Nyquist Stability: Argument Principle, Encirclements, and Practical Application

All flashcards

Open-loop transfer function

The product of the forward-path and feedback-path transfer functions, $G(s)H(s)$.

Negative-feedback closed-loop transfer function

$T(s)=\frac{G(s)}{1+G(s)H(s)}$.

Nyquist plot

A complex-plane trace of the open-loop frequency response as frequency varies.

Critical point for unity negative feedback

$-1$, where the characteristic denominator $1+G(s)H(s)$ becomes zero.

Nyquist method advantage

It can assess closed-loop stability from open-loop frequency response without explicitly solving for every closed-loop pole.

Quiz questions

1. With clockwise encirclements counted as positive, what relation does the argument principle give for a mapped contour around the origin?Answer: $N=Z-P$, where $Z$ and $P$ count enclosed zeros and poles.
2. A negative-feedback system has $P=2$ open-loop right-half-plane poles. What net encirclement count is required for closed-loop stability, using clockwise encirclements as positive?Answer: Two net counterclockwise encirclements of $-1$, so $N=-2$.
3. What should be done when an open-loop pole lies on the imaginary axis during Nyquist analysis?Answer: Indent the contour around the pole and include the mapped indentation in the curve.
4. Which system type is not covered by the basic Nyquist criterion as a general stability test?Answer: Nonlinear systems, which require suitable nonlinear analysis methods.
5. What does a Nyquist curve passing exactly through $-1$ indicate for unity negative feedback?Answer: The characteristic equation has a root on the imaginary-axis boundary.