$7.$ Suppose you can make interconnect sections from a gain medium. An interconnect realized on a gain medium has negative resistance. You can cascade infinitesimal sections of interconnect on normal medium and interconnect on gain medium as shown in the figure below to get a composite interconnect. Normal interconnect is represented by the conventional series resistance per unit$-$length $\backslash ($ R $\backslash ),$ series inductance per unit$-$length $\backslash ($ L $\backslash )$ and shunt capacitance per unit$-$length $\backslash ($ C $\backslash ).$

$($a$)$ If the interconnect on gain medium is represented only by a negative series resistance per unit$-$length $\text{'}\backslash (-$R$\_\{$g$\}^\{\backslash$prime$\}\backslash )\text{'},$ and its inductance and capacitance is ignored, then what should be the ratio of $\backslash (\backslash$Delta x$\_\{1\}\backslash )$ and $\backslash (\backslash$Delta x$\_\{2\}\backslash )$ for the overall composite interconnect to be loss$-$less. Justify your answer mathematically.

$($b$)$ Now the gain medium also has a series inductance per unit$-$length $\backslash ($ L$\_\{$g$\}\backslash )$ and shunt

capacitance per$-$unit length $\backslash ($ C$\_\{$g$\}\backslash ).$ Will the ratio of the infinitesimal sections for getting loss$-$less composite interconnect change if you consider inductance and capacitance of the gain medium? What will be the delay of the composite interconnect of total length $\backslash ($ l $\backslash )$ for the loss$-$less case?

$($c$)$ It is given that $\backslash ($ L$\_\{$g$\}>$L $\backslash )$ and $\backslash ($ C$\_\{$g$\}>$C $\backslash ).$ On comparing normel interconnect with

the compositc interconnect of length $\backslash ($ l $\backslash ),$ will the delay increase or decrease? Or if a general statement about the delay cannot be made, give your comments on the effect on delay.