Machine-learned models for magnetic materials

TL;DR

This work addresses the challenge of modeling magnetic materials under wide-frequency and high-current conditions by proposing a differentiable, physics-informed autoencoder that learns analytical model parameters for a lumped-element impedance circuit. The encoder predicts ladder element parameters $\{L_i, R_i\}$, while the decoder uses the analytic LEEC formula to reconstruct impedance, enabling backpropagation through a physically grounded model. Training on a large set of synthetically generated, diverse impedance families enables the network to generalize to unseen measured data; improvements from a Siamese architecture and a modified Siamese loss further enforce continuity and uniform frequency placement, reducing errors to about $5-7\%$ on average. The approach demonstrates robust, physics-consistent parameter identification across frequency and DC-bias regimes and is applicable to any differentiable analytical model, offering a practical path for fast, physically constrained material modeling in power electronics.

Abstract

We present a general framework for modeling power magnetic materials characteristics using deep neural networks. Magnetic materials represented by multidimensional characteristics (that mimic measurements) are used to train the neural autoencoder model in an unsupervised manner. The encoder is trying to predict the material parameters of a theoretical model, which is then used in a decoder part. The decoder, using the predicted parameters, reconstructs the input characteristics. The neural model is trained to capture a synthetically generated set of characteristics that can cover a broad range of material behaviors, leading to a model that can generalize on the underlying physics rather than just optimize the model parameters for a single measurement. After setting up the model, we prove its usefulness in the complex problem of modeling magnetic materials in the frequency and current (out-of-linear range) domains simultaneously, for which we use measured characteristics obtained for frequency up to $10$ MHz and H-field up to saturation.

Figures (6)

• Figure 1: Impedance of a magnetic ring (a) measured as a function of frequency for various DC current biasings that controls operating point from ($I_\mathrm{DC}=0$ A, red curve) to the full saturation ($I_\mathrm{DC}=27$ A, blue curve), and (b) generated (random sample) using lumped element model with randomly selected outermost characteristics (red and blue one) and continuous transition between different curves). The generated data (b) will serve as a training set for the introduced neural network model, while the measurement ones (a) will be used to test the already trained NN model.
• Figure 2: Four representative families of characteristics $Z_j(I_\mathrm{DC},f)$, where $j=1\dots N_\mathrm{bias}$, from the synthesized training dataset. Solid line denotes impedance amplitude and dashed line denotes impedance phase as in Fig.\ref{['fig:data']}b
• Figure 3: Neural network basic model with an autoencoder structure, which is of fundamental type and used in various fields of machine learning or image processing. The encoder is made of three convolutional layers (CNN) and three fully connected (FC) layers. The CNN layers are used as feature extractors, while the FC layers are used for parameters ($\{L_i,R_i\}$) predicting based on previously found representations. The decoder ($Z$) just implements an analytical formula for the impedance of the LEEC model. The goal of the network is to reconstruct the input characteristics at the output.
• Figure 4: Results for testing the basic neural network model on the measured ring data: (a,b) examples of fitting the analytical model (solid curves), with parameters predicted by the NN model, versus measurements (dots) for the selected DC-bias currents $I_\mathrm{DC}=\{1.8, 7.0, 16.7\}~\mathrm{A}$, (c) relative fitting error for the various DC-bias currents $I_\mathrm{DC}$ plotted versus frequency, and (d) characteristic frequencies $f_i=2\pi R_i/L_i$ of the predicted NN model parameters (each line represent $f_i$ for one LR pair in the LEEC series) versus DC-bias current $I_\mathrm{DC}$.The blue color marks the frequency range where the measurements data are located. .
• Figure 5: Siamese neural network model composed of two copies of the basic model that share the weights and process two neighbor characteristics (that differ in DC-bias current by $\Delta I$) at the same time enabling to train the model to predict similar (or continuously changing) parameters for similar characteristics.