Approximate analytical solutions for the space-charge-limited current in one-dimensional and two-dimensional cylindrical diodes

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Abstract
In high-power microwave diode design, the space-charge-limited current is important because of its relation to the diode impedance, and the formation of the virtual cathode. Although the Langmuir–Blodgett law, as a numerical solution, is helpful, a simple functional expression would be more convenient for practical research. In this paper, a physical approximation has been introduced to analyze the nonlinear Poisson’s equation in a one-dimensional (1-D) cylindrical vacuum diode. With the help of this physical approximation, a solution for the space-charge-limited current for 1-D cylindrical diodes has been investigated and developed. In addition, a comparison between our approximate result and the Langmuir–Blodgett numerical solution shows that the physical approximation method is valid in nonlinear differential equation analyses. This physical approximation can be used to analyze similar nonlinear differential equations. Also, a correction for the space-charge-limited current in a two-dimensional cylindrical diode is obtained within a limitation.

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