What resonance is
An inductor (L) stores energy in a magnetic field and a capacitor (C) stores it in an electric field. Connected together, they pass energy back and forth, like a pendulum swinging between kinetic and potential energy. The circuit oscillates naturally at one frequency, its resonant frequency, where the inductor’s reactance equals the capacitor’s.
The formula
f₀ = 1 ÷ (2π√(LC))
- L in henries, C in farads, f₀ in hertz
- Angular frequency ω₀ = 2πf₀ = 1 ÷ √(LC)
- Characteristic impedance Z₀ = √(L ÷ C): the reactance of L or C at resonance
With a series resistance R:
- Quality factor Q = Z₀ ÷ R
- Bandwidth (−3 dB) = f₀ ÷ Q
Worked example
A 10 µH inductor with a 100 pF capacitor and 10 Ω of resistance:
- f₀ = 1 ÷ (2π × √(10 × 10⁻⁶ × 100 × 10⁻¹²)) ≈ 5.033 MHz
- Z₀ = √(10⁻⁵ ÷ 10⁻¹⁰) ≈ 316 Ω
- Q = 316 ÷ 10 ≈ 31.6
- Bandwidth ≈ 5.033 MHz ÷ 31.6 ≈ 159 kHz
Series and parallel circuits
| Circuit | At resonance | Typical use |
|---|---|---|
| Series LC | Minimum impedance (just R) | Filters that pass one frequency, notch traps |
| Parallel LC (tank) | Maximum impedance | Oscillators, radio tuners, IF amplifiers |
Where LC resonance is used
- Radio tuning: a variable capacitor shifts f₀ to select a station; AM broadcast spans about 530 to 1,700 kHz, FM 88 to 108 MHz.
- Oscillators in transmitters and clocks.
- Filters in audio crossovers, power supplies and RF circuits.
- Wireless charging and RFID tags, where transmitter and receiver coils are tuned to the same frequency.
- Induction cooktops, which drive a resonant tank at tens of kilohertz.
Scaling rules
- Frequency is inversely proportional to √(LC): to double f₀, divide L × C by 4.
- To lower f₀ by half, multiply L × C by 4.
- For a fixed f₀, a larger L with a smaller C raises Z₀ and Q for the same resistance.
Limits
Real components have parasitic resistance, capacitance and inductance. Above a few hundred megahertz, the leads and board traces become part of the circuit, and the measured resonance can differ noticeably from the formula.