An LC tank circuit is the heartbeat of radio transmitters, receivers, oscillators, and bandpass filters. This calculator applies the exact resonance formula — f = 1 / (2π√(LC)) — and solves in all three directions: find the frequency from L and C, find the inductance from f and C, or find the capacitance from f and L. It also reports angular frequency, period, reactance at resonance, characteristic impedance, and — if you supply a series resistance — the Q factor and implied bandwidth.
How it works
An ideal LC circuit contains only an inductor L (henries) and a capacitor C (farads) connected together. When disturbed, charge sloshes between them at a natural rate: the capacitor drives a growing current through the inductor, which builds a magnetic field; when the capacitor is discharged the collapsing magnetic field recharges the capacitor in the opposite polarity; the cycle repeats indefinitely.
The instantaneous behaviour is governed by the second-order differential equation:
L d²q/dt² + q/C = 0
where q is the charge on the capacitor. The solution is sinusoidal with angular frequency:
ω₀ = 1 / √(L × C) (radians per second)
Converting to cycles per second (hertz):
f₀ = ω₀ / (2π) = 1 / (2π × √(L × C))
This is the most important formula in RF and analog electronics. Every term in it matters:
- Larger L → lower frequency (the inductor stores more energy per cycle, so it takes longer)
- Larger C → lower frequency (more charge must slosh back and forth)
- Square-root relationship → doubling both L and C drops the frequency by a factor of 2
Derived quantities
| Quantity | Formula | Meaning |
|---|---|---|
| Angular frequency ω₀ | 1 / √(LC) | Frequency in rad/s |
| Period T | 1 / f₀ | Time for one full oscillation |
| Reactance at resonance | XL = XC = √(L/C) | Impedance each element presents |
| Characteristic impedance Z₀ | √(L/C) | Sets voltage magnification |
| Quality factor Q (series) | (1/R) × √(L/C) | Sharpness of resonance |
| Bandwidth BW | f₀ / Q | Half-power frequency span |
| Free-space wavelength λ | c / f₀ | Relates to antenna sizing |
Worked example
Design a 7 MHz (40-metre amateur radio band) oscillator tank circuit.
Choose a standard 100 µH toroid inductor. What capacitance is needed?
f₀ = 7 MHz = 7 × 10⁶ Hz
L = 100 µH = 100 × 10⁻⁶ H = 10⁻⁴ H
C = 1 / ((2πf₀)² × L)
= 1 / ((2π × 7×10⁶)² × 10⁻⁴)
= 1 / ((4.398×10⁷)² × 10⁻⁴)
= 1 / (1.934×10¹⁵ × 10⁻⁴)
= 1 / (1.934×10¹¹)
≈ 5.17 pFA 5.6 pF standard capacitor would tune the circuit to about 6.75 MHz; a 4.7 pF capacitor gives about 7.35 MHz. A 2–10 pF trimmer spanning those values covers the entire 7.0–7.2 MHz 40-metre CW segment.
Add a 10 Ω series resistance (typical for the inductor winding):
Q = (1/R) × √(L/C)
= (1/10) × √(10⁻⁴ / 5.17×10⁻¹²)
= 0.1 × √(1.934×10⁷)
= 0.1 × 4398
≈ 440Q = 440 is excellent — a sharp, low-loss resonance typical of a well-wound air-core or ferrite toroid inductor.
Formula note
The formula f = 1 / (2π√(LC)) assumes an ideal (lossless) LC circuit. In real circuits, winding resistance, core losses, dielectric losses in the capacitor, and parasitic capacitance all shift the actual resonant frequency slightly and reduce Q. For most practical oscillator and filter designs the ideal formula is accurate to better than 1 % when high-Q components are used. When precision matters — for example in crystal-controlled reference oscillators — small-signal simulation (SPICE) or measured trimming is used on top of this formula.