# Does this Root Locus analysis contradict my understanding of Nyquist stability criteria?

I have been trying to theoretically analyse the operation of a 741 op-amp from a stability point of view by firstly deriving its open-loop transfer function.

Here is the magnitude plot of a 741 op-amp.

From the above plot, I generated the open-loop transfer function.

From the open-loop transfer function I have derived the closed-loop transfer function and to keep things simple I have assumed that the op-amp is configured as a unity gain buffer ( β = H(s) = 1 ).

$$\\$$

Now, the Root Locus method plots the loci of the closed-loop poles on the s plane as some parameter is varied from 0 to infinity. I vary the dc open-loop gain k which is set to 100000 for the 741. The values of the poles at various values of dc open-loop gain can be found by equating the denominator of the closed-loop transfer function (the characteristic equation) to zero and solving for s. That is to say, the roots of the characteristic equation are equal to the poles of the closed-loop transfer function.

Here is the characteristic equation which is equated to zero.

By entering a range of values for k into the above equation and solving for s using the following equation….

….I get the pole plot below on the s plane. The arrows represent increasing k values. I’m pretty confident about this plot because I have seen the same plot for a second-order system on various web sites.

My problem is this. I know from fig.1 above and from Nyquist stability criteria that increasing the dc open-loop gain will reduce gain and phase margins. As open-loop gain k increases, the loop gain will get down to unity at a higher frequency where the loop phase lag is nearer -180ᴼ thereby reducing stability margins and moving the op-amp closer to oscillation. (The phase lag should never quite reach -180ᴼ because this is a theoretical 2 pole amplifier). Why then doesn’t the root locus plot converge on the imaginary axis on the s plane as the dc open-loop gain k is increased. The plot is vertical. I think I am misunderstanding something.

The only thing you're misunderstanding is that those vertical root loci do, in fact, imply and ever-diminishing phase margin. They're describing a system that goes resonant, with a Q that increases (or $$\\zeta\$$ that decreases) as the gain goes up.
• What's important in this case is the angle from \$s = 0$ to the poles, which is getting ever less. Another way of looking at it is to think of the root locus as dimensionless, and just keep zooming it out so that you can see it -- then it starts to look bad. And that "bad" is rightly so -- because with a 2nd-order system like that, the higher in frequency you get, the more that unmodeled poles lurking far off in the left-hand plane will start having an effect, bending the locus out of the stability region. Dec 13 '19 at 16:35