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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

  • 3

    MEC Model of Claw-Pole Machine

    (1) (2) (3) (4) (5) (6)

  • 4

    MEC Sources

    Magnetomotive Force Result of Amperes current Law

    Represents effects of winding currents

    Incorporates winding layout

    Similar to a voltage source

    +_

    Ry

    Rt1

    Rs

    Rt2

    l1

    t2t1y

    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

    ( ) ( )= xAxdx

    R

  • 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

  • 8

    Summary Table

  • 9

    Example System for Research Presented Herein

  • 10

    Nodal Analysis of Example

    P =A u[ ]1 11 12 13 1 2 23 22 21 2 Tm 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 0 Tm m m mF P F P=

    ( )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

  • 12

    Mesh Analysis of Example System

    R =A F

    [ ]11 12 22 21 Tm c c c c =

    [ ]2 0 0 0 0 TmF=F

  • 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

  • 14

    Modeling Magnetic Materials

    B-H Curve B Curve

    Nonlinear Saturation of Material

  • 15

    Solving Nodal Formulation

    ( ) 0Pf u = =A u

    ( ) ( )1111

    11

    1

    111 cm

    m

    ccm

    m

    uuu

    PPP

    u

    fJ

    ++=

    =

    ( ) ( ) 0111111 =+= mmccmcm PFuPuPPuf( )11111 cmcc uuP =

    1

    11

    c

    cc A

    B

    =

    Utilizing Newton Raphson ( ) ( )1J + = n 1 n n nu u u f u

  • 16

    1

    1

    rc

    cB

    ( )1

    1

    2

    1

    210

    1

    1

    ln rc

    c

    clawrc

    c P

    lll

    llNP

    =

    =

    11

    1 1

    cc

    c

    A

    B=

    ( ) 11111

    1

    1

    1ccm

    m

    c

    m

    c Puuu

    P

    u+

    =

    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

    =

    ( )1111

    1

    1

    1 cm

    c

    m

    c

    uu

    P

    u

    P

    =

    from -B curve

    Closed-form Expression for Jacobian

    So, one can establish closed-form expression

  • 17

    Solving Mesh Formulation

    ( ) 0Rg = =A F

    ( ) ( )( )

    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

    = +

    + + + =

    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

    = + + + +

    + + + +

  • 18

    Closed-form Expression for Jacobian

    m

    c

    c

    c

    c

    rc

    rc

    c

    m

    c B

    B

    RR

    = 12

    12

    12

    12

    12

    12

    1212

    12

    12

    rc

    cB

    Obtained from -B curve

    121212

    12 1c

    rcrc

    c RR

    =

    1212

    12 1

    cc

    c

    A

    B=

    112 =

    m

    c

    12

    12

    1212

    1212

    c

    rc

    crc

    c

    m

    c

    BA

    RR

    =

    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 magn

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