# Z Transform Transfer
Functions 2 CC Poles Frequency Response

Cuthbert Nyack

The Z transform of a 2 pole system is shown below:-
The frequency response is obtained by substituting
e^{jwT}
= e^{j2pw/ws}
for z.

The applet below shows the frequency response of a 2 pole system.
The gray curve above the magenta lines show the response between 0 and
½ w_{s} while the
gray curve above the green lines
show the response between ½ w_{s}
and w_{s}. Orange lines show
the phase variation. White disc shows the unit circle, black line
is the real z axis and orange line is the imaginary z axis. Poles are
shown by the "x"s. When the "Theta" parameter is
zero the view is along the imaginary axis from the top. -90° corresponds
to a view along the real z axis from the left and +90° represents
a view along the real z axis from the right.
Note that having a pole near the unit circle produces a peak in the
response. As the angle varies from 0 to 180º the peak response
moves from 0 to half of the sampling frequency. i.e. response changes
from a lowpass to a bandpass to a highpass one.

eg parameters (0.99, 60.0, NA, NA, 62, 0, 10.0, 1.5, 1, 0.1)

show a sharp resonant peak of ~44.679 at ~ 0.165w_{s} = (60/360)w_{s}. At this frequency the phase
goes through zero.

eg parameters (0.8, 60.0, NA, NA, 62, 0, 10.0, 1.5, 1, 1.0)

show a reduced Q resonant peak of ~3.2063 at ~ 0.165w_{s} = (60/360)w_{s}. The phase now
goes through zero at 0.185w_{s}.

*Return to main page*

*Return to page index*

Copyright 2000 © Cuthbert A. Nyack.