# What is the unit for resonant frequency?

What is the unit for resonant frequency? where $\omega_0 = \frac{1}{\sqrt{LC}}$? Is it just $HF^{-1}$?

• Note: Capacitance has units "F," not "C." – Shamtam Jun 12 '13 at 0:23
• There is no "resonant frequency", there is only "resonance frequency": users.ece.gatech.edu/mleach/misc/resonance.html – Alfred Centauri Jun 12 '13 at 1:56
• @AlfredCentauri except that 'resonant' does not apply to the frequency, but to the construct which is resonating - it is the 'frequency at which the construct is resonant', as such, the 'resonant frequency [of the construct]' is perfectly valid and correct. Also, as a side note, our language evolves through the acceptance of idioms which become common usage. Try insisting that no-one calls a 'vacuum cleaner' a 'Hoover' – Matt Taylor Jun 12 '13 at 8:11

In the simplest way possible:

L is in henries (H) - $\Omega \cdot s$.

C is in farads (F) - $\dfrac{s}{\Omega}$

Multiply both and you have $s^2$. Take the square root, you have $s$. Invert it, you have $\dfrac{1}{s}$, that is, $\frac{rad}{s}$.

If the expression is written as $\omega_0 = \dfrac{1}{2\pi\sqrt{LC}}$ the resonant frequency is in hertz.

• It's actually rad/sec, you need to add an extra factor of 2pi to get to Hz. – helloworld922 Jun 12 '13 at 0:25
• yes, rad/s, thanks thats what i initially thought – azza Jun 12 '13 at 0:31
• It's rad per second, but in dimensional analysis radians are the same as no unit at all because a radian is the ratio of two lengths. – The Photon Jun 12 '13 at 2:44
• But dimensional analysis doesn't tell the whole story, it would for example let you confuse torque with energy. Also compare en.wikipedia.org/wiki/Steradian – starblue Jun 12 '13 at 19:43

From Wiki:

$H = \Omega s$ (Wikipedia entry for Henry)

$F = \dfrac{s}{\Omega}$ (Wikipedia entry for Farad)

Thus:

$\dfrac{1}{ \sqrt{HF} } = \dfrac{1}{s} =$ Hz, as expected for a frequency

$\omega$ is "angular frequency" in $\frac{rad}{s}$,

$f$ is natural frequency AKA "frequency" in $Hz$

So your title has an inherent conflict in it, you ask for frequency but talk about $\omega$.

Also Radians are dimensionless so the rad in $\frac{rad}{s}$ is a place holder for a scaling factor.