TEAM Workshop Problem No. 13
3-D Non-Linear Magnetostatic Model



1. General Description

The model is shown in Fig. 1. An exciting coil is set between two steel channels, and a steel plate is inserted between the channels. The coil is excited by dc current. The ampere turns are 1000 and 3000 AT which is sufficient to saturate the steel.
The problem is to calculate magnetic fields at various positions.


2. Analyzed Region and Boundary Conditions

If the symmetrical and periodic conditions [1] can be used, the 1/4 region shown in Fig. 2(a) is enough to be analyzed. The analysis of 1/2 region shown in Fig. 2(b) using only symmetrical boundary conditions is also acceptable.


3. Mesh Description

The mesh is not specified.


4. Nonliniarity

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 Fig. 3. The curve for high flux densities (B>1.8 T) should be approximated by Eq.(1):



where µo is the permeability of free space. The constants a, b and c are -2.381E-10, 2.327E-5 and 1.590 respectively. Ms is the saturation magnetization (2.16 T) of the steel. Equation (1) shows that the steel part is assumed to be completely saturated when B is higher than 2.22 T.


5. Quantities and Distributions to be Calculated

5a. Points where flux densities are compared

To compare results, please complete Tables 1, 2 and 3. Fig. 4 shows the specified positions for average flux density in the steel and flux density in the air. Fig. 5 shows the recommended points to be compared. The points 1 and 4 are for comparison between various numerical methods of analysis. The points where large errors may occur, such as due to large flux density changes, are chosen. The points 5 to 8 show the recommended points to be compared with experiment. Around these, flux densities can be measured accurately because the gradient of flux density is not so high.


5b. Distributions of flux density vectors

Distributions of flux density vectors on the x-y plane at z=63.2 mm, and on the y-z plane at x=0 mm are to be presented.


6. Description of Computer Program

To compare formulations, variables, etc., please complete Table 4. The used memory in the item No. 17 in Table 4 is defined as the sum of dimensions declared in the program.




Fig. 1 3-D non-linear magnetostatic model.





Fig. 2 Analysed region.





Fig. 3 B-H curve of steel.





Fig. 4 Specified positions for flux density (see Tables 1 and 2).





Comparison between various numerical methods:

o the points where the flux densities change suddenly - - - 1, 2
o the point where the permeability changes suddenly - - - 3
o the point where the error due to the cancellation may be large - - - 4

Comparison between calculation and experiment:

o the average flux densities - - - 5, 6, 7 (No. 1, 12 and 25)
o the point where the flux density is high and it does not change suddenly - - - 8 (No. 27)

Fig. 5 Recommended points to be compared
(see Table 1, 2 and 3)


Table 1 - Average flux density |B| [T] in the steel (see Fig. 4)

Table 2 - Flux density |B| [T] in the air (see Fig. 4)

Table 3 - Flux density |B| [T] (see Fig. 5)

Table 4 - Description of computer program


References

  1. T. Nakata, N. Takahashi, K. Fujiwara & A.Ahagon: "Periodic Boundary Condition for 3-D Magnetic Field Analysis and its Applications to Electrical Machines", IEEE Trans. Magnetics, MAG-24, 6, 2694 (1988).
  2. O. C. Zienkiewicz: "The Finite Element Method (Third Edition)", McGraw-Hill (1977).
  3. P. P. Silvester, H. S. Cabayan & B. T. Browne: "Efficient Techiques for Finite Element Analysis of Electrical Machines", IEEE Trans. PA&S, PAS-92, 6, 1274 (1973).
  4. 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).
  5. H. Akima: "A New Method of Interpolation and Smooth Curve Fitting Based on Local Procedures", Journal of ACM, 17, 4, 589 (1970).
  6. C. R. I. Emson: "Methods for the Solution of Open-Boundary Electromagnetic-Field Problems"", IEE Proc., 135, Pt.A, 3, 151 (1988).
  7. P. Tong & J. N. Rossetos: "Finite-Element Method (Basic Technique and Implementation)", MIT Press (1977).
  8. 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).
  9. 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).