S O 6 4 lEEE TRANSACTIONS ON MAGNETICS, VOL 32, NO 5 , SEPTEMBER 1996

~ a l c ~ l a t i o n of the L in EMS-MAG

M. Andriollo, G. Martinelli, A. Morini and A. Tortella Departrneiit of Electrical Engineering - University of Padova

Via Gradmigo, 6/a - 1-35 13 1 Padova - Italy

Ahttract-The paper describes a procedure to determine the propulsion force produced by the linear synchronous motors used in magnetically levitated vehicles of electromagnetic type. The formulation enalbles to compute the instantaneous value of the force starting from the winding inductances calculated via FEM numerical analysis. The results are compared to the ones obtainable by means of two other procedures implemented in a commercial FEM code and show that, under the same accuracy, the proposed procedure requires less calculation time.

I. INTRODUCTION

The magnetically levitated vehicles of electrotnagnetic type (EM§-MAGLEV) are propelled by iron-core linear syn- chronous motors (LSM) with 3-phase armature windings on the guideway and DC excitation coils wound around salient poles on the vehicle [l]. The DC coils work as levitation magnets as well Due to the complexity of the time-varying niagnetic configuration, a fiilly analytical approach enables to obtain only the mean value of the motor propulsion force [2], whereas the determination of the instantaneous value requires the recurrent utilization of a numerical code.

The paper describes a procedure, valid in steady-state running and in conditions of linear niagnetic circuit, which requires at first a reduced nuniber of numerical analyses and then determines the instantaneous value of the propulsion force by means of an analytical formulation.

With reference to a configuration partially derived from the German prototype Traiisrapid [l], Fig.1 shows the longitudinal section of the LSM in one polar pitch 22.

The following simplifying hypotheses are assumed:

- 2D coiliiguration (no transverse edge-effects); - infinite length along x axis (no end-effects); - magnetic field confined inside the motor; - constant permeability of the ferromagnetic material.

These assumptions reduce the coinplexity of the mathemati- cal model and are acceptable in the preliminary design of the LSM. Moreover tlie FEM analysis can be performed on an ordinary PC with reasonable calculation time.

Once the winding inductances have been numerically determined for a suitable number of reciprocal positions between the vehicle and the guideway, the instantaneous and mean values and the harmonic spectrum of the propulsion force are analytically determined by means of tlie virtual work principle.

Manuscript received March 4, 1996. This work was supported by the Italian National Research Council (CNR)

under the Progetto Finalizzato "Trasporti 2".

2 7 ~ 5 1 6 c

Fig.l. LSM longitudinal section [27: polar pitch, 1,2,3: on-ground armature phases; 4: on-board field winding; XO: time-v'uying reciprocal position between the field winding and phase 1; the slots in the salient poles contain the on-board linear generator conductors; all sizes in mm].

The proposed method can be extended to the calculation of the levitation force. Since preliminary FEM analyses showed that the ripple of the levitation force is much lower and its mean value, determined via FEM analysis, is very close to the one determined via analytical formulations, the procedure has been developed with reference only to propulsion.

Tlie obtained force values are compared to the ones available by means of two fully numerical procedures based on the Maxwell's stress tensor method [3] and on the local virtual work principle [4].

11. ANALYTICAL FORMULATION OF THE PROPULSION FORCE

Since the model is linear, the propulsion force per polar pair can be expressed as function of the magnetic energy in the following way:

with xo time-varying reciprocal position between the field winding and the phase 1. At t=O the magnetic axes of such windings are supposed aligned, that is xo=O (Fig.1). The vector [i] consists of the currents in the four windings and [ I ] is the 4x4 matrix of the inductances associated with the armature phases 1,2,3 and the field winding 4:

The [ I ] matrix depends only on the permeability and on the geometrical configuration of the magnetic circuit.

0018-9464196505.00 0 1996 IEEE

5065

In steady-state conditions the velucle runs at constant speed u, the field current 4=14 is constant and the phase current ii (i=1,2,3) is sinusoidal with angular frequency o-rculz:

L J J

1 = - f i I a s i n [ ~ x O + y - - 2n(i - 1) z 3

(3)

The angle y is the displacement of the phase 1 current at t=O. - The generic inductance /ij (i,J=1,2,3,4) can be analytically expressed as a Fourier series expansion with respect to xo, terminating at an appropriate harmonic order n:

(4)

The quantity Ti corresponds to the pitch of the inductances; it is:

T.. = T.. = 22 l I i I 3 , J = 4 I s i, j < 3

i = j = 4 IJ JI

5.. IJ = 5 J * ..

213

511 = 5 2 3 = 5 1 4 = 5 4 4 = O

522 = e 3 1 ='533 =-512 534 = -524 = 27~ 13 Lk,ij = Lk,ji

Lk,14 = Lk,24 = Lk,34

Lk, l l = Lk,22 = Lk,33

Lk,12 = Lk,23 = Lk,31 = 0 k even

0 k odd i The coefficients Lki are expressed by integrals which c,an be

numerically estimated (e.g. by means of the Gauss-Legendre formulas), once 'an appropriate number m of values of the inductances li in the interval OIs&13 ltas been calculated. These numerical values of 1;j can be computed via FEM code by integrating the magnetic energy on the whole domain.

Expanding (I), the force consists of the stun of 3 components:

The components take into account the interaction between the excitation and armature m.1ii.f.s as well as the reluctance variation due to the salient poles on the vehicle and to the armature slots on the guideway.

Bearing in mind (3) and (4), the expression of the force (5 ) can be arranged as follows:

F,( xo,[i]) = XF;tk l cos(6kh so T y) + n'

k=O n"

+x[F:kfl sin(6kh xO 32y)+F tk sin(6kh xO)]+ k=O

n +CFt sin(6kh xO) k= 1

with h=n/z, n'= IF], n"= I IJ n + l and:

Ff; =F!l = O

Fi = -3I:kh Lk,44

The instantaneous value, the mean value @=O) and the harmonic spectrum of the propulsion force can be directly obtained from (6). Since the coefficients Lk,ij are not dependent on the currents, the developed formulation makes possible to determine F, for various load conditions without performing additional FElM analyses, unlike fully numerical calculations [3,4].

111. DETERMINATION OF THE PROPULSION FORCE

The procedure has been applied utilizing the ANSOFT Maxwellm2D FEM code.

With reference to the geometry and sizes given in Fig.1, the other electromagnetic [data are:

- one turn for both the: armature phases and the field

- armature currents r.m.s value Ia=850 A; - field current I4=5500 A; - armature current initial displacement 7'0"; - permeability of the ferromagnetic material cl,=6000; - vehicle speed u=l 10 d s .

The chosen constant value of the permeability is the average value in the range 0.5+1.;!5 T for the magnetic steel TERN1 1035. It has been verified that the force values in the hypothesis of linearity are about 5% different from the ones obtainable performing a ntm-linear FEM analysis.

By increasing the number of the mesh elements, both the magnetic energy and the inductances approach asymptotic values; Tab.1 shows that np3000 gives adequate accuracy.

With reference to an I T M analysis for a given relative position, Tab.11 compares the force values obtained by means of the developed procedure (n=2, m=8) to the ones available via two post-processor capabilities of the FEM code based on the Maxwell's stress tensor method and on the local virtual work principle [5] . The comparison shows that the deter- mination of the propulsion force via analytical formulation is less affected by the mesh refinement; as a consequence a coarser mesh can be used thereby reducing the calculation time. It must be pointed out that the force estimation via Maxwell's stress tensor method is also affected by the choice of the integration path inside the air gap: in order to com- pensate such effect, the force is calculated on four different paths (at a 2 mm distance from one another) and the corre- sponding average value is considered.

winding;

5066

1600

TABLE I MAGNETIC ENERGY w, [J/m] AND INDIJCTANCES 1, [pH/m] AS FIJNCTIONS

OF THE MESH REFINEMENT

"t *E w m 111 122 133 144

(b) - (4 *

- (a) (n=2, m=8) - (a) (n=3, m=12) - ~ - - ~

(a) (n=4, m=16) - - - - -

1100 100 751.9 12.54 11.84 11.76 50.16 1750 220 762.8 12.76 12.05 11.96 50.92 2200 350 764.4 12.81 12.08 12.00 51.08 2700 470 766.4 12.84 12.11 12.04 51.20 3210 610 768 12.88 12.08 12.15 51.31

2500

nt 11 2 l13 l14 l23 l24 134

mean value -

1100 -2.64 -6.36 20.64 -2.15 -3.93 -14.04 1750 -2.73 -6.47 21.00 -2.13 -4.12 -14.26 2200 -2.75 -6.49 21.08 -2.13 -4.16 -14.30 2700 -2.76 -6.51 21.14 -2.13 -4.18 -14.34 3210 -2.77 -6.52 21.20 -2.14 -4.20 -14.36

Values calculated at xo=21 5 mm nt total number of the mesh elements, ng number of elements in the air-gap

TABLE I1 PROPULSION Fc IRCE F, [N/m] AS F U " T I ( IN CIF THE MESH REFDEMENT

nt "a F, (a) F, 6) F, ( 4

1100 100 2331 2346 1750 220 2386 2420 2283 2200 350 2387 2385 2700 470 2379 2410 2370 3210 610 2379 2365 2426

Values calculated at xo=21.5 mi. nt: total number of the mesh elements, ng: number of elements in the air-gap. (a): analytical formulation, ( 1 ~ 2 , m=8); (3): Maxwell's stress tensor method; (c): local virtual work principle.

Fig.2 displays the influence of the number m of the inductance sampling points (in the interval o I s 0 5 d 3 ) on the amplitude of the force harmonics. The first two harmonics (angular frequency Gh and 12h) are obtainable with good accuracy with m=8; on the contrary, a greater value of m is necessary to estimate the high order harmonics with adequate accuracy. It has been verified that the mean value of the propulsion force is almost independent on m, provided m24.

t 4 2000 1 m=8 L%$$ll

t m=12 1500 1 1 m=16

1000 1 i 500

n U 1 2 3 4

harmonic order k

Fig.2. Influence of the number m of the inductance samplings on the amplitude of the propulsion force harmonics.

Fig.3 compares the progress of F, versus x 0 esti the analytical formulation to the discrete values o means of the fully numerical methods. Three c drawn, with reference to different values of the harmonics n and of the inductance samplings m. Th obtained by means of the different methods are accordance. The values obtained via Maxwell's stre method show a great excursion (vertical lines) around the average value (symbol 0) when the integration path varies.

Fig.3. Propulsion force F, versus the vehicle position xo [(a): analytical formulation; To): Maxwell's stress tensor method (c): local virtual work principle].

IV. CONCLLJSIONS

The paper describes an analytical formulation, valid in steady-state running and in conditions of linear magnetic circuit, to determine the propulsion force in iron-core linear synchronous motors. The procedure makes possible to calcu- late the instantaneous value, the mean value and the harmonic spectrum of the force by means of a reduced number of FEM analyses; the demanded accuracy being the same, such analyses require a number of mesh elements smaller than other methods implemented in FEM codes, with a consequent reduction in calculation time. Further- more the procedure does not require additional FEM analyses when the load conditions change.

REFERENCES [ I ] W.J.Mayer, J.Meins and L.Miller, "The high-speed Maglev transportation

system Transrapid, IEEF Trans. on Magnetic.s, vo1.24, n.2, March 1988, pp. 808-811.

[2] S.A.Nasar and LBoldea, Linear Electric Motors: Theory, Design and Practical Applications, Prentice-Hall, 1987, pp.99-107.

[3] N.1da and J.P.A.Bastos, Electromagnetics and calculation of fields, Springer-Verlag. 1992, pp. 188-195.

(41 J.L.Coulomb and G.Meunier, "Finite element implemenkition of virtual work principle for magnetic or electric force and torque computation", IEEE Trans. onMagnetics, vol. 20, n. 5, September 1984, pp. 1894-1896.

[SI Ansoft Corporation, Maxwell@2Dfield simulator - User's guide & User's reference, Versions 4.33 & 6.2, June 1994 &June 1991.