Time-domain Formulation of a Multi-layer Plane Circuit Coupled with Lumped-parameter Circuits using Maxwell Equations

Abstract

We calculate electromagnetic phenomena in the multi-layer plane circuit starting from the Maxwell equations. We present a numerical method of potential and current density in two-dimensional conductors, where their time developments are treated as phenomena of wave propagation. We treat the plane conductors by dividing them into small finite-volume elements, similar to the case of the partial element equivalent circuit method, and the transport equations are then solved by the finite-difference time-domain method. Furthermore, we develop a calculation method for the boundary in a multi-layer plane by applying the method we have used in multi-transmission lines. We formulate the boundary conditions of a multi-layer plane coupled with lumped-parameter circuits and introduce an algorithm to reduce calculation costs that are largely associated with the two-dimensional extension from the multi-transmission-line case. We perform calculations of the wave propagation of potential, current density, and charge density in the time domain for a simple plane circuit. These calculations are presented as supplementary materials of the present paper.

Origin of common‐mode noise in two‐dimensional finite‐size circuit and reduction of the noise using a symmetric three‐line circuit

“…In this section, we briefly describe our calculation method, which uses time-domain MLP circuit equations. 20 We start with Maxwell's equations to capture the electromagnetic noise phenomena in a circuit. We use integral equations of the scalar potential and vector potential derived from Maxwell's equations using the Lorentz gauge.…”

“…At the boundaries, we can connect lumped-parameter circuit by solving Kirchhoff's current and voltage law and branch constitutive equation as conditions of lumped-parameter circuit and Equation 1as condition of MLP, simultaneously. 20,22…”

“…Recently, we extended our calculation method from a one-dimensional multitransmission line to a two-dimensional multilayer plane (MLP) using the finite element method and the finite-difference time-domain method. 20 We can thus consider a two-dimensional finite-size conductor, such as the ground plane, in a printed circuit board.…”

The time domain numerical method of three-dimensional conductors including radiation with lumped parameter circuit

“…The vector potential can be also expressed in the same way using the equations (2), (14), (25), and (26):…”