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From: Cursitor Doom <cd999666@notformail.com>
Newsgroups: sci.electronics.design
Subject: Re: Omega
Date: Sun, 30 Jun 2024 16:38:40 -0000 (UTC)
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On Sun, 30 Jun 2024 13:51:42 -0000 (UTC), Phil Hobbs wrote:

> Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> wrote:
>> On 2024-06-30 03:44, Cursitor Doom wrote:
>>> Gentlemen,
>>> 
>>> For more decades than I care to remember, I've been using formulae
>>> such as Xc= 1/2pifL, Xl=2pifC, Fo=1/2pisqrtLC and such like without
>>> even giving a thought as to how omega gets involved in so many aspects
>>> of RF.  BTW, that's a lower-case, small omega meaning
>>> 2*pi*the-frequency-of-interest rather than the large Omega which is
>>> already reserved for Ohms. How does it keep cropping up? What's so
>>> special about the constant 6.283 and from what is it derived?
>>> Just curious...
>>> 
>>> 
>> As an old colleague of mine from grad school would say, "It just comes
>> out in the math." ;)
>> 
>> The 2*pi factor comes from the time domain / frequency domain
>> conversion, and the basic behavior of linear differential equations
>> with constant coefficients. (That's magic.(*))  For now we'll just talk
>> about LR circuits and pulses.
>> 
>> A 1-second pulse (time domain) has an equivalent width of 1 Hz
>> (frequency domain, including negative frequencies).  That's pretty
>> intuitive, and shows that seconds and cycles per second are in some
>> sense the same 'size'.  The two scale inversely, e.g. a 1-ms pulse has
>> an equivalent width of 1 kHz, also pretty intuitive. (Equivalent width
>> is the mathematical quantity for which this 1-Hz/1-s inverse relation
>> holds exactly, independent of the shape of the waveform.)
>> 
>> Moving gently towards the frequency domain, we have the ideas of
>> resistance and reactance.  Resistance is defined by
>> 
>> V = IR, (1)
>> 
>> independent of both time and frequency.  Actual resistors generally
>> behave very much that way, over some reasonable range of frequencies
>> and power levels.  Either V or I can be taken as the independent
>> variable, i.e. the one corresponding to the dial setting on the power
>> supply, and the equation gives you the other (dependent) variable.
>> 
>> A 1-Hz sine wave of unit amplitude at frequency f is given by
>> 
>> I = sin(2 pi f t),  (2)
>> 
>> and the reactance of an inductance L is
>> 
>> X = 2 pi f L.    (3)
>> 
>> The reactance is analogous to resistance, except that since inductance
>> couples to dI/dt rather than I.  From the definition of inductance,
>> 
>> V = L dI/dt.   (4)
>> 
>> Plugging (2) into (4), you get
>> 
>> V = L dI/dt = L * (2 pi f) cos(2 pi f t) = X_L cos(2 pi f t),    (5)
>> 
>> where X_L is the inductive reactance.
>> 
>> We see that the voltage dropped by the inductance is phase shifted by
>> 1/4 cycle. Since the cosine reaches its peak at 0, where the current
>> (the independent variable) is just going positive, we can say that the
>> voltage waveform is _advanced_ by a quarter cycle, i.e. that the
>> voltage is doing what the imposed current was doing a quarter cycle
>> previously. (This seems like a fine point, but it's crucial to keeping
>> the sign of the phase shift right, especially when you're a
>> physics/engineering amphibian like me--the two disciplines use opposite
>> sign conventions.)
>> 
>> Besides the phase shift, the voltage across the inductance has an extra
>> factor of 2 pi f.  This is often written as a Greek lowercase omega,
>> which for all you slipshod HTML-mode types is ω = 2πf.
>> 
>> Writing the sine wave as
>> 
>> I = sin(ωt) (6)
>> 
>> is faster, but the factor of 2 pi in amplitude keeps coming up, which
>> it inescapably must, and it doesn't even really simplify the math much.
>> 
>> For instance, if we apply a 1-V step function across a series RL with a
>> time constant
>> 
>> tau = L/R = 1 second,    (7)
>> 
>> the voltage on the resistor is
>> 
>> V = 1-exp(-t).    (8)
>> 
>> In the frequency domain, the phase shift makes things a bit more
>> complicated.  If we use our nice real-valued sinusoidal current
>> waveform (6) that we can see on a scope, then (after a small flurry of
>> math), the voltage on the resistor comes out as
>> 
>> V = sin(t - arctan(omega L/R)) / sqrt(1 + (omega L / R)**2). (9)
>> 
>> This is because sines and cosines actually are sums of components of
>> both positive and negative frequency, and which don't behave the same
>> way when you differentiate them:
>> 
>> sin(omega t) = 1/2 * (exp(j omega t) - exp(-j omega t)) (10)
>> 
>> and
>> 
>> cos(omega t) = 1/2 * (exp(j omega t) + exp(-j omega t)). (11)
>> 
>> By switching to complex notation, and making a gentlemen's agreement to
>> take the real part of everything before we start predicting actual
>> measurable quantities, the math gets much simpler.  Our sinusoidal
>> input voltage becomes
>> 
>> Vin = exp(j omega t) (12)
>> 
>> and the voltage across the resistor is just the voltage divider thing:
>> 
>> V/Vin = R / (R + j omega L).  (13)
>> 
>> At low frequencies, the resistance dominates and the inductance doesn't
>> do anything much, just a small phase shift
>> 
>> theta ~= - j omega L/R.
>> 
>> At high frequencies, the inductance dominates.  In the middle, the two
>> effects become comparable at a frequency
>> 
>> omega0 = R/L.
>> 
>> At that frequency, the phase shift is -45 degrees and the amplitude is
>> down by 1/sqrt(2) (-3 dB) and the power dissipated in the resistor
>> falls to half of its DC value.
>> 
>> If we're using the series LR as a lowpass filter, that's the frequency
>> that divides the passband, where the signal mostly gets through, from
>> the stopband, where it mostly doesn't.
>> 
>> So when we think in the time domain, a 1-ohm/1-henry LR circuit
>> responds in about a second, whereas in the frequency domain, its
>> bandwidth rolls off at omega = 1, i.e. at 1/(2 pi) Hz.
>> 
>> With sinusoidal waveforms, we can think of 1 second corresponding to 1
>> radian per second, whereas with pulses, a 1 second pulse has a
>> 1-Hz-wide spectrum (counting negative frequencies).
>> 
>> Thing is, a sine wave varies smoothly and goes through a much more
>> complicated evolution (positive to negative and back) within a cycle,
>> so it just takes longer, by a factor that turns out to be 2*pi.
>> 
>> Cheers
>> 
>> Phil Hobbs
>> 
>> 
>> (*) Kipling, "How the Rhinoceros got his skin"
>> 
>> 
> Belay that last bit—it’s exactly backwards. I’ll fix it when I get back
> from church.
> 
> Cheers
> 
> Phil Hobbs

Jeez, Phil. That's not just a follow-up, it's practically a treatise! 
Thanks for taking all that trouble. It's going to take quite some 
digesting!