# Coupling and magnetic materials with nonlinear B(H) characteristic

## Introduction

For a Steady State AC Magnetic coupled with Transient Thermal application the temperature resulting from the thermal problem solving is a quantity involved in the system of magnetic equations by means of the material physical properties: magnetic permeability, electrical conductivity…

The models proposed for the electric and magnetic properties of materials with respect
to the temperature (resistivity **ρ(T)**, permeability **μ(T)**) are depicted in
chapter Materials:
principles (see § Loi B(H, T) law : models for soft materials).

## Reminder of the topic

In the Steady State AC Magnetic applications the unknown potentials and the physical quantities derive from them (magnetic field strength and magnetic flux density) are sinusoidally varying with respect to the time. Thus, the complex representation with respect to the time is used, and the solution of the magnetic problem, time independent, expressed by complex numbers, is obtained after one solving.

In case of the hypothesis of quantities sinusoidally varying with respect to the time,
it is, a priori, impossible to take into account the magnetic materials with nonlinear
magnetization characteristic. To overcome this difficulty, the real **B(H)**
nonlinear magnetization saturation curve is replaced in Flux computation by a curve
called “equivalent”, obtained from energetic considerations. This method and the models
proposed are described in chapter Materials: principles (see § Modèles Equivalent models for Steady State AC Magnetic applications).

## !!!Attention !!!

For a Steady State AC Magnetic coupled with Transient Thermal application, it is not
possible to apply the same method of the equivalent **B(H)** curve to the** real
B(H, T)** dependence, which depends also on the temperature **T**.

## …

Nevertheless, from the point of view of the utilization of **B(H)** equivalence for
magneto thermal analyses, the following two remarks can be considered:

- in the cold state of the part to be heated, the value of source currents is high
enough so that a very high saturation of the nonlinear magnetic part be achieved. The
magnetic state of the part material is located exactly on the asymptote of the real
**B(H)**curve, where the model of equivalent**B(H)**curve is absolutely valid; - as soon as the temperature of the part increases, we notice a “bend flattening” of
the
**B(H)**curve, and then the approximation of the equivalent**B(H)**curve is acceptable.

The comparison between the results of the simulations and the experimental results has
shown the validity of the models of equivalent **B(H)** curve. The differences are of
the order of the quantity of uncertainty, the measures for the real cases being
particularly difficult to carry out.