ELECTRICAL AND ELECTRONICS ENGINEERING : If all roots of characteristic equation lie on imaginary axis, system is

226. The resonant peak (Mr) of a second-order system is a measure of:

a) Rise time
b) Overshoot
c) Frequency response damping
d) Bandwidth
Answer: c) Frequency response damping
Explanation: Mr is inversely proportional to damping ratio ζ.

227. For a stable second-order system, the damping ratio ζ must be:

a) ζ < 0
b) ζ = 0
c) ζ > 0
d) ζ > 1
Answer: c) ζ > 0
Explanation: Positive damping ensures poles lie in left half-plane.

228. The transfer function of a pure differentiator is:

a) 1/s
b) s
c) 1
d) s + 1
Answer: b) s
Explanation: Differentiator amplifies high frequencies; its transfer function is G(s)=s.

229. The transfer function of a pure integrator is:

a) 1/s
b) s
c) 1/(s+1)
d) s+1
Answer: a) 1/s
Explanation: Integration in Laplace corresponds to division by s.

230. The sensitivity of closed-loop system to parameter variations is:

a) High
b) Low
c) Zero
d) Infinite
Answer: b) Low
Explanation: Feedback reduces sensitivity of system to changes in components.

231. The steady-state error for a unit step in a Type 0 system is:

a) 0
b) 1/(1+Kp)
c) ∞
d) None
Answer: b) 1/(1+Kp)
Explanation: Type 0 → finite steady-state error for step input.

232. The transient response depends mainly on:

a) Zeros
b) Poles
c) Type of input
d) Feedback path
Answer: b) Poles
Explanation: Poles determine natural frequency and damping ratio.

233. A system is said to be critically damped when ζ =

a) 0
b) 0.707
c) 1
d) >1
Answer: c) 1
Explanation: ζ = 1 gives the fastest response without oscillation.

234. The steady-state error for a step input in a Type 1 system is:

a) 0
b) 1/Kp
c) Infinite
d) None
Answer: a) 0
Explanation: Type 1 → one integrator → zero error for step input.

235. A stable system will always have:

a) Positive real poles
b) Poles on left half of s-plane
c) Imaginary poles
d) Zeros on right half
Answer: b) Poles on left half of s-plane
Explanation: Negative real part of poles ensures stability.

236. The rise time of a system decreases with:

a) Decrease in natural frequency
b) Increase in natural frequency
c) Increase in damping
d) Increase in time constant
Answer: b) Increase in natural frequency
Explanation: Higher ωn leads to faster rise time.

237. Phase lag occurs when:

a) Output leads input
b) Output lags input
c) Both in phase
d) Output is zero
Answer: b) Output lags input
Explanation: Phase lag means output occurs later than input in phase.

238. In a stable feedback system, feedback is usually:

a) Positive
b) Negative
c) Zero
d) Infinite
Answer: b) Negative
Explanation: Negative feedback stabilizes and controls gain.

239. The phase margin is defined at:

a) Unity gain frequency
b) Cutoff frequency
c) Resonant frequency
d) Bandwidth frequency
Answer: a) Unity gain frequency
Explanation: Phase margin is measured at the frequency where |G(jω)| = 1.

240. The gain margin is defined at:

a) Phase crossover frequency
b) Gain crossover frequency
c) Resonant frequency
d) None
Answer: a) Phase crossover frequency
Explanation: Gain margin is the amount of gain reduction to reach critical point.

241. The steady-state error constant for parabolic input is:

a) Kp
b) Kv
c) Ka
d) None
Answer: c) Ka
Explanation: Acceleration error constant (Ka) relates to parabolic inputs.

242. A root locus asymptote angle for two branches is given by:

a) ±90°
b) ±45°
c) ±60°
d) ±120°
Answer: a) ±90°
Explanation: Two branches → asymptote angles ±90°.

243. The frequency at which phase = -180° is:

a) Gain crossover frequency
b) Phase crossover frequency
c) Cutoff frequency
d) Resonant frequency
Answer: b) Phase crossover frequency
Explanation: By definition, phase = -180° → phase crossover frequency.

244. The bandwidth of a feedback system is usually:

a) Increased by feedback
b) Decreased by feedback
c) Unchanged
d) None
Answer: a) Increased by feedback
Explanation: Negative feedback broadens frequency response.

245. The steady-state error for ramp input in Type 0 system is:

a) Finite
b) Infinite
c) Zero
d) None
Answer: b) Infinite
Explanation: Type 0 cannot track ramp → infinite error.

246. The Laplace transform of a unit ramp is:

a) 1/s
b) 1/s²
c) s
d) s²
Answer: b) 1/s²
Explanation: L{t} = 1/s².

247. The Laplace transform of δ(t) is:

a) 1
b) 0
c) s
d) 1/s
Answer: a) 1
Explanation: Unit impulse transform is 1.

248. The closed-loop transfer function is $\frac{G(s)}{1+G(s)H(s)}$ . This represents:

a) Open-loop system
b) Closed-loop system
c) Feedback path
d) Error function
Answer: b) Closed-loop system
Explanation: Standard closed-loop transfer formula.

249. For a unit feedback system, error signal =

a) R(s) + C(s)
b) R(s) - C(s)
c) R(s)C(s)
d) None
Answer: b) R(s) - C(s)
Explanation: e(s) = input – output (unity feedback).

250. The natural frequency of $s^2 + 4s + 25 = 0$ is:

a) 5
b) 25
c) 2
d) 4
Answer: a) 5
Explanation: $ω_n = √25 = 5$ .

251. The damping ratio for $s^2 + 4s + 25 = 0$ :

$ζ = \frac{4}{2×5} = ?$
a) 0.2
b) 0.4
c) 0.8
d) 1
Answer: b) 0.4
Explanation: ζ = 4/10 = 0.4 → underdamped.

252. The steady-state error for unit step in Type 2 system is:

a) 0
b) ∞
c) 1
d) Depends on gain
Answer: a) 0
Explanation: Type ≥1 → zero step error.

253. A system with gain margin of 10 dB and phase margin of 60° is:

a) Unstable
b) Stable
c) Marginally stable
d) Uncontrollable
Answer: b) Stable
Explanation: Positive margins indicate stability.

254. Increasing derivative gain (Kd) in PID:

a) Increases overshoot
b) Reduces overshoot
c) Slows response
d) Increases steady-state error
Answer: b) Reduces overshoot
Explanation: Derivative adds damping → less overshoot.

255. Increasing integral gain (Ki):

a) Increases steady-state error
b) Reduces steady-state error
c) Increases overshoot
d) b & c
Answer: d) b & c
Explanation: Ki eliminates error but increases oscillation/overshoot.

256. The PID controller equation is:

a) $u = Kp e + Ki \int e dt + Kd \frac{de}{dt}$
b) $u = Kp e$
c) $u = Ki e$
d) None
Answer: a) $u = Kp e + Ki \int e dt + Kd \frac{de}{dt}$
Explanation: Standard PID law.

257. The root locus moves toward infinity when:

a) Zeros < Poles
b) Zeros = Poles
c) Zeros > Poles
d) Gain is 0
Answer: a) Zeros < Poles
Explanation: Unmatched poles → branches go to infinity.

258. Adding poles near origin makes system:

a) Faster
b) Slower
c) More oscillatory
d) Stable
Answer: b) Slower
Explanation: Pole near origin → increased time constant → slower response.

259. In Nyquist plot, encirclement in clockwise direction means:

a) Unstable closed loop
b) Stable
c) Marginally stable
d) None
Answer: a) Unstable closed loop
Explanation: Clockwise encirclement → right-half-plane poles.

260. If all roots of characteristic equation lie on imaginary axis, system is:

a) Stable
b) Unstable
c) Marginally stable
d) Divergent
Answer: c) Marginally stable
Explanation: Oscillations of constant amplitude → marginally stable.

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Monday, November 3, 2025

If all roots of characteristic equation lie on imaginary axis, system is