RC Time Constants and 2*Pi

[dknguyen]

Now, this is a really dumb question, but in sensor datasheets, I have noticed that all the recommended LP Filter cutoff frequencies never seemed to match up when I calculated the RC time constant using their recommended values of R and C. It's just recently I have realized it's always been off by a factor of 2*Pi.

Now the time constant is T = RC

I have always used f = 1/(RC)

The datasheets use:
f = 1/(2*Pi*RC)

Where does this factor of 2*Pi coming in?

 

[dknguyen]

Nvm. I got it.

 

[ljcox]

What does Nvm mean?

 

[dknguyen]

Shorthand for Nevermind.

 

[Papabravo]

Now, this is a really dumb question, but in sensor datasheets, I have noticed that all the recommended LP Filter cutoff frequencies never seemed to match up when I calculated the RC time constant using their recommended values of R and C. It's just recently I have realized it's always been off by a factor of 2*Pi.

Now the time constant is T = RC

I have always used f = 1/(RC)

The datasheets use:
f = 1/(2*Pi*RC)

Where does this factor of 2*Pi coming in?

For those who are interested in the answer it is the difference between a frequency expressed in Hertz(Hz.) or the arcane cycles per second (cps), and a frequency expressed in radians per second. 2*pi is the constant of proportionality between those two measures of frequency, because there are 2*pi radians in one cycle.

 

[dknguyen]

You sure that's why the 2*Pi is there? It makes some sense, but consider that the units of the time constant:
t=RC has units of seconds.

So since 1/seconds = Hertz, it would seem that f = 1/t = 1/(RC) which apparently isn't true as far as cutoff frequency goes. It would seem that it completely bypasses radians per second. That's the mistake I was making. I think it's because the transfer function of a low-pass filter is

1+(jwRC)^-1

so even though RC is in "seconds" it's in a slightly different context because w is in radians per second. (Not sure what I am trying to say here).

 

[Papabravo]

Consider that 2 is a dimensionless constant and pi is also a dimensionless constant. R and C both have dimensions which when multiplied together have units of seconds.

A time constant often shows up in circuits with exponential behavior like the RC and the RL circuit. In these cases one time constant is the time it takes for the output to have a value of e^-1. Said another way it is when t = RC so that t/RC = 1 and e^-(t/RC) is about 37% of the original value.

If we look at the complex impeadance of an ideal capacitor, it is (jwC)^-1, where j is the imaginary unit, w is the natural or radian frequency equal to 2*pi*f, and C is the capacitance. It is easy to mix these things up and it depends on what you are trying to do.

I took from your initial post that you were interested in the cutoff frequency of a low pass fileter which includes R and C and w.

The 3 dB point of a single pole low pass filter is

| 1 + jwRC | = sqrt(2)

sqrt( 1^2 + (wRC)^2) = sqrt(2)

1 + (wRC)^2 = 2

w = (RC)^-1

and finally

2*pi*f = (RC)^-1  implies that

f = (2*pi*R*C)^-1

Does this help?

 

[dknguyen]

Yes it does. THank you.

 

[santan]

RC Time Constants

Can anyone explain me about T=RC how did it get. i mean to prove it..

 

[Electronman]

Can anyone explain me about T=RC how did it get. i mean to prove it..

**broken link removed**

 

[crutschow]

Some of the confusion is perhaps from going between the time and frequency domain. The RC time constant definition comes from the time domain. The 2PI factor can be considered as part of the transformation from the time domain to the frequency domain.

 

[MrAl]

Hi,


The physical reasoning is that the phenomenon of frequency in nature is different
than the way we want to be able to look at it. Nature came first, we as humans
came second. Nature 'invented' the way the phenomenon of frequency came about,
while we reinvented frequency to make it more convenient to our way of thinking.
This means that when we want to look at nature we have to convert between
nature's true form and our modified form, and that requires a constant equal to 2*pi.

Chronologically i believe we understood time before we understood frequency, and
that we wanted to be able to state things as occurring at a rate per second because
we already knew what a second was by then. We didnt yet know that we needed to
use the constant 2*pi to understand nature itself because we didnt yet have this
deeper understanding of nature. As soon as we became aware of nature more exactly
in this way, we realized we needed to convert between our old understanding and
the new, more natural understanding.

 

[Elahayahu]

I just discovered the reason for the 2pi and I was searching to be sure. This is my answer to why the 2pi is there.

A wave frequency f is the reciprocal of the wave period of oscillation T, where...

f = 1/T or T = 1/f

Frequencies f are often confused with angular frequencies w, where...

w=2*pi*f or f=w/(2*pi) or w=2*pi/T or T=2*pi/w

A system natural frequency is commonly written w_n "omega sub-n", where...

w_n = 2*pi*f_n where f_n is the frequency of the natural undamped oscillation of the system.

The time constant tau makes the negative exponential decay part of a system response take the form...

exp(-t/tau)

A stable underdamped system response to an excitation may take a form containing a negative exponential decay function within, such as...

y(t) = A*exp(zeta*w_n*t)*sin(Bt+C)

where zeta is the damping ratio,

zeta = (critically damped time constant)/(underdamped time constant)
critically damped time constant, tau_cd = 1/(w_n)
underdamped time constant, tau_ud = 1/(zeta*w_n)

overdamped --> zeta>1 (no oscillation)
underdamped --> zeta<1
critically damped --> zeta = 1

The time constant and the period of oscilation are two different things.

Underdamped systems have a natural frequency and a damped natural frequency w_d.

w_d = (w_n)*(sqrt(1-zeta^2)), where 0 <= zeta <= 1

The frequency of oscillation is effected by the damping ratio.

So, the point I am trying to make is that the "natural frequency" of a system is not a frequency. It is 2*pi*f.

The frequency of oscillation of an underdamped signal is something like

f_d = w_d/(2*pi)

And the period of oscillation of an underdamped signal is something like

T_d = (2*pi)/w_d

I hope this helps.