# Loop gain less than one is stable

So I read about Stability/Negative Feedback in amplifiers here https://www.allaboutcircuits.com/technical-articles/negative-feedback-part-4-introduction-to-stability/

I understand that an LTI feedback amplifier has a closed-loop gain of I also understand that the Loop gain is defined to be AB and that for a loop gain magnitude less than one, there will some oscillations but these will die out as time goes on - thus being stable. Now, that contradicts with the equation up there.

If we set the Loop gain AB < 1 to something like AB = 0.5 and then consider at a 180 degree phase shift, which will mean that $$G_{CL}=\frac{A}{1-(0.5)}=2A$$ Now that suggests to me that for a loop gain magnitude less than 1 (0.5 here) and at a phase shift of 180 degrees, the amplifier should be unstable as the oscillations would build up with a CL gain of 2A? That contradicts the previous intuition I had. How come the equation isn't matching up with what I thought?

• Having a closed loop gain greater than 1 does not make a system unstable. But consider what happens if $A\beta=1$. – The Photon May 6 '18 at 16:03
• Obviously, in the given formula Axbeta=1 gives Gcl=A/2. The Photon, I suppose you mean LOOP GAIN=1, right? This is equivalent to Axbeta=-1. – LvW May 6 '18 at 19:30
• Low loop gains may produce unpleasant settling behavior, tho "stable" (does not build to infinite amplitude). – analogsystemsrf May 7 '18 at 2:43

## 3 Answers

I think, some clarification is necessary:

When A is the open-loop gain of the amplifier (without feedback) and beta is the feedback factor, it is important if the feedback network is connected to the inverting (negative feedback) or the non-inverting input of the opamp.

In most cases (amplification) we have negative feedback and the closed-loop gain is

Gcl=A/(1+A*beta)

However, with respect to the correct sign, it is important to realize that the quantity we call LOOP GAIN includes the negative sign at the summing node, hence

Loop Gain LG=-A*beta

It is important to include the negative sign at the summing node in the loop gain definition because that is the way we are measuring/simulating the loop gain function: We open the loop at a suitable point and measure the total gain (and phase) of the whole loop - of course including the minus sign).

Therefore:

Gcl=A/(1-LG)

Note that Barkhausen`s oscillation condition (there is no "stability criterion" from Barkhausen) requires LG=1 for a circuit to oscillate (unity loop gain with a phase shift of zero deg).

That means: All feedback circuits are stable for LG<1 (examples: pos. feedback with LG=0.5 or negative feedback with LG<0).

I hope this clarifies some points.

The Barkhausen stability criterion says that when the loop gain magnitude is equal to 1 and has a phase shift of 180 degrees (+- 360n), the system may be unstable and stably oscillatory.

I think you're trying to say in the first paragraph that when the loop gain is positive, you get a decaying oscillation. This is only true in certain circumstances. In a second order transfer function (you can break down any LTI system into a cascade of 2nd orders and 1st order transfer functions), you can see that you get that decaying oscillation stuff when you have poles that are underdamped, not when your loop gain is simply positive.

Hope this answers the question.

If say a video Amp or some Plant transfer function say is not unity gain stable and has a peaking gain of say 6dB max at the breakpoint of GBW, as long as your sub-unity gain is below the peaking gain, the loop gain can never reach 1 or unity gain Barkhausen criteria even if the phase shift is 3rd order.

• Of course you would never run it at unity gain or less, but at least it would become stable if the loop gain is below peaking factor or Q.

If it is unity gain stable with an integrator representing a 90 deg max phase shift, then sub-unity gains will also be stable since the gain never reaches 1 and 180 deg stability threshold.