**Time Response of Second Order Systems – II**

**1. What will be the nature of time response if the roots of the characteristic equation are located on the s-plane imaginary axis?**

a) Oscillations

b) Damped oscillations

c) No oscillations

d) Under damped oscillations

**Answer:** c

**Explanation:** complex conjugate (non-multiple): oscillatory (sustained oscillations)

Complex conjugate (multiple): unstable (growing oscillations).

**2. Consider a system with transfer function G(s) = s+6/Ks ^{2}+s+6. Its damping ratio will be 0.5 when the values of k is:**

a) 2/6

b) 3

c) 1/6

d) 6

**Answer:** c

**Explanation:** s+6/K[s^{2}+s/K+6/K] Comparing with s^{2}+2Gw+w^{2}

w= √6/K

2Gw=1/K

2*0.5*√6/K =1/K

K=1/6.

**3. The output in response to a unit step input for a particular continuous control system is c(t)= 1-e ^{-t}. What is the delay time Td?**

a) 0.36

b) 0.18

c) 0.693

d) 0.289

**Answer:** c

**Explanation:** The output is given as a function of time. The final value of the output is lim_{n->∞}c(t)=1; . Hence Td (at 50% of the final value) is the solution of 0.5=1-e^{-Td}, and is equal to ln 2 or 0.693 sec.

**4. Which one of the following is the most likely reason for large overshoot in a control system?**

a) High gain in a system

b) Presence of dead time delay in a system

c) High positive correcting torque

d) High retarding torque

**Answer:** c

**Explanation:** Large overshoot refers to the maximum peak in the response of the closed loop system and this is mainly due to the high positive correcting torque.

**5. For the system 2/s+1, the approximate time taken for a step response to reach 98% of its final value is:**

a) 1s

b) 2s

c) 4s

d) 8s

**Answer:** c

**Explanation:** C(s)/R(s) = 2/s+1

R(s) = 1/s (step input)

C(s) = 2/s(s+1)

c(t) = 2[1-e^{-t}] 1.96 = 2[1-e^{-T}] T= 4sec.

**6. The unit step response of a second order system is = 1-e ^{-5t}-5te^{-5t} . Consider the following statements:**

**1. The under damped natural frequency is 5 rad/s.**

**2. The damping ratio is 1.**

**3. The impulse response is 25te**

^{-5t}.**Which of the statements given above are correct?**

a) Only 1 and 2

b) Only 2 and 3

c) Only 1 and 3

d) 1,2 and 3

**Answer:** d

**Explanation:** C(s) = 1/s-1/s+5-5/(s+5)^2

C(s) = 25/s(s^{2}+10s+25)

R(s) = 1/s

G(s) = 25/(s^{2}+10s+25 )

w= √25

w = 5 rad/sec

G = 1.

**7. The loop transfer function of controller Gc(s) is :**

a) 1+0.1s/s

b) -1+0.1s/s

c) –s/s+1

d) s/s+1

**Answer:** a

**Explanation:** The transfer function of the controller is 0.1s+1/s

Gc(s) = 0.1s+1/s.

**8. The peak percentage overshoot of the closed loop system is :**

a) 5.0%

b) 10.0%

c) 16.3%

d) 1.63%

**Answer:** c

**Explanation:** C(s)/R(s) = 1/s^{2}+s+1

C(s)/R(s) = w/ws^{2}+2Gws+w^{2}

Compare both the equations,

w = 1 rad/sec

2Gw = 1

Mp = 16.3 %

**9. Consider a second order all-pole transfer function model, if the desired settling time(5%) is 0.60 sec and the desired damping ratio 0.707, where should the poles be located in s-plane?**

a) -5+j4√2

b) -5+j5

c) -4+j5√2

d) -4+j7

**Answer:** b

**Explanation:** G = 1/√2

Gw = 5

s = -5+j5.

**10. Which of the following quantities give a measure of the transient characteristics of a control system, when subjected to unit step excitation.**

**1. Maximum overshoot**

**2. Maximum undershoot**

**3. Overall gain**

**4. Delay time**

**5. Rise time**

**6. Fall time**

a) 1,3 and 5

b) 2, 4 and 5

c) 2,4 and 6

d) 1,4 and 5

**Answer:** d

**Explanation:** Maximum overshoot, rise time and delay time are the major factor of the transient behaviour of the system and determines the transient characteristics.