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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
2
Outline
• Magnetic Equivalent Circuit (MEC) Modeling
• Alternative MEC Formulations– Nodal-based
– Mesh-based
• Comparison of Numerical Properties
4
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
�
5
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�
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
( ) ( )∫=xAx
dxR
μ
7
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
10
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=
11
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
13
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
15
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
BΦ
=
Utilizing Newton Raphson ( ) ( )1J
−
+ ⎡ ⎤= − ⎣ ⎦n 1 n n nu u u f u
16
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
17
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
∂ ∂ ∂= + + + − +
∂ ∂ ∂∂ ∂ ∂
+ + − + +∂ ∂ ∂
18
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)
19
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
20
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
21
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
22
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
23
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
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
25
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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