The Characteristic-Resistance Method for Grounded Semi-Infinite Grids

A variety of existence and uniqueness theorems for the current flows in infinite electrical networks have been previously established, but there is virtually no information on how to compute those current flows under the practical requirement that the power dissipated and energy stored in the network be finite. This problem is addressed herein. It is shown that the characteristic-impedance method of analyzing lumped transmission lines can be adapted to semi-infinite half-plane resistive grids and to three-dimensional half-space resistive grids. The method yields the one and only current flow within the grid for which the total power dissipation is finite. It also yields a practical procedure for computing the currents and voltages in the grid. That procedure is remarkably efficient and uses very little computer time. Moreover, semi-infinite grids whose branch impedances are positive real functions can also be analyzed in a similar way. This allows the computation of transient behavior in the presence of energy-storage elements, but the required computer time is now considerably longer. Nonlinear grids can also be encompassed by an approximation technique. These results appear to provide a basis for the numerical analysis of certain boundary value problems of practical importance in engineering and the physical science.