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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.
A circuit is at resonance when inductive and capacitive reactance are equal and cancel, which happens at
f0 = 1 / (2π√(LC))
Notably, resistance does not appear: it does not shift the resonant frequency, only how sharp the resonance is. With L = 100 mH and C = 100 µF, f0 = 50.33 Hz regardless of what resistance is present.
At series resonance the two reactances cancel and only resistance remains, so impedance is at its minimum and current peaks — potentially to a dangerous degree, since the voltage across L and across C individually can greatly exceed the supply voltage. At parallel resonance the opposite occurs: impedance reaches a maximum and the current drawn from the supply is minimal, while a large circulating current flows round the LC loop itself.
For a series circuit the quality factor is
Q = (1/R)√(L/C) and bandwidth = f0/Q
Continuing the example with R = 10 Ω: Q = (1/10)√(0.1/0.0001) = 3.16, giving a −3 dB bandwidth of 50.33/3.16 ≈ 15.9 Hz. High Q means a narrow, selective peak — what a radio tuner wants. Low Q gives a broad, gentle response — what a power-supply filter usually wants, because a high-Q filter will ring and amplify any disturbance near its resonant frequency.
Resonance is not always designed in; sometimes it appears uninvited. An input filter's inductance resonating with its capacitance, or cable inductance with a decoupling capacitor, can amplify switching noise instead of attenuating it. Damping — deliberately adding resistance to lower Q — is the standard fix, trading a little efficiency for stability.
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.
f0 = 1/(2π√(LC)) with inductance in henries and capacitance in farads. For 100 mH and 100 µF this gives about 50.33 Hz.
No. Resistance affects the sharpness of the resonance and the peak current, but the frequency at which the reactances cancel depends only on L and C.
At series resonance impedance is minimum and current peaks. At parallel resonance impedance is maximum and supply current is minimal, while a large current circulates inside the LC loop.
How sharp the resonance is. Q = (1/R)√(L/C) for a series circuit, and the −3 dB bandwidth is f0/Q. High Q gives a narrow, selective peak; low Q gives a broad, well-damped response.
Yes, and at series resonance it routinely does — by a factor of roughly Q. A circuit with Q = 10 on a 50 V supply can develop around 500 V across the inductor and the capacitor, which is a genuine safety and component-rating issue.
Because a high-Q filter rings. If switching noise or a load step falls near the resonant frequency, a lightly damped filter amplifies it rather than attenuating it. Deliberately adding damping resistance lowers Q and keeps the filter stable.
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