====== LC resonance ======

**LC resonance** occurs when the reactive effects of an [[inductor|inductor]] and a [[capacitor|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:

<code>
f0 = 1 / (2 π sqrt(L C))
</code>

f0 is in hertz, L in henries and C in farads. A 10 μH inductor with a 100 pF capacitor gives:

<code>
L = 10 × 10^-6 H
C = 100 × 10^-12 F
f0 ≈ 5.03 MHz
</code>

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:

<code>
Q = 2 π f0 L / R
Bandwidth = f0 / Q       (half-power bandwidth)
</code>

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.

===== References =====

  * [[https://openstax.org/books/university-physics-volume-2/pages/15-5-resonance-in-an-ac-circuit|OpenStax: resonance, quality factor and bandwidth]].
  * [[https://openstax.org/books/university-physics-volume-2/pages/14-5-oscillations-in-an-lc-circuit|OpenStax: energy exchange in an LC circuit]].