Root Locus for Distance/Velocity Lag

Cuthbert Nyack
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-bs/(s + a),
C(s)/R(s) = Ke-bs/(s + a + Ke-bs)

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.

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COPYRIGHT 2006 Cuthbert Nyack.