Integrals of waveforms?

[Argus25]

Al,

There is something of interest in that blue table, the rms value of the 50% sine light dimmer.

If you wanted to calculate that by integrating the squares of the wave from pi/2 to pi and from (3/2)pi to 2pi and take the root etc and find the rms value. You can find proofs of this on the net. It is just as though those two 1/4 waves are added like the sine wave has been half wave rectified, in terms of the power delivery to a load. Which leads me to an interesting story.

There was a series of exam questions posed in NZ in an electrical engineering paper that went something like this:

1) You have 230V rms AC sine wave applied to a 5000 Ohm resistor generating heat, what is the power dissipated in the resistor ? (nearly every student gets this right)

2) You place a silicon power rectifier in series with the resistor in question 1. The resistor cools down a little but it is still quite hot. Ignoring rectifier losses, what is the power dissipated in the resistor now ?

3) Considering the half wave rectification situation described in question 2, what DC voltage would you apply to the 5000 Ohm resistor in another circuit to generate the same amount of heat ?

If the student knew off the cuff that the rms value of a half wave rectified sine wave was 0.5Vpeak or 1/root2 of its rms value the second two questions are a breeze. Or perhaps they were slick at math and could calculate the rms value by integrating 0 to pi of a squared sine function and taking the root etc as you will see in the mathematical proofs.

But there is another way to work it out, even if the student didn't know the mathematical tricks:

If half of the waveform has gone missing, there must be half the power delivery. So the student simply solves for a new V using V(squared)/R = half the power value they calculated for question 1.

(Since power is proportional to the square of the voltage, then to get half the power the voltage must have been scaled down by a factor of 1/root2 or 0.7071. So the rms value of the half wave rectified voltage must be 0.7071 x 230 = 163v, and therefore the power in the resistor must be 163(squared)/R or about half what is was on full wave and the DC voltage required 163V to answer the two more difficult questions).

So its good there are a few ways to skin the cat with some of these questions.

[julie_m]

There's certainly no harm in trying to understand the theory a little better, while you are unable to do much in the way of practical work. The worst that can happen is you say "s*d this for a game of soldiers, it's too hard for me" and even then you're no worse off; the best that can happen is, you'll gain a valuable insight through understanding the theory better.

At the very least, knowing multiple routes to the same answer helps you check your working-out.

[Argus25]

There's certainly no harm in trying to understand the theory a little better..... the best that can happen is, you'll gain a valuable insight through understanding the theory better.

At the very least, knowing multiple routes to the same answer helps you check your working-out.

This is very true.

Einstein made the remark about machines something like: A machine should be as simple as possible to perform its function, but no simpler.

I think this also applies to solutions to problems. (provided they give the correct answer that is)

So the remark could be reformatted to something like:

"The solution to a problem should be as simple as possible, but no simpler".

In electronics, if the correct answer can be acquired with a simple energy equivalency, rather than half a page of calculus, the former has to be better.

This is one of the concepts that always astonished and impressed me about Einstein's work. Take E= mC^2. It is so simple in its form but it explains so much. It was known that the energy of a photon was hf, so by making mC^2 equal hf you can calculate the momentum of a photon, which is a notion associated with discrete particles, bringing wave and particle theory closer together.

The other simple formula that produces extremely complex results, resembling all the forms we see in nature is the Z = Z^2 + C where you feed the equation output back into its input (the Mandelbrot set) in the field of fractal geometry. It is hard to believe such a simple equation form produces the complex and lifelike shapes of nature, but it does.

Hugo.

[Radio Wrangler]

Mathematicians are rather fond of 'elagance' even to the point where the elegance makes things harder to visualise.

When having to explain things to people, it's good to be able to show several ways until one clicks and understanding dawns. With a group of victi... er... people, different people click with different methods. At the end, everyone should have 'got it' but there is value in everyone seeing the multiple routes. Life is good when you can see that what you thought were separate things just turn out to be different aspects of one thing. The universe gets a little simpler.

David

[Al (astral highway)]

Originally Posted by julie_m

The worst that can happen is you say "s*d this for a game of soldiers, it's too hard for me" and even then you're no worse off; the best that can happen is, you'll gain a valuable insight through understanding the theory better.

Hey Julie, it's not so much in my case that I find this onerous, it's just that I don't have a formal training in it: for someone like me who likes both precision and accuracy when they are appropriate (another huge discussion here) I can find this frustrating at times and inspiring at others.

This takes us into the realm of different types of 'knowing.' There's the explicit explanation of things according to rules, and then there's a kind of knowing that is able to see deeply into the subtle behaviours that are going on.

The kinds of (pulse) circuits I'm working with can produce devastating behaviours. Eg, in a fault condition, it's possible for everything silicon to be wiped out from the front end of the circuit to the (digital) back end, even with good design, and even with isolation (eg, gate transformers), shielding and separate power supplies.

It is a good idea for a design/ builder of such circuits to know the stress that components are under, especially things like transient voltage suppressors and so on.

Also, we could bring to mind the 'coin shrinking' pulse experiment in which a coin experiences extraordinary forces.

Here is data from one such experiment (approx)

Time: Current:

0s 0A
10us 7KA (yes, Kilo Amps)
20uS 42KA
30us 56KA
40uS 42KA
50uS 35KA

by 80uS, all is flat again.


One physics lab measured a pulse of 138,000A with an induced voltage in the coin of 192V. Total power, 26.5 MegaWatts. Because of Lenz's Law - an induced EMF produces a current whose magnetic field opposes the change in original magnetic flux - the work coil is blown to pieces and is encased in a concrete box.

Back to topic: I need to know exactly where in my circuit to put TVS, where to put decoupling and how much. I don't want to lace the thing with unnecessary components, but it helps me to understand all possible behaviours, including all failure modes. (Einstein's 'A machine should be as simple as possible to perform its function, but no simpler', as pointed out by Hugo.)

Hugo: very interesting comments there on Einstein and esp on his energy mass equivalency insight. That blew my mind when I first encountered it.

[Al (astral highway)]

Originally Posted by Argus25

The other simple formula that produces extremely complex results, resembling all the forms we see in nature is the Z = Z^2 + C where you feed the equation output back into its input (the Mandelbrot set) in the field of fractal geometry. It is hard to believe such a simple equation form produces the complex and lifelike shapes of nature, but it does.

I love fractals and am always looking for obsessively iterative patterns in nature. Thanks for this, Hugo!

I also love infinite sums and approximations.

Srinivasa Ramanujan (a latter-date Euler in some ways) found:

2Pi *SQRT 2= 99^2/1103

And also:

(9^2 + 19^2/22)^1/4 =??

yup, it's 3.14159265262...!

[kalee20]

Originally Posted by astral highway

You might recall that you sent me those amorphous alloy toroids (grey and green) a couple of years ago now, maybe longer - along with some notes you prepared and tech data? Between ops I've been gradually immersing myself in making and testing my own magnetics. This is proving really handy as I can now make a pretty decent gate transformer, and more recently, have built my own current transformers when to buy a commercial one of the same spec would cost silly money!

Glad the cores are useful! (By the way, high-permeability cores make super current transformers, but suffer massively if there's a bit of DC flowing. If there is, you could be fooled into thinking there's less AC than there actually is!)

Looking at the currents is highly educational! You can calculate to your heart's content, but if you've neglected a few nH of inductance, things can change massively. Probing really does teach oneself the sensitivity of fast-switching circuits to physical layout!

[Al (astral highway)]

Originally Posted by kalee20

Glad the cores are useful! (By the way, high-permeability cores make super current transformers, but suffer massively if there's a bit of DC flowing.

Good to be reminded, Peter. Thank you. I've designed DC blocking (cap) into the gate drive circuit, for example. I hadn't considered it in other applications such as RF chokes in the power line; do however recall that when I met Ed (Dinning ) recently he suggested me that air-cored chokes were better in this application for the same reason - saturation .