Root Locus Analysis (3). Hany Ferdinando Dept. of Electrical Eng. Petra Christian University. General Overview. This section explain how to use the Root Locus method to design compensator The compensator is lead and lag compensator Students need some basic geometry for precision drawing
Dept. of Electrical Eng.
Petra Christian University
If R1C1 > R2C2 then it is lag network, otherwise it is lead network
s = -1 ± j√3
Based on the closed-loop poles, z = 0.5, wn = 2 rad/s and Static velocity error (Kv) = 2s-1
It is desired to modify the closed-loop poles so that the wn = 4 rad/s without changing the damping ratio z
For the wn = 4 rad/s without changing the damping ratio z, the desired closed-loop poles are inline with the line between the origin and the closed-loop poles of the system
s = -2 ± j2√3
If G(s) is the open-loop transfer function, then the open-loop of compensated system is
How to find a and T????
Desired closed-loop pole
PB is the bisector between PA and PO
f is the desired angle
at s = -2+j√3 is -210o
Therefore the f is 30o
From the previous slide method, the point C and D is approximately -5.4 and -2.9
You can try it by yourself!!
From the previous slide, you can calculate the T, aT and all Rs and Cs until you get:
where K = 4Kc
K = 18.7916
Compensated open-loop transfer function is
with Gc(s) is
Find the value for all Rs and Cs in the circuit
s = -0.3307 ± j0.5864
Based on the closed-loop poles, z = 0.491, wn = 0.673 rad/s and Static velocity error (Kv) = 0.53s-1
It is desired to increase the Kv to about 5s-1 without appreciably changing the location of the dominant closed-loop poles
The new Kv is about 10 times of the old Kv, therefore the b (a in the Lead Compensator) is set to 10.
The zero of the compensator is also 10 times of the pole.
K = 1.06Kc
The Root Locus plot of the compensated system is very close to the uncompensated one. Therefore, we need Matlab to help us.
The new dominant closed-loop pole with the same z is -0.31±j0.55 (from Matlab). The gain K is 1.0235
To find the desired dominant complex pole in the Root Locus plot, one can use rlocfind(num,den) function. Output of this function is the selected pole and the gain at that pole.
When this function is called, there is a pointer one can use to select a pole in the complex plane. Usually, the Root Locus is drawn first.
Bode diagram as a tool for frequency response analysis is the next topic.
Prepare yourself for this…