Discussion
If the transfer function of the following network is
\[ \frac{V_{o}(s)}{V_{i}(s)} = \frac{1}{2 + SCR}, \]
(Figure: a series resistor R from the input Vi to the output; across the output, a capacitor C and the load resistor RL are connected in parallel. Vo is taken across RL.)
then the value the Load Resistor RL is :
Answer: C
The load resistor is R.
The output is taken across C in parallel with \(R_{L}\), fed through the series resistor R, so the circuit is a voltage divider.
\[ \begin{aligned} Z_{p} &= \frac{R_{L}}{1 + sCR_{L}} \\ \frac{V_{o}}{V_{i}} &= \frac{Z_{p}}{R + Z_{p}} \\ &= \frac{R_{L}}{R(1 + sCR_{L}) + R_{L}} \\ &= \frac{1}{1 + R/R_{L} + sCR} \end{aligned} \]
For this to equal \(\frac{1}{2 + sCR}\), the constant terms must match: \(1 + R/R_{L} = 2\), so \(R_{L} = R\).
With \(R_{L} = R/2\) the denominator would start with 3, with \(R_{L} = 2R\) it would start with 1.5, and with \(R/4\) it would start with 5.
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