Towards 2D Electronic Circuits in the Spatial Domain
N. Ratier · HAL (Le Centre pour la Communication Scientifique Directe) · 2009
Electronic circuits are, by nature, functions of one independent variable in the time domain. They operate (in the sense of differential operators) from an input signal e(t) to an output signal s(t). In this sense, we can speak of the electronic circuits of an ODE (ordinary differential equation) with its associated initial condition. We show that periodic networks of resistances (PNR) can be used to define a kind of 2D circuits in the spatial domain. These circuits operate as a PDE (partial differential equations) from an input signal f(x, y) to an output signal g(x, y). The two independent variables are now the coordinate axes x and y. As to the initial conditions, they are replaced by boundary conditions. We present how to construct an electronic circuit of a linear PDE up to the fourth order. The discrete solution at each voltage node of the center of each cell composing this circuit converges towards the solution of the PDE g(x, y), with its boundary conditions, for a given input signal f(x, y) when the number of periodic cells increases.