Nyquist plot mirrored across imaginary axis

Question

I am reading Foundations of Oscillator Design by Guillermo Gonzales. When an amplifier is connected in a positive feedback configuration, the transfer function is \$A_{vf}(s)=\frac{A_v(s)}{1-\beta (s)A_v(s)}\$

In that case, the book explains that the Nyquist test states that \$N=Z-P\$ where N is the number of encirclements of 1+j0, Z is the number of zeros in the RHP and P is the number of poles in the RHP. Of course, the usual Nyquist criterion deal with encirclements of -1+j0 instead, but my understanding is that we have 1+j0 here because the denominator of the closed-loop transfer function is \$1-\beta(s)A_v(s)\$ rather than \$1+\beta(s)A_v(s)\$

Up to there, this matches what I have learned in control theory courses. I am, however, confused as to how the book draws the Nyquist plots of \$\beta(s)A_v(s)\$.

For example, in Example 1.1, \$\beta(s)A_v(s)=\frac{3}{s(s+1)(s+2)}\$.

When plotting that using WolframAlpha, I obtain the following Nyquist plot:

The book draws them mirrored across the imaginary axis:

Why is this? Is one of them wrong?

Answer 1

The answer is simple: For the Nyquist plot, we need the loop gain.

This is the gain araound the loop when it is open - and it includes the sign at the summing junction. For positive feedback (oscillator case) the loop gain is positive - and the "critical point" is at "+1".

In some books/contributions the loop gain is defined as the product of the blocks/stages within the loop only - without consideration of the sign at the summing junction (that`s what you call "the usual Nyquist criterion"). In case of negative feedback, this results in a critical point which is at "-1".

For my opinion, this last definition is not a good one because each measurement/simulation of the loop gain includes, of course, the sign at the summing junction - in particular, the minus sign within a negative feedback loop.