Re: Omega

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 &omega; = 2&pi;f.
 
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.
 
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!