Resonance Frequency Calculator

In an LC circuit, energy continuously transfers between the inductor's magnetic field and the capacitor's electric field. At resonance frequency f₀ = 1/(2π√(LC)), this energy exchange is most efficient.

The inductor's reactance XL = 2πfL increases with frequency, while the capacitor's reactance XC = 1/(2πfC) decreases. At exactly f₀, XL = XC, and the reactive components cancel.

For a 10μH inductor with a 100pF capacitor: f₀ = 1/(2π√(10×10⁻⁶ × 100×10⁻¹²)) ≈ 5.03 MHz.

The quality factor Q = (1/R)√(L/C) determines bandwidth — higher Q means sharper tuning but narrower bandwidth.

Every radio receiver uses LC resonance to select stations. An AM radio tunes across 530-1700 kHz by varying a capacitor while keeping the inductor fixed. FM radios operate at 88-108 MHz with similar principles.

Bandpass filters combine LC circuits to pass a specific frequency range while rejecting others — cellular base stations use cavity resonators with Q factors exceeding 10,000.

In power supplies, LLC resonant converters operate near resonance for zero-voltage switching, achieving 95%+ efficiency. Crystal oscillators exploit the mechanical resonance of quartz (equivalent to an extremely high-Q LC circuit) for precise clock generation — typical quartz crystals have Q factors of 10,000-100,000 compared to 10-100 for discrete LC circuits.

Component tolerances directly affect resonance accuracy. A capacitor rated at 100pF ±10% could be 90-110pF, shifting resonance frequency by ±5%. For precision applications, use C0G/NP0 ceramic capacitors (±1%) and air-core or powdered-iron inductors with tight tolerances.

Parasitic elements also matter — every capacitor has parasitic inductance (equivalent series inductance, ESL) and every inductor has parasitic capacitance (self-resonant frequency). Above self-resonant frequency, an inductor behaves as a capacitor.

PCB trace inductance (approximately 1nH per mm) and pad capacitance (0.1-0.5pF) become significant above 100 MHz. At GHz frequencies, distributed elements (microstrip lines, striplines) replace discrete LC components entirely.

To calculate the resonant frequency of an LC circuit, use f = 1/(2π√(LC)), where L is inductance in henries (H) and C is capacitance in farads (F). First convert both values to base SI units.

For example, a 100 µH inductor (1×10⁻⁴ H) with a 100 pF capacitor (1×10⁻¹⁰ F) gives LC = 1×10⁻¹⁴, so √(LC) = 1×10⁻⁷, and f = 1/(2π×1×10⁻⁷) ≈ 1.5915 MHz.

This is the frequency where inductive and capacitive reactances cancel. HyperPhysics (Georgia State University) presents this as the standard series-resonance condition.

The natural frequency of an undamped spring-mass system is fₙ = (1/2π)√(k/m), where k is the spring constant in newtons per meter (N/m) and m is the mass in kilograms (kg).

For a spring with k = 200 N/m carrying a 0.5 kg mass: k/m = 400, √400 = 20, so fₙ = 20/(2π) ≈ 3.183 Hz.

Note the identical mathematical form to the LC formula — stiffness k plays the role of 1/C, and mass m plays the role of L. Khan Academy covers this simple-harmonic-motion relationship, which underlies mechanical vibration analysis.

For a series RLC circuit the quality factor is Q = (1/R)√(L/C), and the −3 dB bandwidth is BW = f₀/Q, both derived from the resonant frequency f₀ = 1/(2π√(LC)).

Example: R = 10 Ω, L = 100 µH, C = 100 pF gives L/C = 1×10⁶, √(1×10⁶) = 1000, so Q = 1000/10 = 100. With f₀ ≈ 1.5915 MHz, the bandwidth BW = 1.5915 MHz / 100 ≈ 15.915 kHz.

A higher Q yields a sharper, more selective peak. The IEEE Standard 100 dictionary defines Q as 2π times energy stored divided by energy dissipated per cycle.

Resonant frequency is expressed in hertz (Hz), the SI unit meaning cycles per second, defined by BIPM and NIST as s⁻¹. Inductance uses the henry (H = V·s/A = kg·m²·s⁻²·A⁻²), and capacitance uses the farad (F = A·s/V).

In circuits these appear scaled:

• microhenries (µH, 10⁻⁶)
• picofarads (pF, 10⁻¹²)
• kilohertz (kHz) and megahertz (MHz).

Mechanical systems use N/m for spring constant and kg for mass. Angular frequency ω = 2πf is measured in radians per second (rad/s). Always convert to base SI units before applying formulas so the answer emerges directly in hertz, per NIST guidance on coherent units.

Angular frequency ω relates to ordinary frequency by ω = 2πf, measured in radians per second, while f is in hertz. At LC resonance the angular resonant frequency simplifies elegantly to ω₀ = 1/√(LC), so f₀ = ω₀/(2π) = 1/(2π√(LC)). For a spring-mass system ω₀ = √(k/m).

Engineers often work with ω because reactance expressions — inductive X_L = ωL and capacitive X_C = 1/(ωC) — are cleaner. At resonance ωL = 1/(ωC), which rearranges to ω² = 1/(LC). Encyclopaedia Britannica notes this shared oscillatory form links electrical and mechanical resonance.

Resonance frequency governs technology across many fields. Radio and TV tuners use LC circuits to select one station and reject others; MRI machines tune RF coils to the proton Larmor frequency; and wireless chargers match transmit and receive coil resonance for efficient power transfer.

Mechanically, engineers calculate natural frequency to keep bridges, buildings, turbine blades, and vehicle suspensions away from destructive resonance — the 1940 Tacoma Narrows Bridge collapse is the textbook cautionary tale.

Musical instruments, quartz-crystal clocks, and MEMS gyroscopes all exploit resonance. HyperPhysics and Britannica document these electrical, mechanical, and acoustic examples in depth.

Series and parallel LC circuits share the same resonant frequency, f₀ = 1/(2π√(LC)), but behave oppositely at that frequency.

A series LC circuit reaches minimum impedance at resonance (ideally zero, limited only by resistance), so it passes the resonant frequency — acting as an acceptor circuit. A parallel LC (tank) circuit reaches maximum impedance at resonance, blocking the resonant frequency and behaving as a rejector circuit.

This distinction drives filter topology: series LC forms band-pass paths to ground, while parallel LC forms band-stop or high-impedance loads. HyperPhysics illustrates both impedance curves for tuned-circuit design.

The most frequent error is forgetting to convert prefixes to base SI units — entering 100 (µH) as 100 H instead of 1×10⁻⁴ H throws the result off by millions.

• Another is dropping the 2π factor, which confuses angular frequency ω₀ = 1/√(LC) with ordinary frequency f₀.
• Watch the square root: it applies to the entire product LC, not to L and C separately.
• For Q factor, remember Q = (1/R)√(L/C) is the series-circuit form; parallel circuits use Q = R√(C/L).
• Finally, real components carry parasitic resistance and self-resonance, so measured frequency drifts slightly from the ideal calculation, as noted in NIST measurement guidance.