Bode and Nyquist Plots

Engineers commonly employ Bode and Nyquist plots to visualize the response of an LTI system to sinusoidal input signals of various frequencies. The Bode plots graph the logarithm of the gain and the phase lag against the logarithm of the frequency. The Nyquist plot consists of the trajectory of G(iω) , as ω runs over a range of frequencies, where G(s) is the transfer function of the system.

This applet illustrates these plots for transfer functions of the form A e^-sT r(s) where r(s) is a rational function.

In the main window, a portion of the complex plane is displayed at left. On it zeros of G(s) are marked with blue o's and poles are marked with green x's. A point s = iω on the imaginary axis is marked with a yellow diamond. Segments connect this point to the zeros and poles, and horizontal grey lines extend from the poles and zeros in the positive real direction. These are to be used in measuring the argument of the difference between s and the zero or pole. This angle is indicated by a grey directed arc.

The various objects in this graph can be moved by depressing the mousekey with the cursor nearby, and dragging. Rolling the cursor over this window results in a readout of the complex number. The scale of the window is adjusted using a log zoom slider below it.

Check boxes toggle the ability to create new real poles, complex conjugate poles, real zeros, or complex conjugate pairs of zeros, or to annihilate them. When one of the create boxes is checked, the corresponding object may be placed and created by a mousekey click at the desired location on the complex plane. When one of the annihilate boxes is checked, a mousekey click will annihilate the nearest pole or zero.

Sliders to the right of the complex plane control the value of the delay constant T and the amplitude A .

A checkbox at bottom toggles the display of the formula for the transfer function G(s) .

At upper right, boxes create or foreground spawned windows containing Nyquist or Bode plots.

The Nyquist diagram window displays the trajectory of G(iω). The unit circle is marked in red. The value of ω selected by the yellow diamond on the complex plane in the main window is indicated here by a yellow diamond and strut from the origin. The direction of positive ω is indicated by an arrowhead on the trajectory. Rolling the cursor over this window results in a red strut joining the origin to the cursor point and a readout of the polar coordinates of the point. Depressing the mouse key suppresses this readout. A slider at the bottom of this window allows one to zoom in or out.

The Bode diagram window displays the graphs of log|G(iω)| and Arg(G(iω)) as functions of log(ω) . The current position of iω is indicated by a yellow diamond on each graph.