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.