A Multirate Field Construction Technique for Efficient Modeling of the Fields and Forces within Inverter-Fed Induction Machines

Transcription

1 A Multirate Field Construction Technique for Efficient Modeling of the Fields and Forces within Inverter-Fed Induction Machines Dezheng Wu, Steve Peare School of Electrical and Computer Engineering Purdue University September 30, 2010

2 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/Peare 2

3 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 B n =B ns +B nr B t =B ts +B tr Calculate torque and radial force using the Maxwell Stress Tensor (MST) method under arbitrary stator excitation and rotor speed ft Bn. Bt, fn Bn Bt PR l 2 2 z e 0 t s s r z 0 n s 3 Torque T f d, Radial Force F l f Rd 2

4 Assumptions The flux density in the axial direction is zero rm Hysteresis in the iron is neglected sm rm Thermal conditions are assumed constant No deformation occurs in stator and rotor teeth Linear magnetics 4

5 Stator Basis Function Derivation ns ts sm Bnas / I sm Btas / I0 0 ns ts sm sm 0.02 ns 0.01 ts ns [T/A] ts [T/A] sm [deg] sm [deg] 5

6 Rotor Basis Function ( nr, tr ) Derivation Impulse Response 1. Set a discrete-time impulse input to a transient FEA program i as (t) = I 0 when t = t 0 i as (t) = 0 when t t 0 2. Record the flux density components (B nid, B tid ) for t t Subtract the stator magnetic field B nr = B nid i as ns, B tr = B tid i as ts 4. Divided by I 0 nr = B nr / I 0, tr = B tr / I 0 nr, rm nr tr ns, ts rm Bnr / I rm Btr / I0 tr 0 rm 6

7 Complete Characterization Process 7

8 Magnetic Flux Density Due to Stator The flux density generated by arbitrary stator phase-a current is approximated as B B i i nas sm as ns sm tas sm as ts sm Due to symmetry, the total flux density generated by stator currents Bns sm iasns sm ibsns sm icsns sm P 3 P 3 due to i due to i Bts sm iasts sm ibsts sm icsts sm P 3 P 3 bs cs 8

9 Magnetic Flux Density Due to Rotor Obtain rotor magnetic field using the convolution of stator current signal and rotor basis function due to i as due to i bs due to i cs Bxr rm, t ias txr rm, tibs txr rm, t ics t xr rm, t P 3 P ias t xr sm rm, tibs t xr sm rm, t P ics t xr sm rm, t P 3 where x can be n or t 9

10 Complete Field Construction Stator Current as Model Input Obtain the total flux density in the discrete-time form,,,,,, B t B t B t n sm ns sm nr sm B t B t B t t sm ts sm tr sm In the computer, the discrete convolution of the stator current and rotor basis function, ( ) ( ), B t i t t t t t xr sm as m xr sm rm rm m m m1 where x can be n or t 2 2 i t ( t ) ( t ), t m1 P 3 t 2 4 i t ( t ) ( t ), t m1 P 3 t bs m xr sm rm rm m m cs m xr sm rm rm m m 10

11 Voltage-Input-Based FC Technique Basic idea: v v i i Current-inputbased FC B n, B t Stator voltage equations are used to relate voltage and current: v i r p qs qs s ds qs v i r p ds ds s qs ds v i r p 0s 0s s 0s where is the angular speed of an arbitrary reference frame, and the flux linages are expressed as Li qs ss qs qs, r Li ds ss ds ds, r Li 0s ls 0s Due to the induced rotor current Unnowns: L ss, L ls, λ qs,r, λ ds,r 11

12 Characterization of Rotor Basis Flux Linage Use the same FEA solutions as in the characterization of stator and rotor basis functions. i as = constant Static FEA Solution abcs L ss = L ls = as bs i as as bs i as L ss, L ls i as = impulse I p, when t=0 ={ 0, else Transient FEA Solution abcs (t) (t= qs (tl ss i qs (t 2I p /3 t Impulse response (vector) 12

13 Calculate qs,r, ds,r Procedure: 1. convolution. 2. transformation between reference frames, ( ) n qs r tn iqs ( tm) ( tn, tm) ( tn tm) ds, r ( tn) K m1 ids ( tm) cos( ) sin( ) K( tn, tm) sin( ) cos( ) ( t ) ( t ) n m ( t ) ( t ) r n r m where r is the electric rotor angle, and is the angle of the reference frame 13

14 Voltage-Input Based FC Diagram v qd0s Then i qd0s i abcs, and i abcs are then used in the current-input-based FC i qds = L -1 S i qd0s Inverse Reference Frame Transformation Coupled Stator Circuit i as,i bs, i cs ns, ts B ns, B ts Current-Input-Based FC Convolution nr, tr B nr, B tr + B n, B t Maxwell Stress Tensor Method f n, f t 14

15 An Induction Machine Fed By An Inverter + v dc + + v ag v cg v bg v bs v as v cs + Induction Machine A sine-pwm modulation with 3rd-harmonic injection is used. The duty cycles for the three phases are d dcos t d cos 3 t a e 3 e 2 dc dcoset d3cos 3 et 3 2 dc dcoset d3cos 3 et 3 d3 d / 6, e

16 Challenges 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. 16

17 Computational Burden of FC Dominated by Convolution, ( ) ( ), B t i t t t t t xr sm as m xr sm rm rm m m m1 2 2 i t ( t ) ( t ), t m1 P 3 t 2 4 i t ( t ) ( t ), t m1 P 3 t bs m xr sm rm rm m m cs m xr sm rm rm m m Assume Flux Densities are Calculated at p points in the Airgap with N samples B C i ( px1) ( px[ N ]) ([ N ] x1) xr 2 O( pn ) computations 17

18 Multirate Field Construction 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 1/ hslow In the slow subsystem, FC is used with sampling rate of : Input i as,lf, i bs,lf, i bs,lf Output B n,lf, B t,lf Low Sampling Reduces Dimension of Convolution Matrix 1/ hfast In the fast subsystem, Fast FC is used with sampling rate of : Input i as,hf, i bs,hf, i bs,hf Output B n,hf, B t,hf Truncate Fast Impulse Response at N fast samples Truncated Impulse Response Reduces Dimension of Convolution Matrix Indeed Size of the Matrix Nearly Independent of Switching Frequency 18

19 Multirate Field Construction Total current 20 i as (A) Low-frequency component High-frequency component Time (sec) Low-frequency component i as,lf (time step of ) slow Re-sampling i as High-frequency component i as,hf h (time step of ) i i i h fast as, hf as as, lf 19

20 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 r s = 1.2 sm rm rm rm sm rm Machine parameters Airgap Rotor outer diameter Stator outer diameter Stac length Shaft diameter Value 1.42 mm mm mm 88.9 mm 39.4 mm Lamination material M-19 Stator winding material Rotor bar material Number of turns per coil Number of coils per phase Copper Aluminum 22 6 coils in series connection 20

21 Example Operating Conditions rm =1760 rpm V dc = 280 V Sine-PWM modulation with 3 rd harmonics injected Switching frequency = 1 Hz (set low for FEA computation) Step size of FC = 1 ms (slow subsystem), 0.01 ms (fast subsystem) (oversampled) Nfast = 100 samples B n,lf = O(999 x ) calculations/second B n,hf = O(999 x ) calculations/second If used Single-rate FC = O(999x ) calculations/second Step size of FEA = 0.01 ms 21

22 Result Stator Current FEA ~ 270 hours FC ~ 48 minutes f sw -4f e f sw -2f e fsw+2fe FEA FC 5 4 f sw +4f e FEA FC i as (A) i as (A) Time (sec) i as Frequency (Hz) Frequency spectrum of i as 22

23 Result -- Torque f sw -3f e FEA FC 5 4 f sw+3f e FEA FC Torque (Nm) Torque (Nm) Time (sec) Frequency (Hz) Torque Frequency spectrum of Torque 23

24 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

25 Acnowledgement This wor is made possible through the Office of Naval Research Grant no. N