Poles and Zeros

Theory

The poles and zeros are the locations where a transfer function becomes infinity and zero respectively. Poles and zeros define the characteristics and stability of a transfer function. Namely, like there is shown in stability, if a pole or zero is located in the left half plane it is called stable. Being stable means that the output converges to zero, or to a steady state.

Poles

The poles can be found by solving for the roots in the transfer function's denominator. If the denominator consists out of a polynomial with degree n, then there are also n poles present.

If we have for example a transfer function $H(s)$, then the poles are found by solving for $D(s) = 0$.

Let's solve for $D(s) = 0$ in the case $D(s)$ is of the following form;

There can be seen that the order of s equals three and there are thus 3 poles. In this case, it is rather simple to find the poles, which are in this case simply;

Now by evaluating whether the poles are in the left -or right-half plane, conclusions about stability can be made. Clearly, there is only one pole in the LHP, this pole being $s_2 = -5$. While the others are in the RHP. This means that this system has two non-stable poles, which results in the fact that the system will not be stable either if we take for the zeros $N(s)=1$.

Zeros

The approach taken for the poles can exactly be executed for the zeros as well, however now we only need to set $N(s) = 0$, instead of $D(s)$. If we choose for $N(s)$ and solve for the roots, we find the following results;

Clearly, both zeros are in the LHP, which implies they are both stable.

Complex poles/zeros

Since in the Laplace domain the variable $s = \sigma + j \omega$, there can also be complex poles and zeros. Nothing drastically changes as it is only important where the real part of the pole or zero is located. An example of a complex pole can be given by;

By evaluating the real part of the poles computed above, there can be seen that no matter what, the real part is in the LHP, and therefore these poles are stable. Important to note is that complex poles always come in pairs of two, both a positive and negative complex part.

(Strictly) Proper

In order to have a physical system. The degree of the denominator should always be larger than the degree of the numerator. So there always should be more poles than zeros. This is because of the fact that a transfer function with more poles than zeros will always converge to zero if the frequencies move toward infinity. This behavior is also true for any physical system. If the degree of the numerator is equal to the degree of the denominator then the tf is called proper. However, when $deg(N(s)) < deg(D(s))$ the tf is called strictly proper, which is the case for all physical systems.

Now there is shown how to compute the poles and zeros and with the knowledge of what stability means, there is explained in the next articles how to graph the behavior of a transfer function in a Bode Plot. But also, how the poles and zeros can be found back in the behavior of the Bode plot.