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From: John Woodgate
Subject: Re: Deriving H for magnetic cores
Date: Tue, 17 Dec 2002 20:31:00 +0000
Organization: JMWA Electronics Consultancy
Reply-To: John Woodgate
NNTP-Posting-Date: Tue, 17 Dec 2002 22:23:18 +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 Tue, 17 Dec 2002:
>I have just about had it with inconsistent H and B formulas swirling
>around in catalogs and on the net for computing the magnetic field
>strength and flux density in magnetic cores for inductors and
>transformers. They often are shown with poorly explained units, or
>other conditions of applicablity. Therefore I will make this post
>painfully explicit, as all good science and engineering should be, so
>that others can have a hope of following along.
>I don't want to see any contrary formulas in response to what I am about
>to do, unless they come with full derivations from first principles,
>either Ampere's Law or Biot-Savart Law (not sure if the latter is
>applicable to regions of inhomogeneous permeability, though.)
>SI units please. If you have a formula you think is the right one, and
>it's in Oersteads, inches, bushels, or any other silly units, please
>convert to SI before refuting my results.
>I will derive H using Ampere's Law applied to a toroidal core wrapped
>with N turns carrying I amperes. First off, let's at least agree on
>Ampere's Law for magnetostatics:
> H_vec . dl_vec = I
You've already gone over the heads of half the people here, I suspect.
You are considering a far too sophisticated model to analyse, too. Start
with a single circular conductor.
>where H_vec is the magnetic field strength vector in (A/m), the period
>represents the vector dot product, and "dl_vec" is the vector
>differential unit of length along contour C. I is the current in
>amperes flowing through the closed contour.
>Let's orient the toroid so it's plane is in the x-y plane, and its
>center is at the origin. Now wrap N turns around the toroid and connect
>to a current source so that current is flowing in the positive z
>direction at points where the inner diameter of the toroid intersects
>the x-y plane.
>By the right hand rule and other standard conventions (for a cylindrical
>coordinate system) we should now agree that:
>H_vec = phi_hat*H
>dl_vec = phi_hat*r*d_phi
>phi_hat is the unit basis vector in the phi (angle measured
>counterclockwise from the x axis in the x-y plane) direction, r is the
>radius in meters of a particular circular contour C in the x-y plane and
>centered about the origin, and d_phi is the differential of the angle
>phi. Note that the contours that are sensible are ones within the
>toroidal core, so that if the inner radius of the toroid is a, and the
>outer radius is b, then a<=r<=b.
>Finally we should agree that the current flowing through any contour C
>of radius r, is N*I, in amperes.
>Now we can squabble about the assumptions that must be made in order to
>use Ampere's Law. The most important assumption is that H=|H_vec| is
>uniform over any C. Any counters to this assumption?
>We expect to see variation in H vs. r ; that will be accounted for in
>The other assumption regards the accounting for the magnetic flux
>developed by H as being located almost entirely inside the core. This
>is a subject about which I am still confused. I grew accustomed to
>calculating fields in uniform regions of permeability, using the
>Biot-Savart Law before learning to use Ampere's Law. What happens to H
>and B when a blob of material with a different permeablity is placed in
>the region is not clear from the Biot-Savart Law. It seems to create a
>much more complex problem in which the magnetic boundary conditions and
>Ampere's Law or the Biot-Savart Law (now both almost intractable to
>compute) become all coupled together into a horrible system of equations.
Just leave out such high-level complications. This was supposed to be a
learning process, wasn't it?
>But how I think the argument is supposed to go for our carefully
>simplified problem is like this: Since the core is of high permeablity
>relative to epsilon_0 (the permeability of free space), then virtually
>all of the flux will be within it. Thus, all of the H will be within it
>as well. If someone can explain this better, I'd appreciate it.
No, you are back on track now.
>Finally, let the toroid be thin in the z dimension so that H is
>relatively uniform over variations in z for a given r. (Does this matter?)
> H_vec . dl_vec = I
> phi_hat*H . phi_hat*r*d_phi = N*I
>H*r d_phi = N*I
>and finally we arrive at a formula:
>H=N*I/(2*pi*r) in units of (A/m) of course.
Yes, but a FAR simpler model would have given you that, and we could ALL
>Next we generalize this to a core with an effective path length of Le in
>meters. The point of this is that we are dealing with all sorts of core
>geometries, and so it is necessary to generalize the result to cores
>other than the toroid. We define the effective length Le to mean the
>following: Le is that length of a contour C chosen such that the H over
>that contour, multiplied by the cross sectional area of the core (taken
>perpendicular to dl_vec at the intersection of the area surface with C),
>yields the same value of flux as the result of the general calculation
>of the flux PHI:
>PHI = B_vec . ds_vec
>With Le defined in this manner then we can disconnect the core geometry
>from the above formula with one further step, that is to replace the r
>dependence with a contour length dependence. (Note that this whole
>argument for Le is related to one or another of the mean value theorems
>of mathematics, but I can't explain it rigorously anymore.)
Well, you are still right, but your reasoning is arcane.
>Since the perimeter of a circle is P=2*pi*r, then the contour length
>Le=2*pi*r, so that the formula becomes:
>H=N*I/Le in units of (A/m).
>This should apply to any core geometry, provided that Le is known.
>Note the absence of any other constants in this formula.
>The Ferroxcube catalog lists the following formula:
>H=I*N*sqrt(2)/Le in (A/m)
>I am pretty sure that they are speaking of the peak magnetic field
>strength when I is a sinusoidal current in RMS amperes.
Yes. Because peak H gives peak B and peak B is what might cause
>But they didn't
>make this clear, and the frustration of having to figure out what they
>meant, as well as having to figure out what a whole bunch of other
>magnetics engineering formulas mean is what has led me to decide to
>ignore most of them and derive them on my own.
I agree; there is a lot of 'TRADITION' in the Fiddler on the Roof sense
in this field (pun?, what pun?).
>I will have to deal with B on another day, as I have to get to work.
B = uH. u is in general not constant but a non-linear function of H and
previous history - it has 'memory'. That makes things really exciting.
Forget the vectors; KISS.
Regards, John Woodgate, OOO - Own Opinions Only. http://www.jmwa.demon.co.uk
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