Second order system

As discussed before, the transfer function $H(s)$ of a second order system is

$$H(s)=\frac{N(s)}{s^2+2\zeta \omega_n s+\omega_n^2} =\frac{N(s)}{(s-p_1)(s-p_2)}$$

where $N(s)$ can be either $\omega_n^2$, $2\zeta\omega_n s$, or $s^2$, and $p_1$ and $p_2$ are the two roots of the denominator:

$$p_{1,2} = \left\{ \begin{array}{ll} (-\zeta \pm \sqrt{\zeta^... ...ega_n & \mbox{if $\vert\zeta\vert \le 1$} \end{array} \right.$$

Note that when $\vert\zeta\vert<1$, the poles are a complex conjugate pair which can be written in polar form as well as in Cartician form:

$$p_{1,2}=-\omega_n\zeta \pm j\omega_n\sqrt{1-\zeta^2} =\vert p\vert e^{\mp j\angle{p}}$$

where

$$\vert p\vert=\omega_n,\;\;\;\;\mbox{and}\;\;\;\; \angle{p}=tan^{-1}\frac{\sqrt{1-\zeta^2}}{\zeta}=cos^{-1}\zeta$$

How the magnitude of the frequency response $\vert H(j\omega)\vert$ changes as a function of frequency $\omega$ can be qualitatively determined from the pole-zero plot of $H(s)$. For convenience, we define these vectors in the s-plane:

$$u\stackrel{\triangle}{=}j\omega,\;\;\;\;\; v_1\stackrel{\tri... ...{=}j\omega-p_1,\;\;\;\; v_2\stackrel{\triangle}{=}j\omega-p_2$$

• $N(s)=\omega_n^2$, $H(s)$ has no zeros and two poles and
  
  $$\vert H(j\omega)\vert=\frac{\omega_n^2}{\vert j\omega-p_1\ver... ...ga-p_2\vert} =\frac{\omega_n^2}{\vert v_1\vert\vert v_2\vert}$$
  
  
  As $\omega_n^2/(\vert v_1\vert\vert v_2\vert)$ is some constant when $\omega=0$ and approaches 0 when $\omega \rightarrow \infty$, the system is a low-pass filter.

  

• $N(s)=2\zeta\omega_n s$, $H(s)$ has one zero and two poles and
  
  $$\vert H(j\omega)\vert=\frac{2\zeta\omega_n\;\vert j\omega\ver... ...ac{2\zeta\omega_n\;\vert u\vert}{\vert v_1\vert\vert v_2\vert}$$
  
  
  As $\vert H(j\omega)\vert=2\zeta\omega_n \vert u\vert/(\vert v_1\vert\vert v_2\vert) = 0$ when $\omega=0$ or $\omega \rightarrow \infty$ but it is greater than 0 in between, the system is a band-pass filter.

  

• $N(s)=s^2$, $H(s)$ has two repeated zeros and two poles and
  
  $$\vert H(j\omega)\vert=\frac{\vert j\omega\vert^2}{\vert j\ome... ..._2\vert} =\frac{\vert u\vert^2}{\vert v_1\vert\vert v_2\vert}$$
  
  
  As $\vert H(j\omega)\vert=\vert u\vert^2/(\vert v_1\vert\vert v_2\vert)=0$ when $\omega=0$, but it approaches a constant when $\omega \rightarrow \infty$, the system is a high-pass filter.