Amplifier design: feedback with unclear poles

Question

When designing a closed loop amplifier for audio the poles of the open-loop transfer function are usually only vaguely known.

\$\frac{1}{(s-p_1)(s-p_2)...}\$

These poles can be modeled as \$RC\$ low-passes which attenuate the signal from their break frequency and, more importantly, add phase shift to the signal.

To avoid to much phase shift before gain reaches the 0-dB limit usually a pole is deliberately added at a very low frequency so the open loop gain falls below 0 dB before the other poles kick in.

So far so good. But how can the second, third etc pole be estimated to get a clue where the first pole has to be?

What does the notion of "so much dB feedback is ok to be applied without threatening the stability" mean? How is the applicable amount of dB feedback defined?

An example: I want an amplifier to amplify a signal 30 times and have an open loop gain of say 10000. What does it mean when I say "I apply ...dB of feedback"? Usually I would make a voltage divider of 29/1 and therefore get a gain of 30 (factor, not dB). I don't know how to put this any simpler, but doesn't the applicable amount of feedback depend on how my closed loop gain of the amplifier should be? It's often said that the more feedback the better but when I make a unity gain buffer It is useless since I want to amplify my music, right?

Long story short:

What is meant by the applicable amount of feedback?

How do I estimate the other poles?

Answer 1

I think the keyword is "loop gain" - and the most important point (as far as stability is concerned) is the frequency where the loop gain Alo is unity (0 dB).

Because the loop gain is the product of the amplifier´s open-loop gain Alo and the feedback factor Hr (Alo=Hr*Aol) you can find the BODE diagram for the loop gain response very easily:

Draw Aol=f(w) and 1/Hr(w) as a BODE plot and the difference between both curves gives you the loop gain Alo (in dB). The stability criterion requires that at the frequency where both curves meet (Alo=0 dB) the phase shift of the loop gain must not equal to (or even more than) 180 deg.

That means: The "rate of closure" (slope) of the loop gain at this crossing frequency must not be -40dB/dec. (Rule of thumb: If the second pole is identical to the frequency with Alo=0 dB we have app. 45 deg phase margin).

Based on this requirement you can derive the necessary/desired location of the second pole.

Remark ("It's often said that the more feedback the better"): You strictly must distinguish between DC and dynamic stability. Heavy feedback improves the stability of the operating point but - at the same time - degrades the dynamic stability (against oscillations).

EDIT: Sorry for the large picture. I don´t know if/how the size could be reduced.

Answer 2

Estimating the location of the other poles would require a thorough analysis of the specific circuit you are using. Hard to even crack into that without more information.

Regarding the effect of feedback gain on closed-loop stability: consider that when you put a feedback divider in the loop (1/29 in your case) you reduce the open-loop gain. In other words, the gain of the loop will fall to 0dB earlier. For a typical single-pole dominant loop, this will benefit stability because the unity gain frequency will be further from the detrimental phase shifts introduced by your high-frequency (parasitic) poles. In this sense, unity gain configuration is the most stressful for a high-frequency amplifier. You see some op-amp datasheets that list "unity gain stable" as a feature.

When you see a specification for the amount of feedback you can apply, it is referring to this phenomena. 0dB feedback would be unity (closed loop) gain configuration, while in your case the feedback is -30dB. Note that you introduce an attenuation in the feedback transfer function in order to achieve an amplification in your closed loop transfer function. I.e. your divider is -30dB or 1/30 and your closed-loop gain will be 30dB or 30x.

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It seems you are missing the distinction between the open-loop and closed-loop transfer functions. When we talk about the gain bandwidth product (GPB) we are talking about the product of the DC gain the -3dB frequency of the closed-loop transfer function. When I talk about the unity gain frequency of the open-loop transfer function, that is a different thing.

As for how you specify the feedback, dB is always a relative measure. 0dB is the same as 1. You can specify the feedback gain (actually an attenuation) in dB. 0dB would be unity feedback, a 1:10 attenuation would be -20dB, a 1:100 attenuation would be -40dB and so on. So it would make sense to say that an amplifier is stable with no more than -20dB feedback...

Answer 3

I found the answer to my question by accident:

Just for those who are still interested in what I asked. Now everything makes sense and gets way clearer. Thanks for the help anyways!

Edit : I made up an example to clarify it and make it more tangible: Assume a system which has 3 Amplifiers (x100, x100, x10) and 3 Low Passes which have poles at 1Hz (all for mathematical simplicity). We obtain \$\frac{1e5}{(s+1)^3}\$ and bode plotting it gives a phase shift of 180° at about 1.7 Hz. When looking into the output, the gain has fallen to about 80dB. To stay mathematically stable we need to attenuate the signal by more than 80dB (1/10000) to have the phase-shifted signal which is put into the negative input lower than one (0dB). That makes for \$1+Ab\$ (omitting the 1) \$1e5/1e4\$ which is equal to 100dB (A)- 80dB(b) = 20 dB. There we are: the highest applicable feedback factor is 20dB. Adding more poles makes this even lower (critical phase shift is added up earlier; imagine 4 poles, all a 1Hz, then 180° is reached while gain is still 87dB, so we have to attenuate the outgoing signal by 87dB to stay stable, hence we have 100-87 = 13dB left for feedback.) This concludes to: b = A minus maximum-applicable-feedback. Note that the amount of applied feedback cannot be determined solely from b, you have to know A! Done:-)