Question E7G02:
What is the frequency response of the circuit in E7-3 if a capacitor is added across the feedback resistor?  

Choices:  
A: High-pass filter  
B: Low-pass filter  
C: Band-pass filter  
D: Notch filter    

Correct Answer is  B: Low-pass filter  

When you add a capacitor across the op-amp’s feedback resistor, the stage becomes an active low-pass filter.

Why “Low-pass filter” is correct:
- At low frequencies, the capacitor’s reactance is high, so it is effectively open. The feedback impedance is just $R_f$, and the amplifier has its normal midband gain.
- At high frequencies, the capacitor’s reactance is low, so it shunts $R_f$, reducing the feedback impedance and thus the gain. Output falls as frequency rises → low-pass behavior.

Useful formulas (inverting amplifier case with $R_{in}$ and $R_f$; capacitor $C$ across $R_f$):
- Capacitive reactance: $$Z_C = \frac{1}{j 2\pi f C}$$
- Feedback impedance: $$Z_f = R_f \parallel \frac{1}{j\omega C} = \frac{R_f}{1 + j\omega R_f C}$$
- Gain: $$A_v(j\omega) = -\frac{Z_f}{R_{in}} = -\frac{R_f}{R_{in}}\cdot \frac{1}{1 + j\omega R_f C}$$
- Magnitude: $$|A_v| = \frac{R_f}{R_{in}}\cdot \frac{1}{\sqrt{1 + (\omega R_f C)^2}}$$
- Corner frequency: $$f_c = \frac{1}{2\pi R_f C}$$
- Roll-off: 20 dB/decade (6 dB/octave) above $f_c$.

For a non-inverting stage, the capacitor across the feedback resistor similarly reduces the high-frequency divider ratio, causing gain to drop toward unity at high frequency—still a low-pass.

Why the other choices are incorrect:
- High-pass filter: Would require a series capacitor in the input path or in series within the feedback so low frequencies are attenuated. A capacitor across $R_f$ attenuates highs, not lows.
- Band-pass filter: Needs both a high-pass and a low-pass action (two reactive elements or a resonant network). A single shunt capacitor across $R_f$ provides only one pole (low-pass).
- Notch filter: Requires a frequency-selective cancellation network (e.g., twin-T or resonant notch). A single RC across the feedback resistor cannot create a notch.

Rule of thumb:
- Capacitor across feedback resistor → low-pass.
- Capacitor in series with input (or feedback path) → high-pass.
- Combine both (or use resonant networks) → band-pass or notch.