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.