Resonant Frequency LC Calculator
Resonance (Hz)
159.2
How it works
An LC circuit (inductor and capacitor) resonates at a frequency where inductive and capacitive reactances are equal and cancel: f₀ = 1 / (2π × √(LC)). At resonance, the circuit stores energy oscillating between magnetic field (inductor) and electric field (capacitor).
**Series vs. parallel resonance** Series LC: at resonance, impedance is minimum (theoretically zero for ideal components) — maximum current flows. Voltage across inductor and capacitor are equal and opposite, potentially much larger than source voltage (voltage magnification = Q). Parallel LC (tank circuit): at resonance, impedance is maximum — minimum current drawn from source. Used as bandpass filters and oscillator tank circuits.
**Quality factor (Q)** Q = ω₀L/R = 1/(ω₀CR), where R is the series resistance. High Q: narrow bandwidth, high frequency selectivity, voltage/current magnification. Low Q: broad bandwidth, more damped. Q = f₀ / BW (center frequency divided by 3 dB bandwidth). Radio tuners need high Q to separate closely spaced stations.
**Applications** Radio tuning: variable capacitor adjusts resonant frequency of LC tank to select a station. Crystal oscillators: quartz crystals have very high Q (10,000–100,000), providing precise, stable resonant frequency. RF filters: bandpass and bandstop filters use LC networks to select or reject frequency bands. Wireless power transfer (WPT/inductive charging) uses resonant coupling at matched frequency.
**Parasitic effects** Real components have parasitic elements: capacitors have series inductance (ESL), inductors have parallel capacitance. These parasitics limit usable frequency range and shift actual resonance from calculated value. Above a component's self-resonant frequency, an inductor behaves as a capacitor and vice versa.
Frequently Asked Questions
- Series LC resonant circuit has minimum impedance (theoretically 0) at f₀ = 1/(2π√LC) — maximum current passes. Parallel LC has maximum impedance at f₀ — minimum current from source. For a bandpass filter with f₀ = 10 MHz and Q = 50 (bandwidth = 200 kHz): choose L = 1 µH, then C = 1/(4π²f₀²L) = 1/(4π² × 10¹⁴ × 10⁻⁶) = 253 pF. Q = ω₀L/R gives required series resistance R = 2π × 10⁷ × 10⁻⁶/50 = 1.26Ω (mostly winding resistance). Adjust L and C values while maintaining the ratio L/C for desired Q; component tolerances and parasitic effects require iteration in practice.
- Q (quality factor) = f₀ / BW = ω₀L/R = 1/(ω₀CR). High Q: narrow bandwidth (sharp filter), high voltage/current magnification at resonance, slow energy decay. Low Q: wide bandwidth (broad filter), low magnification, fast decay. Example: f₀ = 1 MHz, Q = 100: BW = 10 kHz. Q = 10: BW = 100 kHz. AM radio needs Q ≈ 50–100 to separate stations 10 kHz apart. Crystal oscillators achieve Q = 10,000–100,000. Air-core inductors have very high Q (>100) at RF; ferrite-core inductors have lower Q (~50) due to core losses. Higher frequency generally allows higher Q for a given inductor size.
- Ferroresonance: in power distribution systems with long underground cables (capacitive) connected to lightly loaded transformers (inductive), the LC resonant frequency can coincide with the power frequency (50/60 Hz) — causing extremely high overvoltages (3–5× nominal) that damage transformers and equipment. Series resonance in power factor correction capacitor banks: if a capacitor bank resonates with the system inductance at a harmonic frequency (250 Hz, 350 Hz for 5th and 7th harmonics), harmonic currents are amplified. Detuned reactors (shifting resonance away from harmonics) prevent this — standard practice in industrial capacitor banks.
- Quartz crystals have Q = 10,000–100,000 because: mechanical vibrations in quartz have extremely low damping (high mechanical Q), piezoelectric coupling converts mechanical resonance to electrical resonance with minimal loss, and quartz's thermal coefficient of expansion is very low (reducing frequency drift with temperature). In comparison, a copper coil inductor has Q of 50–200 — limited by copper winding resistance. Crystal Q is 100–1000× higher than coil Q, giving crystal oscillators frequency stability of ±1–100 ppm vs. ±1,000–10,000 ppm for LC oscillators. TCXO (temperature-compensated crystal oscillators) achieve ±0.1 ppm stability.