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Next: Root locus of Second Up: Evaluation of Fourier Transform Previous: First order system

Second order system

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

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

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:

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

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 Cartesian form:

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

where

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

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:

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


next up previous
Next: Root locus of Second Up: Evaluation of Fourier Transform Previous: First order system
Ruye Wang 2012-01-28