Lecture Set 8 - Purdue University

Lecture Set 8:Induction Machines

S.D. SudhoffSpring 2021

2

About this Lecture Set

• Reading–Electromechanical Motion Devices, 2nd Edition,

Sections 6.1-6.9, 6.12

• Goal–Understand the theory, operation, and modeling of

induction machines

Lecture 68

Preliminaries

3

4

Sample Applications• Low Power:–Shaded pole machines (small fans)–Permanent Split Capacitor Machines

(Residential HVAC)• Medium Power:– Industrial HVAC–Pumps–Vehicle Propulsion–Drive Applications

• High Power:–Ship, Train Propulsion–Wind Power Generation

5

Characteristics

• The Good

• The Bad

• The Ugly

6

Types of IM Machines

• Wound Rotor

• Squirrel Cage

• Solid Rotor

1.5 MW Induction Generator

7

8

150 Hp @ 1800 RPM

9

5 Hp @ 1800 RPM

ECE321 10

1 Hp @ 1800 RPM (Stator)

11

1 Hp @ 1800 RPM (Rotor)

Lecture 69

Theory of Operation: Stator MMF

12

13

Two-Phase IM

14

Stator MMF

15

Stator MMF

16

Stator MMF

17

Stator MMF

18

Stator MMF

19

Stator MMF

Lecture 70

Theory of Operation: Rotor MMF

20

21

Rotor MMF

22

Rotor MMF

23

Rotor MMF

24

Rotor MMF

25

Rotor MMF

26

Rotor MMF

27

Comparison of MMFs

• As viewed from stator

• As viewed from rotor

28

IM Torque Production

Lecture 71

Overview of Machine Models

29

30

Our Analysis

• Machine Variable Model

• Referred Machine Variable Model

• QD Model

• Steady-State Model

31

Machine Variable Model

• Voltage equations

• Flux Linkage Equations

• Inductance Matrices

• Torque equation

abs ( )

abs s srT

abr sr r abr

L L iL L i

λλ

00

lr mrr

lr mr

L LL L

L

cos sinsin cos

r rsr sr

r r

L

L

absabssabs pλ irv

abr r abr abrp v r i λ

as

[( )sin2

(i )cos ]

e sr as ar bs br r

br bs ar r

PT L i i i i

i i i

00

ls mss

ls ms

L LL L

L

32

Referred Machine Variable Model

• Referral of variables

• Voltage equations

• Flux linkage equations

• Torque equation

rr

sr N

N rr2

abr

abs

rT

sr

srs

abr

abs ii

LLLL

)(λλ

00

lr msr

lr ms

L LL L

L

rr

rrmssr

rs

sr LNN

cossinsincos

LL

abrsr

abr NN ii

abrrs

abr NN vv

abrrs

abr NN λλ

absabssabs pλ irv

abrabrrabr pλ irv

]cos)sin)[(

2rarbsbras

rbrbsarasmseiii(i

iiiiLPT

QD Model

33

34

Phasor Model

• Torque: ]~~Re[2

2 *arasmse IIjLPT

Lecture 72

Machine Variable Model:Voltage and Flux Linkage Equations

35

36

Voltage Equations

dtdirv as

assas

dtdirv bs

bssbs

dtdirv ar

arrar

dtd

irv brbrrbr

abrabrrabr

absabssabs

ppλλ

irvirv

][)( bsasT

abs f ff

][)( brarT

abr f ff

37

Flux Linkage Equations – Scalar Form

brasbrarasarbsasbsasasasas iLiLiLiL

brbsbrarbsarbsbsbsasbsasbs iLiLiLiL

brarbrarararbsarbsasarasar iLiLiLiL

brbrbrarbrarbsbrbsasbrasbr iLiLiLiL

38

Flux Linkage Equations – Matrix Form

abrsrabssabs iLiL λ

abrrabsT

srabr iLiL )(λ

abr

abs

rT

sr

srs

abr

abs

ii

LLLL

)(λ

λ

39

Derivation of Magnetizing Inductances

40


41


42


43


44

Flux Linkage Equation Summary

IL ssss

sss L

LL

00

IL rrrr

rrr L

LL

00

rr

rrsrsr L

cossinsincos

L

mslsss LLL

mrlrrr LLL

m

sms

NLR

2

m

rmr

NLR

2

m

srsr

NNL

R

Lecture 73

Machine Variable Model:Torque Equation

45

46

Starting Point

abr

abs

rT

sr

srs

abr

abs

ii

LLLL

)(λ

λ

47

Derivation of Torque Equation

48

Derivation of Torque Equation

Lecture 74

Referred Machine Variable Model:Flux Linkage Equations

49

50

Why

51

Starting Point

abr

abs

rT

sr

srs

abr

abs

ii

LLLL

)(λ

λ

IL ssss

sss L

LL

00

IL rrrr

rrr L

LL

00

rr

rrsrsr L

cossinsincos

L

52

Referral of Flux Linkage Equations

53

Referral of Flux Linkage Equations

54

Summary

abr

abs

rT

sr

srs

abr

abs

ii

LLLL

)(λ

λ

rr

rrr

r

sr L

LNN

002

LL

mslrmrr

slrrr LLL

NNLL

2

rr

rrmssr

r

ssr L

NN

cossinsincos

LL

Lecture 75

Referred Machine Variable Model:Voltage and Torque Equations

55

56

Referral of Voltage Equations

abrabrrabr

absabssabs

ppλλ

irvirv

57

Referral of Voltage Equations

• Finally, we get

• Whereabrabrrabr

absabssabs

ppλλ

irvirv

rr

sr N

N rr2

58

Referral of Torque Equation

• It can be shown that

]cos)(sin)[(2 rarbsbrasrbrbsarasmse iiiiiiiiLPT

Lecture 76

The Stationary Reference Frame

59

60

Stationary Reference Frame

• Stator transformation

• Rotor transformation

bs

ass

ds

sqs

ff

f

f10

01absss

sqds fKf

sqds

ssabs fKf 1)(

br

ar

rr

rrs

dr

sqr

ff

f

f

cossinsincos

abrsr

sqdr fKf

sqdr

srabr fKf 1)(

61

Stationary Reference Frame

62

Geometrical Interpretation

Lecture 77

Transformation of the Voltage Equations to the Stationary Reference Frame

63

64

Transformation of Voltage Equations

65


66


67


68

Summary

sqds

sqdss

sqds pλ irv

sqdr

sdqrr

sqdrr

sqdr pλλ irv

[ ]s s s Tdqr dr qr λ

Lecture 78

Transformation of the Flux Linkage Equations to the Stationary Reference Frame

69

70

Transformation of Flux Linkage Equations

abrsrabssabs iLiL λ

abrrrabsabr iLiL T'sr λ

71


72


73


74


sqdrms

sqdsss

sqds LL ii λ

mslrrr

mslsss

sqdrrr

sqdsms

sqdr

LLL

LLL

LL

''

' iiλ

Lecture 79

Transformation of the Torque Equation to the Stationary Reference Frame

75

76

Transformation of the Torque Equations

sin coscos sin2

r rTe sr abs abr

r r

PT L

i i

77


78


79


)(2

sqr

sds

sdr

sqsmse iiiiLPT

'( )2

s s s se qr dr dr qr

PT i i

)(2

sds

sqs

sqs

sdse iiPT

Lecture 80

QD Equivalent Circuit

80

81


82


83


84

Comments on QD Machine

• When is it valid ?

• What is it used for ?

Lecture 81

Balanced Steady-State Operation and the Phasor Model

85

86

Stator Side

• Steady-state forms

• We can show

2 cos[ (0)]as s e esfF F t

2 sin[ (0)]bs s e esfF F t

)0(~~ esfjsas

sqs eFFF

)(~~ )0(esfjsbs

sds ejFFF

s

dss

qs FjF ~~

Stator Side

87

Stator Side

88

89

Rotor Side

• Rotor relationships

• We can show

2 cos[( ) (0)]ar r e r erfF F t

2 sin[( ) (0)]br r e r erfF F t

sdr

sqr FjF ~~

ars

qr FF ~~

brs

dr FF ~~

90

Rotor Side

91

Rotor Side

92

Rotor Side

Lecture 82

Balanced Steady-State Operation Phasor Equivalent Circuit

93

94

Phasor Equivalent Circuit (2-Phase)

• Torque: ]~~Re[2

2 *arasmse IIjLPT

95


• Voltage Equations

• Slip

• Torque

)~~(~)(~arasmseaslsesas IILjILjrV

)~~(~~

'arasmsearlre

rar IILjILjsr

sV

e

res

]~~Re[2

2 *arasmse IIjLPT

96

Derivation

97

Derivation

98

Derivation

99

Derivation

100

Derivation

101

Chief Problems with Model

• Magnetizing Saturation

• Leakage Saturation

• Distributed System Effects

• Thermal Effects

Lecture 83

Balanced Steady-State Operation of 3-Phase Machines

102

103


• Where

msMarasMe LLIIjLPT23]~~Re[

23 *

104

Delta Connected Machine

105

Wye Connected Machine

Lecture 84

Analyzing Machine Operation

106

107

Typical Operating Situations

• Utility grid– Fixed voltage, fixed frequency voltage source

• Inverter (power electronic control)– Voltage source based inverter (variable voltage, variable

frequency)• Volts-per-hertz control

– Current source based inverter (variable current, variable frequency)• Maximum torque per amp control• Field-oriented control• Direct torque control

108

Derivation of Rotor Current

109

Torque for a Given Stator Current

• We can show

22

22

)()(2

rrsr

rsMs

eLr

rILPNT

110


111


112


Lecture 85

Operation From A Current Source

113

114

Operation from Current Source

• Machine Parameters• 460 V, l-l, rms; 50 Hp @ 1800 rpm– Stator resistance: 72.5 m– Rotor resistance: 41.3 m– Stator and referred rotor leakage: 1.32 mH–Magnetizing Inductance: 30.1 mH

• Operating Conditions– Armature current: 50 A, rms– Frequency: 60 Hz

115

Current Source Torque - Speed

0 100 200 300 4000

50

100

150

200

250216.266

1.508

T e ri

376.9920 ri

116

Maximum Torque Per Amp Control

• Control Summary

• To show this, start with

rrMs

rrsres

rLNPLrT

I

2

22*

||))((||2

rr

rs L

r

22

22

)()(2

rrsr

rsMs

eLr

rILPNT

117


118


119


120

MTPA Control Example

• Consider the 50 Hp example• Suppose we want 200 Nm at a speed of 1000 rpm• Compute the magnitude of the a-phase current, A• Compute the required voltage• Compute the efficiency

121


122


123


Lecture 86

Operation From A Voltage Source

124

125

Operation From Voltage Source: Stator Current

126

Example 1

• Consider our 50 Hp machine• Fed from 460 V l-l rms source• Load torque of 200 Nm• Objective: Compute speed and current

127

Steady-State Operating Point

0 100 200 300 4000

100

200

300

400

500493.596

0.499

T e V s e ri

T L

376.9840 ri ri

128

Steady-State Operating Point

129

Example 2

• Consider our 50 Hp machine• Fed from 460 V l-l rms, 60 Hz source• Let’s look at machine properties versus speed

130

Current vs Speed

0 100 200 300 4000

50

100

150

200

250

300271.655

22.45

I as V s e ri r rs

I as V s e ri r r0

I as V s e ri r rb

376.6150 ri

131

Torque vs Speed

0 100 200 300 4000

100

200

300

400

500493.595

4.981

T e V s e ri r rs

T e V s e ri r r0

T e V s e ri r rb

376.6150 ri

132

Efficiency vs Speed

0 100 200 300 4000

0.2

0.4

0.6

0.8

0.985

0

V s e ri r rs

V s e ri r r0

V s e ri r rb

4000 ri

133

Efficiency vs Output Power

0 1 104 2 104 3 104 4 104 5 104 6 104 7 104 8 104 9 104 1 1050

0.2

0.4

0.6

0.8

0.985

0

V s e ri r rs

V s e ri r r0

V s e ri r rb

9.216 1040 P out V s e ri r rs P out V s e ri

r r0 P out V s e ri r rb

Lecture 87

Parameter Identification

134

135


• DC Test

136


• Blocked Rotor Test

137


• Blocked Rotor Test

138


• No Load Test

Lecture 87

Volts Per Hertz Control

139

140


• The idea:

141


• The idea:

142


0 100 200 300 4000

100

200

300

400

500493.583

0

T e eb ri

T e eb 0.75 ri

T e eb 0.5 ri

T e eb 0.25 ri

376.6150 ri

Lecture Set 8 - Purdue University

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