**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)(s ^{2}+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 s ^{3}+5s^{2}+(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)(s ^{2}+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)(s ^{2}+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 s ^{3}+14s^{2}+(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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