Optimum RC phase-shift network between two arbitrary impedances

The following problem is investigated. Given two terminating impedances
Z_{1}
and
Z_{2}
, design an
RC
ladder network of n sections such that the terminated transfer voltage ratio
|V_{2}/V_{1}(j \omega_{0})|
is maximized at a given phase shift
(\angle V_{2}/V_{1}(j \omega_{0})=\phi_{0})
. This problem is solved explicitly. It was found that the optimum
RC
phase-shift network is a geometrical progression network-properly terminated at both ends. In the distributed case (
n
approaches infinity), the optimum
RC
network is an exponentially tapered line, terminated by four lumped impedances.