In this problem we will match a $\backslash (100\backslash$Omega $\backslash )$ load to a generator impedance of $\backslash (50\backslash$Omega $\backslash )$ using a combination of a transmission line and lumped elements. For all parts below, the load is connected to a transmission line of characteristic impedance $\backslash (50\backslash$Omega $\backslash )$ and unknown length $\backslash ($ l $\backslash ).$ Assume $\backslash ($ v$\_\{$p$\}\backslash )$ is equal to speed of light in free space.

a$)$ For this part, there is a shunt, lumped element after the transmission line of length $\backslash ($ l $\backslash )($between generator and the transmission line$).$ Find the minimum distance $\backslash ($ l $\backslash )$ from the load where the unknown shunt element is to be connected.

b$)$ For part $($a$),$ determine the unknown shunt element and its element value such that the input impedance is matched to the generator at $1$ GHz $.$

c$)$ For this part, replace the shunt element with a series, lumped element. Find the minimum distance $\backslash ($ l $\backslash )$ from the load where the unknown series element is to be connected.

d$)$ For part $($c$),$ determine the unknown series element and its element value such that the input impedance is matched to the generator at $1$ GHz $.$