Question $4$

$4.$ Consider the circuit shown in Figure $4\mathrm{.}1.$ The source, widehat$(V_{in}),$ is a variable frequency sinusoidal source. The output voltage is measured across terminals a and $b.$

$($a$)$ For the circuit shown in Figure $4\mathrm{.}1,$ replace the circuit elements to the left of the dashed line $($the voltage source and the two resistors$)$ with an appropriate Th$$venin equivalent circuit, where the open circuit voltage is expressed in terms c widehat$(V_{ln}).$

$($b$)$ Write an expression for the ratio of the magnitude of the output voltage, widehat$(V_{ab}),$ to $t$ input voltage, widehat$(V_{in}),$ in terms of the inductance, angular frequency and the Th$$venin equivalent resistance you calculated in question $4($a$).$

$[6$ mi

$($c$)$ Calculate the frequency $($in Hz $)$ at which $|A|=\frac{\sqrt[\phantom{2}]{ab}}{\sqrt[\phantom{2}]{s}n}=0\mathrm{.}3$

$[10$ ma

$($d$)$ For $|A|=0\mathrm{.}3,$ express the value of $|A|$ in decibels.

$[2$ me

$($e$)$ For the circuit shown in Figure $4\mathrm{.}1$:

i$)$ Calculate the angular frequency of the "knee" in the voltage ratio $($where $|A|=-3dB).$

$[2$ ma

ii$)$ Sketch the Magnitude Bode plot for $|A|$ against angular frequency.

$[5$ marks$]$