08 - Pole Zero Analysis

The dynamic output \(y(t)\) of the system for changes in the input \(x(t)\) can be written in a polynomial form as: \[\begin{gathered} a_ny^{(n)} + a_{n-1}y^{(n-1)} + \cdots + a_1y^{(1)} + a_oy = b_mx^{(m)} + \\ b_{m-1}x^{(m-1)} + \cdots + b_1x^{(1)} + b_ox \end{gathered}\] where \(y^{(n)}\) is the \(n^{\text{th}}\) derivative of \(y\). With \(n\ge m\), and with zero initial conditions, i.e., \(y^{(n)}=y^{(n-1)}=y=0\) at \(t=0\), we can write the above differential equation using Laplace transform as below: \[\frac{Y(s)}{X(s)} = \frac{b_m s^m + b_{m-1} s^{m-1} + \cdots + b_1 s + b_o}{a_n s^n + a_{n-1} s^{n-1} + \cdots + a_1 s + a_o} = G(s) = \frac{Q(s)}{P(s)}\] This ratio \(\dfrac{Y(s)}{X(s)}=G(s)\) is called as the transfer function of the system.