Article — LC Resonant Frequency Calculator
LC resonant frequency calculator
An inductor-capacitor (LC) circuit reaches resonance when inductive reactance equals capacitive reactance, at frequency f = 1 / (2π√(LC)). A 10 µH coil with a 100 pF capacitor resonates near 5 MHz; a 0.2 µH coil with 16 pF hits 88 MHz, the bottom of the FM broadcast band. At resonance, energy moves between L and C with almost no current drawn from the source.
The formula is the workhorse of radio engineering, filter design, oscillator circuits, wireless power transfer, and impedance matching. Plug in henries and farads, get hertz. Mix units (mH with µF) and you are off by orders of magnitude, so unit hygiene matters more than algebraic skill.
What LC resonant frequency means
An inductor stores energy in its magnetic field; a capacitor stores energy in its electric field. Connect them and the energy sloshes back and forth at a rate determined by their values. Resonance is the frequency at which the sloshing is most efficient: inductive reactance X_L = 2πfL exactly cancels capacitive reactance X_C = 1/(2πfC), leaving only resistance to limit current.
Set X_L = X_C and solve: 2πfL = 1/(2πfC) gives f² = 1/(4π²LC), and taking the square root yields the resonance equation. The 2π factor comes from converting cycles per second (Hz) to radians per second (ω), which is what reactance formulas naturally use.
An LC circuit with no resistance would oscillate forever, transferring energy from inductor to capacitor and back. Real circuits have wire resistance and core losses, so oscillations damp out unless replenished by an active source (an oscillator's amplifier).
The LC resonant frequency formula
One equation, three useful rearrangements.
f = 1 / (2π √(LC)) ω = 1 / √(LC)L = 1 / (4π² f² C) C = 1 / (4π² f² L)Z_0 = √(L/C) (characteristic impedance)A worked AM radio example. A typical receiver uses a 250 µH ferrite-rod inductor and a variable capacitor that swings from 35 to 350 pF. At C = 350 pF: f = 1 / (2π √(250×10⁻⁶ × 350×10⁻¹²)) = 538 kHz. At C = 35 pF: f = 1.70 MHz. The pair covers the 540 to 1,700 kHz AM broadcast band.
Plug a 2.4 GHz Wi-Fi frequency back into the formula to find what LC pair works: f = 2,450 MHz, pick L = 2 nH (a printed loop trace). Then C = 1 / (4π² × (2.45×10⁹)² × 2×10⁻⁹) = 2.1 pF. At these frequencies parasitic capacitance from layout matters more than the deliberately placed C.
Series vs parallel LC resonance
Same components, different topologies, opposite behavior at resonance.
Series LC is used as a passband element: at resonance, impedance is minimum (just the wire resistance R), so signal current peaks. Out of band, X_L and X_C add to a large impedance that blocks the signal. Crystal radio receivers and notch filters use this topology.
Parallel LC is a tank circuit: at resonance, impedance peaks (Q·R, where Q is the quality factor), so signal voltage peaks while current draw is minimum. Oscillators use parallel LC to set frequency; bandpass filters use it as a coupling element. The same L and C resonate at the same frequency in both connections, but the impedance behavior is opposite.
Q factor and bandwidth at resonance
The quality factor Q describes how sharp the resonance peak is. High Q means a narrow, tall peak; low Q means broad, shallow. Q = f_0 / bandwidth, where bandwidth is the frequency range between the -3 dB points.
For an LC tank: Q = (1/R) × √(L/C). Air-core inductors at HF (3 to 30 MHz) hit Q values of 100 to 300. Quartz crystals achieve Q greater than 10,000 because their mechanical resonance is so well defined. Ceramic resonators sit between. The trade-off is that higher Q tunes more narrowly: a Q = 1000 oscillator drifts visibly with temperature; a Q = 50 RF amplifier accepts a wide channel.
The atomic clock that defines the second uses a microwave resonance in cesium-133 at 9.192631770 GHz. The Q of that transition exceeds 1013, which is why the second can be defined to 15 decimal places.
LC resonance in radio tuning
Every analog radio uses an LC tank to select which station to receive. The tuning knob moves a variable capacitor's vanes or rotates a slug inside a coil, changing L or C continuously across the band.
FM tuners use varactor diodes, semiconductor capacitors that change with applied voltage. Sweep a control voltage and the resonant frequency moves smoothly. Modern receivers use phase-locked loops to lock the LC oscillator to a quartz reference, providing the frequency stability of a crystal with the tunability of an LC.
For impedance matching at RF, two LC pairs can transform any source impedance to any load impedance at one frequency. The L-network is the simplest: one series element, one shunt element, two component values, exact match at the design frequency.
Real components vs ideal LC
The textbook formula assumes ideal components. Real inductors and capacitors have parasitic elements that shift the actual resonance.
Inductors have series resistance (ESR, mostly wire and core losses), stray capacitance between adjacent turns, and self-resonance where they stop behaving as inductors. Above SRF, a 10 µH wire-wound part looks like a capacitor; below SRF it looks like an inductor with extra ESR. Capacitors have series inductance from leads and series resistance from dielectric loss. A 1 µF electrolytic at 1 MHz looks more like a 50 nH inductor.
Above their SRF, inductors act as capacitors and capacitors act as inductors. Pick parts rated well above your design frequency. For a 100 MHz LC tank, use inductors with SRF > 300 MHz and ceramic capacitors with SRF > 1 GHz at the chosen value.
Common LC resonance mistakes
The math is simple. The unit hygiene and the parasitics catch most beginners.
- Mixing SI prefixes — µH with µF gives an answer 1,000× off from µH with pF. Convert everything to base H and F before plugging into the formula.
- Forgetting the 2π — 1/√(LC) gives angular frequency in rad/s, not Hz. Divide by 2π for cycles per second.
- Using nominal values above SRF — a 10 mH choke labeled at 100 kHz may have an SRF of 500 kHz. Use the manufacturer's impedance chart, not the catalog value.
- Confusing series and parallel behavior — same L and C resonate at the same f, but a series LC drops minimum impedance there while parallel LC peaks. Pick the topology by what you want to do at that frequency.
- Ignoring stray capacitance — at VHF and above, the layout adds picofarads. PCB trace capacitance, connector capacitance, and probe capacitance shift resonance.