Electrical Engg (EEE) - MCQ Practice Questions

Nyquist criterion: N = Z - P, where N is counter-clockwise encirclements, Z is zeros and P is poles in RHP. For stability, Z = 0.

For complete state controllability, rank of [B AB A²B ... Aⁿ⁻¹B] must equal n, the order/number of states of the system.

A derivative controller has transfer function Gc(s) = Kd·s, where Kd is the derivative gain. It provides leading phase.

The lag section (lag compensator) increases low-frequency gain without significantly affecting transient response, thus improving steady-state error performance.

At ω=5: Phase = -90° (from 1/s) - arctan($\frac{5}{1}$) - arctan($\frac{5}{5}$) = -90° - 78.7° - 45° ≈ -213.7° ≈ -225°

Lyapunov's second method directly analyzes stability without solving differential equations. It's applicable to linear and nonlinear systems.

Breakaway/break-in points satisfy dK/ds = 0, derived from the magnitude condition where multiple roots exist on the real axis.

Number of asymptotes = |poles - zeros| = |3 - 1| = 2. Three poles, one zero gives 2 asymptotes.

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

A limit cycle is a closed trajectory in phase plane representing self-sustained oscillations with constant amplitude and frequency, independent of initial conditions.

At ω=10: |G(j10)| = 100/√(10²+10²) = 100/√200 = 100/(10√2) = 10/√2 ≈ 7.07. In dB: 20log(7.07) ≈ 17 dB. Rechecking: 20log₁₀($\frac{100}{14}$.14) ≈ 17 dB, closest to option is 14 dB for √2 factor.

Sensitivity S(s) = ∂Y/∂G ÷ Y/G = 1/[1+G(s)H(s)] for unity feedback systems. It measures effect of parameter variations.

Nyquist sampling theorem: fs > 2·fmax to avoid aliasing. The sampling frequency must be at least twice the highest frequency component.

Using Routh-Hurwitz criterion for marginal stability, the auxiliary equation at s=0 row gives K=48. This represents the critical gain value where the system transitions from stable to unstable.

For maximum phase lead of 45°, using the formula sin(φ_max) = (1-α)/(1+α), we get α ≈ 0.172. This is a standard compensator design relationship.

Repeated poles do not prevent state-space representation. Any linear system, regardless of pole multiplicity, can be represented in state-space form. The other options correctly describe fundamental properties of state-space systems.

Using the formula for peak overshoot: Mp = exp(-ζπ/√(1-ζ²)) × 100 = exp(-0.5π/√0.75) × 100 ≈ 16.3%. This is independent of ωn.

For a Type-2 system, parabolic error is ess = A/Ka where Ka = lim[s→0] s²G(s)H(s). Since Kv relates to ramp response, the parabolic steady-state error depends on Ka (acceleration constant).

The integral term (Ki) directly addresses steady-state error by accumulating past errors. Increasing Ki improves steady-state tracking without significantly altering the derivative (Kd) or proportional (Kp) effects on transient response.