Discussion
The transfer function of the following differentiating circuit is :
(Figure: an inverting op-amp circuit. The input \(V_{1}(S)\) reaches the inverting input through R1 in parallel with a capacitor C; R2 is the feedback resistor from the output to the inverting input; the non-inverting input is grounded; the output is \(V_{2}(S)\).)
Answer: A
The transfer function is \(-\left[\frac{R_{2}(R_{1}C_{s} + 1)}{R_{1}}\right]\), where \(C_{s}\) means C times s.
For an inverting amplifier the gain is \(-Z_{f}/Z_{in}\). Here \(Z_{f} = R_{2}\), and the input element is \(R_{1}\) in parallel with C.
\[ \begin{aligned} Z_{in} &= \frac{R_{1}}{1 + sR_{1}C} \\ \frac{V_{2}}{V_{1}} &= -\frac{R_{2}}{Z_{in}} \\ &= -\frac{R_{2}(sR_{1}C + 1)}{R_{1}} \end{aligned} \]
Option B swaps \(R_{1}\) and \(R_{2}\). Options C and D drop the minus sign of the inverting amplifier, and D also changes the sign inside the bracket.
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