Series LCR circuits and resonance

In a series LCR circuit, the same current flows through R, L and C. Their voltages are out of step: V_R is in phase with the current, V_L is 90° ahead, and V_C is 90° behind. So the voltages do not simply add; they add like vectors (phasors).

V_L and V_C point in opposite directions, so they partly cancel. The result:

Z = √(R² + (X_L – X_C)²), the impedance, and I = V / Z.

The current is out of phase with the supply by an angle φ, with tan φ = (X_L – X_C) / R. If X_L is bigger, the circuit acts inductive (current lags); if X_C is bigger, capacitive (current leads).

Because of the phases, the rms voltages across R, L and C can add up to more than the supply voltage. That is not a mistake.

Resonance: when X_L = X_C, the L and C effects cancel completely. Then:

• Z = R, the smallest possible, so the current is the largest.
• The current is in phase with the voltage.
• The resonant frequency is ω₀ = 1/√(LC).

Example: L = 0.1 H and C = 10 µF: ω₀ = 1/√(10⁻⁶) = 1000 rad/s, about 159 Hz.

At resonance, V_L and V_C are each large but equal and opposite: together they add to zero, though each alone may be bigger than the supply.

Uses: a radio is tuned by changing C until the circuit resonates with one station's frequency, which then gives a large current while others do not. Metal detectors also work on resonance.

An LC circuit on its own, once charged, swaps energy between the capacitor's electric field and the inductor's magnetic field, oscillating at ω₀ = 1/√(LC).