# Numerical field computations

## Why Numerical field computations?

Electromagnetic problems are described by just four equations: Maxwell’s equations. The most appropriate way to solve an electromagnetic problem is to find an analytical solution for these equations. This analytical solution is exact (not an approximation) and can usually be evaluated very fast.

In most cases however, analytical solutions are available only for simple geometries such as plates, cylinders and spheres and for materials with idealized magnetic characteristics. In practice, the majority of the problems deal with much more complicated geometries and materials. Here, numerical computations are a powerful alternative to provide (an approximation of) the solution.

For example, an analytical solution exists to calculate the magnetic field distribution around an infinitely long cylinder in a uniform alternating magnetic field. Fig. 1 shows this field distribution for a conductive and a ferromagnetic cylinder. If the cylinder is not infinitely long, if the field is not uniform or if the cylinder has a nonlinear magnetic behaviour, it is difficult or even impossible to find an analytical solution.

Figure 1: Magnetic field lines and magnetic field amplitude for an infitely long cylindrical shield in a uniform imposed field of 10 uT and 1 Hz. Left: ferromagnetic cylinder with relative permeability 175 and conductivity zero. Right: conductive cylinder with relative permeability 1 and conductivity 59 MS/m. In both cases, the field inside the cylinder is 31 times weaker than the imposed field.

## Finite Element Method

The finite Element Method (FEM) discretizes the geometry into a large number of triangles, called “mesh”. The solution is calculated in every node of the mesh. To do this, interpolation functions are used as shown in Fig. 2a: the function is 1 in one node and zero in all other nodes. The eventual solution is a superposition of all node functions, resulting in the “landscape” of Fig. 2b: the FEM solution is a surface consisting of many flat surfaces, an approximation of the exact solution. The higher the number of nodes/triangles, the more accurate the solution, and the higher the required computation time.

Figure 2: (a) The interpolation function Nk(x,y) for the Finite Element Method is 1 in the point (xk,yk) and zero in all other nodes of the grid (b) each function A(x,y) can be approximated by a linear combination of interpolation functions.

Finite Element Methods can deal with almost any geometry. Nevertheless, very long and thin objects may be problematic. Problems containing such objects can be modelled by applying special boundary conditions or by using the network or circuit method. Other techniques exist to deal with moving objects and to couple electromagnetic problems with thermal or structural problems.

Figure 3: Magnetic field lines and magnetic field amplitude for a permanent magnet synchronous machine.

## Circuit method

The idea of the circuit method is to model objects such as the thin plate in Fig. 4 as a grid of connected conductor filaments in Fig. 4b. Each filament has its own resistance, self inductance and mutual coupling with all other conductors, including the source conductors. Once the resulting electrical circuit shown in Fig. 5 is assembled, the only thing to do is to solve it. The magnetic field is calculated a posteriori by using the currents in the branches in combination with the law of Biot and Savart.

The circuit method is easy to understand and easy to implement. Another advantage is the possibility to model complex 3D geometries and sources. As the number of unknowns is much smaller than for 3D Finite Element Methods, the calculation time is low if the resistances and inductances in the circuit can be determined in an analytical way.

Although the solution of the electrical circuit is exact, the circuit method yields an approximate solution only, depending on the grid refinement. However, the accuracy is good even for a coarse grid of filaments. As the grid is refined, both the accuracy and the required computation time increase. Modelling ferromagnetic objects in the circuit method is possible but not so easy. Consequently, the method is most suited for problems containing few ferromagnetic objects (e.g. no electrical machines).

Figure 4: A conducting plate with given height, width and thickness in a uniform alternating field B0 (a) is replaced by a grid of conductors in the circuit method (b).

Figure 5: One cell in the grid of filaments in Fig. 4b results in the green electrical circuit. The source field in Fig. 4b is represented by the purple circuit, mutually coupled with the green circuit.

Example: shielding of buried high voltage cables.

Fig. 6 shows six plates in copper above three buried high voltage cables (in red). The aim is to study the effect on the magnetic shielding performance of the number of sheets, the size of the sheets and the contact resistance between adjacent sheets.

The thin blue lines in Fig. 6 illustrate the grid of cells used by the circuit method. For a high contact resistance, Fig. 7a shows that the currents flow along the edges of each plate. For a low contact resistance between adjacent plates, the current flows along the edge of the entire shield (Fig. 7b). Fig. 8 depicts a Finite Element simulation of the situation of Fig. 7b.

Figure 6: A shield consisting of 6 copper plates (thick black lines) above three buried high voltage cables (red lines). The thin blue lines illustrate the grid of filaments used by the circuit method to evaluate the magnetic shielding efficiency above the plate.

Figure 7: Current density in the filaments of the copper plates. Red means high current and blue means low current. In (a), the six copper plates don’t have any electrical contact; in (b) the contact resistance between all six plates is zero.

Figure 8: Finite Element simulation of the field distribution near the six copper shields, caused by the three phase electrical power system in the buried cables.

## Network method

In the network method, the electromagnetic system is replaced by a magnetic network instead of an electrical circuit in the circuit method. The impedances are reluctances and the sources are magnetomotoric forces. As the reluctances are not always easy to determine, the network method yields less accurate results, but the computation time to solve a network with few elements is short.

Example: magnetic network of the switched reluctance  motor

The 6/4 switched reluctance motor (SRM) is approximated by a magnetic network and is constructed in such a way that every geometrical part of the magnetic circuit, in which we a assume a uniform field pattern, is substituted by a certain reluctance.

Fig. 9 shows the reluctances of the network model. The reluctances of the stator pole, the stator yoke, the air gap, the rotor pole, the rotor yoke, are indicated with subscripts sp, sy, δ, rp, ry respectively. The reluctances R are calculated by means of geometrical dimensions and magnetic permeability.

As shown in Fig. 9, each elementary part of the magnetic circuit of the SRM is described by 3 parallel reluctances. This is due to the fact that the magnetic material degradation due to the manufacturing process is incorporated in the model. The reluctance in the middle, is the reluctance of the non degraded material. The two reluctances placed parallel to the middle reluctance, describes the degraded material and uses different magnetic material properties.

Figure 9: Magnetic network of the 6/4 switched reluctance motor.

## Comparison of numerical and analytical techniques

 Analytical FEM Circuit Network Computation time Very short High Average Short Magnetic properties Non-linear, hysteretic Non-linear*, hysteretic Linear Non-linear Very thin objects Yes No Yes Yes Geometry Simple Complex 2D Complex 3D* Complex 3D, few ferromagnetic objects Complex 3D Assets Fast parameter studies to gain insight in e.g. influence of material properties Accurate modelling of 2D electromagnetic systems (e.g. motors, actuators,...) Models of complex 3D geometries with lots of air and thin objects (e.g. shielding) Fast alternative for FEM Drawbacks No realistic geometries Slow: optimizations requiring many FEM evaluations may consume too much time Ferromagnetic materials strongly increase the complexity of the model Not as accurate as FEM

* Computation time may become huge