Note: this document describes both time (step) response and frequency (Bode) response concepts. If you have not learned about frequency response yet you can either ignore the frequency response information, or you can click on this check box to hide it:
This page lets you explore how the location of poles (or values of ζ and ω_{0}) changes the behavior of a system. This page is best viewed full screen on a larger screen - I've made no effort to accomodate devices with smaller screens, like phones.
Use the sliders and text boxes to change pole locations, or you can use the pole-zero zero diagram (below) to drag the poles with your mouse. You will see the effect of your choices on the step response and Bode plot on the graphs below. You can change the type of damping (over, under, critical) either by using the radio buttons or by changing ζ.
Under the two graphs you will find some explanatory text that describes some of what you see in the graphs.
For the overdamped case:
\[\begin{array}{c} H\left( s \right) = \frac{{\omega _0^2}}{{{s^2} + 2\zeta {\omega _0}s + \omega _0^2}} = \frac{{\omega _0^2}}{{{s^2} + 2\zeta {\omega _0}s + \omega _0^2}} = \frac{{{\alpha _1}{\alpha _2}}}{{\left( {s + {\alpha _1}} \right)\left( {s + {\alpha _1}} \right)}}\\ {\rm{distinct}}\;{\rm{real}}\;{\rm{poles}}\;{\rm{at}}\;s = - {\alpha _{1,2}} = - \zeta {\omega _0} \pm {\omega _0}\sqrt {{\zeta ^2} - 1} \\ {y_\gamma }\left( t \right) = 1 - \frac{{{\alpha _2}{e^{ - {\alpha _1}t}} - {\alpha _1}{e^{ - {\alpha _2}t}}}}{{{\alpha _2} - {\alpha _1}}} \end{array}\]For the critically damped case
\[\begin{array}{c} H\left( s \right) = {\left. {\frac{{\omega _0^2}}{{{s^2} + 2\zeta {\omega _0}s + \omega _0^2}}} \right|_{\zeta = 1}} = \frac{{\omega _0^2}}{{{{\left( {s + {\omega _0}} \right)}^2}}} = \frac{{{\alpha ^2}}}{{{{\left( {s + \alpha } \right)}^2}}}\\ {\rm{double}}\;{\rm{pole}}\;{\rm{at}}\;s = - \alpha = - {\omega _0}\\ {y_\gamma }\left( t \right) = 1 - {e^{ - \alpha t}}\left( {1 + \alpha t} \right) \end{array}\]For the underdamped case:
\begin{gathered}© Copyright 2005 Erik Cheever This page may be freely used for educational purposes.
Erik Cheever Department of Engineering Swarthmore College