determine the locations of all the poles and zeros (including zeros at s = infinite). Make an S-Plane plot of the infinite poles and zeros

Question

Determine the locations of all the poles and zeros (including zeros at $S = \infty$). Make an $S$-Plane plot of the infinite poles and zeros. $$G(s) = \dfrac{5S^2 + 20S + 15}{S(S + 3)(S^2 + 4S + 4)}$$

Answer 1

We can find the non-infinite poles and zeros as follows:

First of all, factor the top and bottom, canceling out any common factors. We have $$ G(s) = \frac{5s^2 + 20s + 15}{s(s+3)(s^2 + 4s + 4)} = \frac{5(s+1)(s+3)}{s(s+3)(s+2)^2} = \frac{5(s+1)}{s(s+2)^2} $$ The zeros are the values of $s$ for which $G(s)=0$. $G(s)$ will be zero whenever our simplified numerator is zero. That is to say $s$ will be a zero if $$ 5(s+1) = 0 $$ The poles are the values of $s$ for which $G(s)$ has an infinite discontinuity. $G(s)$ will be undefined whenever our simplified denominator is zero, so we have to find the zeros of the bottom, which is to say that $s$ will be a pole if $$ s(s+2)^2 = 0 $$ Now, to find the behavior of $G$ at infinity, you need to find $$ \lim_{s\to \infty} G(s) = \lim_{s \to \infty} \frac{5s^2 + 20s + 15}{s(s+3)(s^2 + 4s + 4)} $$ If this limit is $0$, then $G$ has a zero at $\infty$. If this limit is $\pm \infty$, then $G$ has a pole at $\infty$. If this limit is any non-zero constant, then $G$ has neither a zero nor a pole at $\infty$.