A Comparison of Nodal- and Mesh-Based Magnetic Equivalent Circuit Models

Transcription

1 A Comparison of Nodal- and Mesh-Based Magnetic Equivalent Circuit Models H. Derbas, J. Williams 2, A. Koenig 3, S. Pekarek Department of Electrical and Computer Engineering Purdue University 2 Caterpillar, 3 Hamilton-Sundstrand April 7, 2008

2 Outline Magnetic Equivalent Circuit (MEC) Modeling Alternative MEC Formulations Nodal-based Mesh-based Comparison of Numerical Properties 2

3 MEC Model of Claw-Pole Machine Φ Τ(1) Φ Τ(2) Φ Τ(3) Φ Τ(4) Φ Τ(5) Φ Τ(6) Φ 3

4 MEC Sources Magnetomotive Force Result of Ampere s current Law Represents effects of winding currents Incorporates winding layout Similar to a voltage source R y y t1 t2 R t1 R t2 Ni ss l1 R s + _ Ni ss 4

5 MEC Flux Tubes Flux Tubes Shape determined by engineering judgment Establish topology of MEC network Incorporate geometry of the machine dx A( x) u 2 u 1 5

6 Node Potentials and Reluctance Magnetic Scalar Potentials Represent node potentials Reluctance Calculated from geometry of flux tube Similar to resistance Allows for effects of magnetic saturation R μ dx x ( ) A( x) 6

7 Example MEC-Based Design Program % % Stator Input Data % OD ; % STATOR OUTER DIAMETER, m ID ; % STATOR INNER DIAMETER, m GLS 26.97e-3; % STATOR STACK LENGTH, m DBS 4.98e-3; % STATOR YOKE DEPTH, m SFL 0.99; % STACKING FACTOR H0 0.64E-3; % STATOR SLOT DIMENSION, m H1 0.0; % STATOR SLOT DIMENSION, m H2 1.3e-3; % STATOR SLOT DIMENSION, m B e-3; % STATOR SLOT DIMENSION, m SYNR 2.4e-3; % STATOR YOKE NOTCH RADIUS, (weight calculation only), SLTINS 2.997e-4; % SLOT INSULATION WIDTH, m G e-3; % MAIN AIR GAP LENGTH, m SAWG 13.75; % WIRE GUAGE OF ARMATURE WINDING, 1.29e-3 TC 11.0; % NUMBER OF TURNS PER COIL ESC 2.54e-3; % ARMATURE WINDING EXTENSION BEYOND STACK RSC 7.62e-3; % ARMATURE WINDING RADIUS BEYOND STACK CPIT 3.0; % COIL PITCH IN TEETH STW 3.86e-3; % WIDTH OF TOOTH SHANK, m DENS ; % DENSITY OF IRON, ROTOR & STATOR, kg/m^3 SLTH 0.828e-3; % STATOR LAMINATION THICKNESS, m % % Rotor Input Data % TED 12.0e-3; % ROTOR END DISK THICKNESS, m DC 50.0e-3; % ROTOR CORE DIAMETER, m CL 28.1e-3; % ROTOR CORE LENGTH, m GLP 27.0e-3; % LENGTH OF ROTOR POLE, m CID 51.5e-3; % FIELD COIL INNER DIAMETER, m COD 74.0e-3; % FIELD COIL PLASTIC SLOT OUTER DIAMETER, m COILW 28.0e-3; % FIELD COIL WIDTH, m WPT 7.39e-3; % ROTOR TOOTH WIDTH AT TIP OF TOOTH, arclength, m WPR 27.0e-3; % ROTOR TOOTH WIDTH AT ROOT OF TOOTH, arclength, m HPT 2.997e-3; % ROTOR TOOTH HEIGHT AT TIP OF TOOTH, m HPR 11.38e-3; % ROTOR TOOTH HEIGHT AT ROOT OF TOOTH, m TRAD 1.19e-3; % ROTOR TOOTH BEND RADIUS, m RP 12.0; % NUMBER OF POLES TRC 306.0; % FIELD WINDING NUMBER OF TURNS RAWG 19; % WIRE GUAGE OF FIELD WINDING, 0.813e-3 TFLD 48.8; % HOT FIELD TEMPERATURE, C TA 20.0; % AMBIENT TEMPERATURE, C SD e-3; % SHAFT DIAMETER, m DD 58.6e-3; % DISK DIAMETER (claw V to claw V), m CW 7.697e-2; % CHAMFER WIDTH, rad CD 3.0*G1; % CHAMFER DEPTH, m 7

8 Summary Table 8

9 Example System for Research Presented Herein 9

10 Nodal Analysis of Example ( F Ht ) m c mag u Au ϕ P [ u u u u u u u u u u ] m s1 s2 c23 c22 c21 m2 [ FP FP] ϕ m m m m T 10 T

11 Nodal-Based Matrices A P AP1 AP2 T AP2 A P4 A P1 Pm + Pc 1 Pc P P + P + P P 0 P 0 P P + P + P P P ag c2 c2 ag c2 ag c2 c3 c3 ag 0 0 Pc3 Pag + Pc3 Pag 0 Pag Pag Pag 3Pag + P s A P P s A 3Pag + Ps Pag Pag Pag 0 Pag Pag + Pc 3 Pc P P P + P + P P 0 Pag 0 Pc 2 Pag + Pc 1+ Pc 2 Pc P P + P P4 ag c3 ag c2 c3 c2 m 11

12 Mesh Analysis of Example System ϕ A R ϕ F [ φ φ φ φ φ ] F m 1 2 c22 c21 [ 2F ] T m T 12

13 Mesh-Based Matrix A R R R R R R + R R Rc 12 Rc Rag Rag 0 0 R R R R + 2R 0 0 M 2 3 ag c23 ag c22 3 ag ag 3 ag Rc 23 + Rag 0 0 Rc Rag Rag R 0 0 R R + 2R c22 ag c22 ag 13

14 Modeling Magnetic Materials Nonlinear Saturation of Material B-H Curve μ B Curve 14

15 Solving Nodal Formulation f J f u ( ) A u ϕ 0 1 Utilizing Newton Raphson u u J ( u ) f( u ) f P ( u) ( P + P ) u P u F P 0 1 m m1 1 m m Φ P ( u u ) m1 1 + n 1 n n n P ( P + P ) + ( u u ) 1 11 m m1 1 um 1 um 1 B Φ A 15

16 Closed-form Expression for Jacobian P P μ B Φ u μ B Φ u r m1 r m1 P μ μ r r B c 1 ( l l ) μ0 N 1 2 P l l ln 1 μ claw l 2 from μ-b curve r B Φ 1 A Φ u m1 P u m1 ( um 1 uc 11) + Pc 1 So, one can establish closed-form expression P u m1 α β γ P 1 α β γ m1 1 ( u u ) 16

17 Solving Mesh Formulation ( ) A ϕ F 0 g ϕ R ( ) ( + ) g ϕ R φ R φ R R φ 1 M m ag 2 ( ) + R 23 + R φ 22 + R 22φ 21 2F 0 c ag c c c m R1, R2, R3, Rc21, Rc22, and Rc23 are flux-dependent reluctances J R R R R + φ + φ + φ φ + ( ) 1 c M m m m 1 φm φm φm R R R ( φ + φ ) + ( φ φ ) + ( φ + φ ) c22 3 c23 m c21 m 2 m c22 m m m φ φ φ 17

18 Closed-form Expression for Jacobian R 2 2 r2 2 Φ 2 φ m R μ r2 μ B 2 B Φ 2 φ m R μ μ 2 1 r2 r2 B c 12 μ r2 R 2 Obtained from μ-b curve R φ 2 m R r2 2 A 2 B Φ Φ φ μ B A So, one can establish closed-form expression (much less tedious than permeance-based) μ m r2 2 18

19 Comparison of Nodal and Mesh-Based Formulations H c 10 5 A/m Implementation Nodal Model Approx. Jacobian Nodal Model Analytic Jacobian Mesh Model Analytic Jacobian Convergence 3 iterations 3 iterations 4 iterations 19

20 Comparison of Nodal and Mesh-Based Formulations H c 8*10 5 A/m Implementation Nodal Model Approx. Jacobian Nodal Model Analytic Jacobian Mesh Model Analytic Jacobian Convergence 5 iterations DNC 5 iterations 20

21 Comparison of Nodal and Mesh-Based Formulations H c 16*10 5 A/m Implementation Nodal Model Approx. Jacobian Nodal Model Analytic Jacobian Mesh Model Analytic Jacobian Convergence 53 iterations DNC 6 iterations 21

22 Interpretation of Results x x n+ 1 n J [ ( )] 1 x f ( x ) n n Implementation Nodal Model Approx. Jacobian Nodal Model Analytic Jacobian Mesh Model Analytic Jacobian Condition Number ~ 2 x 10 5 ~ 2 x 10 8 ~ 500 Ill-conditioning mainly due to difference in airgap permeance and claw permeances 22

23 Scaling to Help? A P1 Pm + Pc 1 Pc P P + P + P P 0 P 0 P P + P + P P P ag c2 c2 ag c2 ag c2 c3 c3 ag 0 0 Pc3 Pag + Pc3 Pag 0 Pag Pag Pag 3Pag + P s A 3Pag + Ps Pag Pag Pag 0 Pag Pag + Pc 3 Pc P P P + P + P P 0 Pag 0 Pc 2 Pag + Pc 1+ Pc 2 Pc P P + P P4 ag c3 ag c2 c3 c2 Not in this case m 23

24 Challenge of Mesh-Based Implementation Physical movement between magnetic materials Permeances go to zero with non-overlap Reluctances go to infinity with non-overlap Loops are position dependent Not a challenge for stationary magnetics Can use discrete rotor positions for machine design and create set of stationary magnetics Recent research has shown promising algorithmic method to automate loop changes 24

25 Conclusions Nodal-based and Mesh-based MEC have different numerical properties Mesh-based has better convergence in Newton Raphson formulations Exact nodal formulation of Jacobian highly ill-conditioned Approximate Jacobian better conditioned, but still poor relative to mesh-based In cases where motion represented, Mesh-based formulation must deal with infinite reluctance Algorithmic means of dealing with infinite reluctance is being considered 25