Re: Omega

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 &omega; = 2&pi;f.  The
factor &omega;L 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(&omega;t) (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
 
Phil Hobbs
 
(*) Rudyard Kipling, "How the rhinoceros got his skin", in *Just So
Stories*, Macmillan, 1902.
 
(**) 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--in physics an advance is a negative phase shift,
whereas in EE it's positive, owing to different sign conventions.
 
(***) Here I've adopted the EE sign convention, and so am using 'j' for
the corresponding square root of -1.  I use 'i' in physics and math
discussions, and that helps keeps everything straight.

Many thanks, Phil. I've printed this out as I have a funny feeling that
despite your sterling attempts at simplifying things, I'm going to need to
read it over and over and over again.....