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From: Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net>
Newsgroups: sci.electronics.design
Subject: Re: Omega
Date: Sun, 30 Jun 2024 13:51:42 -0000 (UTC)
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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 ω = 2πf.
> 
> 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.
> 
> 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