Resonance, Coupled Coils and Three-Phase

This lesson covers three AC topics that GATE EE tests every year: resonance, magnetically coupled coils, and three-phase circuits.

Series resonance

At ω₀ = 1/√(LC), X_L = X_C, so Z = R (minimum) and the current is maximum, in phase with the voltage.

• Quality factor Q = ω₀L/R = 1/(ω₀CR) = (1/R)√(L/C).
• Bandwidth BW = ω₀/Q = R/L (rad/s). Half-power frequencies: ω₁ω₂ = ω₀².
• Voltage across L or C at resonance = Q × source voltage (voltage magnification).

Parallel resonance

An ideal parallel LC tank has maximum impedance (infinite) at ω₀; with a coil of resistance R, ω₀ = √(1/(LC) − R²/L²) and the dynamic impedance is L/(CR). Current magnification instead of voltage.

Coupled coils

• Mutual inductance M = k√(L₁L₂), with coupling coefficient 0 ≤ k ≤ 1.
• Series aiding: L = L₁ + L₂ + 2M; series opposing: L₁ + L₂ − 2M.
• Dot convention: current entering both dots gives aiding fluxes (+M).
• Ideal transformer: V₁/V₂ = N₁/N₂, and an impedance Z_L on the secondary looks like (N₁/N₂)² Z_L from the primary.

Balanced three-phase systems

• Total power P = √3 V_L I_L cos φ (φ = phase angle of the load impedance).
• A balanced delta load Z_Δ is equivalent to a star load Z_Δ/3.
• Phase sequence ABC: V_B lags V_A by 120°.

Two-wattmeter method (three-wire balanced load):

• P = W₁ + W₂.
• tan φ = √3 (W₁ − W₂)/(W₁ + W₂).
• At pf 0.5, one wattmeter reads zero; below 0.5, one reads negative.

Worked example

A balanced star load of (8 + j6) Ω per phase on a 400 V line supply.

• V_ph = 400/√3 = 230.9 V; |Z| = 10 Ω; I = 23.09 A; pf = 0.8.
• P = √3 × 400 × 23.09 × 0.8 = 12.8 kW.

Practice