The Applet below show the root locus plot of a first order type 0 system with and without distance-velocity or transport lag. The open and closed loop transfer function of the system is:-

G(s) = Ke-

C(s)/R(s) = Ke-

Because of the exponent in the numerator, the root locus has several branches although only the primary branch(angle condition = 180°) is relevant here. The primary branch has one pole at -a, the root locus starts at -a and -∞ and end at +∞ ±jp. The parameters (0.0, 1.0, 0.33) show the primary branch. Parameters (0.0, 1.0, 1.0) show the first 2 branches.

With the resolution of the plot the eg Parameters (2.3, 1.0, 1.0) show the step response of the system with D/V lag being delayed by "b" and oscillatory(pink plot) with angular frequency = ~ 2.03rad/s.

The plot shows the closed loop poles sit on the real axis. It also shows that the closed loop poles of the higher branches reach the imaginary s axis for larger values of K.

Gif image below shows how applet should appear.

COPYRIGHT © 2006 Cuthbert Nyack.