LC resonance
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LC resonance occurs when the reactive effects of an inductor and a capacitor balance. Energy exchanges between the magnetic and electric fields. Resonant circuits are used for radio tuning, oscillators and filters.
Resonant frequency
For ideal lumped components:
f0 = 1 / (2 π sqrt(L C))
f0 is in hertz, L in henries and C in farads. A 10 μH inductor with a 100 pF capacitor gives:
L = 10 × 10^-6 H C = 100 × 10^-12 F f0 ≈ 5.03 MHz
Doubling capacitance lowers the resonant frequency by a factor of sqrt(2), rather than a factor of two. To halve the frequency while keeping inductance constant, capacitance must increase fourfold.
Series and parallel circuits
In a series RLC circuit, inductive and capacitive reactance cancel at resonance. The input impedance is then the series resistance, so current is greatest for a fixed driving voltage.
An ideal parallel LC circuit instead has zero net susceptance at resonance and an infinite impedance in the lossless model. Real losses and external loading produce a finite impedance. These two arrangements therefore have different uses.
Quality factor and bandwidth
For a simple series RLC circuit with total series resistance R:
Q = 2 π f0 L / R Bandwidth = f0 / Q (half-power bandwidth)
Q is dimensionless. With the 5.03 MHz circuit above and R = 10 Ω, Q is about 31.6 and the half-power bandwidth is about 159 kHz. Loading by a source or a following stage can change these values.
Stray capacitance, coil construction and component tolerances shift a real circuit's resonance. At sufficiently high frequencies, a distributed model may be needed instead of treating every component as a point.