Energy-based modeling for field-circuit coupling

TL;DR

The paper develops a unified energy-based framework, extending port-Hamiltonian concepts to field–circuit problems with DAEs that may include algebraic components. It demonstrates that the generalized representation preserves a dissipation inequality and remains closed under power-preserving interconnections, enabling consistent coupling of linear MNA circuits with magnetoquasistatic conductor models (stranded, solid, foil). For quadratic Hamiltonians, the implicit midpoint rule ensures a discrete energy balance, while implicit Euler can violate energy conservation in both simple oscillators and more complex transformers. Numerical experiments validate theoretical results and highlight practical performance, showing good energy balance with midpoint discretization and highlighting the need for structure-preserving time integration in engineering applications. The work lays the groundwork for extending energy-based field–circuit models to nonlinear electromagnetic problems encountered in transformers and electrical machines.

Abstract

This paper presents a generalized energy-based modeling framework extending recent formulations tailored for differential-algebraic equations. The proposed structure, inspired by the port-Hamiltonian formalism, ensures passivity, preserves the power balance, and facilitates the consistent interconnection of subsystems. A particular focus is put on low-frequency power applications in electrical engineering. Stranded, solid, and foil conductor models are investigated in the context of the eddy current problem. Each conductor model is shown to fit into the generalized energy-based structure, which allows their structure-preserving coupling with electrical circuits described by modified nodal analysis. Theoretical developments are validated through a numerical simulation of an oscillator circuit, demonstrating energy conservation in lossless scenarios and controlled dissipation when eddy currents are present. The applicability of the methodology towards engineering applications is studied through a numerical simulation of a nonlinear three-phase transformer.

Figures (8)

• Figure 1: Computational domain with one representative of each conductor model.
• Figure 2: \ref{['fig:oscillator_circuit']} Schematic of the oscillator circuit. \ref{['fig:oscillator_fe_geometry']} 2D axisymmetric domain of the inductor, where $\Omega_{\star}$ denotes either the domain of a solid $\Omega_{\mathrm{sol}}$ or a stranded conductor $\Omega_{\mathrm{str}}$. The dimensions are given in mm.
• Figure 3: The energy of the oscillator with a nonconducting core when \ref{['fig:oscillator_str_noncond_core_euler']} stranded conductor with the implicit Euler method, \ref{['fig:oscillator_str_noncond_core_trap']} stranded conductor with the trapezoidal rule, \ref{['fig:oscillator_sol_noncond_core_euler']} solid conductor with the implicit Euler method, and \ref{['fig:oscillator_sol_noncond_core_trap']} solid conductor with the trapezoidal rule are used. The Hamiltonian contains the contributions from the magnetic and the capacitor energies.
• Figure 4: The energy of the oscillator with a conducting core when \ref{['fig:oscillator_str_cond_core_euler']} stranded conductor with the implicit Euler method, \ref{['fig:oscillator_str_cond_core_trap']} stranded conductor with the trapezoidal rule, \ref{['fig:oscillator_sol_cond_core_euler']} solid conductor with the implicit Euler method, and \ref{['fig:oscillator_sol_cond_core_trap']} solid conductor with the trapezoidal rule are used. The Hamiltonian contains again the contributions from the magnetic and the capacitor energies.
• Figure 5: The convergence of \ref{['fig:osc_convergence_solution']} the solution and \ref{['fig:osc_convergence_energy']} the energy conservation for different time integrators. Error measures are defined in \ref{['eq:errors']}.

Theorems & Definitions (26)

• Remark 1
• Remark 2
• Theorem 3: Energy dissipation
• proof
• Theorem 4: Structure-preserving interconnection
• proof
• Remark 5
• Theorem 6: Discrete energy dissipation
• proof
• Remark 7