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An inductor and capacitor together form a resonant circuit that naturally oscillates at one frequency, f0, where their reactances cancel. The quality factor Q measures how sharp that resonance is — a high Q means a narrow, selective peak with low losses; a low Q means a broad, damped response. The bandwidth is the width between the two half-power (−3 dB) frequencies.
| Quantity | Formula |
|---|---|
| Resonant frequency | f0 = 1/(2π√(LC)) |
| Quality factor (series) | Q = (1/R)√(L/C) |
| Bandwidth | BW = f0/Q = R/(2πL) |
| Half-power frequencies | f1, f2 = f0(√(1+1/4Q²) ∓ 1/2Q) |
At resonance a series RLC circuit has minimum impedance (just R), so current peaks; a parallel LC "tank" has maximum impedance, so it rejects that frequency.
The frequency f0 = 1/(2π√(LC)) at which an inductor's and capacitor's reactances are equal and cancel, so the LC circuit naturally oscillates and responds most strongly.
The quality factor measures how sharp and low-loss a resonance is. High Q gives a narrow, selective peak; low Q gives a broad, heavily damped response.
Bandwidth = f0/Q. A higher Q means a narrower −3 dB bandwidth around the resonant frequency, so the circuit is more selective.
The two frequencies (f1 and f2) either side of resonance where the power drops to half (the voltage to 0.707). The gap between them is the bandwidth.
At resonance a series RLC has minimum impedance (current peaks), while a parallel LC tank has maximum impedance (it rejects that frequency). The resonant frequency formula is the same.
Decrease L or C. Since f0 ∝ 1/√(LC), halving either component raises the frequency by about 41%.
Lower series resistance, or a higher L/C ratio (Q = (1/R)√(L/C)). Low-loss components and good layout give higher Q.
A high-Q tuned circuit has a narrow bandwidth, so it selects one station and rejects nearby ones. Too high, though, and it may cut the signal's sidebands.
In a series RLC the reactances cancel, leaving only R, so impedance is minimum and current is maximum. In a parallel tank the impedance is maximum at resonance.
Yes — stray inductance and capacitance can form parasitic resonances that ring and radiate EMI. Damping resistors or snubbers tame them.
For light damping the shift is negligible, so f0 = 1/(2π√(LC)) is used. Very high resistance (low Q) slightly lowers the peak frequency.
Simple LC circuits have Q of a few to a few hundred; high-quality RF resonators and crystals reach thousands or more.
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