Electrical Engg (EEE) - MCQ Practice Questions

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°

Type of system is determined by the number of poles at origin. Option A has s² in denominator, making it Type-2

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

Number of root locus branches = max(number of poles, number of zeros) = max(3, 1) = 3

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

For first-order system, bandwidth ωBW = 1/τ = $\frac{1}{0}$.1 = 10 rad/s (where -3dB occurs)

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

For Type-1 system, Kv = lim s→0 s·G(s) = 20/(5·10) = $\frac{20}{50}$ = 0.4

Controllability requires the controllability matrix to have full rank n. This ensures all states can be moved from origin to any desired state

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

Corner frequency = pole frequency. Rewriting: H(s) = 10/(s+10), so pole at -10, corner frequency = 10 rad/s

Lead compensator adds phase lead at its designed center frequency ωm. For transient improvement, it should coincide with gain crossover frequency to increase phase margin

Factoring: (s+1)(s+2)(s+3) = 0 gives poles at s = -1, -2, -3. All in LHP, so system is stable

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

The type of a system is determined by the number of poles at origin. Here, there is one pole at origin (s term), making it Type 1.