A Semianalytical Method for Predicting Primary and Secondary Current Density Distributions: Linear and Nonlinear Boundary Conditions

Potential distributions and their associated current density distributions (primary and secondary) are typically obtained by solving Laplace’s equation. 1-3 The methods used to solve Laplace’s equation include analytic and numerical methods. Analytic methods ( e.g., conformal mapping 4 ) provide the maximum insight into the problem and usually yield closed-form potential and current density distributions. Unfortunately, analytic techniques are system specific and are often difficult to obtain. Numerical techniques are very general but usually give a numerical value for the potential at a particular location. We present here a semianalytical method (or analytic method of lines), which is for a two-spatial-coordinate problem analytic in y and numerical in x; thereby the technique is more general than a particular analytic solution technique and gives better insight than numerical techniques for a certain class of problems (Laplace’s equation, which has constant coefficients in at least one of the independent variables). It is important to note that the method presented here for solving Laplace’s equation in two spatial coordinates with nonlinear boundary conditions does not require iterations for interior node points as is usually the case. 5 The reason for this is that our method does not require interior node points, but instead, only has node points in the boundaries. The nonlinearities of the boundary conditions are removed by solving for the constants that appear in the solution of Laplace’s equation as explained in the following discussion. The technique is an extension of the method presented by De Vidts and White, 6 who presented the semianalytical method for solving the one-dimensional, unsteady-state diffusion equation. The semianalytical method presented by De Vidts and White consists of using the method of lines 7 to solve the diffusion equation with finite differences used in the spatial direction. (This method was mentioned by Smith et al., 8 but they did not present any results.) The resulting system of linear ordinary differential equations is then solved analytically using the matrix-exponential method. 9 This technique is extended here to solve Laplace’s equation with two spatial coordinates. The second derivative of the potential in the x direction is cast into finite differences accurate to order h 2 (h 5D x), and the second derivative in the other direction is replaced by two first-order-derivative equations. The resulting system of ordinary differential equations is solved analytically using the matrix-exponential approach. The method requires determining constants of integration in a manner described previously by Subramanian et al. 10 The method is illustrated by first solving Laplace’s equation for a rectangle in which a cathode faces an anode between two insulators. The method employed for this simple case is then extended to other current-distribution problems. Semianalytical Method Consider the rectangle ABCD shown in Fig. 1 of dimensionless length 1 (AB 5 CD 5 1) and dimensionless height b (AC 5 BD 5 b). The dimensionless governing equation for the dimensionless potential inside the rectangle is given by Laplace’s equation [1]