## 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.

Be patient it can take several seconds to load, and several seconds each time a change is made.

## 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 thicker gray 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.

• For magnitude, start at the left side of the graph. All of terms will be equal to 0 with the exception of the constant term and any poles and zeros at the origin. This gives us a place to start drawing the sum. From here on we simply pay attention to the slope. If there are no poles and zeros at the origin, the slope is zero, else it is determined by those poles and zeros at the origin. We continue with this slope until we get to the first finite pole or zero at which point the slope increases or decreases by a multiple of 20 dB/decade. For example a 1st order pole decreases the slope by 20 dB/dec, a second order pole by 40.... We keep moving to the right, increasing or decreasing the slope as we come to zeros or poles.
• We do the same for phase. At the left side only poles and zeros at the origin, and the sign of the constant contribute to the initial phase. As we move to the right the slope will increase (for a zero), or decrease (for a pole).

## Show Exact Bode Plot (and a time domain example)

#### Exact plot

On the Bode plot, the gray lines represent the asymptotic plot, adn the black 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:
(Note: it can take several seconds to refresh after changing variables.)
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

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