LCR Circuits & Resonance

With several elements in series, a neat vector trick gives the impedance and phase at once. It reveals resonance: the frequency at which a circuit responds most strongly, the principle behind every radio tuner.

The vector (phasor) method#

For a series circuit with ℰ = ℰ₀ sin ωt:

1. Draw R as a vector along the x-axis.
2. Draw the capacitive reactance 1/ωC along +y (π/2 ahead) and the inductive reactance ωL along −y (π/2 behind).
3. Add them. The resultant’s length is the impedance Z; its angle with the x-axis is the phase factor φ.

On switching on, the current behaves in a complicated way, but soon settles into this steady state at the source’s frequency.

CR and LR circuits

CR: Z = √(R² + 1/ω²C²), tan φ = 1/ωCR, current leads. LR: Z = √(R² + ω²L²), tan φ = ωL/R, current lags.

The LCR circuit#

The two reactances point in opposite directions, so the net reactance is X = 1/ωC − ωL:

• X_C > X_L: φ > 0, current leads (capacitive).
• X_C < X_L: φ < 0, current lags (inductive).
• X_C = X_L: net reactance zero, current in phase, Z = R.

Resonance#

Vary ω and the peak current i₀ = ℰ₀/Z is greatest when 1/ωC = ωL:

A small R makes the resonance sharp: the current is large only very close to ν₀. Such a circuit used near ν₀ is a resonant circuit.

LC oscillations#

With R = 0 the current at resonance would be infinite for any emf: a finite current can flow with no source at all. Connect a charged capacitor to a pure inductor: it discharges through the inductor, whose emf then recharges it the other way, and so on, at ν = 1/2π√(LC). Energy swaps between the capacitor’s electric field and the inductor’s magnetic field.