Phase Lag 0T1T2 Transient, Frequency Analysis and Root locus

Cuthbert Nyack

Applets below illustrate the application of the Lag Compensator to a third order Type 1 system. As a third order system, it can become unstable. The increased phase lag of the compensator tends to make the system unstable for small TG.

The aim of this compensator is to reduce the gain in the region of the crossover frequency, therefore reducing the crossover frequency to a frequency where the phase margin is acceptable.

For the parameter values (K, 1.0, 0.5, TG, β, 4.0, 1.0), the uncompensated system is unstable for K > 3.
For the parameter values (3.0, 1.0, 0.5, 20.0, 8.0, 4.0, 1.0), the data(approximate values) on the applet indicates that:-
The uncompensated crossover frequency "ucof" is 1.4147 with a slope of 40.08dB/dec.
The uncompensated phase "upm" and gain "ugm" margins are -0.02° and 0.011dB, effectively zero.

The 3dB frequency of the uncompensated system "3dbu" is 2.082rad/s.

Assuming a phase margin of 50° and gain margin > 15dB is desirable, then passible values of TG and β are (32.1, 6.0), cross over frequency = 0.4467rad/s, slope 24.21dB/dec. Peak closed loop gain = 1.461dB, 3dB freq = 0.803rad/s
(17.8, 7.0), cross over frequency = 0.3950rad/s, slope 23.99dB/dec. Peak closed loop gain = 1.508dB, 3dB freq = 0.715rad/s
(13.5, 8.0), cross over frequency = 0.3554rad/s, slope 23.62dB/dec. Peak closed loop gain = 1.745dB, 3dB freq = 0.643rad/s
(12.0, 9.0), cross over frequency = 0.3234rad/s, slope 23.64dB/dec. Peak closed loop gain = 1.948dB, 3dB freq = 0.583rad/s
(11.3, 10.0), cross over frequency = 0.2968rad/s, slope 23.68dB/dec. Peak closed loop gain = 2.119dB, 3dB freq = 0.531rad/s
Larger values produce better looking responses but can be difficult to implement because of the long time constant of the implementing circuit.

The effect of the Lag compensator is also shown with the Nyquist plot below. The parameters on the plot show the values of the curves where the "x" is located. The position of the "x" is changed by w on the applet.

Parameter values (0.0, 1.0, 0.5, 12.0, 10.0 , 4.0, 2.0) show one of the possible shapes of the root locus. There are 4 poles at 0, -0.0083, -1.0 and -2.0, one zero at -0.083 and therefore 3 asymptotes. The locus which starts at -2 moves to - infinity along the -ve real axis. The locus which starts at 0 meets the one which starts at -0.0083, move off the real axis, recombine, one then moves to the right to end at the zero, while the other moves to the left to meet the one which started at -1.0. They meet at ~ -0.32, move off the real axis and towards the asymptotes.

Parameter values (0.0, 1.0, 0.5, 10.0, 10.0 , 4.0, 2.0) show another possible shape for the locus. Here the locus which starts at -1 ends at the zero at -0.1. The loci which starts at -0.01 and 0.0 are now the ones which move towards the asymptotes.

For the parameter values (3.0, 1.0, 0.5, 20.0, β , 4.0, 1.0), the poles are real for β > ~ 14.0 and complex conjugate for smaller values of β. For the parameter values (0.0, 1.0, 0.5, 12.0, β, 4.0, 3.0) Increasing β from 1.0 which introduces phase lag compensation produces little change in the shape of the locus but the value of Kv at a point on the locus changes. For β = 1.0, Kv = 3.0 the closed loop poles are at ~±j1.32. With β = 10, Kv must be increased to 26.6 for the poles to be at ~±j1.32.

Design by root locus techniques means choosing TG and β to locate the dominant closed loop poles at a specified location.
eg For Kv = 3.0, T1 = 1.0, T2 = 0.5. TG = 20.0, β = 7.0 sets the dominant closed loop poles at ~ -0.33 ± ~ j 0.47 with magnitude ~ 0.58 and angle ~ 55° to the real axis. This system has a crossover frequency of 0.3943rad/s with slope -23.91dB/dec, a 3dB frequency of 0.713rad/s, a phase margin of 51.13°, a gain margin of 16.32dB and a peak closed loop response of 1.376dB.

Gif images below shows how applet should appear.

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