King Size Air Coil Project

Why do I need a Godzilla style, *** KING-SIZE *** coil in my woofer crossover ?

Why use 6,1 kilograms of sheer copper per a single coil ?
Multiply that by two, for a stereo system ?
Why not “Go with the Flow” ?
Why not use an iron core and be ... “done” with it ?
Simple and easy ?

Well, the main reason for this is that I am just a bit ... crazy.
The other reason is HYSTERESIS. Translated into English, that simply means: nasty distortion.

As mentioned earlier, I am not fully satisfied with the iron core based woofer coils, specifically, ones with an inductance of 5,2 mH and with a DC resistance of 0,2 ohms, the ones that till recently I had installed in my DIY speakers.
The 0,2 ohms, related to the nominal 8 ohms of the speaker woofer, constitutes 2,5% of the actual speaker impedance. This is a significant portion thereof, in comparative terms. But putting that aside, there is also the issue of memory effects within the the iron cores. There is a rumor within the audiophile world, that any and all transformers, inductors, chokes, ones that are equipped with iron based cores, that all these have a nasty little attachment that always comes along with them, irrespectively of the their form and material used, be it mu-metal magnetic stuff, baked powder ferrite stuff, or otherwise. These cores will ALWAYS introduce a certain amount of HYSTERESIS:
{ the above graph - sourced from Wikipedia }

Now, I don’t know about you guys, but I do ** NOT ** like hysteresis, as it distorts my audio signal.
If you wish to read more about this "Hysteresis" thing, here is a good place to start:
Some shall say, that within the lowest registers, within the BASS frequencies, such hysteresis does not mean a thing, as you “ ** Can not hear it ** ”.
Well guess what:  What if I told you that that is an “Accountants” version of cost optimization ?  That it is a rude lie that is directly associated with the "Let's make MONEY, and not MUSIC" principle ?  The deal is as follows:  It is fairly easy to make an air coil,  i.e. an coil that has AIR as the CORE (meaning: none, null, nichts, nothing, zap, zero, ….), so long as you produce it using *** THIN WIRE ***.   
An air coil, meaning a coil without any core whatsoever, one with free “air" as it's only core,  is a very *** specific *** product.  Specific in the sense that it’s relative permeability is fairly low, mainly:  M_r = 1,003 or something.  The 1,0000 is reserved for absolute vacuum, whereby, in comparison, natural air is at a just slightly “higher” value than that vacuum, so they say. Whatever. But the permeability of air has one positive side to it: it does not introduce any memory effect. No magnetizing effect. It does not introduce any ** hysteresis **.
Big deal ?
Well, …. yes.
It is a big deal.

Why ?   
Because a low magnetic permeability of air implies that you now have to use lots and lots of turns of wire, within the coil winding, so as to achieve the same result, i.e. the same required level of inductance that you need. But more wire means a bigger DC resistance. So now we fall into a crazy cycle: We need to choose a fat wire, one with more cross section area.  But a fat wire shall result with a coil that is bigger in size. This in turn means even more meters of our fat wire.  This translates into much more material being used. As mentioned earlier, the accountants from those big time companies at this point shall say: " *** STOP ! *** ".
They shall say: “… HEY!!! .... Wait a minute. If you need lots and lots of wire, that implies that you intend to spend lots and lots of money, cost-wise, for the purchase of material, for the purchase of that long, fat, copper wire that is necessary to produce such a coil. But we are in the business of making money, and not in the business of making music !….”
So, what is the first, most obvious "Accountant’s" solution to the cost / benefit problem ?

Obviously, … a  “To Hell with the Benefit and Optimize the Cost” type of approach.  Now, how can we optimize the cost of a bucket-full of coil winding turns, an extensive amount of windings of a fairly long copper wire, an amount that is necessary to obtain the required inductance ?   Any of the two following 'Uncle Scrooge' methods will do:

A). Reduce the diameter of the wire used. This way, the coil will be less bulky. The required number of turns will fit on a smaller carcass. The overall weight of the coil will be smaller, so the cost of the material will also be smaller. But what about the fact that a smaller cross section of wire imposes a much higher overall DC resistance ?  Well, …. let the "Audiophiles" worry about that one. This “Reduce the cross-section” method is the most obvious one and the most used method within the “smaller coils”, those that are typical of the midrange and tweeter crossover circuits. The circuits, where any distortion would be more than apparent and Uncle Scrooge would not get away with it. But he CAN get away with an increased DC resistance of these coils.  Such increased DC resistance implies greater damping, and more often than not, lower quality of those filter circuits, and / or lower efficiency of the whole speaker system. In other words, the speaker system, especially the midrange, where some serial inductance applies, may be not as efficient as it could be.  
B). Apply an iron core of high magnetic permeability, so as to reduce the necessary number of  turns within the winding, so as to shorten the required length of wire, but at the same time, achieve the same “inductance”. You get “the same result”, but use up much less wire.  The Accountants are Happy. In other words: keep the audiophiles Semi-Happy. Since the length of required wire is now much shorter, the Accountant may now be "generous". He says: give them what they like best. Give them a “sniff” of some thicker cross section of wire – just to keep them semi happy in terms of a fairly low DC resistance, a parameter which seems to bother them most when considering the woofer crossover circuits. So now you have the Accountant generously "allowing" a higher cross sectional area of the wire used, so as to cut down the DC resistance.  Albeit, this thicker cross section in view of the controversial “use some iron core” strategy, comes as a compromise of doubtful benefits. It comes at a price. To be perfectly clear: a price to be payed by the Audiophiles, and not by Uncle Scrooge. The price is called Hysteresis.  As a general rule of thumb, high values of magnetic permeability, which are possible to achieve with iron or ferrite based core materials, tend to introduce “hysteresis”, or magnetic “memory” and hence, they tend to distort the sound. And do not believe those marketing fliers of some of those supposedly highly advanced magnetic materials.  A high magnetic permeability shall always come at a price of hysteresis. Sometimes it shall be huge, sometimes it shall be "decent".  But it will always be there. Even in those most advanced super duper ferromagnetic materials, that are bluntly pushed down your throat.
Hysteresis, according to Wikipedia, is a natural phenomenon of soft magnetic materials: ”…
Hysteresis is the dependence of a system not only on its current environment but also on its past environment. {{ i.e.: Its HISTORY.}}  This dependence arises because the system can be in more than one internal state. To predict its future development, either its internal state or its history must be known.[1] If a given input alternately increases and decreases, the output tends to form a loop as in the figure. However, loops may also occur because of a dynamic lag between input and output. Often, this effect is also referred to as hysteresis, or rate-dependent hysteresis. However, this effect disappears as the input changes more slowly.
Hysteresis occurs in ferromagnetic materials and ferroelectric materials, as well as in the deformation of some materials (such as rubber bands and shape-memory alloys) in response to a varying force. In natural systems hysteresis is often associated with irreversible thermodynamic change. Many artificial systems are designed to have hysteresis: for example, in thermostats and Schmitt triggers, hysteresis is used to avoid unwanted rapid switching. Hysteresis has been identified in many other fields, including economics and biology.
Magnetic hysteresis
Starting at the origin, the upward curve is the initial magnetization curve. The downward curve after saturation, along with the lower return curve, form the main loop. The intercepts hc and mrs are the coercivity and saturation remanence.
When an external magnetic field is applied to a ferromagnet such as iron, the atomic dipoles align themselves with it. Even when the field is removed, part of the alignment will be retained: the material has become magnetized. Once magnetized, the magnet will stay magnetized indefinitely. To demagnetize it requires heat or a magnetic field in the opposite direction. This is the effect that provides the element of memory in a hard disk drive.
The relationship between field strength H and magnetization is not linear in such materials. If a magnet is demagnetized (H=M=0) and the relationship between H and M is plotted for increasing levels of field strength, M follows the initial magnetization curve. This curve increases rapidly at first and then approaches an asymptote called magnetic saturation. If the magnetic field is now reduced monotonically, M follows a different curve. At zero field strength, the magnetization is offset from the origin by an amount called the remanence. If the H-M relationship is plotted for all strengths of applied magnetic field, the result is a hysteresis loop called the main loop. The width of the middle section is twice the coercivity of the material.[16]A closer look at a magnetization curve generally reveals a series of small, random jumps in magnetization called Barkhausen jumps. This effect is due to crystallographic defects such as dislocations.[17]..”  
{ End of Wikipedia citation }.
As can be seen, this graph above is definitely *** NOT a straight line *** .  
Actually, it is *** not even a curve *** .
It is an ugly piece of non-linear stuff,  with *** internal memory of past signals **** , past magnetizations, past events.  Very ugly.
So, now you ask me: what can we do about it ?
Simple answer:  
*** GET RID of the MAGNETIC CORE *** . 
*** Build an AIR COIL *** .
OK, no big deal, at least within the midrange and the tweeter sections ….
It just so happens that the inductances that are necessary in those sections are not to extravagant, and actually can tolerate the DC resistances involved, allowing you to create farily decent air coils. Franky speaking, apart from the not-so-decent diameters of the wires used, air coils,  as a rule, can be found in such midrange- and high-frequency crossovers most of the time. 
But how do you obtain a woofer AIR COIL ?
A Coil, say, 5,2 mH, of 0,2 ohms for the BASS ?  Will it cost a fortune ?  Frankly speaking: it will. But even worse than that: where shall you purchase it ???
Even if you find a place to “source” it, indeed, it may cost a small fortune, due to the quantity of material and the workmanship. So, my simple answer to you is: BUILD it by yourself. And that is exactly what this "little" project is all about. 
I have actually built such a woofer coil, but not 0,2 ohms …. initially I was targeting for much less, if at all possible. 
So, what we do now is as follows:  We purchase 12 kilograms of copper winding wire. The exact amount should be calculated with the help of air-coil web calculators, which can be found on the web. Some are listed below. Do the calculations. Having the result, add a hefty “spare” amount of wire length to the calculated result, as the accuracy of these calculators and the precision of the results is a function of the accuracy of our workmanship, further down the road. Mainly: a function of how neatly shall you execute this small DIY project.
Actually, the wire that I purchased is not even a wire, because wires have a round cross section. The stuff that I obtained is more of a flat profile, with a rectangular cross section. It is 2mm high and 3,5 mm in width, totaling 7 mm2 of cross section area.
With this “wire”, or profile, I am in the process of creating AIR COILS with an inductance of 5,2 mH and less than 0,2 Ohms. The trouble is, that such a profile is very stiff …. so I have supplemented myself with the help of additional tools, such as presses, clutches (or whatever they are called), those used by carpenters.
This is what the Work-In-Progress production of my first coil looked like: ….

The FIRST layer of the winding, the very first turns (the very hook-up method, how to literally "start" the winding, actually speaking), are the most difficult ones to execute, as you have to make sharp bends on the bobbin, almost at straight angles, and these bends need to be applied very neatly and evenly. Considering the stiffness of the winding profile, I am happy with the results achieved thus far.  I am just starting the second layer, so now the bends shall not be as sharp and generally it should be much easier to proceed now.
The tricky thing is to “start” the whole operation. After a few frustrating failed attempts, I came up with a very non-obvious, albeit very successful strategy. As the “profile” does not easily bend “sideways” – I gave up the concept of trying to “route it in” via the normal wire-insertion “slots” that are readily available on the side of the bobbin. Instead, I drilled a hole in the floor of the bobbin, close to one of the corners – a logical place for the winding to start. I routed the profile through this corner hole in the bottom surface of the bobbin (the bottom that would normally “touch” the central iron core of a transformer – albeit: we have no such core – just a hole full of air in the center section of the bobbin). After threading an ample amount of starting length through the floor, I bent the profile to a U_TURN shape (along the "wide", 3,5mm edge, a fairly easy task, with pliers). The bend took place along the thinner axis, so I was bending 2mm worth of the thickness of the material, along the wide axis, by which it is much easier to bend.  This "U_TURN" type of hinge creates a pirates hook that firmly holds to the floor and gives you a good starting point of reference for the winding process.
The winding takes place by ONE QUARTER of a turn. After EACH and EVERY BEND of the wire (which actually is preceded by a contra-bend of the next following length, so as to avoid the belly-up syndrome…), I press, squeeze, secure, the QUARTER WINDING via a piece of wood (so as not to rupture the delicate isolation varnish) with the FIRST carpenters press.  Once that this first quarter turn is SECURE and TIGHT, I have the freedom to perform the NEXT BEND. After performing the next BEND of the wire (which actually is preceded by a contra-bend of the next following length in the opposite direction, so as to facilitate a nice contact of the wire to the bobbin, and to avoid the belly-up syndrome…), I press / squeeze / secure the QUARTER WINDING via a piece of wood (so as not to destroy the delicate isolation varnish) with the SECOND carpenters press.
So now this consecutive quarter turn is also SECURE and TIGHT.
Now, here we go ....
Now, the FIRST carpenters press is free to be dismantled and REUSED for the next quarter turn. So I dismantle it. With the SECOND one intact, I have the freedom to perform the NEXT BEND, the next quarter-turn. After performing the next bend, ….. blah, blah, blah, …. repeat the above by 4 times the number of turns that you need to wind ….. In my case it shall be 300 full turns or so. That means that I have to repeat these activities 1200 times. … Off we go ...
…. <<< time passes, my blisters are growing >>> ....
And for those of you, who wonder how to assess / calculate things like the geometries and numbers of turns and the necessary amounts of copper that need to be purchased, there are lots of helpful “calculators” available on the web. These are just some of those that I use:
As for the copper itself, I sourced it from these guys:
It comes at something like 10 EUR per kilogram, give or take, depending on the thickness of the profile.
My coil, after it has been finally finished, came up with a total weight of 6 kg, give or take. That means that the cost of the material was in the order of 60 EUR. Even slightly more. 

…. <<< time passes, my blisters are growing >>> ....

I am gradually progressing with the process of winding the air core woofer coil. 

Although it is being wound on a transformer bobbin, please do not be misled. This shall indeed be an air coil. Nothing to be inserted inside  :)

The process of creating this coil is a very time consuming and a mundane process, as it requires the use of the carpenters clutches, that secure each and every QUARTER turn that is accomplished. I use a pair of such clutches, so that whilst creating a new quarter turn, I “un-clutch” the one that is further off, whereby the closer one stays intact.
This way, I have full control with regards to the tightness of the winding. And indeed, believe me  (... and if not me – than at least the blisters on my hands)  it ** IS ** tight.

Winding a coil with a 2 mm x 3,5 mm = 7 mm2 copper profile turns out to be no trivial feat, but luckily with the clutches it is still within the possible.   As can be seen above, the coil has already exceeded a weight of 4 kilograms.  The stiffness of the 7mm2 profile is similar to the stiffness of a bicycle wheel spike.  Just imagine. 

The inductance achieved thus far is 2,3 mH, so things are looking good. Each doubling of the number of turns essentially quadruples the inductance, so I am not that far off to go.  Over 67% of the winding is complete by now. The bobbin turns out to be a tad too small, but we will manage.

The pieces of wood are used so as to isolate and protect the insulation varnish of the copper profile from the metal jaw of the carpenters clutch.  This way, I am sure that there shall be no internal short circuit within the coil, when it is ready.

After each and every new quarter turn, I leave the clutches in place, tightly squeezed, so that the winding maintains a very dense and tight profile.

The only thing that was a bit way-off in this project was the estimation of the required length of copper profile.  Initially, I purchased 6 kilograms of copper and this amount was "supposedly" enough, supposedly sufficient for the creation of a pair of coils, 3 kg each.
Oooops !   I slightly miscalculated on that one. An error of 200%. 
It turns out that the whole 6 kg of copper will be used up for the creation of a single coil.  And hopefully it will turn out to be enough. Ouch !  I am not even considering the option that it would not be enough).
Well, … nobody is perfect.

…. <<< time passes, my blisters are growing >>> ....
After some first initial experiences, I would rather introduce a small correction. When purchasing the material, I would now rather go for a profile that is “more flat” … something like say “5mm x 1,4mm” or similar. So as to make the bending easier, at least along the main “axis” of interest …
…. <<< time passes, my blisters are growing >>> ....
So far, so good.  I’m up at layer number 10, with 20 turns within a layer.  That is somewhere in the whereabouts of 200 turns. This means that I have already repeated the quarter-turn cycle 800 times over.  The coil is now a beefy 2 kg in weight, give or take. Thanks to the “flat” surface of the profile, it is actually fairly easy to maintain a “flat” surface of the whole winding, even at layer number 10. The winding tends to 'even itself out' by itself. The winding process takes place at quarter-turn steps. Each step involves the “un-clutching” of the previous carpenters clutch, and “re-clutching” it in front of it’s peer – each time a new quarter turn is executed and performed. Under the steel jaw of the clutch I put a piece of wood, so as not to injure the fragile isolation varnish that is on the copper profile. In essence, what I am saying here is that I have already performed 4×200 = 800 cycles of clutching and un-clutching, each time using a piece of wood as a separator. Crazy.  But the clutching effort is well worth it, because the 10-th layer is almost as smooth and even as the very first one. But what about the inductance ?   Well, the inductance meter reading states, that I am at a level of 1,5 mili-Henries.  But having said that, one needs to keep in mind that two small metal jaws of the two metal clutches protrude into the inside cavity of the winding, actually constituting and "unwanted steel, magnetic core". So this value of inductance reading may be a bit over-optimistic, say 10%. As for the DC resistance, it goes without saying, for the time being the readings state that the resistance is literally unmeasurable.

…. <<< time passes, my blisters are growing >>> ....
Finally ! The big coil is now ready.
I must admit, the amount of work associated with its production turned out to be more than initially anticipated.
Weight:  6.1 kg.

Inductance: 5.68 mH

DC Resistance: 0.26 ohms   ( You have to subtract the 0.08, which is the resistance of the measurement leads, from the 0.34, which is the supposed "result"). 

Core:  AIR
Hysteresis:  NONE.
Unfortunately, I did not quite achieve the goal of hammering the DC resistance down below 0.2 Ohms. But the result of 0.26 is also impressive and very satisfactory. 
The amount of copper rectangular profile  ( 2mm x 3,5mm = 7mm2 )  that was supposed to be sufficient for making two such coils – actually turned out to be just about sufficient to produce one coil.
The bobbin that I used is from a company called Weisser, ans specifically the model EI-150a,  whereby the “a” means that there is no middle separator wall subdividing the winding area into left and right partitions / halves. As you may have noticed, the bobbin is a bit too small.   Also, if you are to consider the optimal “geometry” of the air-core winding, it turns out that for best results, you should strive to obtain a “square” cross-section of the winding area. 
Therefore, a more optimal bobbin would be something like two sizes bigger, say an EI-174 or maybe an EI-196, but with the “b” designator. Having one of those, you would only use one HALF of the bobbin winding area, leaving the other one totally unused. This would allow to obtain a more optimal geometry.
By “optimal” I mean the “most” inductance with the “least” length of wire. This would translate into the same target inductance, achieved by less wire, and hence with a resulting even lower DC resistance.
The following is a conceptual drawing of the “b” type bobbin and with brown I depicted the “active winding area”.  the other half of the bobbin should not be used. This might seem as a “waste” ... but actually, it is the opposite. The bobbin itself - this you get for the price of peanuts. But the copper wire or profile – this, price wise – is a  TOTALLY  different story.
You are better off obtaining your inductance with less wire – it simply turns out cheaper. Considering the fact that 1 kg of copper winding profile is at or even above 10 EUR / kg, it tends to make a difference.
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The upper half of the bobbin would be left unused, in such a scenario.  Please note that the actual “shape” of the brown winding area cross sections is a regular square. This is the shape that you should strive to target and obtain, if maximum inductance-for-buck is to be realized, with minimum DC resistance.
The  resulting optimal coil has a somewhat “flat” profile  (This can be verified with any of the air coil inductance winding calculators that are available on the web).
If you are considering making / winding your own coils, there is one other thing that you need to consider, need be aware of.  The resulting inductance of the coil is a function of the number or turns SQUARED.  This means, for example, that if you “completed” winding 50% of the intended number of turns, you shall observe that the intermediate inductance is only but 25% of the target value ( 50% of 50% = 25% ).
On the other hand, if you have measured your inductance achieved thus far, and see that it is only but 50% of what it “should be” – you may get nervous that the space on the bobbin is running out and that you will not “fit” all the necessary windings onto the bobbin. Fret not.  50% of inductance actually means, that you have completed more than 70,7% of the project. 70,7% of the total number of turns is already in place. 70,7% of the total “thickness” of the winding layers is also already there.   70,7% is equal to square_root( 50% ).  Or in other words ... 0.707 x 0.707 = 0.500. 
If you don’t believe me, check it out on Wiki:

Inductance of a solenoid

solenoid is a long, thin coil, i.e. a coil whose length is much greater than the diameter. Under these conditions, and without any magnetic material used, the magnetic flux density
<Mail Attachment.png> within the coil is practically constant and is given by
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<Mail Attachment.png> is the magnetic constant,
<Mail Attachment.png> the number of turns,
<Mail Attachment.png> the current and
<Mail Attachment.png> the length of the coil.
Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying
<Mail Attachment.png>the flux density by
<Mail Attachment.png>the cross-section area:

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When this is combined with the definition of inductance,
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it follows that the inductance of a solenoid is given by:
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{ End of Wikipedia Citation }
In the last formula, you clearly see that the “N squared” is your number of turns.
You can also clearly see that the “shorter” the solenoid, the “better”.
That is why it is of benefit to use just one half of a bobbin.
I rest my case.