Question $1$

$($a$)$ Write down a formula in terms of $V_o$ and $R$ for the voltage at node, $e_1,$ in the circuit shown in Figure $1\mathrm{.}1.$

$[5$ marks$]$

$($b$)$ The complicated voltage pulse shown in red in Figure $1\mathrm{.}2$ is to be applied to a circuit. Write an equation for the voltage as a function of time for this pulse using unit step functions as apprr$--.-.-$

$[5$ marks$]$

$($c$)$ For the circuit shown in Figure $1\mathrm{.}3,$ find an expression for the Th$$venin equivalent resistance, in terms of $R.$ across the terminals a and $b.$

$[5$ marks$]$

$($d$)$ For the circuit shown in Figure $1\mathrm{.}3,$ find an expression for the Th$$venin open circuit voltage in terms of $R$ and $I_o,$ across the terminals a and $b.$

$[5$ marks$]$