Nonuniform TEM transmission line. Part 1: Lossless and log-periodic properties

The nonuniform lossless TEM line, with frequency-independent per-unit-length inductance and capacitance, is studied by transforming the differential equation for voltage into Schrödinger form, in terms of a transformed voltage ¿ and scattering potential U. The advantages of this formulation are outlined. An equivalence theorem for two lines is stated, based on the input-output-impedance relations. These relations for a cell are developed from expansions for ¿ in terms of open-circuit and short-circuit functions of the cell. The input impedance is shown to be stationary with respect to small errors in these expansions, according to variational expressions for input impedance and admittance. The latter yield a simple formula for input impedance of a slightly nonuniform line. The periodicity theorem is stated and illustrated for a log-periodic line. Various semi-infinite log-periodic lines of frequency-independent inductance and capacitance are classified and illustrated by their l.f. behaviours. The analytic properties of expressions for the l.f. admittance are examined. The necessary and sufficient conditions for constant input conductance are discussed.