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Grasp negative feedback in audio amps (Part 1)

29 Dec 2015  | Bruno Putzeys

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The siren song of negative feedback is that maximising A makes the ETF approach zero and makes the STF approach -1/B(s). The quantity A(s) • B(s) is called "loop gain." This is the more precise term for what we loosely call "the amount of negative feedback." Let's take a hypothetical audio amplifier with exactly one integrator (see Box 1) and plot the various functions and values in figure 2.

Figure 2: Relationships between gain and loop functions in a feedback loop.

Higher-order loops
The 20dB/decade gain slope becomes quite limiting if you want to have really good audio performance. Suppose you want to achieve 0.001% THD at 20kHz and your power stage's open-loop distortion is a third harmonic at 0.3%. The obvious thing to do would be having 300 times = 50dB loop gain at 3x20kHz = 60kHz. A unity gain turnover point (more or less the closed-loop bandwidth) of 18MHz would do. The challenge would be futile only if the plan was to succeed.

Linearizing the output stage first looks more promising. What is sometimes called linearizing is really the use of some form of local feedback.

Figure 3: "Linearizing" the output stage.

This (figure 3) is called a "nested loop." For simplicity's sake I'm presuming that we're making an amplifier with a closed-loop gain of 1 and that the two control blocks are perfect integrators—that is, their frequency response is a 20dB/decade slope that continues all the way to DC. Once you get the hang of it you can insert more accurate transfer functions.

So A/s and B/s are integrators, the constant factors A and B being two times their respective gain-bandwidth products. The unity gain frequency of the global loop would be set a good bit lower than the closed-loop bandwidth of the local loop. In this particular example this simply means A<We can write the system as:

Solving for y:

Without giving it much thought we've just constructed a feedback loop with second-order behaviour (figure 4). For low frequencies (remember, s=2πjf), what remains of the error is now directly proportional to the square of frequency. For every halving of frequency, distortion drops by a factor four.

Figure 4: Loop functions with the added "linearizing" of the output stage.

A normal three-stage amplifier with local feedback around the power stage would be a suitable implementation. Note that ETF(s) shows only what happens to the distortion contribution of the power stage. Errors outside the local loop are corrected only by A/s and errors outside the global loop are not corrected at all.

Cascading several stages
Another method of constructing a higher-order loop is cascading several stages. There's an old rumour about amplifiers constructed of several stages to make a lot of gain that were then compensated to death so as to make the loop stable again. The necessary compensation capacitor, they say, is then so large that you have no more slew rate left.

It's subtler than that. Doing it like that just makes a perfectly normal first-order loop response and normal slew rate, albeit with spectacularly improved gain at 0.01Hz if fancy took you there. It's just another way of ineffectively building something so impressively complicated that having gotten it to work at all makes you feel all warm and fuzzy inside.

Getting improved loop gain from a cascade of integrators is done like this (figure 5):

Figure 5: Cascading integrators to improve loop gain.

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