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From: Cursitor Doom <cd999666@notformail.com>
Newsgroups: sci.electronics.design
Subject: Re: Omega
Date: Mon, 1 Jul 2024 17:52:17 -0000 (UTC)
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On Sun, 30 Jun 2024 21:17:53 -0400, Phil Hobbs wrote:

> On 2024-06-30 09:51, 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...
>>>>
>>>>
> Okay, once more with feelin'.  Hopefully this is a bit more coherent
> throughout.
> 
> 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.(*))  Pardon my waving my arms a
> bit--that way we can avoid DEs and Fourier integrals.  Here goes.
> 
> 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.
> 
> 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 inductance couples
> to dI/dt rather than I.  This comes right out of 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).    (5)
> 
> 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.
> (**)
> 
> 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 supercool HTML-mode types is ω = 2πf.  The
> factor ωL comes in exactly the same way as resistance, except for
> the frequency dependence and quarter-cycle phase shift, so it's called
> _reactance_, as noted above.
> 
> 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.
> (Once you get to Fourier transforms, keeping the 2*pi explicit saves
> many blunders, it turns out.)
> 
> 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)
> 
> This rises from 0 to ~0.63 in 1s, 0.9 in 2.3s, and 0.95 in 3.0s.
> 
> 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, 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 linear phase shift with frequency
> 
> theta ~= - j omega L/R.  (14)
> 
> 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 resistance
> falls to half of its DC value.
> 
> If we're using the series LR as a lowpass filter, omega_0 is the
> frequency that divides the passband, where the signal mostly gets
> through, from the stopband, where it mostly doesn't.  It's worth noting
> that if you extrapolate the low-frequency straight line (14), it passes
> through 1 radian at omega_0 as well.
> 
> 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.
> 
> <Correcting brain fart due to trying to do too many things at once>
> 
> Thus 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). Weird, right? What's
> up with that?
> 
> One way of understanding it is that a square pulse has a lot more
> high-frequency components than a sine wave.  To make our 1-s decaying
> exponential resemble a 1-s pulse a bit more closely, we need it to start
> from 0 and go back to 0.  A first try would be making it symmetric.
> That costs you in bandwidth, because going up takes as long as going
> down, so you have to speed up the time constant.
> 
> If we speed it up by a factor of 2, the amplitude reaches 1-exp(-1) ~=
> 0.63V before starting down again.  By the inverse scaling relation, that
> doubles the bandwidth, getting us to 2 rad/s.  To make it a bit more
> square-looking, we could speed it up some more.  Getting up to 90% of
> full amplitude takes 2.2 time constants, which notionally takes us to
> 4.4 rad/s.
> 
> Someplace in there we have to start using Fourier integrals, because
> otherwise we'll start thinking that a perfectly square pulse has
> infinite bandwidth, which it doesn't.  To avoid that, perhaps you'll
> take my word that some more math will show that an actually rectangular
> pulse gets you up to 2*pi rad/s, i.e. 1 Hz.
> 
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