Electrical Engg (EEE) - MCQ Practice Questions

For minimum phase systems, magnitude and phase are related through the Kramers-Kronig relations. Knowledge of magnitude plot uniquely determines phase plot.

The derivative term acts as damping in the system. Increasing Kd increases damping, which reduces overshoot and oscillations without affecting steady-state error significantly.

Gain margin is the factor by which the system gain at phase crossover frequency can be increased before the system becomes unstable (phase reaching -180° at unity gain).

Lag compensator (zero at -0.1, pole at -0.01, with pole closer to origin) adds gain at low frequencies without significantly affecting phase at crossover, improving steady-state accuracy.

The polynomial factors as (s+1)(s+2)(s+3) = 0, giving poles at -1, -2, and -3. All poles are in the left half-plane, making the system stable.

Closed-loop gain = G(s)/(1+G(s)). At DC, this becomes 100/(1+100) = $\frac{100}{101}$ ≈ 0.99, showing the effect of feedback on reducing overall gain.

The number of sign changes in the first column of the Routh array equals the number of poles in the right half-plane. A sign change indicates at least one pole in RHP, making the system unstable.

For ramp input, ess = 1/Kv where Kv = lim s→0 s·G(s) = $\frac{10}{2}$ = 5. Therefore ess = $\frac{1}{5}$ = 0.2

At the gain crossover frequency, magnitude is 0 dB. When this equals phase crossover frequency, the phase is -180°, giving phase margin = -180° - (-180°) = 0°

Peak overshoot = exp(-ζπ/√(1-ζ²))·100% = exp(-0.5π/√0.75)·100% ≈ 16.3%

Adding a pole at origin adds -90° phase shift at all frequencies, decreasing phase margin. It also reduces high-frequency gain, decreasing gain margin

For lead compensator, maximum phase lead occurs at ωm = 1/√(τ₁τ₂) where τ₁ = 1/a and τ₂ = 1/b, giving ωm = √(ab)

Stability depends only on eigenvalue locations (all in LHP = stable). Controllability requires rank[B AB A²B] = n, which eigenvalues alone don't determine

Settling time ts ≈ 4/(ζωn) for 2% criterion, thus ts ∝ 1/(ζωn)

Lag compensator increases DC gain significantly, improving steady-state error, while its phase lag is restricted to lower frequencies, minimizing transient effects

ωn = √(4+9) = √13 ≈ 3.6 rad/s, ζ = 2/√13 ≈ 0.55

Higher K reduces ess proportionally but shifts root locus rightward, potentially crossing into RHP, thus reducing stability margins

PM = 30° is acceptable (typically 30-60°), GM = 8 dB (>6 dB threshold) indicates good gain stability. Both margins suggest satisfactory performance

Kv = lim(s→0) s·G(s) = lim(s→0) s·10/[s(s+5)] = $\frac{10}{5}$ = 2 sec⁻¹

For ramp input, ess = 1/Kv. Given ess = 0.2, therefore Kv = $\frac{1}{0}$.2 = 5 sec⁻¹