Conductor geometry independence of phase velocity in TEM-mode transmission lines

Based on Caratheodory's (1952) generalization of the Riemann mapping theorem, it is shown that a proof is possible for the conductor geometry independence of phase velocity in transverse electromagnetic (TEM)-mode transmission lines. Specifically, an effort is made to show that a proof of the invariance of phase velocity with conductor geometry is possible from essentially circuital considerations. The argument implies that it is obvious that any TEM-mode transmission line which consists of two conductors, each of arbitrary section, one totally enclosed within the other, can be transformed into a concentric, right circular, coaxial line. Since solutions of Laplace's equation are invariant under conformal transformation, the inductance and capacitance per unit length of the original and transformed lines must be the same. To establish that fact generally, it remains then only to observe that, for a normal coaxial line, a problem to which there is a simple analytic solution, phase velocity is independent of geometry.<
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