An alternative method for calculating the space-charge-limited current for a cylindrical diode

Summary form only given. Estimates of the space-charge-limited current density in cylindrical and spherical geometries are not readily available. The classic method for evaluating the limiting current between two coaxial electrodes was originally developed by Langmuir; the essence of the method is a series expansion of the solution as a power series in ln r/r/sub i/ where r/sub i/ is the inner (cathode) radius. This series converges slowly, a large number of terms must be evaluated for reasonable results, and it is not applicable when relativistic effects are significant. Direct numerical solution of the governing differential equation tends to give poor results; the differential equation is numerically "stiff." An alternative is to non-dimensionalize the governing differential equations, transform coordinates so that the solution is well-represented by low-order polynomials, and generate simple approximate solutions. An additional benefit of this alternative method is that a planar diode, an infinitesimally thin wire in a coaxial cylinder, and a point emitter in a concentric shell, can be shown to be special cases of the more general solution; these special cases are explicitly solved.