What is the correct statement of the Cauchy residue theorem used in the Nyquist criterion (m and n are the number of poles and zeros respectively)?

Concept:

The Cauchy Residue Theorem is a fundamental result in complex analysis and is used in control systems through the Nyquist criterion to relate the encirclements of a point in the complex plane to the poles and zeros of a function.

According to the theorem, for a closed contour in the complex plane enclosing singularities, we have:

\( \oint G(s) ds = 2\pi j (m - n) \)

Where,

m = number of poles of the function inside the contour

n = number of zeros of the function inside the contour

Calculation:

Given a transfer function G(s), its logarithmic derivative \(\frac{G'(s)}{G(s)}\) has poles at each zero and pole of G(s).

By applying the Cauchy Residue Theorem to this function around a closed contour, we get:

\( \oint \frac{G'(s)}{G(s)} ds = 2\pi j (m - n) \)

This gives the relationship used in the Nyquist stability criterion.