# Showing $T(s) = R(s)/(1+R(s))$ is stable using Niquist's Criterion. Help!

$R(s) = \frac{1}{s^2+2s+5}$ and $R(i\omega) = \frac{5-\omega^2-2i\omega}{(5-\omega^2)^2 + 4\omega^2}$.

The Nyquist criterion states:

Let $R$ be strictly proper and stable. Suppose that the contour $R(i\omega)$ does not pass through or wind around -1. Then $T$ has all its poles in the open left half plane, so is also strictly proper and stable.

How can I show that $R(i\omega)$ does not cross or wind around -1?

Hint: We were meant to find the real part of $R(i\omega)$ which is: $$\frac{5-\omega^2}{(5-\omega^2)^2 + 4\omega^2}$$

I am not sure if this helps or not.

Thank you!

• Niquist criterion is often shown by plotting. – Arash May 16 '18 at 1:56