Ash
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Why is the gap useful ?
L = n^2 / R
Gap increases R very significantly. Without it, it would take less turns to get to the same inductance value, and that means, the saved space (and Copper cost) would be better put into larger cross section of the core, lowering the hysteresis losses and keeping it away from saturation
Am i missing something ?



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Medved
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The missing thing in your thinking is the limited magnetic flux the core could handle  exceeding the limit, it saturates. With that, the inductance drops ~100..1000x, so such inductor become practically just a resistor (just the wire resistance). So in real life the number of turns is steered not by the required inductance, but by the required voltage across the winding just before the core start to saturate. On high frequencies a further limitation may be the losses in the core (hysteretic, eddy currents), this may impose the maximum useful flux density to become even lower. Without any gap, that would lead way too high inductance (well, except when we are talking about pure transforemer, where we would like the inductance to be as high as possible). So a gap is added (either dedicated, like with steel plate or ferrite core, or distributed across the core volume as with the iron powder cores) to just bring the inductance down to a level, leading to an inductance you want.
Most magnetic materials used for transformers, chokes and so on saturate when the magnetic flux density reaches about 1.5 to 2T. When assuming the same density is across the whole core (or at least across its minimum cross section part), we get: Phi = B * S where the B is the flux density, S the core area.
The magnetic flux change is, what generates the voltage across the coil (if we neglect the resistive drop  anyway the idea is to keep that minimum to limit losses): Vcoil = N*dPhi/dt With AC it means an AC flux amplitude vs voltage amplitude: Vpeak = N * Phi * 2*pi*Freq. The rms voltage is then (expect sinewaves) Vrms = 1/sqrt(2) * N * Phi * 2*pi*Freq For the maximum voltage before the core saturates it means Vrms = sqrt(2)*pi * N * Bsat * S * Freq
So you get the first equation for any magnetic design using given core: N = Vrms / (sqrt(2) * pi * Bsat * S * Freq). Then you get the conductor and winding window sizes, calculate the copper losses and based on that you are either done (losses and core size and cost are reasonable) or select different core size and redo the calculation, so iterate till you get some usable combination.
Then after that you calculate the gap size: Because the relative permeability of the core material uses to be in 100's or above, practically all the magnetic tension is across the gap, you may simplify the inductance equation to: L = N^2 * S / g where the g is the gap width, the S is the cross section area at the place of the gap.
Just one interesting consequence: With typical European reactor ballast designs using one common core plate size for many fluorescent ballasts, what differs among individual types is then the number of turns, wire gauge and the core stack height, what usually remains the same is the gap thickness as well. It is a consequence of basic equations: Given the fixed plate geometry:  the core width is constant, so the cross section area is proportional to the stack height.  the winding window is the same for all, so the copper crosssection is fixed as well. For the same losses density it means the total current (N * I product) remains constant as well. So the consequences for the ballast design:  The higher current, so thicker wire you need, less turns fit into the window. The number of turns is inverse proportional to the required current.  Less turns means you need higher core stack, as you need larger cross section (again, inverse proportional relation).  The last two means the core stack height become proportional to the required current.  The inductance required is inverse proportional to the current. But it is proportional to N^2 and core stack height. If you add all that together, you get:  Stack height is proportional to the current  Number of turns is inverse proportional to the current  The gap size is all the time the same (for the same ballast efficiency category), so you will end up with one, common gap spacer insert component for all ballast designs using that size of plates.
So with the given materials winding technique (= achievable copper fill ratio) you can easily converge to given "constants" for a straight forward new ballast design (StackHeight = Volt * Current * StackHeightConstant; N = Volts / Current * Nconstant; WireCrossSection = Volts * Current * GaugeConstant)



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Ash
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So :
The requirement coming from Bmax defines a maximum allowable Volts/Turn for a given cross section, therefore minimum turns
Then the gap is made to lower the inductance as needed
But then : There are older ballasts for HID and FL, some with effectively no gap (made same way as transformer, each 2nd plate is reversed so they overlap) and some with massive and poorly controlled air gaps (the 60's Fluorescent choke). How come they work well without concerns about all this ?



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The fact the plates are stacked interleaved does not mean there is no gap. All is "orbiting" just around the accuracy, cost and available assembly technology.
Actually making the gap in the way the plates are stacked all in one direction forming two core parts, with some insert between defining the gap size is one of the least accurate methods  you need really a machinery for most of the process (from plate cutting till core assembly and clamping) to get reasonable stability in the results. The thing is, such gap effective size is dependent not only on the clamping pressure, but on the exact shape of the plate cut edge as well. And to have all that so the result is within 5..10% tolerance is impossible without a fully automated assembly. So this method become the common way only rather recently.
When such automation is not available, you have to resort to different approach: One option is to adjust each individual piece (control the clamping by a feedback from electrical measurements). But that is really expensive and mainly there is high risk the thing will shift parameters after that step. Or another method was used: Instead of making the gap by the spacer when assembling the core, the gap get stamped into the plates itself. Of course, it leaves some bridges across the gap, which then saturate during operation, but the gap size, so the final inductance, becomes completely immune towards the further assembly tolerances. With that the plates are then assembled so it does ot form any other gap, hence the interleaving.



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Ash
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In Eltam/VS ballasts they use 2 approaches :
In the HID and "Euro FL choke" they use a flattened Aluminum wire as the spacer, and then do individual pressing (tapping with a hammer) to each unit. Apparently it does not shift anymore during Polyester varnish dipping and final assembly stages
In the "early 90s FL choke" the gap is fixed size and is made by the I plate being cut actually as a square bracket shape with indentation ( like "[" ), so the center leg of the E plate does not fully reach it, and then they are pressed with full force to minimize any other gaps. They are not interleaved
But in old ballasts i seen :
 HID ballast : Interleaved, no idea if there is any "saturation window" inside
 Fluorescent ballast from 60s : Looks handmade and awfully inaccurate (tolerances on the order of millimeters), with the air gap being on the order of 10mm. But even those served well !! (except buzzing very audibly !)



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There is no reason, why even a crude way made ballast should not serve well, if it is done properly. The thing is, such crude way usually requires individual adjustments, which can not be automated and requires quite some skill and working consistency from the workers, so they have to be paid better. So it would remain as a very expensive step in otherwise mechanized mass production.
The window cut ways, or the use of "M" shaped plates (those, where the gap height is dictated by the cut away section) was one way of achieving the required accuracy without any individual adjustments, but such plates were more expensive (more waste from cutting the plates from the large sheet raw material, more complicated manipulation, so more expensive assembly  it needs to be manual, but less skill is required).
The most modern machinery allowed to get the accuracy good enough even using a spacer, when all assembly becomes 100% mechanized. All that was motivated just by an effort to lower the production cost (fewer, even less skilled operators are sufficient)... But the drawback is (when there is no manual adjustment), the setup needs to be adjusted first, yielding quite a lot (10's till some 100) pieces outside of the spec limits.
The only principal difference among these methods is their cost partitioning (setup vs production), not the achieved results (provided all are done well)... For low volume production you need a worker really knowing what he is doing anyway, because the products vary so much, while the productivity is not that much a problem compare to really mass production. So then even the manual adjustment becomes very viable option  as it allow easy adoption to broad product variants  the manual labor is unrivaled in flexibility...



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Ash
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What i thought is, is it possible to make more compact lowloss chokes, by eliminating the gap and then less turns of wire are needed for the same inductance (and also, then thicker wire can be used till the window is full)
So, a "lowloss" choke like that would actually just saturate or atleast be way more lossy than the original on core losses ?



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Well, that is not that straight forward. It is true, with normal designs the copper losses are really the majority of the losses and that you may reduce by using larger cores. But with the core size, the core losses will increase as well. And with 1T flux density the core losses per kg of the core will be all the time the same. So if you create a 20kg beast for a F36T8, the core losses would be 40x higher than they are today with the standard design. So you have to really find a good balance:  Small (no) gap and very large core operated close to saturation means low copper losses, but higher core losses and very large and heavy result.  Larger gap and small core operated close to saturation means relatively low core losses (small core) high copper losses (present chokes), but lightweight result.  Larger gap and large core operated at way lower than saturation flux density means copper losses still significant (but smaller), but low core losses. Still very large and heavy result.
From these three you have to find the best compromise, optimizing all together (to get low enough losses, yet not too oversized and overpriced result). And that is the main task of the ballast design engineer...



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