From: "Christopher R. Carlen"
Subject: Re: Deriving H for magnetic cores
Date: Wed, 18 Dec 2002 18:32:53 -0800
Organization: Sandia National Laboratories, Albuquerque, NM USA
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NNTP-Posting-Date: Thu, 19 Dec 2002 01:31:43 +0000 (UTC)
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John Woodgate wrote:
> I'm in favour of using sophisticated math if it's necessary to solve a
> problem, even if I don't understand the math. But I don't understand why
> you choose a model with 3 dimensional geometry and then get tied in a
> knot about interpretation.
I don't think I got tied in knot. I brought up the assumptions that are
needed to take the general case toward a simple case. If I can't
recognize and grapple with the assumptions that need to be made, how can
I ever have confidence in the final result? Perhaps the only thing that
wasn't needed is the z axis. But eventually to deal with the concept of
Le, it is necessary to deal with the flux, which requires a surface
integral in 3D. To eliminate the surface integral requires an
application of the mean value theorem, even if done only conceptually.
If you can do it all in a simpler manner, then I will counter with
questioning your assumptions. You will ultimately be forced to pull out
the math in order to put down my questions. Example: prove that lim
x->1 x = 1. "Just substitute 1 for x" isn't good enough, and you know
it. The rigorous proof using epsilons and deltas leaves students heads
spinning. But one day I sat there and played with epsilons and deltas
until the light bulb went on.
> I like to use the simplest models that are adequate. In this case,
> consider a very long, thin, straight conductor. Ampere, B-S and Laplace
> all show you that the H contours are circles centred on the conductor,
> and in SI units H = I/2[pi]r.
What is Laplace in this context? Are you talking about the relatively
easier method of calculating B using the vector magnetic potential, as
opposed to Biot-Savart (which isn't helpful with our magnetic core anyway):
del^2 A_vec = -mu J_vec
B_vec = curl A_vec ???
where del^2 is the Laplacian applied to the vector magnetic potential
A_vec, and J_vec is the current density?
> Now bend, twist, roll up, whatever, your conductor. Can you 'break' the
> H-contours by any such process? No.
> You can go from the straight conductor to a single-turn loop, and, with
> a bit of juggling, you can do rectangular and even triangular loops
> (although that made my brane 'urt!). The you can go to long solenoids
> and thin multi-turn loops. NONE of this needs vector calculus. With
> sufficient confidence gained, THEN you can tackle toroids and things.
What is Ampere's Law or the Biot-Savart Law in a form other than vector
calculus? Truly the measure of elegance of any derivation is in being
able to reduce the vector calculus to a special case quickly, or to be
able to make some nice geometric arguments that eliminate the ugly terms
in what would have resulted from a "brute-force" computation.
Your conceptualizing is very nice, and your mental pictures have just
helped me to picture all this more clearly. But what am I going to do
about it? I'll go back to the vector calculus to make it quantitative.
John, isn't it normal for EEs to take a course in applied E&M which is
fully vector calculus based, but with an engineering calculation
emphasis, rather than a theoretical derivation emphasis?
I think that is normal, as that is the type of course I have just
completed, which is part of a BS EE curriculum at a typical university.
Some call it the "weed out course" for the EE program, but whatever
it is, it was awfully useful albeit rather formidable at times.
The question seems to be, is it necessary to start at such a high level
of mathematical generality, working downward toward simpler special
cases, as opposed to starting from some other place? For the derivation
of magnetic fields, where else is there to start from? Even the
simplest case of a straight conductor, requires vectors. At least you
have to start with them, until you can make the geometrical arguments or
whatever to eliminate the ugliness.
I'm afraid I just don't understand how I could understand better, by
using less math.
In the beginning, I thought all this div curl grad stuff was truly
bonkers, even though I could easily carry out the mechanics of
calculating the answers. But no insight, except for the grad, which is
pretty easy to get a feeling for. I was worried that I would go through
the course producing numbers that I would have no idea what they meant.
But after a few months of heavy hitting with the EM homework, and the
many cases in which I went off exploring beyond the scope of the
homework and the course's objectives, the questions that arose about the
key topics like Gauss' Law and Ampere's Law, the displacement current,
etc., eventually light bulbs started going on in my head.
Deriving the telegrapher's equations for transmission lines from the
RLCG circuit model, then taking Maxwell's equations in their full
generality, and tediously developing the phasor representations, then
deriving the wave equations for plane wave propagation, and solving them
for plane wave functions.
The insight gained from having the patience to really sit through all
the steps of the processes listed above, to arrive at the final formulas
we all know, I can't imagine how I would understand things better had I
just been handed the formulas and told to crunch numbers with them.
Well hopefully you know what I'm trying to say. I never regret a heavy
dose of mathematics. If I ultimately arrive at a simple and elegant way
to understand something or derive a formula, I expect it was made
possible because of starting from the first principles at some point,
and letting the insight mature from repeated application.
And one final thing: I *do* appreciate and consider carefully your
responses. So thank you.
Christopher R. Carlen
Principal Laser/Optical Technologist
Sandia National Laboratories CA USA