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Calculate resonant frequency, angular frequency and characteristic impedance for LC tank circuits.
f₀ = 1 / (2π√(LC))
An LC circuit — also called a tank circuit — consists of an inductor and capacitor connected together. Energy oscillates between the magnetic field of the inductor and the electric field of the capacitor at a natural frequency determined by f₀ = 1/(2π√(LC)). At this frequency the inductive and capacitive reactances are equal and opposite, so they cancel.
In a series LC circuit, impedance is minimum at resonance — the circuit passes maximum current at f₀ and acts as a bandpass element. In a parallel LC circuit, impedance is maximum at resonance — it blocks current at f₀ and acts as a notch or trap. Both configurations resonate at the same frequency; the difference is how they interact with the rest of the circuit.
The characteristic impedance Z₀ = √(L/C) determines the Q factor when combined with circuit resistance: Q = Z₀/R for series, Q = R/Z₀ for parallel. Higher Q means a narrower bandwidth and a sharper resonance peak. In radio receivers, LC tanks with Q > 50 can select a single station from a crowded band.
Resonant Frequency (f₀)
f₀ = 1 / (2π√(L·C))Angular Frequency (ω₀)
ω₀ = 1 / √(L·C)Characteristic Impedance (Z₀)
Z₀ = √(L/C)Tune an LC circuit to 1 MHz AM band using a 500 µH ferrite loopstick antenna inductor. Find required capacitance.
C = 1/(4π² × f² × L) = 1/(4π² × 10⁶² × 500×10⁻⁶) = 50.7 pF
Transmitter output tank circuit for the 20 m amateur band (14.0–14.35 MHz). Using L = 2 µH, find C.
C = 1/(4π² × (14.2×10⁶)² × 2×10⁻⁶) = 63 pF · Use 10–100 pF variable cap
LC Resonance Formulas
Resonant frequency: f₀ = 1 / (2π√LC)Quality factor: Q = (1/R) × √(L/C)Bandwidth: BW = f₀ / QCharacteristic impedance: Z₀ = √(L/C)At resonance: XL = XC → ωL = 1/(ωC)L = 100 nH → C = 1/(4π²×(98×10⁶)²×100×10⁻⁹) = 26.4 pF
Use variable capacitor 10–50 pF to tune 88–108 MHz band
L = 10 mH → C = 1/(4π²×(1000)²×10×10⁻³) = 2.53 µF (use 2.2 µF standard)
Actual f₀ = 1/(2π√(10×10⁻³ × 2.2×10⁻⁶)) = 1.072 kHz
Inductive load 100 mH → resonating C = 1/(4π²×50²×0.1) = 101 µF
Use 100 µF / 250 VAC motor-run capacitor
Design Tip
High-Q circuits (Q > 10) have narrow bandwidth and are sensitive to component tolerances. Use 1% tolerance inductors and capacitors for RF applications. Parasitic resistance in real inductors limits achievable Q (typically 20–200 for air-core coils).
The resonant frequency tells you where the peak sits. The Q factor tells you how narrow it is. For a series LC, Q = (1/R)·√(L/C), and the bandwidth between the −3 dB points is BW = f₀/Q. High Q means a tall, narrow, selective peak; low Q means a broad, gentle one. A radio front-end wants high Q to pull one station out of a crowded band; a power-supply filter often wants low Q so it doesn't ring.
Real parts cap your Q whether you like it or not. The inductor's series resistance and the capacitor's ESR both dissipate energy, dragging Q down, and stray capacitance from the layout shifts f₀ off where you calculated it. Air-core RF coils land around 20–200; a lossy iron-cored choke can be far worse.
Q factor and bandwidth (series LC)
Q = (1/R)·√(L/C) BW = f₀ / QA small VHF-range tank built from an air-core coil and a fixed NP0 cap.
f₀ = 1/(2π√(10×10⁻⁶ × 100×10⁻¹²)) = 5.03 MHz
A larger LC combination landing in the low-RF / IF region.
f₀ = 1/(2π√(100×10⁻⁶ × 10×10⁻⁹)) = 159 kHz
Related: RLC circuit adds the resistance that sets Q, capacitor impedance covers the cap's own self-resonance, and you can size parts with the inductor calculator or tune a radiator with the antenna calculator. Reference: LC circuit (Wikipedia).
Did you know? LC circuits were used in early radio sets (1890s–1910s) to tune to specific station frequencies. A variable capacitor mechanically changed capacitance to tune the resonant frequency — the same principle as a modern digital PLL, just analog.