Force and torque computation from 2-D and 3-D finite element field solutions

A. Benhama A.C. Wi I I ia mson A.B.J. Reece

Abstract: The Coulomb virtual work (CVW) method of force and torque calculation is a relatively recent development which has received little publicised appraisal or validation particularly for three-dimensional situations. The authors outline the CVW method and describe its application to a range of force calculation problems, showing that the CVW method can be superior in accuracy and ease of implementation to the Maxwell stress tensor (MST) method. Results of the two-dimensional CVW, two- dimensional MST, and three-dimensional CVW are compared against either analytical or measured data. The 2-D CVW calculation produces discrepancies against analytical or test results similar to those obtained with the MST method with centroid paths. Comparisons between 2-D, 3-D and analytical or experimental data show that the force or torque computed from the 3-D analysis is in much better agreement with the analytical or experimental data than that obtained from 2-D.

1 Introduction

The operation of some electromechanical and electro- magnetic devices depends upon the forces that act upon either current-carrying conductors or magnetised parts. In such devices, therefore, the calculation of forces is a subject of great importance. This paper is concerned with the calculation of total or net forces.

There are four basic ways of predicting electromag- netic forces, namely the Lorentz, Maxwell stress tensor (MST), classical virtual work, and Coulomb virtual work (CVW) methods.

The Lorentz method can be used to predict electro- magnetic forces acting on current carrying structures only. Although it is easy to use with the finite element (FE) field formulation, it cannot be applied to deter- mine forces acting on ferromagnetic structures, except where the reaction force of a single coil appears on the iron.

The MST method is the most popular method of force calculation, and its implementation in the finite element context, especially in 2-D problems, can appear to be relatively straightforward. It requires only a sin-

OIEE, 1999 IEE Proceedings online no. 1999021 9 Paper first received 8th June and in revised form 13th August 1998 The authors are with the Department of Electrical Engineering and E k e tronics, UMIST, PO Box 88, Manchester M60 lQ0, UK

IEE Proc-Electr. Power Appl., Vol. 146. No. 1. Junuury 1999

gle field solution and necessitates the integration of a simple force density expression over any closed surface surrounding the structure (ferromagnetic, winding, or both) on which the force is needed. There are, however, accuracy problems associated with the MST method which are well documented [ l , 61, particularly with regard to the dependency of the computed force on the type of elements crossed by the integration path, the placement of this path inside the elements and its loca- tion within the finite element mesh. Very often in 3-D cases, finite element field analysis software produces the mesh by assembling or generating 3-D building blocks. The unit of construction is therefore a defined volume, which is in the nature of a scalar quantity, rather than an assembly of surfaces each of which requires a vector for complete specification. In the 3-D case, the surface integrations required for the MST method are therefore a great deal more complex than the volume integrations required for the alternative methods described below.

One major advantage of the MST method is that contributions to total force coming from different sec- tions of the moving part may be estimated provided that the integration surface coincides with the surface of the body under force. In this case Maxwell’s tensor integration over a closed surface, is carried out over several surfaces separately.

This application is, however, limited when used in the FE context because of the discontinuities of the tangential component of the magnetic field across ele- ments of the body/air interface where the surface of integration for Maxwell stress is selected [l]. Although theoretically impossible to calculate the force applied in cases where the body under force is not completely sur- rounded by air (e.g. force applied to a tooth) this situa- tion may be accommodated, as explained in [2], where the tooth is slightly separated from its stator or rotor core.

In the classical virtual work method the calculation of the electromagnetic force is based on the variation of the total magnetic energy of the system when its moving part is physically displaced in the direction of the component of force required. At least two FE field solutions are required, one before and the other after the displacement. If two, or even three components of force are required then the number of solutions needed at each point increases accordingly.

Although there may be other factors, without doubt the biggest disadvantage of the classical virtual work technique is that it requires two solutions for different meshes and the subtraction of two quantities with simi- lar values for each desired component of force.

A more refined version of the classical virtual work method makes use of calculation of the coenergy at a

25

set of closely spaced positions of the moving part, together with a curve fit, to produce a polyncmial function; the polynomial is then differentiated to give the force or torque as a continuous function of position 131. This application of virtual work is most convenient in problems where the moving part moves in only one direction, such as the determination of the torque-angle characteristic of a rotating machine, or the force dis- placement of an axisymmetrical actuator.

The CVW method, like the classical virtual work method, is based on the principle of conservation of energy and the principle of virtual displacements. Unlike the classical virtual work method, which uses a finite difference approximation to evaluate the c oen- ergylenergy derivatives, in the CVW method the global force acting on a moving structure is evaluated by direct, closed form differentiation of magnetic c oen- ergylenergy of the free space region between the mova- ble and the fixed parts of the system under consideration. Like the MST method only one field solution is required to produce all force or torque com- ponents. In a FE field solution the CVW emplcys a volume integral to determine the global force, thus, for the reason given previously, making it more straight- forward to implement in a 3-D FE electromagnetic package than the MST method.

It has been shown that the CVW method is exactly equivalent to the MST method for the case of first- order finite elements [4] if the path for integraticn of the Maxwell stress is formed by joining midpoints of element edges. In other words, the CVW method avoids the dependency of the computed force ori the position of the integration path within the element [4]. Like the MST method, the CVW method is also sensi- tive to the mesh discretisation, due to the fact that the computed force is computed using only a tiny amount of finite elements. Therefore, the computed force has been found to depend on the position of the onc-ele- ment-thick sheared layer within the finite element mesh [5 , 61. This problem can be alleviated by the particular implementation of the CVW method considered in detail elsewhere for 2-D problems [5] .

It may be concluded from the above review that the CVW approach of force calculation provides a general algorithm which is easier to apply in both 2-D and 3-D numerical electromagnetic field analysis software. This paper illustrates this advantage by reference to seteral examples.

2 Maxwell stress method

The Maxwell stress approach computes local stre:ss a t all points of a bounding surface then sums the local stresses, by means of a surface integral, to find the glo- bal magnetic force. Thus, if the total field surrounding a body is known, then the force exerted on the body in terms of the field alone can be expressed by

where s is the surface enveloping the body under force, B is the flux density and n and t are unit vectors in the normal and tangential direction on the surface s.

In 2-D problems, the surface integral of the force density is reduced to a line integral, making its iniple- mentation relatively easy in these problems. In con- trast, in 3-D problems the surface integral must be

26

evaluated over all, or selected surfaces of elements defining the structure where the force is needed. This selection and identification of surfaces makes the implementation of the MST method in a 3-D computer program far from straightforward.

3 Coulomb virtual work method

The CVW method is based upon the law of conserva- tion of energy and the principle of virtual displacement. The global magnetic force acting on the moving part of an electromechanical device is calculated as the deriva- tive of the magnetic energy with respect to the virtual displacement at constant flux linkages, or the derivative of magnetic coenergy with respect to the virtual dis- placement at constant current. The first method (energy) is more convenient when the magnetic vector potential is used in the finite element formulation, whereas the second (coenergy) applies when the mag- netic scalar potential is used.

Mathematically, the global magnetic force in the q direction, when the magnetic vector potential is used in the FE formulation, is given by [7, 81

where the summation is over all the finite elements in a sheath of air surrounding the part on which the force is to be calculated. Qe is the finite element volume and B is the flux density in the element.

The first term concerns the change in energy arising from the change in flux density with the volume unchanged. The second term describes the change aris- ing from the change in volume with the flux density constant .

The general direction q can refer to: z or y 0, 5 , g , or z

z for axisymmetric linear

for 2-D linear translations for 2-D rotation around the z-axis for 3-D linear translations

‘= 0, , OY, 0, for 3-D rotation 1 translations To evaluate the derivatives involved in the equation of the force, the finite element mesh of the device is divided into three regions, the movable region, the fixed region, and the intermediate region. The movable part of the device which undergoes the virtual displace- ment and the fixed region are not affected geometri- cally by the virtual displacement. The intermediate, or the sheared, region is the part of free space between the movable and the fixed parts that is selected to absorb the virtual displacement of the movable region.

The derivatives dB2/dq and d&,/dq require a knowl- edge of the co-ordinate derivatives of the nodes in each of the three mentioned regions. Numerical values are given to these nodal derivatives by assigning a factor between 0 and 1 to each node of the mesh. Nodes of the movable region are assigned a virtual displacement of 1. Nodes which are unaffected by motion, such as those attached to the fixed part, are assigned a factor of 0. Intermediate factors between 0 and 1 could be assigned to nodes in the free space surrounding the movable part in such way that the virtual displacement is absorbed by the finite elements in the air region.

One or many sheared shells, one element thick, of free space surrounding the movable part of the device

IEE Pruc.-Elecfr. POW.PI. Appl., Vol. 146, Nu. I , Junuarj 1999

can be identified at the time of mesh generation by giv- ing them separate labels. As illustrated in Fig. 1, each sheared layer is one element thick and can be of any shape. To simplify the programming, virtual distortion was confined to a shell formed by one layer of air ele- ments only. In this case the nodal co-ordinate deriva- tives have two distinct values: zero for the outer nodes and one for the inner nodes of the sheared or distorted shell. It has been shown that, in the case of 2-D analy- ses, little benefit accrues from shearing shells more than one element thick, although averaging results for sev- eral such shells can improve accuracy [5]. The use of a one-element-thick layer was found to give good results when it was selected away from the corners.

I I n l 2 1 1 I

U Fig. 1 Body under force surrounded by sheured layers

The force or torque algorithm in the finite element field computation therefore involves computation of the nodal values of the potential using the normal finite element method, determination of the nodal co-ordi- nate derivatives with respect to the virtual displacement of the movable part, and then integration over the vir- tually distorted finite elements.

t l

4 2L b

Fig. 2 Outline of two purdk.1 current-currying conductors

4 Validation

4.1 Problem I: Parallel current-carrying conductors The 2-D system considered consisted of two parallel current-carrying conductors of square section placed in free space (Fig. 2) [5] . An analytical solution is availa- ble for the force between them, permitting a check on the results of the MST and the CVW methods of force calculation. The distance between the centres of the conductors is lm and the dimension of each conductor

IEE Proc.-Electr. Power Appl.. Vol. 146. No. I , January 1999

is 5 x 5". The conductors carry equal and opposite currents of 1 A.

The forces computed by both the MST and the CVW methods change with the mesh discretisation and the boundary of the mesh so that several meshes were pre- pared to investigate these problems. To investigate the effect of mesh discretisation, convergence of the com- puted force with the number of nodes was carried out. The result is shown in Fig. 3 , where it can be seen that the force computed by both the CVW and the MST methods converges to the analytical value as the number of nodes is increased. Agreement within 1% is obtained with a finite element mesh containing 5680 nodes.

F c -1.90

& -2.00 a

-2.05 4000 4500 5000 5500 6000

number of nodes Fig. 3 Q--0 MST method _____ analytical method 0-0 CVW method

Computed force uguinst number of nodes in mesh

O r

Y

-1.0 ._ t L

-1.5 (U

E -2.0 E4 0.25~10 050x1 0 0.75~10 1 .OOxlO

distance L, m Fig.4 V- - - D analytical method U- ---U MST method 0-0 CVW method

Computed force uguimt distance L to boundury

To investigate the effect of the boundary, the com- puted force was plotted against the distance L from the conductor to its remote boundary (Fig. 4). A remote boundary of 10m used in the earlier calculations was found to be adequate.

Having determined satisfactory discretisation and boundary positions, several sheared layers of free space finite elements surrounding the conductor were selected for the CVW method. Integration paths within each sheared layer were also selected for the MST method.

Two different cases were considered for the MST method; in the first, the integration path is selected arbitrarily and randomly within the sheared layer but in the second, it is selected to pass through the centroid of the elements of the sheared layer (centroid paths).

The force computed by the MST method was plotted against the position of the integration path for the two

21

cases, as shown in Fig. 5. The computed force using an arbitrary integration path through an element is very inaccurate and highly dependent on the position of the path within the mesh. The force computed using (:en- troid paths, although still dependent upon the element chosen, yields better results than the case of the a.rbi- trary paths. The accuracy of the computed force may be improved by taking an average value of the forces computed from several centroid paths.

-3 0 5 10 15 20 2!5

layer position (see fig.1) Fi .5 a.i%n [uyer V-V arbitrary path within layer 0- --O optimised path within layer 0-0 analytical

Coniputed force ,from MST niethod against position of path

-1.50 r

t IC -1.75

-2.25 b

-2.50t ' ' ' " " ' ' ' ' " ' ' " ' ' ' ' ' ' ' J 0 5 10 15 20 25

layer position (see fig.1) Computed f o r c e , f i o m CVW method uguinst position and number Fig. 6

of sheared luyer 0-0 using several layers 0-0 analytical 0-0 using single layer

The force computed by the CVW method against the layer position is plotted in Fig. 6. A comparison of Figs. 5 and 6 shows that when the integration path is the centroid path within the sheared layer, the com- puted force by the MST method agrees closely with the CVW method for that layer. From Figs. 5 and 6, it can be seen that the force computed by both the MST and the CVW methods is multivalued, i.e its value is not the same when various integration paths and sheared layers are used. This unreliability in the computed force was alleviated by the implementation of the CVW method over several layers as illustrated in Fig. 6.

4.2 Problem 2: Coils in air The objective in this problem was to determine the interaction force between two identical rings of current with dimensions as shown in Fig. 7. An accurate value of force was computed from an analytically derived expression for the force between two coxial circular fil- aments [12]. The cross-section of each conductor was subdivided into equal filaments and the forces on efery

28

filament of one coil, due to all the filaments of the other coil were computed and summed to give total force. The subdivision was carried out in a progressive manner until the resultant total force converged to a value of 76.49" for equal coil currents of 300A, cor- responding to the assumed uniform current distribution of 3Amm-2.

Fig. 7 R , = 50mm, R2 = 60mm, h = IO", d = 40". J = 3Aimm'

Interaction force between two identical rings of current

Although axisymmetric, this problem was solved using both a 2-D axisymmetric model and a full 3-D model, and the interaction force was computed in three different ways namely by 2-D CVW, 2-D MST with centroid paths, and 3-D CVW. In this way the axisym- metric model provided an initial check on the 3-D algo- rithms.

In each formulation several different meshes were generated for the problem discretisation. The CVW method was applied to evaluate the z-directed interac- tion force between the rings. During the mesh discreti- sation process several sheared shells of free space finite elements surrounding the rings were labelled and num- bered according to the distance from the ring. Advan- tage was taken of symmetry in setting up the model. A sufficiently remote boundary was determined by trial.

To evaluate the results of the CVW method and the MST method, the former was applied to the first 10 shells of free space finite elements surrounding the modelled ring, whereas the latter was applied to cen- troid paths within the shells.

, I

m a r

0 4 8 12 shell number

' 70

Fig. 8 lutions - analytical V-V computed 2-D MST (centroid paths) 0-0 computed 2-D CVW M-¤ computed 3-D CVW

Computed force uguinst sheured shell position for different forniu-

Fig. 8 shows the comparison between the computed forces in terms of the location of the sheared shell and its analytical value. This Figure indicates that, for a given shell, the force computed by the 2-D CVW method agrees closely with the 2-D MST method with centroidal path within the shell. The results of both methods and the 3-D CVW method agree with the ana- lytical value to within 2% and 30/0.

The dependency of the computed force on the loca- tion of the sheared shell of the CVW method and the integration path for the MST method is not as

IEE Proc.-Eleclr. Power Appl.. Vol. 146. No. I , Junuury 1999

pronounced as in the previous example, especially with regard to the first shell. This is due to the fact that the mesh descretisation around this region was made extremely fine.

z-axis t Nyloil support I

aluminium extension MS moving

cylinder MS lid

z=o

........ ....... - R

The actuator was modelled using a 2-D axisymmetric model and a full 3-D model. One octant of the total geometry was modelled in the 3-D analysis. Results of the 2-D axisymmetric study have already been reported in [5] , where the CVW method was applied to eight concentric sheared layers in the airgap around the cyl- inder. In the full 3-D model, the CVW method was applied to a one-element-thick shell selected far from the cylinder edges. Therefore the same force was com- puted in three different ways, namely, by 2-D MST, 2- D CVW and 3-D CVW, and these computed values could be compared with the measured values.

10.0

z 7.5 ai e ,o 5.0

Fig.9 Geometry of the uctuutor

2.5 4.3 Problem 3: Axisymmetric actuator The test system considered here is one used in a previ- ous publication [5 ] . It consists of an axisymmetrical experimental actuator with radial cross-section as shown in Fig. 9. The actuator is composed of three dif- ferent parts: the fixed part, the moving part, and the coil winding. The fixed part is made of mild steel and is in the form of a cylinder with a base, a lid, and a cen- tral pillar around which is wound a coil of 1450 turns. The moving part consists of a cylindrical liner of nyloil which carries a mild steel cylinder which is acted on by electromagnetic forces. The nyloil allows very low fric- tion displacement of the cylinder along the central pil- lar. An aluminum extension is added to the central pillar in order to maintain the axisymmetrical nature of the actuator and to improve stability during the motion of the moving cylinder. An aluminum collar surrounds the bottom end of the central pillar in order to support the moving cylinder. A hole exists on the axis of the pillar in order to hang weights from the moving cylin- der.

A direct current in the coil produces a magnetic field which creates an upward or downward force on the moving cylinder, the direction of the force depending upon the position of the cylinder.

Force against current characteristics for various posi- tion were obtained as follows: (i) The position of the cylinder was fixed by non-mag- netic washers, and measured by an LVDT. (ii) By means of a known mass, a known gravitational force was applied to the cylinder. (iii) Demagnetisation of the actuator was carried out, to remove any residual magnetism, by subjecting the actuator to a sequence of hysteresis cycles of decreasing amplitude (the demagnetisation state B = H = 0, is approached by a series of decreasing cycles). Then the coil current was increased slowly until a balance between the magnetic force and the gravitational force was obtained. At each position the cylinder movement was limited to 0.016 mm to keep its magnetic history relatively simple, and then the current was reduced until the initial reading of the transducer was obtained. Satisfactory and convincing repeatability of results was achieved.

0.3 0.6 0.9 1.2 current, A

a

z

9 -

ai e 6: "..

3 -

0 " ' ~ " ' ~ " ' ~ " ' ~ 0.3 0.5 0.7 0.9 1.2

current, A b

Fig. 10 positions of the niovuble cylinder a . measured

Compurison between computed und meusured force for two

0 4 computed 2-D MST (axisymmetric) 0-0 computed 2-D CVW x--x computed 3 - D CVW

V-V computed 2-D MST (axisymmetric) 0-0 computed 2-D CVW -0 computed 3-D CVW

h . measured

Fig. 10 illustrates the comparison between the com- puted force in terms of the applied current for two positions. The computed results from both 2-D CVW and 2-D MST with centroid paths and 3-D CVW are in very good agreement with the experimental data. The 3-D result appears less accurate than the 2-D results at one point but the error (always less than 4%) can be attributed to the use of a 3-D mesh for an object with 2-D symmetry. It is considered that Fig. 10 represents a stern test of any method of force computa- tion. The field in the moving cyclinder has a complex distribution and does not have a dominant component along one of the axes. At higher excitation currents sat- uration is high and surface tangential field components are significant.

Further comparisons between the computed 2-D results and experimental data, not reproduced here, show that although good agreement is obtained when

29 IEE Proc.-Electr. Power Appl. , Vol. 146, No. I , January 1999

seven sheared layers of elements are used to compute force (discrepancies are between 0.5% and 6.5'%), the discrepancies increased (to between 9.5% and 15.5%) for computation with the first sheared layer only [5]. This large departure between the measured and the computed forces may be attributed to the large errors in the computed fields of elements near or adjacent to the sharp edges of the moving cylinder.

4.4 Problem 4: Switched reluctance motor The SR motor considered in this paper has the same specifications as the one used by Sangha [9, 10, 111. It is a three-phase motor with six stator poles and l-our rotor poles. Other parameters are given in Table 1.

Table 1: Details of SR motor

Number of phases

Number of turns per pole

Core axial length

Shaft material

Core material

Core lamination thickness

Nominal stacking factor

Air gap length

Stator outer diameter

Rotor outer diameter

3

18

24.1 m m

EN9 steel

Transil 315

0.35"

0.98

0.3"

90"

46"

The position where the reluctance of the motor is minimum, i.e. when a stator pole is facing a rotor pole is defined as the aligned position, and the position of maximum reluctance as the unaligned position. When one of the stator coil sets is excited, a magnetic flux is produced around the coil and the rotor. The rotor will experience a torque, so as to move the rotor from the unaligned to the aligned position. With appropsiate switching and excitation of the stator coils, the rotor can be made to rotate at any desired speed with con- trolled torque. The aim in this paper is to predict the torque-angle characteristics for various phase currents using both 2-D and 3-D finite element analysis, and to compare the computed results with test data. Advan- tage is taken of the motor symmetry, so as to reduce the problem size to one half and one quarter in the 2-D and 3-D FE analyses respectively. Because of the large changes in flux density around the overlapping region, the finite element meshes were made finer in this region.

Fig. 11 3-0 mesh used io nzodel the SR motor

In the 2-D FE analysis the SR motor is implicitly assumed to be infinitely long in the axial direction, so

30

neglecting end effects. The air gap was represented by three layers of elements. In the 3-D analysis, the 2-D meshes with around 3300 nodes were imported into the 3-D mesh generator where they were projected into the axial direction over a distance equal to half the core length. To include the end effects (end windings and fringing at the core ends), the mesh was extended beyond the core end of the motor. Fig. 11 shows a 3-D model mesh used: the air region discretisation is not shown for clarity. Having generated the 2-D and the 3- D meshes, the Magnetostatics solver was used to deter- mine the nodal magnetic potentials in the 2-D and 3-D models for a given rotor position and phase current. Then the CVW and the MST methods were used to determine the torque. The mid-layer and the mid-shell of air finite elements surrounding the rotor were used in the torque computation.

0 10 20 30 40 50 angle, deg

a

0.3

E 2

g 0.2 I g

t I I

0.1

0 0 10 20 30 40 50

angle, deg b

Fig. 12 a measured V-V 2-D MST (centroidal paths) 0-0 2-D CVW 0- - -0 3-D CVW u I = 10A 11 I = 20A

Torque-ungle cliuructerisiics for dij'erent currents

Results of the 2-D CVW, the 2-D MST (with cen- troid paths) and the 3-D CVW are compared against the measured data in Fig. 12 for different phase cur- rents. The Figure indicates that for an angle less than about 15 degrees, i.e. as the stator and the rotor start to overlap, the computed torques from the 2-D CVW and 2-D MST underestimated their measured values by an average error of lSu/o. This error was reduced to 7% when substantial overlap was established between the rotor and the stator poles. These differences are mainly attributed to the end effects which were not taken into account in the 2-D FE analysis. The significance of these end effects varies as the rotor moves from the unaligned to the aligned position and depends upon both the thickness of the air gap, the saturation and

IEE Pro< -Electr Poiirr Appl , Vol 146. No I , Jurzirorj 1Y9Y

the extent of the end winding. In the unaligned position where saturation is not severe, the end effect leakage is mainly caused by the large air gap thickness. However, in the aligned position, the airgap thickness is the smallest but effective saturation is much more severe, causing flux to escape from the core ends. In the inter- mediate positions between the aligned and the una- ligned positions both saturation and air gap thickness contribute to the end leakage.

The torque results computed from the 3-D CVW were in a very good agreement with the measured data, only 4% difference existing between the two. This good agreement may be attributed to the fact that the 3-D FE analysis and the 3-D CVW method virtually elimi- nate the error introduced by neglect of end effects. The 4% difference is probably largely attributable to inher- ent errors associated with the finite element method, and to some extent experimental errors.

5 Conclusion

Following a review of some difficulties in obtaining forces and torques from finite element solutions, partic- ularly for 3-D problems, the potential advantages of the CVW method have been investigated.

The use of the MST method with an arbitrary inte- gration path within a layer was found to be inaccurate and unreliable.

Application to an axisymmetric linear problem has shown good agreement between CVW and analytically derived force values, particularly when the CVW results for several layers were averaged. Application to an axisymmetric actuator with severely nonlinear behaviour gave good agreement with measured forces over a wide current range. Whereas a 2-D calculation of torques for a switched reluctance motor gave dis- crepancies against test results similar to those obtained using Maxwell stress, a 3-D CVW calculation gave very good agreement. The CVW method has been found to be relatively easy to implement for 2-D, axisymmetric and 3-D problems, and for both magnetic vector and magnetic scalar formulations. It is much easier to use than the Maxwell stress method for 3-D problems.

The CVW method (which yields all force components from a single 3-D solution) can easily be extended to electric field problems, so its speed and ease of applica- tion could prove valuable in the increasingly important area of nanotechnology. Its advantages might also be significant in coupled magneticimechanical problems.

It is concluded that the CVW method can be recom- mended as a valuable addition to engineers’ computa- tional aids, particularly for three-dimensional problems.

6 Acknowledgement

The authors would like to thank GEC ALSTHOM Engineering Research Centre, EPSRC for supporting the present work, Dr P.S. Sangha for providing meas- ured data of the SR motor described in this paper and Lucas Advanced Engineering Centre for carrying out tests to provide valuable magnetic characteristics for materials used in experiment.

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References

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