## Saturday, January 30, 2021

### Audio Amplifier Feedback - Two-Pole Compensation

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

As I noted in my posts on limitations of dominant pole compensation and on nested loops, dominant pole compensation locks a lot of gain inside the local feedback loop and makes it unavailable to the global feedback loop. The simplest way to unlock (part of) that gain is two-pole compensation.

Two-pole compensation implemented with a C-R-C “T” compensation network is well known since 1980s and was featured in Douglas Self's comprehensive article on frequency compensation in the February 1994 issue of Electronics World - see FIG 1D on the right:

An intuitive explanation of how the C-R-C compensation network works is simple:

• $R_P$ and $C_{P2}$ form a frequency-dependent voltage divider that feeds $C_{P1}$ with a portion of the output voltage
• At low frequencies, the impedance of $C_{P2}$ is high compared to $R_P$,  so $C_{P1}$ sees a only a small portion of the output voltage, the local feedback's loop gain is reduced, and more gain is available for the global feedback loop
• At high frequencies, the impedance of $C_{P1}$ and $C_{P2}$ is low compared to $R_P$,  so both capacitors in series work as a Miller compensation.
Between those two extremes, the amplifier with two-pole compensation has two poles (hence the name), so the loop gain falls at a rate of 40dB/decade, and a zero, which decreases both the rate of loop gain change by 20dB/decade and the phase lag. Compared to Miller compensation with the same bandwidth, two-pole compensation provides a welcome boost of global loop gain at audio frequencies:

Despite being around for 30+ years and probably because of its simplicity, a formal analysis of two pole compensation was published only in 2010 by Harry Dymond and Phil Mellor from University of Bristol, who presented their paper at the 129th AES convention. A brief summary:

• Similarly to Miller's single dominant pole, the location of the two poles depends on the amplifier's stage inside the local feedback loop
• The zero is located at the frequency $f_z = {1 \over {2 \pi R_P (C_{P1}+C_{P2})}}$
• Unity loop gain frequency is $f_0 = B \times g_m {(C_{P1}+C_{P2}) \over 2 \pi C_{P1} C_{P2}}$, where $B$ is the transfer function of the feedback network and $g_m \approx {1 \over {r_e + R_e}}$ is the transconductance of the input stage

Practically speaking, let $C_{P1}$ have the same value as for Miller compensation, choose $C_{P2}$ twice as big, and make $R_P$ as low as you can without affecting stability.

There is a so-called "output inclusive" version of the two-pole compensation where the bottom end of $R_P$ is connected not to the ground but to the output of the amplifier. The output-inclusive version has almost identical design equations to what is discussed above.