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A magnetic oriented approach to the systematic coupling of field and circuit equations

Herbert Egger, Idoia Cortes Garcia, Vsevolod Shashkov, Michael Wiesheu

TL;DR

The paper develops a magnetic-oriented field-ciber coupling framework for low-frequency electromagnetic devices, unifying a magneto-quasistatic vector-potential model with magnetic oriented nodal circuit analysis. It proves a general power-balance relation for the coupled system and shows that the energy functional can be treated in a gradient-like form, enabling structure-preserving time integration and potential model-order reduction. A numerical example with a rectifier, nonlinear diodes, and a linear transformer demonstrates second-order convergence and verified discrete energy balance, validating the approach. The work highlights nonlinear extensions, potential DA index reductions, and practical implications for stable, energy-aware field-circuit simulations and design optimization.

Abstract

A novel strategy is proposed for the coupling of field and circuit equations when modeling power devices in the low-frequency regime. The resulting systems of differential-algebraic equations have a particular geometric structure which explicitly encodes the energy storage, dissipation, and transfer mechanisms. This implies a power balance on the continuous level which can be preserved under appropriate discretization in space and time. The models and main results are presented in detail for linear constitutive models, but the extension to nonlinear elements and more general coupling mechanisms is possible. The theoretical findings are demonstrated by numerical results.

A magnetic oriented approach to the systematic coupling of field and circuit equations

TL;DR

The paper develops a magnetic-oriented field-ciber coupling framework for low-frequency electromagnetic devices, unifying a magneto-quasistatic vector-potential model with magnetic oriented nodal circuit analysis. It proves a general power-balance relation for the coupled system and shows that the energy functional can be treated in a gradient-like form, enabling structure-preserving time integration and potential model-order reduction. A numerical example with a rectifier, nonlinear diodes, and a linear transformer demonstrates second-order convergence and verified discrete energy balance, validating the approach. The work highlights nonlinear extensions, potential DA index reductions, and practical implications for stable, energy-aware field-circuit simulations and design optimization.

Abstract

A novel strategy is proposed for the coupling of field and circuit equations when modeling power devices in the low-frequency regime. The resulting systems of differential-algebraic equations have a particular geometric structure which explicitly encodes the energy storage, dissipation, and transfer mechanisms. This implies a power balance on the continuous level which can be preserved under appropriate discretization in space and time. The models and main results are presented in detail for linear constitutive models, but the extension to nonlinear elements and more general coupling mechanisms is possible. The theoretical findings are demonstrated by numerical results.
Paper Structure (5 sections, 1 theorem, 17 equations, 2 figures)

This paper contains 5 sections, 1 theorem, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Magnetic oriented modeling
  3. Field-circuit coupling
  4. Numerical results
  5. Discussion

Key Result

theorem thmcountertheorem

$$ Let $(\psi,q_C,q_V,q_M,a)$ be a solution of shashkov:eq:10--shashkov:eq:14 and set $v_I=A_I^\top \partial_t \psi$, $i_V = \partial_t q_V$. Further define $H(\psi,q_C,a)=\frac{1}{2}(\|A_L^\top \psi\|^2_{L^{-1}} + \|q_C\|_{C^{-1}}^2 + \|a\|_{K_{\nu}}^2)$. Then The total energy stored in the system thus changes only by dissipation due to resistive losses and by power supplied to or drawn from the

Figures (2)

  • Figure 1: Schematic sketch of a full wave rectifier circuit (left) and geometry of transformer modeled by the field equations (right). Based on schops2011multiscale.
  • Figure 2: Left: input voltage $v_{src}$ and rectified output voltage $v_R$. Right: convergence history for the magnetic potential $\psi_4$ and maximal discrepancy in the energy balance.

Theorems & Definitions (4)

  • remark thmcounterremark
  • theorem thmcountertheorem: Power-balance of the coupled system
  • proof
  • remark thmcounterremark