📘 AC Circuits – Resonance, Power Factor & Advanced RLC Analysis
🔹 1. Series Resonance
In a series RLC circuit, resonance occurs when:
XL = XC
Since:
XL = ωL XC = 1/(ωC)
At resonance:
ωL = 1/(ωC)
Resonant frequency:
ω₀ = 1 / √(LC) f₀ = 1 / (2Ï€√(LC))
🔹 2. Characteristics of Series Resonance
- Impedance is minimum (Z = R)
- Current is maximum
- Power factor = 1 (Unity)
- Voltage across L and C can be very high
🔹 3. Worked Example – Series Resonance
Given: L = 0.2H C = 50µF R = 10Ω Find resonant frequency.
Step 1: Calculate f₀
f₀ = 1 / (2Ï€√(LC))
= 1 / (2Ï€√(0.2 × 50×10⁻⁶))
= 50.3 Hz (approx)
🔹 4. Quality Factor (Q)
Quality factor indicates sharpness of resonance.
Q = ω₀L / R or Q = 1 / (ω₀CR)
Higher Q → Sharper resonance.
🔹 5. Bandwidth
Bandwidth (BW) is:
BW = f₂ − f₁
Relation with Q:
Q = f₀ / BW
🔹 6. Parallel Resonance
In parallel RLC circuit:
Impedance is maximum at resonance.
- Line current is minimum
- Power factor = 1
🔹 7. Power Factor
Power factor = cosφ
- Lagging → Inductive circuit
- Leading → Capacitive circuit
- Unity → Purely resistive
🔹 8. Worked Example – Power Factor
Given: R = 8Ω XL = 6Ω Find power factor.
Impedance:
Z = √(R² + XL²) = √(64 + 36) = 10Ω
Power factor:
cosφ = R/Z = 8/10 = 0.8 lagging
🔹 9. GATE Important Points
- Resonance frequency formula must be memorized
- Quality factor frequently tested
- Parallel resonance trick questions appear
- Power factor improvement concept important
🎯 Final Summary
Resonance and power factor form the core of AC circuit analysis. Understanding Q-factor and bandwidth improves conceptual clarity. These topics are extremely important for GATE and IES.
Strong AC Fundamentals = Strong Power System Foundation
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