Equivalent transformations for the mixed lumped Brune-type section and uniform transmission line

As an analytical method for nonuniform transmission lines (NTLs) equivalent transformations are extended to a more general case, namely a mixed lumped Brune-type section and a uniform transmission line (unit element, UE). Circuits consisting of a cascade connection of a lumped Brune section and a UE are equivalent to one consisting of a cascade connection of a nonuniform transmission line whose characteristic impedance distribution is expressed with a trigonometric function and a lumped Brune section. This equivalent transformation method is easily applied to a circuit consisting of a lumped C section and a UE. The equivalent circuit is a circuit consisting of an NTL and a lumped C section. In this case, the characteristic impedance distribution of the NTL may be expressed in terms of a hyperbolic function. Exact network functions of the NTLs are easily obtained from the equivalent circuits without solving the telegrapher´s equation. By considering the limiting case of these equivalent transformations, equivalent transformations for circuits consisting of a cascade connection of a lumped resonance circuit and a circuit and a uniform transmission line are derived.<>