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

In my previous post, I discussed how distortion can be corrected by cascading multiple opamps:

Obviously, in such a structure the stability problem arises. One
possible solution was devised by John D. Yewen and described in his
article "High-precision composite op-amps", published by *Electronics & Wireless World* in February 1987.

Yewen observed that, if every opamp in the schematic above is a perfect integrator (i.e. its open-loop gain falls with frequency at 20dB/decade), the closed-loop transfer function of the above arrangement is simple enough to calculate and to test for stability using the algebraic Routh-Hurwitz stability criterion, without ever looking at Bode plots. Further, the closed-loop transfer function can be manipulated to stability by adding voltage dividers at the outputs of the error correcting opamps:

As for choosing the dividers for stability, Yewen mentions the binomial series (1,1; 1,2,1; 1,3,3,1; ...) and the Butterworth coefficients (1,1; 1,1.41,1; 1,2,2,1; ...) and gives one worked example of a three-opamp unity-gain invertor.Turns out, the general rule for Yewen's dividers is quite simple for:

- two- or three-opamp composite amplifiers...
- with the Butterworth coefficients...
- built from identical, single pole compensated opamps.

Say, we need an inverting amplifier with gain $(-A)$. For example, in Yewen's worked example above, gain is $(-1)$, so $A=1$. To achieve that, the feedback network $R1R2$ should divide the output signal by ${A+1} = {{R1+R2} \over {R1}} = 2$. Here is the rule:

- To build a two-opamp composite, add the opamp
*C1*and divide its output voltage by $(A+1) \times 2$. - To build a three-opamp composite, add another opamp
*C2*and divide its output voltage by $(A+1) \times 4$.

For example, for an inverting composite with $A=2$, the following values will work:

All three circuits above are stable and have phase margin of about 65° (loop gain on the top, closed-loop transfer function at the bottom):

The price for such stability is diminishing returns on extra opamps - each additional error correcting opamp contributes less to the overall loop gain because of the voltage dividers. Also, the closed-loop transfer function peaks around the crossover frequency, so a low-pass input filter is required.

Does it work in practice? Yes, as long as the opamps are (i) identical and (ii) sufficiently close to single-pole, at least around the crossover frequency. Here is a practical composite with the negative gain of 10 and total loop gain of over 90dB, built from four OPA134:

It is quite stable and clips ok:

Can we make it even better? Of course we can. Stay tuned...