SYNCHRONOUS MACHINES

Two-axis model

du

fu

fi

di

Di

Qi

quqi

d

q

d

q

Q D

f

Synchronous machines

8 MW diesel-generator 270 MVA turbo-generator

Rotor windings

Field winding Damping bar Field winding Damping bar

A PM machine with 30 poles and 36 slots+

--

+-

++

-+

--

+-

++

-+

--

+-

++

-

q = 2/5

dd

ddd

dd0

dd

0d

dd s d q

qq s q d

ff f f

DD D

QQ Q

u R it

u R it

u R it

R it

R it

d d d df f dD D

f df d f f fD D

D dD d fD f D D

q q q qQ Q

Q qQ q Q Q

L i L i L i

L i L i L iL i L i L i

L i L i

L i L i

Voltage, flux-linkage and motion equations

3 d2 d

dd

q q md dJp i i Tp t

t

Rotor frame of reference

Space-vector model for induction motorRotor frame of reference

Voltage and flux linkage

The axes in the rotor frame of reference denoted by d and q

dj

ddd

rsr r r

s s s s

rrr r

r r r

u R it

u R it

r rrs s m rs

r rrm s r rr

L i L i

L i L i

j

j

rs sd sqrr rd rq

i i i

i i i

dd

dd

dd

dd

sdsd s sd sq

sqsq s sq sd

rdrd r rd

rqrq r rq

u R it

u R it

u R it

u R it

sd s sd m rd

sq s sq m rq

rd m sd r rd

rq m sq r rq

L i L iL i L iL i L iL i L i

i sd

i sd

u sd

u sd

i sq

i sq

u sq

u sqq

d

32e sd sq sq sdT p i i

Load angle

ddt

dds t

is replaced by

where the load angle is the angle between the excitation voltage vectorup (on q-axis) and stator voltage vector ur

s

ˆ sinˆ cos

d s

q s

u uu u

For synchronous machines, the position angle of the rotor is defineddifferently from the angle typically used for induction machine analysis

d

q

rsu

pu

Assumptionwhere Lmd is the direct-axis magnetising inductance

Direct-axis equivalent circuit

df dD fD mdL L L L

ddd

dd0

d

dd s d q

ff f f

DD D

u R it

u R it

R it

d d d md d f D

f f f md d f D

D D D md d f D

L i L i i i

L i L i i i

L i L i i i

d dd d

d dd d

d d0d d

dd s d q d md d f D

ff f f f md d f D

DD D D md d f D

iu R i L L i i it t

iu R i L L i i i

t tiR i L L i i it t

Direct-axis equivalent circuit

di

dd

dt

ddmdL

tddDL

tddfL

t

DR fR

d D fi i i Di fi

dddL

tfu

d dd d

d dd d

d d0d d

dd s d q d md d f D

ff f f f md d f D

DD D D md d f D

iu R i L L i i it t

iu R i L L i i i

t tiR i L L i i it t

More accurate equivalent circuit for direct axis

Assumptionwhere Lmd is the direct-axis magnetising inductance

di

dd

dt

ddmdL

tddDL

tddfL

t

DR fR

d D fi i i Di fi

dddL

t fuddkL

t

df dD md fDL L L L

Quadrature-axis equivalent circuit

Notationwhere Lmq is the quadrature-axis magnetising inductance

qQ mqL L

dd

d0

d

qq s q d

QQ Q

u R it

R it

q q q mq q Q

Q Q Q mq q Q

L i L i i

L i L i i

d dd d

d d0d d

qq s q d q mq q Q

QQ Q Q mq q Q

iu R i L L i i

t ti

R i L L i it t

Quadrature-axis equivalent circuit

qi

dd

q

tddmqL

tddQL

t

QR

q Qi i Qi

ddqL

t

d dd d

d d0d d

qq s q d q mq q Q

QQ Q Q mq q Q

iu R i L L i i

t ti

R i L L i it t

Notations and abbreviations

Leakage factors and other abbreviations

2

2

2

2

1

1

1

1

dfdf

d f

dDdD

d D

fDfD

f D

qQqQ

q Q

LL L

LL L

LL L

LL L

1

1

df fDf

dD f

dD fDD

Ddf

L LL LL LL L

Synchronous machine in steady stateSteady state => a) Space vectors are constants in the rotor frame of reference

j constant

j constant

rs d qrs d q

u u u

i i i

b) The time-derivatives of flux linkages vanish

dd 0d d

qdt t

c) The currents of the damper windings are zero

0D Qi i

Space-vector diagram fora synchronous generator

rsi

rsu

pu

df fL i

q qL i

j d dL i

j rs

r

s sR i

fid

q

rs

di

qi

j rd q sL L i

j rq sL i

s qd d

q s q d

f f f

u R iu R iu R i

d d d df f

q q q

L i L i

L i

j

ˆj j

r rs s s q q pd d

p p df f

u R i L i L i u

u u L i

ˆ sinˆ cossd

q s

i i

i i

Electromagnetic torque3 32 2e q q q q qd d d d df fT p i i p L L i i L i i

Neglecting the stator resistance

sinˆcos

s q qd

q s pd d

u u L i

u u L i u

ˆcos

sin

s pd

d

sq

q

u ui

Lui

L

=>

2 2

2

ˆ ˆ ˆ ˆ ˆcos sin sin32

ˆ ˆ ˆ3 1 1 sin sin 22 2

q s p sd p se

q qd

s p s

qd d

L L u u u u uT p

L L L

u up uX X X

Operator inductances of synchronous machine

Laplace transformation of the voltage and flux-linkage equations

0

0

0

0

0

0

0

ds qd d d

ff f f f

DD D D

qq s q q d

QQ Q Q

u R i ss

u R i ss

R i ss

u R i ss

R i ss

d d d df f dD D

f df d f f fD D

D dD d fD f D D

q q q qQ Q

Q qQ q Q Q

L i L i L i

L i L i L iL i L i L i

L i L i

L i L i

0 0

0 0

0 0

d dd d

d df dDf f

f df f fD f

DdD fDD DD D

iis sL L L

iL L L i

s sL L L ii

s s

The flux differences expressed using the current differences

Typical initial values: 00 0 00; f

D Q ff

ui i i

R

0 0

0 0

q qq qq qQ

qQ QQ QQ Q

iiL Ls s

L L ii

s s

Typical initial values: 00 0 00; f

D Q ff

ui i i

R

0 0

0 0

q qq qq qQ

qQ QQ QQ Q

iiL Ls s

L L ii

s s

We try to express the stator flux as a function of the stator current. The rotor quantities should be eliminated from the equations.

Quadrature-axis operator inductance

0

00

qq s q q d

QQ Q Q

u R i ss

R i ss

Laplace transformed voltage and flux-linkage equations for the q-axis

Quadrature-axis operator inductance II

0 00 => Q Q Q QQ Q Q Q

R iR i s

s s s

0 00 => q qQ qQQ qQ q Q Q Q q

Q Q

i L s iL i L i i i

s s R L s s

Voltage equation for the damper winding

Flux-linkage equation for the damper winding

0 0 0 0q q q qQ qq q q qQ Q q q qQ q

Q Q

i i L s iL i L i L i L i

s s s R L s s

=>

Flux-linkage equation for the stator winding

20 0 0q qQ q q

q q q q qQ Q

L s i iL i L s i

s R L s s s

Quadrature-axis subtransient inductance

2 2'' lim lim qQ qQq q q q qQ qs s Q Q Q

L s LL L s L L L

R L s L

At the beginning of a transient process

'' 11 1

Q mqq s s

Q mqQ mq

L LL L L

L LL L

L’’q is called quadrature-axis subtransient inductance

2qQ

q qQ Q

L sL s L

R L s

Quadrature-axis operational inductance

Quadrature-axis operator inductance III2 2

2

2

''

''0

''''

''0

1 1

11

11 1

1

1

qQ qQQ Qq q

qQ Q Q Q Qq q q

Q QQ Q

Q Q

qQ Q Qq q qQ

Q Q qQq q

Q Q qQ Q

qq

q

L LL Ls L L s s

L s R R L RL s L L L LR L s s s

R R

L L LL L s sL R T sRL LL L T ss s

R R

sT

Ls

T

where''

'' '' '' ''0 0 0 0; qQ

q Q q qQ Q QQ q

LLT T T T T

R Lopen-circuit short-circuittime constant time constant

Quadrature-axis operator inductance IV

''0

'' ''

'' '' ''

11 1 1 1 1 1

1 1 1

q

q q qq q

q q q

sT A B

sL s s sL LL Ls s s s

T T T

An equation needed later for the inverse Laplace transformation

Short circuit at the terminals of a 1 MW permanent-magnet motor

B

H

rB

cH

Magnetic characteristic of the permanent magnets

Magnetization curve for a NdFeB material

Neorem Magnets Oy http://www.neorem.fi/

-1000

-800-600-400-200

0

200400600800

1000

0 50 100 150 200 250 300 350 400 450 500

Time [ms]

Line

vol

tage

s [V

]

-15000

-10000

-5000

0

5000

10000

15000

0 50 100 150 200 250 300 350 400 450 500

Time [ms]

Line

cur

rent

s [A

]

Line voltage and current in a 3-phase short circuit

-350000

-300000

-250000

-200000

-150000

-100000

-50000

0

50000

100000

0 50 100 150 200 250 300 350 400 450 500

Time [ms]

Torq

ue [N

m]

Torque and minimum flux density in permanent magnets

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0 50 100 150 200 250 300 350 400 450 500

Time [ms]

Min

imum

flux

den

sity

[T]

-12000-10000

-8000-6000-4000-2000

02000400060008000

10000

0 50 100 150 200 250 300 350 400 450 500

Time [ms]

Line

cur

rent

s [A

]

-1000

-800-600-400-200

0

200400600800

1000

0 50 100 150 200 250 300 350 400 450 500

Time [ms]

Line

vol

tage

s [V

]

Line voltage and current in a 2-phase short circuit

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0 50 100 150 200 250 300 350 400 450 500

Time [ms]

Min

imum

flux

den

sity

[T]

-150000

-100000

-50000

0

50000

100000

150000

0 50 100 150 200 250 300 350 400 450 500

Time [ms]

Torq

ue [N

m]

Torque and minimum flux density in permanent magnets