Re: Omega

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.
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com