LC Resonant Frequency

LC Resonance

When an inductor and capacitor are connected together, they form a resonant circuit — energy oscillates between the magnetic field (inductor) and electric field (capacitor).

The Resonant Frequency

f₀ = 1 / (2π · √(L · C))

Or in angular frequency:

ω₀ = 1 / √(L · C)    (rad/s)

At resonance:

• The inductive reactance equals the capacitive reactance: X_L = X_C
• In a series RLC: impedance is minimized → maximum current
• In a parallel RLC: impedance is maximized → minimum current

Physical Intuition

Think of a pendulum: energy alternates between kinetic (inductor: ½LI²) and potential (capacitor: ½CV²). The resonant frequency is the natural "swing rate."

Derivation

At resonance: ωL = 1/(ωC) → ω² = 1/(LC) → ω = 1/√(LC) → f = 1/(2π√(LC))

Bandwidth and Q Factor

In a real RLC circuit with resistance R:

Q = (1/R) · √(L/C)    (quality factor)
BW = f₀ / Q           (bandwidth)

A high Q means a narrow, sharp resonance peak — good for radio tuning.

Examples

Applications

• Radio tuning: adjust C to select a station
• Filters: bandpass and notch filters
• Oscillators: crystal oscillators, LC oscillators
• Wireless charging: resonant coupling

Your Task

Implement double lc_resonance(double l, double c) that returns the resonant frequency f₀ in Hz.

Use #include <math.h> for sqrt and define PI.