Re: Omega

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

--
Dr Philip C D Hobbs  Principal Consultant  ElectroOptical Innovations LLC /
Hobbs ElectroOptics  Optics, Electro-optics, Photonics, Analog Electronics