Saturday, March 27, 2021

Audio Amplifier Feedback - Estimating Poles in Lead-Lag Compensation Scheme

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

In the last post, I postulated that a feedback network combining lead and lag compensation:

has a transfer function with 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 $T_{p2} \approx (R_f || R_g || R_n)C_f$.

The actual transfer function, calculated from the circuit theory, is$$B(s)={R_g \over {R_f+R_g}}{{(s T_{z1}+1)(s T_{z2}+1)}\over{(s(R_f||R_g)C_f+1)(s R_n C_n +1)+s(R_f||R_g)C_n}}$$As usual, it can be transformed into many equivalent forms as needed. The exact poles are the roots of the quadratic equation $${(s(R_f||R_g)C_f+1)(s R_n C_n +1)+s(R_f||R_g)C_n} = 0$$It can be solved algebraically, but the result is unwieldy and obscures, rather than clarifies, the placement of the poles.

Instead, it is easier to estimate the poles as follows. The time constant of the pole associated with $C_n$ is calculated by assuming that $C_f$ is an open circuit at the frequencies of interest and that the signal source (here, the output of the opamp) has zero impedance, then computing the equivalent resistance “seen” by the $C_n$, which is $R_f || R_g +R_n$.

The time constant of the pole associated with $C_f$ is calculated by assuming that $C_n$ is short circuit at the frequencies of interest and, again, that the output of the opamp has zero impedance, then computing the equivalent resistance “seen” by the $C_f$, which is $R_f || R_g||R_n$. 

The approximation is based on a number of assumptions that I may look at in a future post, but is surprisingly accurate (within 1% of the actual frequency for a reasonable audio band network). 

In any case, the exact roots are not that important. The approximate formulas help to see what shapes the loop gain, and what values can be changed to optimize it. For an experienced designer, the Bode plot by itself becomes sufficiently informative to skip the formulas altogether.

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