WAVE PROPOGATION ON LOG-PERIODIC TRANSMISSION LINES

The propagation of waves along a transmission line, loaded log- periodically with impedance elements, is considered. By definition, the element spacing forms a geometric series, and the element magnitudes are proportional to their respective distances from the input end of the line. It is shown that the over-all effective impedance per unit length and admittance per unit length are represented by logarithmic functions. A general orthogonal series expansion is then found for such functions. Next, the general solution of the transmission line equations is considered for impedance and admittance per unit length. Although the solution for the purely log-periodic line cannot be factored into a simple form, a variational method is presented by means of which it can be approximately solved in terms of logarithmic waves. The variational technique makes use of formulas being stationary with respect to the current and voltage distributions on the line. The distributions are assumed and the stationary property allows one to calculate the valves.