📘 AC Circuits – Advanced Worked Examples (GATE Intensive Practice)
This section contains deep, multi-step numerical problems designed for serious GATE aspirants. All problems require careful phasor and impedance analysis.
🔹 Problem 1 – Phasor Conversion
Convert the following into phasor form:
v(t) = 50 sin(ωt + 45°)
Phasor = 50∠45°
🔹 Problem 2 – Series RLC Current
Given: R = 5Ω L = 0.05H C = 100µF f = 50Hz V = 200∠0° V
Step 1: Angular Frequency
ω = 2πf = 314 rad/s
Step 2: Reactances
XL = ωL = 314×0.05 = 15.7Ω XC = 1/(ωC) = 31.8Ω
Step 3: Impedance
Z = R + j(XL − XC) Z = 5 − j16.1
Magnitude:
|Z| = √(5² + 16.1²) ≈ 16.85Ω
Step 4: Current
I = V/Z I = 200/16.85 I = 11.87A
🔹 Problem 3 – Power Calculation
Using previous result, Find real power.
cosφ = R/|Z| = 5/16.85 = 0.297
P = VI cosφ = 200 × 11.87 × 0.297 = 705 W (approx)
🔹 Problem 4 – Parallel RLC
R = 10Ω XL = 20Ω XC = 20Ω V = 100V
Since XL = XC → Resonance.
Impedance maximum Current minimum
🔹 Problem 5 – Resonant Frequency
L = 0.1H C = 25µF
f₀ = 1/(2Ï€√(LC))
= 100.6 Hz (approx)
🔹 Problem 6 – Quality Factor
R = 5Ω L = 0.2H f₀ = 50Hz
ω₀ = 314 Q = ω₀L / R = 314×0.2 / 5 = 12.56
🔹 Problem 7 – Bandwidth
If Q = 10 f₀ = 100Hz
BW = f₀ / Q = 100 / 10 = 10Hz
🔹 Problem 8 – GATE Trap (Phase Angle)
R = 3Ω XL = 4Ω
Z = √(3² + 4²) = 5Ω φ = tan⁻¹(4/3) φ = 53°
Power factor = cos53° = 0.6 lagging
🎯 Final Summary
Advanced AC circuit analysis requires strong phasor handling. Most GATE errors happen due to complex number mistakes. Practice repeatedly for speed and accuracy.
Master Phasors → Master AC Circuits
.png)
No comments:
Post a Comment