When to scale by 2pi when doing time dependent analysis ?

[MathWizard]

When doing basic fixed frequency analysis with phasors, I'd always be watching out for when to convert/scale from freq f in pure 1/s to w(omega) in rad/s. I'm used to rads as a ratio of lengths and so the units can cancel, but I have no problem just using them as a unit [rad]. I know putting units into infinte series like expotentials can make problems too.

But now when doing time dependant analysis, I found I'm supposed to call a time constant say tau=RC or L/R in seconds , and so it's inverse to me should of course be it's inverse with 1/s=Hertz. But then right away I went to scale with 2pi and got the wrong answer. So I just dropped doing it, for TDAnalysis. (I'm learning EE at home, not in school or anything)

But in online lectures it's not f, it's omega, the angular freq, and they stressed that, but I don't have to scale with 2pi or 1/2pi anymore ? ?

And what about if you take a function like Q(tau)=exp(tau)*[sin(tau) - exp(j*tau) +tau]
Will there be mixed equations where in 1 term , you would calulate 1 omega , but in another term , it would be scaled  2pi ?

[Doctorandus_P]

I'll make an amazing revelation:
pi has to do with circles.

That means you use it for converting between rad/s and Hz. Hz is the number of circles per second.

Not just any RC time constant has to do with circles. Time does not run in circles. Last time I checked it just went one way.

Do not try to remember formulas and how to apply them, but think about them until you understand them.

[MathWizard]

Sure and planes and complex geometry, and changing things.

[pwlps]

The phase (in radians) of a periodic signal can be written as omega*t or 2pi*f*t.  If you define a time constant like tau=RC  it should be used exactly like t in these expressions, hence 1/tau is used like omega.

I'm used to rads as a ratio of lengths and so the units can cancel, but I have no problem just using them as a unit [rad]. I know putting units into infinte series like expotentials can make problems too.

Actually all this has nothing to do with the radian definition, rather with the fact that all trigonometric functions take arguments in radians. You could in principle define them with arguments e.g. in degrees and define omega in degree/s, although this wouldn't be practical and many formulas (Euler, series expansions) would get more complicated.  (it was just a digression for math wizards    )

And what about if you take a function like Q(tau)=exp(tau)*[sin(tau) - exp(j*tau) +tau]
Will there be mixed equations where in 1 term , you would calulate 1 omega , but in another term , it would be scaled  2pi ?

This function of course doesn't make sense, the argument of a sin can only be a phase. 

[MathWizard]

 I'll have to try them more and side by side, and I yeah I know we don't take sin of secs, I forgot to say w was 1

[MathWizard]

Hmmm so 2pi*f litterally gives me the phase shift between Vo and Vin of an RC filter, as a time, or it's inverse, an angular frequency. I'm applying 5KHz to it on the scope. I would have learned that before, but long since forgetten, and so thats that phase in RLC 2nd ODE's when you put it in magnitude phase form.

Ok thanks that helps, in the past I would work out the time between to crests of different waves. Hmmmm