Resonant Frequency Calculator

The resonant frequency of an LC circuit is the frequency at which inductive and capacitive reactances are equal and opposite, causing them to cancel. At this frequency, the circuit oscillates most freely — with minimum impedance in series configuration and maximum impedance in parallel configuration. The resonant frequency formula is: f₀ = 1 / (2π√(LC)), where L is inductance in henries and C is capacitance in farads.

This fundamental formula is one of the most important in electrical engineering, governing the operating frequency of radio transmitters and receivers, oscillators, filters, resonant power converters, wireless charging systems, musical instrument pickups, and seismometers. The formula reveals that resonant frequency is inversely proportional to the square root of both inductance and capacitance — decreasing either L or C increases f₀, and vice versa.

The physical picture: energy oscillates between the inductor's magnetic field and the capacitor's electric field. When the capacitor is fully charged, all energy is electric (½CV²). The capacitor then discharges through the inductor, transferring energy to the magnetic field. When fully discharged, all energy is magnetic (½LI²). The inductor's collapsing field recharges the capacitor with opposite polarity, and the cycle repeats. Without resistance, this would continue indefinitely — with resistance, oscillations damp out exponentially at rate α = R/(2L).

The angular resonant frequency ω₀ = 2πf₀ = 1/√(LC) in radians per second is often more convenient for mathematical analysis involving Laplace transforms and phasor diagrams. Many circuit equations are simpler expressed in terms of ω than f.

The characteristic impedance Z₀ = √(L/C) in ohms is a property of the LC network that determines the voltage-to-current ratio of oscillations and the impedance transformation properties of the resonant circuit. In LC transmission line models, Z₀ equals the characteristic impedance of the equivalent transmission line. For LC matching networks, Z₀ determines the transformation ratio between source and load impedances.

Practical applications: AM radio tuning (f₀ varies from 535 kHz to 1,605 kHz by varying C with a variable tuning capacitor); LC oscillator circuits in clocks and communication systems; medical imaging (MRI resonant frequency f₀ = γB₀/2π depends on static magnetic field B₀ and gyromagnetic ratio γ); and inductive power transfer (Qi charging operates at 87–205 kHz, resonant at f₀ of the coil+capacitor tank).