LC Resonant Frequency
Reference for LC resonant frequency f = 1 / (2π√LC).
Covers inductance, capacitance, and applications in radio tuning, bandpass filters, and oscillators.
The Formula
f = 1 / (2π√(LC))
An LC circuit oscillates at its natural resonant frequency. At this frequency, energy swings back and forth between the inductor's magnetic field and the capacitor's electric field.
Variables
| Symbol | Meaning |
|---|---|
| f | Resonant frequency (Hz) |
| L | Inductance (Henrys, H) |
| C | Capacitance (Farads, F) |
| π | Pi (approximately 3.14159) |
Example 1
L = 10 mH, C = 100 nF. Find the resonant frequency.
f = 1 / (2π√(0.01 × 10⁻⁷))
f = 1 / (2π√(10⁻⁹)) = 1 / (2π × 3.162 × 10⁻⁵)
f ≈ 5,033 Hz ≈ 5.03 kHz
Example 2
Design an FM radio tuner for 100 MHz with L = 0.1 μH. What capacitance is needed?
C = 1 / (4π²f²L)
C = 1 / (4 × 9.87 × (10⁸)² × 10⁻⁷)
C ≈ 25.3 pF (picofarads)
When to Use It
Use the LC resonant frequency formula when:
- Tuning radio receivers and transmitters
- Designing bandpass and notch filters
- Building oscillator circuits
- Selecting components for a specific operating frequency
Limitations
- Real LC circuits include resistance (wire resistance, core losses) that damps oscillation — the Q-factor (Q = (1/R)√(L/C)) determines how sharp the resonance peak is
- Component tolerances (±5–20% for typical capacitors) mean the actual resonant frequency will differ from the calculated value — variable capacitors are used in radio tuning circuits for fine-frequency adjustment
- Series and parallel LC circuits have the same resonant frequency formula, but their behavior at resonance is opposite: a series LC has minimum impedance at resonance, while a parallel LC has maximum impedance
Key Notes
- Formula: f₀ = 1 / (2π√(LC)): The resonant frequency depends only on inductance L (henries) and capacitance C (farads). At f₀, the inductive reactance X_L = 2πf₀L exactly equals the capacitive reactance X_C = 1/(2πf₀C), so they cancel.
- Series vs parallel resonance: In a series LC circuit, resonance means minimum impedance (Z = R_series) and maximum current. In a parallel LC circuit (tank circuit), resonance means maximum impedance and minimum current from the source — the energy circulates between L and C.
- Quality factor Q = f₀/(f₂−f₁) = (1/R)√(L/C): Q measures the sharpness of the resonance peak. High Q means narrow bandwidth, sharp frequency selectivity. Radio tuners exploit high-Q LC circuits to select one station while rejecting adjacent frequencies.
- Energy oscillates between L and C: At resonance, energy shuttles continuously between the magnetic field of the inductor and the electric field of the capacitor — like a frictionless pendulum. Real resistive losses (R) cause the oscillations to decay unless energy is replenished.
- Applications: LC resonance is the basis of radio and TV tuners, oscillator circuits (clock generation), bandpass and band-reject filters, wireless power transfer (resonant inductive coupling), and the initial design step for RF amplifiers and transmitters.