.5<

\“! & ), ,

1S-

Ifi :2.0a

~ w1$2’0r~=o.ol

1.5 1.50.4

\ \\

1.0— —~ : & -

0.5 0.5

0 0025102050- 025102050 -

A A

I I 1 I I 1 I 1 1 I ! I0 0.2 0.4 0.6 0.8 1.0 0 0.2 0,4 0.6 9.0 1.0

10+A*

FIG, 8.6. FIG. 87.

FIG. S.6.—Minimum mean-square er ror of servo with tacbomcter -feedback looP as afu nct ion of t ach om et er -loop ga iu A, when T~ = O.

FIG. S7.-Minimum mean-square er ror of servo with tachometer -feedback loop

as a funct ion of t achometer -loop gain A, when r3 = 0.05. Numbers on curves a re values

of Tna (~ (32/N) 8.

Figure 87 shows how the minimum C2depends on Tfi. The curves

a re plot t ed for four differen t va lues of Z’- when rs = 0.05, tha t is, when

r = 0,M8. The dependence on 1“~ becomes impor tan t on ly for va luesof A less than 2.

Example.—We r etu rn to the example of Sec. 8.3 and assume tha t

;2 = zGzo mil~ SCC–2,

N = 0.636 m ilz sec.

p = 0.1 I (87)see–l,

Tm = 0.316 sec.



356 APPLICATIONS OF THE NEW DES IGN METHOD

Then

1

= 0.281,

1

F = 12”7”

Taking A > l/r ’ [see Eq. (86)], we choose

A = 14.

Then we find that

Z2 = 0.009,

r = 1.187,

K* = 158 see-l,

Ta = 0.252 See,~~ = 3.11 milsz,

fin = 1.76 roils.

[SEC. 8.8

(88)

(89)

(90)

Thus the rms er ror for the t achometer -feedback-loop servo is 35 per cent

less than the rms er ror for the propor t iona l-cont rol servo. It will be

seen from Fig. 8.7 that the choice of T~ = 0.316 sec has increased themean-square er ror on ly negligibly above the minimum given by T - = O.

8.8. Decibel-log-frequency Diagram.—We now examine the form

taken by t he decibel–log-fr equency cu rve for th e ser vo with tach om et er -

feedba rk loop, wh en t he best ch oice is m ade of t he pa ramet er s.

From Eq. (43) we see that

O* =K“

(91)

(Tmp+l)p+AT;~ip6”

oThk may also be wr it ten as

A T aju

00 _ K.(T~ju + 1) (T~ju + l)(!l’.jw + 1) .— —E – .4 ‘rOwz A Tajw

(92)

1 + ~T.ju + l)(T~_)

Now

AT&(T”ju + l)(Z’@ + 1) = ‘T ti””

if o is small,

A

1

@3)

= T.j~’if u is la rge.

Ther efor e, if A is large, A T=jW/ [(TJ ti + 1) (Taji + 1)] is the dominant

t erm in the denomina tor of Eq. (92) provided that l/AT. < ~ < A/ T~.

Thus, if A is large,



SEC. 88] DECIBEL-LO(-FREQUENCY DIAGRAM

~=–j+,

1

e ‘f “ < m.’

Km 1=——

A Tau2’ ‘f AT .<u<;,

jKv 1 A“=——,Au

if —<u <—,T . T%

Kvs——

T~u2’if.>;.

m

The cent ra l segmer ts of the segmen ted approximate

357

(94a)

(94b)

(94C)

(94d)

decibel–log-fre-

quency diagram-meet at a poin t where w = 1~-7’=and It70/ c I = (KuT~) /A.

This complet es t he gen era l cha ra ct er iza tion t o t he t ra nsfer locus, it being

assumed ordy that A is la rge. We have now to consider the effect of

t he best ch oice of con st an ts, a ccor din g t o t he t heor y developed a bove.

If A is sufficien t ly la rge, it follows from Eq. (75) tha t z = O a nd from

Eq. (78b) tha t z = 1. By Eqs.

~9) and (80), the best choice of

con st an ts makes

KVTO— = 2(1 + Z’)x’ = 2. (95)A

Thus one should have I&/cl = 2

where the cent ra l segments of the

decibel–log-fr equency d iagr am in -

ter sect ; in other words, t he ve-

locit y-er ror con st an t Ku sh ou ld be

so chosen tha t the in tersect ion of

the segments of t he decibel-log-

frequency diagram at u = 1 / Ta

will be about 6 db above feedba ck

cutoff. The loop gain 100/cl be-

+60

+50

% +40

~

. +30.—

‘ii

-1 0

-2 0

w

lclG, 88.-Dccibcl-log-f r cque l,cl, dia-gram for ser vo wit h t ach om et er -feedbu ckloop,

comes zero db in the 6-db-per -octave segment , 6 db from the a lmvc-

men tion ed in ter sect ion ; by Eq. (94c) t his occu rs for

(96)

(97)

The theory, as illust ra ted by Fig. 8.7, has shown tha t the exact value of A

does not much mat ter if it is only grea t enough. Change in A \ vill notmat er ia lly ch ange t he cent ra l pa rt of t he decibel–log-fr equen cy dia gram

but on ly the length of the cent ra l srgments; all that is r equired is tha t

these shall be sufficien t ly long. [From the steacfy-sta te poin t of view

it is clea r tha t the hmgth of the ( 1,/T,,, ,4/ 7’,,,)-s(J gmunt cont ru ls the



358 APPLICATIONS OF THE NEW DES IGN METHOD [SEC. 88

stability of the servo.] The best filter t ime constan t 2’. is determined by

Eq. (80) or , if .4 is la rge enough, by

(98)

This t heor et ica lly der ived decibel–log-fr equ en cy dia gram is simila r

to tha t commonly believed desirable, except that the velocity-er ror

const ant is smaller than might ha ve been expect ed; usually K“ is assumed

to be so chosen that O db is a t or below the middle of the 6-db-per-octave

middle segment . In pract ice, however , t he amplifier gain is frequent ly

tu rned down considerably below that assumed in the steady-sta te design

of the servo. Thus we can claim substant ia l agreemen t with the usual

design pr act ice, if n ot wit h its t heory.

The approximate peak amplificat ion is found as follows: If A is

su fficien tly la rge a nd if1

AT.<u<;) (99)

.then by Eq. (92)

_ ~ ICJT& + 1).

e – A Tau2(loo)

ande.

This approximation is rough unless A is very la rge; otherwise it tends to

underest ima te t he ma gnit ude of 00/61. H owever , if we so a pproxim at e

60/’61, t h e squ ar e of t he amplifica tion is

To simplify t he compu ta tion we su bst it ut e t he dim ension less pa ramet er s

~=!y

/(103)

~ = T~@~J

and obtain

00 t 1+X

K= !72~z

– 2qx + q’ + q%’(104)

The peak amplificat ion occur s for the value of z at which 180/0,]2 ismaximum—the va lue of z for which

(105)

4



SEC, 89] N YQUIS T DIAGRAM 359

is a min imum. This va lue of z is determined by

-(Z’ – 2qz

dx )+ z + ‘2 = ‘1=2*IW=29’) = 0 ‘106”)

orx2+2x —2y=o; (106b)

solvin g t his, we obt ain

Z=–l+dmq (107a)

or

1

(d

K T.

“ = F: )1+2*–1. (107b)

Subst itut ing Eq. (107a) back in to Eq, (104), we find that the peak

amplifica t ion is

d!& . __ q’ /1 + Zq

01-. 2+4q+(q’ –2q–2)<l+2q”(108)

In Fig. 8.9 the peak amplifica t ion is show+nplot ted against q. This graph

is to be used with caut ion, since, as ment ioned above, for reasonable

values of A the peak amplifica t ion is underest imated. If A is very large,

q = 2 [see Eq. (95)]; the peak2.0

amplifica tion is t hen

60

z ma .

= 1.27. (109) ,

8.9. Nyquist Diagr am.—Theg. 1.5

form of the Nyquist diagram of

t he ser vo wit h t ach omet er -feed-

back loop can be inferred from the 1,00

form of the decibel–log-frequency1 2 3

diagram. However, since the ~=y

Nyquist diagram is much used in Fm. 89,-Pernk amplifica t ionof servowith tachomet er -feedba ckoop whenta chom.

servo design , it may be wor th eter -loopgain is lar ge.

while to give an example of a

Nyquist diagram of a servo designed in accordance with the rms-er ror

criterion.

Let us assume the same input signal as in the example of Sec. 8.3 or

6.12 and assume that the noise ar ises only from a potent iometer wound

with ~ turn per mil (see Sec. 6.13). The spect ra l density of the noise isthen fa ir ly flat ou t to 3 cps; we assume tha t it can be fa ir ly represented

by GN(.f) = 0.05 roil’ see [see Eq. (6182)]. We choose then

~ = ‘fj’” mils2/sec2,

N = 0.05 milz see ,

)

(110)

~ = 0.1 see–l,



360 APPLICATIONS OF THE NEW DESIGN METHOD [SEC. 8 .10

and h en ce

r = 0.227. (111)

If A were zero, we should wish to take 7’~ < 0.4r_’ [Eq. (21)]—in thh

case T~ <0.207. Since A is not t o be t aken as zero, we ch ooseT~ = 0.3 sec. (112)

Now

1– = 19.4,~2 (113)

and we wish to choose A enough la rger than th is to a llow for the effect

-2 -1 0 of the la rger T*. We t herefor eo take

A = 25. (114)

/Then

y- -J 22 = 0.009,

z = 1.162,

yq = 2.72,

Km = 511 see–l, (115)/ _2j T . = 0.133 see,

~,~. &10.-Nyqu is t d iagramof servo d~ = ().68 roil,with tachometer -feedbackoop. Thecircleis drawnfor [Oo/6’~1 1.28. j. = 3.13 Cps.

The Nyquist diagram for th is servo is shown in Fig. 8.10. The circle

in Fig. 8.10 is the plot of 160/0~]= 1.28. The peak amplifica t ion is near ly

1.28, which is la rger than t he value 1.22 tha t can be read from t he approxi-

mate cu rve of Fig. 8.9. It is notewor thy tha t doubling the velocity-

er ror constan t and thus magnifying th is Nyquist plot in the ra t io 2/1

will cu t down t he peak amplifica t ion and thus improve stability.

Thus we see again (cf. Sec. 88) that the rme-er ror cr it er ion calls for a

lower gain than a naive, examinat ion of the Nyquist diagram would indi-

ca te. In actual pract ice a servo to be used in the presence of poten t iom-

eter noise is designed with the help of a Nyquist diagram, and the gain is

t h er ea ft er a rbit r ar ily r educed . Our th eor y is thus in accord with exist ing

practice.

MANUAL TRACKING

8.10. In t roduct ion .-In manual t racking the human opera tor can beconsidered as par t of a servo loop, in which he per forms the funct ions of a

power element and a servomotoi (Fig. 8.11). The opera tor observes

the misalignment between the telescope and the ta rget and turns a hand-

wheel in the direct ion tha t tends to reduce this misalignment , The

h an dwh eel dr ives t he t ra ckin g u nit , wh ich , in t ur n, posit ion s t he t elescope,

t hu s closin g t he feedba ck loop.



SEC. 8.11] THE AIDED-TRACKING UNIT 361I

In the simplest type of manua l-t racking system the human opera t or

is the only power source in the closed loop. In more complex systems,

where torque amplifica t ion requ ires the use of a dr iving motor , the

human opera tor is a secondary servomotor . Limita t ions on the speedand torque available apply to both the power motor and the human

opera tor ; in addit ion , t he per forman ce of t he human oper at or is gover ned

by condit ions of fa t igue and menta l and physica l comfor t . It is charac-

t er ist ic of the human opera tor tha t ther e is a t ime lag between the instan t

when an er ror is observed and the instan t when cor r ect ive act ion on the

par t of “he oper at or M st ar ted,

ScweTarget

mot!on

@“

. E Operator .@ Tr~nJng

@f Handwheeldisplacement

I

%00

Fm. S.11.—Manual-tracking loop.

As in any servo system, in order to obta in good per formance the loop

must have high gain and stability. The equa liza t ion is achieved by the

judicious choice of the available parameter s: the handwheel ra t io, the

opt ica l magnifica t ion , and the t racking t ime constan ts. The present

chapter includes a theoret ica l discussion of a t racking system based on

t he assumpt ion t hat t he human opera tor behaves like a linear mechanism.

The rms-er ror cr it er ion is applied to obta in the best t racking t ime con-

stant . The theoret ica l resu lt s compare favorably with those found by

exper imen t . The ent ire discussion is limited to an a ided-t racking

system with a handwheel input .

8.11. The Aided-t racking Unit . -Aided t racking is a combinat ion of

dkplacement and ra te t racking. In pure displacement t racking the

opera tor has a direct connect ion , either mechanically or elect r ica lly,

with the con t rolled member . In t racking a ta rget moving a t constan t

angula r ra te, the opera tor must tu rn his handwheel a t a constan t ra t e.

If he is lagging the ta rget , he will tu rn faster unt il the er ror is cor r ected;

if h e is lea din g t he t ar get , h e will t ur n m or e slowly.In pure ra t e t racking it is the speed of the output tha t is determined by

the posit ion of the input handwheel; in t racking a ta rget moving a t a

constan t r a t e the handwheel need not be tu rned aft er the proper adjust -

ment has been made.

When these two types of t racking are combined, an er ror in ra t e and

the resu lt ing dkplacement er ror a re cor rected simultaneously; a change



362 APPLICA TIO.VS OF THE NEW D17SIGAT METHOD [SEC. 811

in the handwheel posit ion changes the ra te of mot ion of the output a t

the same t ime that the displacement er ror is cor rected. This is a ided

tracking.

A basic design for the aided-t racking unit is shown in F ig. 8.12. The

outpu t of the differen t ia l 90 is a linear combinat ion of the handwheel

dkplacement o and the displacement of the var iable-speed-dr ive outpu t

0;00 = K14 + KZ’J , (116)

where K1 and Kt a re gear ra t ios. The speed of the var iable-speed dr ive

is pr opor tion al t o t he h an dwh eel displa cemen t;

() = K, Q’. (117)

The two foregoing equat ions combine to give the equat ion of the aided-

t ra ck in g unit ;

(118)

(119)

is the a ided-t racking t ime constan t . The aided-t racking t ime consfan t

h as t he followin g ph ysica l in ter pr et at ion : If a given ch an ge in t he posit ionof th e han dwheel result s in chan ges in posit ion and velocit y of t he ou tput

by Ad. an d AO~,r espect ively, t hen

(120)

In the a ided-t racking unit shown in Fig. 8.12, the opera tor ’s hand-

u

FIG. S. 12.—Basic mechanica l design of an

a ided -t r ack ing un it .

wheel is connected through gear-

ing direct ly to the load, as well as

to the movable member of thevar iable-speed dr ive. Such an ar -

rangement is sa t isfactory only

when the speed and torque re-

quirements a t the load can be met

without the applica t ion of high

torques at the handwheel and it is

mechanica lly convenien t to gear

dir ect ly fr om t he h an dwh eel in pu t

to the load. F requent ly, how-

ever , t he oper at or ’s posit ion is r emot e fr om t he loa d t o be con tr olled, or t he

outpu t torque required is high; in either case, some form of elect r ica l

power dr ive must be used. A t racking unit suitable for this purpose is

shown in F ig. 8013.



SEC. 8.12] BELS T AIDED-TRACKING TI,!fE CONS TANT 363

There are severa l common types of manua l-t r acking con t rols, (1)

a handwheel, (2) a handle bar or some modifica t ion of a handle bar , (3)

a joy st ick, The choice of input con t rol depends la rgely on the type of

t racking to be per formed, tha t is, whether it is t r acking in one or two

coordka tes and whether the t racking is done by one or more opera tor s.

The ch oice depen ds a lso on wh eth er t he oper at or is in mot ion or is sta t ion-

ary, on whether he can use one or both hands for t racking, and on the

space ava ilable for the opera tor and the t racking cont rols

jPjentiornetj ~

SpeedWferential

*control /

Load

Position——*control

Synchro

FIG. 8.1:3 .—llcmote-?ontrol aided-tracking unit.

If th e oper ator is sea t ed at a con sole and t r acks in only on e coor din ate,

the hand~vheel type of input seems to be prefer r ed. The rela t ionship

between handwheel speed and ou tpu t should be such that the opera tor is

not required to turn much more slowly than 10 rpm nor much faster than

200 rpm. The handwheel shou ld have sufficien t iner t ia and be large

enough to permit smoothness in turn ing but shoukf be as fr ee of fr ict iona l

drag as possible to preven t t ir ing of the opera tor by long per iods of

tracking,

The t ime required for an er ror in t racking to become percept ible, if

the t racking is done th rough a telescope, can be reduced by increasing

the magnifying power of the telescope. The opt ica l magnificat ion tha t

can be used is limited, however , by the size of field. If the field of viewis t oo small, it is very difficu lt t o get on ta rget ; in addit ion , the apparen t

velocity of the t a rget in the field of view is so grea t as to make t racking

arduous.

8.12. Applica t ion of the Rms-er ror Crit er ion in Determin ing the Best

Aided-t racking Time Constant .—It will be the purpose of the present

sect ion l to give a quant ita t ive t rea tmen t of the a ided-t racking system

with handwheel con t rol. We shall determine the values of the a ided-

t racking t ime constan t for which the system is stable, as well as the bestvalue in the rms sense.

For the purpose of th is invest iga t ion it is assumed tha t the human

oper at or beh aves like a lin ea r m ech an ism . Mor e pr ecisely, it is t hou gh t

reasonable (on t he basis of admit tedly cr ude exper im en ts) t o assume tha t

1 Section 812 is a revision of a paper by R. S, Phillips ent it led “Aided Tracking,

Par t II, ” which was published as RL Repor t No. 453, Nov, 3, 1943.



364 APPLICATIONS OF THE NEW DES IGN METHOD [SEC. 8.12

at a ll t imes the opera tor turns the handwheel a t a ra te propor t iona l to the

magnitude of the t racking er ror . It is well known that there is a t ime lag

between the st imulus and the opera tor ’s react ion . In fact , it seems to

be reasonable to assume that a t any t ime t he turns the handwheel a t a

2 .0

1 .8

1.6

1.4

1.2

~Eg 1.0

~

50.8

0.6

0.4

0.2

0

//

“-’Y(%“’y”Itl~T!me delay eLp

f

1’/

1 2 3

t/ L —

Fm. 8. 14.—Delayed response to a unit-step

function tiinput.

ra te propor t ional to the t rackinger ror which exist ed at the t ime

(t – L), where L is the magni-

tude of the t ime lag. The ra te of

tu rn ing can then be expressed as

~’(t ) = v,e(t – L). .4 reasonable

va lue for the t ime delay L wa s

found to be 0.5 sec.

An exper ien ced oper at or m igh t

a lso ant icipa te t he er ror by ta king

in to account the ra te of change of

er ror . This would add a der iva -

t ive term to the above equat ion ,

wh ich t hen becomes

d’(t ) = v,e(t – L)

+ ,,,’(t – L), (121)

wher e vI is t he pr opor tion al-con -t rol constan t in (see)–l and VZis

the der iva t ive-con t rol c o n s t a n t

(dim en sion less). Th e equ iva len t

expression for 1#1’(t )n the Laplace-

t ra nsform term in ology is

(122)

For mathemat ica l convenience, the term e’p has been replaced by

the expression [(Lp/3) + 1]3. An idea of the goodness of this approxi-

mat ion can be gained from Fig. 814, wh er e there is plot t ed the response of

an opera tor with VZ= O to a unit -st ep funct ion e-input , for the two

condit ions of t ime lag eL2 and [(.LP /3) + 113. The full curve gives the

response for a react ion lag of the form [(Lp/3 + II a; t his is

[ 01

2

y (t) = ; – 1 + e-a~/L 1+2;+; ; . (123)

The dashed cu rve illust ra tes the type of response to be expected on the

assumpt ion that the opera tor does noth ing until a t ime L a ft er h is st imu-

lus and then turns a t a ra te propor t ional to the er ror .

I See Chap. 2 of t h is volume.



SEC. 8.12] BEST

The equat ion of

as

The approximat ion

AIDED-TRACKING TIME CONSTANT 365

the a ided-t racking unit [Eq. (118)] can be r ewr it t en

()%=K, p+~~. (124)

a

to Eq. (122) tha t we shall use is

()p~ + 1 3 pl#l = (., + V,p)c. (125)

Combining Eqs. (124) and (125) then gives

R+1)3P””=“P2+(V+4P+21’ (126)

Here we have replaced K1 VIby v, and K,vz by w This does not rest r ict

the genera lity in any way, since \ ve can only obtain condit ions on the

ra t io v1/vz by a linear type of theory. Fina lly, making use of the rela -

t ion 00 = OJ– c, we obta in an equat ion relfit ing the er ror and the input ,

[f++ Y“+2P2+P+ap++l’=(?+’)’p” ‘127)

The above chain of physica l considcmtions has thus brought us to

t h e a ided-t r ack ing equa tion . Th er e rem ains t he pr oblem of det erminin g

the best va lue of the t racking constant , i“. and the opera tor parameters

(PI,Y2). In or der to accomplish this, the rms (r ror ]vill first be computed

and then minimized; th is proccdurc follm~s the method developed in

Chap. 7.

The input O,t o be assumed is onc of const :mt velocity over a sequence

of intervals, with abrupt changes in velocity a t the cnd of each in terva l.

These changes a re all ind~pcndeut , because in a ided tracking it is the

change in velocity, and not the angukw posit ions, tha t is significant .

Let x(t ) be a unit -ramp funct ion,

z(t ) = 0, (t ~ 0),

I (128)r (t ) = (, (t > 0),

and let the a. be mdcpendcnt r :m(lom var i:ll)lcs Ir ith zer o means. Then

O, is defin ed a s

e, =z

a,,,r(l — f,,), (129)

r,

where the t . a r e dist r ibuted in some random fashion with density C. A

sample of Ofis sh own in F ig. 8.15. This t ype of input was chosen beca use

it represen t s, a t least roughly, the typica l input in an aided-t racking



366 APPLICATIONS OF THE NEW DESIQN METHOD [SEC. 8.12

system, in which the opera tor cor rect s mainly for changes in the angular

velocit y of t he t ar get .

Alth ough Oris n ot a st at iona ry r an dom pr ocess, its secon d, der iva tive

is a sequence of independent delta funct ions of the type descr ibed in

I‘%

I

I

t. tn+l tn+ztn+3Time—

Fm 8.16.—Graph of L%.

Sec. 6.11. The spect ra l density of

the second der iva t ive has the value

N for all frequencies, wh er e

N = ~2u. (130)

Sin ce t he t ra nsfer fu nct ion con ta in s

a factor of pz, no difficu lt y r esu lt s

from working with the improperspect r a l densit y

NIW) = ~“ (131)

Th e mean-squ ar e er r or obt ain ed

by u sin g t he a bove in pu t is pr ecisely

the same as the in tegra ted-square er ror for an input W2 x (t ). Hence

the answer also gives a measure of the t ransient response to a ramp-funct ion inpu t.

As in Sec. 7.2, the er ror spect rum is simply,

G,(j) = lY(20rj,f)lW(j9, (132)

wh er e Y(2r jf) is t he t r ansfer fu nct ion ,

The mean-square er ror is

This can be eva lua ted by the table of integra ls in the appendix. One

obtains

L2N

2=4PX

8 + 77 – y’ – 3a + 3P – 3-’f17 + 10 X 3-SW9 – 2 X 3-a~2 + 3-ac@7

&x–8@+2x 3-’a@+7ay+w’3– 3c22-3-’@2-a7t ‘

(135)



SEC. 812] BEST AIDED-TRACKING 1’IME CONSTANT

whereL

a = VIL + V~—-,T .

/ 3 = v,L ;, y = v~.a

The system is stable for the set of all those con t rol

367 I

(13ti)

parameters

(v,, v,, 2’0) which lie in the region of stability. This region is bounded by

the sur face over which the denominator of Eq. (135) vanishes. It is

clea r that the plane L/ Ta = O and the plane vlL = O bound the region of

stability. F igure 8.16 is a three-dimensiona l sketch of th is region .

d

$

A

Vz

FIG. 8.16.—Stabilit y r egion .

va lue 0.0825 NL2, athe mean-square error , ~ at tains its min imum

the poin t L/ T . = 0.55, v,L = 2.25, VZ= 4. This means tha t the best

t r acking can be done with an a ided-t racking constan t T . = 1.8L and

with vZ/vl = L/O.56. For a t ime lag L of 0.5 sec thk gives T . = 0,9

sec and v~i’vl = 0.9 sec; the opera tor should thus use about as much

der iva tive con tr ol as pr opor tion al con tr ol. H owever , in t he exper im en ts~

per formed the opera tor v-as not able to in t roduce very much der iva t ive

con t rol. We therefore set VZequal to zero.

When .2 = O, Eq. (135) becomes

L2N 8 – 3a + 36 + 10 X 3-343 – 2 X 3-3~2,;2=_4B 8a – 8fl + 2 X 3-’@ – 3a2 – 3-3&

(137)

J A. Sohczyk, “Aided Track ing,” RI , Repor t No, 430, Sep t . 17, 1943.



368 APPLICATIONS OF THE NEW DES IGN METHOD [SEC. 812

where

a = VIL and R = ,,L & (138)a

The region of stability becomes the area shown in Fig. 8“17. The mean-square er ror a t tains it s minimum value 1.1NL2 for L~Ta = 0.2 and

I~1

7=

Io 1 2

*,L3

FIG. 8.17.—Stability region for X2 = O.

VIL = 1.7. With L = 0.5 see, thk cor responds to a value of 2.5 sec for

t he a idin g con st an t T.. The rms er ror is then about 3.6 t imes as large

a

7

~6~

*5

34s1+3

32

1

005101520253035404550

7rackingonstsnt in units of L

FIG. S.18.—Aided-tr a&ingerr or for a rau -dom-velocityinput . vz _ O.

as tha t obta ined with opt imal de-

r iva t ive cont rol. “

The full line of Fig. 8.18 is a

dot of the minimum value of ~

a s a fu nct ion of t he a ided-t ra ckin g

constan t for v, = O. This curve

of mean-squ ar e er ror a s a fu nct ion

of t he a ided-t ra ckin g con st an t cor -

responds very closely to what is

obta in e d exper imenta lly. The

circled dot s on this graph are ex-

per iment al a ver age mean -squa re

er r or e for t h e cor r esponding a ided-

t rackine t ime constan ts. These

data were obta ined with a handwheel t rac~ig unit .] The ordinate

sca le for t he exper imenta l points has been so adjust ed tha t t he t wo minima

coincide. The theoret ica l opt imal value for T. with VZ= O is 5L = 2.5

aec; the opt imal value as found exper imenta lly was between 2 and 3 sec.

1Sobczyk op . cd., p . 21.



APPENDIX

TABLE OF INTEGRALS

Th e followin g is a table of integra ls of t he t ype

/l.=%$ ‘dz

9.(Z)

—. h.(z) h.(-x)’

where

hn(z) = a@ + alz’-’ + . ~ ~ +am,

g.(z) = boxz”–z + b,xz”–’ + , .+ L,,

and the root s oj hn(x) all lie in the upper halj plane. Th e table lists the

integra ls In for va lues of ?2 from 1 to 7 inclusive. 1

b~I,=—

2aOal

.b, + a+

1, =2aoal

aoalbt—a~bo + aobl — —

a:Ia =

2aO(aOa j– a la ,)

b,( –a,a, + a,a~) – a,a,b, + a,a,b, + a+ (am – ala,)

1, =2a0(aoa~ + a?a4 – a1aza3)

Ms16=—

2aOA5

Mb = bo( –aoaAas + a,a~ + a~as – ata,ai) + aObl( –a,as + a2aJ

+ a0b2(a0cbb – a1a4) + a0~3(–a0a3 + a1a2)

+ a+ ( –a0a1a5 + aOa~ + a?a4 – a1a2a3)

AS = a~a~ – 2@a la ia h – a0u2a3ab + @a4 + C@!

— a1a2a3a4

L=*2aOA6

ii’f6= bO( – aOu3&a6 + a0a4a~ – u?ai + ‘hla2f%a6 +

— ala~as — a~a~ — aia~aa + aia3a4a6)

+ aob,( ‘alaKa6 + a2a~ + a%a6– a3a4aK)

+ ala~a~

a1a30.4a6

1 This table wae compu ted by G. R. MacLane, foiiowin g th e’ m et hod developed

in sec. 7.9.

369



370 TABLE OF IN TEGRALS

+dz(-aoa~ – @a@,+(3@@5)+d)3(@I@a5+ @36 – @a2@)

+a&4(aw1a5 – aoa i – a!a4+a,a2a3)

+~(@d+@i@@a6 – 2aOa1a4a5 –a0a2a3a5 + aoa~at

M,17 =—,

2aOATwhere M? = bOmO+ aOblm l + a~b~m t + . . + UObGm&

m . = a~a~a~ — 2a0a1a~aT – 2a0a2a4a~ + a0a2a5a6a1 + a0a3a5a~

+a0a ja5a7 – aOa4a~a6 + a~a j + sa1a2a4a6a7 — za1a2a6a~

— ala~a4a~ — a la~a7 + a la~aba .2 + a~a$ — za~a3a~ay

—a~a4a5a7 + afa~a f, + a2a3a~a7 — a2a3a4a5a6 + a2a~a i

m l = aOa4a~ — a0a5a6a7 — a1a4a6a7 + a1a5a~ — a~a? + za2a3U6U7

+ a2a4a5a7 – a2a~a s – a~a~ – a3a~a7 + a3a4a5a6

m t = aoa,a~ — aoa3a6a7 — aoad abaT + aoa~a~ — alaza~a~

+ ala@~ + ala~a7 – ala4a5a6

m3 = ‘&~ + za0a1a6(37 + aOa3a4@ — aOa3@,a6 — a~a~ — (_31a2(34a7

+ i3@2a5a6

??34 = (@5a7 — aOa]a4a7 — a0a1a5a6 — a0a2a3a7 + a0a&6

+ @a4a6+ al&7 – @a2a3a6

??35 = U&Z@7 — &~ — aOa@2(37 — a0a1a3a6 + ‘k0a1a4a5

+ aOa2a3a5 – a0aja4+ a~a2a6 – a!aj – ala~a5 + ala2aSa4



Index

.4

Acceler at ion -er ror con st an t, 145

defin it ion of, 66

Aided t racking, defin it ion of, 361

Aided-t racking t ime constant , rms-er ror

cr it er ion in det ermin in g, 363–368

defin it ion of, 362

Aided-tracking unit , 361-363

Amplidyne, 106

Amplifier , motor and power, 103

rota ry magnet ic, 106

Amplit ude, defin it ion of, 40

Arbit rary input , response to, 48

Attenuat ion and phase diagrams, con-

st ruct ion and interpreta t ion of, 171

.It tr nuat ion-phase analysis, 163

At ten ua tion -ph ase r ela tion sh ips, 169

:Iu tucor rela t ion funct ion, defin it ion of,

273

of filt er ed sign al, 288–291

n or m~hzed, defin it ion of, 274

B

llack-emf constant , 104

IIarnes, J L,j 9

l}odc, H TV , 17, 169

Br :mrh point , 13,5

Br idge, spwd-fccdl,a ck, 214

l)r id~c T, symmet r ica l, 123

Brolvn, G. S., 16, 158

Buildup t ime 7,, 142

Bu rn slde, W’. S., 337

Bnsh, V., 9, 208

c

Ca ldwcll, S. H ., 280

Cbllmder , A.; 158

Ca rsla w, H , S., 51

Chandra sckhar , S ., 266

Clamping, 245-246

definition of, 245

Commutated signal, 293

Commutator , 292

Constant (see type of constant )

Con ver gence, absolute, abscissa of, 51

region of defin it ion of, 51

d efin it ion of, 23>234

Ckmvolu tion , defin it ion of, 54

Coradi harmonic analyzer , 283

Correlat ion funct ions , 273-277, 283-288

cross-, defin it ion of, 277

and spe~t r a l densit y, r ela t ion between ,

28.%288

Cor r ela t ion ma tr ix, defin it ion of, 277

Cross-cor rela t ion funct ion , defin it ion of,

277

Crossover , defin it ion of, 304

Cros s-spect r a l densit y, defin it ion u f, 279

D

D-c ser vo mot or s, 103

Der ihcl, 163

Dccil ,cl–r]l lase-:,l ]glc diagrams, 179, 346,

357

Dday t ime, 142

Delta funct ion , 279

definit ion of, 30

Der ivat ive of 8,, 328

Der ivat ive cont rol, 197

Design rncthod, new,

3zKL368

applica t ions of,

J )c.ign tmhniqurs, h istory of, 15

IMwtors, phase-sensit ive, 111

I)ickson, I.. E., 334, 335

l)] ffer cn t ia l, 134

Diffcr cn tia ] equat ion analysis, 152

Dirac delta funct ion (see Delta funct ion)

Doctsch , G,, 51, 151

E

E-tr ansfor mer , 102

12mde, F., ]55

371



372 THEORY OF S ERVOMECHAN ISMS

Ensemble, defilt ion of, 266

Ensemble averages, 271-273


Equa liza t ion , in t egra l, 203

wit h s ubsidia r y loops , 208

Equa lizer , br idged -T, 227

fr equ en cy r es pon se of, 227

proport ional -der ivat ive , 199

Ergod ic hypothes is , 271-273


Error coefficien ts , 147

Er ror -measur ing elemen t , 134

Er ror -measur ing sys tems, 77

pot en t iomet er , 95

Evans, L. W., 155

F

F adin g, defilt ion of, 265

F eedba ck , s yst ems wit h , 62-75

F eedba ck cu t off fr equ en cy, 160

Feedback fa lt er (see F ilt er , feedback)

F eedba ck fun ct ion , 134

F eedba ck t r an sfer fu nct ion (s ee Tr ans fer

function, feedback)

Feedback t r ans fer 10CUE,66-68

Fer rell, E . B., 169

Fifter , 24-30

a nt ih un t, u se of, 215

clamped , st abilit y of, 251

wit h clamping, 245–253

feedback , 124

h igh -pa ss, for d -c feedba ck , 126

lin ea r (s ee Linea r filt er )

low-pass, ideal, 141

lumped-constant, 2*2S

frequency-response function of, 42-

43

normal modes of, 26-28

t ra nsfer funct ion of, 59-61

normalized, de6nit ion of, 236

pulsed, defin it ion of, 232

st abilit y of, 233-236, 242–244

t ra nsfer fun ct ion of, 24@242

with pulsed data , 232–245

EC, . two-sect ion high-pass, graph for

design in g, 127

st able, 3840

definit ion of, 3A

wit h swit ch es, 253-254

t ransfer funct ion of, definit ion of, 241

Filter, unst able,38-40

definitionof, 38Filter responseto sinusoidalinput, 238-

240

Follow-up system, 1Fourierintegral,43–48Fourierseries,43Fouriert ra nsform ,definitionof, 45Frequency-responsecharacter ist ics,179

Frequency-responsefunction, 40–51definitionof, 41of lumped-const an t filter ,42–43and weigh t ing funct ion , re la t ion be .

tween,4*5O

G

G, e lementsof, reduct ion of, as ra t iona lfunctions,317–319

genera lredu ction of, 32&321reduct ion of, by exper imen t al da t a,

319-320G (f), 313

G, 313r ., as eum of a ll symmet ric fu nct ion s,

337-338Gain,antihunt,217infinite,idea l ca se of, 34$352

Gardner , M. F., 9

Gear t ra ins, 130

various, iner t ia of, 133

Gen er at in g fu nct ion , 244245

definit ion of, 245

Godet , S., 212

Graham, R. E., 169

Guillemin, E. A., 9, 24, 142

H

Hall, A. C., 16, 17, 21, 158, 163, 309

Harr is, H., 16, 195

Har t ree, D. R., 158

Hazen, H. L., 16

Herwald, S. W., 158

I

In pu t, a rbit ra ry, r espon se t o, 48

rep resen t a tive, r esponse t o, 138

s inusoida l, filt er response to, 238–240

response to, 4@-42

unit -s tep (s ee Unit -s tep in pu t )



INDEX 373

Inpu t -noise spect ra l-dens ity mat r ix, defi-

n it ion of, 313

I npu t sign a l a nd n oise, 34&342

In tegr als, eva lua tion of, 33>339

Int egr at ed-squa re er ror , defin it ion of, 314

J

J aeger , J . C., 51

J ahnke, E., 155

K

Khin tch ine, A., 274

L

La gu er re fu nct ion s, 319

Laplace t ra nsform, 51-58

of convolu t ion of two funct ions, 54

definit ion of, 51

of der iva t ive, 53

of e-”’g(t ), 54

of integra l, 54

limit ing values of, 55

use of, in solu t ion of linear differen t ia l

equ at ion s, 56-58

Lauer , H., 163

Lead, 197

Lee, Y. W., 17

Lesn ick, R., 163

[,in ea r filt er , 28-30

definit ion of, 28

normal response of, to arbit ra ry inpu t ,33-35

to un it -impulse input , 3(P33

Linear funct ion , defin it ion of, 29

Linville, T. M,, 88

Liu, Y. J ., 155

hop, zero-sta t ic-er ror , 140

I.oop t ransfer funct ion, 134

exper imen t a l decibel–log-fr equency

plot of, 228

hfc

hlacColl, L. A., 169

MacLane, G. R., 369

M

Mapping theorem, 68-70

Marcy, H. T,, 195

Matson, L. E., 163

Mean-square, definit ion of, 273

Mean-squa re er ror , min imizing of, 32S

328

Minorsky, h’,,16Modulators, 108

Moments, definition of, 273

Motor and power amplifier (see Ampli-

fier , motor and power )

Mult iloop servo systems (see Servo

sys tem, mu lt iloop )

N

h’etwork, adding, 96

for a -c er r or -signal, 117

der iva tive-plu s-pr opor tion al, 114

equ alizin g, 114

in tegr al-plu s-pr opor tion al, 115

ph ase-la g, 227

frequency response of, 227

Nichols, N, B,, 131

,Yoise, spect ra l den sit y of, 294–295h-or mal modes, defin it ion of, 27

Normal response, definit ion of, 26

Notch in terval, 121

Null devices, 101

Nyquist , H., 16, 71

Nyquist cr iter ion , 7@72

modified form of, 256

Nyquist diagram, 66, 158

of servo wit h t a chomet er -feedback

loop, 359

Nyquist t est , 134, 136, 138

0

Outpu t disturbances, 145

Over-a ll t ransfer funct ion , 62, 225

P

Panton, A. W., 337

Parallel T with equal condensers, 121

Parameters, best cont rol, for fin it e am-

p lifica t ions , 352–360

Parseval theorem, 61

P er forma nce specifica tions, 17

P er iodicit ies, h idden , 279

Phase, defin it ion of, 40

Phase angle, maximum, 199



374 THEORY OF SEE VO.~4ECHA,VI,9MS

Phase margin , 160

Phillips, R. S., 21

Plain, G, J ., 212

Poisson dist r ibu t ion , defin it ion of, 302

Por ter , .4., 158Posit ive sense, defin it ion of, 69

Poten t iometer er r or -m ea sur ing system s,

95

P ot en tiomet er n oise, 305–307

Potent iometer resolu t ion , 9S

Proba bility-dist r ibu t ion funct ions, 269

270

P ulsed filt er (see F ilter , pu lsed)

Q

ouadra t ic factor , 175, 176

R

Radar automat ic-t racking example, 291-

298, 32=333

Rando,T, p roce ss , 266-270


pu re ly, 298-300


st at ion ar y, 270–273


h zwmon ic a na lysis for , 27&291

Random ser ies, 267

Rmdom-walk problem, discret e, 267, 269

R, (a ), 47

Itcmote cont rol, 1

Repet it ion fr equency, defin it ion of, 232

Ikpet it ion per iod, 281


Itesiducs, method of, 46–48, 333It icc, S, 0,, 266

Itms, defin it ion of, 273

I{m s er ror , definit ion of, 309

mathematica l formulat ion of, 3 12–315

It In s-er ror cr it er ion in ser vom ech anism

des ign , 308-339

ltobinson , G., 283

Ib,ot -m ea n-squ ar e (see Rm s)

Itoots, pairs of, product of all sums of,

335-336

s

Servo, best , with pr opor t ional pr oper ties

of, con tr ol, 34>347

clamped, with propor t ional con t rol,

2.59-261

cont rolled by filter with clamping,257-259

wit h pr opor tion al con tr ol, 342–348

decibel–log-frequency diagram for ,

34!5

fl’~ = O, 347-348

pulsed, genera l t heory of, 254-257

st abilit y of, 255–257

with pulsed input , 254261

with tachometer -feedback loop, deci-

bel-log-fr equ en cy dia gr am for , 357with two-phase motor , 224

Servo loops, equa lizat ion of, 196

Servo motor s, d-c, 103

Servo system, 2

m ult iloo p, 7>75, 186

t hr ee-differ en tia l, 137

two-differ en tia l, 135

Servomechanism design , RMS-er ror cr i-

t er ion on , 308-339

Ser vom echa nism in put , t ypica l, 300-304

Ser vom ech an ism s, con tin uou s-con tr ol, 7

defin it ion cor rect ion , 7

r elay-type, 7

Sgn b, defin it ion of, 235

Sin usoida l s equ en ce, 236–238

dct in it ion of, 236

Sobcz:,-k, .4., 118, 124, 367, 368

Spect ra l den sit y, 278–288

and a utocor rela tion funct ion of t ilter ed

sign al, 288–29 1

ca lcu la tion of, 284–286

and cor rela t ion funct ions, rela t ion be-

tween , 283–288


er ror , in tegr at ion of, 323–325

r edu ct ion of, 317–321

for cxpcr im cn ta l da ta , 282-283

of input a nd noise, 2!34-295

norma lized, definit ion of, 280

of r an dom ser ies, 2S1–282

of sta tionary ra ndom pr ocess, 280–281

Speed-volta ge constan t , 104

Stability, r egion of, boundary of, 316

mit radiaton lab series v25 theory of servomechanisms

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