The Approximate Solution of Electric-Field Problems with the Aid of Curvilinear Nets

A rough field plot ascertained graphically can be used as the basis of a computational method of considerable accuracy. The plot is replaced by a net of orthogonal trajectories without sudden breaks of direction, the individual meshes being bounded by arcs of circles or straight lines. With the nodes of this net as the junction points of conductances, the 2-dimensional continuum within the problem boundaries can be converted into a network of lumped conductances. This network is analysed by measurement or computation. The paper deals with the graphical and computational layout of a net whose contours follow the assumed trajectories closely, and explains the graphical and computational determination of the lumped conductances. A large number of formulae and computing routines are derived, making possible the employment of semi-skilled workers or computing aids. The determination of the gradient and other special problems, such as computing the areas associated with Poisson´s equation or dealing with sharp corners, are discussed and the relevant formulae are given. The advantages of the method, compared with the use of regular or irregular nets with straight-line inter-mesh boundaries, lie in the considerable reduction in the number of meshes for the same accuracy, and in the possibility of fitting the problem boundaries very closely. Two practical examples are given: using 4 × 6 meshes, the field gradient at the surface of an edge rounded by a non-circular curve is ascertained accurately within 0.6%; using 2 × 2 meshes, the cut-off frequency of the H11 mode in a circular waveguide is determined with an error of less than 1.6%; this is an eigenvalue problem.