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Multiple Choice Questions

Construction of Root Loci

Construction of Root Loci MCQ

1. With reference to root locus, the complex conjugate roots of the characteristic equation of the O.L.T.F. given below

G(s)H(s) =K(s+3)/(s+1)2, lie on

a) Straight line
b) Parabola
c) Circle
d) Semi-circle

Answer: c

Explanation: Complex conjugate roots of the characteristic equation of the O.L.T.F.lie on circle.

2. Determine the centroid of the root locus for the system having G(s)H(s) = K/(s+1)(s2+4s+5)

a) -2.1
b) -1.78
c) -1.66
d) -1.06

Answer: c

Explanation: Roots of the open loop transfer function are -1,-2+j, -2-j then centroid =Σreal part of open loop pole-Σreal part of open loop zeroes/P-Z
Centroid =(-1-2-2)-0/3 =-5/3 =-1.66.

3. The loop transfer function of an LTI system is G(s)H(s) =K(s+1)(s+5)/s(s+2)(s+3). For K>0, the point on the real axis that does not belong to the root locus of the system is

a) -0.5
b) -2.5
c) -3.5
d) -5.5

Answer: c

Explanation: The points present on the root locus are right to the odd number of poles and zeroes.

4. The angles of asymptotes of the root loci of the equation s3+5s2+(K+2)s+K=0 are:

a) 0° and 270°
b) 0° and 180°
c) 90° and 270°
d) 90° and 180°

Answer: c

Explanation: P-Z =2
Angle of asymptote = (2q+1)180°/P-Z
Angle are 90° and 270°.

5. The intersection of asymptotes of root loci of a system with open loop transfer function G(s)H(s) = K/s(s+1)(s+3) is

a) 1.44
b) 1.33
c) -1.44
d) -1.33

Answer: d

Explanation: The intersection of asymptotes of root loci of a system is same as the centroid which is centroid =Σreal part of open loop pole-Σreal part of open loop zeroes/P-Z.
Centroid = -4/3=-1.33.

6. If a feedback control system has its open loop transfer function G(s)H(s) = K/(s-2)(s2+3s+5) has the root locus plot which intersects the imaginary axis at s =0, then the value of K at this point will be

a) -5
b) 10
c) 5
d) -10

Answer: b

Explanation: The intersection point on the imaginary axis at s =0 is obtained by Routh Hurwitz criteria making s^0 row zero and getting the value K = 10.

7. The open loop transfer function of the feedback control system is given by G(s) =K(s+3)/s(s+4)2(s+5)(s+6). The number of asymptotes and the centroid of asymptotes of the root loci of closed loop system is

a) 4 and (-4,0)
b) 3 and (-12,0)
c) -4 and (-4,0)
d) -3 and (-12,0)

Answer: a

Explanation: Number of Poles = 5
Zeroes =1
Asymptotes =P-Z =4
Centroid =Σreal part of open loop pole-Σreal part of open loop zeroes/P-Z
Centroid = -4-4-5-6+3/4 =-4.

8. The characteristic equation of a control system is given as 1+ K(s+4)/s(s+7)(s2+2s+2)=0. The real axis intercept for root locus asymptote is:

a) -2.25
b) -1
c) -1.67
d) 0

Answer: c

Explanation: Real axis intercept =centroid
Zero =-4 and Pole = -7, -1, -1, 0
Centroid =Σreal part of open loop pole-Σreal part of open loop zeroes/P-Z
Centroid = -7-1-1+4/3 = -1.67.

9. The OLTF of a unity feedback system is K(s+2)(s+4)/(s+5)(s+6) the angle of arrival of the root loci as s =-2, and s =-4 respectively are:

a) 0°,180°
b) 180°,0°
c) 90°,180°
d) 180°, 90°

Answer: b

Explanation: As it is type zero system therefore the angle of arrival can be either 180°, 0°.

10. The characteristic equation is s3+14s2+(45+K)s+K =0, centroid is located at (-x,0) then the value of x is ____________

a) 1
b) 2
c) 3
d) 4

Answer: b

Explanation: Differentiating the equation of K with respect to s and equating it to zero.Breakaway points are -2, -2+1.414j,-2-j1.414. so 2 is complex breakaway point.

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