LC Resonance and Frequency-Selective Circuits

An inductor and capacitor form a resonant system because energy can move back and forth between the inductor's magnetic field and the capacitor's electric field. In an ideal lossless circuit, this exchange continues indefinitely at a natural frequency. The resonant frequency is f0 = 1 / (2π√LC), where L is inductance in henries and C is capacitance in farads. This relationship appears in RF tanks, oscillators, filters, impedance matching, wireless power, power converters, and sensor interfaces.

At resonance, the magnitude of inductive reactance and capacitive reactance are equal. Inductive reactance is XL = 2πfL and increases with frequency. Capacitive reactance is XC = 1 / (2πfC) and decreases with frequency. When the two magnitudes match, their energy storage effects balance. Whether the circuit behaves as a low impedance or high impedance at resonance depends on whether the inductor and capacitor are arranged in series or parallel and how losses are modeled.

This calculator accepts inductance in microhenries and capacitance in nanofarads, then converts both to base SI units for the formula. It reports frequency, period, and reactance at resonance. The result is ideal because it does not include equivalent series resistance, winding resistance, dielectric loss, core loss, parasitic capacitance, PCB trace inductance, or load coupling. Those effects determine the real bandwidth and amplitude.

Manual Calculation Steps

For L = 10 uH and C = 100 nF, convert to 10 x 10^-6 H and 100 x 10^-9 F. The product LC is 1 x 10^-12. The square root is 1 x 10^-6. The frequency is 1 / (2π x 1 x 10^-6), or about 159.155 kHz. The period is the inverse of that frequency, about 6.283 microseconds. The inductive reactance at resonance is 2πfL, about 10 Ohm, and the capacitive reactance magnitude is also about 10 Ohm.

Quality Factor and Bandwidth

Real resonant circuits have loss, usually described by quality factor Q. A higher Q means lower loss and a narrower bandwidth. In a filter, Q affects selectivity and ringing. In an oscillator, Q affects phase noise and frequency stability. In a power circuit, high Q can create large circulating currents or voltage magnification. Resonance is therefore useful but not automatically safe; stored energy and damping must be reviewed.

Component tolerance can shift resonance substantially. Inductors may be 10 percent or 20 percent tolerance, and ceramic capacitors can lose capacitance under DC bias. Temperature also changes both values. A circuit tuned in simulation with nominal components may land outside the desired band in production unless trimming, calibration, or tolerance analysis is included.

Engineering Applications

LC resonance is used in radio front ends, crystal oscillator load networks, class-D and class-E amplifiers, wireless charging coils, EMI filters, buck-converter output filters, antenna matching networks, and proximity sensors. In RF work, a resonant tank can select one band while rejecting others. In power electronics, resonance can be used intentionally for soft switching or unintentionally create ringing that violates EMI limits.

Layout is part of the calculation at high frequency. PCB traces add inductance, pads add capacitance, and nearby copper changes fields. The ideal LC formula gives the starting point, but a final design should include parasitic extraction, simulation, and measurement with the actual components and board geometry.

Resonance can also appear unintentionally. A long supply trace, decoupling capacitor, cable, connector, or motor winding can form an LC network that rings when excited by a fast edge. The ringing frequency often reveals the effective inductance and capacitance in the path. Damping can be added with resistance, snubbers, ferrites, or layout changes, but the right fix depends on whether the problem is voltage overshoot, EMI, conducted noise, or stress on a switching device. Calculating the ideal frequency is a useful first clue during that debugging process.

Measurement technique matters. A long oscilloscope ground lead can add inductance and create or exaggerate ringing. Use short ground springs, appropriate probes, and bandwidth limits deliberately when validating resonant behavior. Otherwise the measurement setup may become part of the circuit being diagnosed.