How To Draw a Bode Plot

There are two ways to draw a bode plot. One is calculating the magnitude and phase of the system transfer function at each frequency and drawing the plot with those points. The other, called asymptotic bode plot, considers straight lines between poles or zeros and has some simple rules for the slopes of those lines. Given its simplicity, these can be drawn by hand. This article focuses on the asymptotic bode plot.

A transfer function is expressed by a DC gain, roots in the numerator (zeros) and roots in the denominator (poles):

H(s) = DC gain * (s/z1 + 1) *… * (s/zn + 1) / ((s/p1 + 1) *… * (s/pm + 1))

with zeros from z1 to zn and poles from p1 to pm. We can have negative and positive poles and zeros.

Magnitude plot

The rules for drawing the magnitude plot are the following:

• The plot starts with a horizontal line at a magnitude equal to the DC magnitude of the system (H(0) = A)
• For every pole, the slope of the line decreases by 20 dB/decade of frequency at that pole’s frequency
• For every zero, the slope of the line increases by 20 dB/decade of frequency at that zero’s frequency
• For every pole or zero at zero frequency, the plot starts with effect of that pole/zero on the slope. Poles and zeros at zero frequency are represented as
  Zero: H(s) = s

  Pole: H(s) = 1/s

  which means that at frequency 1 rad/s, the magnitude must be equal to the DC magnitude A. Then, the trace of the plot must cross A at 1 rad/s and be backtracked to the starting frequency of the plot.

• For multiple zeros or poles at the same frequency, the slope of the line changes according to that number

Phase plot

The rules for drawing the phase plot are the following:

• Start the plot with a horizontal line at 0º phase if the gain is positive or -180º if the gain is negative (negative gain corresponds to a 180º phase between input and output)
• For each negative pole or positive zero, decrease the slope by 45º/decade one decade before the pole/zero, and increase by the same amount one decade after the pole/zero. After two decades, the phase will be -90º than before for each pole/zero.
• For each positive pole or negative zero, increase the slope by 45º/decade one decade before the pole/zero, and decrease by the same amount one decade after the pole/zero. After two decades, the phase will be +90º than before for each pole/zero.
• For each pole or zero at zero frequency, the plot starts with effect of that pole/zero on the phase. That means that poles subtract 90º to the initial phase and zeros add 90º to the initial phase.

For detailed drawings, the complex roots case and an online bode plot generator, visit bode plot in OnMyPhd.