What is the unit for resonant frequency? where \$\omega_0 = \frac{1}{\sqrt{LC}}\$? Is it just \$HF^{-1}\$?
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\$\begingroup\$ Note: Capacitance has units "F," not "C." \$\endgroup\$– ShamtamCommented Jun 12, 2013 at 0:23
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\$\begingroup\$ There is no "resonant frequency", there is only "resonance frequency": users.ece.gatech.edu/mleach/misc/resonance.html \$\endgroup\$– Alfred CentauriCommented Jun 12, 2013 at 1:56
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2\$\begingroup\$ @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' \$\endgroup\$– Matt TaylorCommented Jun 12, 2013 at 8:11
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3 Answers
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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.
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5\$\begingroup\$ It's actually rad/sec, you need to add an extra factor of 2pi to get to Hz. \$\endgroup\$ Commented Jun 12, 2013 at 0:25
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\$\begingroup\$ yes, rad/s, thanks thats what i initially thought \$\endgroup\$– azzaCommented Jun 12, 2013 at 0:31
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\$\begingroup\$ 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. \$\endgroup\$ Commented Jun 12, 2013 at 2:44
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1\$\begingroup\$ 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 \$\endgroup\$– starblueCommented Jun 12, 2013 at 19:43
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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
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\$ \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.
answered Jun 12, 2013 at 2:29