Calculate resonant frequency, inductance or capacitance for LC tank circuits and filters using the Thomson formula: f₀ = 1 / (2π√LC).
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Select what you want to solve for, then enter the two known values.
Thomson Formula — All Three Forms
Rearranging f₀ = 1/(2π√LC) lets you solve for any of the three variables:
LC Resonance Formulas
Resonant Frequency
f₀ = 1 / (2π × √(L×C))
Inductance
L = 1 / (4π² × f₀² × C)
Capacitance
C = 1 / (4π² × f₀² × L)
Angular Frequency
ω₀ = 2π × f₀ = 1/√(LC)
Period
T = 1/f₀ = 2π×√(LC)
Wavelength (RF)
λ = c / f₀ (c = 3×10⁸ m/s)
Worked Examples
1AM radio tank circuit
L = 250 µH
C = 100 pF
f₀ = 1/(2π√(250µ×100p))
≈ 1.006 MHz
2Find C for 100 kHz
Target f₀ = 100 kHz
L = 10 mH
C = 1/(4π²×(100k)²×10m)
≈ 253.3 pF
3Find L for 88 MHz FM
Target f₀ = 88 MHz
C = 10 pF
L = 1/(4π²×(88M)²×10p)
≈ 32.7 nH
4Crossover filter 3 kHz
L = 2.7 mH
C = 1.05 µF
f₀ = 1/(2π√(2.7m×1.05µ))
≈ 2.99 kHz
Frequently Asked Questions
What is the Thomson formula? ▾
The Thomson formula gives the natural resonant frequency of an LC circuit: f₀ = 1 / (2π√(LC)). At this frequency, inductive reactance XL equals capacitive reactance XC, and the circuit oscillates freely.
What happens at LC resonance? ▾
At resonance XL = XC. In a series LC circuit, impedance is minimum (ideally zero) and current is maximum. In a parallel LC tank circuit, impedance is maximum and current from the source is minimum. Energy oscillates between the inductor and capacitor.
What is an LC tank circuit used for? ▾
LC tank circuits are used in: radio receivers (tuning to a station), oscillators (generating a specific frequency), bandpass/bandstop filters, impedance matching networks, and induction heating systems.
How do I choose L and C for a target frequency? ▾
Pick a convenient capacitor value first (e.g. 100 pF for RF), then calculate L = 1/(4π²f²C). Or pick L first and solve C = 1/(4π²f²L). This calculator does both rearrangements automatically.
What is the difference between series and parallel resonance? ▾
Both have the same resonant frequency f₀ = 1/(2π√LC). Series LC resonance gives minimum impedance (short at f₀). Parallel LC (tank) resonance gives maximum impedance (open at f₀). Series is used in bandpass filters; parallel in oscillators and RF amplifiers.