Forced LCR oscillation and resonance

Resonance is the condition in which a system driven by a periodic external force responds with maximum amplitude. It happens when the driving frequency matches the natural frequency of the system.

Differential equation

A series L, C, R circuit is driven by an AC source E = E0 cos ωt.

 ~E0 cos ωt ---- L ---- R ---- C ---+
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By Kirchhoff's law, L di/dt + Ri + q/C = E0 cos ωt. With i = dq/dt:

L d²q/dt² + R dq/dt + q/C = E0 cos ωt

This matches the driven mechanical oscillator m x'' + b x' + kx = F0 cos pt with m → L, b → R, k → 1/C.

Current amplitude

Steady state: try q = q0 cos(ωt − φ). Substitute, write cos ωt = cos(ωt − φ) cosφ − sin(ωt − φ) sinφ, and compare the coefficients of cos(ωt − φ) and sin(ωt − φ):

q0 (1/C − Lω²) = E0 cosφ
q0 R ω         = E0 sinφ

Squaring and adding: q0 = E0 / √[(1/C − Lω²)² + R²ω²].

The current is i = dq/dt = −ωq0 sin(ωt − φ), so the current amplitude is I0 = ωq0:

I0 = E0 / √[R² + (ωL − 1/ωC)²]

The current lags the emf by an angle ψ = φ − π/2, where tanψ = (ωL − 1/ωC)/R. Here X = ωL − 1/ωC is the net reactance and Z = √(R² + X²) is the impedance, so I0 = E0/Z.

Resonant frequency and condition

I0 is maximum when the denominator is minimum, i.e. when X = 0:

ωL = 1/ωC   →   ω_r = 1/√(LC) ,   f_r = 1/(2π√(LC))

At this frequency:

• Z = R (minimum), so I0(max) = E0/R.
• ψ = 0: the current is in phase with the applied emf.
• The circuit behaves as purely resistive; the voltages across L and C are equal and opposite.
• The resonant frequency equals the natural frequency of the undamped LC circuit. A smaller R gives a taller, sharper peak.