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