Stability of op-amps: “Oscillations if phase = 180° and loop gain > 1” question

I'm trying to understand the stability against oscillations for op-amps in a feedback circuit. A common argument I've seen on many sites goes as follows: The closed-loop gain is given as

$$A_{\text{cl}} = \frac{A_0}{1 + BA_0}\tag{1}$$

where $$\A_0\$$ is the open-loop gain of the op-amp, and $$\B\$$ is the fraction fed back to the negative input. Clearly, if the loop gain $$\BA_0\$$ becomes $$\-1\$$, then $$\A_{\text{cl}}\$$ diverges; this is taken to mean that the output is oscillating (a finite output with zero input).

This leads to what appears to be called the "Barkhausen criterion," that an op-amp circuit will oscillate if the magnitude of the loop gain equals 1 when the phase is -180°.

However, it is just as often stated that an circuit will oscillate if, at the frequency at which the phase = -180°, the loop gain is greater than or equal to 1. How is this reconciled with Equation 1? If I let $$\BA_0\$$ equal, say, 3.0 with phase shift of -180° (or really, any combination of $$\|B_A0| > 1\$$ and phase < -180°), Equation 1 has a perfectly well-behaved solution. Is this equation not really the whole picture?

I looked at the data sheets for a number of uncompensated op-amps (so that the phase would reach 180° while the gain was still > 1). Their Bode plots look nothing like the textbooks, and none of them had a magical frequency at which the loop gain was 1 and the phase was -180°.

However, it is just as often stated that an circuit will oscillate if, at the frequency at which the phase = -180°, the loop gain is greater than or equal to 1. How is this reconciled with Equation 1?

If the loop gain is greater than 1 with a phase shift of -180°, if the op-amp remains linear then in principle you could produce an oscillator with a constantly increasing output amplitude.

But of course the output amplitude can't increase indefinitely. There will be some nonlinearity in the op-amp (or feedback network) response that limits the output amplitude. For example, the op-amp could enter saturation mode operation.

Often this will effectively limit the op-amp gain so that $$\\beta A_0\$$ is reduced to 1, and you have the Barkhausen criterion fulfilled after all.

• If nothing else limits the growth of amplitude, the finite power supplies are a limit. And for certain circuits, the opamp has diodes back-to-back between Vin+ and VIn-, to protect the emitter-base junction against breakdown and degraded noise floor; these diodes may be turning off. And for larger signal swings, the finite SlewRate may be the limiting factor. – analogsystemsrf Oct 21 '18 at 4:02
• @analogsystemsrf, the limit from the finite power supplies is the saturation mode of operation. – The Photon Oct 21 '18 at 4:56

As far as I can see, you have detected a conflict between "-1" and "+1", correct?

In this context, the definition of the term "loop gain" plays a major role. That means: It is important if you include the sign at the incoming node into the definition - or not. This is inportant because Barkhausens oscillation criterion requires a POSITIVE loop gain - hence, positive feedback!

The closed-loop transfer function for a circuit with positive feedback is

H(s)=Ao/(1-k*Ao) (feedback factor k).

When the loop gain LG=kAo=1 the circuit becomes unstable. If this condition is fulfilled (or over-fulfilled with LG>1) for one single frequency only, the Barkhausen oscillation criterion is fulfilled and the circuit will - most probably - oscillate (not necessarily, but this is a specific question). Note that LG=1 means |LG|=1 without any phase shift (phi=0 deg).

However, if you are using the closed-loop formula for the special case of negative feedback only (as in your case), we have

H(s)=Ao/(1+k*Ao)

and the circuit becomes unstable if kAo=-1 (as in your post).

(Remark: For an oscillator circuit, it is impossible to exactly realize LG=1 at one single frequency due to tolerances and other uncertainties. Therefore, we realize LG>1 and use some kind of non-linearity within the gain determining circuitry - for example: diodes - which automatically bring the loop gain back to LG=1 for rising amplitudes before hared limiting occurs).

• In different books/websites, differing conventions for the +/-1 are used. I used "+", meaning that at low frequencies there is correct negative feedback and everything is stable. Other conventions use "–", implying that the output needs to be fed back to an inverting input. My question centers on your statement about "or over-fulfilled with LG>1". Nothing in the equation H(s)=Ao/(1-k*Ao) diverges or has any other pathology for LG > 1. Why does this equation remain well-behaved when the circuit is evidently having badly? – amtravco Oct 21 '18 at 17:18
• Negative feedback for low frequencies ensures a stable DC bias point -not more! In contrary, negative feedback always degrades the phase margin!! More than that, a "-" in the formula implies feedback to the "+" input (and NOT to the "-"). For practical oscillator circuits we always design for LG>1 and use in addition a sort of amplitude control. See my comment to Chu`s answer above!. – LvW Oct 22 '18 at 6:13
• amtravco...in your comment, the second to the last sentence is wrong! Of course, there is a severe "pathology". For LG=1 the whole expression goes to infinity and for LG>1 the expression changes its sign. This is connected with a 180 deg jump in phase and leads to instability. – LvW Oct 22 '18 at 14:47

If loop gain is -1 and the phase shift is -180 then a sinusoid at the output will appear unchanged when fed back to the input (ie negative feedback). This is an oscillator.

If the loop gain magnitude is greater than 1 then the sinusoid will grow in amplitude. This is unstable.

In the latter case the sinusoid will stop growing when saturation limits the loop gain magnitude to 1.

• Chu.....typing error? Unity loop gain means: Magnitude of "1" and phase shift within the loop of 0 deg (360 deg). The term "loop gain" must always contain the sign at the summing junction - otherwise, misunderstanding or misinterpretations cannot be avoided (see the question under discussion). Background: Each measurement (or simulation) contains the sign at the summing junction. This can be positive or negative!! – LvW Oct 21 '18 at 13:10
• @LvW Yes, I originally wrote it that way, then changed my mind. I've re-written now. Thank you. – Chu Oct 21 '18 at 13:18
• You are welcome! – LvW Oct 21 '18 at 13:24
• My question is about the statement, "If the loop gain magnitude is greater than 1 then the sinusoid will grow in amplitude. This is unstable." How does this follow from the equation Acl = A0/(1 + B*A0)? There is nothing in that equation that blows up or acts oddly for LG > 1 and phase = -180. – amtravco Oct 21 '18 at 17:27
• If there's a phase lag of -180, the forward path is not a simple gain, but must be, at least, a 2nd order order transfer function. Hence $\frac{A}{1+AB}$, is not valid. – Chu Oct 22 '18 at 6:23