Mathematical modeling of triode system on the basis of field emitter with ellipsoid shape

In this paper the mathematical modeling of the triode emission axially symmetric system on the basis of field emitter is considered. Emitter is an ellipsoid of revolution, anode is a confocal ellipsoidal surface of revolution. Modulator is a part of the ellipsoidal surface of revolution, confocal with the cathode and anode surfaces. The boundary-value problem for the Laplace's equation in the prolate spheroidal coordinates with the boundary conditions of the first kind is solved. The variable separation method is applied to calculate the axisymmetrical electrostatic potential distribution. The potential distribution is represented as the Legendre functions expansion. The expansion coefficients are the solution of the system of linear equations. All geometrical dimensions of the system are the parameters of the problem.