From: John Woodgate
Subject: Re: Deriving H for magnetic cores
Date: Thu, 19 Dec 2002 11:24:03 +0000
Organization: JMWA Electronics Consultancy
Reply-To: John Woodgate
NNTP-Posting-Date: Thu, 19 Dec 2002 11:59:07 +0000 (UTC)
X-Newsreader: Turnpike (32) Version 4.01 <5Z8C9wtxbnpWyFnyfFzqmVF739>
I read in sci.electronics.design that Christopher R. Carlen
wrote (in ) about
'Deriving H for magnetic cores', on Wed, 18 Dec 2002:
>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.
To deal with all your points, this is going to be ONE LONG POST!
To begin with, this exchange on 18th December looks like a 'knot' in
gestation to me:
Roy McCammon wrote:
[nicely worded discourse snipped]
> 1. A contour such that H (a vector) = constant.
> Well that doesn't work even in the special case.
> 2. A contour such that | H | = constant. You probably
> mean it in this sense. That seems plausible, (requiring
> only continuity?), but that sense doesn't pertain to the
> discussion because in the special case we needed the
> fact that H . dL = constant on the contour of interest.
> 3. A contour such that H . dL = constant
> OK, #3 is what I mean. I should have expressed the
> idea more clearly. That type of contour exists in
> the special case, but it is not obvious to me that it
> exists in all cases. The special case is easy to
> work because H . dL = constant. And that is why
> I consider the special case to be not representative.
> In fact, I consider it pathological.
Wouldn't it be actually true that the "contour of constant H" in which
we are interested is the one in which H=|H_vec| is constant (your #2),
since then you could take H (scalar magnitude) out of the integral,
which is the necessary mathematical step to make use of Ampere's Law.
Then you would *just* have to be able to express the unit vector of H in
terms of r, phi, and perhaps z. Not that this would be possible of
Aren't integrals like this parametrized so that you would integrate a
parametric vector function that would follow the contour over which H is
constant and in which the unit directional component is expressed in
terms of the parameter?
>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?
It depends on exactly what you are trying to do. If you really want to
prove everything ab initio, you are probably right, but that is physics,
not engineering. Starting from Biot-Savart is not, however ab initio,
and if you start with that, I see no objection to going from the simple
to the complex geometry, as I outlined, rather than considering the
complex geometry first.
> 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.
I can see no reason at all why determining the average magnetic path
length requires a 3D surface integral. Maybe I misunderstood what Le
>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.
I'm afraid we're not communicating. Your toroid model is one in which
almost the only simple thing is the evaluation of Le (if Le is magnetic
path length). And the pure mathematical approach to finding the Le of an
EI stack must be horrendously difficult in the extreme. Pure math for
field problems doesn't like rectangular structures. (;-)
> 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.
That is pure^2 math, if you rule out the possibility that it's a sort of
definition of the 'lim' operator. I don't see it as relevant to
designing transformers. But perhaps you don't actually want to design
>> 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?
It's yet another way of expressing Ampere and B-S mathematically. All
three ways can be shown to be equivalent, but I don't have the proof.
IIRC, Laplace and B-S differ only in which angle is used in the
equation, and the angles in question are complementary, so cost [theta]
>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?
No, I don't want or need to use anything so sophisticated. I'm not
saying you CAN'T get results that way, but for probably over 90% of
electrical engineers, it isn't the method of choice.
>> 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
I'm not sure what you mean. Is that a rhetorical question or do you want
me to post a *simple* statement of Laplace's theorem? If the latter,
here it is from the text book:
Magnetic field dH at a point A of slant distance r from, and due to, a
conductor element dx carrying a current I:
dH = Icos[theta]dx/4[pi]r^2,
where [theta] is the angle between r and the perpendicular y from A to
Integration over theta gives the field at A due to a long straight
conductor as I/2[pi]y.
>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.
Yes, this is good if you can do it, and everyone you want to communicate
with can understand it. Those conditions are not by any means always
>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?
Well, I don't know whether it's normal or not, but I'm pretty sure that
MOST EEs never touch a grad or a div after graduating, although they no
doubt enjoy touching a curl or three. (;-)
>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.
Well, you are fortunate in having strong math skills. I don't, and it
has been a source of difficulty for me, BUT I can often get help with
the math if I really can't find a way to do it myself.
>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.
I suppose strictly you are correct, but they don't need to 'show' as
vectors in the analysis.
> 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.
YOU might not be able to, because math is the communication medium of
choice for you, but that doesn't apply to most EEs, let alone most
people in general.
>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.
I think that's generally the case. It's easy to visualise a field
weakening or strengthening over an area, but to visualise the vector
elements diverging or rotating *at every point, in general*, is much
more difficult. CAD vector field plots, those with the variable-size
arrows, help a bit now, but we didn't have those in 1957!
>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.
Maybe News 2020 can help with that last problem. (;-)
>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.
But there are many halfway houses between what you did, a tour de force,
and the technician-level thing of, 'We use this equation to solve that
sort of problem.'
>Well hopefully you know what I'm trying to say. I never regret a heavy
>dose of mathematics.
Bully for you!
> 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.
I agree, but in most cases, your 'first principles' are more 'first'
than mine. (;-)
>And one final thing: I *do* appreciate and consider carefully your
>responses. So thank you.
Well, I'm glad we are communicating to a large extent, even if not
Bend your math round this one, for a little light entertainment.
A circuit has a transfer function H(s) = F(s)/G(s).
s = [sigma] + j[omega]. What is the condition for maximal flatness of
the amplitude response |H(s)|, i.e. it has only one maximum, at s = 0
(low-pass) or as s-> infinity (high-pass), or is a maximum for all s
(all-pass)? The band-pass and band-stop cases are left as an exercise
for the student.
Regards, John Woodgate, OOO - Own Opinions Only. http://www.jmwa.demon.co.uk
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