Module II: AC Circuits - Solved Questions

Solution

ω = 2πf = 2π × 50 = 314.16 rad/s

(i) Instantaneous equation:

i = I_msin(ωt) = 12 sin(314.16t) A

(ii) Time to reach 9.6A:

9.6 = 12 sin(314.16t)

sin(314.16t) = 0.8 → 314.16t = 53.13° = 0.927 rad

t = 0.927/314.16 = 2.95 ms

Solution

Power factor: cos φ = P/S = 1000/2000 = 0.5 lagging

Phase angle: φ = cos⁻¹(0.5) = 60°

Current: P = I²R → I = √(1000/10) = 10A

Voltage: V = S/I = 2000/10 = 200V

Impedance: Z = V/I = 200/10 = 20Ω

Reactance: X_L = √(Z²-R²) = √(400-100) = 17.32Ω

Inductance: L = X_L/2πf = 17.32/314.16 = 55.1 mH

Equations: v = 282.84 sin(314.16t) V; i = 14.14 sin(314.16t - 60°) A

KVAR: Q = S×sin φ = 2000×0.866 = 1.732 KVAR

Solution

Branch 1: Z₁ = 12+j16 Ω

|Z₁| = √(144+256) = 20Ω

Y₁ = 1/Z₁ = (12-j16)/400 = (30-j40) mS

I₁ = 230/20 = 11.5A

pf₁ = 12/20 = 0.6 lagging (inductive)

Branch 2: Z₂ = 10-j20 Ω

|Z₂| = √(100+400) = 22.36Ω

Y₂ = 1/Z₂ = (10+j20)/500 = (20+j40) mS

I₂ = 230/22.36 = 10.29A

pf₂ = 10/22.36 = 0.447 leading (capacitive)

Solution

Voltage: V_L = V_Ph (Line = Phase voltage)

Current: I_L = √3 × I_Ph

Power: P = √3 × V_L × I_L × cos φ = 3 × V_Ph × I_Ph × cos φ

Solution

Condition: X_L = X_C (Inductive reactance = Capacitive reactance)

Derivation:

ωL = 1/(ωC)

ω² = 1/LC

ω = 1/√(LC)

2πf_r = 1/√(LC)

f_r = 1/(2π√LC)

At resonance: Z = R (minimum), I = V/R (maximum), pf = 1, φ = 0°

Solution

Solution

Phase Impedance: Z_ph = √(6²+8²) = 10Ω

Phase Voltage: V_ph = V_L/√3 = 400/1.732 = 231V

Phase Current: I_ph = V_ph/Z_ph = 231/10 = 23.1A

Line Current: I_L = I_ph = 23.1A (star connection)

Power Factor: cos φ = R/Z = 6/10 = 0.6 lagging

Solution

Reactance: X_L = 2πfL = 2π×50×0.04 = 12.57Ω

Impedance: Z = √(10²+12.57²) = 16.06Ω

Current: I = V/Z = 200/16.06 = 12.45A

Power Factor: cos φ = R/Z = 10/16.06 = 0.623 lagging

Power: P = I²R = 12.45²×10 = 1550W

Solution

Reactances: X_L1 = 2π×50×0.04 = 12.57Ω; X_L2 = 2π×50×0.2 = 62.83Ω

Total: R_T = 40Ω, X_T = 75.4Ω

Impedance: Z_T = √(40²+75.4²) = 85.35Ω

Current: I = 230/85.35 = 2.695A

Power in Coil 1: P₁ = I²R₁ = 2.695²×25 = 181.6W

Power in Coil 2: P₂ = I²R₂ = 2.695²×15 = 108.9W

Total Power Factor: cos φ = R_T/Z_T = 40/85.35 = 0.469 lagging

Solution

Reactance: X_L = 2π×50×0.02 = 6.28Ω

Phase Impedance: Z_ph = √(25+39.44) = 8.03Ω

Phase Voltage: V_ph = 440/√3 = 254V

Line Current: I_L = I_ph = 254/8.03 = 31.64A

cos φ = 5/8.03 = 0.623; sin φ = 6.28/8.03 = 0.782

Active Power: P = √3×440×31.64×0.623 = 15.02 kW

Reactive Power: Q = √3×440×31.64×0.782 = 18.86 kVAR