Visualising magnetic fields: numerical equation solvers in action

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Article First Online: 11 March This is a preview of subscription content, log in to check access. Asai, S.

CrossRef Google Scholar. Berger, P. Gatzen, M.

Visualising Magnetic Fields - 1st Edition

Xu, Z. Schneider, A. Zhang, Y. Vollertsen, F. Matsunawa, A.

1st Edition

Tang, Z. Bachmann, M.

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Zhou, J. Marcel, B. Avilov, V. Google Scholar. A similar circuit but with 60 60 meshes was analysed to give the resultant loop currents shown in Figures and With the exception of a slight cramping of the contour lines towards the edge of the picture, this does indeed provide an analogy to the way that magnetic fields propagate due to a flux source in this case the flux from a current carrying wire perpendicular to the plane of the paper.

A circuit with two sets of mesh loop voltages, similar to Figure but with a 60 60 mesh, was then generated and analysed.

Note that the directions of the sets of loop voltages are of opposite sign. Figures and show the results of analysing this mesh, and this time we get a contour plot that shows an analogy to the magnetic field pattern of a coil or permanent magnet. The basic technique 23 Further work showed that the resistor values could be taken as analogies to magnetic permeability, with a value of unity for free air or vacuum and values less than unity for more permeable material, e. This then, is the basis of the present visualisation technique. The mesh of resistors. The mesh equations are solved to give the loop currents and a contour plot then gives a visualisation of the magnetic field pattern.

An example for a mesh is given in Figure Within the righthand rectangle in Figure is a set of voltage sources representing a coil in air. Within the other enclosed areas the resistors are set to 0. Characteristics of the technique A characteristic of this technique is that the mesh regions are of fixed size, so defining the spatial resolution of the analysis. Finite element techniques, on the other hand, allow variable size meshes, so that complex regions and regions of rapidly changing magnetic flux density can be modelled more finely. The method does not pretend to produce an analytically correct result, but rather places an emphasis on speed of model creation, with a primary.

The basic technique 25 purpose to produce a representation or visualisation of magnetic field lines and magnetic flux density plots to allow insights to be gained into different model configurations. The method also does not model non-linear effects such as magnetic saturation or permanent magnet demagnetisation. Generating the equations Reference [1] gives a rigorous derivation of the mesh loop analysis technique, and it is not necessary to repeat this here, since we are primarily concerned with the application of the results. It is sufficient to give an example of the equations resulting from a typical mesh.

Consider the 3 3 resistor mesh with a single voltage source in one branch, shown in Figure The meshes are numbered from 0 to 8. The resistor subscripts are Rh for horizontally drawn resistors and Rv for vertically drawn resistors, which is convenient here as this is how they will eventually be numbered and handled in the computer programs. This 3 3 mesh results in the following 32, i. The subscripts have the following meanings: R00 means the self-resistance of mesh 0, i. R11 means the self-resistance of mesh 1, i.

R01 means the mutual resistance between mesh 0 and mesh 1, i. R10 means the mutual resistance between mesh 1 and mesh 0, i. V1 etc. Now some of these coefficient terms, for example R38, are zero, since there is no mutual resistor shared between mesh 3 and mesh 8, and many of the loop voltages are zero also. With the zero terms in the equations a diagonal matrix form emerges, with the mesh self-resistance as the centre diagonal, the mesh vertical resistor mutual resistance as the next diagonal upper or lower and the mesh horizontal resistor mutual resistance as the outer diagonal upper or lower.

The matrix form If we look at a large matrix we find the form shown in Figure Loop currents Loop voltages. The coefficients of the matrix, derived from self and horizontal and vertical mutual resistor values, lie on the diagonal lines and the form is known as a tridiagonal banded sparse matrix. The sparsity comes from all the coefficients not on the diagonal lines shown being zero.

Making use of the technique To produce an acceptably fine resolution, there have to be a reasonably large number of meshes in the modelling space. We might expect to use, say, 50 50 meshes for small models, and many times this for more complex ones. Even 50 50 meshes results in simultaneous equations to solve, and larger models can require or more. Before proceeding any further, therefore, we must demonstrate that this is possible in a sensible computation time, and the next chapter examines numerical algorithms for matrix solution.

Magnetic field

To obtain the loop current distribution in an N N resistor mesh, N2 simultaneous equations must be solved. The solution may be obtained numerically by a direct method based on Gaussian elimination, or by an iterative method usually based on the conjugate gradient technique. From the users point of view the difference between the two methods shows as a trade-off in the areas of memory size and computation time. The direct method requires substantial memory moving into virtual memory for large N but works with a predictable number of computational steps as a function of N.

The conjugate gradient method requires a minimum amount of memory, but has a variable number of iteration steps and hence computation time. Both methods will be described here, together with typical timings and storage requirements, and both are implemented in the software. Gaussian elimination The Gaussian elimination algorithm has two phases, the first of which is forward elimination and is the most computationally intensive. In this phase all terms in the coefficient matrix below the centre diagonal become zero.

In the second phase, backward substitution, the loop current values are extracted. As an example take the 3 3 mesh shown in Figure , and the associated nine simultaneous equations. An operation that can be performed directly on an equation is to multiply both sides by a constant. Another operation that can be performed on sets of such simultaneous equations without altering the solution is to add two and replace one of these by the sum the reader may try a simple example for proof of this. For example, the first two equations of our example are:.

Now add 2 and 3 to eliminate I0: R Equation 4 can then replace 2 in the original set. In doing this we have eliminated I0 from the second equation, and we can similarly manipulate the first and fourth equation of the original set to eliminate I0 from this, so leaving I0 only in the first equation.

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We can repeat all these manipulations to gradually eliminate all the equation terms below the central diagonal line. This completes the forward elimination phase, and the backwards substitution phase is clear. Starting with the last equation we can solve for I8 directly and then substitute the newly found I8 value into the previous equation to solve for I7 and so on. It will be found that a division by R00, R11, R22 , etc. If we remember that this is the self-resistance of a mesh i. The amount of computation and storage necessary for the solution must be considered. When a larger network is used, the matrix form shown in Figure is obtained, and two points should be noted. First, all coefficients not marked as lines are zero. Second, the mutual coefficient lines above the centre diagonal are identical to those below the centre diagonal and therefore do not need separate storage. In the forward elimination stage of the Gaussian method all terms below the centre diagonal become zero, and no storage need be allocated for them.

Figure Computation block required during forward elimination phase, fill in of terms above the centre diagonal and elimination of terms below the centre diagonal. However, all terms between the centre diagonal and the upper horizontal resistor mutual coefficient line become non-zero, and therefore storage must be allocated to these during the analysis.

Also some non-zero terms are created below the centre diagonal, before being set to zero, and a storage block must be allocated as shown in Figure It is mainly the filling in of the upper diagonal terms that causes the larger memory requirement in the direct Gaussian method. In the source code procedure ANALYSE, the lines of code in the forward elimination phase that occupy the most computation time are commented.

Conversion of these lines to machine code can be expected to show an appreciable speed improvement.

Visualising magnetic fields: numerical equation solvers in action Visualising magnetic fields: numerical equation solvers in action
Visualising magnetic fields: numerical equation solvers in action Visualising magnetic fields: numerical equation solvers in action
Visualising magnetic fields: numerical equation solvers in action Visualising magnetic fields: numerical equation solvers in action
Visualising magnetic fields: numerical equation solvers in action Visualising magnetic fields: numerical equation solvers in action
Visualising magnetic fields: numerical equation solvers in action Visualising magnetic fields: numerical equation solvers in action

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