There are two ways to draw a Bode plot. One calculates the magnitude and phase of the system transfer function at each frequency and plots the graph using 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. Due to their simplicity, these can be drawn by hand. This article focuses on the asymptotic Bode plot.
A transfer function is expressed in terms of 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.
The rules for drawing the magnitude plot are as follows:
The chart starts with a horizontal line at a magnitude equal to the system’s DC magnitude (H(0)=A).
For each pole, the slope of the line at that pole’s frequency decreases by 20 dB/frequency decade
For each null, the slope of the line increases by 20 dB/frequency decade at that null’s frequency
For each pole or zero at zero frequency, the plot begins with that pole/zero’s effect 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 a frequency of 1 rad/s the magnitude will be equal to the DC magnitude A got to. Then the trace of the graph must cross A at 1 rad/s and trace back to the start frequency of the graph.
If there are multiple zeros or poles at the same frequency, the slope of the line changes according to that number
The rules for drawing the phase diagram are as follows:
Start the graph with a horizontal line at 0º phase if gain is positive, or -180º if gain is negative (negative gain corresponds to 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 it by the same amount one decade after the pole/zero. After two decades, the phase for each pole/zero point is -90º than before.
For each positive pole or negative zero, increase the slope by 45º/decade one decade before the pole/zero and decrease it by the same amount one decade after the pole/zero. After two decades, the phase for each pole/zero is +90º than before.
For each pole or null at zero frequency, the plot begins with the effect of that pole/zero on the phase. This means that poles subtract 90º from the initial phase and zeros add 90º to the initial phase.
For detailed drawings, the case of the complex roots, and an online Bode plot generator, see Bode plot in OnMyPhd.
Thanks to Hugo R Goncalves | #Draw #Bode #Plot