Root Locus for 3 Real Poles and 1 Real Zero
Cuthbert Nyack
Applet shows the root locus for 3 real poles and 1 real zero. The presence
of the zero means that one of the locus ends on the zero and there
are now 2 asymptotes at ±90°. The asympotes appear to
intersect the real axis at point
p = (sum of poles - sum of zeros)/(number of poles - number of zeros).
Parameters (0.0, 1.0, 0.75, 0.25, 10.0, 1.0) show that when the zero is far to the left, then the locus around the origin appears similar to that of 3 poles with asymptotes at ±90° appearing to
intersect the real axis at 4.0. The locus crosses the imaginary axis and the system is unstable for K > 2.8.
Parameters (0.0, 1.0, 0.75, 0.25, 2.0, 1.0) show that the asymptotes
now correspond to the imaginary axis and the system is always stable.
Parameters (0.0, 1.0, 0.75, 0.25, 1.0, 1.0) show that the zero
cancels the pole at -1 and the locus now has the characteristics
of a 2 pole system.
Parameters (8.0, 1.0, 0.75, 0.25, 1.5, 1.0) show that the step response
has transient oscillations at w = ~2.4
and the frequency response is peaked at that frequency. Increasing
the gain increases the transient oscillation frequency but it
does not go away.
Root locus for different pole zero locations are illustrated by the
following paramaters:-
(0.0, 3.0, 2.0, 1.0, 3.5, 1.0) z-p-p-p
(0.0, 3.0, 2.0, 1.0, 2.5, 1.0) p-z-p-p
(0.0, 3.0, 2.0, 1.0, 1.5, 1.0) p-p-z-p
(0.0, 3.0, 2.0, 1.0, 0.5, 1.0) p-p-p-z
Gif image below show how applet should appear.
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COPYRIGHT © 2006 Cuthbert Nyack.