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EE595S: Class Lecture NotesChapter 14: Induction Motor Drives

S.D. Sudhoff

Fall 2005

Fall 2005 EE595S Electric Drive Systems 2

Overview of Strategies

• Volts-Per-Hertz Control• Constant Slip Control• Field-Oriented Control

Fall 2005 EE595S Electric Drive Systems 3

Volts-Per-Hertz Control

• History

• Advantages

• Disadvantages

Fall 2005 EE595S Electric Drive Systems 4

Volts-Per-Hertz Control

• Basic IdeaMachine will operate close to synchronous speed, thus speed controlled with frequencyWe must avoid saturation

• On Avoiding Saturation(14.2-1)(14.2-2)

asassas pirv λ+=

sesV Λ=ω

Fall 2005 EE595S Electric Drive Systems 5

Volts-Per-Hertz Control

• Elementary Volts Per Hertz Control

Fall 2005 EE595S Electric Drive Systems 6

Volts-Per-Hertz Control

• Example System50 Hp MachineRated for 1800 rpm, 460 V l-l rms, P=4, 60 Hzrs=72.5 mΩ; rr’=41.3 mΩLM=30.1 mH; Lls=Llr=1.32 mH

(14.2-3)⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+=

29.0)(1.0

bmrm

rmbL STTωωω

Fall 2005 EE595S Electric Drive Systems 7

Volts-Per-Hertz Control

• Performance Characteristics

Fall 2005 EE595S Electric Drive Systems 8

Volts-Per-Hertz Control

• Comments on Steady-State Performance

Fall 2005 EE595S Electric Drive Systems 9

Volts-Per-Hertz Control

• Problem 1: Resistance At Low Speed

• Solution: Set Voltage So Slope is Invarient

(4.9-19)

(14.2-4)2,

22,

2,

22,

estssbests

estsseestsbs

Lr

LrVV

ω

ω

+

+=

222

22

22

)()(

~2

3

rrsssrbe

rrssMbe

rs

asrbM

be

e

XsrXrXXXsrr

VsrXP

T

′+′⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡′−⎟⎟

⎠

⎞⎜⎜⎝

⎛+′

′⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

=

ωω

ωω

ωωω

Fall 2005 EE595S Electric Drive Systems 10

Volts-Per-Hertz Control

• Further Improvement – Current FeedbackLet’s approximate torque as

• (14.2-5)• (14.2-6)

Using (14.2-4) we have

• (14.2-7)

• This is not a function of synchronous speed

)( retve KT ωω −=

erre

tvTK

ωωω =∂∂

−=

)(2

3

2222

22

ssbsr

brMtv

Lrr

VrLP

Kω+′

′⎟⎠⎞

⎜⎝⎛

=

Fall 2005 EE595S Electric Drive Systems 11

Volts-Per-Hertz Control

Next, consider torque. We have• (14.2-8)

For steady-state conditions• (14.2-9)

• (14.2-10)

Thus• (14.2-11)

Where• (14.2-12)

)(22

3 eds

eqs

eqs

edse iiPT λλ −=

e

eqss

eqse

dsirv

ωλ

−=

e

edss

edse

qsirv

ωλ

−−=

)2(122

3 2*ss

eqs

eqs

ee IrivPT −=

ω

222

1 eds

eqss iiI +=

Fall 2005 EE595S Electric Drive Systems 12

Volts-Per-Hertz Control

Equating (14.2-7) and (14.2-11)

• (14.2-13)

Or• (14.2-14)

Where• (14.2-15)

• (14.2-16)

2/)2(3 2*2**

tvsseqs

eqsrr

eKIrivP −++

=ωω

ω

2),0max( 2**

corrrre

Χ++=

ωωω

corrcorr sH LPF χ)(=Χ

tvsseqs

eqscorr KIrivP /)2(3 2* −=χ

Fall 2005 EE595S Electric Drive Systems 13

Volts-Per-Hertz Control

Fall 2005 EE595S Electric Drive Systems 14

Volts-Per-Hertz Control

• Performance of Compensated Volts-Per-Hertz Control

Fall 2005 EE595S Electric Drive Systems 15

Volts-Per-Hertz Control

• Start-Up Performance of Compensated Volts Per Hertz Control

Fall 2005 EE595S Electric Drive Systems 16

Overview of Strategies

• Volts-Per-Hertz Control• Constant Slip Control• Field-Oriented Control

Fall 2005 EE595S Electric Drive Systems 17

Constant Slip Control

• Basic IdeaDefinition of Slip Frequency

(14.3-1)Our Strategy

res ωωω −=

Fall 2005 EE595S Electric Drive Systems 18

Derivation of Constant Slip Control

• Consider Our Expression for Electromagnetic Torque

• (14.3-2)

• Solving for the Current Yields

• (14.3-3)

• The Plan

22

22

)()(2

3

rrsr

rsMse

Lr

rILP

T′+′

′⎟⎠⎞

⎜⎝⎛

=ω

ω

estrrestMs

estrrsestres

rLP

LrTI

,2

,

2,

2,

*

||3

))((||2

′

′+′=

ω

ω

Fall 2005 EE595S Electric Drive Systems 19

Avoiding Saturation

• Before Proceeding …

• Consider the Steady-State Equivalent Circuit

(14.3-4)(14.3-5)

Thus(14.3-7)

)~~(~~arasMarlrar IILIL ′++′=λ

srLjLjII

rrreMe

asar /~~

′+′−=′

ωω

222'

rrrs

rMsr

rL

rLI′+′

′=

ωλ

Fall 2005 EE595S Electric Drive Systems 20

Avoiding Saturation

Taking the Magnitude(14.3-7)

Combing (14.3-7) and (14.3-2)

(14.3-8)Thus

(14.3-9)So For Large Torque Commands

(14.3-10)

222'

rrrs

rMsr

rL

rLI′+′

′=

ωλ

rrs

rePT ′=

2'

23 λω

estr

rsetsthreshe r

PT,

2max,,

, 23

′=

λω

2max,

,*

3

2

r

estres

P

rT

λω

′=

Fall 2005 EE595S Electric Drive Systems 21

Constant Slip Control

Fall 2005 EE595S Electric Drive Systems 22

Selection of Slip Frequency

• Control for Maximum Torque Per AmpFrom (14.3-1)

(14.3-11)

Optimizing with Respect to Slip Frequency

(14.3-12)

2,

2

2,

2 )()(2

3

rsetsr

rMsets

s

eLr

rLP

IT

′+′

′⎟⎠⎞

⎜⎝⎛

=ω

ω

estrr

estrsets L

r

,

,, ′

′=ω

Fall 2005 EE595S Electric Drive Systems 23

Maximum Torque Per Amp Control

Fall 2005 EE595S Electric Drive Systems 24

Maximum Efficiency Control

• To Optimize Efficiency Note That(14.3-13)

So(14.3-14)

Comparison to (14.3-14) to (14.3-2)(14.3-15)

Now (14.3-16)

)~Re(3 assin VIP =

22

222

)(33

rrsr

rsMesssin

LrrLIIrP

′+′

′+=

ω

ωω

eessin TP

IrP ω23 2 +=

erout TP

P ω2=

Fall 2005 EE595S Electric Drive Systems 25

Maximum Efficiency Control

So(14.3-17)

Finally

(14.3-18)

Minimizing Yields(14.3-19)

sessoutin TP

IrPP ω23 2 +=−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

′+

′= s

mr

rrss

ms

sreloss

LrLr

LrrT

PP ωω

ω 2

2

22

1

1

,

,2

,

2,,

,,

+′′

′

′=

estr

ests

estrr

estmestrr

estrsets

rr

L

LLr

ω

Fall 2005 EE595S Electric Drive Systems 26

Maximum Efficiency Control

Fall 2005 EE595S Electric Drive Systems 27

Comparison of Constant Slip Controls

• Maximum Efficiency

• Maximum Torque Per Amp

Fall 2005 EE595S Electric Drive Systems 28

Speed Control with Constant Slip Control

Fall 2005 EE595S Electric Drive Systems 29

Transient Performance of Constant Slip Controlled Drive

Fall 2005 EE595S Electric Drive Systems 30

Overview of Strategies

• Volts-Per-Hertz Control• Constant Slip Control• Field-Oriented Control

Fall 2005 EE595S Electric Drive Systems 31

14.4 Field-Oriented Control

• Objective

• Challenges

Fall 2005 EE595S Electric Drive Systems 32

The Basic Idea

θsin2BiNLrTe −=

Fall 2005 EE595S Electric Drive Systems 33

Torque Production in Induction Motor

• Now(14.4-2)

(14.4-3)

)(22

3qrdrdrqre iiPT ′′−′′= λλ

θλ sin||||22

3qdrqdre iPT ′′−=

Fall 2005 EE595S Electric Drive Systems 34

The Plan

• Objective

• ObservationIn Steady-State, This is Always True

• (14.4-4)

• (14.4-5)

• (14.4-6)

Thus

edrre

reqr ri λωω )(1

−′

−=

eqrre

redr ri λωω )(1

−′

=

edr

edr

eqr

eqr

eqdr

eqdr iii ′′+′′=′⋅′ λλλ

Fall 2005 EE595S Electric Drive Systems 35

The Plan (Continued)

• But.. We need flux and current to be perpendicular all the time. To do this ..

(14.4-7)(14.4-8)

• Achieving (14.4-7)By choice of θe

0=′eqrλ

0=′edri

Fall 2005 EE595S Electric Drive Systems 36

The Plan (Continued)

• Achieving (14.4-8)All we need to do is to keep d-axis stator current constant. Proof:

• Consider(14.4-9)

• The q-axis rotor flux linkage is zero (by choice of reference frame)

(14.4-10)• Substitution of d-axis rotor flux linkage equation

into (14.4-10)(14.4-11)

• Thus the d-axis rotor current goes to zero

edr

eqrre

edrr pir λλωω ′+′−+′′= )(0

edr

edrr pir λ′+′′=0

eds

rrMe

drrrre

dr piLLi

Lrip −′′

−=′

Fall 2005 EE595S Electric Drive Systems 37

Some Observations

• Since the d-axis rotor current is zero

(14.4-12)

(14.4-13)

• Combining these results with our flux linkage equation

(14.4-14)

edsss

eds iL=λ

edsM

edr iL=′λ

eqr

edre iPT ′′−= λ

223

Fall 2005 EE595S Electric Drive Systems 38

Some Observations

• Since the q-axis rotor flux linkage is zero

(14.4-15)

• Thus

(14.4-16)

eqs

rrMe

qr iLLi −=′

eqs

edrrr

Me i

LLPT λ′=

223

Fall 2005 EE595S Electric Drive Systems 39

Generic Rotor Flux-Oriented Control

Note: θe must be selected so as to keep q-axis rotor flux equal to zero.

Fall 2005 EE595S Electric Drive Systems 40

Some “Minor” Details

• Determination of position of synchronous reference frame

• Determination of the rotor flux

Fall 2005 EE595S Electric Drive Systems 41

Aside Types of Field Oriented Control

• OrientationsStatorMagnetizing (Air-Gap)Rotor

• MethodsDirectIndirect

• ModificationsRobust

Fall 2005 EE595S Electric Drive Systems 42

Direct Field Oriented Control

• Basic IdeaMeasure the flux directly (more or less)

• Consider

(14.5-1)

• We need to set q-axis rotor flux to zero. Thus

(14.5-2)

⎥⎥⎦

⎤

⎢⎢⎣

⎡

′

′⎥⎦

⎤⎢⎣

⎡ −=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

′

′sdr

sqr

ee

eee

dr

eqr

λ

λθθθθ

λ

λcossinsincos

2)( πλλθ +′−′= s

drsqre jangle

Fall 2005 EE595S Electric Drive Systems 43

Direct Field-Oriented Control

• As a Side Effect

(14.5-3)

• The problem: We can measure the rotor flux linkages. We can measure the stator flux linkages.

22 )()( sdr

sqr

edr λλλ ′+′=′

Fall 2005 EE595S Electric Drive Systems 44

Estimation of Rotor Flux Linkages

• Consider(14.5-4)

• Thus(14.5-5)

• Recall(14.5-6)

)( sqr

sqsM

sqm iiL ′+=′λ

M

sqsM

sqms

qr LiL

i−′

=′λ

)( sqr

sqsM

sqrlr

sqr iiLiL ′++′=′λ

Fall 2005 EE595S Electric Drive Systems 45

Estimation of Rotor Flux Linkages

• So(14.5-7)

(14.5-8)

sqslr

sqm

Mrrs

qr iLLL ′−′

=′ λλ

sdslr

sdmM

rrsdr iL

LL ′−′

=′ λλ

Fall 2005 EE595S Electric Drive Systems 46

Rotor Flux Estimator

Fall 2005 EE595S Electric Drive Systems 47

Direct Rotor Field-Oriented Control

Fall 2005 EE595S Electric Drive Systems 48

Robust Direct Field-Oriented Control

• Sensitivity of Rotor Flux Estimator(14.6-1)

• Sensitivity of Q-Axis Current Calculation

(14.6-2)

sqdsestlr

smqdestM

estrreqdr iL

LL

,,,

,ˆ ′−′

=′ λλ

edrestrr

estMee

qs

LLP

Tiλ′

=ˆ

223

,

,

**

Fall 2005 EE595S Electric Drive Systems 49

Robust Direct Field-Oriented Control

• Sensitivity of the D-Axis Current Injection

(14.6-3)estM

edre

ds Li

,

** λ=

Fall 2005 EE595S Electric Drive Systems 50

Flux Control Loop

• Since we have a flux estimator, consider

• It can be shown that

(14.6-5)1

1,*

+

+=

′

′

sL

Ls

M

estMedr

edr

λ

λ

τ

τ

λ

λ

Fall 2005 EE595S Electric Drive Systems 51

Torque Control Loop

• Torque EstimationRecall

(14.6-6)Now

(14.6-7)So

(14.6-8)Which Suggests

(14.6-9)

)(22

3 sds

sqs

sqs

sdse iiPT λλ −=

sqdm

sqdsls

sqds iL λλ +=

)(22

3 sds

sqm

sqs

sdme iiPT λλ −=

)(22

3ˆ sds

sqm

sqs

sdme iiPT λλ −=

Fall 2005 EE595S Electric Drive Systems 52

Torque Control Loop

• Now that we can estimate torque, how about

Fall 2005 EE595S Electric Drive Systems 53

Torque Control Loop

• Analysis of Torque Control LoopDefine

(14.6-10)(14.6-12)

Now, assuming the control works(14.6-11)

This, assumption, coupled with our torque control loop yields

(14.6-13)

*,

,, 22

3 edrestrr

estMestt L

LPK λ′′

=

edrrr

Mt L

LPK λ′′

=22

3

*eqste iKT =

1

1,*

+

+=

sK

Ks

TT

t

esttt

t

e

e

τ

τ

Fall 2005 EE595S Electric Drive Systems 54

Robust Direct Field-Oriented Control

Fall 2005 EE595S Electric Drive Systems 55

Performance of Robust Direct Field Oriented Control

Fall 2005 EE595S Electric Drive Systems 56

Performance of Constant Slip Drive

Fall 2005 EE595S Electric Drive Systems 57

Performance of Volts-Per-Hertz Drive

Fall 2005 EE595S Electric Drive Systems 58

Indirect Field Oriented Control

• Consider (14.7-1)

Since the q-axis rotor flux is zero

(14.7-2)

Which may be expressed

(14.7-3)

eqr

edrre

eqrrr pir λλωω ′+′−+′′= )(0

edr

eqr

rrei

rλ

ωω′

′′−=

ede

eqs

rrr

rei

iLr′′

+= ωω

Fall 2005 EE595S Electric Drive Systems 59

Indirect Field Oriented Control

• ThoughtWe know what the electrical frequency will be if we have field orientation (i.e. field orientation causes (14.7-1))Does it work backward ? Will using (14.7-1) cause field-orientation ?The answer: Yes !

Fall 2005 EE595S Electric Drive Systems 60

Indirect Field Orientation

• ProofConsider the rotor voltage equations

(14.7-5)(14.7-6)

Now substitute in our expression for electrical frequency

(14.7-7)

(14.7-8)

eqr

edrre

eqrr pir λλωω ′+′−+′′= )(0

edr

eqrre

edrr pir λλωω ′+′−−′′= )(0

eqr

edre

de

eqs

rrre

qrr pi

iLrir λλ ′+′′′

+′′= *

*0

edr

eqre

de

eqs

rrre

drr pi

iLrir λλ ′+′′′

−′′= *

*0

Fall 2005 EE595S Electric Drive Systems 61

Indirect Field Orientation

• Continuing On …Now put in terms of q-axis rotor flux and d-axis rotor current (and stator currents)This yields

(14.7-9)

(14.7-10)

[ ] eqr

edsM

edrrre

ds

eqs

rr

r

rr

eqsM

eqr

r piLiLi

iLr

LiL

r λλ

′++′′′′

+⎥⎥⎦

⎤

⎢⎢⎣

⎡

′

−′′= *

*

**0

[ ]**

*0 e

dsMe

drrreqre

ds

eqs

rr

redrr iLiLp

i

iLrir +′′+′′′

−′′= λ

Fall 2005 EE595S Electric Drive Systems 62

Indirect Field Orientation

• Now, keeping the d-axis stator current fixed we have

(14.7-11)

(14.7-12)

• Comments

edre

ds

eqs

reqr

rrre

qr ii

ir

Lrp ′′−′′′

−=′*

*λλ

eqre

ds

eqs

rr

redrrr

redr i

i

Lri

Lrip λ′

′

′+′

′′

−=′*

*

2)(

Fall 2005 EE595S Electric Drive Systems 63

Indirect Field Oriented Control

• Thus, our control becomes

Fall 2005 EE595S Electric Drive Systems 64

Indirect Field Oriented Control

• Advantages of Indirect Field Oriented Control over Direct Field Oriented Control

• Disadvantages of Indirect Field Oriented Control over Direct Field Oriented Control

Fall 2005 EE595S Electric Drive Systems 65

Performance of DetunedIndirect Field Oriented Control

• Overestimate magnetizing inductance by 25% (we started to saturate machine)

• Underestimate rotor resistance by 25% (rotor resistance increased with temperature)

Fall 2005 EE595S Electric Drive Systems 66

Performance of Detuned Field Oriented Control