IEEE TRANSACfIONS ON MAGNETICS, VOL. 24, NO. I, JANUARY 1988 447
FINITE ELEMENT FORCE CALCULATION: COMPARISON OF METHODS FOR ELECTRIC MACHINES
J.Mizia* K.A.damiak** A.R.Eastham* G.E .Dawson*
• Department of Electrical Engineering Queen's University Kingston, Ontario K7L 3N6
Abstract
This paper presents results from torque and thrust calculations using the finite element method. The torque and thrust have been calculated by the Maxwell stress tensor, the virtual work principle and the Lorentz formula. Calculations for two types of machines have been considered: the switched reluctance motor and linear induction motor. Computational results have been validated by comparison with experimental data. Analysis of these results reveals some practical conclusions about the shortcomings and the advantages of the different methods .
Introduction
Global output values, rather than the details of the magnetic field distribution, are the final aim of most computer calculations in electric machine design. The most important values are: torque, force, power, terminal impedance and voltage or current. Mechanical output values, such as thrust and torque, can be calculated by means of the Maxwell stress tensor, the virtual work principle, and the Lorentz formula. These techniques are well known and have been extensively applied [l·�). However, when combined with finite element analysis, unforeseen discrepancies can sometimes result .
The use of the Maxwell stress tensor has been pred.ominate in the literature [2-4J, but this method requires a very fine discretization. Some improvement can he achieved by proper treatmenl of the solution, which smooths the field distribution in the airgap I�J or by using complementarv pairs of energy functionals [3J. This method allows evaluation of the upper and lower limits of the solution, but the problem has to be solved twice by two different teChniques .
The virtual work principle formulation has been applied frequently as well. but many authors emphasize its weak point, namel� numerical differentiation of the coenerg y (or energy). However, this can be avoided by application of special techniques [lJ.
This paper provides further insights into the three different methods for force and torque calculations through comparison of computed and experimental results . .
Finite Element Analysis of Switched Reluctance Motor
Switched reluctance motors (SRM) are receiving increasing attention in recent years due to their simplicity and controlleabiliry. Investigations have shown that system efficiency of �R�I as a variable-speed-drive is better than comparable variable-speed induction motor drives. Exact evaluation of SRM performances needs a rel iable model for th.e magnetic field calculation. Because of deep saturation of the magnetic circuit and complex geometry, the finite element method (FEM) seems to be most suitable . This method is widely used by designers , but a very small airgap and a salient pole structure cause some difficulties during the computation of torque.
The outline and equipotential lines distribution of SRM is shown in Fig. I. The magnetic field is the first step to performance calculations. In the case of SRM, the torque can be calculated by means of the Maxwell stress tensor or from the virtual work principle . The former method requires the calculation of a contour integral to obtain torque from
where
T = [_1_ J r B,B, dl'l X stack width Jio r
T· torque B" B,,· radial and azimuthal components of
the flux densiry, respectively f' - closed contour around stator.
(1)
** Department of Electrical Engineering Kielce l'nhersity of Technology Kielce. Poland
Fig. 1 Outline and equipotential lines of the magnetic vector potential for one half of the switched reluctance motor
In the latter method, the torque is calculated from the rate of change of coenergy with respect to angular displacement
(2)
where W . total coenergy of the system
Numerical results from virtual work principle method and experimental data have been compared in the paper [5J. However. application of the Maxwell stress tensor method to this same problem gave very inaccurate results. Further investigations revealed that the results were extremely sensitive to mesh discretization and integration contour location. To investigate this problem more systema tically, five different meshes were prepared. They vary from very coarse, Fig. 2a, to very fine. Fig. 2b. Data for each model are summarized in Table L For each model both the coenergy and the torque from the virtual work principle are calculated. The coenergy varies by a small amount for the five different models with the torque value being almost the same in all cases. On the other hand, the Maxwell stress tensor method resuits, shown in Table 1, are very sensitive to discretization densiry and integration contour position. The reason for this sensitiviry can be linked to the very high irregulariry of the magnetic flux densiry tangential component. �ear the pole edges , as a result of small airgap, it has the shape of an abrupt impUlse. To obtain an accurate evaluation of the field in this region a very fine local mesh is required. Fig. 3 presents distribution of the tangential component of the magnetic flnx densiry computed from models 2 and 5. The n ormal component distribution for this same condition is less sensitive to discretization density and contour location (Fig. 4). In the application of the Maxwell stress tensor, it is the
Table 1: Data of the mesh model s and summary of the re sults for the switched reluc tance motor
Angle=14°; Current=10A; Texp
=12.06 Nm; Tcoen=11.02 Nm
�O.Of No.of No.of Co- Torque from Maxwell stress foodel nodes elem. energy Tensor (Nm)
Radius (mm) of contour Avg. for integration Torque
Fig Discretization of the area near pole edges a) model 2 bJ model 5
product of both components that is important. Unfortunately, the maximum sensitivity of the tangential component on discretization density occurs m the area where the normal component has maximum gradient. For this reason, the product of normal and tangential components is even more sensitive to discretization density and contour location than either component separately, Fig. S.
Comparison of calculated torque results from the Maxwell stress tensor (model 5), the virtual work principle (model 2J and experimental torque are shown in Fig. 6. Accuracy of results for both methods is similar, though in some areas the Maxwell stress tensor clearly gives more accurate results.
Performance Analysis of lO-Pole Switched Reluctance Motor
The finite element method has also been applied to the another configuration of SRM for which one segment is shown in Fig. 7. The torque has been calculated for two different mesh models. The data are summarized in Table 2. Torque results, calculated by means of the Maxwell stress tensor. as a function of integration contour location, for different angles and for model 1 are shown in Fig. 8. Values of torque obtained from the virtual work principle is shown as a continuous line. Results for different integration contour locations are close to each other when the. contour doe.s not lie on the iron surface or on the boundary dividing layers of elements in the airgap. The suggestion to locate the integration contour as far as possible from iron parts can be found in the book hy Lowther and Silvester �6]. Our results confirm this conclusion and further indicate that surfaces dividing layers of elements should also be avoided.
Fig. 6 Comparison of numerical and experimental torque for the switched re luctance motor .
Fig. 7 Outline and equipotential lines of the magnetic vector potential for the to-pole SRM
Table 2: Data of the mesh models for the 10-pole SRM
nodes elements ---------- ---------- -----------
Model 1 2289 4475 --------- ---------- ---,--------
Model 2 3132 6171
Calculations for mesh model 2 are more lime consuming, but results are more stable \\'ith respect to integration contour location Fig. 9, even for "badly chosen'l contours.
'
449
ANGLE 4 DEGREES 29.3r-----�----�--____ --__ _r--------_. virtual work prlnclple--
Maxwell stress tensor- If
r- radiuS of element * 0.1 vlding layer
"
It
Torquc/pole/sta�k width (krI)
ANGLE 7 OE:GREES
"
29. 3 r-----�--.,.....--�-----------, virtual work princlple- ..
It "
M-- Maxwell stress tensor
I\radlus of ele;cnt di vidlng layer
It 29.2�---�----�� __ � __ ----__ --__ �----�
1 '1'orque/nole/stack width n:!r)
Fig. 8 Torque versus integration contour position for the lO·pole SRM (model 1) at a rotor angle of 4 and 7 degrees.
Fig. 9 Torque versus integration contour position for the lO-pole SR� (model 2) at a rotor angle of 4 degrees.
Performance Anaiysis of Linear Induction :'>lotors
Analysis of the field and performance of linear induction motor
(LIM) IS more difficult than the rotary, because of the end effects.
Unlike the rotary induction motor were it is sufficient to analyze one pole
pitch, the whole LIM is required because the symmetry and periodicity
conditions are not fulfilled. Discretization of the whole model yields a very large mesh and its refinement to a finer mesh may be limited by
computer memory size or computer CPU time costs. Thus, finding the
force calculation method, which would be efficient even for coarse mesh
is of great practical meaning.
450
2000
1000
� 0 ::> a: I I--
-2000
-3
K M K - EXPERIMENT + + + - LORENTZ
FORMULA
- MAXWELL
STRESS
TENSOR
* * •
-2 - 1 o SLIP
mm
mm mm
mm
2 3
Fig. 10 LIM thrust force experimental data and calculated values from the Lorentz formula and the Maxwell stress tensor
The FEM has been applied to the two dimensional model of single-sided LIM with a double layer reaction rail described in [71. Influence of the transverse end effect has been taken into account by altering the aluminum cap conductivity according to the Russel and Norsworthy factor. The most important parameter of LIM, thrust force, can be calculated by means of the Maxwell stress tensor
where
F = [+- J Bv Bx dl] X WIdth "Po r
dosed contour in the airgap.
(3)
As another possibility the Lorentz formula can be used to used to calculate the thrust force
(4)
where S - crOss section of the secondary conductor
Comparison between the results obtained from the Maxwell stress tensor and the Lorentz formula is presented in Fig. 10. Throughout the whole region of operation. the difference between results obtained for different contours is rather large. To show in detail the variation of ca1culated thrust and normal force as a function of contour position, Fig. 11 has been prepared. Both components of force have been calculated at velocities corresponding to their peak values. If the mesh model has only one layer of elements in the airgap (model 1), the thrust and normal force vary linearly with contour location. For mesh model :: a jump occurs when the integration contour crosses the boundary dividing layers of elements. Deterioration of results for contours near iron parts is not noticeable in this case.
Conclusions
The three examples of force and torque calculations using the finite
element technique presented in this paper show that care must be taken
when discretizing the problem and establishing the position of the
integration contour. Computation results for a small airgap machines indicate the integration contours should be as far as possible from magnetic elements and the contours should not lie on surfaces dividing
layers of elements. To obtain the sam� accuracy of torque results , the
virtual work principle can use a much coarser mesh than that required for the Maxwell stress tensor method, This is an advantage in this study where a new mesh is required for each angle. However, numerical differentiation of the ('oenergy can introduce significant errors.
THRUST (kN)
I-t .,
EXPERIMENT/
v - 5.5 m/s
1,8
1,4
+ ". M •
".
v - 2 m/s
MODEL 1 M M M - MST
+
"
� LORENTZ
MODEL 2
B.5 .. +
+ MODEL 2 + + + - MST + + + MST
121+---+ - LORENTZ +---+ COMBINED
o 3 B 9 -'-2--'-'5 55 O:--�3---=B--9-INTEGRATION CONTOUR POSITION
12 15
IN THE 4IAGAP Cmm)
Fig. 11 Dependence of thrust and normal force, calculated by the Maxwell stress tensor, on the integration contour position
References
1. 1.L. Coulomb and G. Meunier, "Finite element implementation of virtual work principle for magnetic or electric force and torque computation", IEEE Trans. on Magnetics, vol. MAG 20, pp. 1894-1896, 1984.
K.J. Binns, C.P. Riley and M. Wong, "The efficient evaluation of torque and field gradie�t in permanent-magnet maclunes with small airgap", IEEE Trans. on �agnetics, vol. MAG:l, pp. 24,5-2438, 1985.
J. Penman and �I.D. Grieve. "Efficient calculation of furce in electromagnetic devices", lEE Proc., vol. 133, pt. B, pp. 212-210, 1986.
4. J.B. Ayasse, F. Goby, J.P.Pascal and A. Razek, "Two models for force calculation in electromagnetic devices", Proc. Int. Conf. on Electric Machines, Munich 1986, pt. 2, pp. 798-800.
5. G.E. Dawson, A.R. Eastham and J. Mizia, "Switched reluctance motor characteristics: finite element analysis and test results'"
Conference Record of the 1986 IEEE Industry Applications Society Annual Meeting, part I, 86CH2272-3, 1986, pp. 864-869.
6. D.A. Lowther and P.P. Silvester, Computer-aided-design in magnetics. Springer Verlag 1980.
7. G.E. Dawson, A.R. Eastham. J.F. Gieras, R. Gng and K.
Ananthasivam, "Design of linear induction drives by field analysis and finite element techniques", IEEE Trans. on Ind. Appl., vol. IA 22, pp. 865-873, 1986
8. K. Adamiak , 1. Mizia, G.E. Dawson and A.R. Eastham, "Finite element force calculation in linear inductlOn machines". IEEE Trans. on Magnetics (in press)
