Topic: Control Systems
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)?
- A. $\int G(s)ds=2\pi j$
- B. $\int G(s)ds=2\pi j(m-n)$
- C. $\int G(s)ds=2\pi j(n-m)$
- D. $\int G(s)ds=2\pi jm$
Answer: C. $\int G(s)ds=2\pi j(n-m)$
Explanation: The Principle of the Argument, which stems from Cauchy's Residue Theorem and is the foundation of the Nyquist criterion, deals with contour integrals of the logarithmic derivative of a function. The integral evaluates to $2\pi j (Z - P)$, where Z and P are the number of zeros and poles enclosed by the contour. Substituting $n$ for zeros and $m$ for poles gives $\int G(s)ds = 2\pi j(n - m)$.