Question

1. Given a unity feedback system that has the forward transfer function: Ks(s +10) G(s)= 4s +5 do the following: a) Sketch the root locus. b) Find the imaginary-axis crossing (if relevant). c) Find the breakaway or break-in point (if relevant). d) Find the value of K at the breakaway or break-in point (if relevant). e) Find the angle of departure (if relevant).

Answer 1

G(s) = ks(s + 10)/s^2 + 4s + 5 poles are implies -2 plusminus I, p = 2 zeros are implies 0, -10, z = 2 (a) root locus (b) root locus is crossing the imaginary axis at w = 0 rad/sec at k = infinity \r\n k = -(s^2 + 4s + 5)/s^2 + 10s dk/ds = 0 (s^2 + 10s) (25 + 4) - (s^2 + 4s + 5) (2s + 10) = 0 2s^3 + 4s^2 + 20s^2 + 40s = 2s^3 + 8s^2 + 10s + 10s^2 + 40s + 50 24 s^2 + 40s = 18s^2 + 505 + 50 6s^2 - 10s - 50 = 0 s = 3.837, -2.171 k is positive break in point s = -2.171 k at break in point s = -2.171 = (-2.171)^2 + 4(-2.171) + 5/2.171(10 - 2.171) = 0.0605 angle of departure theta theta/s = -2 + 1 = angle -2 + I angle -2 + I + 10/angle (-2 + i)

Matlab code:

clc;clear all;close all;
k=1;
z=[0,-10];
p=[-2-i,-2+i];
sys=zpk(z,p,k)
rlocus(sys)
grid on
Output: