Which configuration yields a differentiator, and what is the transfer function and a practical limitation?

Which configuration yields a differentiator, and what is the transfer function and a practical limitation?

• A. Inverting differentiator with input capacitor Cin and feedback resistor Rf: Vout(s) = - Rf Cin s Vin(s); practical limitation: high-frequency noise and potential instability; stabilization by adding series R or C in feedback
• B. Inverting differentiator with input capacitor Cin and feedback capacitor C; Vout(s) = - Cin C s Vin(s)
• C. Inverting differentiator with input resistor Rin and capacitor Cf; Vout(s) = - (1/Rin Cf) s Vin(s)
• D. Non-inverting differentiator with input capacitor Cin; Vout(s) = Cin s Vin(s)

Answer: (A)

A differentiator in this op-amp setup comes from the capacitor in the input path and a resistor in the feedback path. With the input capacitor Cin, the input impedance is Z_in = 1/(s Cin); with the feedback resistor Rf, the feedback impedance is Z_f = Rf. The closed-loop gain is Vout/Vin = -Z_f/Z_in = -Rf Cin s. This means the output is proportional to the time derivative of the input, with a gain equal to Rf Cin. A key practical limitation is that the gain increases with frequency, so the circuit boosts high-frequency noise and can become unstable due to the op-amp’s finite bandwidth and phase shift. To make it usable, you add stabilization that limits high-frequency gain—for example, a small resistor in series with the input capacitor or a small capacitor in parallel with the feedback resistor—to create a band-limited differentiator that differentiates only over a chosen frequency range.

A differentiator in this op-amp setup comes from the capacitor in the input path and a resistor in the feedback path. With the input capacitor Cin, the input impedance is Z_in = 1/(s Cin); with the feedback resistor Rf, the feedback impedance is Z_f = Rf. The closed-loop gain is Vout/Vin = -Z_f/Z_in = -Rf Cin s. This means the output is proportional to the time derivative of the input, with a gain equal to Rf Cin.

A key practical limitation is that the gain increases with frequency, so the circuit boosts high-frequency noise and can become unstable due to the op-amp’s finite bandwidth and phase shift. To make it usable, you add stabilization that limits high-frequency gain—for example, a small resistor in series with the input capacitor or a small capacitor in parallel with the feedback resistor—to create a band-limited differentiator that differentiates only over a chosen frequency range.