Contents

This page demonstrates the techniques described previously to take a transfer function defined by the user, identify the constituent terms, draw the individual Bode plots, and then combine them to obtain the overall transfer function of the transfer function. Finally, an example is given of how to use the Bode plot to find the time domain response of the system to a sinusoidal input.

Define transfer function

Define `H(s)=C\frac{N(s)}{D(s)}`





Identify individual terms and convert transfer function to standard form

We start with the transfer function (note: the process here follows that on the second page of the file BodeRules.pdf):

Show numeric transfer function; this gets replaced.

We rewrite it by factoring into real poles & zeros, complex poles & zeros and poles & zeros at the origin.

Show original symbolic transfer function; this gets replaced.
Define all symbols; this gets replaced.

Next we write all the poles and zeros is our standard form.

Show modified symbolic transfer function; this gets replaced.

Rewrite the constant:

Define K; this gets replaced.

So

Transfer function in standard form; this gets replaced.

Now the transfer function is in the form we need to apply our rules to draw the Bode plot.


Draw individual terms

In this section we draw the Bode plots of each of the indivuidual termas enumerated above. Select one of the terms by selecting the corresponding radio button. The selected term will be highlighted on the graphs with a thicker line. An explanation that describes how to draw the asymptotic magnitude and phase plots for the selected term is written out below.


Combine individual terms to get final asymptotic plot

The graphs below show the individual plots (with the same color scheme as above), along with the combined plot (a dotted black line) which is the sum of the individual plots for both magnitude and phase; in other words the asymptotic plot of the complete transfer function. This addition of terms can be complicated, but is conceptually quite easy.


Show Exact Bode Plot (and a time domain example)


Exact plot

On the Bode plot, the dotted lines represent the asymptotic plot, the solid line is the exact solution.
The pink dots show the magnitude and phase of the Bode plot at a frequency chosen by the user (see below).


Time domain (sinusoidal steady state) response from Bode plot


Input sinusoid:
The input to the system is A·cos(ω·t + φ°)




Input = 1.0 · cos(1.0 · t + 0°); replace this.

Output sinusoid
The output is M·A·cos(ω·t +(φ + θ)°).
We get M and θ from the Bode plot.
cos(ω · t + (φ + θ)°); replace this.


References

© Copyright 2005 to 2020 Erik Cheever    This page may be freely used for educational purposes.

Erik Cheever       Department of Engineering         Swarthmore College