# MCQs on Control Systems

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01․ Find the relation of damped natural frequency (ωd)
n) √(1-2ζ²)
n) √(1-4ζ²)
n) √(1-1/ζ²)
n) √(1-ζ²)

Damped natural frequency (ωd) = (ωn) √(1-ζ²)

02․ A second order control system has F(jω) = 100/(100 - ω² + 10√2 jω). Find the resonant peak?
0.707
0.5
1
1.5

Resonant peak is the maximum value of magnitude occurring at resonant frequency ωr. Resonant peak µr = 1/(2ζ√(1-ζ²)) Given transfer function F(jω) = 100/(100 - ω² + 10√2 jω) Compare to standard second order equation, Damping ratio ξ = 1/√2 Resonant peak = 1

03․ For a second order under damped system subjected to unit step input the time response has first peak value to be 4 times the second over shoot. Determine the damping ratio ξ ?
0.087
0.87
0.707
0.07

First peak value = 1 + Mp1 Second peak overshoot = Mp2 Where, Mp = Peak over shoot = e-nξπ/√(1-ζ²) Mp1 = e-ξπ/√(1-ζ²) Mp2 = e-3ξπ/√(1-ζ²) From give data, 1 + Mp1 = Mp2 By solving above equation, Damping ratio ξ = 0.087

04․ Find the bandwidth of the system, when rise time of the system is give as 1msec?
1 kHz
500 Hz
350 Hz
750 Hz

Bandwidth is the range of frequencies over which the magnitude has a value of 1/√2. Band width indicates the speed of response of the system. Wider bandwidth indicates that, faster response. Bandwidth = 0.35/Rise time = 0.35/(1*10-3) = 350 Hz

05․ If the bandwidth of the system is very large, the system response is
faster
slower
independent of bandwidth
none of the above

Bandwidth is the range of frequencies over which the magnitude has a value of 1/√2. Band width indicates the speed of response of the system. Wider bandwidth indicates that, faster response. Bandwidth = 0.35/Rise time

06․ The magnitude of a transfer function G(jω)H(jω) at gain cross over frequency is
0
1
infinity
any of the above

Stability from frequency response plots 1 + G(s)H(s) = 0 G(s)H(s) = -1 + j0 Put s = jω G(jω)H(jω) = -1 + j0 |G(jω)H(jω)| at gain cross over frequency = 1 ∠G(jω)H(jω) at phase cross over frequency = -180°

07․ Find the conditions for the system to be stable?
Gain margin and phase margins are positive
Gain margin and phase margins are negative
Gain margin is positive but phase margin is negative
Gain margin is negative but phase margin is positive

Stable system: Gain margin and phase margins are positive and Gain cross over frequency is less than phase cross over frequency. Unstable system: Gain margin and phase margins are negative and Gain cross over frequency is greater than phase cross over frequency. Marginally stable system: Gain margin and phase margins are zero and Gain cross over frequency is equal to phase cross over frequency.

08․ A second order under damped system has damping ratio of 0.3. Find the phase margin of the system?
20°
30°
60°

Phase margin is the allowable phase lag. ∠G(jω)H(jω) at gain cross over frequency = φ Phase margin = 180° + φ And also Phase margin ≈ 100*ξ Therefore, for given system Phase margin = 30°

09․ If the magnitude of G(jω)H(jω) at phase crossover frequency is 0.5. Find the gain margin of the given system?
0.5
2
1
4

Gain margin is the allowable gain. |G(jω)H(jω)| at phase crossover frequency = X Gain margin = 1/X Gain margin in db = 20 log(1/X) For given system gain margin = 1/0.5 = 2

10․ A second order system has peak over shoot = 50% and period of oscillations 0.2 seconds. Find the resonant frequency?