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Abstract
An exact and explicit solution for the reflection and transmission problem of a lossy exponential transmission line of finite length is derived in both the frequency and the time domain via a wave-splitting approach. The problem is formulated first in the frequency domain. Wave-splitting is defined and the solution for internal voltages and currents is derived. Closed-form solutions for the reflection and transmission coefficients are obtained. The explicit solution for a transmission line consisting of several exponential pieces (each with different taper) in series is then easily written down. To obtain the transient response of the lossy exponential line, a time domain Green function approach based on wave-splitting is used. Linear partial differential equations for the Green functions together with the initial and boundary conditions are derived and solved explicitly. The exact and explicit solutions for the transient reflection and transmission up to arbitrary large times are given.