ELECTRICAL AND ELECTRONICS ENGINEERING : The number of integrations in open-loop transfer function gives

59. For a second-order underdamped system, the peak overshoot $M_p$ is given by:

a) $e^{-\pi ζ / \sqrt{1-ζ^2}}$
b) $1 - e^{-ζ}$
c) $e^{-ζ \pi}$
d) $1 - e^{-πζ}$
Answer: a) $e^{-\pi ζ / \sqrt{1-ζ^2}}$
Explanation: Overshoot depends exponentially on the damping ratio ζ.

60. The rise time $t_r$ for a second-order system is approximately:

a) $\frac{1.8}{ω_n}$
b) $\frac{3}{ω_n}$
c) $\frac{2.3}{ω_n}$
d) $\frac{0.5}{ω_n}$
Answer: a) $\frac{1.8}{ω_n}$
Explanation: Approximate relation for 0–100% rise time in an underdamped system.

61. The term “bandwidth” of a control system refers to:

a) Frequency range of zero gain
b) Frequency range of unity phase
c) Frequency range over which system responds satisfactorily
d) Phase crossover frequency
Answer: c) Frequency range over which system responds satisfactorily
Explanation: Bandwidth is the frequency limit of significant response (typically -3 dB).

62. A proportional plus derivative (PD) controller improves:

a) Steady-state error
b) Transient response
c) Steady-state accuracy
d) None
Answer: b) Transient response
Explanation: Derivative term enhances damping, improving transient behavior.

63. The function of a tachogenerator in control systems is:

a) To measure displacement
b) To measure speed
c) To measure position
d) To amplify signal
Answer: b) To measure speed
Explanation: Tachogenerators produce voltage proportional to rotational speed.

64. The transfer function of a DC motor (armature control) is:

a) $\frac{K}{s(Ts+1)}$
b) $\frac{K}{Ts+1}$
c) $\frac{K}{s^2 + sT + 1}$
d) $\frac{K}{s^2(Ts+1)}$
Answer: a) $\frac{K}{s(Ts+1)}$
Explanation: One integration from speed to position, and one lag due to motor time constant.

65. The error detector in an automatic control system compares:

a) Output and input
b) Feedback and reference
c) Desired output and actual output
d) All of these
Answer: d) All of these
Explanation: Error is difference between desired (reference) and actual (feedback) signals.

66. The transfer function of a second-order system is:

$\frac{ω_n^2}{s^2 + 2ζω_n s + ω_n^2}$ .
Its natural frequency is:
a) $ζω_n$
b) $ω_n$
c) $2ζω_n$
d) $ω_n^2$
Answer: b) $ω_n$
Explanation: $ω_n$ represents natural frequency of oscillation.

67. The steady-state error for a unit parabolic input is zero for:

a) Type 0
b) Type 1
c) Type 2
d) None
Answer: c) Type 2
Explanation: Two integrators eliminate error for parabolic input.

68. The open-loop poles determine:

a) Static error
b) Dynamic response
c) Gain margin
d) Phase margin
Answer: b) Dynamic response
Explanation: Pole locations dictate system speed and oscillation characteristics.

69. The characteristic equation of a unity feedback system is:

a) $1 + G(s) = 0$
b) $1 + G(s)H(s) = 0$
c) $G(s) = 0$
d) $H(s) = 0$
Answer: b) $1 + G(s)H(s) = 0$
Explanation: Derived from closed-loop transfer function denominator.

70. The steady-state error constants are:

a) $K_p, K_v, K_a$
b) $K_d, K_i, K_p$
c) $K_1, K_2, K_3$
d) None
Answer: a) $K_p, K_v, K_a$
Explanation: They correspond to step, ramp, and parabolic input errors respectively.

71. The Nyquist plot of a Type 1 system passes through:

a) Origin
b) (-1,0)
c) (1,0)
d) (0,1)
Answer: a) Origin
Explanation: Type 1 (one integrator) introduces -90° phase lag at low frequencies, passing through origin.

72. The main advantage of state-space over transfer function is:

a) Easier Laplace computation
b) Handles MIMO systems
c) Requires fewer equations
d) None
Answer: b) Handles MIMO systems
Explanation: State-space can describe multi-input, multi-output systems directly.

73. The transient response of a second-order system is primarily determined by:

a) Zeros
b) Damping ratio and natural frequency
c) Gain
d) Feedback
Answer: b) Damping ratio and natural frequency
Explanation: ζ and ωₙ define rise time, overshoot, and settling time.

74. A system with characteristic equation $s^2 + 6s + 25 = 0$ is:

a) Overdamped
b) Underdamped
c) Critically damped
d) Unstable
Answer: b) Underdamped
Explanation: $ζ = 6 / (2√25) = 0.6 < 1$ → underdamped.

75. A phase lead compensator improves:

a) Stability margin
b) Steady-state error
c) Low-frequency response
d) All
Answer: a) Stability margin
Explanation: Lead compensators add positive phase → improved phase margin.

76. The polar plot of a first-order system is a:

a) Circle
b) Straight line
c) Parabola
d) Spiral
Answer: a) Circle
Explanation: For $G(jω) = \frac{1}{1+jωT}$ , the locus is a circle in complex plane.

77. The Bode magnitude of $\frac{1}{s}$ is:

a) +20 dB/dec
b) -20 dB/dec
c) 0 dB
d) None
Answer: b) -20 dB/dec
Explanation: Integrator causes -20 dB per decade slope.

78. For stability using Routh-Hurwitz, all elements of the first column must be:

a) Positive
b) Negative
c) Alternate sign
d) Zero
Answer: a) Positive
Explanation: All positive → no sign changes → all poles in LHP.

79. If any row of Routh array becomes zero, the system has:

a) Imaginary roots
b) Repeated real roots
c) Complex roots
d) None
Answer: a) Imaginary roots
Explanation: Zero row indicates symmetric root pairs on imaginary axis → marginal stability.

80. The condition for controllability is:

a) Rank [B AB A²B … Aⁿ⁻¹B] = n
b) det(A) ≠ 0
c) All poles distinct
d) Trace(A) ≠ 0
Answer: a) Rank [B AB A²B … Aⁿ⁻¹B] = n
Explanation: Kalman controllability criterion.

81. For observability, the matrix used is:

a) [C; CA; CA²; …; CAⁿ⁻¹]
b) [B AB …]
c) [A B C D]
d) [A C D]
Answer: a) [C; CA; CA²; …; CAⁿ⁻¹]
Explanation: Kalman observability test.

82. The open-loop transfer function $\frac{K(s+3)}{s(s+2)(s+5)}$ has how many asymptotes?

a) 1
b) 2
c) 3
d) None
Answer: b) 2
Explanation: 3 poles − 1 zero = 2 asymptotes.

83. For the same system, centroid of asymptotes =

$\frac{(-2-5+0)-(−3)}{2}$ = ?
a) -2
b) -1
c) -2
d) -2
Answer: a) -2
Explanation: Centroid = (sum of poles − sum of zeros)/(p−z) = (-7+3)/2 = -2.

84. When feedback gain is increased:

a) Stability increases
b) Bandwidth increases
c) Sensitivity increases
d) Gain margin increases
Answer: b) Bandwidth increases
Explanation: Higher feedback improves response speed but may reduce stability.

85. The number of encirclements in Nyquist plot equals:

a) Number of open-loop poles in LHP
b) Number of closed-loop poles in RHP
c) Difference between right-half open and closed-loop poles
d) None
Answer: c) Difference between right-half open and closed-loop poles
Explanation: $N = Z - P$ , where N = encirclements, Z = RHP closed-loop poles, P = RHP open-loop poles.

86. Servo systems are primarily used for:

a) Speed control
b) Position control
c) Current control
d) Temperature control
Answer: b) Position control
Explanation: Servo systems precisely control angular/linear position.

87. A stepper motor is used in control systems for:

a) Continuous control
b) Discrete position control
c) Speed control
d) None
Answer: b) Discrete position control
Explanation: Stepper moves in fixed angular steps → discrete control.

88. In frequency response, the phase lag increases with:

a) Frequency
b) Gain
c) Damping ratio
d) Time constant decrease
Answer: a) Frequency
Explanation: Lag increases as system’s phase angle drops at higher frequencies.

89. The final value theorem is used to find:

a) Initial value
b) Steady-state value
c) Transient response
d) Time constant
Answer: b) Steady-state value
Explanation: $\lim_{t→∞} f(t) = \lim_{s→0} sF(s)$ .

90. The initial value theorem is used to find:

a) Steady-state response
b) Starting value of response
c) Average value
d) None
Answer: b) Starting value of response
Explanation: $\lim_{t→0} f(t) = \lim_{s→∞} sF(s)$ .

91. In a unity feedback system, steady-state error for step input is given by:

a) $1/(1+K_p)$
b) $1/K_p$
c) $K_p/(1+K_p)$
d) $1+K_p$
Answer: a) $1/(1+K_p)$
Explanation: Derived from position error constant $K_p = \lim_{s→0} G(s)$ .

92. If damping ratio ζ = 1, the system is:

a) Overdamped
b) Underdamped
c) Critically damped
d) Unstable
Answer: c) Critically damped
Explanation: ζ = 1 → fastest response without oscillation.

93. For a second-order system, the damping ratio ζ = 0 gives:

a) Overdamped
b) Critically damped
c) Undamped
d) Oscillatory stable
Answer: c) Undamped
Explanation: ζ = 0 → no damping → continuous oscillations.

94. A unity feedback system has $G(s) = \frac{10}{s(s+2)}$ . Find $K_v$ .

Answer: $K_v = \lim_{s→0} sG(s) = \frac{10}{2} = 5$
Explanation: Velocity constant for Type 1 system.

95. The steady-state error for ramp input is $e_{ss} = \frac{1}{K_v}$ . For $K_v = 5$ :

a) 0.2
b) 5
c) 0.1
d) 1
Answer: a) 0.2
Explanation: $e_{ss} = 1/5 = 0.2$ .

96. The phase margin of 45° gives:

a) Highly stable system
b) Marginally stable
c) Satisfactory stability
d) Unstable
Answer: c) Satisfactory stability
Explanation: Phase margin between 30–60° → good transient response.

97. The gain margin of 10 dB means:

a) Gain can be increased 10 times before instability
b) Gain can increase by factor ≈ 3.16
c) Gain can increase by 10%
d) Gain decreases 10×
Answer: b) Gain can increase by factor ≈ 3.16
Explanation: 10 dB = 20 log₁₀(K) → K ≈ 3.16.

98. The number of integrations in open-loop transfer function gives:

a) Order
b) Type
c) Damping
d) Gain
Answer: b) Type
Explanation: Type equals number of integrators (poles at origin).

99. A system with gain margin = 0 dB and phase margin = 0° is:

a) Stable
b) Unstable
c) Marginally stable
d) Highly stable
Answer: c) Marginally stable
Explanation: Both zero margins → on stability boundary.

100. Increasing derivative gain $K_d$ :

a) Increases overshoot
b) Reduces overshoot
c) No effect
d) Increases rise time
Answer: b) Reduces overshoot
Explanation: Derivative action adds damping → reduces oscillation.

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Monday, November 3, 2025

The number of integrations in open-loop transfer function gives

59. For a second-order underdamped system, the peak overshoot MpM_pMp​ is given by:

60. The rise time trt_rtr​ for a second-order system is approximately: