And only to be used in a zero-energy house. Since I can't turn on the amp because of energy outflow the cap would not have been destroyed.
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Hmm, I never liked courses from the math department. Too short on practical examples. My favorite math course in college was taught by an engineer. The text still sits on my shelf (and gets used). "Advanced Engineering Mathematics" by C.R. Wylie.
Still, I wish I had a better understanding of integration in the complex plane. The theory of residues and all that. I've been wading through Penrose's tome, "The Road to Reality". I=92m in chapter 7 where he talks about complex integration and I find my understanding is lacking.
George H.
Exactly -- I've been told the results of, e.g., Cauchy's Residue Theorem and used it, but it would be nice to go through the derivation someday as well. Granted, there's nothing stopping me from finding and cracking open an appropriate book if I really wanted to do that, so apparently I'm only motivated to the level of, "I'd sign up for a course where then I'll show up and probably learn something" rather than to the level of actually tracking down the material myself. :-)
Here's an alternative when your brain is starting to boil? -->
---Joel
No problem. Wrap a few turns of your antenna around the cap. Be sure to go _counter_ clockwise. (Ask me how I know.) And use magnets, plenty of them, positioned _just so_. That way, the energy you put into your antenna will fill up the cap and thus power the amp, with plenty left over to radiate around the world.
The counterclockwise antenna-magno-restrictive-field will hold the cap together, preventing the dreaded phoomp. Warning: You must get the cap from the Place Joel is talking about.
Ed
It's one of the more remarkable proofs in mathematics. The complex plane is a plane, of course, so you do path integrals, and you get the subject of path independence. The remarkable part is, you get this very specific unit when you integrate around a pole, regardless of what distance you did it at.
This makes sense mathematically, but intuitively, it's like taking the highway from a restaraunt to a bar, and suddenly realizing your position is off by pi/2 miles from when you took the surface streets last time.
On the real line, of course, you can either integrate a function, or you can't. Functions that you can integrate don't have poles in the range (you can't integrate 1/x from -1 to 1 -- how do you know how much infinity you've integrated in each direction?), and have a finite number of holes (points where the limit exists but the function is undefined, like (x^2 +
2x + 1) / (x + 1) has a hole at x = -1; infinite holes are excluded, which takes care of some pathological cases).It makes sense to me to think of a flow field, with rotational flow (curl, if you're in 3D). Complex numbers can be "implemented" by 2x2 matrices, at some expense in redundancy (since half the numbers in the matrix are "unused" in a certain sense). For instance, i ==> [0, 1; -1, 0] is also a
90 degree rotation on a 2D vector. Now, when you integrate around something tangential, regardless of what distance you do it at, you get the total amount of 'circulation'. That makes sense, because the rotation weaker at a distance, but it's a proportionally longer path. If you're thinking of an apparently one-dimensional function, there is no rotation, and you miss the point. So the trick is, the phasor representation of that function rotates about the pole. It's not even a pole -- yes okay, it sticks up towards infinity, but it's more like... a tornado than a pole. That's a far better name for it!Tim
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Hmm I bet there's some nice lecture video's on the web.
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Oh that looks fun, I put Penrose aside for a few days and read some trashy novels.
Oh BTW the conjugate matching is pretty easy to see. I made a voltage source with complex source impedance Z driving a load with complex impedance L. Then the power in the load is something like Vin^2 * (L/Z
+L)^2 * 1/(L). (I need to be a little careful about the 1/(L) term, I think this should be the magnitude, but I'm not sure.) Anyway the term L/Z+L has a maximum when the two complex terms cancel. ie conjugate matching.George H.
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,Thanks Tim, You might enjoy Penrose's book. Lots of math in it. I got mine for only $15, for a 1000+ page book that's a lot of words per dollar.
George H.
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