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 +

Ifd ias WRSM

ibs ics

vdc

_

WRSM/active rectifier

WRSM/passive rectifier

design   d rc l d rt g d st dbs fwss fhrt fwrt fwrp N s I s  N fd I fd Pp

I as  I s cos( r   )

ftipw ftiph 

T

design   d rc l d rt g d st dbs fwss fhrt fwrt fwrp N s N fd I fd Pp ftipw ftiph rdt d num dcon Vdc  f ( I fd , L"q , L"d ) 2

T

MEC Model for Active Rectification • A steady-state mesh-based MEC model for WRSMs

3

Performance Calculation • Electromagnetic torque P Te  , r     2

• Power loss

3 Pres  rfd I  2 2 fd

Resistive loss: Core loss:

2

 agj  Pagj    j 1  Pagj   r

2 na



2

rs ias2 r  dr

0

Pcore  Pld ,TVST  Pld ,YVSY 

where Pld  B   kh f eq

 1

T  Bmax  ke f  dB    f  2   dt and B B dt  b 0  b  2

Total loss:

Pcond  3Vdrop

T

Eddy Current Loss

Hysteresis Loss

Conduction loss:

2

 dB  f eq   dt 2 2  dt  Bmax  Bmin   0   2

1 2



2

0

ias r  dr

Ploss  Pres  Pcore  Pcond 4

Dynamic MEC Network for Passive Rectification l - axial length of machine

RY

db

d st

h

RSH

rtt

w

rt

h

rp

rp

RTT Rag

d

rc

g

as-axis

h

w st

r

ro

rd

rto

r so

t

Φag1 idp1 Φrp1 RRTO1*

h rsi

rd

RRTE

s

Φag2 RRLO1

w stt

w ss

rs

ics' Φst2

d-a x

is

ics' Φst3

X

X

X

ias Φst4

ias Φst5

ias Φst6

ibs' Φst7

ibs' Φst8

ibs' Φst9

RTL

s q-axi θ rm

d

w

rtb

ics' Φst1

RRFB

RFL

Φag4 λdp,1 RRLO2

Φag6

RRLI1

Φag8 λdp,2 RRLO3

idp2 Φrp2 RRTI1 Φrp3 RRTI2 Φrp4 RRTO2*

RRTO3

ifd'

RRSH1

Φrt1

RRSH2

*

ifd X

RRYP

Φrt2

Φag10

Φag12

Φag15

Φag17

Φag20

λdp,3

RRLO4

idp3

Φrp5 RRTO4

RRTE

*

RRTE RRF

RRFB

RRFB

RRPL RRY

5

Rotor Pole Flux Tubes RRTO1

RRTI1

RRTI2

RRTO3

RRTO4 RRTE2

hRTO1

RRTE1

RRTO2

lRTI1 hrtm lRTE1

lRTO1

lRTO2 RRSH1

RRTO1_1 RRTO1_2 RRTO1_3

fwrto(x)

RRTO1 

RRSH2 -x

x

lRTO1  2rdt   lhRTO1 2l

 2rdt   tan 1 (  2 2 2 2 l hRTO1  4rdt  2 hRTO1  4rdt hRTO1

6

 )  

Rotor Pole Tip Leakage Flux Tubes

Stator tooth

g

P3 l - axial length of machine dt

d dp   dp (hRTO1  2rdt )

r P2

P1: leakage inside the bar. P2: leakage in the steel. P3: leakage in the airgap.

P1

1 hRTO1

ddp

RRLO1



0l l d dp  rdt  ln( ) 8 2 rdt P1

Leakage path RRLO1

P2

2 g (d dp  rdt )  g 2  g  d dp  rdt 0l  ln( ) 2 d dp  rdt

Main path RRTO1*

P3

lRTO1

1 * RRTO 1



1 RRTO1



1 RRLO1 7

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:

damper_rtip  [... rdt 3

rdt 2

rdt1 rdt 2

rdt 3 ...]

• Arbitrary vertical depth of the damper bars can be assigned by adjusting the scaling factor αdp.

8

Steady-State KVL MEC Model A(Rnlnl )φ(lnl1) =Fl(nl1)

System equation: where and

φl  st1

stns rt1

rtnr ag1

Fl = Fst(ns1) Frt(nr1) 0(na1) Frp(np1)    T

T

Stator and rotor MMF source: Damper winding MMF source: ( npnd )  j, k   1 where Ndp

1 0 (51) Frp = 0 0 0 

T

rpnp 

T

T

ns3) (31) Fst( ns1)  N(abc i abcs , Frt( nr1)  N(rtnr1) I fd   1 1 0 Nfd I fd T

( nd 1) Frp( np1) (j )=N(dpnpnd ) (j, k )i dp (k )

, if the kth damper winding current is in the jth rotor pole loop.

N(dpnpnd )  j, k   0

In this case,

T

agna rp1

, otherwise. 0 0 1 0 0

0 0 (31) 0  i dp 0 1  9

Dynamic System Structure KVL MEC Model i

MEC System of Equations

 A R (φl ) φl =Fl =Ni

φ

Post-processing l

T λ abcs =PNabc φst

v abcs =f (i,λ )

Dynamic MEC Model

λ (k )

MEC System of Equations

 AR W  2

W1  φl  I  =B  fd     0  i  λ

i (k )

External Circuit Model (e.g. Passive Rectifier)

v =f (i )

i (k )

v(k ) Numerical Integration

pλ =f (v,i ) λ (k  1)=g(pλ ,λ (k ))

OR: user-input v(k )

10

Restructuring the MEC Model 1) Expand the KVL MEC system: AR φl -Nl,abc i abcs -Nl,dp i dp =Nl,fd Ifd 2) Relate the flux and flux linkage: nd nl ) ( nl 1) nd nd )  φ(l nl1) λ abcs =PNl,abcφst , λ (dpnd 1) =M(l,dp φl  0( nd ( nl nd ) M(l,dp_sub

In this case,

 dp ,1  1 1 0  rp1     0 1 1    dp ,2  1 0 1   rp 3   rp 5   dp ,3   M l,dp_sub

3) Dynamic MEC system of equations:

 A R -N l,abc -N l,dp   φ   N 0 0   Ifd  NT   l   l,fd i abcs = 0 I /P 0   λ abcs   l,abc  0      M l,dp   i dp   0 0 I   λ dp  11

Dynamic System with Scaling Apply qd transformation: -1  AR - f scale N l,abc  K s   T  f scale K s N l,abc 0  f M  scale l,dp

-f scale N l,dp   φl   N 0 0   Ifd  l,fd      i qd0s,scl  =  0 f scale I /P 0   λ qd0s  0 f scale I   λ dp    i dp,scl   0     

Adyn

where

i qd0s =f scalei qd0s,scl , i dp =f scalei dp,scl , fscale  103

Compare to the structure block diagram, W1   fscale Nl,abc  K s  

1

 f scale Nl,dp  

0 0   N l,fd B   0 f scale I /P 0   0 0 f scale I  

T  f scale K s N l,abc  W2    f M scale l,dp  

12

State Equations of Damper Bars re,3

re,1

re,2

re,3

ie idp,1 λdp,3

X

re,3

rdp,1 idp,2 λdp,1 re,1

rdp,2 idp,3 λdp,2 re,2

1 3 i e   i dp,k 2 k=1

rdp,3 λdp,3 re,3

From Ohm’s and Faraday’s laws, pλ dp =Tdp i dp

where

 rdp ,1  re,1 rdp ,2  re,1 re,1  Tdp   re,2 rdp ,2  re,2 rdp ,3  re,2    r  r r r  r e ,3 dp ,3 e ,3   dp ,1 e,3 13

State Equations of Stator Windings Rearrange the voltage equation:

 0 1 0 pλ qd0s =v qd0s -rs i qd0s - -1 0 0  λ qd0s  0 0 0 Calculation of resistive loss:

3 Pres  2



2

0

rs ias2  r  d r  rfd i 2fd stator  field

P  2

nd

2

k 1

0



 rdp ,k idp2 ,k  r   re ,k ie2,k  r   d r damper

14



C

Hardware Validation

15

Open Circuit Voltage

Field currents:0-10.2 A Rotor speed: 1800 rpm Maximum error: 5% MEC

Hardware 16

Balanced 3-Phase Load Test 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: Tin _ avg 

Load

Output power:

Pout  Term  Pres

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

MEC Load

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

Term  Pmech  Pcore

rm

RMS values of Phase current (A)

17

Voltage Waveforms

The error of RMS values is approximately 5%. 18

Standstill Frequency Response -At low frequencies (< 0.4 Hz) Match closely. -At mid frequencies (> 0.4 Hz, 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.

Low-frequency asymptote: magnetizing impedances. High-frequency asymptote: subtransient impedances. 19

Different Depth of Damper Bars

αdp=0.5

αdp=0.0001

20

Influence of the Leakage Path Between Poles

21

25 MW WRSM/Rectifier Design +

Ifd ias WRSM

ibs ics

vdc

_

WRSM/active rectifier

WRSM/passive rectifier

θ   drc l drt g d st dbs fwss fhrt fwrt fwrp N s I s  N fd I fd Pp

ftipw ftiph 

θ   drc l drt g d st dbs fwss fhrt fwrt T

fwrp N s N fd I fd Pp ftipw ftiph rdt d num dcon

22

T

Pareto Front

23

Genes Distribution

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). 24

Example Machines

WRSM/active rectifier

WRSM/passive rectifier 25

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