Consider the two$-$port network in the figure below, consisting of two Wilkinson power dividers defined by $[S{]}^a$ and $[S{]}^b,$ a quarter$-$wavelength transmission line defined by $[S{]}^c$ and an unknown two$-$port device defined by $[S{]}^d.$

$($a$)$ By tracking signals $($as was done in class when studying the reflectometer$),$ find the overall $S$ parameters of the network in terms of the $S$ parameters of the device.

$($b$)$ Suppose the overall $S$ parameters are measured to be

$[S]=\left[\begin{array}{cc}0& j0\mathrm{.}1464\\ j0\mathrm{.}1464& 0\end{array}\right]$

Find the $S$ parameters of the unknown device? What is the device? $($Hint: Determine if the device is lossless, and if yes, find the fraction of power dissipated in the device when an incident wave is applied to either port of the device$).$

$($c$)$ Suppose the overall $S$ parameters are measured to be

$[S]=\left[\begin{array}{cc}0\mathrm{.}5& j0\mathrm{.}5\\ j0\mathrm{.}5& 0\mathrm{.}5\end{array}\right]$

Find the $S$ parameters of the unknown device? What is the device? $($Hint: What do the $S$ parameters say about reflection and transmission through the device?$)$