Field-circuit simulation of permanent magnets synchronous motor dynamic

ZESZYTY NAUKOWE POLITECHNIKI ŚLĄSKIEJ 2001

Seria: ELEK TR Y K A z. 176 N r kol. 1500

Yvan LEFEVRE1*, Mieczysław R O NKO W SKI21, Andrzej TRYBULL3)

FIELD-CIRCUIT SIMULATION OF PERMANENT MAGNETS SYNCHRONOUS MOTOR DYNAMIC

Sum m ary. A simplified field-circuit model of a permanent magnet synchronous motor (PMSM) hase been presented in the paper. The motor circuit parameters evaluated by field methods are input to the circuit model in appropriate look-up tables to carry-out simulation of dynamic motor states. Some new features of the proposed approach have to be noted while compared to the reported, ones i.e. the appropriate algorithm has been elaborated to generate the discretized inductance matrix of the motor and its angular displacement derivatives. The presented simulation results have sufficiently proved that the set-up assumptions are reasonable for formulation of the proposed field-circuit model.

Key w o rd s: synchronous motors, permanent magnets, field-circuit simulation

1. IN TR O D U C TIO N

G enerally, two basic approaches to the sim ulation o f electrical m achine (EM) dynam ic can be distinguished: circuit approach and field approach. The circuit approach consists in describing the physical processes involved in term s o f integrated m agnetic-field effects throughout a space to obtain a set o f ordinary differential equations. W hile form ulating the equations, i.e. the circuit m odels o f the m achines, som e m ore or less sim plified assum ptions are usually taken into consideration [8, 10]. T he m athem atical (circuit) m edel fo r m agnetic-field effects is set in term s of the lum ped-param eters o f the m agnetic field and electric circuits in associations. The param eters m ust be known in an appropriate form , based on the field analysis, design estim ate or experim ent.

The m ost im portant lum ped-param eters are the s e lf and m utual inductances, which are functions of the: relative positions o f the fixed and m ovable parts o f the m achine, hysteresis m agnetic saturation, skin effects and power dissipation. Presently, the m achine inductances are usually com puted using eith e r 2-dim ensional (2-D) or 3-dim ensional (3-D) field m ethods; however, to lim it the com putation to a reasonable tim e som e o f the m entioned effects have to be neglected [1, 5],

For som e applications, e.g. synthesis o f control system o f electrical drives, a transform ation of the equations from natural (physical) reference fram e to other reference fram es is used. However, com m only used orthogonal tw o-axis fram e im plies further sim plification: i.e. assum ption o f the m ono-harm onic field distribution in the m achine air-gap, and also sim plified m odels o f saturation and skin effects. Recently, som e efficient m ethods have been reported for solving m ore effectively the problem s o f saturation in m agnetic circuit and skin effects in windings o f EMs [1, 6, 7].

The field approach consists in sim ultaneous solution o f the field equations and the equations o f equilibrium o f e le ctric circuits and m echanical system o f EM. T his is an advanced approach allowing to lim it the num ber o f the sim plified assum ptions, and there are m ethods (e.g. finite e lem ent techniques) being continuously developed and reported [2, 5, 9, 11,13]. However, on the one hand, by using the finite e lem ent techniques m ore and m ore com plex problem s, involved In analysis and design o f pow er e le ctro n ic converter-electrical m achine system s, m ay be solved as the com putational power increases. On the other hand, in m any research and developm ent tasks (e.g. the whole system studies) the sim ulation run tim e is at a prem ium . So in such cases,

11 PhD, DSc, EE, charge de recherche au CNRS, INPT- ENSEEIHT- LEEI, 2, rue Camichel, 31071 Toulouse, France, (+33) 561588359, fax (+33) 561638875, e-mail: lefevre@leel.enseeiht.fr

21 PhD, DSc, EE, associate prof. at Politechnika Gdańska, Katedra Energoelektroniki i Maszyn Elektrycznych, 80- 952 Gdansk, ul. Narutowicza 11/12, Poland, tel. (+48)(+56) 3472087, fax (+48)(+58) 3410880, e-mail:

mronk@.ely.pg.gda.pl

31 MSc, EE,1999-2000 DEA Génie Electrique student at INPT- ENSEEIHT- LEEI, presntly PhD student at Politechnika Gdańska, 80-952 Gdansk, ul. Narutowicza 11/12, Poland, tel. (+48)(+58) 3472534, fax (+48)(+58) 3410880, e-mail: bjosek@ wat3.ely.pg.gda.pl

p articularly fo r high-frequency and high-velocity system s, a trad e -o ff fo r a sensible computing speed vs. high a ccuracy can be provided w hen using m odels o f EM s form ulated in term s of lum ped-param eters. In particular, the constant-topology m ethod used to describe the converter circuits introduces extraneous eigenvalues, i.e. setting-up stiff-problem s. To insure a converging and accurate solution o f a s e t o f differential equations describing the overall system the integration step length m ust be o f the ord e r o f the shortest tim e constant [3, 4 ,1 2 ].

It seem s th a t fu rth e r searches are necessary to find m ethods allowing to com bine more e ffe ctively the field approach and circuit approach to sim ulation o f power e le ctro n ic converter- electrical m achine syste m s dynam ics, particularly with EM s o f lim ited m agnetic circuit saturation effects.

In the paper a sim plified fie ld -circu it m odel o f a perm anent m agnet synchronous m oto r (PMSM) is considered. D ue to the lim ited space o f the paper m ost o f the attention is devoted to the discretisation algorithm fo r the circu it param eters and th e ir derivatives, w hich are needed for solving the ordinary d ifferential equations describing the m oto r dynam ics. The m otor circuit param eters are evaluated using the field calculation, presented as a function o f the rotor angle position, and under assum ption th a t the m agnetic circuit saturation effects are negligible. In the end part o f the paper, som e chosen sim ulation results o f a PM SM d ynam ic are presented. Deeper studies o f the problem , in p a rticular the consideration o f num erical solution o f the equations of equilibrium o f e le c tric circuits and m echanical system o f EM in association, have been presented in the w ork [14].

2. G E N E R A L M O TO R C IR C U IT E Q U A TIO N S IN NATU R A L R E FER E N C E-FR AM E VA R IAB LE S

It is assum ed th a t a PM SM has a three-phase w inding uniform ly distributed in stator slots;

m agnets m ounted on the surface o f the rotor, or there can be m agnets buried inside the rotor; the sta to r w indings currents are o f arbitrary w aveform s and frequency; there are no dam p e r windings on the rotor; the e le ctric fields, m ag n e tic saturation, hysteresis and skin effects, and iron losses can be neglected.

In natural reference-fram e variables (i.e. m otor variables) the equations o f equilibrium of electrical and m echanical system s m ay be expressed in a vector-m atrix form [8]:

electrical system

V = R I + — T , (1)

dt m echanical system

T e = 4 “ m + T L , (2)

dt

where, the e le ctro m a g ne tic torque

T — f ^ ^ c O . b c , ’ i f t b c r >

' V aer

In the above equations the voltage sources V and load torque Tl are general and can be independent o f or dependent on som e variables, and the quantity W c represents the co-energy stored in the m ag n e tic coupling field as a function o f the sta to r current vector I, PM flux-linkage ve cto r T m and ele ctrica l rotor angular d isplacem ent 0e. It should be noted th a t 0 , = p0m and co, = pcom, w here, 0m and com represent m echanical angular displacem ent and velocity o f the rotor, respectively. A lso in the above equations the used sym bols denote: J - com bined rotor and load m om ents o f inertia, p - num ber o f pair poles.

T he used space vectors and m atrices are o f the form : V T = [v, V2 V3] - term inal stator axis voltages; I T = [ii I2 1 3] - sta to r axis currents; T*7 = [ t , 4>2 T 3] - stator axis flux-linkages;

'•'m = K n i '•'m i '•'■1.3] - PM axis flux-linkages; R = diag[R! R2 R3] - a x is resistance m atrix.

If the m o to r structure is restricted to m aterials whose m agnetisation densities are linear with field quantities, then the m oto r becom es an electrically linear system (but electrom echanically

Field-circuit simulation o f permanent magnets synchronous motor dynamic 233

nonlinear) w hose flux-linkages and th e ir derivatives, as w ell as electrom agnetic torque equations can be expressed in term s o f inductances as follows:

4'( t,e e) = L (e e) I ( t ) +4' m(0e) , (4)

¿ T ( t , 0 e) = L(0 e) - ^ I (t) + coe I ( t ) - i - L ( 0 e) + coe - ¿ - 4 'm( 0 . ) , (5)

at at a0e a0e

T - - 4 , T s : L W , * i , , T s : ' , ' - ( e ' ) ( 6 )

w here: L(0e) - sta to r w inding s e lf and m utual-coupling inductance matrix, and 4 ^(0 ,,) - flux- linkage vector (produced by PM and linked with stator w inding) as functions o f electrical rotor angular d isplacem ent 0e. According to Eq. (4) the stator axis flux-linkages m ay be expressed in a vector-m atrix form as:

V i " L n ( 0 e ) M 1 2 ( 0 e ) M1 3 ( 0 c ) ' • i V m l ( 0 e ) V 2 =

M

,(ee) L

(

e)

V

.

L

(

e) .¡

.

In the next section o f the paper the m otor circuit param eters contained in Eqs. (4)-(7) are evaluated using the field calculation m ethod, which is perform ed in term s o f the rotor angular displacem ent and under assum ption that the m agnetic circuit saturation effects are negligible.

3. EVALU ATIO N O F M O TO R C IR C U IT PARAM ETERS

A ccording to Eqs. (4)-(7) the follow ing elem ents have to be evaluated:

L (0 e) , — L (6 e) .

d 0 e e /

(8)

Referring to the general definition o f EMF induced in a single coil linked with the total flu x 4', which are in relative m otion, one can get the follow ing relationships:

d , d T d 0 e d 4 >

e = vF ( t , 0 . ) = --- = ---e u .

d t e d 0 e d t d 0 e e

( 9 )

- = -eco. (10)

from w hich it follow s th a t having the values o f EMF e — com puted by a field m ethod — and knowing the electrical angular speed coa one can evaluate the flux-llnkage derivative:

d4<

d0t

In sim ila r w ay the above relationship can be used to evaluate the circuit elem ents given In Eqs. (4)-(7).

F or exam ple if the sta to r circuits are open, i.e. i1 = 0, i2 = 0 and i3= 0, and the rotor rotates with speed e>m = co n st then one can evaluate the angular displacem ent derivatives o f the PM axis flux-linkages as follows:

d d 0 e

^ mr e l

v m2 ^{II I} e 2

^ 3 . e 3.

(11)

In turn, assum ing vym1 = 0, vym2 = 0, \j/m3= 0 (i.e. rem oving PM from the rotor), o m = const, and supplying only axis “ 1” by d c current i1 = 1 A (currents of other axis i2 = 0 and i3= 0), and next com puting the sta to r axis flux-linkages vy.,, vp1 and y 3 (produced due to current ¡1 = 1 A ) by a field m ethod one can evaluate, according to Eq. (7), the following inductances:

v |/,= L „ (0e), V2 = M2i(0e) , V3= M3l(0e). (12)

R epeating such a com putation fo r o ther axes, w hile assum ing first i1 = 0, i2 = 1 A and i3= 0 and next i1 = 0, i2 = 0 and i3 = 1 A respectively, one can evaluate all the elem ents o f inductance matrix in Eq. (7).

Finally, using the sim ila r procedures as m entioned above, the values o f the angular disp la ce m e n t derivatives o f the e lem ents o f inductance m atrix in Eqs. (5) and (6) can be calculated.

For exam ple, considering the case fo r assum ed y m1 = 0, v m2 = 0, v|/m3= 0 (i.e. rem oving PM from the rotor), com = const, and suppling only axis “ 1” by d c current l1 = 1 A (currents o f other axes i2 = 0 and i3= 0), and com puting by field m ethod the values o f e i, e2, e3 one w ill get:

d d0.

L„(0e) V

M 2,(9 e) = -

e2

M 31(0e) _«3.

(13)

R epeating such a com putation fo r o ther axes, while assum ing first i1 = 0, i2 = 1 A and i3= 0 and next i1 = 0, i2 = 0 and i3 = 1 A respectively, one can evaluate all the elem ents o f the angular d isp la ce m e n t derivatives o f the inductance matrix.

It should be noted that in the above equations the EM Fs e ^ e 2 and e 3 cannot be considered as the sam e quantities, since they represent values evaluated across term inals o f the stator axis w indings fo r each case o f field com putation, respectively, as described above.

a) b)

Fig. 1. Field analysis of synchronous motor with pem anent magnets buried inside th e ro to ra ) geometrical model (PM shown by magnetic flux density B); b) finite element model — mesh generated by EFCAD software

A s an application exam ple o f the above described procedures an evaluation o f the circuit elem ents o f a 2.2 k W and 1500 rpm PM SM is presented. T he longitudinal cross-section o f the m otor is given in Fig. 1. Using EFC AD softw are [1] the m otor m agnetic field distribution and the values o f the circu it e lem ents have been evaluated. The m oving band spanned over one pole pitch has been divided into 180 intervals, i.e. the discretized interval is equal to 0.5 m echanical deg.

C hosen com puting results are shown in Figs. 2, 3, 4 and 5.

Field-circuit simulation o f permanent magnets synchronous motor dynamic 235

Vm1, Vm2. Vit* [Wb] Si, ej, e j [V]

0m [deg] Omtdeg]

Fig. 2. Plots o f stator axis flux-linkages produced by PM Fig.3. Plots of stator axis EMF Induced by Vm vs.

vs. mechanical angular displacement mechanical angular displacement at speed 1500 rpm

L i 1, M 21, M31 [H] d L n /d 9 0 , dM21 /d 0 e , dM31 /d 0 8 [H/deg]

Fig. 4. Plots of PMSM stator Inductances vs. mechanical Flg.5. Plots of PMSM stator Inductance derivatives angular displacement (y m = 0) vs. mechanical angular displacement (v m = 0)

4. S IM P LIF IE D FIE LD -C IR C U IT M O D EL FOR M O TO R DYNAM IC SIM ULATIO N

The flo w ch a rt o f the algorithm fo r sim plified field-circuit sim ulation o f the PM SM dynam ic is show n in Fig. 6. It can be seen that in the first step the circuit m odel param eters o f the motor, evaluated by the m ethod already outlined in Section 2 o f the paper, are read from an input data file w ith the field com putation results, and next they are stored in com puter m em ory using an appropriate procedure. T he procedure is built-up as a separate m odule o f the algorithm and is based on the elaborated functions which generate discretized data — depending upon the rotor angular d isplacem ent — fo r the m odule setting-up and solving the set o f ordinary differential equations. T he equations describe the PM SM dynam ic states. The solving m odule Is linked with procedures generating the m otor supply voltage and the load toque. The procedures have options to support reading the s p e cific supply voltage or load torque form the data file. The equations are num erically solved using R unge-Kutta m ethod [4] (considerations on other m ethods which can be used are reported in the w ork [14]).

W h ile the iteration loop is perform ed the com puting results are stored in the com puter m em ory using appropriate procedure; it decreases the sim ulation run-tim e. W hen the iteration loop is finished the sim ulation results are stored in an output data file as w ell as an option fo r plotting the results.

START

Inputing data:

flux-llnkage vector and its derivative inductance matrix and its derivative Formulation the discretized look-up tables

Reading initial conditions and control data for simulation

Starting computation loop

Formulation of the motor OD equations Equations solution by Runge-Kutta 4th order method

Using the above outlined algorithm and m ethods the sim ulation o f 2.2 kW PM SM dynam ics has been carried-out. Self-controlled operation o f the m otor is considered, and some results are provided fo r a drive starting on fan load (shown in Figs. 7- 10). The sinusoidal supply voltage is assum ed and its phase angle is defined with respect to the angular d isplacem ent o f the rotor. The initial am plitude is equal to 40 V and next it is varied linearly with rotor speed until its rated value is reached.

To validate the proposed algorithm and m ethods the following procedure has been applied. In the first step, using the field method (EFCD software), the steady-state analysis o f the PM SM hase been perform ed. For this purpose the m otor has been supplied by the sinusoidal current at the constant rotor speed with the torque angle equal to 90 deg. In the next step, using the above outlined algorithm and m ethods, the m oto r winding resistance and the supply voltage am plitude have been increased appropriately to reduce the influence o f back EMF on the shape o f m otor currents. For com parison some results obtained by these two ways are shown in Figs. 11-12.

5. CO NCLUSIO NS

The sim plified field-circuit m odel of a perm anent m agnets synchro­

nous m otor (PM SM ) have been presented in the paper. The m otor circuit param eters, evaluated by field m ethods, are input to the circuit m odel in appropriately discretized look-up tables to carry-out sim ulation o f d yn a m ic m oto r states. Som e new features o f the proposed approach have to be noted w hile com pared to the reported, ones i.e. the appropriate algorithm has been elaborated to g enerate the discretized inductance m atrix o f the m otor and its angular displacem ent derivatives.

Using FO R TRAN 77 and the outlined algorithm a sim ulation program m e for UNIX environm ent has been developed. The program m e uses the sam e form ats o f the input and output files as the EFC AD softw are [11] and the DSN post-processor elaborated in IN PT-ENSEEIH T-LEEI.

H

Outputlng and ploting the computing results

STOP

Fig. 6. Flowchart of the algorithm for field-circult simulation of PMSM dynamic states

V

Field-circuit simulation o f permanent magnets synchronous motor dynamic 237

Fig. 7. Plots of the PMSM supply voltage vs. time it. ¡a [A]

Fig. 8. Plots of the PMSM stator axis currents vs. time

T0 [N m] om [rad/s]

Fig. 9. Plots of the PMSM electromagnetic torque Fig. 10. Plots of the PMSM mechanical angular speed

vs. time vs. time

A ls o a batch processing option is possible. T he proposed approach effectively accelerates the analysis o f the PM SM dyn a m ic states, and also it allows further developm ent o f m ore advanced softw are fo r com bined fie ld -circu it sim ulation o f EMs. T he presented sim ulation results have su fficie n tly proved th a t the set-up assum ptions are reasonable fo r form ulation o f the proposed field- circu it m odel.

M/1[Wb) v 1[Wb]

Fig.11. Plots of the PMSM stator axis flux-linkage vs. Fig. 12. Plots of the PMSM stator axis flux-linkage vs.

mechanical angular displacement: used EFCD time: used outlined algorithm software

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Recenzent: Dr hab. inż. W ojciech Szeląg

W p łyn ę ło do Redakcji dnia 1 m arca 2001 r.