This Mathlet illustrates the relationship between the amplitude response as a function of frequency and the pole diagram of the transfer function, using as an example a system, like the spring/mass/dashpot system driven
by through the spring, that is controlled by the differential equation shown at the top.

Grab the [k] slider or the [b] slider to set these system parameters.

Grab the [ω] slider to set the circular frequency of the input signal cos(ωt).

The graph at upper right shows the amplitude of the sinusoidal response as a function of the input circular frequency. A yellow strut marks the chosen input frequency.

The window at lower right shows a part of the complex plane. The real axis is green, the imaginary axis yellow.
Red diamonds mark the roots of the characteristic polynomial, which in this configuration are the poles of the transfer function. Yellow diamonds mark the complex numbers ±iω.

The window at left shows the graph of the modulus of the transfer function, in shades of blue-green. Grey lines edge the same portion of the complex plane as is shown in the lower right window, and the real and imaginary axes are shown in green and yellow. Rectangles above these axes are bordered in the same color. The poles of the transfer function are marked by red diamonds and surmounted by vertical red arrows. The intersection of the plane above the imaginary axis with the graph of the modulus of the transfer function is marked by a yellow curve. Yellow diamonds mark the complex numbers ±iω, and yellow struts connect them to the graph of the transfer function.

The three-dimensional diagram can be rotated by dragging the cursor across that window. Buttons below the graph set the view to a standard top view or side view.

© 2006 H. Miller and J.-M. Claus