# Is natural frequency of an LC circuit equal to angular frequency ? Why don't the units match?

When I was reading the L-C circuit in my textbook I came across the derivation of equations of instantaneous charge and current. Which is no problem, but when I got to the derivation to current . It said something like -

Which I don't quite understand. It says in the third line, Im = Wn Qm

.Where Wn (omega naught) => natural frequency of the LC circuit . And Im (I max) => maximum current. Qm ( Q max ) => maximum charge.

So, in other words it's trying to say that - Wn = 1/T Which is certainly not correct, as W(omega) is angular frequency and not just frequency. I.e. it has the unit of radians per second and not (sec.)^-1. BUT we know that -

So this could help us as by substituting the units of L(coefficient of self-inductance) and C (coefficient of capacitance), we can get the unit of Wn ... BUT SURPRISINGLY We would get - Wn = 1/ (sec.^2)^1/2 Which ultimately is Wn = (sec.)^-1

So basically it's saying that angular frequency is equal to just "frequency". Which contradicts the fact their units are different. HOW IS THIS TRUE ?

Then again, one might ask, how current can oscillate "angularly" ... ?

So my question is , are both natural frequency and angular frequency the same thing ? And if they are, then can someone plz explain this to me -

And also of course...

https://en.m.wikipedia.org/wiki/LC_circuit

To summarize, I just wanted to know few things, -

1. How is angular frequency (radians per second) ( has kind-of "physical meaning") be equal to just frequency(cycles per second) [has all meanings {physical and electrical (oscillation of electrons)} but here, In our case, just focusing to electrical] ?

2. If they are both equal, then, how can electrons oscillate angularly ?

3. If angular frequency is not equal to frequency , then, is my textbook wrong ?

Even though I am poor in English, I am desperately trying to get the reader the "feel" of my question .

Also, I think I'm missing something, probably my basics are not so clear. Anyways,

Thank you very much for the answers. And sorry for poor English.

• Radians aren't a real dimension. "rad/s" is the same thing as "1/s". Angular frequency is just defined as $2 \pi$ times frequency. – knzhou Jan 10 at 22:15
• – The Photon Jan 10 at 22:20
• If Wn = 1/T (i.e. = Fo ) , then how can it be, at the same time,. Wn = 2π Fo ? – DEEKSHANT Jan 11 at 11:49
• Why do you think it's trying to say that $\omega_0 = 1/T$? I don't see that anywhere. – Javier Jan 11 at 14:54
• Can anyone tell me that, is my textbook justified by doing : Im = Wo Qm ? Wouldn't it be Im = Wo Qm/ 2π ? – DEEKSHANT Jan 11 at 15:16

This is an angular frequency not a frequency. This is common practice in all areas of physics/math/engineering etc. The frequency f0 = w0/(2*pi) and this is equal to 1/T, where T = period of oscillation. When dealing with oscillating or periodic equations governed by a term like sin(2*pift) the scale factor of 2*pi ensures that the function complete one cycle when t completes one frequency. f, not 2*pi*f is the quantity that most naturally relates to a physical measure of something (exception = rotational motion where w = d(theta)/dt).

So my question is , are both natural frequency and angular frequency the same thing ?

No. A resonant system has a natural frequency which is characteristic of the system. From the Wikipedia article Natural frequency:

Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving or damping force.

The natural frequency of system may be expressed in units of ordinary frequency $$(\mathrm{Hz})$$ or angular frequency $$(\mathrm{\frac{rad}{s}})$$

Both ordinary frequency and angular frequency have the same dimension of inverse time $$[\mathrm{T^{-1}}]$$ but ordinary frequency express the number of cycles per second while angular frequency expresses the number of radians per second. Since there are $$2\pi$$ radians in one cycle, it follows that angular frequency is just the ordinary frequency multiplied by $$2\pi\mathrm{\frac{rad}{cycle}}$$.

Typically, the symbol for ordinary frequency is $$f$$ or $$\nu$$ while the symbol angular frequency is $$\omega$$ thus

$$\omega = 2\pi\nu$$

You can see this clearly in the expression for the energy of a photon of frequency $$\nu$$ which can be expressed as

$$E_\gamma = h\nu = \frac{h}{2\pi}2\pi\nu = \hbar\omega$$

Radians are defined as the arc length over the radius - $$\theta=arc length/radius$$. But both can be kept in any units and you would have said that this is dimensionless. Essentially it is.

People could measure $$\theta$$ this way, or by degrees, which are two completely different methods, explaining the same logic. In physics, angles are generally measured in radians.

So when you say $$f=\omega/2\pi$$, $$\omega$$ is describing the same logic $$f$$ does, except in an angular context.

In this way, you could relate $$\theta$$ measured in radians and degrees as follows:

$$\theta_r$$ = $$\theta_0$$ $$\pi/180$$. These are not fundamentally different, it is just a form of definition.