TITLE INDEX Volume 44 (1962)

PACE Biotin-Folic Acid Interrelationship : a Re-evaluation. (F. F. Dias, M. H.

Bilimoria and J. V. Bhat) . . . . . . 121

Corynebacferium Barkrri, nov.spec., A PectinoIytic Bacterium Exhibiting a Biotin-Folic Acid Interrelationship. (F. F. Dias, M. H. Bilimoria and J. V. Bhat) . . . . . . 59

Equal Area Method of Evaluating the Describing Function. (B. L. Deekshatulu) . . . . . . 77

Physico-chemical Investigations on the Basic Carbonate of Cobalt. (D. S. Bbaradwaj and A. K. Vasudeva Murthy) . . . . 68

Propagation of Microwaves on a Single Wire-Part 111. (v. Subrah- manyam) . . . . . . 27

Propagation of Microwaves on a Single Wire-Part IV. (v. Subrah- manyam) . . . . . . 122

Microbial Decomposition of Pectic Substances-Part 11. The role of yeasts in the process with particular reference to a polygalacturonase producing Cryptococcus laurentii. (M. H . Bilimoria and J. V. Bhat). I5

Reaction between Disulphur Monoxide and Piperidine in Organic Solvents. (bliss. I<. Sharada and A. R. Vasudeva Murthy) . . . . 49

Secondary Flows Induced in a Non-Newtonian Fluid between Two Parallel Infinite Oscillating Planes. (P. L. Bhatnagar and Miss. G. K. Rajeswari) 219

Some Investigations on Dielectric Rod Aerials-Part V. (H. R. Ramanujam i n d Mrs. R. Chatterjee) . . . . . . 164

Some Investigations on Dielectric Rod Aerials-Part VI. (H. R. Ramanujam and Mrs. R. ~hatterjee) . . . . . . 203

Some Theoretical Investigation in Dielectric Rod Wave Guide. (v. ~ubrahmaniam) . . . . . . 118

Studies in Cosmic Radio Noise a t Bangalore. (M. Krishnamurthi and S. ~amakr ishna) . . . . . . I

Studies on Soil Bacteria Decomposing Glycerol-Part I. Isolation, identification and test for glycerol decomposition by different bacteria. (Y. I. Shethna and J. V. Bhat) . . . . . . 8

241

Studies on Soil Bacteria Decomposing Glycerol-Part I1 : Metabolism of Streptornyces and Azotobactot species (Y. I. Shethna and J V. hat) 141

Study of the Transient Stability Problem. (H N. Ramachandra Rao and N. Dharma ~ a o ) . . . . . . 91

Torqne Roll Axis and its influence on Automotive Engine Mountings. ailas ash Nnth Gupra and M. R. Krishnamurthy Rao) . . . . 104

JOURNAL OF THE

INDIAN INSTITUTE OF SCIENCE

VOLUME 44 JULY 1962 NUMBER 3

A STUDY OF THE TRANSIENT STABILITY PROBLEM

H. N. RAMACHANDRA RAO AND N. DHARMA RAO (Eiecirical Engineering Seerion, Indian Insritate of Science, Bangalore-12)

Received on February 2,1962

This paper presents a simple graphical method of determining the-critical clearing angle making use of the fundamental stability theorems of Lagrange and Liapounoff for a conservative system. A graphical construcfion is also given to find the data for the swing curve without numerical computation. The methods proposed are illustrated by two typical examples and the results obtained are compared with those of the existing methods.

The differential equation characterising the dynamic behaviour of a synchronous machine, known as the swing equation, is 3.'

under the usual assumptions of constant input, n o damping and constant voltage behind transient reactance, where ,

M - inertia constant

Pi = shaft power input corrected for rotational losses

P, - P, sin 8 - electrical power output corrected for electrical losses

P, = amplitude of t h e power anglecurve

S - rotor angle with respect to a synchronously rotating reference. 9 1

92 H. N. RAMACH~NDRA RAO AND N. DHARMA RAO

By introducing a new variable, modified time T defined by the equation

T- r 2 / [ ( ~ / 1 8 0 ) ( & / ~ ) ]

equation 1 reduces to

d2S/d7" - P- sin S

where 8 is in radians and P-SIP,,,. I n the following analysis, the swing equation will be made use of in t he form of equation 2 only.

Let the swing equation during fault and that after the fault is cleared be respectively

d 2 8 / d ~ : - P, - sin F 131

where TI = r d[(n / 180)(~,, / M ) ] , TI = t 1/[h / 180) (P,, / M ) ] , PI - E lPm1 and P2- E~P",,, P,,,, and P,, being respectively the amplitudes of the power angle curve during the fault and after clearing. The post fault swing equation 4 describes the motions of an autonomous conservative system with a nonlinear restoring force ~ ( 6 ) - s in6 - P2 Setting d ~ / d ~ ~ - w,, equation 4 becomes

with the initial conditions, S - So, wl = ojl0 a t T2 = 0. Separating variables and integrating, equation 5 becomes

3

where E ir the total energy of the syarem. The quantity ~ ( 8 ) - / ~ ( 6 ) 6 6 re-

presents the work done by ~ ( 6 ) and so it is the potential energy. w:/2 represents the kinetic energy. In other words, equation 6 means : kinetic energy + Potential energy -constant, thus expressing the law of conservation of energy. For a conservative system there are two fundamental stability theorems2 : the theorem of Lagrange which states. " If the potential energy is a minimum a t the state of equilibrium, the equilibrium is stable", and the converse theorem of Liapounoff, "If the potential energy is not a minimum at the state of equilibrium, then the equilibrium is unstable ". For 0 < S < n, the potential energy v(S) is a minimum at S - sin-' pz and a maximum a t 8 - .rr -sin-' p2. The first equili- brium point of equation 5 is located at 6 - sin-' P2 and the second equilibrium

A Study of the Transient Stability Problem 93

point at 6 = n -sin-'&. Hence by virtue of the theorems stated above, the first equlibrium point is a stable one known as a vortex and the second an unstable one known as a saddle. Solving equation 6 for wl we get

In the (6,wl) plane, called the phase plane, equation 7 specifies a definite phase trajectory for a definite value of the total energy E. For E= V,,,(S), equation 7 describes the particular phase trajectory, known as the separatrix, passing through the unstable equilibrium point. The critical clearing point is located on the separatrix, its co-ordinates being fixed by initial conditions. The initial conditions are given by the velocity versus displacement curve of the sustained fault swing equation with proper adjustment of time scale. Setting w - dS/dTI in equation 3 and integrating there results

where 6 (0) is the rotor angle at the instant of fault inception. The locus of initial conditions is given by

where K- d T , / ~ T ~ . The critical clearing angle can be obtained by superposing the w,,, Vs6 curve on the phase portrait for the post fault swing equation, the point of its intersection with the separatrix giving the critical clearing angle. The following alternative procedure results in a simple and elegant graphical construction. The equation of the separatrix curve is

Equating (9) and (10) a transcendental equation of the type f (6 ) = A6 + B +cos 8 = O results. The solution of this equation can be effected graphically by

plotting the straight line ( - A 8 - B ) and the cos 6 curve, the intersection between the two giving the critical clearing angle 8,. A refined value 8,, of 6, can be obtained by means of the Newton-Raphson formula

To summarise, the procedure for finding the critical clearing angle by the methods discussed above involves the following steps.

94 8. N. RAMACHANDRA RAO AND N. DHARMA RAO

Method I (a) The separatrix curvegiven by equation 10 is sketched in the (6, w , ) plane.

(6) The locus of initial conditions given by equation 9 is also drawn on the same graph sheet. The abscissa of the intersection point between the two curves gives the critical clearing angle.

Method 2

( a ) Equating ( 9 ) and (10) a transcendental equation of the form A; + B + cos 6 r; 0 is found. If the approximate solution of this equation found graphically is S,, a better approximation a,,, can be obtained by using equation I I.

Time corresponding to the critical cleaaring angle. Let the ru Vs S curve given by (8) of the sustained fault swing equation 3 be drawn. The increment in T, needed to traverse the increment AS isL5

where w,, - ( w , + u2)/2. w1 and w2 being the values of w a t the beginning and and at the end of the increment AS. If ATl is small, i t is nearly true that

w,, - w (0) + A w/2 [I31

where w ( 0 ) is the value of w at the beginning of the increment A TI and A w is the change in w during this increment. A combination of the equations 12 and 13 gives

A w - ( ~ / A T , ) A ~ - ~ u ( o ) . [14]

'AXES FOR AS, Am

FIG. I Graphical Construction for finding time

A Study of the Transient Stahiliry Problen~ 95

If a fixed value of A TI is chosen, equation 14 represents a straight line of slope AT,) and A w intercept of-2w(0) where A w and A 6 are measured from o (0) and 6 (0) existing a t the beginning of the increment. The intersection of this line with the w Vs 6 curve locates the point satisfying simultaneously the original differential equation 3 and also equatiou 14. The process of finding time can be mechanized as shown in Fig. I. The point [8(0), w(O)] a t TI - TI (0) is first located. A protractor is then pz t at the point [8(0), - w(0)] with its horizontal and vertical lines aligned with the coordinate axes. The intersection between the w Vs 6 curve and the line with the slope ( 2 1 8 3,) locates the point T,(o) + A TI. The process can be repeated locating rather quickly a series of points equally spaced in time. The methods presented above will now be illustrated by two examples.

Example I.

A 25 MVA, 60 cycle water wheel generator delivers 20 MW over a double circuit transmission line to a large metropolitan system which may be regarded as an infinite bus. A 3-phase fault occurs a t the middle of one of the trans- mission lines which is subsequently cleared. Find the critical clearing angle and t i n ~ e . ~

DATA : Initial angle, 8(0) = 18.1'

Input power, Pi = 0.8 P.u.

Inertial constant, M = 2.56 x 1 0 ' ~ p u

Prefault power angle equation - 2.58 sin 6

Power angle equation during fault - 0.936 sin 8

Post fault power angle equation = 2 06 sin S

TI = St, T2- 11.4t, K- 0.702

Method I

T h e potential energy ~ ( 6 ) for the post fault swing equation is

v (6) = (sin S -0.388) d 6 - 1 -cos 6 -0.388 6. j The maximum value of ~ ( 6 ) , V,,(6) - 0.86 and it occurs at 6 - 157.2'. The equation for the separatrix curve is

a, c. J{z [0.86 - ~(6)]:.

9G H. N. RAMACH~NDRA R40 AND h'. DNARMA RAO

E = l

FIG. I1 Phase Trajectories for the post fault swing equation of example

A Sludy of the .7Fansient Stability Problrm

Fn;. III Finding Critical Clearing Angle

98 R. P]. RAMACHANDRA RAO AND N. DII~RMA RAO

The locus of initial conditions is given by

wlo - 0.702 OJ

where w = d(1.71 5 + 2 cos S - 2.44). [IS]

The intersection of the separatrix curve and the locus of initial conditions occurs a t Fj - 138', which i c the critical clearing angle. The details of the procedure are shown in Fig. 2.

Method 2

Equating the expressions for w , a?d wlo we get

0.063 6 - 0.905 - cos S.

The solution of this equation obtained graphically in Fig. I11 is 8, = 138". A better value S,, is obtained using equation 11.

S,, - 2.41 +0.01/0.7321 - 2.4237 radians or 138.9'.

This value of critical clearing angle compares favourably with 139' obtained by the equal area method.

The graphical construction for finding time from the w VsS curve given by equation 15 is clearly illustrated in Fig. IV. The increment A T, is chosen as 0.4 sec. corresponding to a value for At equal to 0.05 sec. Thus the angle to be used in the graphical construction is

In using the graphical construction indicated in this paper either same scale is to be used on both the 6 and w axes or , if the scales on the two axes are different, i ts effect on the angle tan-'(2/ATl) must be properly taken care of as follows :

Let 1 unit of 6 be represented by 6, mnz

Let 1 unit of w be represented by w , nzm

Let AT, = h secs.

Then theangle to beused in the graphicalcons~ruction is tan- ' [(2/h)(~,/w,)l. In Fig. IV the scales on both the axes are chosen equal. The times corresponding to various angles are shown across the positive half of the w Vs 5 curve. The first point corresponding to 0.11 sec. (28') is located using the formula given by equation 12, for it is very diflicul~ to locate intersection point between the line

A Sludy of the Tramient Stability Problem 99

Frc. l'i Time scale on the Vsa curve

with the slope 5 and the initial portion of the 10 Vs6 curve. Thereafter graph~cal construction is used to find time. By interpelat~on, the time correspond~ng to the angle of 139' is found as 0.62 see, which is the critical clearing time. The critical clearing time ohtained by step-by-step m e t h o d ~ i " ~ is 0.61 sec. The swing curves obtained by the two methods are shown in Fig. V.

Example 2 A generator supplies power through parallel high voltage transmission

lines to a large metropolitian system considered as a infinite bus. A 3-phase fault occurs on one of the transmission lines which is subsequently cleared. Find the critical clearing angle and time.4

DATA : lni tial angle, 6 (0) - 35.2'

Input power, Pi. - 1p.u Inertial constant, M - 2.78 x p.u Prefault power angle equation = 1.735 sin 6 Power angle equation during fault - 0.42 sin 6

Post fault power angle equation - 1.25sinS

T, - 5 15 1, T? = 8.86 t , K = d T,/dTz -0.582

100 H. N. RAMACHANDRA RAO AND N. DHARMA RAO

FIG. V Swing Curve for Example 1 with Sustained Fault

The potential energe V ( S ) for the post fault swing equation is

Y (8) - S - 0.8) d8 = l - cos 8 - 0.8 S.

0

The maximum value o f v(S), v,,(s) = - 0.17 and i t occurs S - 126.8'. The equation for the separatrix curve is

w , - d{2[ -0 .17 - v ~ E ) ] ] and that for the locus OF initial conditions is

wlo -0.582 w where w - d(4.76 6 + 2 cos S - 4 55) .

.A Study of the Transient Stability Problem 101

The intersection of the separatrix curve and the locus of initial conditions occurs a t 6 = 52', which is the critical clearing angle. The details of the procedure are shown in Fig. VI.

Phase portrait for thepost fault swing Equation of Example 2

102 H. N. RAMACHANDRA RAO AND N. DHARMA RAO

Method 2

Equating the expressions for wl and u11o we get

cos 6 - 0.0075 6 + 0.604.

The solution of this equation obtained graphically in Fig. 111 is 8, -50". A better value of 8, is obtained using equation 11 as 6,, = 52", which compares favourably with the value of 51.6" obtained by the equal area method. Using the graphical construction illustrated clearly in the previous problem, the time corresponding to t he critical clearing angle of 52' is found as 0.11 sec. Step- by-step method I1 also gives 0.11 sec.

DISCUSSION

The method presented in this paper identifies the critical switching angle as a point on the separatrix curve, whicb separates the region of stable motions from that of unstable motions. I n addition, the method described in the paper introduces the analysis of the potential energy, stored in the generator rotor, as a tool in the determination of the critical switching angle. This interpreta- tion is made possible by the fundamental stability theorems of Lagrange and Liapounoff for a conservative system. The fundamental difference between the topological method presented here and the conventional equal area method is that the former is based on the concept of energy, while the latter is based on the concept of power ; in the former, t he critical clearing angle is obtained by equalising the maximum value of the potenlial energy with the total initial energy and in the latter by equalising the area representing acceleration power with that representing deceleration power. While, in the phase plane method, it is clearly shown that the various possible motions of the system take place along paths of constant energy, this is not placed in evidence in the equal area method. Nevertheless, the two methods complement each other bringing out the important fact that the stability of a nonlinear system, for a given type of excitation, is dependent on the initial conditions unlike a linear system which is either stable or unstable, the driving function and initial conditions having no effect on stability.

The graphical construction given in this paper finds time increments by assuming a constant average velocity during a small angular increment As, while the point-by-point method 1 1 ~ , ~ finds the incremental angles during a small time interval At, assuming the velocity t o be constant at the value computed a t the middle of At : the former method finds time increments from angular increments whereas the latter method finds angular increments from time increments. There is good agreement between the results obtained by the two methods eventhough they proceed on different lines. If an analytical solution is t o be found for the swing equation, i t is necessary first of all to approximate sin 6 by a polynomial i n 6. Reference 1 approximates sin 6 by a 8 + b s3 such

A Study o f the Transient Stability Probkm 103

that the integral square error is a minimum. Minimization of the integral square error essentially means that emphasis is placed on the errors according to the square of the error magnitude. In other urords, the approximation resultingfrom the minimization of the integral square error attempts to cut down Iarge errors a t the cost of many small errors. Other types of error criteria may be chosen, but their mathematical treatment becomes very difficult, if not impossible. Therefore, analytical solutions are of little value.

The authors believe that the cosine 6 curve method, based on the energy concept, for finding the critical clearing angle in conjunction with the step-by-step graphical metthod for finding time form a good combination for the solution of the transient stability problem.

1. Dharma Rao, N. . . A new approach to the transient stability problem, A. I. E. E., Transaction paper No. 62-103 (to be published).

2. Andronow, A. A. and Chaikiu, C. E. . . Theory of oscillations, (New Jersey : Princeton University Press. 1949, 63-69.

3. Kimbark, E. W. . . Power System Stability, Val. I, Elements of stability calculaitons. (New York: John Wiley), 1957, 30-50.

4. Stevenson, W. D.

5. Cunningham, W. J.

. . Power System Analysis, (New York: McGraw- Hill), 1955, 338-345.

. . Introduction fa nonlinear analysis, (New York: McGraw-Hill), 1958, 37-39.

TORQUE ROLL AXIS AND ITS INFLUENCE ON AUTOMOTIVE ENGINE MOUNTINGS*

BY KAILASH NATB GUPTA AND M. R. KRISHNAMURTHY RAO (Department of Infernal combustion Enginebring, Indian Instirule of Science, Bangalore.12)

Received on March 24, 1962

ABSTRACT In automotive engines the torque excitation does no t occur about an axis

parallel to any of the principal axes of the engine. This causes the roll movement of the engine to occur about an axis, known as Torque Roll Axis, and its accurate location becomes essential for the determination of the most favourable disposition of the engine mountings. The paper deals with the theoretical analysis to predict the location of Torque Roll Axis. The results of the analysis are applied to locate the TorqueRoll Axis of seven automotive multi-cylinder engines for experimental verification. I t i s noticed that a judicious combination of the analytical and experimental methods would reduce considerably the time and effort involved in locating the torque roll axis. Further experimental work confirms the fact that the arrangement of mountings about the torque roll axis leads to the maximum isolation of engine vibrations.

INTRODUCTION An engine mounted on resilient supports has six natural modes of vibration.

But under the influence of its inherent disturbances it will get excited in three of its natural modes of vibration, provided these disturbances are occuring along and about the principal axes of the engine and all the natural modes are decoupled. In a modern automotive engine the location of all the three principal axes xx, yy and zz is as shown in Fig. I, and the ideal condition of decoupling of modes of vibration can be obtained by arranging mountings of equal stiffness symmetrically with respect to these principal axes. In practice it is not possible to attain this ideal arrangement and the mountings are generally fitted symmetrically about the vertical plane passing through axis xx.

This arrangement leads to some complications regarding isolation of rolling motion, because of the fact that torque excitation does not occur about axis xx. The interesting phenomenon arising out of this is discussed below.

In reciprocating engines of the automotive type the longitudinal principal axis .XX is inclined t o the crankshaft axis about which torque excitation occurs. Consequently the engine rolls about a third axis, called Torque Roll Axis, which lies between the crankshaft axis and the principal axis xx. If the foregoing arrangement of the mountings is modified so as t o locate them about the Torque Roll Axis the engine mass will be excited in only one mode under the

"Extracted from the Thesis submitted to the Indian Institute of Science by Kailash Nath Guptafor theDegree of Master of Science of the Institute, July 1961. 104

Torque roll axis and its influence on automotive engine mountings 105

influence of rocking torque and there will be complete decoupling. Further if the natural frequency of the engine mass on its mountings about this axis is substantially low as compared to firing frequency at low speeds, a very good engine installation whl result.

Y Elevation

1 Plan FIG. I

Symmetrical Mounting arrangement of a Power Unit

It is the intention of the present investigation to devise a method for the easy location of Torque Roll Axis and to study its impact on enginevibration.

REVIEW OF PREVIOUS WORK

There was a belief a t one time that roll took place about an axis paraliel to crankshaft axis and passing through the centre of gravity of the engine. Hence emerged the practice of arranging the mountings about this axis. This was giv& up in favour of ' Floating Power Arrangement ' as shown in Fig. I I , where the longitudinal principal axis was assumed to be the roll axis and rhe mountings were arranged about it.

106 KAILASH NATH GUPTA AND M. R. KRISHNAMURTHY RAO

The ' Floating Power Arrangement ' did not, however, take into considera. tion the effect of non-coincidence of torque axis with principal axis xx on the engine roll movement. Den Hartogl and 11iffe2 studied this effect and showed that torque excitation about an axis different from prin?cipal axis of inertia. would also induce rotation about an axis at right angles to the torque axis, and the resultant motion would take place about an axis different from torque axis,

The Chrysler Floating Power System

~ i e s i l i g ~ was the first t o notice that engine roll took place about the Torque Roll Axis, which was in between the torque axis and the principal axis xx and recommended the location of the mountings about the Torque Roll Axis. But other workers such as h o n 4 , Harrison5 and Horovitz6 recom- mended the location of the mountings about the principal axis xx, since they considered that in a conventional engine (incorporating the gear box) the torque component along the principal axis xx would be much greater than that along the other principal axis yy. Though this was partly true yet ~ i cc l a i sen~showed that the torque component along the principal axis yy amounted to 30-40% of the total unbalanced torque and so argued that i t was not justifiable to neglect i t . Therefore if t he unbalanced torque components along axes xx as well as yy were considered, the roll must take place about another axis namely Torque R ~ l l Axis.

Insview oP the foregoing controversy regarding the roll movement of the engine and its influence on the location of engine mountings it was deqided to carry oat an analytical and experimental study of the disposition of the Torque Roll Axis and i ts impact on isolation of engine vibrations.

Torque roll axis and its influence on autontotivc engine mountings 107

Position o f the Torque Roll Axis of an engine located in space :-Consider an engine located in space, as shown in F i g 111. Let xx, yy and zz (zz being perpendicular to the plane of paper) represent the principal axes of

FIG. 111 Torsue roll axis location

inertia and G the centre of gravity. Let under the influence OF torque Ma cos o t acting about the torque axis AA the engine roll about an axis BB, making an angle ,B with the principal axis xx. Then the following differential equations are derived.

where a is the angular displacement of the engine about the axis BB at any instant, and I, and Iy represent principal moments of inertia of the engine about axes x x and yy respectively. The equations [I] and [2] have a solution of the type a - a. coswt, which on substitution in these equations yields

Mo cos +/cos f l a. ' - -- r, oZ

Mo sin $/sin and 00 - - -

I, id ' [41

108 KAILASH NATH GUPTA AND M. R. KRISHNAMURTHY RAO

Equating equations [3] and [4] we have

or tan ,8 - ( I ~ / I ~ ) tan $.

The above relation establishes the location of axis BE, known as torque roll axis, with respect to the principal axis xx. For a given engine I,,l, and fi are constant. Therefore /3 is also a constant. The equation [5] can be written as

where C=(I,/I,) tan i,b is a constant for a given engine.

The following inferences can be drawn on the basis of equation [5] for an actual engine where the possibility of I, - I;, $ - 0 or TT!~ is remote.

(a) The torque roll axis is independent of the magnitude and frequency of the exciting torque. This indicates that its position would remain unchanged under the influence of any periodic torque.

(b) Since I, is always the minimum moment of inertia (Iz < I,), p will be less than 4, i.e., torque roll axis will lie between the principal axis xx and torque axis AA.

(c) The inclination of torque roll axis to the torque axis depends upon the ratio of two principal moments of inertia I,, I, and the angle between the principal axis and the torque axis.

Position of Torque Roll Axis of an engine supported on flexible Mountings :-Now consider the engine to be suspended resiliently such that all possible mode$ of oscillations are decoupled. Let Kx and K, represent the torsional stiffness of the mountings about axes xx and yy respectively. Then, under the influence of periodic torque Mo cos wt the engine mass will be subjected to rolling motion about an axis BB. Resolving the torques acting on the engine mass along the principal axes, the following equations can be written for dynamic equilibrium of the engine mass.

and & a s i n P +Kua sin p = M o sin $cos wt.

For steady state conditions the solution of the above equations may be assumed to be a - ao cos wt, which on substitution in eqns. [7] and [8] gives

Torque roll axis and its influence on automotive engine mounfings 109

' M sin $/sin and o 0 - L - .

~ ~ - l y w ~

Equating equations [9] and [lo] we obtain

Mocos $/cos ,8 Mo sin $/sin B --=-- K* - Ix m2 Ky - I, wZ

where w, and a, represent the natural frequencies (rad./sec.) of the system about the axes xx and yy respectively. The eq. [ll] can be written as

(:T- 1 t a n p I; - . - - - -- - tan + 1, ($1

where C is a constant for the engine equal to (I,/I,) tan $.

The following inferences can be drawn on the basis of q. [12]-

(a) The position of torque roll axis for a particular engine installation is independent of the magnitude of the exciting torque, but it does depend upon its frequency.

(b) At w,/m = 1, it occupies the position of the principal axis xx and a t w,/w - 1 , it occupies the position of the principal axis yy.

(c) If mounting stiffnesses be so adjusted that w,-cu, the position of torque roll axis will be same as given by the relation 161.

( d ) For values of w,/w and w y / o sutliciently small compared to unity the eq. 1121 reduces approximately to eq. [6].

110 ICAILASI-I NATH LIUPTA AND IVI. K, h R l S H N h M U R T I l Y KAO

( e ) In actual engine installations where vibration isolation is an important consideration, the ratio of any natural frequency of the system to the exciting frequency is less than 1/3, which on squaring becomes suificieo~!y small Compared to 1. Consequently eq. [12] reduces approximate!y to eq. [6], and eq. [6] defiiler approximately the position of the torque roll axis in actual engine ins!allation.

A graphical study of the eq, [!2] is also made. In Fig. IV the dimension- less quantity ( 1 / ~ ) tan ,3 is plotted against r.u,/w with w,/w as parameter. The dotted line in the figure represents eq. [6]. Within the practical limitations for values of and w,,!w namely 0 ,33 one can conclude that the results obtained

FIG. I V Position of torque roll axis in a resilently suspended

engine for different frequency ratios

from eq. [12] d o not deviate much from those from eq. [ 6 ] . The percentage error introduced by the use of eq. [6] in place of eq. [12] can be given as

Torque roll nxis and ils influaice on azltarnorive engine rnol~~iing~ 1 1 I

The percentage error is plotted against o, /w with o,/w as parameter in Fig. V for values w,/w arid w,,/w ranging from 0 to 0.33. The maximum error as indicated by these graphs is 12.2274 which occurs for values of w,/w - 0.33 and ,o,,/u = 0.

0.0 040 0.20 0.30 ' .*/a

FIG. V Percentage error in assuming the engine to be suspended freely

in space for different frequency ratios

112 KAILASH NATH GUPTA AND M. R. KRISHNAMURTHY RAO

In actual engine installations w,/w cannot be zero, and it is a design consideration to keep all possible natural frequencies as low and as close as possible, under which condition the error becomes negligibily small.

On the basis of foregoing it can be concluded that the position of torque roll axis given by equation [6] would be in good agreement with practical results.

The experimental investigntion centred round several automotive engines vailable i n t he Internal Combustion Engineering Department and was conducted n the following stages :

(a) Determination 'of centre of gravity and principal axes and principal moments of inertia of the engine mass.

( b ) -With the help of the data obtained from (a) analytical determination of the position of torque roll axis.

( c ) Experimental determination of the location of torque roll axis and comparison of the results with those got by the analytical method.

( d ) A comparative study of the engine movement for different arrangements of the mountings.

a b c FIG. VI

Ddfermination of centre of gravity of engine mass

Determination of Centre of Groxily : - The method for determining the centre of gravity of the engine ir illustrated in Fig. VI. The point of

intersection of lines A, and 8, Bz would be the centre of gravity of the engine.

QuadriHlrr p;niiolum with enqinz in suslxnded pociiion

PLATE I

KAII.ASH NATH GLPTA, er. ol. J. l t~dinn 111sr. Sci., Vul. 44 , NO. 3

Test-rig for locatins tho lorque roll axis

PLATE 11

, Torque roll axis and its infiuence on automotive engine mountings 113

Determination of Moments of Inertia and Principal Axes :-A quadri- filar pendulum was used to determine the moment of inertia of the eugine mass. The photograph of the test-rig is shown in Pate I. The test-rig had a provision to suspend the engine in such a way that the centre of gravity might lie on the centre line of oscillation of the pendulum formed with the eiigine and wires.

Since the centre of gravity of the engine mass lies generally on the plane containing the centre lines of the cylinders, a symmetry could be assumed about this plane, and an axis zz perpendicular to this plane and passing through the centre of gravity of the engine would he one of the principal axes. The other principal axes xx and yy would lie on the plane of symmetry.

By using the engine as a quadrifilar pendulum the moments of inertia of the engine about any three axes in the plane of symmetry were determ'ined. From these values the location of the principal axes xx and yy and themoments of inertia about them could easily be determined. The procedure is explained in Ref. 4.

Determinrrtion of Torque RON Axis :-

(a) by the analytical method:-The torque roll axis making an angle 0 with the principal axis xx was determined from the formula [ 5 ]

and then its inclination to the crankshaft axis was obtained.

( b ) by the experiment:-The photograph of the rig for locating the torque roll axis is shown in Plate XI. The engine was suspended on four helical springs of suitable and equal rating such that the plane containing either the rear springs or the front springs was perpendicular to the torque roll axis already determined by the analptical method, and their mid-pointsadjnsted to the level of the torque roll axis. The engine was started and its speed was adjusted to obtain large amplitude of oscillation in roll. The axis about which the engine was rolling was determined in the following way.

A plate painted white was attached to the front end of the engine, and , a vertical line lying on the vertical plane containing crankshaft axis was marked

on the plate. Under engine running conditions the vertical linewas illuminated by a stroboscope and the point about which the line was rolling wuld be easily located and marked. The plate was then moved to rear end, and as before the point about which the roll was occuring was marked. This was further verified by mounting a stand on the floor which was carrying a pointer facing the end plate, as shown in Plate 11. Under engine running conditions the relative movement between the vertical line and the stationary pointer end was observed. The position of the pointer was shifted so as to obtain least relative movement and the point opposite the pointer was marked on tbe plate.

114 KAILASH NATH GUPTA AND M. R. KRISHNAMURTHY RAO

The points thus located on the end faces of the engine would lie on torque roll axis. The location of the springs was adjusted with respect to this new torque roll axis if necessary. The engine was started and the foregoing procedure was repeated so as t o get consistent results for the location of torque loll axis In most cases it was found that the correct location of the torque roll axis was obtained in the first attempt.

To know the influence of lubricating oil mass on moments of inertia and torque roll axis experiments were conducted on two engines with and without wax,substituti~n for lubricating oil. With the rest of the engines the experi- ments were conducted with wax substitution'for lubricating oil.

Comparative study of Engine Movement for D!firent Arrangements o f Zngine Mountings :-The layout of the experimental set-up is shown in Fig. VI1. Engine mountings were located symmetrically about the torque roll axis and

(IMIYUT

FIG. VII Measurement of amplitude at the mounting point

displacements both in vertical and horizontal directions were measured a t each mountings point for two speeds of the engine-630 and 1050 r.p.m. For measurement of displacements Bruel and Kjaer equipment was used, which consisted of an accelerometer, a preamplifier and an analyser.

Similar sets of readings were taken for the following arrangements of the mountings. Springs located

(a) symmetrically about longitudinal principal axis xx .

(b) symmetrically about an axis through the centre of gravity of the engine hut parallel to crankshaft axis,

(c) a t the points recommended by the manufacturer.

In all cases the vertical planes in which front and rear mountings were located remained unaltered.

Torque roll axis and its infiuence on automorive engine mountings 115

RESULTS AND DISCUSSION

Centre o f Gravity:- Table I presents the pasitions of centre of gravity and of the end points of crankshaft axis of various engines. While

TABLE I Centre of Gravity of Engine Mass

N.B. : The units are in inches The order in which the values of co-ordinates are X, Y, Z

Y Co-ordinates are in bold for clarity

Co-ordinates of Engine -- -

Centre of Crankshaft Crankshaft gravity front end 0 , rear end O,

1. Cheverolet a aster 85) engine with gear box assembly

2. Mercedes Benz (OM 312) engine

3. Mercedes Benz engine with wax being substituted for lubricating oil

4. Perkins (P,) engine

5. Perkins engine with wax being sub- stituted for lubricating oil

6. Leyland (Commet 3) engine with wax being substituted for lubri- cating oil

7. Meadows (4DC330 MKZ) engine with wax being substituted for lubricating oil

8. Fiat (1 100) engine with wax being substituted for lubricating oil

9. Deutz (F4 ~ 5 1 4 ) engine with wax being substituted for lubricating oil

measuring the co-ordinates of the points care was taken to keep the plane containing the centre-lines of the cylinders vertical, so that Y-coordinates of the centre of gravity might give an idea of its relative position with respect to this plane. A scrutiny of the Y-coordinates (in bold in the Table I) of the centre of gravity and the end points of crankshaft axis for each engine shows that these values are almost identical and therefore the centre of gravity does lie very close to the plane containing the centre lines of the cylinders. Hence the assumption of the symmetry about this plane, stated earlier, is justified. A study of the coordinates of centre of gravity for engines 2-5 also indicates that lubricating oil mass has little effect on the position of centre of gravity.

Principal Axes :- The determination of principal axes in a three dimensional body is very difficult and laborious. Rut the assumption of symmetry about the vertical plane containing crankshaft axis and passing through the centre of gravity simplifies the matter.

Table I1 presents the principal moments of inertia of various engines and the inclination of the longitudinal principal axis to the torque axis. The longitudinal principal axis is found to be inclined to torque axis in all the engines, and the variation in inclination ranges from 9.3' to 19.4' except in Fiat Engine, where the inclination is 42'. This high value in the case of Fiat Engine is due to the fact that Fiat Engine is very compact in longitudinal direction as compared to other engines.

Torque RON Axis:-The inclination of torque roll axis to torque axis as obtained by the analytical and experimental methods is presented in Table 11. There is a good agreement between the two results. Table I1 also shows thirt lubricating oil mass does affect appreciably the disposition of the. longitudinal principal axis with respect to torque axis, and the consideration of its effect has enhanced the accuiacy of the results obtained by the analytical method. But the change in the results thus brought about is very small. This can be explained on the following grounds :

(a) The mass of the lubricating oil is appreciably small compared to that of the engine, and hence the position of centre of gravity and moment ofinertia values are not much changed.

(6) The inclination of the torque roll axis depends upon the ratio of the principal moments of inertia, and as is evident that if the numerator and denominator values change in the same direction by same magnitude, the quotient gets affected to a very little extent.

Engine Movement for Different Arrangements o f Mountings : -- Corn- parative values of amplitudes measured both in vertical aod horizontal directions at mounting points are shown in Fig. VIII at two engine speeds 630 and 1050 r.p.m. for different arrangements of the mountings. It will be seen

Torque roll axis and its influence on autamofiw engine mountings 117

TABLE 11 Principal axes, principal moments of inertia and torque roll axis

Principal moment of inertia ~~~~~$$"W~~~~'~ Ib. in. Sec2 axis as determined by

-- Engine Inclination

L I, of axis xx Calcula- Experiment with tion torque axis

1. Cheverolet (Master 85) en- 87.45 339.05 12.4' 9.2" gine with gear box assembly

2. Mercedes Benz (OM 312) 139.32 265.78 9.3' 4.4' engine

3. Mercedes Benz engine with 175.5 310 5 15.1' 6.4' wax being substituted for lubricating oil

4. Perkins (P,) engine 136.1 295.3 13.2" 7.0"

5. Perkins engine with wax 147.0 297.5 18.1' 9.0' being sustituted for lubri- cating oil

6. Leyland (cornmet 3) engine 262.5 541.0 19.4' 9.64' with wax being sub'stituted for lubricating oil

7. Meadows ( 4 ~ ~ 3 3 0 ) MK2 223.9 457.9 14.7~ 7.4O engine with wax being sub- stituted for lubricating oil

8. Fiat (1100) enginee with wax 22 7 30.53 42.0' 8.3O being substituted for lubri- cating oil

9. Deutz (F4 ~ 5 1 4 ) engine with 502.2 29.8.2 10.2~ 4.1' wax being substituted for lubricating oil.

from the figure that the arrangement of locating the mountings about torque roll axis results in producing minimum amplitudes of vibration at all stations in vertical and horizontal directions.

118 KAILASH NATH GUPTA AND M. R. KRTSHNAMURTHY RAO

T h e horizontal displacement indicates the side thrust coming on the mountings. T h e mountings are, generally, less stiff i n this direction and they should be subjected t o least side thrust. The arrangement of the mountings about torque roll axis fulfils this requirement t o t h e maximurn extent.

I t is observed from Fig. VIII that 'at front mounting points t h e amplitudes of vibration for different arrangements of t h e mountings differ apprecmbiy. But it is not s o a t t h e rear mounting points. T h e axes about which mountings are arranged for various arrangements deviate t o a great extent a t the plane

Station I

Vert. Disp.

Station I

Hor. Disp.

FIG. VIII .Comparison of amplitudes at stations 1, 2, 3 and 4

for different arrangements of spring mountings

Station 1 : Left hand front mounting point Station 2 : Right hand front mounting point Station 3 : Left hand rear mounting point Station 4 : Right hand rear mounting point

( a ) Springs located about the Torque Roll Axis ( b ) Springs located about the longitudinal principal

Axis xn (c) Springs located about an axis passing through

C.G. and parallel to the crankshaft axis ( d ) Springs located at the points recommended by

the manufacturer

Torque roll axis and its ir!fluence on automotive engine mountings 119

Station 2

Vert. Disp.

Station 2

Hor. Disp.

Station 3

Vert. Disp.

0.15

Station 3 0.01

Hor. Disp. 0.00

Station 4

Vert. Disp.

Station 4

Hor. Disp.

FIG. VIII (conid.) Comparison of amp!itude at stations I , 2 , 3 and 4

for different arrangemen!s of spring mountings

120 KAILASH NATH GUPTA AND M. R. KRISHNAMURTHY RAO

containing front mountings, as this plane is at a greater distance from the centre of gravity of the engine than the plane containing rear mountings. This indicates that the arrangement of mountings about torque roll axis becomes more and more critical as the distance of the plane containing these mountings increases from the centre of gravity of the engine.

From the foregoing it is evident that maximum benefit is obtained by locating the mountings about the torque roll axis.

In all the automotive engines the torque axis does not coincide with any of the principal axes. Under such conditions roll takes place about an axis which lies in between the longitudinal principal axis and torque axis. This axis is called the Torque Roll Axis.

The position of torque roll axis can be located easily and quickly, by following the analytical cum experimental method used in this investigation.

The flexible mountings if located about the torque roll axis are subjected to least dynamic load and hence provide maximum isolation of vibration from the chassis.

1. Iliffe, C. E.

2. Den Hartog. 1. P.

3 . Riesing, E. F .

4. Anon

5. Harrison, H. C.

. . 'The theory of Flexible Mpun1ing.s for Internal Combustion Engines Insr. Auto. Engrs. Proc., 1939-40,34, 7 7 . '

. . 'Mechanical Vibrations', (New York : McGraw- Hill), 1956, 76.

. . 'Resilient Mountings for Passanger Car Power- plants , S. A. E. Quar~erly Trans., 1950, 4, 38.

. . 'Engine Mounting ', Aulo Engr. 1953, 43, 87.

. . 'Engine Installation ', Auto Engr., 1956, 46, 380.

6. Horovitz, M. . . 'Suspension of Internal Combustion Engines in Vehicles ' Inst. Mech. Engrs. Proc. Aulo Div. 1957-58, 1 7 .

7. Nicolaisen, J. . . Discussion on Ref. 6, Insr. Mech. Engrs. Proc. Auro Div. 1957-58, 37.

BIOTIN-FOLIC ACID INTERRELATIONSHLP - A REaEVALUATION

BY F. F. DIAS, M. H. BILIMORIA AND J. V. BHAT ' (Fermeniarion Technology Laborarory, Indian Instirute of Science, Bangalore-12)

An interrelationship between biotin and folic acid in the nutrition of Corynebacterium barkeri, nov. spec., was recently described.' Subsequent investigations carried out on the nutrition of a few unidentified species of Brevibacterium, Arthrobacier and Corynebacterium also pointed to the possible interrelationship between the two vitamins. It was difficult to believe that such an interesting and widespread interrelationship should have remained unnoticed so far and to ascertain therefore the purity or otherwise of the folic acid sample employed (Hoffmann-La ~ o c h e ) a fresh series of experiments were carried out with strain 7 of C. barkeri and as many as four other samples of folic acid derived from different sources and manufactured in three different countries. It was indeed a revelation to notice that the five folic acid samples fell into two categories, two of which repeatedly gave identical results reported earlier and the rest (3) behaving differently in the sense that they did not meet the partial requirements of the organism to biotin. Does this mean that the folk acid obtaining commercially are, microbiologically speaking, different or that the two samples giving partial biotin response though procured from two different laboratories got coataminated with biotin or some substance replacing biotin? In the meantime, the authors are sending their sample of folic acid to Switzerland for its re-evaluation and would seek the cooperation of scientific workers in not accepting as final the conclusions drawn with regard to the interrelationship described earlier for C. barkeri with respect to the biotin-folk acid. This does not, however, invalidate the rest of the results presented in that paper.1 The authors wish to thank Drs. A. Sreenivasan. H. R. Cama and D. V. Rege for the generous supply of folic acid samples.

REFERENCE Dias, F. F., Rilimoria, M. H., and Bhat,J. V., . . J. Indim InN Sci , 1962. 44, 59.

PROPAGATION OF MICROWAVES ON A SINGLE WIRE Part IV

BY V. SUBRAHMANYAM (Department of Electrical Communication Engineering, Imlian Instimfe of Science, Banglore-12)

Received on March 29, 1962

A theoretical study of the reduction in phase velocity and rndial field spread of the Harms-Goubau line, as a function of the wire radius, coating thickness and dielectric conslant for different wavelengths, has been carried out. It is observed that the dielectric coating thickness has more control over the spread of the field in the radial direction than the dielectric constant of the coating. The percentage of the total power flowing in the dielectric coating, as a function of the coating thickness and dielectric constant for three wavelengths has been calculated. These results when compared with that of the Sommerfeld line show clearly the superiority of the Harms-Goubau line as a surface wave guide.

In an earlier communication,' the effect of t he wire radius on the field spread and propagation parameters for the Sommerfeld and Harms-Goubau line was reported. The object of the present paper is to study the variation of the phase velocity and radial field spread, with respect t o the wire radius in the case of Sommerfeld line and wire radius, dielectric coating thickness and dielectric constant in the case of the Harms-Goubau line respectively. I t is found that a dielectric coating thickness of even 0.002 cm reduces the field spread to more than half its value without the dielectric coating. To obtain the same reduction in field spread for the Sommerfeld line, the wavelength of excitation has to be reduced to one third its original value.

Having found the superiority of the Harms-Goubau line over that of the Sommerfeld line, i t is of interest to calculate the percentage of the total power flowing in the dielectric coating. Detailed calculations have been done to determine this percentage of the power, a s a function of the dielectric coating thickness and dielectric constant for wavelengths Xo = 4.00, 3.45 and 1.25 cms and wire radius 0 - 0.05, 0.10, 0.15 and 0.20 cm. I t is obserded that increasing the dielectric coating thickness increases t he percentage of power flowing in the coating, where as, i t decreases as the dielectric constant of the coating is increased. If we consider cases, where the reduction in phase velocity is less than 4%, and the dielectric constant of the coating 6 > 2.0, the power flowing in the coating is less than 8% of the total power propagated.

Propagation of microwaves on n single wire 123

(i) Sommerfeld line:-The radial decay factor y2, as derived from the boundary conditions is given by2

and 17 1 - 1.70 x x a ho-3'2 for a copper wire immersed in air. a is the radius of the wire and XO, the free space wavelength. The real part of yl, as derived from equation [I] is

The real part of the decay factor is plotted as a function of the wire radius for different wavelengths in Fig. I. It is observed from the graphs that the decay factor does not increase appreciably, within the wavelength range A0 = 4.00 cms to 3.2 cms. Secondly, for the wavelength region plotted, the rate of increase of the value of the decay factor is not much for wire radii a - 0.30 cm to 0.10 cm. However, it is much faster as the radius of the line decreases below 0.10 cm. The slow rate of increase of the decay factor is responsible for the slow rate of reduction of radial field spread in the Sommerfeld line.

(iij Harms-Goubau Line:-Using the field components as reported elsewhere.' imposing the boundary condition that the tangential component of the electric field on the surface of a perfect conductor is zero and matching the transverse impedance Ez/H+ at the inrerface between the dielectric coating and air, the following equation is

where yd and kd refer to the value of the decay factor and wave number inside the dielectric and y', the valm of the decay factor ontside the dielectric, ie., in air. In the above expression, the ratio of the Hankel Functions [ ~ ~ ( ' ) ( j ~ ' d ) / ~ ~ ( ' ) ( j ~ ' n ' ) ] has been taken to be equal to j In (0.89 y'a'), as a small argument approximation.

If we consider wires, whose radii are very small compared with the bee space wavelength ho and the dielectric coating thickness very small or af the

same order of magnitude as the wire radius, the small argument approximate relations are used for the Bessel and Neumann functions and hence equation [3] reduces t o

r ' Z { ~ n ( 0 . 8 9 y'a') - (11s) In (a f /u) ] - (116 - 1)(2x/~o)~/n(o'/a) [4]

where 6 represents the relative dielectric constant of the coating.

RADIUS OF THE LOUMERFELD LINE IN CM

FIG. I Radial decay factor, as n function of the wire radius, for differcot wavelengths

The radial decay factor -f', as calculated from the above expression, is plotted in Fig. I1 as a function of r. for different values of Xo and two different d u e s of dielectric coating thickness a" - 0.002 cm. and 0.01 cm. The variation

Propagation o f microwaves on a single wire

DIELECTRIC CONSTANT

FIG. TI Rndial deeny factor of the Harms-Goubau line as a function of thc dielectric constant of the coating for dnfferent waxlengthr and coating th~chesxres,

Wire radiur oiO.10 an.

of y' as a function of the dielectric coating thickness for different wavelengths and two different value of the dielectric constant E =2.0 and 4.0, is presented in Fig. 111. A study of these two sets of graphs show that the rate of increase of y' with increase in the value a" is faster than the rate of increase w ~ t h the increase in value of 6. In other words, the coating thickness i s more effective,

in increasing y ' , than the dielectric constant. For example, increasing a" from 0.01 cm to 0.02 cm at Xo - 3.45 crns and 6 = 2.0, the value of y' is increased by a factor 1.5, where as increasing E from 2.0 to 4.0 a1 the same wavelength with a"- 0.01 cm, increases y' from 0.20 to 0.25 only.

DIELECTRIC COATING THICKNESS IN CM

FIG. 111 Radial Decay factor as a function of the dielectric coating thickness, for

different wavelengths. Wire radius n,0.10 cm.

Plopagation of microwaves on a single wire 127

The percentages of power flow contained withm a certain radial distance from the transmiss~on line, as calculated for both the Sonimerfeld and Harms- Goubau line, are represented graphically for the sake of comparison in Fig. IV. The curve a" = 0 corresponds to the Sommerfeld line. The radius of the wire a - 0.10 cm and the free space wave length A,, - 4.00 cnls and the dielectric con- stant e = 2.40 for the Harms-Goubau line. The variation of the rad~us of the area

RAOUL DISTANCE IN CM

FIG. IV Radial fidd spread as a. function of the percentage of power flow, for different

dielectric coating thicknesses a". Wire radius a=0.10 m ' X0=4.00 cms. *-2.4

around the wire within which 90% of the power is propagated as a function of the dielectric constant of the coating for diflerent values of wavelengths and

Radius of the area around the wire, with in which 90% of the power is propagated as a function of the dielectric constant, for different wavelengths and coating thicknesses.

Wire radius a=0.10 cm.

coating thicknesses is plotted in Fig. V. From these two sets of graphs, the superior influence of the dielectric coating thickness in reducing the radial field spread than the dielectric constant is brought out more clearly.

Propagation o J microwaves on a single wire 129

PHASE VELOCITY ald fine :-The phase velocity b, of the Sommerfeld line is :ssion2

"P = C [1 - 0.63 1 f I Cos 0/(k a)'] locity of light

is1

0 = - ( ~ / 4 ) [ 1 - l / ( l n / f l + I)]

RADIUS OF THE SOMMERFELD LINE IN CM

FIG. VI ge reduction in phav velocity of the wave as a functioo of wire radius,

Fo; different wnvclenglhr

The percentage reduction in phase velocity as a function of the wire radius a for different values of wavelength A,, is plotted in Fig. VI. It is seen that the percentage reduction in the phase velocity of the wave is very small for increasing values of wire radius ; and as the conductivity of the conductor increases with decreasing values of wavelength, the reduction in the phase velocity of the wave decreases with increasing frequency.

FIG. VII Percentage reduction in phase velocity of the Harms-Gouhau line, as a function of the dielectric constant for different wavelengths and coating thicknesses.

Wire radius a=0.10 cm.

Propagation of microwatvs on a single tvire

DIELECTRIC COATING THICLNESS IN CM

Ro. VlII Percentage duction in phase velocity of the HarmsGoubau linr, an a functkx of the dielectric caating thielmerr, fsr diffrrrnt ivavelcngrh.

n=O.lOcm. r-2.4

(ii) Harms-Goubau Line :-With a knowledge of the radial decay factor $, the propagation constant h of the guided wave can bd derived. I t is given by

hZ - - [ (y ' )Z + kZ] . [61

If the line is assumed to be lossless, hZ - - b2 , where /3 is the phase constant of the guided wave. For the wire radius and coating thickness considered earlier, [y'2/k2] < < 1 . Hence we get

/3 = k [ I + 112. y f2 /kZ] [TI and the phase velocity v, of the wave is given by

vD = , d l / = C [l - yt2/2kZ]

The percentage of reduction in v, as a function of the dielectric constant e

for different values of wavelength ho and dielectric coating thickness a" is shown in Fig. VII. Fig. VIII shows the variation of 6 v,/v,,% with respect to the thickness a" at e = 2.40 for different wavelengths. In both these graphs, the wire radius a is taken to be 0.10 cm.

For sake of comparing the relative values of the reduction in v, between the Sommerfeld line and Harms Goubau line, Fig. IX is plotted for variations in wire radius a, for A0 - 4.00 and 1.25 cms. I t is seen clearly that 6v,/v,% is appreciably large in the case of the Harms-Goubau line than the Sommerfeld line.

Distribution o f Power in the Harms-Goubau Line : - The power transmitted around the line is given by4

where lo is the peak value of the current and eo is the free space dielectric constant.

The power flowing in the dielectric coating is given by

where ed is t he absolute dielectric constant of the coating. The above two equations have been derived with the assumption that the arguments of the cylindrical functions are very small and hence the small argument approxima- tions for the. Hankel and Bessel functions are valid.

Propagution o f micron~aves on a single wire

RADIUS Of T W W 8 E IN CU

FIG. IX Percentage rcduetion in phase velocity of the wave as a function of the wire radius for different wavelengths.

s z 2 . 4 tor she Harms-Goubnu line.

The percentage of the total power flowing in the dielectric coating is then given by

[c/(P, + pa)] 7; :,- (P;/&)% - ln(o'/o)/{ln (a'/*) - E [In (0.89 y'a') + 0.51 j [I I]

where c - < ~ / E O , is the relative dielectric constant of the coating.

This percentage of the power decreases as the dielectric constant of t h e coating increases i.e., a part of the energy goes into the air medium from the dielectric medium. The radial decay factor y' increases at a lower rate for

COATING THICKNESS :

2 . 0 2.5 3.0 3.5 4 .0

DIELECTRIC CONSTANT

FIG. X Percentage of the total power Rowing in the dielccrric coating as a function of the dielectric constant, far different wire radii

ho=4.00 crns.

increasing values of E and so P~/P,% decreases. Figures X and XI show the variation of % as a function of C, for wavelengths A,, - 4.00 crns and 3.45 cms respectively. Curves are drawn for two different coating thicknesses a" - 0.01 cm and 0.03 cm.

Propagation of microwaves on a single wire

FIG. XI Percentage of the total power flowing in the dielectric coating as a function of the dielectric constant for different wireradii.

~0'3.45 CmS.

The effect of the dielectric coating thickness fl on the percentage of fill', is brought out in Figure XI1 and XI11 for two different wire radii a = 0 05 cm and 0.20 cm. The curves are drawn for ha - 4.00 crns and 3.45 crns. respectively. The slow rate of variation of this with the dielectric constant is also seen in the intermediate curves drawn for E -2.0 to 4.0. TO bring out the relation between the wavelength and the percentage of the power more explicitly, Figure XIV has been presented for three different wire radii a - 0 05 cm, 0.10 cm, and 0.20 cm.

It is observed from this family of curves that P,/P,% increases as (i) the thickness of dielectric coating increases (ii) the wire radius is made smaller and ( i i i ) the wavelength is decreased. However i t decreases with increasing valuc o f the dielectric constant.

DIELECTRIC COATING THICKNESS I N CM

Percentage of the total power flowing in the dielectric coating as a function of the coating thicknes, for difierent dielectric constants.

X~=C.OO cms.

Propagation of microwaves on a single wire

DIELECTRIC COATING THICKNESS IN CM

FIG. XI11 Percentage of the total power flowing in the dicleetric mating as D

Amctim of the coating thickncn, for different didcctric connants. ho-3.45 CmS.

DIELECTRIC COATING THICKNESS IN CM

FIG. XIV Percentage of the total power flowing in the dielectric coating as a function of the coating thickness, for different wire radii.

e=2.0.

Propfl~aiion of micro#.nses o n a single wire 139

COMPARATIVE STUDY

To enable a comparative study of the two lines, namely Sommerfeld aqd Harms-Goubau line, with particular reference to the reduction in phase velocity of the guided wave and the radial field spread, the following two tables are presented. It is seen from the respective figures, the superiority of tile Harms-Goubau line, over that of Sonmerfeld line, as a surface wave guide for single wire transmission. .

TABLE I Radial field spread for different percentages of power flow.

Wire radius for both the lines, a=0.10 cm. Dielectric constant r=2.40 and coating thickness 0"=0.01 cm. for the Harms-Ooubau line.

-- - -- - - . - Percentage Sommerield line Harms-Goubau Line

of - - - - .-- .. . - -. .. . - -. - Power flow Xa=4 cms. Xa=1.25 cms. ha=4cms. ho=1.25cms.

-- - - - ... . -- . - - - --- 90% 28.44cms. 12.19cms. 3.61cms. 1.21cms.

75:;. 11.10 ,, 5.42 ,, 1.74 ,, 0.68 ., 50% 2.30 ,, 1.44 ,, 0.68 ,, 0.34 ,. 25% 0.43 ,, 0.38 ,, 0.27 ,, 0.20 ,, 10% 0.18 ,, 0.17 ,, 0.15 ,, 0.13 ,,

- . - -- . - - . -- -

Percentage reduction in phase velocity of the surface wave. Wire radius for both the lines, a-0.10 cm.

-- . -- - ~ - -. . . - --- cp

d e lo= 4.00 C ~ S . 3.45 cms 1.25 cnm.

m 0 e,E u Harms-Goubau line 2 Harms-Goubau line 5 Harrn~Goubau line sg $ * * Y

8: g ,E . - -- . 1 8 . ~ . 8 ." r-2.0 e-4.0 - ~ ~ 2 . 0 r-4.0 gr' E-2.0 1 - 4 4

m Y1 p.-.p-.-.-.. - - - . .-

The author wishes to express his gratitude t o Sri S. K. Chatterjee for suggesting the problem and for guidance and to Dr. ( ~ r s . ) R. Chat ter jeef~~ many useful discussions. The author is also grateful to Prof. S. V. C. Aiya far his kind encouragement and for the facilities provided during the work. He is also thankful to the Ministry of Scientific and Cultural AlTairs for the award of a National Research Fellowship.

REFERENCES 1. Subrahmanyam, V. . . 3. Indian Ilirr. Sci., 1962, 44, 27. 2. Caubau, G . . . J. Appl. Phys., 1950, 21, 1 119.

3. Borgnis, F.E. and Papas, C. H. . . Handbuclr derPIrl.sik, 1958, 16, 37R.

4. Berceli, T . . . Proc. IEE, IBb l , 108 C . 386.