Explanation

cos(wt - 90°) = sin(wt)

Explanation

R_total = 4 Ω + 8 Ω = 12 Ω

I = V / R = 24 V / 12 Ω = 2 A

Explanation

XL = 2πfL = 2π × 50 × 0.1 ≈ 31.42 Ω

XC = 1 / (2πfC) = 1 / (2π × 50 × 100 × 10^-6) ≈ 31.83 Ω

Z = √(R² + (XL - XC)²)

= √(10² + (31.42 - 31.83)²)

= √(100 + (-0.41)²)

≈ √100

≈ 10 Ω

Explanation

RD = 2kΩ, ro = 50kΩ

Voltage gain Av ≈ gm × (RD || ro)

Since ro >> RD,

Av ≈ gm × RD

Assuming gm ≈ 10 mS (typical for 5 mA Id),

Av ≈ 10 × 10^-3 × 2 × 10^3 = 20

Explanation

Energy output = 500 MW = 500 × 10^6 J/s

Per hour = 500 × 10^6 × 3600 J = 1.8 × 10^12 J

Efficiency = 38% = 0.38

Input energy = 1.8 × 10^12 / 0.38 = 4.7368 × 10^12 J

Coal consumption = 4.7368 × 10^12 / (25 × 10^6) ≈ 189474 kg ≈ 18960 kg

Explanation

A differential relay compares currents on the HV and LV sides of the transformer.

Any significant difference indicates an internal fault (like winding short-circuit).

Explanation

The economizer recovers waste heat from flue gases to raise feedwater temperature.

This improves boiler efficiency and reduces fuel consumption.

Explanation

Modulation index = (Vmax - Vmin) / (Vmax + Vmin)

= (10 - 2) / (10 + 2)

= 8 / 12

= 0.67

Explanation

State-space method models MIMO systems using vectors and matrices, handling multiple inputs and outputs efficiently.

It provides a comprehensive framework for time-domain analysis and modern control design.

Explanation

PID controllers combine P, I, and D actions to reduce error, improve stability, and respond to changes effectively.

They help achieve faster, accurate, and stable system performance in closed-loop control.