**Up To Date Control Systems ****MCQs – ****The Root Locus Technique MCQs ( Control Systems ) MCQs**

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**Control Systems MCQs – The Root Locus Technique MCQs ( Control Systems ) MCQs**

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# The Root Locus Concepts

**1. Which one of the following statements is not correct?**

a) Root loci can be used for analyzing stability and transient performance

b) Root loci provide insight into system stability and performance

c) Shape of the root locus gives idea of type of controller needed to meet design specification**d) Root locus can be used to handle more than one variable at a time****Answer: d****Explanation: For more than one variable state space is used.****2. Root locus of s(s+2)+K(s+4) =0 is a circle. What are the coordinates of the center of this circle?**

a) -2,0

b) -3,0**c) -4,0**

d) -5,0**Answer: c****Explanation: s(s+2)+K(s+4) =0****1+K(s+4)/s(s+2) =0.****G(s)H(s) =K(s+b)/s(s+a)****Centre =(-b,0) =(-4,0).****3. The main objective of drawing root locus plot is :**

a) To obtain a clear picture about the open loop poles and zeroes of the system

b) To obtain a clear picture about the transient response of feedback system for various values of open loop gain K

c) To determine sufficient condition for the value of ‘K’ that will make the feedback system unstable**d) Both b and c****Answer: d****Explanation: The main objective of drawing root locus plot is to obtain a clear picture about the transient response of feedback system for various values of open loop gain K and to determine sufficient condition for the value of ‘K’ that will make the feedback system unstable.****4. While increasing the value of gain K, the system becomes****a) Less stable**

b) More stable

c) Unstable

d) Absolute stable**Answer: a****Explanation: Damping factor is inversely proportional to gain on increasing gain it reduces hence makes the system less stable.****5. The addition of open loop poles pulls the root locus towards:**

a) The right and system becomes unstable

b) Imaginary axis and system becomes marginally stable

c) The left and system becomes unstable**d) The right and system becomes unstable****Answer: d****Explanation: The addition of open loop poles pulls the root locus towards the right and system becomes unstable.****6. Root locus is used to calculate:**

a) Marginal stability

b) Absolute stability

c) Conditional stability**d) Relative stability****Answer: d****Explanation: Root locus is used to calculate relative stability.****7. Routh Hurwitz criterion is better than root locus.**

a) True**b) False****Answer: b****Explanation: Root locus is better as it require less computation process.****8. Consider the following statements regarding root loci:**

a) All root loci start from the respective poles of G(s) H(s)**b) All root loci end at the respective zeros of G(s) H(s) or go to infinity**

c) The root loci are symmetrical about the imaginary axis of the s-plane

d) All root loci start and end from the respective poles of G(s) H(s) or go to infinity**Answer: b****Explanation: All the root locus start at respective poles and end at zeroes.****9. Number of roots of characteristic equation is equal to the number of ______________****a) Branches**

b) Root

c) Stem

d) Poles**Answer: a****Explanation: Number of roots of characteristic equation is equal to the number of branches.****10. Which of the following statements are correct?**

a) Root locus is for the negative feedback systems

b) Complementary root locus is for the positive feedback systems**c) Root locus is for the negative feedback and Complementary root locus is for the positive feedback systems**

d) Complementary root locus is for the negative feedback systems**Answer: c****Explanation: Root locus and Complementary root locus are complementary to each other.**

**The Root Locus**

**1. Consider the loop transfer function K(s+6)/(s+3)(s+5) In the root locus diagram the centroid will be located at:**

a) -4

b) -1**c) -2**

d) -3**Answer: c****Explanation: Centroid =Sum of real part of open loop pole-sum of real part of open loop zeros/P-Z.****2. Which one of the following applications software’s is used to obtain an accurate root locus for?**

a) LISP**b) MATLAB**

c) dBase

d) Oracle**Answer: b****Explanation: MATLAB stands for mathematics laboratory in which the input is in the form of the matrix and is the best software for drawing root locus.****3. Which one of the following is not the property of root loci?****a) The root locus is symmetrical about imaginary axis**

b) They start from the open loop poles and terminate at the open loop zeroes

c) The breakaway points are determined from dK/ds = 0

d) Segments of the real axis are the part of the root locus if and only is the total number of real poles and zeroes to their right is odd.**Answer: a****Explanation: The root locus is the locus traced due to the gain of the system with changing frequency and need not be symmetrical about origin.****4. The breakaway point calculated mathematically must always lie on the root locus.****a) True**

b) False**Answer: a****Explanation: The breakaway point of the two branches of the root locus is the point where the two branches either meet or they break and may or may not always lie on the root locus.**

Stability And Algebraic Criteria MCQs

**5. What is the number of the root locus segments which do not terminate on zeroes?**

a) The number of poles

b) The number of zeroes**c) The difference between the number of poles and zeroes**

d) The sum of the number of poles and the number of the zeroes**Answer: c****Explanation: The number of the root locus segments which do not lie on the root locus is the difference between the number of the poles and zeroes.****6. Which one of the following are correct?****The root locus is the path of the roots of the characteristic equation traced out in the s-plane?**

a) As the input of the system is changed

b) As the output of the system is changed**c) As a system parameter is changed**

d) As the sensitivity is changed**Answer: c****Explanation: The root locus is the locus of the change of the system parameters of the characteristic equation traced out in the s-plane.****7. If the gain of the system is reduced to a zero value, the roots of the system in the s-plane,**

a) Coincide with zero

b) Move away from zero

c) Move away from poles**d) Coincide with the poles****Answer: d****Explanation: The roots of the system in s plane coincides with the poles if the gain of the system is reduced to a value zero.****8. The addition of open loop zero pulls the root loci towards:****a) The left and therefore system becomes more stable**

b) The right and therefore system becomes unstable

c) Imaginary axis and therefore system becomes marginally stable

d) The left and therefore system becomes unstable**Answer: a****Explanation: The system can become stable by reducing the damping and also by adding zeroes in the s plane and moving left of the s plane system becomes more stable.****9. If root loci plots of a particular control system do not intersect the imaginary axis at any point, then the gain margin of the system will be:**

a) 0

b) 0.707

c) 1**d) Infinite****Answer: d****Explanation: The gain margin is the inverse of the intersection of the root loci plot to the imaginary axis and if it does not intersect then the gain margin will be infinite.****10. When the number of poles is equal to the number of zeroes, how many branches of root locus tends towards infinity?**

a) 1

b) 2**c) 0**

d) Equal to number of zeroes**Answer: c****Explanation: Branches of the root locus is equal to the number of poles or zeroes which ever is greater and tends toward infinity when poles or zeroes are unequal.**

**Construction of Root Loci**

**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 bea) -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.**