This demonstration shows how a second order pole (complex conjugate roots) expressed as:

$${1 \over {{{\left( {{s \over {{\omega _0}}}} \right)}^2} + 2\zeta \left( {{s \over {{\omega _0}}}} \right) + 1}} = {1 \over {1 - {{\left( {{\omega \over {{\omega _0}}}} \right)}^2} + j2\zeta {\omega \over {{\omega _0}}}}},$$is displayed on a Bode plot. To change the value of ω_{0}, you can either change the value in the text box, below, or drag the vertical line showing ω_{0} on the graphs to the right. Similarly, to change ζ you can use either the text box, or the slider. The exact values of magnitude and phase are shown as black dotted lines and the asymptotic approximations are shown with a thick magenta line. The value of ω_{0} is constrained such that 0.1≤ω_{0}≤10 rad/second, and 0.01≤ζ≤0.99.

Enter value for ω_{o}: , and ζ: , or click on graph to set ω_{0} and use slider, below, for ζ.

ζ: 0.01 0.99

The asymptotic magnitude plot starts (at low frequencies) at 0 dB and stays at that level until it gets to ω_{0} (1 rad/sec). At that point the gain starts dropping with a slope of -40 dB/decade. Note: it is -40 dB per decade because there are two poles in the denominator. df

The asymptotic phase plot starts (at low frequencies) at 0° and stays at that level until it gets to ω_{0}/10^{ζ} (0.1 rad/sec). At that point the phase starts dropping at -90°/decade until it gets to -180° at ω_{0}·10^{ζ} (10 rad/sec), at which point it becomes constant at -180° for high frequencies. Phase goes through -90° at ω=ω_{0}. If ζ<0.02 the phase transition between 0 and -180° can be approximated by a vertical line.

© Copyright 2005 to 2019 Erik Cheever This page may be freely used for educational purposes.

Erik Cheever Department of Engineering Swarthmore College