Transcript
Page 1: A Comparison of Nodal- and Mesh-Based Magnetic Equivalent ...cemeold.ece.illinois.edu/seminars/CEME408PurduePekarek.pdf · Mesh-Based Magnetic Equivalent Circuit Models H. Derbas,

A Comparison of Nodal- and Mesh-Based Magnetic

Equivalent Circuit ModelsH. Derbas, J. Williams2, A. Koenig3, S. Pekarek

Department of Electrical and Computer Engineering

Purdue University2Caterpillar, 3Hamilton-Sundstrand

April 7, 2008

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Outline

• Magnetic Equivalent Circuit (MEC) Modeling

• Alternative MEC Formulations– Nodal-based

– Mesh-based

• Comparison of Numerical Properties

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MEC Model of Claw-Pole Machine

ΦΤ(1) ΦΤ(2) ΦΤ(3) ΦΤ(4) ΦΤ(5) ΦΤ(6)

Φ

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MEC Sources

• Magnetomotive Force– Result of Ampere’s current Law

– Represents effects of winding currents

– Incorporates winding layout

– Similar to a voltage source

+_

Ry

Rt1

Rs

Rt2

�l1

�t2�t1

�y

N is s

N is s

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MEC Flux Tubes

• Flux Tubes– Shape determined by engineering judgment

– Establish topology of MEC network

– Incorporate geometry of the machine

u

u

1

2

dx

A( )x�

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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

( ) ( )∫=xAx

dxR

μ

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Example MEC-Based Design Program%------------------------------------------------------------------------------% Stator Input Data%------------------------------------------------------------------------------OD = 0.12985; % STATOR OUTER DIAMETER, mID = 0.09662; % STATOR INNER DIAMETER, mGLS = 26.97e-3; % STATOR STACK LENGTH, mDBS = 4.98e-3; % STATOR YOKE DEPTH, mSFL = 0.99; % STACKING FACTORH0 = 0.64E-3; % STATOR SLOT DIMENSION, mH1 = 0.0; % STATOR SLOT DIMENSION, mH2 = 1.3e-3; % STATOR SLOT DIMENSION, mB0 = 2.4617e-3; % STATOR SLOT DIMENSION, mSYNR = 2.4e-3; % STATOR YOKE NOTCH RADIUS, (weight calculation only), SLTINS = 2.997e-4; % SLOT INSULATION WIDTH, mG1 = 0.305e-3; % MAIN AIR GAP LENGTH, mSAWG = 13.75; % WIRE GUAGE OF ARMATURE WINDING, 1.29e-3TC = 11.0; % NUMBER OF TURNS PER COILESC = 2.54e-3; % ARMATURE WINDING EXTENSION BEYOND STACKRSC = 7.62e-3; % ARMATURE WINDING RADIUS BEYOND STACKCPIT = 3.0; % COIL PITCH IN TEETHSTW = 3.86e-3; % WIDTH OF TOOTH SHANK, mDENS = 7872.0; % DENSITY OF IRON, ROTOR & STATOR, kg/m^3SLTH = 0.828e-3; % STATOR LAMINATION THICKNESS, m

%------------------------------------------------------------------------------% Rotor Input Data%------------------------------------------------------------------------------TED = 12.0e-3; % ROTOR END DISK THICKNESS, mDC = 50.0e-3; % ROTOR CORE DIAMETER, mCL = 28.1e-3; % ROTOR CORE LENGTH, mGLP = 27.0e-3; % LENGTH OF ROTOR POLE, mCID = 51.5e-3; % FIELD COIL INNER DIAMETER, mCOD = 74.0e-3; % FIELD COIL PLASTIC SLOT OUTER DIAMETER, mCOILW = 28.0e-3; % FIELD COIL WIDTH, mWPT = 7.39e-3; % ROTOR TOOTH WIDTH AT TIP OF TOOTH, arclength, mWPR = 27.0e-3; % ROTOR TOOTH WIDTH AT ROOT OF TOOTH, arclength, mHPT = 2.997e-3; % ROTOR TOOTH HEIGHT AT TIP OF TOOTH, mHPR = 11.38e-3; % ROTOR TOOTH HEIGHT AT ROOT OF TOOTH, mTRAD = 1.19e-3; % ROTOR TOOTH BEND RADIUS, mRP = 12.0; % NUMBER OF POLESTRC = 306.0; % FIELD WINDING NUMBER OF TURNSRAWG = 19; % WIRE GUAGE OF FIELD WINDING, 0.813e-3TFLD = 48.8; % HOT FIELD TEMPERATURE, CTA = 20.0; % AMBIENT TEMPERATURE, CSD = 17.575e-3; % SHAFT DIAMETER, mDD = 58.6e-3; % DISK DIAMETER (claw V to claw V), mCW = 7.697e-2; % CHAMFER WIDTH, radCD = 3.0*G1; % CHAMFER DEPTH, m

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Summary Table

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Example System for Research Presented Herein

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Nodal Analysis of Example

P ϕ=A u

[ ]1 11 12 13 1 2 23 22 21 2

T

m c c c s s c c c mu u u u u u u u u u=u

[ ]0 0 0 0 0 0 0 0T

m m m mF P F P= −ϕ

( )m c magF H t=

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Nodal-Based Matrices

1 2

2 4

P PP T

P P

A A

A A

⎡ ⎤= ⎢ ⎥⎣ ⎦

A

1 1

1 1 2 2

2 2 3 31

3 3

0 0 0

0

0

0 0

0 3

m c c

c ag c c c ag

c ag c c c agP

c ag c ag

ag ag ag ag s

P P P

P P P P P P

P P P P P P

P P P P

P P P P P

+ −⎡ ⎤⎢ ⎥− + + − −⎢ ⎥⎢ ⎥− + + − −=⎢ ⎥− + −⎢ ⎥⎢ ⎥− − − +⎣ ⎦

A

3 3

4 3 2 3 2

2 1 2 1

1 1

3 0

0 0

0

0

0 0 0

ag s ag ag ag

ag ag c c

P ag c ag c c c

ag c ag c c c

c m c

P P P P P

P P P P

P P P P P P

P P P P P P

P P P

+ − − −⎡ ⎤⎢ ⎥− + −⎢ ⎥⎢ ⎥= − − + + −⎢ ⎥− − + + −⎢ ⎥⎢ ⎥− +⎣ ⎦

A2

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0

P

sP

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦

A

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Mesh Analysis of Example System

R =A Fϕ

[ ]11 12 22 21

T

m c c c cφ φ φ φ φ=ϕ

[ ]2 0 0 0 0T

mF=F

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Mesh-Based Matrix

12 13 23 22

12 12

13 13

23 23

22 22

2 0 0

2 0 0

0 0 2

0 0 2

M c c ag c ag c

c c ag ag

c ag ag c agR

c ag c ag ag

c ag c ag

R R R R R R R

R R R R

R R R R R

R R R R R

R R R R

− − − +⎡ ⎤⎢ ⎥− + −⎢ ⎥⎢ ⎥− − − +=⎢ ⎥+ + −⎢ ⎥⎢ ⎥− +⎣ ⎦

A

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Modeling Magnetic Materials

B-H Curve μ –B Curve

Nonlinear – Saturation of Material

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Solving Nodal Formulation

( ) 0Pf u = − =A u ϕ

( ) ( )1111

11

1

111 cm

m

ccm

m

uuu

PPP

u

fJ −

∂∂++=

∂∂=

( ) ( ) 0111111 =−−+= mmccmcm PFuPuPPuf

( )11111 cmcc uuP −=Φ1

11

c

cc A

=

Utilizing Newton Raphson ( ) ( )1J

+ ⎡ ⎤= − ⎣ ⎦n 1 n n nu u u f u

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1

1

rc

cB

μ∂∂

( )1

1

2

1

210

1

1

ln rc

c

clawrc

c P

lll

llNP

μμ

μ=

⎟⎠⎞⎜

⎝⎛

−=

∂∂

11

1 1

cc

c

A

B=

Φ∂∂

( ) 11111

1

1

1ccm

m

c

m

c Puuu

P

u+−

∂∂

=∂Φ∂

1 1 1 1 1

1 1 1 1 1

c c rc c c

m rc c c m

P P B

u B u

μμ

∂ ∂ ∂ ∂ ∂Φ=

∂ ∂ ∂ ∂Φ ∂

( )111

1

1

1

1 cm

c

m

c

uu

P

u

P

−−=

∂∂

γβαγβα

from μ-B curve

Closed-form Expression for Jacobian

So, one can establish closed-form expression

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Solving Mesh Formulation

( ) 0Rg ϕ = − =A Fϕ

( ) ( )( )

1 12 11 13 12

23 22 22 21 2 0

M m c c c ag c

c ag c c c m

g R R R R

R R R F

ϕ φ φ φ

φ φ

= − − +

+ + + − =

Rc11, Rc12, Rc13, Rc21, Rc22, and Rc23 are flux-dependent reluctances

( )

( ) ( ) ( )

11 21 1211 11

22 13 2321 12 22

c c cM m m m c

m m m

c c cm c m c m c

m m m

R R RJ R

R R R

φ φ φ φφ φ φ

φ φ φ φ φ φφ φ φ

∂ ∂ ∂= + + + − +

∂ ∂ ∂∂ ∂ ∂

+ + − + +∂ ∂ ∂

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Closed-form Expression for Jacobian

m

c

c

c

c

rc

rc

c

m

c B

B

RR

φμ

μφ ∂Φ∂

Φ∂∂

∂∂

∂∂

=∂∂ 12

12

12

12

12

12

1212

12

12

rc

cB

μ∂∂

Obtained from μ-B curve

121212

12 1c

rcrc

c RR

μμ−=

∂∂

1212

12 1

cc

c

A

B=

Φ∂∂

112 −=∂Φ∂

m

c

φ

12

12

1212

1212

c

rc

crc

c

m

c

BA

RR

∂∂=

∂∂ μ

μφ

So, one can establish closed-form expression (much less tedious than permeance-based)

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Comparison of Nodal and Mesh-Based Formulations

4 iterationsMesh Model Analytic Jacobian

3 iterationsNodal Model Analytic Jacobian

3 iterationsNodal Model Approx. Jacobian

ConvergenceImplementation

Hc = 105 A/m

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Comparison of Nodal and Mesh-Based Formulations

5 iterationsMesh Model Analytic Jacobian

DNCNodal Model Analytic Jacobian

5 iterationsNodal Model Approx. Jacobian

ConvergenceImplementation

Hc = 8*105 A/m

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Comparison of Nodal and Mesh-Based Formulations

6 iterationsMesh Model Analytic Jacobian

DNCNodal Model Analytic Jacobian

53 iterationsNodal Model Approx. Jacobian

ConvergenceImplementation

Hc = 16*105 A/m

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Interpretation of Results

( )[ ] ( )nnn1n xfxxx 1−+ −= J

~ 500Mesh Model Analytic Jacobian

~ 2 x 108Nodal Model Analytic Jacobian

~ 2 x 105Nodal Model Approx. Jacobian

Condition NumberImplementation

Ill-conditioning mainly due to difference in airgappermeance and claw permeances

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Scaling to Help?

1 1

1 1 2 2

2 2 3 31

3 3

0 0 0

0

0

0 0

0 3

m c c

c ag c c c ag

c ag c c c agP

c ag c ag

ag ag ag ag s

P P P

P P P P P P

P P P P P P

P P P P

P P P P P

+ −⎡ ⎤⎢ ⎥− + + − −⎢ ⎥⎢ ⎥− + + − −=⎢ ⎥− + −⎢ ⎥⎢ ⎥− − − +⎣ ⎦

A

3 3

4 3 2 3 2

2 1 2 1

1 1

3 0

0 0

0

0

0 0 0

ag s ag ag ag

ag ag c c

P ag c ag c c c

ag c ag c c c

c m c

P P P P P

P P P P

P P P P P P

P P P P P P

P P P

+ − − −⎡ ⎤⎢ ⎥− + −⎢ ⎥⎢ ⎥= − − + + −⎢ ⎥− − + + −⎢ ⎥⎢ ⎥− +⎣ ⎦

A

Not in this case

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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

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Conclusions• Nodal-based and Mesh-based MEC have different

numerical properties

• Mesh-based has better convergence in Newton Raphsonformulations– ‘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


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