**By R. Belmans**

## Electromagnetic fields

**The notation and basic laws of the electric** and magnetic fields are explained in this section. It is not intended to present the complete electromagnetic field theory. Only a limited set is given, necessary to understand the types of physical problems treated in this topic, enabling the modelling of technical devices to be studied by numerical simulations of such fields. In this topic, fields used for energy conversion are discussed only. The high frequency fields that, except those for microwave heating, are used to transfer information are not considered.

**In general**, two classes of electromagnetic fields can be distinguished, the time independent static and time varying fields (Fig. 3.1). They car be scalar and vector fields. A typical scalar field for example is the electrostatic potential distributionbetween charged electrodes and the magnetic field strengthsurrounding a current carrying conductor is a typical vector field. In the group of slow varying fields we can find the same types.

**Fig, 3.1. Classification of electromagnetic fields.**

**We have to distinguish between** the slow and fast varying electrical current flow field with regard to the geometrical dimensions of the current carrying conductor. The slow varying fields are understood to be fields not leading to current redistributions. This means that there are no eddy current effects as the dimensions of the current carrying conductor are smaller than the penetration depth of the field. The current at those frequencies is distributed as in the DC case, uniformly over the whole surface of the conductor. Eddy current effects are considered in the fields with fast varying time dependency, due to the low frequency treated as quasi-stationary. High frequency fields as focussed in antenna problems, leading to the electromagnetic waves, are not considered in this topic.

**Most of the physical issues in electrical** energy engineering can be described by quasi-static phenomena. Slowly varying and periodic fields up to 10 kHz are considered to be quasi-stationary. Electrical energy devices such as electrical motors and actuators, induction furnaces and high-voltage transmission lines are operated at low frequency. Exceptions are microwave devices for electroheat applications, where inherently the displacement current is not negligible.

**Typical examples of quasi-static electromagnetic** fields are the fields excited by coils in rotating electrical machines, transformers and inductors. Inside these conductors the displacement current is negligible and the magnetic field H outside the coil is exclusively excited by the free current density J. For those quasi-static fields, AMPERE’S law is applicable (Binns et al.’3).

To decide whether the displacement current can be neglected or not, depends on the wavelength X of the problem considered in the frequency domain. If it is large, when compared to the physical dimensions of the problem /, the displacement current is negligible. To consider this phenomenon in the time domain, the rise time Ta of a step function must be large inside the problem compared to the transit timeField problems are quasi-static if eq.(3.2) is valid.

In generalrespectivelyis sufficient.

**For this class of problem, the interesting fields vary slowly and can be periodic. Then three categories of problems are distinguished:**

**• static**

**• slowly varying transient**

**• time-harmonic eddy current.**

**In time-harmonic problems sinusoidal** varying field quantities are assumed. In theory, a time-harmonic solution is only valid for a linear system as a sinusoidal excitation does not yield a single frequency response in the non-linear case.