Omega

Watch this (your question is addressed at 2:07)

piglet

Thanks, Erich. I did wonder if radians had something to do with it. However, knowing that 2 pi radians = 360 degrees or a full wavelength doesn't help me understand why this figure multiplied by the frequency multiplied by the inductance gives us the reactance of a coil. Small omega therefore equals one second's worth of signal and I don't get how multipying that by the inductance amounts to the reactance!

If a thing 1 unit in radius rotates one time per second, its circumference is 2pi so its rim velocity is 2pi/second.

A capacitor current depends on the rate of change of the voltage.

I = CV/T

A 1 volt peak 1 Hz sine wave has its max rate of change at the zero crossing, and that rate is 2pi volts/second. So 1 farad driven with a

You can Spice all that to get a feel for things.

You'd probably have to master Maxwell's equations before you got it. My second year university math course was all about differential equations and how to integrate them. It never made a great deal of sense to me, but by the time I'd got through it, the differential equation involved in electronics were familiar enough that I didn't have any trouble understanding that radians per second made sense, and it didn't prompt any bursts of curiosity.

That was the point in my career where I noticed for myself that magnetism made sense as the relativistic consequence of the forces between moving electric charges, and got told that Wheeler was writing a textbook around that idea.

If you don't get the right education, some idea can be inaccessible.

Thanks, John. This is the fundamental stuff I just overlook since the need for it so rarely crops up. It's easier to take an abstraction level higher and lose all the vital detail in the process. You're up early today. Off to church are we?

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

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

It's good to understand the basics, but I mostly use Spice these days, even for simple things like voltage dividers.

No, I just woke up early with a zillion ideas. That happens some times.

Today is San Francisco Pride Day (Dikes on Bikes and such) so it's a good day to stay close to home. Traffic and parking and everything will be a nightmare.

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!

I went to one of those once in Berlin when I lived there. God almighty! I've never seen such a complete and utter freak show in my entire life. I just stood there gaping open-mouthed at the procession. It must have been pretty obvious to any of those weirdos on the floats that I wasn't there to celebrate them! I just find it bizarre that parents actually take their kids to see this sort of thing and they're all waving their rainbow flags and whatnot. No kids of mine would be allowed within a million miles of an event like that. No wonder the God-fearing folks reckon the world's going to hell in a handbasket. :(

You've had a number of answers - but not really answering at the "gut" level. Why is 2 pi so important - how does omega get involved in so many aspects of RF?

Every one of the formulas you mentioned has to do with frequency. The unit of measurement for that is Hertz which is CYCLE(s) per second. A cycle's length is 360 degrees regardless of frequency. A CIRCLE's length is 360 degrees regardless of frequency. A circle's length is also 2*pi*r regardless of frequency. Therefore a CYCLE's length (a.k.a wavelength a.k.a. omega) is also 2*pi*r long.

So 2*pi is used in the conversion between the number of degrees (time) and distance (length displacement) or "How much happened ?" (length displacement) "and how long did it take?" time (frequency).

That's what some call the "gut level" understanding aas to why 2*pi appears so often. If you use the math a lot over time it becomes less mysterious - if that's the right term. I guess you develop an intuitive understanding or something like that.

Ed

There's something about electronics and its associated mathematics that I've always found challenging to be honest. Many years ago when I studied medicine and then law I didn't have to do any real work at all to speak of. I just absorbed those subjects easily without even trying. I was often baffled as to why so many students struggled with the topics in those subjects and why they clearly had to work so hard for so long to keep up with the class. I'd skip lectures and assignments with gay abandon but still get top marks. Not so with the hard sciences! Now I can finally empathise with my contemporaries back in those days who had to really make a serious effort to grasp the principles and who just about scraped through. They would never go on to excel in those fields just as I have never and will never excel at electronics. Still, no one can be outstanding at everything in today's world and the days of the true polymaths are long gone.

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

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 ω = 2πf. The factor ωL 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(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. (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.

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

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) ~=

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.

Far from it. In Sydney we have a gay pride march down Oxford Street every year, which I've never watched (though it does show up on the TV news every year). A couple of my neighbours decorate a truck for the procession just outside the building ever year and I say hello to them if I go past them while they are working on it.

It keeps them happy, and NSW is a much better place now that the cops have been persuaded that gay-bashing isn't one of their duties, and now have a contingent marching in the procession.

I'm a bit surprised that freaks like you, with your unbounded passion for fatuous conspiracy theories, don't get together in the same sort of way to show off your unique talents.

Donald Trump's MAGA rallies probably qualify as much the same sort of celebration of rampant irrationality, but they don't get reported that way - or haven't so far.

Could always move to places that tolerate that sort of stuff. Saudi Arabia. Moscow. Abilene, Texas. Lots of options.

But Mr. Larkin seems to prefer it in San Fran and you seem to prefer flitting about western Europe, interesting.

Well, I did warn John about moving to SF many years ago as I could tell the way it was going, but he went ahead anyway and seems to have an uncanny ability to only notice the nice side of the place (mainly by not venturing downtown, it seems).

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.....

A Surprising Collision Phenomenon

... It's because most people are only taught that Pi applies to circles, but it also applies to things that are periodic (like the weights in the video perfectly bouncing off each other and the wall).

But to go deeper, why would something periodic have Pi showing up in it? Because the math we invented to explain periodic things is called Trigonometry. Turns out trigonometry is based on (here it comes) circles.

So, just about anything that has some repeating aspect to it can be described with math that relates it to a circle, and that is why Pi shows up. ...

Danke,

You warned me? What's your real name?

When I decided to move to SF, my friends said "But it's full of gay guys" to which I replied "and lots of smart beautiful women."

Do the math on that.

Right, we avoid the tourist traps and homeless encampments. The hills separate SF into a collection of villages, with even different weather.

It's a great place to be weird. People get it.

But 2*pi is still 2*pi even here.