1

Electric Machines and Drives

Leila Parsa Rensselaer Polytechnic Institute

2

Reference Frame Theory

Introduced by R.H. Park in 1929 to model synchronous machines

3

Three-Phase Transformationto the Arbitrary Reference Frame

abcsssqd fKf 0

cs

bs

as

abcs

fff

f0

0

qs

qd s ds

s

ff f

f

f = voltage, current, or flux linkage

a = a-phaseb = b-phasec = c-phase

q = q-axis (quadrature axis)d = d-axis (direct axis)0 = zero sequence

4

The Reference Frame Transformation

21

21

21

32sin

32sinsin

32cos

32coscos

32

sK

= reference frame speed (rad/sec)= reference frame position (rad)

5

13

2sin3

2cos

13

2sin3

2cos

1sincos1

sK

sqdsabcs fKf 01

The Inverse Transformation

6

2 cosas s e vv V

22 cos3bs s e vv V

22 cos3cs s e vv V

f

ee

sVv

- electric frequency (Hz)- electric radian frequency (rad/sec)- electrical position (rad)- rms Voltage (V)- phase shift (rad)

e e t

2e f

Example: Three-Phase Set of Voltages

7

0qd s s abcsv K v

0

2 2cos cos cos2 cos3 3

2 2 2 2sin sin sin 2 cos3 3 3 3

1 1 1 22 cos2 2 2 3

s e vqs

ds s e v

s

s e v

Vvv Vv

V

Transform to the Arbitrary Reference Frame

8

2 2 2 23 3 3 3

3cos cos cos cos cos cos cos2

x y x y x y x y

using the identity,

vesqs Vv cos2

Voltages in Arbitrary Reference Frame

2 2 2 23 3 3 3

2 2 cos cos cos cos cos cos3qs s e v e v e vv V

q-axis voltage

2 2 2 23 3 3 3

3sin cos sin cos sin cos sin2

x y x y x y x y

vesds Vv sin2

2 2 2 23 3 3 3

2 2 sin cos sin cos sin cos3ds s e v e v e vv V

d-axis voltage

zero sequence voltage

using the identity,

2 20 3 3

1 2 cos cos cos 03s s e v e v e vv V

9

208 V, 3-phase, f = 60 Hz (208 V line-to-line rms)

208 V 120 V3sV 0v

1. Stationary reference frame 0

2 cossqs s ev V

2 sinsds s ev V

2. Synchronous reference frame e

svseqs VVv 2cos2

0sin2 vseds Vv

Numerical Example

10

Axis Sketch with =0

essqs Vv cos2

essds Vv sin2

11

Axis Sketch with =e

2 cos 170Veqs s vv V

0sin2 vseds Vv

12

=e and v=30o

2 cos 147 Veqs s vv V

2 sin 85Veds s vv V

13

Arbitrary

Stationary

Synchronous

Rotor

0

e

r

sdsqs fff 0,,

ss

dss

qs fff 0,,

se

dse

qs fff 0,,

sr

dsr

qs fff 0,,

sK

rsK

ssK

esK

csbsass ffff 31

0

abcsssqd vKv 0

abcses

esqd vKv 0

compact notation Notes: In all reference frames

Commonly Used Reference Frames

14

abcsTabcs

cs

bs

as

csbsascscsbsbsasasin iviii

vvvivivivP

1 1 1 10 0 0 0

T TTin s qd s s qd s qd s s s qd sP K v K i v K K i

1 1

3 0 02

30 02

0 0 3

T

s sK K

ssdsdsqsqsin ivivivP 00323

Real Power in the q-d Reference Frame

15

Reference Frame Transformation

Developed by R.H. Park in 1929 for analysis of synchronous machines.

Allows treatment of balanced three-phase ac systems as two-phase dc systems.• This leads to application of classical control theory• Also simplifies control equations of some systems

16

asasas pRiv

asas Li

bsbsbs pRiv

bsbs Li

cscscs pRiv

cscs Li

Transforming Circuit Elements: R-L Example

voltage equations

flux linkage equations

note:dpdt

17

abcsabcs Li

abcssabcssabcss iLKLiKK

sqdsqd Li 00

1. Compress equations

Transform Flux Linkage Equations

qsqs Li

dsds Li

ss Li00

2. Transform equations

3. Expand equations

18

abcsabcsabcs pRiv

s abcs s abcs s abcsK v K Ri K p

}{ 01

00 sqdsssqdsqd KpKRiv

sqdsssqdsssqd KpKpKKRi 01

01

0 }{

Transform Voltage Equations

1. Compress equations

2. Transform equations

IKK ss 1

19

03

2cos3

2sin

03

2cos3

2sin

0cossin

}{ 1

sKp

0000000

}{ 1

ss KpK

dqssqdsqdsqd pRiv 000

0

ds

dqs qs

3. Expand equations

dsqsqsqs pRiv

qsdsdsds pRiv

sss pRiv 000

20

qsqs Li dsds Li 0 0s sLi

dsqsqsqs pRiv qsdsdsds pRiv sss pRiv 000

Equivalent Circuit

q-axis d-axis zero sequence

21

abcsabcsabcs pRiv

dqssqdsqdsqd pRiv 000

0qs

ds

dqs

Coupled Inductors

voltage equations

22

cs

bs

as

mslsmsms

msmslsms

msmsmsls

cs

bs

as

iii

LLLL

LLLL

LLLL

21

21

21

21

21

21

flux linkage equations

Coupled Inductors

abcssabcs iL

abcsssabcss iKK L

sqdssssqd iKK 01

0 L

23

ls

msls

msls

sss

L

LL

LL

KK

00

0230

0023

1L

qsmslsqs iLL

23

dsmslsds iLL

23

0 0s ls sL i

expanded form

24

Transformation of Circuit Elements

Balanced three-phase ac circuits transformed to two-phase dc circuits (neglecting the zero sequence and assuming analysis in the synchronous reference frame)

Flux linkage equations for inductive circuits were used for generality to other circuits; including electric machines

Coupling terms between the q- and d-axes result from the transformation. These will later be viewed as back-emf terms when observing electric machinery in the synchronous reference frame.

25

vjas sV V e

~

as s vV V

23 vj j

bs sV V e

23 vj j

cs sV V e

Balanced Steady-State Voltages

2 cosas s e vv V t

22 cos3bs s e vv V t

22 cos3cs s e vv V t

26

2 coseqs s vV V

2 sineds s vV V

2 22 e e

s qs dsV V V

1tane

dsv e

qs

VV

Synchronous: = e

note:

Synchronous Reference Frame q-d Voltages

27

edse

eqs

eqs LIRIV

eqse

eds

eds LIRIV

Steady-State Calculation (=e)

steady-state equations

28

e

ds

eqs

e

e

eeds

eqs

VV

RLLR

LRII

222

1

solve for currents

eds

eqs

e

ee

ds

eqs

II

RLLR

VV

steady-state equations in matrix form

29

Steady-State q-d Calculations

In the synchronous reference frame, the q-d circuits are supplied from dc and the corresponding dc solution is steady-state (inductors treated as short-circuit, capacitors treated as open-circuit)

Can be used to analyze steady-state operation of electric machines

Equations can be linearized about the dc operating point for application of control theory

30

PMSM Rotor and Stator

31

Rotor and Stator Designs

32

Some PMSM Applications

Electric bike Electric airplane

Remote operated vehicle All-terrain vehicle

Permanent magnet AC motors (PMAC) :• quasi-rectangular fed BLDC Motor• Sinusoidally fed PMSM

BLDC motors:• Two phases are conducting at each instant of time• A low resolution position sensor (Hall Sensor) is enough for providingcommutation instants• Higher torque density compared to PMSM at low and medium speeds• Commutation torque pulsation• The performance deteriorates at high speeds

PMSM :• Position information is needed at each instant of time• More advanced control techniques with faster transient response is applicable to this kind of drive over the whole speed range.

Permanent Magnet Motor Drives

BLDC Motors

BLDC Current Waveform

Back-EMF and phase current

BLDC Motors

c

b

a

c

b

a

c

b

a

c

b

a

eee

iii

dtd

MLML

ML

iii

RR

R

vvv

0 00 00 0

0 00 00 0

)(1ccbbaa

re ieieieT

rr

Le Bdt

dJTT

Equations governing three phase BLDC motor:

where ea , eb , and ec are trapezoidal back-EMFs.

The electromagnetic torque is expressed as:

And, the interaction of Te with the load torque determines how the motor speed builds up:

High Speed Operation of BLDC Motors

• The BLDC operation above rated speed is being performed by advanceangle technique• At a given torque or a given speed, it is difficult to find the exact advance-angle to be applied.• At high speeds the stator winding inductance's can cause the phase currentto deviate significantly from the ideal rectangular waveform which will reducethe torque production at high speeds

37

Permanent Magnet Synchronous Machines (PMSMs)

assume: 1. Round rotor 2. Sinusoidal flux linkages

msmdmq LLL23

motor construction

winding connection

38

mslsasas LLL

lsL

msL

o 1cos 1202asbs ms msL L L

mslssbsbsasas LLLLL csc

mscsbscsasbscsbsasascsasbs LLLLLLL21

- stator leakage inductance (H)

- stator magnetizing inductance (H)

Inductance Termsconsider a current in the a-phase winding

self inductance Lasas

mutual inductance Lasbs (determines the component of as due to ibs)

remaining inductance terms

39

rmasm ' sin

32sin rmbsm '

32sin rmcsm '

Magnet Fluxmagnet flux linking the a-phase

magnet flux linking the b- and c-phase

40

asassas pirv

bsbssbs pirv

cscsscs pirv

PMSM Equationsmachine coil variables

flux linkages

1 1

sin2 21 1 2sin2 2 31 1 2sin2 2 3

as asls ms ms msr

bs ms ls ms ms bs m r

ms ms ls ms rcs cs

iL L L L

L L L L i '

L L L L i

41

Reduced Flux Linkage Equations

1 1 sin2 2

3 1 sin2 2

as ls ms as ms bs ms cs m r

ls ms as ms as bs cs m r

L L i L i L i '

L L i L i i i '

for a wye connection,ias + ibs + ics = 0

sinas s as m rL i '

where Ls = Lls + (3/2)Lms

cosas s as s as r m r

s as s as as

v r i L pi 'r i L pi e

eas - a-phase back-emf

42

asassas pirv

bsbssbs pirv

cscsscs pirv

r r r rqs s qs r ds qsv r i p

r r r rds s ds r qs dsv r i p

0 0 0s s s sv r i p

Coil Voltage Equationsmachine variables

rotor reference frame, q-d variables

43

1 1

sin2 21 1 2sin2 2 31 1 2sin2 2 3

as asls ms ms msr

bs ms ls ms ms bs m r

ms ms ls ms rcs cs

iL L L L

L L L L i

L L L L i

abcmabcssabcs i L

Flux Linkage Expressions

compress equations

44

r r rs abcs s s abcs s abcmK K i K L

1

0 0 0r r r r r

qd s s s s qd s qd mK K i

L

ls

msls

msls

rss

rs

L

LL

LL

KK

00

0230

0023

1L

mslss LLL23

define

Transform Flux Linkages

inductance

magnet flux linkage

0

0

0

r rqd m s abcm mK

45

rqss

rqs iL

'iL mr

dssrds

0 0s ls sL i

Flux Linkage Equationsin the Rotor Reference Frame

compared to a-b-c variables:no coupled terms and no rotor position dependence

0 0

0 0 00 00 0 0

r rqs s qsr r

ds s ds m

s ls s

L iL i '

L i

46

'iLpiLirv mrrdssr

rqss

rqss

rqs r

qssrr

dssr

dssrds iLpiLirv

Equivalent Circuit Model

substitute flux linkage equations into voltage equations

q-axis circuit d-axis circuit

47

r - electrical rotor position (rad)

rm - mechanical rotor position (rad)

r - electrical rotor speed (rad/sec)

rm - mechanical rotor speed (rad/sec)

rmrP2

rmrP2

General Machine with P Poles

48

rqsmr

rds

rqssr

rqsmr

rqs

rdssrout i'iiLi'iiLP

23

23

rermeout TP

TP 2

r

oute

PPT2

rqsme i'PT

223

Torque Equation from Power

49

torque control

in q-d-0 variables

rqsme i'PT

223

Torque Control

50

dtd

JT rm

rmLrmme JpTBT

Mechanical Equations

Te - electrical torque (N·m)Bm - friction constant (Kg·m2/sec)J - total inertia (Kg·m2)rm - mechanical speed (rad/sec)TL - load torque (N·m)rm - mechanical rotor position (rad)

also rmrm p

51

PMSM Model

The PMSM can be modeled with a fairly straightforward resistance, inductance, and back-emf term in each phase.

If the zero sequence current can be neglected, the q-d model represents the machine as a two-phase circuit with dc quantities in the steady-state. The two circuits are coupled by back-emfterms.

The torque equation in the q-d model is simplified compared to the machine-variable (a-b-c) model. This property will be used for developing a torque control.

The q-d model also leads to simple control in other systems such as induction machines, active rectifiers, and active filters.

52

round rotor machine sdq LLL

rqsv r

dsv rqsi r

dsi

rqsV r

dsV rqsI r

dsI

constant

use notation

'ILIrV mrrdssr

rqss

rqs

rqssr

rdss

rds ILIrV

Lrmmrqsme TB'IPT

223

PMSM Steady-State Equations

with 0dtdp

53

solving for voltages

0

'II

rLLr

VV mr

rds

rqs

ssr

srsr

ds

rqs

222srs

rds

mrr

qs

ssr

srs

rds

rqs

LrV

'VrL

Lr

II

solving for currents

222

'

srs

rdssrmr

rqssr

qs LrVLVr

I

222srs

rdssmr

rqssrr

ds LrVr'VL

I

Steady-State Solutions

54

inverter voltages resulting currents

vrsas Vv cos2

32cos2 vrsbs Vv

32cos2 vrscs Vv

vsr

qs VV cos2

vsr

ds VV sin2

22

21 r

dsr

qss VVV

rqs

rds

v VV1tan

irsas Ii cos2

32cos2 irsbs Ii

32cos2 irscs Ii

isrqs II cos2

isrds II sin2

22

21 r

dsrqss III

1tanrds

i rqs

II

PMSM with Drive (Steady-State)

55

Ideal Drive Calculations with v = 0

Te 0.33N mTe32

P2 'm Iqsr

i 68 degi atanIdsr

Iqsr

2 Is 3.79AIs 2.7AIs1

2Iqsr

2 Idsr2

Idsr 3.51A

Iqsr 1.42AIqsr

Idsr

rs

r Ls

r Ls

rs

1Vqsr r 'm

Vdsr

Vdsr 0VVdsr 2 Vs sin v

Vqsr 127VVqsr 2 Vs cos v

q- and d-axis voltages and currents

r 628rads

rP2rm

rm 628rads

rm 6000 RPM

v 0 radVs 90 V

operating conditions'm 0.156 V sLs 11.4 mHrs 2.9 P 2

motor parameters

56

PMSM Steady-Sate Example v=0

57

Ideal Drive Calculations with i = 0

put all current in the q-axisi 0 deg

Iqsr 2 Is cos i Iqsr 3.79A

Idsr 2 Is sin i Idsr 0A

Vqsr

Vdsr

rs

r Ls

r Ls

rs

Iqsr

Idsr

r 'm

0

Vqsr 109V

Vdsr 27.1 V

Vs1

2Vqsr

2 Vdsr2

Vs 79.4V 2 Vs 112V

v atanVdsr

Vqsr

v 14deg

Te32

P2 'm Iqsr Te 0.89N m

58

PMSM Steady-Sate Example with i = 0

Five-Phase PM Motor

0 50 100 150 200 250 300 350-1

0

1

0 50 100 150 200 250 300 350-1

0

1

0 50 100 150 200 250 300 350-1

0

1

Rotor position

i bemf

i

bemf

i

bemf

The proposed five phase motor is supplied with combined fundamental and thirdharmonic of current. It takes advantage of higher torque density of BLDC andbetter controllability of PMSM. Besides it benefits from other advantages of multi-phase machines

•Reducing the amplitude and increasing the frequency of torque pulsation• Increased reliability• Reducing the stator current per phase without increasing the voltage per phase• Higher power density• Providing the possibility of injecting harmonics of current and improving torque density• The possibility of independently controlling multi motor from a single inverter• Existence of more space vector compare to three-phase system•Improvement of noise characteristics

Advantages of Multi-Phase Machines:

Mathematical Model

dtdIRV s

sss

msss

The stator voltage equations are given by:

where the airgap flux linkages are presented by :

)5

2(3sin

)5

4(3sin

)5

4(3sin

)5

2(3sin

)3sin(

)5

2sin(

)5

4sin(

)5

4sin(

)5

2sin(

)sin(

31

r

r

r

r

r

m

r

r

r

r

r

mm

mssss IL

Mathematical Model

21

21

21

21

21

)5

2(3cos)5

4(3cos)5

4(3cos)5

2(3cos3cos

)5

2(3sin)5

4(3sin)5

4(3sin)5

2(3sin3sin

)5

2cos()5

4cos()5

4cos()5

2cos(cos

)5

2sin()5

4sin()5

4sin()5

2sin(sin

52)(

T

The transformation matrix for this system:

1d

1q

3d

3q

33

and 11qd 33qd space

Mathematical Model

)33(2.25

3333111.1 iiii dsqsqsdsdsqsqsdsPT

Finally, torque equation:

Stator voltages in d1, q1, d3, q3axes

Stator fluxes in d1, q1, d3, q3axes

1111 )25( mdsmslsds iLL

iLL qsmslsqs 111 )25(

3333 )25( mdsmslsds iLL

333 )25( qsmslsqs iLL

dtdirV ds

qsdssds1

111

dtd

irV qsdsqssqs

1111

dtd

irV dsqsdssds

3333 3

dtd

irV qsdsqssqs

3333 3

Control Block Diagram