Akihisa Kameari

In electromagnetic field analysis, the Darwin model enables us to consider the effects of inductance and capacitance. In this study, we examine two interrelated Darwin model methods for accelerating the convergence of an iterative solution. One method introduces redundant variables, enlarging the size of the coefficient matrix, whereas the other method, the subspace correction method, does not. Numerical results reveal that both methods significantly improve the convergence of the Darwin model equations. In particular, the subspace correction method is more advantageous in terms of computational cost than solving by introducing redundant variables. Moreover, it is estimated that even with redundant variables, block preconditioning can reduce the overall computational cost.

The Darwin model for electromagnetic quasistatic (EMQS) field analysis is of particular interest for the analysis of power electronic devices, such as power transformers and wireless power transfer systems. This model is an approach that neglects the second derivative term in time of the displacement current and considers the electrostatic field. This approximation allows conventional techniques for magneto-quasistatic (MQS) field analysis, such as considering nonlinear magnetic material properties and coupled circuit analysis, to be appropriated, and capacitance effects to be included. Several methods have been proposed for frequency-domain analysis. This is because it is difficult to properly derive a formulation of time domain analysis. We have already proposed a potential formulation in frequency domain analysis and succeeded in obtaining appropriate calculation results. In this study, we propose a time-domain analysis of the low-frequency stabilization formulation in the finite element method. The proposed formulation is based on the Coulomb-type gauge condition, which defines no additional variables and redundant variables to improve the convergence characteristics. In addition, the system matrix can be made symmetric for solving by general preconditioners with iterative solvers.

As an extension of the Cauer ladder network method, this paper presents the equivalent circuit for an eddy current field including a translational mover. The speed voltages for the inductors in the equivalent circuit are obtained by using the directional derivatives of the inductors in the direction of mover displacement. These directional derivatives are determined by using the fact that the electromagnetic force can be represented in two forms, namely, Maxwell’s stress tensor and the virtual work principle. This result is also verified by using the theory of continuum mechanics. Finally, numerical examples of an electromagnet are presented.

Significant advances has been made to the Cauer ladder network method, an attractive method recently proposed for the model order reduction of linear eddy current problems. However, these developments are often discussed in the context of simple two-dimensional (2D) problems with solid conductors only. The three-dimensional (3D) implementation of the method with general coil connections requires further treatment and deserves detailed discussions. This paper reformulates the method for 3D implementation with stranded coils (with no eddy current) as well as solid conductors with arbitrary connections between and coils and external resistors.

A method for motor model order reduction is developed using the Cauer ladder network. The analyzed domain is decomposed into stator and mover domains, where the multiport Cauer ladder network of each domain is constructed independently from the other. The two domains are connected through electromagnetic field modes at the boundary. The boundary condition is derived from the coordinate transformation. The reduced model accurately reconstructs the induction motor property affected by slot harmonics.

To achieve efficient multi-port model order reduction, a multi-port Cauer ladder network (CLN) method is formulated that directly yields resistance and inductance matrices that consist of the network elements in the matrix Cauer form. The eddy current field driven by multiple power sources is accurately reconstructed using a small number of network elements. The matrix Cauer form achieves faster convergence of the transfer function than a single-port CLN method and almost the same convergence as a block PVL method.

A matrix formulation of the Cauer ladder network (CLN) method is derived using the finite-element method to clarify the mathematical aspects of the CLN method. The CLN method directly yields orthogonal expansions of electric and magnetic fields that ensure the equivalence of the Cauer network to the eddy-current field. The CLN method is as exact as the Padé approximation via the Lanczos (PVL) process but more efficiently provides the circuit parameters and the orthogonal expansion than the PVL.

In this paper, open boundary techniques are employed to represent the Cauer ladder network (CLN) for unbounded eddy-current problems. Recently, a novel and effective model order reduction method, called the CLN method, has been proposed in which linear eddy-current fields are represented by a CLN. The CLN method was initially formulated for closed domain eddy-current problems. This paper demonstrates that the CLN method can be extended to unbounded-domain problems when combined with conventional open boundary techniques. The CLN network parameter extractions of two parallel busbars are demonstrated as a numerical example.