Question E7G05:
How can unwanted ringing and audio instability be prevented in an op-amp audio filter?  

Choices:  
A: Restrict both gain and Q  
B: Restrict gain but increase Q  
C: Restrict Q but increase gain  
D: Increase both gain and Q    

Correct Answer is  A: Restrict both gain and Q  

To prevent a filter from “ringing,” you want it well damped and with ample phase margin. Both high closed-loop gain and high Q reduce damping and phase margin, so restricting both keeps the filter stable.

Why A is correct (Restrict both gain and Q)
- Higher Q means lower damping, which increases overshoot and ringing in the time domain.
- Higher op-amp gain (or noise gain in active filter topologies) reduces phase margin and can push the loop toward oscillation.
- Limiting both keeps the filter well-damped and the op-amp comfortably stable.

Why the other choices are incorrect
- B: Restrict gain but increase Q — High Q still causes peaking and ringing even if the op-amp doesn’t oscillate.
- C: Restrict Q but increase gain — High gain can still reduce phase margin and cause ringing or instability; in some topologies, increasing gain also increases Q.
- D: Increase both gain and Q — This is the most likely to ring or oscillate.

Useful formulas and rules of thumb
- Damping ratio and Q:
  $$Q = \frac{1}{2\zeta}$$
  - Underdamped (ringing): $\zeta < 1$ (i.e., $Q > 0.5$)
  - Critically damped: $\zeta = 1$ ($Q = 0.5$)
  - For flat passband (Butterworth, 2nd order): $Q \approx 0.707$; peaking grows for $Q > 0.707$.
- Step-response overshoot for a 2nd-order underdamped system ($Q > 0.5$):
  $$M_p = \exp\!\left(-\frac{\pi}{2Q}\sqrt{4Q^2 - 1}\right)$$
  Higher $Q$ → larger $M_p$ → more ringing.
- Example (Sallen–Key low-pass with equal R and equal C):
  $$Q = \frac{1}{3 - K}$$
  where $K$ is the DC gain. As $K \to 3$, $Q$ explodes, causing severe peaking/ringing. Keeping $K$ modest helps keep $Q$ modest.
- Op-amp selection: use an op-amp with unity-gain stability and sufficient gain–bandwidth (GBW). A simple rule of thumb is to choose
  $$GBW \gg f_0 \cdot K \cdot Q$$
  so the op-amp maintains phase margin across the filter’s operating range.

Bottom line: To avoid ringing and audio instability in active audio filters, keep both the filter Q and the op-amp gain modest.