Rules for Drawing Bode Diagrams


The table below summarizes what to do for each type of term in a Bode Plot.  This is also available as a Word Document or PDF.

The table assumes ω>0. If ω<0, magnitude is unchanged, but phase is altered.

 

Term Magnitude Phase
Constant: K 20log10(|K|) K>0:  0°       
K<0:   ±180°
Pole at Origin

(Integrator) $\frac{1}{s}$

-20 dB/decade passing through 0 dB at ω=1 -90°
Zero at Origin

(Differentiator) $s$

+20 dB/decade passing through 0 dB at ω=1
(Mirror image, around x axis,of Integrator)
+90°
(Mirror image, around x axis, of Integrator about )
Real Pole \[\frac{1}{\frac{s}{\omega_0}+1}\]
  1. Draw low frequency asymptote at 0 dB
  2. Draw high frequency asymptote at -20 dB/decade
  3. Connect lines at ω0.
  1. Draw low frequency asymptote at 0°
  2. Draw high frequency asymptote at -90°
  3. Connect with a straight line from 0.1·ω0 to 10·ω0
Real Zero \[\frac{s}{\omega_0}+1\]
  1. Draw low frequency asymptote at 0 dB
  2. Draw high frequency asymptote at +20 dB/decade
  3. Connect lines at ω0.
(Mirror image, around x-axis, of Real Pole)
  1. Draw low frequency asymptote at 0°
  2. Draw high frequency asymptote at +90°
  3. Connect with a straight line from 0.1·ω0 to 10·ω0
(Mirror image, around x-axis, of Real Pole about 0°)
Underdamped Poles

(Complex conjugate poles)

\[\begin{gathered}
\frac{1}{{{{\left( {\frac{s}{{{\omega _0}}}} \right)}^2} + 2\zeta \left( {\frac{s}{{{\omega _0}}}} \right) + 1}} \\
0 < \zeta < 1 \\
\end{gathered} \]
  1. Draw low frequency asymptote at 0 dB
  2. Draw high frequency asymptote at -40 dB/decade
  3. If ζ<0.5, then draw peak at ω0 with amplitude
        |H(jω0)|=-20·log10(2ζ), else don't draw peak
  4. Connect lines
  1. Draw low frequency asymptote at 0°
  2. Draw high frequency asymptote at -180°
  3. Connect with straight line from \[\omega = \frac{{{\omega _0}}}{{{{10}^\zeta }}}{\text{ to }}{\omega _0} \cdot {10^\zeta }\]

You can also look in a textbook for examples

Underdamped Zeros

(Complex conjugate zeros)

\[\begin{gathered}
{\left( {\frac{s}{{{\omega _0}}}} \right)^2} + 2\zeta \left( {\frac{s}{{{\omega _0}}}} \right) + 1 \\
0 < \zeta < 1 \\
\end{gathered} \]

  1. Draw low frequency asymptote at 0 dB
  2. Draw high frequency asymptote at +40 dB/decade
  3. If ζ<0.5, then draw peak at ω0 with amplitude
         |H(jω0)|=+20·log10(2ζ), else don't draw peak
  4. Connect lines
(Mirror image, around x-axis, of Underdamped Pole)
  1. Draw low frequency asymptote at 0°
  2. Draw high frequency asymptote at +180°
  3. Connect with straight line from \[\omega = \frac{{{\omega _0}}}{{{{10}^\zeta }}}{\text{ to }}{\omega _0} \cdot {10^\zeta }\]

You can also look in a textbook for examples.

(Mirror image, around x-axis, of Underdamped Pole)

For multiple order poles and zeros, simply multiply the slope of the magnitude plot by the order of the pole (or zero) and multiply the high and low frequency asymptotes of the phase by the order of the system.

For example:

Second Order Real Pole \[\frac{1}{{{{\left( {\frac{s}{{{\omega _0}}} + 1} \right)}^2}}}\]
  1. Draw low frequency asymptote at 0 dB
  2. Draw high frequency asymptote at -40 dB/decade
  3. Connect lines at break frequency.

-40 db/dec is used because of order of pole=2.  For a third order pole, asymptote is -60 db/dec

  1. Draw low frequency asymptote at 0°
  2. Draw high frequency asymptote at -180°
  3. Connect with a straight line from 0.1·ω0 to 10·ω0

-180° is used because order of pole=2.  For a third order pole, high frequency asymptote is at -270°.


References

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

Erik Cheever       Department of Engineering         Swarthmore College