# Poles and zeros in a sum of expression with poles and zeros

Maybe the title of my question is a quite confusing but I am going to explain better. I have an expression like this: $$\frac{Z_1}{Z_2+Z_3+Z_4+Z_5+Z_6},$$ where every parameter $Z_x$ is an expression whit poles and zeros, like $\dfrac{sa+b}{sc+d}$ with $s=j\omega$.

Can I find all the poles and zeros of the first expression without resolve it if I know the poles/zeros of the singles $Z_x$?

Zeros can only occur where the numerator is $0$ or the denominator is $\infty$. And the denominator can only be $\infty$ at a pole of one or more terms in the denominator. You do have to check these points to make sure cancellation does not occur that will result in something non-zero. But that is easier than fully "resolving" the denominator.
Poles can only occur when the numerator is $\infty$ or the denominator is $0$. Unfortunately, identifying where the denominator is $0$ will require "resolving" it. Therefore these potential poles (they too will need checked) are much harder to find.
• If each of the $Z_i$ in the denominator is finite, then so is their sum. So the only way for the entire denominator to have a pole is at a pole of one of the individual $Z_i$. So if you know the poles of the each of the $Z_i$, you also know where the poles of the denominator as a whole have to be. But you do have to be careful of the possibility of poles of two terms canceling out. For example $$\frac {x-1}x - \frac{x^2 - 1}x$$ has no pole, though each term does. – Paul Sinclair Jun 23 '18 at 17:17