Nonlinear Multi-Scale Model Order Reduction of Eddy-Current Problem

Abstract

1. Introduction:Optimal operation of electromagnetic drive systems requires coupled analysis of electromagnetic equipment with their control units. Accurate equivalent circuit models of the electromagnetic equipment are essential and are expected to include the high-frequency eddy currents in nonlinear laminated iron cores.Model order reduction (MOR) is a powerful methodology providing reduced models based on finite element analysis. Some of the MOR approaches are capable of generating equivalent circuit models such as Cauer Ladder Network (CLN). The CLN method is one of the promising MORs proposed for eddy current problems in linear mediums [1]. It was extended to the magnetically nonlinear problems using the first-order approximation method (FO-CLN) [2]. Another extension concerns the analysis of laminated cores using homogenized eddy current field as a multi-scale MOR for linear mediums [3].This paper makes a progress by introducing the nonlinear multi-scale MOR based on a new piece-wise linearization technique. Homogenization on its nonlinear form has been always a big challenge and is a fortiori so in multi-scale MOR. The main contribution of this work is to linearize peace-wise the nonlinear homogenization using the first-order approximation methodology of FO-CLN to find the equivalent circuit model for nonlinear laminated iron cores.2. Multi-Scale MOR:The CLN method is a "Macroscopic-Scale" MOR, which aims for efficient electromagnetic field calculation over the whole machine on the macroscopic scale. On the other hand, "Microscopic-Scale" MOR deals with the homogenization and expressing the components of electromagnetic equipment with fine structures, such as laminated steel sheets. Fig. 1 illustrates how the microscopic-scale MOR approximates a stacked laminated-core Fig.1(a) into a homogenized bulk material Fig.1(b); see e.g. [4] for a homogenization method that considers eddy currents and skin effect in laminated stacks using the Legendre polynomials. The homogenized eddy-current field is expressed as K'a=-σ'da/dt+j0, where K', σ', a and, j0 are the homogenized stiffness and conductivity matrices, magnetic vector potential, and external current vectors, respectively. On the multi-scale MOR, the standard CLN procedure is adapted with K' and σ'.3. FO-CLN:Using the CLN, the time-dependent electromagnetic fields in a linear eddy-current problem are decomposed into time-independent modes weighted by time-dependent coefficients.On linear case, these modes are constant and the coefficients are derived from the network currents and voltages in Fig. 1(c).When the core has nonlinear magnetic characteristics, the modes vary with respect to saturation in the core with field-dependent permeability μ(B)In the FO-CLN method, the first magnetic mode is considered as the main and only source of the core saturation.The saturation level is measured by the current h1 in Fig. 1(c).The CLN is performed at different degrees of saturation which results in parametric CLN elements as in Fig. 1(c).In Fig.1 the FO-CLN method is shown as the transition between Fig. 1(b) to Fig. 1(c).4. Nonlinear Multi-Scale MOR:The FO-CLN methodology is applied to the multi-scale MOR since it is based on the frozen permeabilities obtained during the calculation of the first magnetic mode. More details will be given in the full paper.5. Numerical Example:The accuracy of the proposed method is verified over a laminated inductor with sheet thickness of 0.5mm, shown in Fig. 2(a). The currents from the conventional transient finite element method and it's equivalent CLN compared with the 0th order homogenization in Fig. 2(b), demonstrates the accuracy of the proposed method. **