# Linearity and speed tradeoff with closed-loop feedback

This article describes some tradeoffs involved with methods to increase an amplifier's linearity. One of the ways to increase a circuit's linearity is by applying negative feedback, which converts an amplifier's uncontrolled open loop gain $$\a\$$ to approximately $$\1/\beta\$$, a value which can often be much better controlled independent of the input and so the amplifier is linearized. The article describes a tradeoff with high-frequency operation:

To make the closed-loop system stable, we have to apply frequency compensation techniques to sufficiently reduce the loop gain to below 1 when the loop phase shift is 180°. This is conceptually equivalent to placing a low-pass filter in the loop. Low-pass filters suppress high-frequency signals, therefore, we expect the operation frequency of the closed-loop system to be much more limited than that of the original uncompensated amplifier. To summarize, applying negative feedback increases linearity at the cost of reducing the frequency of operation.

I understand that when considering the open-loop system on its own, in order for it be stable when put in the feedback loop, generally its bandwidth will need to be significantly reduced. However, another main benefit of closed-loop operation is bandwidth enhancement. For example, with a simplified first-order system, the bandwidth of the closed-loop system is $$\1+T\$$ times better than the open-loop system, where $$\ T\$$ is the loop gain.

In light of that, is the article saying that in practice, the factor by which the open-loop system's bandwidth will have to be reduced for stability reasons is much larger than $$\1+T\$$, therefore overall reducing the overall bandwidth?

What the article is trying to communicate is that by compensating an amplifier it results in the open loop gain rolling off at a lower frequency (lower open loop bandwidth). This results from the compensation inserting a dominant pole which results in a lower open loop gain at each frequency above the dominant pole.

The closed loop gain magnitude response plot is limited by the open loop gain magnitude response plot. The closed loop response rolls off as it approaches the open loop response as frequency increases. Therefore the more severely the amplifier is compensated, the earlier the open loop magnitude response rolls off limiting the closed loop bandwidth.

The closed loop gain curve cannot increase above the open loop gain curve except perhaps where there is peaking at the top end of the closed loop bandwidth in a situation where the phase margin has a low value.

The value (1 + Loop gain) is the factor (open loop gain)/(closed loop gain) and, when calculated at dc, represents the bandwidth improvement factor achieved by adding negative feedback.

These principles are what is at play when an amplifier is decompensated (under compensated). By reducing the severity of the compensation with the resulting increase in open loop bandwidth, it is possible to extend the closed loop bandwidth but the trade-off is that the amplifier must be used at a higher closed loop gain in order to keep the amplifier stable.

So, for example halving the value of a Miller compensation capacitor will double the open loop bandwidth (double the frequency of the dominant pole) which will double the closed loop bandwidth but we must then double the closed loop gain (halve beta) in order to maintain stability margins. AC open loop gain has increased, we reduce beta by the same factor keeping loop gain (beta * Aol) and therefore stability margins constant.

With regard to increasing the closed loop bandwidth by reducing the value of the Miller compensation capacitor in order to increase the open loop bandwidth - increasing the open loop bandwidth in this way will increase the closed loop bandwidth by the same factor leaving the ratio of open loop bandwidth to closed loop bandwidth unchanged and also leaving the ratio of open loop gain at dc to closed loop gain at dc unchanged. This means that when changing the compensation, the bandwidth improvement factor, 1+beta*Aol, remains unchanged when calculated at dc. This all assumes that beta is being held constant.

The distortion is reduced by the factor 1+beta*Aol but in this this case the value for Aol at the frequency of interest is used. Aol reduces with increasing frequency and therefore distortion increases with increasing frequency.

This theory assumes that the 1/beta response crosses the open loop gain response at a frequency where the open loop gain is falling at -20 dB/decade (-90 degrees open loop phase shift) which for a large part of its frequency response it approximately will be.

Response to comment

Yes, if I understand you correctly, in your second paragraph you are suggesting making an open loop amplifier with a gain aol equal to the required gain Ad. This amplifier would then have a bandwidth equal to Wol which is the open loop bandwidth. As you say, this wouldn't reduce distortion. To reduce the distortion you need to design an amplifier with very high low frequency open loop gain and then throw most of it away by applying negative feedback to get the much lower gain which you actually need.

Yes, if you haven't got closed loop stability then you could compensate more heavily (reduce the frequency of the dominant pole) which will reduce high frequency open loop gain stabilizing the amp, the problem being that this will reduce closed loop bandwidth as you say. But as I mention in my answer, one way around this is to increase the open loop gain by lightening the compensation (increase open loop bandwidth) but also reduce beta (increase closed loop gain). Doing both these things should stabilize the amp. Increasing the open loop gain makes the amp more unstable but increasing closed loop gain (reducing beta) improves stability margins. Stability depends on the magnitude and phase of the loop gain which is equal to beta*Aol. By using this technique you can keep a high closed loop bandwidth with high open loop gain with the possible disadvantage that you may have to use the amp with a higher closed loop gain than you need.

I'll rewrite that last point to try and clarify it.

If you are dealing with a unity gain stable amplifier which you want to use with a closed loop gain of greater than unity (say 5) then when you set beta to 1/5 you are improving the stability to be greater than that if the gain was set to unity (you have more stability than required). This excess stability can be made use of by lightening the compensation and increasing the open loop bandwidth 5 fold. This has the effect of giving the same stability margins as the previously unity gain stable version of the amplifier, when configured for unity closed loop gain, but has increased the closed loop bandwidth 5 fold compared to when the closed loop gain was set to 5 for the unity gain stable version. But now the closed loop gain must not be reduced below a value of 5 or the amp will go unstable. This is refered to as decompensating the amplifier.

Increasing the open loop gain degrades stability margins but reducing beta (increasing the closed loop gain) improves stability margins. Stability depends on the magnitude and phase of the loop gain which is equal to beta*Aol.

But what you wrote makes sense and seems correct to me.

• Thanks for the comment, I think I follow your points. To try to put it together to illustrate the bottom line of why there is a speed-linearity tradeoff, I tried to summarize it here (not in comments due to space). Does this look accurate? Commented Sep 24, 2023 at 18:58

The problem with saying:

For example, with a simplified first-order system, the bandwidth of the closed-loop system is 1+T times better than the open-loop system, where T is the loop gain.

Is that many things change at the same time, besides the bandwidth enlargement.

1. The bandwidth has been enlarge, but the gain has been reduced. So the bandwidth enlargement is only valid for an smaller gain.
2. The linear region of the amplifier has been extended, i.e. it can behave as a linear amplifier for a wider input range of voltages/currents.

Therefore, it is not precise to say that feedback has, somehow, increased the speed of your amplifier. I had a similar discussion with another user in this answer I wrote.

Feedback, to properly function, needs that your amplifier corrects the error signal by sending a signal around the loop. This implies that there will be a delay in your signal path, the settling is now governed not only by your load, but also by your feedback network.

You might not see the speed degradation at low frequencies, but at frequencies beyond your -3dB point, an open-loop amplifier will theoretically be much faster than the closed-loop one because it only needs to drive a load.

When it comes to compensation, this is required when the amplifier is unstable for a given feedback network. This effectively reduces the closed-loop bandwidth of the amplifier for a given gain and you have an slower amplifier but stable amplifier.

This is the reason why people complain that compensating wideband high-speed amplifiers is difficult because of one wishes to keep the wideband characteristics as much as possible. Any junior engineer can slap a resistor between the inputs of an op-amp and claim they have compensated, but at the high expense of bandwidth and gain.

The noise and linearity trade-off they explain in the article is a bit weak. There could be a mild trade-off for the design choices, but I don't know of a type of circuit where this is a hard trade-off. He uses a very small example where this trade-offs apply, but I cannot imagine this applying to more realistic amplifiers with several stages and large loop gains. Typically, the linearity is governed by the last stage of the amplifier, which happens to be the one contributing the least to the input-referred noise.

Your title says "noise and speed trade-off", but there's no mention of it in the article. Nonetheless, there exists such a trade-off in feedback amplifiers. For instance, usually the first stage is designed to have the largest gain, and as such, the largest noise contribution. In integrated circuits, a very simple way to get lots of gain is to increase the W/L ratio of your input stage. However, this comes with the penalty of increase parasitic capacitance, specially at the input. This translates into more capacitance your feedback amplifier has to drive, thus becoming slower.

• Sorry for the confusion, I updated the title (got mixed up with different question). For your point starting with: “You might not see the speed degradation…” I think I see what you’re getting at, since instead of just driving some cap load you have the cap load and e.g. some feedback capacitance. The phrasing is a bit confusing to me though, “at frequencies beyond -3dB, open-loop will be faster”. What does it mean to be faster “at a given frequency”? Do you mean “at frequencies beyond -3dB point, the open-loop amplifier will have higher gain”? Commented Sep 24, 2023 at 17:38