Mathematical models for Induction

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    M A T H E M A T I C A L M O D E L S O R I N D U C T I O NMACHINES

    P. Pillay, S e n i o r Member IEEE and V . Levin

    Depar t ment of El ec t r i c al Engi neer i ng

    Uni ver s i t y of New Or l eansNew Or l eans L 7 0 1 4 8Ph ( 5 0 4 ) 2 8 6 - 7 1 6 1 ; Fax: ( 5 0 4 ) 2 8 6 - 3 9 5 0

    Abstract-Diffe rent mathematical models have been usedover the years to examine different problems associated withinduction motors. These range from the simple equivalent circuitmodels t o more complex d,q models and abc models which allowthe inclusion of various forms of impedance and/or voltageunbalance. Recently, hybrid models have been developed whicha l low the inclusion of supply side unbalance but with thecomputational eco nom y of the d,q models. This paper presentsthese models with typical results and provides guidelines for theiruse.

    I.INTRODUCTION.

    The well kn ow n equivalent circuit model of the inductionmotor [11 has been widely used over the years to examine thesteady state behavior of induction motor. Both ABC and variousforms of d,q models have been used to s tudy transient behavior [21.Lately, several hybrid models have also been developed for themodeling of machines for particular motor drive or transientoperation [31. This is a review paper, with the aim of presentingthese models together wi th results and guidelines for their use. Thepaper is organized as follows: Section II presents the transformertype equivalent circuit model. Section 111 presents the ABC model.Section IV presents the d,q models. Sections V and VI presentABC/dq and DQ/abc models. Section VI1 has the conclusions.

    21 22

    I

    frequency f f r e q u e n c y sf0 )

    f r e q u e n c y f (b) f r e q u e n c y s f

    Fig.1. The conventional (a) and transformer type (b) equivalentcircuits of the induction motor.

    II CONVENT1 ON AL AN D TRAN S FORMER TYPE

    EQUIVALENT CIRCUITS OF THE INDUCTION MOTOR.

    The conventional equivalent circuit (EC) for an inductionmotor is sh own in Fig.la. This circuit has been widely used forstudying the steady state operation of induct ion motors. It can giveerroneous results wh en either the stator or rotor circuits have powerelectronic devices connected t o them, if the machine itself has anyphase unbalance or during severe transients created during startingor autoreclosing.

    The transformer type EC [4,51, a variation of theconventional circuit (Fig.lb), can be used with fair accuracy forrectifier calculations, while including the effect of source impedanceinduced overlap. Parameters of the transformer typ e EC are relatedt o those of the conventional EC in the following way:

    - - - - -z,=Q+s; q,=q +z k=nz,,J z,+z,);

    where the s ymbols have their usual meanings.

    of operation w ith sinusoidal cur rents are as follows:From Fig.1 b the moto r equations f or the steady state mode

    - - _ _Z =Z,,+k Z2

    The application of the conv entional EC in a wide variety ofapplications is well kno wn and not included here. As an example o fthe use of the transformer type model, its application to a slipenergy recovery induction motor drive with a step-down chopperbetween the rotor rectifier and inverter is given in Fig.2 [41. Fig.2shows the predicted stator, supply and rotor current waveforms ofa 4-pole, slip- energy recovery induction m otor drive together wit htheir measured counterparts at a speed 1300 rpm. The resultsindicate that the drive performance can be fairly accuratelycalculated with the EC circuit. This model does not work well at aslip o f 116 where a peculiar harmonic effect takes place. A moredetailed model is needed for this purpose which is discussed later.

    Both the transformer type model and conventionalequivalent circuit of the induction mot or neglect mutual inductanceeffects and therefore cannot be applied for an accurate prediction ofthe transients in the moto r. Hence more rigorous models should beused for the analysis of the motor, particularly when driven byvariable speed drives, when the machine has impedance unbalanceor subjected to certain forms of supply unbalance.

    0-7803-3008-0195 $4.00 0 1995 IEEE 606

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    Ca,L

    ;?

    aa3

    CO

    --

    T i m e i n s )

    t10

    I5t Time m s )

    Measuredio e

    Fig.2. Measured and calculated waveforms at speed 1300 rpm.

    111 THE ABCiabc MOTOR MODEL.

    A . Basic ABC/abc model.

    All moto r models for the transient analysis of an inductionmotor are based on the so-called ABC/abc model. The equations ofthis model are derived under assumptions that the MMF in the airgap of the motor is sinusoidal, there is negligible saturation andnegligible losses in the core of the machine. Then the simplifiedschematic of the stato r and rotor windi ngs in Fig.3 can be used toobtain the electromagnetic equations.

    Fig.3. The schematic diagram of a 3-phase induction mot or f o rABC/abc model.

    The most cieneral form of these equations i s as follows:

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    0 0 r,, 0 0 0

    0 0 O r , O

    0 0 0 0 rrb 0

    0 0 0 0 0 r , c i

    r - 0 0 0 0

    O r , O 0 0

    i,

    O i

    i ,

    O i

    O ii+

    wherev ,..., ,, are applied phase voltages;lea, ..., IC are currents of each rotor and stator phase;r ..., ,, are resistances of each phase;L,,,, ..., ,,, are the leakage inductances of each phase;8 is electrical angle betwe en the axis of stator phase A

    MAA,..,Mcc are self inductances or mutual inductances

    . .

    and axis of rotor phase a;

    P i1

    between each phase of the stator (rotor)and each of the otherstator or rotor phases when 8 = 0 ;

    p i s the symbol for differentiation;fAA(8),.., CcW re functions of 8;f m( 8 ) , . . . , f cc ( 0 ) are derivatives of the above functions;U - is electrical angular speed of the m otor

    If both stator and rotor windings are electrically andmagnetically symmetrical, a more well kn ow n fo rm of equation 12)results:

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    where M - is mutual inductan ce betwee n a stator phase and a rotorphase when O = O ;

    L, = LIS+Ma,: L', = L,, +Ms,:a, = cos@: b, =sin@;a2 = cos( @ 2n/3) ; b, = sin@ + 2n/3);a3 =cos(@-2n/3); b3=sin(O-2n/3)

    The expression for electromagnetic torque is as follows:

    P2

    T,= -MJ(imim +iSbirb+i,i,)sin8 +(isairb+ isbimiJJsin(e +2x/3)+

    where P is the number of poles in the motor.

    induction motors,In the case of Y or A stator connected squirrel cage

    and the parameters of the ABC/abc model are related to theparameters of the EC of Fig.1a in the following simple way:

    Thus the ABC/abc model allows the tracking of the"natural" phase currents directly at any time of a transient. Also themodel of the for m in (2) is not l imited by conditions of symmetry ofeither supply voltages or phase impedances. Therefore the ABC/abcmodel, is suitable for a study of complex unbalance in the motor(operation of the mot or wi th a nonuniform air gap, operation of themotor under bo th unbalanced voltages and unbalanced impedances,etc.) w hen the simp lifyi ng assumptions of alt ernative models 161render the m inapplicable. A s an example of the application of theABC/abc model, transients during start up of a 2 2 KW inductionmotor (with parameters given in the Appendix I) are presented inFig.4.

    The inherent defect of the direct application of theABC/abc model for digital simulation of the motor transients i s thelarge computation time required for inversion of the time-varyinginductance matr ix in (3) during each step of integration. Many othermodels and approaches were developed to avoid this time-consuming operation.

    0.g

    8.3

    u.7

    2 5.03.4

    E 1.a2 0.2sg-1 '4

    -3.1

    -4.7

    fi 7-0 25 50 75 100 125 150 175 200 225 250

    Time (ms)

    5.20 r

    b) Start up electric 'torqueFig.4. Start u p st,ator current and torque of a 22 k W induction

    motor (using ABC/abc model).

    B. ABC/abc moldel without inversion o f the inductance matrix

    One straiightforward approach of avoiding the problem ofthe inversion of the variable inductance matrix, is to invert theinductance matrix analytically before numerical integration of theequations. In general, such a procedure is extr emely difficult. But ifthe phase impedances are symmetrical and conditions in (5) resatisfied, then explicit expressions for derivatives of the currents canbe obtained relatively easily. Now the ABClabc model has thefollowing form:

    where [il s the column of stator and rotor phase currents; [VI s thecolum n of applied voltages: [ AI, [ Bl are 6 b y 6 and 6 by 3 matricesrespectively

    The entries of [AI and CBI are given in Appendix II.Equation (7) can be integrated much faster than equation 3). nFig.5 the results of transient analysis of the same motor as inpreceding chapter but using (7) are shown. The current and torquewaveforms are identical with those o f Fig.4.

    However one can see that this m odel still has 7 equations.Reference frame theory allows the reduction from six equations tofour, with a constant inductance matrix.

    IV. D-Q MODELS OF THE INDUCTION M OTOR.

    A . The theory of d q models.

    Park's transformation can be applied to the ABC/abcmodel of any symmetrical induction machine. In the general case,such a transformation leads to the dqO reference frame L6.71 whichcan be used for studying certain types of unbalanced operation ofthe motor as well as for stability analysis and controller design.

    Fig.7 shows a schematic of a 3-phase induction motorwit h the q,d axes superimposed. The d axis lags the q axis by 90"(electrical). Coils QS, DS, qr, dr replace the real phase coils AS, BS,CS, ar, br, or. d,q variables are obtained f ro m abc variables byapplication of the Park transformation be low:

    (a) Start up stator current

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    9 0

    a 37

    7 5 0

    3.4

    1.8

    ?

    a

    ; .2S -1 .4

    -3.1

    4.7

    - 6 7.-0 25 50 75 100 125 150 175 2W 225 250

    Time (ms)

    5 2

    4.5

    -201 v ' ' ' ' ' ' ' ' ' ' 'a w 50 75 100 125 150 175 zoa 2 2 5 250

    Time(ms)

    (a) Start up stator current (b) Start up torque

    Fig.5. Stator transients of the motor using the transformed ABC/abc model.

    +q o x i s#

    and

    Fig.6. The schematic diagram of a 3-phase induction motor for d-qmodel. where Wqs, W ,,, W q , , W,, denote flux linkages of t he coils in the d-q

    frame; Os, p are the angles betwe en the q axis and stator phase Aand rotor phase A respectively; L is the same as in 6 ) (apparentmutua l inductance of the motor); L=L,,+L,=L',+(1/2)M,, is theapparent self inductance of a stat or phase; L =L,, + L= L', + ( 1 2)M,,is the apparent self inductance of a rotor phase.

    The choice o f the angle 8, and hence the speed of therotation of the d-q frame, defines the type of the d-q model. Theexpression for electromagnetic torque however does not depend onthe particular reference frame and has the f ollow ing general form:

    se cos(e-y) cos(e+y)

    where y=2 rr/3 , while abc variables are obtained by the inverse Parktransform:

    The general form o f the voltage balance equations for the In this case O,=O and the q,d axes are stationary. Henced-q model is as follows: the equations are as follows:

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    . 13)

    where [ R I~

    I S 4 by 4 matr ix of stator resistances rm and rotorresi sta nces r,; v = V,,cos(w,t);v,, =-V,,sin(w,t); V - is the peakvalue of the stator voltage; U - is electrical synchronous speed ofthe motor

    The expressions for vqr and v depend on the frequencyand the phase of the voltage applied to the rotor. Usually for mos tpractical applications of the stationary frame, v =vd, =O.

    Fig.7 shows the results of the computer simulation of startup currents of t he 22 k W nduction motor in the stationary referenceframe. The q-axis stator variables of the stationary reference framebehave in the same way as do the physical stator variables. Inparticular, the iq current coincides with the actual stator phase Acurrent (Fig.7).

    Therefore this mode l is advantageous when only transientsin the stator are of interest as in the case of sttidying statortransients of squirrel cage induction motors connected to the bus,or stator fed variable-speed induction motor drives.

    4

    1 0 . ~ HASE A STATOR CURRENT

    - 1 0 . 4 I.a m

    56.m100.00 150.00 200.00 2 5 8 0 0

    T I M E (a) m i 1 SEC

    3d

    zW(LrrU 1 I

    - 1 0 . a .0. 00 50.00 100.00 150.00 200.00 250. 00

    T I M E b ) m i 1 1 i-SEC

    Fig.7. Start up stator current using the stationary frame.The d-q stati-'iary reference frame enables o n e t o obtain

    relatively simple but accurate equivalent circuits o f the motor for

    PSpice implementation and studying electromechanical transients.In Fig.8 the PSpice d-q equivalent circuits based on the stationaryreference frame and the torque equivalent circuit are also given.

    Rs

    (3 '~ R E v3 H2-- -1ds + idrL Vd s + v4

    Fig.8. PSpice equivalent circuits.In Fig.8, V1 throu gh V 5 are dummy (zero) voltage sources tomeasure stator and irotor currents. H1 and H Z are Current ControlledVoltage Sources (CCVS) where

    HTel and HTe2 are also CCVS to represent the electromagnetictorque:

    HTel =(--)l( Lmirqird;' 3 HTe2=(--)(-)LmiJ,3 (15)2 2

    R and Lj are the friction coefficient and the inertia of the motorrespectively. Parameters L L L are the same as in 6) nd as inthe conventional equivalent circuit. The starting up torque o f the

    22k W motor obtained b y the PSpice circuits simulation i s given inFig.9.

    D l l l l l l l cmn: C l l V l V S 14:S1:111

    a) c:\inimmrsn.nr-

    l .......... . ..............th t bU

    11-D 1l11.1l

    11 : 1 2 7 2 : 1 5011.: *, 11. O V 5 p v

    Fig.9. Start UP torque of the mot or (using PSpice).

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    C. Ro t o r reference frame.

    N o w Oq = 8 = w,dt and the q,d axes rotate at rotor speedand the q-axis position coincident with the rotor phase A axis.Hence fr om Fig.6, the equations of the model have the followingform:

    where [ R I and [ L ] coincide with the corresponding matrices ofequation (1 3); vq,=V,,cos(w,t-8); v,,=-V,,sin(w,t-O); 8 is the anglebetween stator and rotor phase A axes

    Since in this model the rotor q-axis variables are at slipfrequency, the y behave in the same way as the rotor phase Avariables (Fig.10). Hence the ro tor reference frame is convenient forstudying transient phenomena in the rotor.

    PHASE A ROTOR CURRENTt

    -10. 0d 10. 40 50.00 100.00 150.00 200.00 250.00

    TIME a ) m i 1 11 - S E C

    Q-AXIS ROTOR CURRENT : :10-OBt- :

    I I

    5 0 . 0 0 1 8 0 . 0 0 1 5 0 . 0 0 200.00 --&. 00- 10 . .GI 0 . 0 0T I M E b l m i 1 1 1-SEC

    Fig.10. Start up rotor current using rotor reference frame

    D. Synchronously rotating reference frame.

    In this case the q,d axes rotate at synchrondus speed andO,=w,t. Th e equ at ion s o f the model are as follows:

    wh er e v=V v=O I n t hi s model, t he st at or d -q v olt ages a ndcurrents are DC quantities (Fig.1 1 in the steady state.

    PHASE A STATOR CURRENT10.0%.

    -1 0. 0 0 .a 0 0 5 8 i m 0 0 1~0.00 ~ 0 0 . 0 0 ZSI

    T I M E ( a ) m i 1 I-SEC

    Q -AXIS STATOR CURRENT

    00

    -1 a 001 1a 0 50.00 100.00 150.00 200.00 250.00T I M E b l m i 1 1 i-SEC

    Fig.1 1 . Start up stator current using synchronously rotatingreference frame

    It is thus possible to use a larger step l ength in the digitalintegration routine to obtain a reduced computati on time when usingthis frame. This frame is also often used for stability analysis andcontroller design, because o f the ability t o linearize the d,q variables.One important area o f application o f rotating d-q reference frame

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    theory i s in the field oriented control of AC motor drives.While d-q reference frames have a very wide area of

    application, they are not appropriate for studying unbalancedoperation of the motor especially when conditions (5) are not met.One particular example is the case of motor operation duringunbalanced phase faults or autoreclosing operation when thetracking of individual phase current s is necessary to simulate circuit

    breaker performance. This leadst o

    the ABC/dq model of the nextsection.

    a / . ABC/dq MOTOR MODEL.

    In man y practical problems one faces the situation whereonly variables o f the stat or (rotor) and electromagnetic torque are ofinterest while the variables of the rot or (stator) are of no significantimportance. The d-q model cannot be easily applied if there iscomplex unbalance of the stator (rotor) circuit. A typical example ofsuch a problem is bus transfer or autoreclosing operation ofinduction motors.

    When studying dri ve hot or nteractions during bus ransferand autoreclose operations it is important t o consider the opening ofthe breaker and overlap effects in the rectifier in detail [ 8 ] . n thiscase the so-called hybrid ABC/dq reference frame can be used whichpreserves the stator states in their original form, while onlytransforming the rotor states to d,q axis variables. A schematic ofthe induction motor with ABC/dq axes shown is in Fig.12.

    tq axis +d

    s t a t o r phase

    c ax i s +

    Fig.12. Schematic diagram of a 3-phase induction motor forABC/dq model.

    In the model, the d-axis coincides with phase A of thestator while the q-axis leads the d b y 90" (electrical). The differentialequations of the ABC/dq model can be obtained by applying twotransformations in cascade to the ABC/abc impedance matrix. Atfirst, the balanced three phase rotor winding is transformed t o a twophase d'q' equivalent frame wh ich is stationary relative to the rotor(axis d' coincides wit h the r otor phase A axis). Then the d'q' frameis transformed to the d-q reference frame which s stationary relativeto the stator yielding the following equations:

    where

    M

    l o

    0 0

    0 0

    0 0

    0 @M2

    12

    -M -M

    -@M 0 L :2

    1-M -E 0 '2

    Expressions for vq, vd depend on the frequency of appliedrotor voltage. If the frequency is equal to the slip frequency then

    where V - is the peak rotor voltage;s - is the slip of the motor

    The expression for electromagnetic torque in the ABC/dqmodel is as follows:

    Unlike the ABC/abc model, the induct ance matrix of (1 8 )is time invariant and does not need to be inverted during each stepof integration. Therefore transients c an be simulated much fasterwith the advantage that stator variables coincide with physicalstator variables of the motor .

    In Fig.13 an induction moto r and induction moto r driveconnected to t he sarne bus are shown, while Fig. 1 4 gives the resultof a simulation of .the system.

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    Fig.13. The schematic of induction motor and induction motor drive.

    MOTOR TORQUE(MOTOR CONNECTED TU T H EB U S DIRECTLY) MOTOR STATOR CURRENTMorOR CONNECTEO 70 T H E aus DIRECTLY

    Fig.14. Current and torque of the mot or connected to the bus.

    In Fig.14 autoreclosing takes place at 1.5 sec. of thetransient. Thus the ABCidq model allows tracking o f all statorvariables of the motor and drive during opening and reclosingoperation of the breaker.

    VI.DQ/abc MOTOR M O D E L .

    A similar idea can be applied for st udying phenomena inthe rotor circuit. If for example, a detailed study of the slip energyrecovery induction motor drive of Fig.3, including the effe ct ofoverlap in the rectifier [91 is carried out, a complex configuration inthe rotor circuit results while a detailed knowledge of the stator

    variables may not be necessary. Hence the rotor variables shouldbe preserved in their natural for m. A schematic diagram of theDQ/abc model is sh own in Fig.15 where the DQ/abc axes aresuperimposed.

    In this model, the d axis coincides with phase A o f therotor, while the q-axis leads the d by 90 (electrical). The equationsof the DQ/abc model can be obtained by applying twotransformations in cascade. A t first, the three phase stator windingsare transformed to a tw o phase d'q' system stationary relative tothe stator (axis d' coincides wit h stator phase A axisi Then the d'q'frame is transformed to a d - q reference frame

    f +q a x i s +d axis

    rotor phaseC oxi5

    Fig.15. The schematic diagram of a 3-phase induction motor forDQlabc model

    stationary relative to the rotor. The result of the transformation is asfollows:

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    where M is the same as in ( 1 9);

    The expression for electromagnetic torque has thefollowing form:

    T =-[ f i .-Ml&,+-Mi&,+Mi i --Miqirb--Miqim]1 (a)2r a 22 2 2

    Like the ABC/dq model, the inductance matrix of theDQlabc model is time invariant. Hence its inversion during eachstep of integration is avoided while the rotor states are retained intheir original form.

    Fig.16 shows the stator, supply and rotor currentwaveforms of the slip energy recovery drive at a speed of 1 25 0 rpm(slip = 1 6 ) predicted with DQ/abc model. The waveforms ofmeasured and predicted rotor current in Fig.16 are almost identical.Indeed the DQ/abc model enables one to consider the overlap effect

    in detail and gives the cor rect value of the overlap angle. Thesimple equivalent c ircuit is unable to predict this result, particularlyat this slip of 1 / 6 .

    machines," Sir Isaac Pitman & Sons LTD, London, 1968.2. P. Krause and C . Thomas,"Simulation of symmetrical inductionmachinery," Iff rans. PAS-84, 1965, pp.1038-1053.3 . J.E. Brown, W. Drury, B.L. Jones and P. Vas, "Anal ysis o f theperiodic transient slate of a static Kramer drive," Proc lff, ~01.133,Pt.B. no 1, Ja n 1 386, p p.2 1-3 0.4. P. Pillay and IL. Refoufi, "Calculation of slip energy recoveryinduction motor drive behavior using the equivalent circuit," /ffTrans. lnd. Ap p l . , vo1.30, no. 1, Jan/Feb 199 4, pp. 154- 1 63.5. D.G.O. Morris," ;ome test s of an exact practi cal theor y of theinduction motor," F roc. Iff, vol. 97, Pt.11, p p. 767 -7 78 .6. P. Krause, "Analysis of electric machinery," McGraw-Hi//, 1986.7. R. Lee, P. Pillay and R. Harley, "D,Q reference frames fo r thesimulation of induction motors," fPS R Journal, vo1.8. October

    8. T. Higgins, P. Young, W. Snider, H. Holley, "Report on bustransfer studies,'"lEff Trans. Energy Conversion, vo1.5, no.3,September 1990, pp. 470-484.9. E. Akpinar, P. Pillay, "Modeling and performance o f slip energyrecovery induct ion motor drives," Iff rans. Energy Conversion,vol. 5, no. 1, March 1990, pp. 203-210.

    1984, pp. 15-25.

    APPENDIX I.

    2 2 k W Induction moto r parameters

    VII. CONCLUSIONS.

    This paper has reviewed and presented the details ofseveral different types of mathematical models suitable for theinduction motors and drives. Guidelines for the use of each modelhas been provided.

    ACKNOWLEDGMENT.

    The authors a cknowle dge EPRl and Entergy Corporation forfinancial support.

    Base powerBase stator voltageBase stator currentBase stator impedanceBase torqueNumber of polesStator resistanceRotor resistanceStator leakage reactanceRotor leakage reactanceMagnetizing reactanlceMoment of inertia

    27.91 8kVa220V (phase)42.3A (phase)5 .21 Ohm177.8"

    0.021p.u.4

    0.057p.u.0.049p.u.0.132p.u.at 50 Hz3.038p.u.0.29kg m2

    REFERENCES.

    1. M. G. Say,"The performance and design of alternating current

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    A

    U 4P r e d i c t e d M e a s u r e d

    . . .. ... . . .. . ......... . .. . ... .. .......... .. ... . .. ........ .... .. ...... .... ....... ... .............

    c01t 4

    CI

    0

    jut v v vv v vv v vv v vA .

    A

    ffi ;D.

    Fig.16. Measured and calculated waveforms o f the slip-energy recovery induction motor drive at a speed 12 50 rpm (using DQ/abcmodel)

    APPENDIX II.ABC/abc MOTOR MODEL WITHOUT INVERSION OF THE

    INDUCTANCE MATRIX.I1 A12 A13 A14 A15 A

    The equation (7) has the following form:

    where

    A1 1 =(P1 *r,)/det[cllA1 2=( Q1 *r,+CS(Pl-Q ))/det[C11A1 3 =( Ql *r,+CS(Ql-P )) /de t [CI]A1 4 = - (P1 *S1 + Ql (S2 + S3 ) ) / d e t [C l ]A1 5 = - (P1 * S3 + Q 1 (S1 +S2))/det[C11A1 6 = - (P1 * S2 + Q l ( S l + S3 )) / de t [C 11A 2 1 = A 1 3 A 3 1 = A 1 2A 2 2 = A 11 A 3 2 = A 1 3A23 = A1 2 A3 3 = A1 1A 2 4 = A 1 6 A 3 4 = A 1 5A 2 5 = A 1 4 A 3 5 = A 1 6A 2 6 = A 1 5 A 3 6 = A1 4A41 =-(P2*R01 +Q2(R02+R03))/det[C21A4 2 = - (P2 * R 0 2 + Q2(R01 + R03))/det[C21A43 =-(P2*R03 + Q2(R01 + R02))/det[C21A44=P2*r,/detlC21A4 5 = (Q2*r, +CR(Q2-P2))/det[C2]A4 6 = (Q2*r, +CR(P2-Q2))/det[C2]A 5 1 = A 4 3 A 6 1 = A 4 2A 5 2 = A 4 1 A 6 2 = A4 3

    A5 3 = A4 2 A6 3 = A4 1A 5 4 = A 4 6 A 6 4 = A4 5A5 5 = A4 4 A6 5 = A4 6A 5 6 = A 4 5 A 6 6 = A 4 4B1 1 =-Pl /de t [Cl ]8 4 1 = (P2 * a l +Q2(a2 +a3))M,/(LSdet[C2I)B12 = -Q1 /det[C11B42 = (P2*a3 +Q2 (a l +a2))M,,/(LSdet[C21)5 1 3 = B 1 2843 =(P2*a2 +Q2(al +a3))M,,/(LSdet[C21)821 =B13 B51 =B43B22=B11 B52 = 841B 2 3 = 8 1 2 8 5 3 = 8 4 2B31 =B13 861 =B428 3 2 = B13 862 = B43B33 =B11 863 = B41C1 = (3M,,2/2L,)-L,C2 =-3M,,2/4L,S1 =(M,,* al *r,/Lr ) +M,,w,blS2 =(MS,a3*r,/L,) + M,,w,b3S3 =(M,,a2*r,/L,) +M,,w,b2CS =-3d3M,,2~,/4L,

    616

    P1 =C12-C2'Q1 =C2?-Cl *C2det[C 1 1= C 1 +2 *C23-3 *C 1 *C2'C3 = (3Ms,'/2L,)-LrC4 =-3M,,'/4LSKO1 =(M,, *al *r,/L,)+M,,*w,*blR 0 2 = (Ms, a3 *rJLJ +M, * w e b 3R 0 3 = (Ms, a 2 * rJLJ +M,, * w e *b 2CR =-W3M,,2~,/4L,P2 = c 3 2 - c 4 20 2 = C4'-C3 *C4det[C2] =C33 +2*C43-3*C3*C42a1 =cos(@a 2 = cos(@ 2n/3)a3 = cos(&2n/3)b l =sin(@b2 =sin(@ + 2n/3)b3 =sin(B-2n/3)L, = L,, + (3/2)M,,L, = L,, + (3/2)M,,