Question $2(40$ points$)$: Transmission Line Model

The simple transmission line model yields the least accurate results. However, it is well suited to get some physical insight about the radiating elements. This model represents the microstrip antenna by two slots, separated by a transmission line of length $\backslash ($ L $\backslash )$ and a low$-$impedance $\backslash ($ Z$\_\{$c$\}\backslash )$ as shown in Fig$\mathrm{.}2.$

$2.$a Design a rectangular patch antenna with a length of $\backslash ($ L $\backslash ),$ width of $\backslash ($ W$=7\backslash$mathrm$\{$~mm$\}\backslash )$ on top of a RT$/$duroid $5870$ substrate with dielectric constant of $2\mathrm{.}33$ and thickness $\backslash ($ h $\backslash )$ of $1\mathrm{.}570$ mm so as to resonate at $\backslash ($ f$\_\{6\}=12\backslash$mathrm$\{$GHz$\}\backslash ).$

$2.$b Determine the input impedance and the position of the inset feed$-$point where the input impedance is $150$ ohms.