TEAM Workshop Problem No. 20
3-D Static Force Problem
1. General description
The model is shown in Fig. 1. The center pole and yoke are made of steel.
The coil is excited by a dc current. The ampere-turns are 1000, 3000,
4500 and 5000 which is sufficient to saturate the steel.
The problem is to calculate the magnetic field and electrmagnetic force.
2. Analysed Region and Boundary Conditions
If the symmetrical boundary conditions can be used, the ¼ region
shown in Fig. 2 is a sufficient for analysis.
Mesh Description
The mesh is no specified.
4. Nonlinearity
The B-H curve of the steel shown in Fig. 3 is to be used. The
typical values of B(T) and H(A/m) are also shown in Table 1.
The curve at high flux densities (B>2.3 T) cannot be measured and is
approximated by the following equation:
B = µoH + Ms,
where µo is the magnetic constant and Ms is the
saturation magnetization (2.16 T).
5. Quantities to be Calculated
To compare results, please complete Tables 2, 3, 4 and 5. Fig. 4 shows the
positions at which the flux density should be calculated.
6. Description of Computer Program
To compare formulations, variables, etc., please complete Table 6. The
used memory in item No. 16 in Table 6 is defined as the sum of dimensions
declared in the program.

Fig. 1 3-D model for verification of force calculation.

Fig. 2 Analysed region.

Fig. 3 B-H curve of steel.

Fig. 4 Positions at which the flux density should be calculated
(see Tables 2, 3 and 4).
Table 1 - Typical data of B-H curve
Table 2 - z - directional
components Bz of flux densities at points P1 and P2 (see
Fig.4)
Table 3 - z - directional
component Bz of average flux densities in center pole
(alpha-beta) and yoke (gamma-delta) (see Fig. 4)
Table 4 - x - directional
components Bx of flux densities along lines a-b
and c-d (see Fig. 4)
Table 5 - z - directional
components Fx of force
Table 6 - Description of computer
program
References
- O. C. Zienkiewicz: "The Finite Element Method (Third Edition)",
McGraw-Hill (1977).
- P. P. Silvester, H. S. Cabayan & B. T. Browne: "Efficient Techniques
for Finite Element Analysis of Electrical Machines", IEEE Trans.
PA & S, PAS-92, 6, 1274 (1973).
- J. H. Hwang & W. Load: "Finite Element Analysis of the Magnetic
Field Distribution inside a Rotating Ferromagnetic Bar", IEEE Trans.
Magnetics, MAG-10, 4, 1113 (1974).
- H. Akima: "A New Method of Interpolation and Smooth Curve Fitting
Based on local Procedures", Journal of ACM, 17, 4, 589 (1970).
- C. R. I. Emson: "Methods for the Solution of Open-Boundary
Electromagnetic-Field Problems", IEE Proc., 135, Pt. A, 3, 151
(1988).
- P. Tong & J. N. Rossetos: "Finite-Element Methood (Basic Technique
and Implementation)", MIT Press (1977).
- P. Sonneveld: "CGS, a Fast Lanczos-Type Solver for Nonsymmetric
Linear Systems", Report 84-16, Department of Mathematics and
Informatics, Delft University of Technology, The Netherlands (1984).
- A. Bossavit & J. C. Verite: "The 'TRIFOU' Code: Solving the 3-D
Eddy-Currents Problem by Using H as State Variable", IEEE Trans.
Magnetics, MAG-19, 6, 2465 (1983).