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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B-H</i> 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 serves as a basis for the frequency calculation. In favor of the accuracy of frequency calculations, an iterative approach is implemented. The calculated frequency from one iteration is used as the input excitation frequency of the subsequent iteration. It is shown that if the strip is insufficiently excited, the estimation of the parameters of the effective <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B-H</i> curve becomes inaccurate. Therefore, the possibility of a non-uniform power distribution of a supply circuit with two power inverters was analyzed. Such non-uniform power distribution can be exploited to increase the local strip excitation in a way that favors the identifiability of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B-H</i> curve. As a main finding, it is demonstrated that the estimation accuracy can be significantly improved. 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B-H</i> curve known from laboratory measurements.