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

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°.

Answer 1

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.

Answer 2

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).

Answer 3

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.