LC Resonant Frequency

Equivalent forms

\omega_0 = \frac{1}{\sqrt{LC}}

T_0 = 2\pi\sqrt{LC}

Resistance only damps the ringing; the frequency itself is set purely by L and C — the geometry, not the loss.

Symbol	Name	SI unit	Dimension	Range
$f_0$	Resonant frequencyoutput Natural oscillation frequency	Hz	T⁻¹	1 – 1000000000
$L$	Inductance Tank inductance	H	M·L²·T⁻²·I⁻²	1e-9 – 0.01
$C$	Capacitance Tank capacitance	F	M⁻¹·L⁻²·T⁴·I²	1e-12 – 0.000001

Unit systems

Symbol	SI
$f_0$	Hz
$L$	H
$C$	F

Where it holds

Describes the undamped natural frequency; with resistance R the damped frequency is slightly lower,

\omega _d = \surd (\omega _{0}^{2} - (R/2L)^{2})

. Breaks down once R/2L approaches

\omega _{0}

(overdamped, no oscillation).

Dimensional analysis

$1/(2\pi \sqrt{[L][C]}) = 1/\sqrt{H\cdot F} = 1/\sqrt{s^{2}} = s^{-1} = Hz \checkmark$

Discovery

William Thomson (Lord Kelvin) · 1853

Kelvin showed mathematically that a capacitor discharging through an inductor oscillates, deriving the √(LC) period years before Hertz built spark resonators around the idea.

Try this

Connect just an inductor and a capacitor and the energy sloshes back and forth forever — an electrical pendulum. How fast does it ring?

An LC tank uses L = 1 mH and C = 1 nF. Find its natural oscillation frequency.

Research status: stable

Real-world applications

Radio oscillators and tuners
Clock and reference oscillators
Wireless power and RFID coupling
Induction heating frequency setting

Common misconceptions

More resistance changes the resonant frequency a lot — light damping barely shifts it
Energy is lost each swing in an ideal LC — it only sloshes between E- and B-fields; loss requires R

Experimental verification

Charge a capacitor, close it onto an inductor, and a scope shows a decaying sinusoid whose frequency matches

1/(2\pi \sqrt{L}C)

; the ring frequency is independent of the initial charge.

Derivation

For an ideal loop,

V_L + V_C = 0 \Rightarrow L d^{2}Q/dt^{2} + Q/C = 0

.

This is simple harmonic motion with

\omega _{0}^{2} = 1/LC

,

so \omega _{0} = 1/\sqrt{LC}

and

f_{0} = \omega _{0}/2\pi = 1/(2\pi \sqrt{LC})

.

Limiting cases

L\cdot C

→ large⟶

f_{0} \to 0

Bigger components ring more slowly — a heavy, soft electrical pendulum.

L\cdot C

→ small⟶

f_{0} \to GHz

and upTiny L and C give the microwave/RF resonances used in radios and clocks.

What if…

What if you quadruple the capacitance?

Frequency halves — $f_{0}$ scales as $1/\sqrt{C}$ , so $4\times C$ gives $\tfrac{1}{2} the$ frequency.

What if there's some series resistance?

The tank rings at a slightly lower, damped frequency and the oscillation decays — a real, lossy resonator.

1

1 mH with 1 nF

Given ·

L:: 0.001
C:: 1e-9

Find · f_0

Steps

$LC = 10^{-3} \times 10^{-9} = 10^{-12}$
$\sqrt{LC} = 10^{-6} s$
$f_0 = 1/(2\pi \times 10^{-6}) \approx 1.59\times 10^{5} Hz = 159\,\mathrm{kHz}$

Result · About 159 kHz.

RLC Series Resonance→Inductive Reactance→reactance capacitivespeed of light em