Noncylindrical Conductors

Author: Edmund A. Laport

We shall now apply the theory for cylindrical wires to the calculation of empirically shaped conductors. By a series of successive steps, accurate determinations of the capacitance and characteristic impedance can be made for lines made of conductors of any arbitrary shape. In practice one must often use a certain structural member as a transmission line. The calculations are tedious since they sometimes involve sets of several simultaneous equations to solve for the charge ratios. The urgency of the situation would determine the actual need to submit to this procedure. The method is based on the premise that a sufficient number of small wires would eventually assume the complete outline of the empirical surfaces. This process automatically solves the charge-distribution problem over all the surface in terms of those on the discrete wires assumed in the development. Actually, one need not pursue this development through many steps to have adequate information for extrapolation to the limit and the final result.

An example of this process is the case of two parallel flat strips of half thickness p with spacing a, used as a balanced transmission line.

Fig. 6.6

The numerical results of each step only are included. The dimensions used are as follows (see Fig. 6.6):

Thickness 0.031 inch ( ρ = 0.0155 inch)

Width 0.500 inch

Spacing 0.500 inch between center lines

In step 1, the extremities of each strip are replaced by four wires of diameter equal to the thickness of the strips and having the same outside dimensions. Solving this as a balanced line,

In step 2, another wire is placed midway between each of the above pairs and given the same potentials, and for this step Z₀ = 185 ohms. The charge ratios for step 3 (Fig. 6.6) for the inner wires are found to be Q₃/Q₁ = 0.418 and Q₂/Q₁ = 0.600. The computed value Z₀ = 178 ohms. These values can now be plotted in Cartesian coordinates. If we call N the number of wires in one side of the circuit and plot the computed values of Z₀ against 1/N, as in Fig. 6.7, the ultimate value of a continuous strip of wires conforming to the originally desired flat strips can be found at the intersection of this curve with the value 1/N = 0. This value is seen to be about 175 ohms.

The process can be applied to complex surfaces such as angles, channels, or other structural forms which are much more complicated than this simple example.