# Asymptotic Bode: Underdamped Pair of Poles

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 ω0, you can either change the value in the text box, below, or drag the vertical line showing ω0 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 ω0 is constrained such that 0.1≤ω0≤10 rad/second, and 0.01≤ζ≤0.99.

Enter value for ωo: , and ζ: , or click on graph to set ω0 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 ω0 (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 ω0/10ζ (0.1 rad/sec). At that point the phase starts dropping at -90°/decade until it gets to -180° at ω0·10ζ (10  rad/sec), at which point it becomes constant at -180° for high frequencies. Phase goes through -90° at ω=ω0. If ζ<0.02 the phase transition between 0 and -180° can be approximated by a vertical line.

References

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

Erik Cheever       Department of Engineering         Swarthmore College