LC Resonant Frequency – Thomson's Formula, Q and Bandwidth
Put an inductor and a capacitor together and you have built an electrical pendulum. Energy sloshes from the capacitor's electric field into the coil's magnetic field and back again, and it does so at one particular rate set by Thomson's formula f₀ = 1 / (2π · √(L · C)). This LC resonant frequency calculator evaluates that expression with L in henries and Cin farads, then rearranges it for whichever quantity you leave blank — so it also answers “what capacitor do I need for 13.56 MHz?” and “what coil pairs with this 470 pF part?”
Why the reactances cancel
Resonance is not a mysterious property of the pair; it is simply the one frequency where the two reactances are equal in magnitude. Inductive reactance X_L = 2πfL rises with frequency while capacitive reactance X_C = 1/(2πfC) falls, so they must cross exactly once. Setting 2πfL = 1/(2πfC) and solving for fgives Thomson's formula directly. Because they are opposite in sign, at that crossing they cancel and the circuit looks purely resistive — which is why the tool also reports the characteristic impedance Z₀ = √(L/C), the common magnitude each reactance holds at resonance.
A 100 µH coil with 220 pF resonates at 1.0730 MHz, with ω₀ = 6.742 × 10⁶ rad/s, a period of 932 ns and Z₀ = 674.2 Ω. Halving either component multiplies the frequency by √2, never by two — the square root is the single most common source of mental arithmetic errors in tank design.
Q, bandwidth and damping
The frequency tells you where the peak sits; the quality factortells you how sharp it is. Add the coil's equivalent series resistance and a series tank gives Q = (1/R)·√(L/C); supply a parallel load and the relationship inverts to Q = R·√(C/L). From Q follow the −3 dB bandwidth BW = f₀/Q, the half-power edges f₀·(√(1 + 1/(4Q²)) ∓ 1/(2Q)), and the damping ratio ζ = 1/(2Q) that classifies the response as underdamped, critically damped or overdamped.
For that same tank with R = 2.5 Ω, Q works out at 269.68, giving a bandwidth of just 3.9789 kHz between 1.0710 and 1.0750 MHz and a damping ratio of 0.0018540. Q is also a magnification factor: drive that series tank with 1 V and roughly 270 V appears across the inductor and across the capacitor. Component voltage ratings, not the supply rail, are what fail first in a high-Q resonator.
Tuning, stray capacitance and tolerance
Swap the fixed capacitor for a variable one and the covered band follows f ∝ 1/√C, so the tuning ratio is exactly √(C_max/C_min) and the coil cancels out. A 10–365 pF broadcast gang therefore gives 6.04 : 1 whatever it is wired to — with a 240 µH loopstick that spans 3.2487 MHz down to 0.5377 MHz.
Board traces, coil self-capacitance and a scope probe each add a few picofarads across the tank. Folding 5 pF of stray into a 220 pF tank pulls 1.0730 MHz down to 1.0610 MHz — a 1.1 % shift from 2.3 % more capacitance, because frequency follows the inverse square root.
Tolerance compounds the same way. With ±5 % parts and that 5 pF of stray, the worst-case window runs from 1.0105 to 1.1169 MHz — over ten percent wide, which is why RF tanks are built with a trimmer rather than a calculated part alone. The tool names the tolerance model it used beside the window, since applying the spread to the effective capacitance and applying it to the ordered part alone give visibly different answers.
Where lumped modelling stops
Every result here assumes an ideal lumped inductor and capacitor. Real coils have self-resonance, real capacitors have lead inductance, and above roughly a gigahertz the board itself becomes part of the circuit. Treat the calculated resonant frequency as the starting point a trimmer moves from, and the reported band label — LF, MF, HF, the 455 kHz IF, the AM broadcast band, NFC at 13.56 MHz — as the sanity check that you are designing in the right decade.