**By R. Belmans**

## Modelling of electrostatic and magnetic devices

**The modelling of a technical device** is a process of neglecting and simplifying in order to describe the physical device in a mathematical model. The type and quantity of the simplifications influences the accuracy of the solution. In most cases, a choice has to be made between the required accuracy and the numerical size of the model.

**Fig. 9.1. Assumptions.**

**To define an appropriate field problem,** the model of a real electromagnetic device, simplifications have to be made on three levels (Fig. 9.1). The recommended predictions concerning the behaviour of the real device must be translated into a model representation. Thus, assumptions have to be made concerning different issues.

**The time dependence of the problem can be of the type:**

**• static**

**• stationary changing**

**• periodically changing, or may be**

**■ transient.**

**The problem can be considered geometrically as a**

**• 2D problem that is constant in the z-direction, or as an**

**• axially symmetrical problem, only consisting of a tangential component, or as an entire**

**• 3D model.**

**In the mathematical formulations, simplifications are already embedded. The most important effects of the device must be present in the particular potential formulation for example such as:**

**• non-linearities,**

**• hysteresis effects**

**• couplings to other field types**

**• couplings of external circuits,…**

Decisions have to be taken to define the model and thus to describe the real device with a maximum of accuracy.

**To choose the appropriate** solver module of a numerical field computation software package the dependence with respect to the time of the field problem has to be considered. Table 9.1 represents a systematic for choosing an appropriate solver. Time-harmonic or periodic problems can be handled with a transient solver as well. The disadvantage of this approach is the huge computational cost.

**Particular attention must be** paid to the analysis of electromagnetic devices such as electrical machines. By studying different modes of operation, different problem types have to be defined.

In the following sections the model properties of the various field types are discussed.

**Table 9.1. Selection of appropriate solver modules.**

## Static problems

**A static field is a field with** constant boundary conditions and is excited by DC currents or voltages. If a DC voltage is applied, the DC current depends on the voltage only by the DC resistance. No induced effects such as eddy currents or induced voltages are present in the model. In a static model, the flux is constant in time and thus the reluctivities are constant as well.

Regarding the static computation as an experiment in the laboratory, a probably varying temperature during the experiment can represent a ‘time’ dependent factor, if the stationary temperature is not reached yet. Material properties such as the conductivity of a field-exciting conductor change with varying temperature and thus the resulting field as well if a voltage is imposed as excitation. This effect can be taken into account by defining the material at the particular temperatures in the different materials assuming a stationary temperature during the simulation.

## Quasi static problems

A quasi-static field is a time-varying field where no eddy currents are involved. The field solution does not depend on the time-derivative term in the differential equations. It can be regarded as a static field for a particular instant of time. The calculation of the field is performed for a certain instant of time, and therefore the flux density and the resulting inductances are calculated for this specific instant of time.

**The problem is excited by** imposed currents. After the calculation of the coil inductances out of the solution, the induced voltages can be derived.

If the field is driven by imposed voltages, a time-dependent solver has to be chosen to fulfil the voltage law. The applied voltage is the sum of all induced and resistive voltage drops.

An example of a quasi-static field calculation is an instantaneous no-load calculation of a device operated by time-dependent currents where eddy current effects can be neglected, such as an induction machine at synchronous operation.

## Time-varying problems

When eddy currents are involved and have to be considered, a time-derivative term appears in the differential equations. The varying field generates induced voltages and currents. The eddy currents are influencing the field.

**Each form of time variation** can be modelled either in the time domain or in the frequency domain. A time-stepping solver is called a transient solver. When the field is periodic with one or a limited number of frequencies, it is more efficient to perform a field calculation in the frequency domain instead of in the more convenient time domain. If only one frequency is involved, this solver is called a time-harmonic solver.

## Transient, time domain

**A transient solver starts from given** starting conditions. Depending on actual field quantities, considering the imposed sources and boundary conditions, the differences of the field quantities are calculated. Adding these differences to the previous field quantities leads to the field at the new instant of time. The time dependence of the field is approximated by a Taylor expansion.

**Frequency domain **A solver in the frequency domain represents the field quantities, applied sources and boundary conditions as a summation of phasors rotating at a given angular frequency

**From each phasor**, the magnitude and the phase, or the real and imaginary component is calculated. When non-linear materials are considered, the material characteristic varies at given intermediate frequencies.

**Time-harmonic problems** In this problem class, all field quantities, imposed sources and boundary conditions are assumed to be sinusoidal varying with respect to the time.

For all material properties it is assumed to be constant in time.

For non-linear material, an effective material characteristic is used. This characteristic gives a sinusoidal averaged value in terms of the rms value of the field quantity.

**Regarding the magnitude of the magnetic potentials,** it can be noticed that independent sources will cause a potential peak whereas short-connected conductors will try to keep a constant potential. The boundary conditions and the external conditions for the conductors determine the relative magnitudes of the potentials. The potential is smoothly distributed over the regions without current.