Induction Heating System as a Soft-Sensor for the $B-H$ Curve of Steel Strips in a Continuous Induction Furnace

Continuous in-line identification of magnetic properties of ferromagnetic steel strips is developed and investigated. It is demonstrated that estimates of the magnetization curve of the strip material can be made if the operating frequency of the induction heating (IH) system is known. In particular, the nonlinear function of the effective magnetization curve (effective $B-H$ curve) of the strip material is estimated by a moving horizon estimator (MHE) when heating the strip in the IH furnace. The magnetization curve is modeled using a linear and an exponential function and parametrized by two parameters. The MHE is based on frequency measurements and the current power of the electric supply circuit. The complex coil impedance derived from a distributed-parameter formulation of time-harmonic Maxwell's equations is used for the frequency calculation. It is shown that if the strip is insufficiently excited, the estimation of the parameters of the effective $B-H$ curve becomes inaccurate. Therefore, the possibility of a non-uniform power distribution of a supply circuit with two power inverters was analyzed. As a main finding, it is demonstrated that the estimation accuracy can be significantly improved if the local strip excitation can be increased. It is also proven that a suitable unequal power distribution does not change the strip exit temperature compared to a uniform power distribution. The estimation results are validated in a scenario for a steel strip with the effective $B-H$ curve known from laboratory measurements.