A Dynamic Magnetic Equivalent Circuit

Model for the Design of Wound Rotor

Synchronous Machines

Research of Xiaoqi (Ron) Wang

Advisor: Steve Pekarek

Department of Electrical and Computer Engineering

Purdue University

WRSM/Rectifier Design

2

WRSM/active rectifier

+

_

vdcWRSM

ias

ibs

ics

Ifd

WRSM/passive rectifier

rc rt st bs ss rt rt

T

rp s s fd fd p

design d l d g d d fw fh fw

fw N N I P ftipw ftipI h

rc rt st bs ss rt rt

T

rp s fd f dt num ond p c

design d l d g d d fw fh fw

fw N N I P ftip r d dw ftiph

cos( )as s rI I " "( , , )qdc dfdV f I L L

• A steady-state mesh-based MEC model for WRSMs

3

MEC Model for Active Rectification

Performance Calculation

• Electromagnetic torque

• Power loss

4

22

agj agj

e r

1 agj r

,2

na

j

PPT

P

loss res core condP P P P

2

2 2

0

3

2res fd fd s as r rP r I r i d

2

1 max

2

0

T

eld h eq

b b

Eddy Current LossHysteresis Loss

B k f dBP B k f f dt

B B dt

, ,core ld T ST ld Y SYP P V P V

2

0

13

2cond drop as r rP V i d

Resistive loss:

Core loss:

where

Conduction loss:

Total loss:

2

2 20max min

2T

eq

dBf dt

dtB B

and

Dynamic MEC Network

for Passive Rectification

5

d-axis

db

dst

q-axis

as-axis

θrm

drp

drc

rsh

wrt

wrp

hrtt

wstt

g

rds

rdt

l - axial length

of machine

rro

hrtb

rsi r

so

hrto

wss

wst

RY

RTT

RTL

RSH

ics'

Φst1

ics'

Φst2

ics'

Φst3

ias

Φst4

ias

Φst5

ias

Φst6

ibs

'

Φst7

ibs

'

Φst8

ibs

'

Φst9

X X X

RRLO1

RRTI1

Rag

idp1 i

dp2idp3

RRSH1

RRSH2

RRYP

ifd' i

fdX

RFL

RRY

RRFB

RRFB

RRF

RRPL

Φag1

Φag2

Φag4

Φag6

Φag8

Φag10

Φag12

Φag15

Φag17

Φag20

Φrt1 Φ

rt2

λdp,1

λdp,2

λdp,3

RRTO1

*

RRTE

RRLO2 R

RLI1R

RLO3R

RLO4

RRTO2

* RRTO3

* RRTO4

*

RRTE

RRTI2 R

RTEΦ

rp1

RRFB

Φrp2

Φrp3

Φrp4

Φrp5

Rotor Pole Flux Tubes

6

2 2 2 2

11

1

11

1 1

2

2

2tan ( )

24 4

RTO dtRTO

RTO

RTO dt

RTO dt RTO dt

l rR

lh l

h r

l h r h r

RRTE1

RRTO1

RRTI1

RRTE2

RRTO2 R

RTO3 RRTO4

RRTI2

fwrto

(x)

-x x

hrtm

lRTE1

lRTO2

lRTO1

RRTO1_1

RRTO1_2

RRTO1_3

hR

TO

1

RRSH1

RRSH2

lRTI1

Rotor Pole Tip Leakage Flux Tubes

7

Main path

Leakage path

lRTO1

hR

TO

1

r dt

RRTO1

*

ddp

l - axial length

of machine

P1

P2

P3

Stator tooth

g

RRLO1

12

3

1

0

1

2

0

*

1 1 1

( 2 )

1ln( )

8 2

2 ( ) ln( )

2

1 1 1

dp dp RTO dt

dp dt

RLO dt

PP

dp dt dp dt

dp dt

P

RTO RTO RLO

d h r

d rl l

R r

g d r g g d rl

d r

R R R

P1: leakage inside the bar.

P2: leakage in the steel.

P3: leakage in the airgap.

Damper Bar Placement

• Odd number damper bars: one of the bars is located in the center of the

most inner two RRTI sections.

• Even number damper bars: there is no hole in the center of the most

inner two RRTIi sections, but they are symmetrically distributed on the

two sides of the rest of the rotor pole sections.

• Arbitrary number and radius of damper bars can be applied:

• Arbitrary vertical depth of the damper bars can be assigned by

adjusting the scaling factor αdp.

8

3 2 1 2 3[... ...]dt dt dt dt dtr r r r rdamper_rtip

Steady-State KVL MEC Model

9

System equation: ( ) ( 1) ( 1)

R l l=nl nl nl nl A φ F

where l st1 st rt1 rt ag1 ag rp1 rp

T

ns nr na np φ

and T T T T T

( 1) ( 1) ( 1) ( 1)

l st rt rp= ns nr na np

F F F 0 F

Damper winding MMF source: ( 1) ( ) ( 1)

rp dp dp( )= ( , ) ( )np np nd ndj j k k F N i

where ( )

dp , 1np nd j k N , if the kth damper winding current is in the jth rotor pole loop.

( )

dp , 0np nd j k N , otherwise.

Stator and rotor MMF source: T( 1) ( 3) (3 1) ( 1) ( 1)

st abc abcs rt rt fd fd fd, 1 1 0ns ns nr nr I N I F N i F N

In this case,

(5 1) (3 1)

rp dp

1 0 00 0 0

= 0 1 00 0 00 0 1

F i

Dynamic System Structure

10

MEC System of Equations

R l l l( ) = =A φ φ F Ni

i lφPost-processing

T

abcs abc st=λ N φP

abcs = ( , )v f i λ

KVL MEC Model

MEC System of Equations External Circuit Model

(e.g. Passive Rectifier)

= ( )v f i

( )i k

Numerical Integration

= ( , )λ f v ip

( 1)= ( , ( ))λ g λ λk p k

( )λ k

( )i k

( )v k

Dynamic MEC Model

OR: user-input ( )v k

R 1 l fd

2

I=

A W φB

W 0 i λ

Restructuring the MEC Model

11

1) Expand the KVL MEC system: R l l,abc abcs l,dp dp l,fd fd- - = IA φ N i N i N

2) Relate the flux and flux linkage:

( 1) ( ) ( 1) ( ( ) ( ) ( 1)

abcs l,abc st dp l,dp l l,dp_sub l= , =nd nd nl nl nd nl nd nd nd nlP λ N φ λ M φ 0 M φ

In this case,

l,dp_sub

,1 1

,2 3

,3 5

1 1 00 1 11 0 1

dp rp

dp rp

dp rp

M

3) Dynamic MEC system of equations:

R l,abc l,dpfdl l,fdT

l,abcabcs abcs

dp dpl,dp

- - I0 0

= 0 / 00 0 0

P

A N N φ NN i I λ

Ii λM

Dynamic System with Scaling

12

Apply qd transformation:

dyn

-1

R l,abc s l,dp fdl l,fdT

s l,abc qd0s,scl qd0s

dp,scl dpl,dp

- - I0 0

= 0 / 00 0 0

scale scale

scale scale

scalescale

f f

f f Pf

f

A

A N K N φ N

K N i I λIi λM

where 3

qd0s qd0s,scl dp dp,scl scale= , = , 10scale scalef f f i i i i

Compare to the structure block diagram,

1

1 scale l,abc s l,dpscalef f

W N K N

T

s l,abc2

l,dp

scale

scale

f

f

K NW

M

l,fd 0 0

0 / 00 0

scale

scale

f Pf

N

B II

State Equations of Damper Bars

13

λdp,1

re,1

re,2

re,3

re,3

re,1 r

e,2re,3

re,3

rdp,1 i

dp,2rdp,2 i

dp,3rdp,3

ie

λdp,2

λdp,3Xλ

dp,3

idp,1

3

e dp,k

k=1

1i i

2

From Ohm’s and Faraday’s laws,

dp dp dp=pλ T i

where ,1 ,1 ,2 ,1 ,1

dp ,2 ,2 ,2 ,3 ,2

,1 ,3 ,3 ,3 ,3

dp e dp e e

e dp e dp e

dp e e dp e

r r r r r

r r r r r

r r r r r

T

State Equations of Stator Windings

14

Rearrange the voltage equation:

qd0s qd0s qd0s qd0s

0 1 0= - - -1 0 0

0 0 0sp r

λ v i λ

Calculation of resistive loss:

22 2

0

22 2

, , , ,0

1

3

2

2

res s as r r fd fd

stator field

nd

dp k dp k r e k e k r r

k

damper

P r i d r i

Pr i r i d

Hardware Validation

15

C

Open Circuit Voltage

16 MEC Hardware

Field currents:0-10.2 A

Rotor speed: 1800 rpm

Maximum error: 5%

Balanced 3-Phase Load Test

17

Conditions:

- Rotor speed: 1800 rpm

- RMS line-line voltage: 480 V

- Power factor: 0.8 lagging

(parallel RL load)

- Pmech = 303 W

- rbrush = 1 Ω

- Loss of exciter is not modeled

Average input torque:

_e rm mech core

in avg

rm

T P PT

Output power:

out e rm resP T P

Load

RMS values of

Phase current (A)

Average input

torque (Nm)

Output power

(kW)

MEC Test MEC Test MEC Test

1 4.8 4.5 21.01 19.98 3.1775 2.9858

2 8.0 7.6 33.81 32.26 5.3641 5.0707

3 12.0 11.4 49.95 47.78 7.9783 7.5915

4 15.9 15.2 66.11 64.16 10.4445 10.1030

Load

MEC Test

Ps+f

(W)

Pcore

(W)

Pdp

(W)

Pcore+dp

(W)

Ps+f

(W)

Pcore+dp

(W)

1 235.5 232.3 12.5 244.7 229.5 247.9

2 431.2 241.8 32.9 274.7 414.5 292.8

3 793.2 254.2 86.4 340.6 757.6 354.4

4 1267.4 265.3 181.9 447.2 1211.2 477.0

Voltage Waveforms

18

The error of RMS values is approximately 5%.

Standstill Frequency Response

19

Low-frequency asymptote: magnetizing impedances.

High-frequency asymptote: subtransient impedances.

-At low frequencies (< 0.4 Hz)

Match closely.

-At mid frequencies (> 0.4 Hz, <20 Hz)

Additional conduction path that exists

between the copper plates and the rotor

shaft.

-At higher frequencies (> 100 Hz)

Error between MEC and FEA comes

from the modeling of rotor pole tip

leakage flux path.

Lower test-q due to eddy currents.

Different Depth of Damper Bars

20

αdp=0.5 αdp=0.0001

Influence of the Leakage Path Between Poles

21

25 MW WRSM/Rectifier Design

22

WRSM/active rectifier

+

_

vdcWRSM

ias

ibs

ics

Ifd

WRSM/passive rectifier

rc rt st bs ss rt rt

T

rp s s fd fd p

d l d g d d fw fh fw

fw N I N I P ftipw ftiph

θ rc rt st bs ss rt rt

T

rp s fd fd p dt num con

d l d g d d fw fh fw

fw N N I P ftipw ftiph r d d

θ

Pareto Front

23

Genes Distribution

24

WRSM/active rectifier: larger height of rotor teeth (HRT), airgap length (G), and pole pair (Pp).

WRSM/passive rectifier: larger stack length (GLS), stator turns (Ns) and field turns (Nfld).

Example Machines

25

WRSM/active rectifier WRSM/passive rectifier

Conclusions

• Developed a voltage-input dynamic MEC model that

includes damper bar currents dynamics.

– Enables exploration of alternative damper configurations

– Enables multi-objective design of machine/passive rectifiers

– Readily extended to multi-phase machines

• Initial study utilized to compare Passive/Active designs

– Surprising result is that the machine for the passive designs are less

massive for a given specified loss

26