Author Topic: Application of "Novel approach for 3D air gap reluctance calculation"  (Read 3453 times)

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Offline e-pirate

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Hello, folks!
Long story here. I need to wind about a dozen of inductances for driving 8W CFL UV lamps for UV lightbox. I did a small research on the topic and came to a conclusion, that using Iron powder or MPP cores is to expensive in that particular case. On the other hand, there are many types of regular toroidal ferromagnetic cores that are dirt cheap and widely available. They come with initial permeability of 1000 to 2500, making them useless for relatively high current (in my case 0.7-0.8 A according to rough calculations) application due to core saturation. As everybody knows, this problem can be solved by introduction of an air gap into the magnetic path. So, I decided to go that way and wind some gapped inductances. To do so, I need some kind of solution-finding tool to calculate core and winding parameters. The software for such calculations is mostly stick to the E-shaped cores and some standard EPCOS types of core materials like N30, N27, N87. I did some research and wrote a simple program (see the attached file) that can calculate inductance specification from arbitrary toroid size/shape, desired inductance, initial permeability, wire turns and diameter, current, core saturation, fill factor, current density and any combination of this parameters. Not a rocket since, actually, just an implementation of standard inductance related equations with some validity checking. This program let me punch in just any combination of parameters and immediately see the result.

But this not even a half way to the solution! The real problem starts right at the point where the Air Gap break the core (and magnetic path) in two pieces. I fount, that the well known reluctance equation, that can be found in famous Wm. T. McLyman's "Transformer and Inductor Design Handbook 4th Ed" is not accurate enough cause of fringing flux effect and it's impact on the final reluctance. The calculation of the fringing flux effect is simplified, giving lower results in compare to what is really measured. Sure, I can get away with that result and than adapt calculated parameters to what I get on practice by experimenting (furthermore I have a good RLC-meter and can build a rig for measuring saturation current), but this is not a true engineering way (as I will not be able to distribute the result if I need to produce just s slightly different inductance) and currently I have a plenty of time for analytical solution before my home lab will became functional back again.

Fortunately, after a long and hard search, we found a recent article that describes a modern approach for calculation air gap reluctance with improved fringing flux effect calculation. This is achieved by dividing air gap to typical sections and treating the whole thing as a sum of several pre-defined elements, rather than a simple gap. See the attached document: "A Novel Approach for 3D Air Gap Reluctance Calculation" by J.Muhlethaler, J.W.Kolar and A.Ecklebe. Accidentally, I found that Mr. Jonas Mühlethaler is the chair man at the Gecko Simulations - the developer company of the GeckoMAGNETICS simulation tool, implementing that method for core calculations, meaning some serious math and since is involved here.
The problem with the method above is in two basic things. First of all, I was not able to figure out how to use it even for described cases. This is due to my math. It is more than a decade since I did math and physics in university. It seems, that I'm way to stupid for this to be implemented in code and I need help in calculation chain to be implemented in code. The other problem is in core shapes. The described cores are... yap, orthogonal/rectangular: O, E and E-I cores and I, from initial point of view, see no way how to use this methods on toroidal cores. Another dead end. In my opinion, the solution is somewhere near to "Air gap type 1", described in section II (Fig. 4), but I have no clue how to handle the 'high' of the affected core side and it's shape on sides as they are round.

Here is the toroidal core with a gap and magnetic flux for such core. It is a little bit rainbowish, but hope this will help.

Dimensions in Fig. 1:
Cw - core width
Ch - core heigh
Lg - gap length
Ho - affected heigh of the outer side of the core
Hi - affected heigh of the inner side of the core
OD - core outer diameter
ID - core inner diameter

So, help needed.

P.S. In case the solution will be found, I will make the program and source code available under something like GPLv2, so everybody can use this method for accurate gapped toroid inductance calculations.
« Last Edit: October 14, 2015, 03:28:47 pm by e-pirate »
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Offline T3sl4co1l

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Re: Application of "Novel approach for 3D air gap reluctance calculation"
« Reply #1 on: October 13, 2015, 02:02:00 pm »
Ooh, nice ref.

So the secret sauce is (8), where you use it to find a correction factor (from your core's proportions), which then is used as a ratio against the simplified formula, extended to 3D as appropriate.  These multiple steps are necessary because each type of joint considered (I, II, III) has different properties in different directions, which use the different corrections (sigma).

So it's making the assumption that all the fringing flux flows on the four sides of the core, neglecting the exact corners.  Which I would guess loses approximately pi*l / (2*w) of the total fringe contribution.  Not enough that you'd see the difference on the log scale of Fig.6.

(Corners would be that many times worse to account for, because the 3D structure of the joint must be considered, and there probably is not a reasonable formula for calculating them.)

For cut toroids, it would probably be reasonable to assume h is either (I.D.) / 2, or some fraction less (maybe around 4/pi or sqrt(2)/2?).  As long as (I.D.) >> l, it won't be a big difference.

Tim
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Offline e-pirate

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Re: Application of "Novel approach for 3D air gap reluctance calculation"
« Reply #2 on: October 13, 2015, 03:38:56 pm »
I will post a 3D model of a gapped toroidal core with dimension marks for clearing sizes and variables. This will let us speak "same language of math".
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Offline e-pirate

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Re: Application of "Novel approach for 3D air gap reluctance calculation"
« Reply #3 on: October 14, 2015, 03:19:01 pm »
OK. I did a drawing with all possible dimensions and sections.
So, as far as I can see, the closest method of calculation is described in for air gap type 1 according to a "novel approach". As I mentioned above, I see two major problems to be solved:
1. The "heigh" of the affected core side for calculation portion of the fringing flux on the outer and inner sides of the core (violet and orange parts marked Ho and Hi respectively). I can see at least two variants: the outer/inner length of a sector length or outer/inner diameters. I think diameters are closer to what it should be.
2. The "heigh" of the top/bottom sides of the core as shown on Fig. 2. As it seems to me, the problem is with the affected area (Sa). I'm not sure that the Sa, shown on the Fig. 2, is the only affected area and we can calculate the portion for the Cw * ha area as it was just a regular rectangular part and then drop the portion for "non exciting" part.
« Last Edit: October 14, 2015, 03:27:25 pm by e-pirate »
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Offline T3sl4co1l

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Re: Application of "Novel approach for 3D air gap reluctance calculation"
« Reply #4 on: October 14, 2015, 07:32:20 pm »
Drawing..??

Tim
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Offline e-pirate

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Re: Application of "Novel approach for 3D air gap reluctance calculation"
« Reply #5 on: October 14, 2015, 08:07:07 pm »
The "drawing" is in the original/first post :-)
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Offline T3sl4co1l

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Re: Application of "Novel approach for 3D air gap reluctance calculation"
« Reply #6 on: October 14, 2015, 08:08:14 pm »
Oh, well that wasn't there last I looked..!

Better to put it in the latest post, FYI..

Ed: OK so, I'd suggest smooshing the toroid into a square shape.  This could be done in a couple of simple ways, none of which is probably quite correct for the fields, but likely one is closer than the rest.

If we assume it's a square shape, where the inner and outer dimensions are given by ID and OD, it will probably be just a little "large": the h parameter is clearly too high, because some amount of that first 1/4 turn of the toroid will be "lower" to the gap.

We could then assume it's very similar, but a fraction smaller, perhaps pi/4 (equal circumference), or sqrt(2)/2 (actually the size of the inscribed diagonal diamond in the square, not related to circles; but perhaps a close enough "magic number" to be reasonable).

We might further expect that it depends on the ratio OD/ID, because a very "fat" core has much less path length, or alternately, much more material closer to the gap on the inside edge (where the 'ha' is labeled in Fig.2), and so we should perhaps have an adjustment where the equivalent h is smaller when OD/ID is large.

Perhaps that's even a sufficient compensation, i.e., pi/4 is a 23% reduction, which perhaps applies when ID is 23% less than OD, which seems a fairly typical geometry.  And for thicker cores (ID << OD), the compensation will accordingly be stronger.

Likely, a complete description would be very involved, including ID, OD and gap in a nontrivial manner; one should expect, as ID itself becomes closer to the gap length (and the question of planarity matters more and more: is the gap itself two parallel planes, or are the planes always radial sections as pictured?) that the path length nearest ID will be much more heavily weighted, and all kinds of nonuniformities will pop up.  Not that we are necessarily interested in those complexities, but for 1% level accuracy, we must take those effects into account, even if they are a small contribution (presumably, ~1%, imagine that!) for average geometries.

Tim
« Last Edit: October 14, 2015, 08:26:05 pm by T3sl4co1l »
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Offline e-pirate

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Re: Application of "Novel approach for 3D air gap reluctance calculation"
« Reply #7 on: October 15, 2015, 09:46:07 am »
Do not rely on the picture, I just used the first one that look close enough. The gap planes are completely parallel and not a radial type of thing. I don't even think it is possible to cut a sector of the toroid to make a gap. In reality, the toroid will but cut using a thin diamond disk. Or just split/broke into two pieces and then glued back again with a desired gap width. So, gap should be treated as completely parallel searches.

Back to our assumptions. What if we assume toroid as a square with outer perimeter equal to length of the outer circle and inner perimeter of inner square equal to length of the inner circle and same cross section? Or we can assume it same size as it will make same "magnetic path" length with the toroid and same cross section parameters. Wouldn't that be a more close to toroid?
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Offline T3sl4co1l

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Re: Application of "Novel approach for 3D air gap reluctance calculation"
« Reply #8 on: October 15, 2015, 11:06:44 pm »
Back to our assumptions. What if we assume toroid as a square with outer perimeter equal to length of the outer circle and inner perimeter of inner square equal to length of the inner circle and same cross section? Or we can assume it same size as it will make same "magnetic path" length with the toroid and same cross section parameters. Wouldn't that be a more close to toroid?

If you do the math, you will see that that is equivalent to my first suggestion. ;)

Tim
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Offline e-pirate

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Re: Application of "Novel approach for 3D air gap reluctance calculation"
« Reply #9 on: October 16, 2015, 09:06:00 am »
Finally, I got the method! I mean the whole thing. It's all about planes. So, now I'm able to make calculations for our assumption.
P.S. I've found Mr. Jonas Mühlethaler on Facebook and wrote to him. But frankly speaking, I don't expect much of help from him. Hope. hi will give me a right direction.
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Offline e-pirate

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Re: Application of "Novel approach for 3D air gap reluctance calculation"
« Reply #10 on: October 22, 2015, 09:12:57 am »
I did calculations. For the toroid 20x10x5 I found Rclassic = 33.125 H^-1 and with a "new approach" it is to be Rna = 17.375 H^-1 . Which is 1.9 times less and well correspond to the graph at Fig. 6 for the lg/w ratio of 0.2 (rightmost position). For my calculation I use an assumption of the equal magnetic path length of toroid and square shaped cores and equal gap shape. But I have no idea how accurate the assumption is.
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Offline e-pirate

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Re: Application of "Novel approach for 3D air gap reluctance calculation"
« Reply #11 on: October 29, 2015, 10:00:33 pm »
Finally, Mr. Jonas Muhlethaler - the head author of the "New Approach" responded in LinkedIn and provided me with his e-mail. Hope, he will help me with analytical solution of the problem and I will be able to release the program.
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