# Bode Plots

Cuthbert Nyack

In early experiments on hearing, it was found that the human ear was
logarithmically sensitive to sound pressure levels. This led to
the concept of the bel and decibel. For voltages 2 signals
V_{1} and V_{2}, V_{1} is said to be greater
than V_{2} by D decibels where
D = 20 log(V_{1}/V_{2}).
The bode plot represents the transfer function of a circuit or system
by a log(amplitude or power) vs log(frequency) and a phase vs
log(frequency) plot. Some of these are illustrated in the applets below.

Applet above shows a Bode plot of a system with transfer function of
the form a/(jw + a). The phase plot is in green.
Vertical scale for phase is - p/2 to +
p/2. With hgain = 1 horizontal scale is in
angular frequency and 0.01 to 100 covering 4 decades. The orange line
shows the log log Bode plot. Vertical log axis is from -60dB to +20dB.
Red line is linear plot of the same function using the same horizontal
log scale but a vertical linear scale(0 to 4/3). Notice that the log log
plot can be closely approximated by 2 straight lines.
One line at low frequencies is flat with zero slope while the other at
high frequencies has slope of 20dB per decade. The transition point
is at w = a.
The Bode plots
also make it easier to combine the responses of systems in series and
for analysing the stability of systems.

The above applet is for a system with transfer function
(a^{2} + b^{2})/(a^{2} + b^{2} -
w^{2} + 2 j a w)
This is the transfer function of a resonant circuit. Plots are similar to
the above but now the slope of the high frequency line is 40dB per decade
and the resonant behaviour gives rise to the characteristic peak in the
response. The vertical scale for phase is from
- p to + p.

This third applet is for a transfer function (a b c)/((j
w + a)( jw + b)(
jw + c)). Vertical phase scale is from
- 3p/2 to + p/2.
Vertical log scale is 60dB per decade with vgain = 1. Horizontal log
scale is from 10^{-2} to 10^{6} rad s^{-1}.
Note the phase drops by p/2 and the slope
decreases by 20dB per decade at w = a,
w = b and at w
= c.

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COPYRIGHT © 1996 Cuthbert A. Nyack.