## Saturday, March 20, 2021

### Audio Amplifier Feedback - Combining Lead and Lag Compensation

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

The lead and lag compensation schemes can be combined in one feedback network:

The transfer function of this lead-lag feedback network has two poles and two zeros:$$B(s)={R_g \over {R_f+R_g}}{{(s T_{z1}+1)(s T_{z2}+1)}\over{(s T_{p1}+1)(s T_{p2}+1)}}$$where $T_{z1}=R_f C_f$, $T_{z2}=R_n C_n$, $T_{p1} \approx (R_f || R_g +R_n)C_n$ and $T_{p2} \approx (R_f || R_g || R_n)C_f$.

For given $R_f$ and $R_g$, both zeros and one pole can be placed freely, which makes the combined compensation scheme quite flexible.

Compare the $1/B$ feedback factor for the lead-lag compensation scheme (blue) with both lead (red) and lag (light blue) alone; green is the open-loop gain:
It is clear that unlike the pure lead compensation, the combined scheme doesn't extend the bandwidth; unlike the pure lag compensation, it preserves the loop gain in the audio band; and from the rate-of closure, it appears that the phase margin is similar for all three compensation schemes.

One possible design procedure for the combined compensation scheme is:

1. Choose $R_f$ and $R_g$ from the desired DC gain $(R_f+R_g)/R_g$ and the expected parasitic capacitances
2. Choose the time constants for the first pole ($T_{p1}$) and both zeros ($T_{z1}$ and $T_{z2}$)
3. Calculate $C_f=T_{z1}/R_f$
4. Calculate $C_n=(T_{p1}-T_{z2})/(R_f || R_g)$
5. Calculate $R_n=T_{z2}/C_n$