LC Oscillation Frequency

Equivalent forms

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

T = 2\pi \sqrt{LC}

Exactly the harmonic-oscillator formula ω = √(k/m) with L playing the role of mass (inertia of current) and 1/C the role of stiffness — a perfect mechanical–electrical analogy.

Symbol	Name	SI unit	Dimension	Range
$f$	Resonant frequencyoutput Natural oscillation frequency of the LC circuit	Hz	T⁻¹	0 – 100000000
$L$	Inductance Inductance of the coil	H	M·L²·T⁻²·I⁻²	0.000001 – 0.01
$C$	Capacitance Capacitance of the capacitor	F	M⁻¹·L⁻²·T⁴·I²	1e-12 – 0.000001

Unit systems

Symbol	SI	CGS	Imperial
$f$	Hz	Hz	Hz
$L$	H	abH	H
$C$	F	cm	F

Where it holds

Valid for ideal lossless LC circuits. Real circuits add a damping term from resistive losses; high-Q radio circuits approach the ideal but never reach it.

Dimensional analysis

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

Discovery

William Thomson (Lord Kelvin) · 1853

Kelvin analyzed the discharge of a Leyden jar through a coil and derived the oscillation period 2π√(LC) — the result Hertz later used to predict the frequency of his radio-wave experiments.

Try this

How does a radio tuner pick one station out of dozens — using only a knob, a coil, and a capacitor?

An LC tank circuit has L = 50 μH and C = 200 pF. What is its resonant frequency?

Research status: stable

Real-world applications

AM/FM radio receivers tune the variable C to select stations
Wireless RFID tags and NFC coils resonate at 13.56 MHz
Crystal-oscillator alternatives in low-cost RF design
Magnetic-resonance imaging (MRI) RF coils tuned to the Larmor frequency

Common misconceptions

Energy is conserved (in the ideal case), only its form alternates between E-field and B-field
f does not depend on the initial amplitude — purely on L and C
Doubling both L and C halves the frequency (factor of 2 inside the square root)

Experimental verification

Excite the tank with a brief pulse and observe ringing on an oscilloscope; the measured period matches

2\pi \sqrt{LC}

to within component tolerance. Frequency-meter readings of Colpitts/Hartley oscillators reproduce the same result.

Derivation

Kirchhoff's loop:

L dI/dt + Q/C = 0

with

I = dQ/dt

gives

L d^{2}Q/dt^{2} + Q/C = 0

.

This is SHM with

\omega ^{2} = 1/(LC)

, hence

f = 1/(2\pi \sqrt{LC})

.

Limiting cases

C \to 0

⟶

f \to \infty

Vanishing capacitance gives huge restoring 'stiffness' — oscillation becomes infinitely fast (limited in practice by parasitic capacitance).

L \to \infty

⟶

f \to 0

Huge inductance gives huge 'inertia' — oscillations become arbitrarily slow.

Resistance present⟶ f slightly lower, amplitude decaysAn RLC circuit oscillates

at \omega _d = \surd (1/LC - R^{2}/(4L^{2}))

and dies away as energy dissipates.

What if…

What if C is reduced by a factor of 4?

Frequency doubles — f goes as $1/\sqrt{C}$ .

What if a resistor is added in series?

The oscillation decays exponentially; quality factor $Q = (1/R)\sqrt{L/C}$ measures how many cycles survive.

What if the inductor's core saturates at high current?

L drops, so f rises — the circuit becomes nonlinear and generates harmonics (an undesirable distortion in RF amplifiers).

1

Radio tank circuit

Given ·

L:: 0.00005
C:: 2e-10

Find · f

Steps

Compute $LC = 5\times 10^{-5} \times 2\times 10^{-10} = 10^{-14}$
$\sqrt{LC} = 10^{-7}$
$f = 1/(2\pi \times 10^{-7}) \approx 1.59\times 10^{6} Hz$
That's solidly inside the AM broadcast band (530–1700 kHz region)

Result ·

f \approx 1.59\,\mathrm{MHz} (AM

broadcast band)

2

Capacitor for a 1 MHz tank

Given ·

L:: 0.0001
f target:: 1000000

Find · C

Steps

Rearrange: $C = 1/((2\pi f)^{2} L)$
$(2\pi \times 10^{6})^{2} \approx 3.95\times 10^{13}$
$C = 1/(3.95\times 10^{13} \times 10^{-4}) = 1/3.95\times 10^{9}$
$C \approx 2.53\times 10^{-10} F = 253\,\mathrm{pF}$

Result ·

C \approx 253\,\mathrm{pF}

Self-Inductance of a Solenoid→Energy Stored in a Capacitor→Energy Stored in an Inductor→RC Circuit Charging→