## Introduction

Usually, the term of coupling is used when, for a given device, we want to study several physical phenomena: electric, magnetic, thermal, mechanic…

Each phenomenon is described by means of equations (Maxwell's equations, Fourier's equation, circuit equation, mechanical equation…). Thus, the coupling requires the solving of a complex system of equations.

## Strong/weak coupling: definitions

• a strong coupling :

when a simultaneous solving of the systems of equations which characterize the coupled phenomena is carried out. (For example, in case of magnetic field/circuit coupling, the magnetic field equations and the circuit equations are solved simultaneously);

• a weak coupling :

when the systems of equations characterizing each phenomenon are solved separately, this requiring the transfer of the results between systems. (For example, in case of magnetic field/kinetic coupling solved by using the step by step in time domain method, the equations of the magnetic field and the mechanical equations are solved successively for each time step)

## Details about the magnetic/ thermal coupling

Although the strong coupling of the systems of thermal and magnetic equations is very interesting from the theoretical point of view, in many cases this is completely uninteresting from the computational point of view, due to the important difference between the values of the thermal and magnetic time constants.

Indeed:

• from the electromagnetic point of view, in induction heating the range of frequencies is comprised between 50 Hz and 1 MHz; consequently, the period of the electromagnetic phenomena has values lower than 0.02 seconds;
• from the thermal point of view, the value of the time constant of a transient thermal phenomenon is of the order of a second, often higher, practically never lower than 0.1 s.

However, to treat a strong coupling, there is only one “time” variable for all the phenomena involved in the coupling. The reduced value of the time step which must be adapted to the smaller time constant, will lead to prohibitive computation times, as the time of the study must be related to the greatest time constant.

Thus, in order to obtain acceptable values of the computation time in case of magnetic/thermal coupling, it is necessary to decouple the thermal and the magnetic solving processes.