RLC Series Resonance

Equivalent forms

f_0 = \frac{1}{2\pi\sqrt{LC}}

X_L = X_C \;\text{at resonance}

Geometry of the components alone fixes the resonant frequency — resistance only sets how sharp the peak is, not where it sits.

Symbol	Name	SI unit	Dimension	Range
$\omega_0$	Resonant angular frequencyoutput Angular frequency of peak response	rad/s	T⁻¹	1000 – 1000000000
$L$	Inductance Series inductance	H	M·L²·T⁻²·I⁻²	0.000001 – 0.001
$C$	Capacitance Series capacitance	F	M⁻¹·L⁻²·T⁴·I²	1e-12 – 0.000001

Unit systems

Symbol	SI
$\omega_0$	rad/s
$L$	H
$C$	F

Where it holds

Valid for linear, lumped, time-invariant RLC elements. Departs at frequencies where stray capacitance/inductance and radiation matter, and for nonlinear or saturating components.

Dimensional analysis

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

Discovery

Heinrich Hertz / Oliver Lodge · 1887

Lodge demonstrated electrical resonance ('syntony') between tuned circuits, and Hertz's spark-gap resonators that proved Maxwell's waves were themselves tuned LC loops — the seed of all radio tuning.

Try this

Tune a radio and one station leaps out of the static. What makes a circuit pick one exact frequency and ignore the rest?

A series RLC circuit has L = 100 μH and C = 100 pF. Find the resonant frequency where the current peaks.

Research status: stable

Real-world applications

Radio/TV tuning
Band-pass and notch filters
Wireless-power coupling
Metal detectors and RFID

Common misconceptions

Resistance shifts the resonant frequency — it doesn't; R only broadens or narrows the peak
At resonance the inductor and capacitor voltages are zero — they are individually large (Q times the source) but opposite and cancel

Experimental verification

Sweep a signal generator across a series RLC and watch the current (or resistor voltage) peak sharply at

f_{0}

; measured peaks match

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

within component tolerance.

Derivation

Series impedance

Z = R + j(\omega L - 1/\omega C)

.

The magnitude is minimized (current maximized) when the imaginary part vanishes:

\omega L = 1/\omega C \Rightarrow \omega ^{2} = 1/LC \Rightarrow \omega _{0} = 1/\sqrt{LC}

.

Limiting cases

L \to

large or

C \to

large⟶

\omega _{0} \to 0

Heavier 'electrical mass' and 'softer spring' both slow the oscillation.

R \to 0

⟶

Q \to \infty

, peak → infinitely sharpWith no loss the resonance becomes a razor-thin spike (ideal tuner).

What if…

What if you halve the capacitance?

Resonant frequency rises $by \sqrt{2} (\approx 41$ %) — turning a variable cap is exactly how you tune a dial.

What if you add series resistance?

The peak frequency stays put but the response broadens (lower Q) — you hear more adjacent stations.

1

100 μH with 100 pF

Given ·

L:: 0.0001
C:: 1e-10

Find · f_0

Steps

$LC = 10^{-4} \times 10^{-10} = 10^{-14}$
$\sqrt{LC} = 10^{-7} s$
$f_0 = 1/(2\pi \times 10^{-7}) \approx 1.59\times 10^{6} Hz (AM$ band)

Result · About 1.59 MHz — right in the AM broadcast band.

LC Resonant Frequency→Inductive Reactance→reactance capacitivebandwidth quality factor