A Multirate Field Construction Technique for Efficient Modeling of the Fields and Forces

within Inverter-Fed Induction Machines

Dezheng Wu, Steve Pekarek

School of Electrical and Computer EngineeringPurdue University

September 30, 2010

Motivation for Research

• Fields-based modeling of machines valuable analysis tool– Investigate slot geometries, material properties– Calculate force vector (radial and tangential)– Readily model induced currents in magnetic material

• Limitation as a design tool– Numerical computation expensive

• Field construction– Attempt to establish a fields-based model while minimizing

computation requirements– FC of induction machine initially considered by O’Connell/Krein in

parallel with Wu/Pekarek2

Field Construction – Basic Idea

• Use a minimal number of FEA solutions to characterize machine behavior – Create basis functions for stator and rotor magnetic fields

• ‘Construct’ the magnetic field in the airgap using stator field and rotor basis functions under arbitrary current

Bn=Bns+Bnr Bt=Bts+Btr

• Calculate torque and radial force using the Maxwell Stress Tensor (MST) method under arbitrary stator excitation and rotor speed

2 2

0 0

1 1. ,

2t n t n n tf B B f B B

2

2 2

0 0Torque , Radial Force

2z

e t s s r z n s

P R lT f d F l f R d

3

Assumptions

• The flux density in the axial direction is zero

• Hysteresis in the iron is neglected

• Thermal conditions are assumed constant

• No deformation occurs in stator and rotor teeth

• Linear magnetics

sm

rm

rm

4

Stator Basis Function Derivation

MagnetostaticFEA Solution

ias=I0 Bnas, Btas smts

smns

k

k

0

0

/

/

IBk

IBk

tassmts

nassmns

0 50 100 150 200 250 300 350-0.02

-0.01

0

0.01

0.02

deg][sm

[T/A

]nsk

kns

0 50 100 150 200 250 300 350-0.01

-0.005

0

0.005

0.01

deg][sm

k ts[T

/A]

kts

5

Rotor Basis Function (knr,ktr) Derivation

Impulse Response1. Set a discrete-time impulse input to a transient FEA program ias(t) = I0 when t = t0

ias(t) = 0 when t ≠ t0

2. Record the flux density components (Bnid, Btid) for t ≥ t0.

3. Subtract the stator magnetic field Bnr= Bnid – iaskns , Btr= Btid – iaskts

4. Divided by I0

knr= Bnr / I0 , ktr= Btr / I0

TransientFEA

Solution

ωrm=0

ias= impulse

Bnr, Btr

+-

rmtrrmnr kk ,

Bnid Btid

tjtj-1 tj+1 tj+2...

I0

tsns kk ,

0

0

/

/

IBk

IBk

trrmtr

nrrmnr

6

Complete Characterization Process

7

Magnetic Flux Density Due to Stator

• The flux density generated by arbitrary stator phase-a current is approximated as

• Due to symmetry, the total flux density generated by stator currents

nas sm as ns sm

tas sm as ts sm

B i k

B i k

2 2 2 4

3 3

due to due to

2 2 2 4

3 3

ns sm as ns sm bs ns sm cs ns sm

csbs

ts sm as ts sm bs ts sm cs ts sm

B i k i k i kP P

ii

B i k i k i kP P

8

Magnetic Flux Density Due to Rotor

2 2 2 4, , , ,

3 3

2 2, ,

3

2 4,

3

xr rm as xr rm bs xr rm cs xr rm

as xr sm rm bs xr sm rm

cs xr sm rm

B t i t k t i t k t i t k tP P

i t k t i t k tP

i t k tP

• Obtain rotor magnetic field using the convolution of stator current signal and rotor basis function

where x can be ‘n’ or ‘t’

due to ias due to ibs due to ics

9

Complete Field Construction – Stator Current as Model Input

, , ,

, , ,

n sm k ns sm k nr sm k

t sm k ts sm k tr sm k

B t B t B t

B t B t B t

Obtain the total flux density in the discrete-time form

1

1

1

, ( ) ( ),

2 2( ) ( ) ,

3

2 4( ) ( ) ,

3

k

xr sm k as m xr sm rm k rm m k mm

k

bs m xr sm rm k rm m k mm

k

cs m xr sm rm k rm m k mm

B t i t k t t t t

i t k t t t tP

i t k t t t tP

In the computer, the discrete convolution of the stator current and rotor basis function

where x can be ‘n’ or ‘t’10

11

Voltage-Input-Based FC Technique

v iCurrent-input-based FC

iv Bn , Bt• Basic idea:

• Stator voltage equations are used to relate voltage and current:

0 0 0

qs qs s ds qs

ds ds s qs ds

s s s s

v i r p

v i r p

v i r p

,

,

0 0

qs ss qs qs r

ds ss ds ds r

s ls s

L i

L i

L i

where w is the angular speed of an arbitrary reference frame, and the flux linkages are expressed as

Due to the induced rotor current

Unknowns: Lss, Lls, λqs,r, λds,r

12

Characterization of Rotor Basis Flux Linkage

Use the same FEA solutions as in the characterization of stator and rotor basis functions.

Impulse response (vector)

Static FEASolution

TransientFEA Solution

ias = constant

ias= impulse

={

abcs(t)

abcs

Lss , Lls

kt

Lss=as bs

ias

Lls=as2bs

ias

k(t=qs(t Lssiqs(t

2Ip/3I

13

Calculate lqs,r, lds,r

Procedure: 1. convolution. 2. transformation between reference frames

,

1,

( ) ( )( , ) ( )

( ) ( )

nqs r n qs m

n m n mmds r n ds m

t i tt t k t t

t i t

K

cos( ) sin( )( , )

sin( ) cos( )

( ) ( )

( ) ( )

n m

n m

r n r m

t t

t t

t t

K

where qr is the electric rotor angle, and q is the angle of the reference frame

14

Voltage-Input Based FC Diagram

Then iqd0s iabcs, and iabcs are then used in the current-input-based FC

Bn , Bt Maxwell StressTensor Method

fn, ft+

Bns, Bts

Bnr, Btr

InverseReference FrameTransformation

iqd0s

vqd0s

ias, ibs, ics Current-Input-Based FC

CoupledStator Circuit

kns, kts

knr, ktr

Convolution

An Induction Machine Fed By An Inverter

15

3

3

3

cos cos 3

2cos cos 3

3

2cos cos 3

3

a e e

c e e

c e e

d d t d t

d d t d t

d d t d t

3 / 6, 120ed d

A sine-PWM modulation with 3rd-harmonic injection is used. The duty cycles for the three phases are

v

Induction Machine

dc

vag

+

vbg

+

vcg

+

+

+vas

vbs

v

Challenges

16

• Wide Range of Time Scales – (Switching Frequency versus Rotor Time Constant)

• Resolution of n Hz requires a discrete-time simulation of 1/n second

• For a simulation with step size h, the maximum frequency obtained using a discrete-time Fourier transform is 1/(2h)

• Total number of sampling steps in the steady state that is required is 1/(nh)

• Example:

• Desired frequency resolution is 1 Hz

• Step size is 10 μs

• Total number of simulation steps required in steady state is 100,000.

• The large size of rotor basis function and amount of sampling steps add difficulties to computer memory and the computational effort.

Computational Burden of FC

• Dominated by Convolution

• Assume Flux Densities are Calculated at p points in the Airgap with N samples

1

1

1

, ( ) ( ),

2 2( ) ( ) ,

3

2 4( ) ( ) ,

3

k

xr sm k as m xr sm rm k rm m k mm

k

bs m xr sm rm k rm m k mm

k

cs m xr sm rm k rm m k mm

B t i t k t t t t

i t k t t t tP

i t k t t t tP

( 1) ( [ ]) ([ ] 1)

2( ) computations

px px N N xxr

O pN

B C i

17

Multirate Field Construction

18

• In the slow subsystem, FC is used with sampling rate of : Input ias,lf , ibs,lf , ibs,lf

Output Bn,lf , Bt,lf

Low Sampling Reduces Dimension of Convolution Matrix

• In the fast subsystem, ‘Fast’ FC is used with sampling rate of : Input ias,hf , ibs,hf , ibs,hf

Output Bn,hf , Bt,hf

Truncate ‘Fast’ Impulse Response at samples Truncated Impulse Response Reduces Dimension of Convolution MatrixIndeed Size of the Matrix Nearly Independent of Switching Frequency

1/ slowh

1/ fasth

fastN

• Partition Currents into Fast and Slow Components Use ‘slow’ impulse response to calculate ‘slow’ component of flux density Use ‘fast’ impulse response to calculate ‘fast’ component of flux density

Multirate Field Construction

19

, ,as hf as as lfi i i

High-frequency component ias,hf

Low-frequency component ias,lf

Re-sampling ias

0.22 0.225 0.23 0.235 0.24 0.245 0.25 0.255 0.26−40

−30

−20

−10

0

10

20

30

40

Time (sec)

i as (

A)

Total current

Low-frequency componentHigh-frequency component

(time step of )slowh (time step of )fasth

Example Induction Machine Studied

• 3-phase 4-pole squirrel-cage induction machine• 36 stator slots, 45 rotor slots• Rated power: 5 horsepower• Rated speed: 1760 rpm• rs = 1.2

sm

rm

rm

sm rm rm

Machine parameters Value

Airgap 1.42 mm

Rotor outer diameter 136.92 mm

Stator outer diameter 228.6 mm

Stack length 88.9 mm

Shaft diameter 39.4 mm

Lamination material M-19

Stator winding material

Copper

Rotor bar material Aluminum

Number of turns per coil

22

Number of coils per phase

6 coils in series connection

20

Example Operating Conditions

21

wrm=1760 rpm

Vdc = 280 V

Sine-PWM modulation with 3rd harmonics injected

Switching frequency = 1 kHz (set low for FEA computation)

Step size of FC = 1 ms (slow subsystem), 0.01 ms (fast subsystem) (oversampled)

Nfast = 100 samples

Bn,lf = O(999 x 10002) calculations/secondBn,hf = O(999 x 1002) calculations/second

If used Single-rate FC = O(999x1000002) calculations/second

Step size of FEA = 0.01 ms

0.2 0.21 0.22 0.23 0.24 0.25 0.26−40

−30

−20

−10

0

10

20

30

40

Time (sec)

i as (

A)

FEAFC

0 500 1000 1500 2000 2500 3000 3500 40000

1

2

3

4

5

Frequency (Hz)

i as (

A)

FEAFC

Result – Stator Current

22

FEA ~ 270 hoursFC ~ 48 minutes

ias Frequency spectrum of ias

fsw-2fe

fsw+2fefsw-4fe

fsw+4fe

0 500 1000 1500 2000 2500 3000 3500 40000

1

2

3

4

5

Frequency (Hz)

Tor

que

(Nm

)

FEAFC

Result -- Torque

23

Torque Frequency spectrum of Torque

fsw-3fe

fsw+3fe

0.2 0.21 0.22 0.23 0.24 0.25 0.26

10

15

20

25

30

Tor

que

(Nm

)

Time (sec)

FEAFC

Conclusions

• Method to efficiently model fields and forces in inverter-fed induction machines presented– Requires Minimal FEA Evaluations (at Standstill)

• Multi-rate Leads to Relatively Low Computation Burden – Does Not Increase with Switching Frequency

• Can be Applied to Flux Density Field Construciton in Iron, i.e. Calculate Core Loss

• Requires a Partition of Time Scales

24

Acknowledgement

• This work is made possible through the Office of Naval Research Grant no. N00014-02-1-0623.

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