Resonance frequency

Prof. Dr. Thorsten Jungmann (last updated 2026-03-02)

Resonance frequency: The frequency at which inductive and capacitive reactance cancel each other out – the resonant circuit oscillates freely.

The resonance frequency (German: Resonanzfrequenz) is the frequency at which a resonant circuit made up of an inductor and a capacitor shows its greatest tendency to oscillate. At resonance, the inductive and the capacitive reactance cancel each other out: in the series resonant circuit, the impedance becomes minimal (purely real); in the parallel resonant circuit, it becomes maximal.

1 Quantity symbol and unit

The quantity symbol of the resonance frequency is $f_{\mathrm{res}}$ (also $f_0$).
The unit is the hertz:

$$[f_{\mathrm{res}}]=\mathrm{Hz}$$

Alternatively, the resonance angular frequency ${\omega }_{\mathrm{res}}=2\pi f_{\mathrm{res}}$ is given in ${\mathrm{s}}^{-1}$.

2 Derivation

The resonance condition is that the inductive reactance equals the capacitive reactance:

$$X_L=X_C\Rightarrow {\omega }_{\mathrm{res}}L=\frac{1}{{\omega }_{\mathrm{res}}C}$$

Solving for ${\omega }_{\mathrm{res}}$:

$${\omega }_{\mathrm{res}}^2=\frac{1}{LC}\Rightarrow {\omega }_{\mathrm{res}}=\frac{1}{\sqrt{LC}}$$

This yields Thomson’s oscillation equation (German: Thomson’sche Schwingungsgleichung):

$$f_{\mathrm{res}}=\frac{1}{2\pi \sqrt{LC}}$$

where:

• $L$ is the inductance in henry (H)
• $C$ is the capacitance in farad (F)

The formula applies ...

… exactly to the undamped resonant circuit ($R=0$). In real resonant circuits with a finite $R$, the resonance frequency deviates slightly; for typical quality factors $Q>10$, the difference is negligible.

3 Practical and worked examples

3.1 RLC series resonant circuit

A resonant circuit consists of $L=10\mathrm{mH}$ and $C=100\mathrm{nF}$.

$$f_{\mathrm{res}}=\frac{1}{2\pi \sqrt{10\cdot 10^{-3}\mathrm{H}\cdot 100\cdot 10^{-9}\mathrm{F}}}=\frac{1}{2\pi \sqrt{10^{-9}{\mathrm{s}}^2}}=\frac{1}{2\pi \cdot 31.62\cdot 10^{-6}\mathrm{s}}=\underline{5.03\mathrm{kHz}}$$

At this frequency, the impedance of the series resonant circuit is minimal and purely real: ${\underline{Z}}_{\mathrm{res}}=R$.

3.2 VHF receiver

A radio receiver is to be tuned to $f_{\mathrm{res}}=100\mathrm{MHz}$. The inductor has $L=50\mathrm{nH}$. Required: the capacitance.

$$C=\frac{1}{(2\pi f_{\mathrm{res}}{)}^2\cdot L}=\frac{1}{(2\pi \cdot 100\cdot 10^6\mathrm{Hz}{)}^2\cdot 50\cdot 10^{-9}\mathrm{H}}=\underline{50.7\mathrm{pF}}$$

4 Typical orders of magnitude

Application	Resonance frequency	Typical components
Mains filter	$50\mathrm{Hz}$	$L\sim \mathrm{mH}$, $C\sim \mathrm{mF}$
Audio frequencies	$20\mathrm{Hz}$ to $20\mathrm{kHz}$	$L\sim \mathrm{mH}$, $C\sim \mathrm{nF}$ to $\mu \mathrm{F}$
AM radio (medium wave)	$0.5$ to $1.6\mathrm{MHz}$	$L\sim \mu \mathrm{H}$, $C\sim \mathrm{nF}$
FM radio (VHF)	$87.5$ to $108\mathrm{MHz}$	$L\sim \mathrm{nH}$, $C\sim \mathrm{pF}$

5 Example phrasings

• ‘The resonance frequency of the series resonant circuit is $f_{\mathrm{res}}=5.03\mathrm{kHz}$.’
• ‘At resonance, the inductive and capacitive reactance cancel each other out.’
• ‘The quality factor $Q$ of the resonant circuit determines the sharpness of the resonance and thus the bandwidth.’
• ‘By changing the capacitance, you can shift the resonance frequency – every tunable receiver uses this principle.’

6 Related quantities

• Frequency: $f_{\mathrm{res}}$ is a particular frequency
• Angular frequency: ${\omega }_{\mathrm{res}}=2\pi f_{\mathrm{res}}=\frac{1}{\sqrt{LC}}$
• Inductance and capacitance: determine $f_{\mathrm{res}}$
• Reactance: at resonance $X_L=|X_C|$, total reactance $X=0$
• Impedance: purely real at resonance ($\underline{Z}=R$)
• Quality factor: $Q=\frac{f_{\mathrm{res}}}{\Delta f}=\frac{1}{R}\sqrt{\frac{L}{C}}$, determines the bandwidth
• Bandwidth: $\Delta f=\frac{f_{\mathrm{res}}}{Q}$, frequency range around the resonance

7 Related components

• Inductor: stores energy in the magnetic field, determines $f_{\mathrm{res}}$ together with $C$
• Capacitor: stores energy in the electric field, determines $f_{\mathrm{res}}$ together with $L$
• Resistor: determines the damping and the quality factor of the resonant circuit