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 V1 and V2, V1 is said to be greater than V2 by D decibels where D = 20 log(V1/V2). 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 (a2 + b2)/(a2 + b2 - w2 + 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 106 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.