Question

Q3. Consider the feedback system in Figure 3. In the case when 2L G(s) Figure 3: Block diagranm G(s) and when k is positive: (a) Sketch the root locus of the closed loop system (10) To assist in this (to indicate on root locus diagram) ) Compute the open loop poles and zeros (i) ealeulate the portion of the locus lying on the real axis; (iii) calculate the angles of asymptotes make with the real axis arad also the value of centroid; (iv) caleulate the point at which the loeus beeaks away from the real axis (v) caleulate the angle of departure of the locus from the complex poles; 5) (b) Find the gain at which the elosed-loop poles have a damping ratio of 0.7071 Hint: Use step of 0.1+0.1j, and for relining an interval use a step of 0.02 +0.02j 40]

Answer 1

T (s) = G (s) k = k (s^2 + 4)/s (s + 1) (s^2 + 16) open loop pole (p) s (s + 1) (s^2 + 16) = 0 So, s = 0, s = -1, s = plusminus I 4 open loop zeros (z) s^2 + 4 = 0, Rightarrow s = squareroot - 4 s = + j 2 Rightarrow only those port lies in root on real axis which contains odd number of pole and zero right side to it Angle of asymptotes = (2 q + 1) times 180/12 z, where q = p - z - 1 = 4 - 2 - 1 = 1 I . e, q = 0, 1 p = 4, z = 2 Rightarrow