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From: Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net>
Newsgroups: sci.electronics.design
Subject: Re: Omega
Date: Sun, 30 Jun 2024 21:17:53 -0400
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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.

Cheers

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