Efficient transformer design is a vast and complex subject: beyond my understanding.
This information provided me with a reasonable example suitable for Tesla coil use only.

Please Remember:
These transformers with their high current and high voltages are lethal. There are no second chances with units of this size and the much larger so called Pole 'Pigs', or overhead distribution transformers.




1: The original transformer: 8.25KV 0.25 Amps

The original transformer had a burnt-out secondary winding when I was kindly given it for free. My aim in this project was to provide a robust power source, allowing me to move away from the fragility and low amperage of NST's.




2: The bare cores once removed



You may wish to skip this first section if your not interested in the theory

Originally the secondary winding was wound conventionally around the primary coil on the middle leg. The all important cross-sectional area of the centre leg where all four pairs of cores meet is 8 square inches. Larger cores being harder to saturate and better at dissipating heat.
The classic textbook explanation is that the output voltage depends on the primary to secondary turns ratio. However, there has to be sufficient flux created in the primary at the outset, for this condition to be true (achieved by the core exciting current). Even then there may not be a straight Turns/pri:Turns/sec relationship.
The amount of flux created depends on things like: core size and material, input voltage, number of primary turns, primary inductance (primary current).
As you ultimately want a high turns ratio, you will want the minimum number of primary turns capable of creating sufficient flux, but also without saturating the core, which will happen when too few turns are used.
Transformer design is complex, too complex for me, and the usual textbook explanation leads you to wrongly think it's easy to design one. Aware of this from my own teenage college years I took the easier path and used the existing primary specifications. If you also do this don't stray too far from the original specifications, as manufacturers design the primary to be as close to saturation as possible from the outset, without it actually occurring.


If you need a completely new primary there are several standard formulas for calculating both the core size needed and the number of turns, but these all depend on you knowing the magnetic permeability of the core and also assume that the coefficient of coupling is close to '1'.
If you want to know the core size needed for a particular power level, an approximation can be found by taking the square root of the expected power and multiply that figure by 0.14. This 5800 watt DIY transformer should have a core size of sqr(5800) * 0.14 = 10.66 sq inches. (It actually has only 8 sq inches)
A transformer for electronics or particularly audio use has to be built to a high standard. But Homemade Transformers for Tesla use only really need to fulfil two requirements. A high output voltage, and the ability to supply as much current as possible.
So assuming you are using a proper laminated core and not an old pipe packed with welding rods, an acceptable method that will give a starting point for DIY construction, is to measure the cross sectional core size in sq inches where it passes through the primary. This measurement we will call 'A', the input voltage is 'E', and 'K' equals 6.5 for a 60Hz system or 7.507 for 50HZ.

Number of primary turns = (K * E) / A.

Using this figure on my core I get a figure of 222 turns needed to create sufficient flux.

The number of secondary turns is worked out like this:
222 turns / 240 volts = 0.925 volts per turn on the primary. Assuming a coefficient of '1', a secondary of 10,000 turns will develop 10,000 * 0.925volts = 9,250 volts.
It would then be advisable to wind 222 primary turns and connect it to the mains to check that your primary current draw 'at idle' (core exciting current) is not too high. Allow up to ~5% to 10% of the max expected primary current when the secondary will be short-circuited for this figure. Remember though that the bigger the core, the more exciting current will flow.
If this is Ok then wind a temporary secondary coil of ten turns. Then measure the induced voltage in that and divide by ten. In a perfect world this would equal the 0.925 volts that the earlier calculation gave you. If instead you get [say] 0.7 volts and you still want a 10,000 volt output, you will now need 10,000 / 0.7 = 14,285 turns.

Although the formula actually gave 222 turns for my own core, in reality I found the original had 130 turns. This may be due to my core material, on which I have no data and also to the fact it uses a bifilar winding. So on my primary of 130 bifilar turns, (130*2) 260 / 240v input means each primary turn will carry 1.083v. I am also using two secondaries wound on each outside leg (for reasons explained below) Therefore each secondary of 5432 turns will develop 1.083 * 5432 = 5884 volts and their combined voltage is 11,769v. Due to losses and less than perfect coupling I actually have ended up with ~10,900 volts (actually 10.87K. One side giving 5.51K the other 5.36K).

As mentioned the coefficient of coupling is generally assumed to be close to '1', but this may not be attainable with home wound transformers even when using a winding jig. This is because the layers may not lay flat on top of one another, which can also make the windings larger than you had originally planned for.

As an example:
If you used 20 thou wire, in a winding that is 10 inches wide you 'should' in theory get 500 turns per layer (10/ 0.02) In practice you may end up with a 20 thou wire followed by a 2 or 3 thou gap before the adjacent winding. So each winding in effect takes up on average 22.5 thou. 10/ 0.0225 = 444 windings instead of 500.
So if you had originally planned on 30 layers of 500 equaling 15,000 turns in total, you will now need 15,000 / 444 = 34 layers.
Each layer also needs insulation paper, and this you will find is the main cause of a bulky winding.
The very first layer of 20 thou conductors with 5 thou of insulation will most likely become 26 or even 27 thou as the paper will not lay dead flat.
But the next layer, in addition to the problem just mentioned, may also not sit level on their underlying layer, so subsequent layers may be up to 30 thou, instead of 25. Also remember that any bump or unevenness on an inner layer gets far worse by the time it has got to the outside.

As mentioned already I simply rewound the primary with new wire using the same specifications as the original. You may even be able to use the existing primary wire without unwinding it.

The secondary wire that I used was 0.4mm / AWG #26 / SWG #27). After a lot of time looking at different tables for wire amperage I used a figure of ~500 circular mills per Amp (cir mil/amp) but only because the windings are under oil.(use 1000 to 750 cir mil/amp in air. A 'cir mil' = diameter of wire in thou squared)

The calculation is: 0.4mm = 15.748 thou, squared this equals 248 cir mil, 248/500 = 0.5 (0.495) Amps. The ballast actually gives 537 m/a with the primary drawing 21 amps.

The bifilar primary winding is AWG #14 / SWG #16 wire. The current capability using 500 cir mil/amp is 8.25 Amps. The bifilar winding though means it will carry 16.5 amps.
As the depth of the winding is not too deep the oil should be able to circulate easily, so for this reason I run at a max of 21amps. The oil and the actual winding only getting warm to physical touch.

The weight of the whole assembly in the box with oil is 31 Kgs or 68 pounds.




4: The cores are in matched pairs.
It is most important that you keep the pairs (top and bottom) together, as they will have the best fit on their mating surfaces.





4a: Making the two secondary bobbins

I was also given a large inductor that used the same cores and centre leg bobbin as the radar transformer. As I was going to wind the new transformer with two secondaries, one on each outside leg, I now needed two extra bobbins. I decided therefore to modify the bobbin that came from the inductor, by cutting it in half and making two, as can be seen above on the right.

The reason for winding two secondaries in this unorthodox and inefficient manner, was because it was the only way I could get enough windings to increase the transformers output voltage from its original 8.25KV.
My arrangement meant that each coil should output just under 5.9 kv, but most importantly both windings were wound out of phase to each other. This means that when one coil has its outer end at 5.9Kv positive in respect to its inner end at zero volts, the second coil will have its outer end at 5.9Kv negative to its inner end. As the two inner ends are always at zero volts they can be linked together. This means a connection to the two outer ends will result in having a combined voltage of 11.8kv at 50hz.




The phases relationship of the two secondaries

Another reason for this arrangement is less voltage stress on the windings. NST's also use this system to keep their construction costs down, but they also ground the centre tap to the core.
To wind this out of phase arrangement, imagine you are looking at the coils from the top or plan view. Starting the windings at the top I wound one winding clockwise and the other anti-clockwise, with the primary sitting in the middle. This arrangement, shown below, allows you to have the inner low voltage ends of both windings at the same physical end, meaning you only need a short link between them to join the zero points together.



Some transformers will automatically limit their current when the secondary is short-circuit. This feature occurs if there is a flux path created by the primary that doesn't interact with the secondary. This then causes a current limiting effect to occur that would be the same as placing an inductor in series with the secondary. Transformers that have this feature built in include NST's, welders, MOT's, and it is achieved by having so called 'shunts' (metal inserts), bridging the gaps between the core's legs. This characteristic also occurs if there is excessive leakage flux, which it seems is the case with my transformer.
I know this because at max volts on the variac (267v) and with the secondary output shorted I have 185v dropped across the ballast, and 82v across the transformer primary still. If it did not have a high leakage flux I would have virtually zero volts across the primary during short circuit conditions, giving the ballast more work to do.




5: The edges are milled flat

I glued a flat rectangular piece of phenolic sheet to the freshly milled edges of the old bobbin to make the fourth missing side.




6: Both wooden mandrels

These mandrels are the same size as the legs of the core. The left hand one corresponding to the larger middle leg, while the thinner right hand one is used for the two outside legs. The wooden mandrels are gripped in the chuck of the lathe, and a plywood side piece is slipped on (see below). Next the bobbin is slid on, followed by another plywood side piece. The latter being held in place by a couple of small tacks.




6: Wooden mandrels

The plywood side pieces were used to contain and also position the insulation paper that is applied between each layer. I used insulation paper on the primary as well but the side pieces were not used.
I then used the tailstock (see #8 - modified because of the raised chuck) to steady the far end of the mandrel. This is particularly important when winding the primary as that needs quiet a lot of pressure applied to get it to lay flat. Without the tailstock the mandrel would rip itself out of the chuck.




7: Paper insulation

This is used between each layer and serves two purposes.

a: It acts as insulation, as the adjacent layer above and below can be several hundred volts higher or lower respectively. The wider each layer is, the more the voltage difference is. With for example a 10,000v NST with two 50 layer coils of 5,000 volts each, then each layer handles 100 volts. The voltage difference for a point at the end of a coil to the end of the layer immediately above or below could be 200 volts though!
Also remember that peak voltages are 1.414 times the RMS value. This is why you have the paper half an inch each end wider than the actual winding of the wire (see #13 below).

b: It allows you to get a much neater winding, which is very important if you want to pack as many turns as possible in.





8: The setup on my lathe

This rather cumbersome setup was needed as the centre height of the lathe is not enough to allow for a fully wound coil. The chuck of the lathe is driving another chuck which is raised above the lathe bed, allowing more room for the coil to swing around in.




9: The Primary bifilar winding. AWG #14
Because the thick wire needed some two-handed work I left the side pieces off when winding this for better access, and just positioned the paper by eye.




10: One of the Secondary AWG #26 windings

In each case the lathe's screw feed was engaged and a suitable gear employed that would on each turn of the mandrel, move the saddle a distance that matched the diameter of the wire.
The lathe was run at around 30rpm to 60 rpm. I am fortunate to have a variable frequency drive with a foot operated switch, making the whole procedure easier to handle. I also did a couple of test runs with some old magnet wire to 'fine tune' my procedure. It pays dividends in the long run.




11: Adding the paper at the end of a layer
Here you can see how the paper makes sure that the wire is correctly positioned to allow it to start its next layer by winding on top of its last turn. Also try and arrange that the paper joins are as close to the ends as possible. This avoids adding unnecessary bulk to the winding in its centre region.




12: Fitting the paper
There should only be a minimal overlap, but there must be at least some, ideally 0.5 inch or 1 cm.




13: Guide lines

On each new layer I marked the extent each side that the winding would extend to. I left a clear half an inch at each end. This is to avoid a flash-over occurring at the edges, which can occur if there is insufficient paper. Do not use a pencil or black marker, both will conduct the HV.




14: Securing the Primary with thread

This is not really needed, but it may stop the primary turns from slipping down.




15: End of secondary wire being anchored


The method I adopted here was to use a simple cable tie taped across the coil. The secondary wire was then wound on top securing it. The raised end of the cable tie (to the right) will stop it from pulling out. I then made a small hole in the cable tie with a hot pin and threaded the wire through it and tied it off. You can then solder a heavier wire to the magnet wire. I had the thicker wire inside a piece of plastic tube which I was able to slide down over the tapered end of the cable tie. This not only covered the soldered joint, but also made a stiff fixing to stop undue flexing of the delicate connection.




16: Ready to assemble

Originally the cores were held together by metal bands. I found that very large Jubilee clips made a first class substitute. I also found that the mating surfaces had to be a really good clean fit. I do not know if this is because of these particular cores, or the unusual arrangement I have adopted.
But any old varnish or bits of bobbin 'dust' from when you slide them down onto the lower cores, before putting the top core on, will affect the mating of the surfaces and ultimately the output voltage.




17: Ready for the oil
I re-used the old transformer oil again. I also had a large 6H oil cooled inductor which also 'donated' some of its oil.




18: The picnic box was a perfect fit

I had spent quiet a bit of time trying to find a container that was exactly the right size. At the last moment I spotted our picnic box languishing under the stairs. In a flash out came the tape measure - perfect fit. Only then to be told by my wife that the box would be handy for a picnic.
Perfect, now I had two good reasons for using it.




19: The 'Guinea Pig' transformer
Finn Hammer calls his homemade transformer a 'Pork Chop', explaining that "This transformer is not a neon, and it is too small to be called a pig. Porkchop perhaps?"
As the same reasons apply to me also I therefore have followed in the tradition and chosen the name "Guinea Pig" (known as a Cavy elsewhere in the world)

___________________________________________________________









20: The homemade inductor I use as ballast with the transformer



Was it worth it?
As an exercise, most definitely yes. I learnt a lot about transformer design and AC theory thanks to the web and support from various news groups. Also you don't realise the work involved till you build one yourself. They are not a quick day's work. You need to get all the materials ready to hand. Insulation paper pre-cut. Bobbins and mandrels made. You most definitely need some form of winding machine that has a lead screw arrangement as getting the wires close together not only allows the maximum number of turns, but results in a better flux pattern.

Was it worth it financially ?
Absolutely - NO.
Magnet wire, especially the big stuff for the primary, is not cheap, and you need a lot of it. But we don't build these things to make money do we. I used approx' 2.25 miles or 3576 metres of secondary wire (0.4mm / AWG #26 / SWG #27) current capability is 580 m/a under oil.

The primary is a bifilar winding of AWG #14 / SWG #16. The normal current capability of this gauge is around 9 amps under oil. But the bifilar winding means it will carry 18 amps easily, with short runs at around 21 amps.

The maximum output is 10,900 volts at 537 m/amps this gives 5.85 kva. This draws up to ~21 amps at 276 volts (Variac on maximum) on the primary through the current limiting ballast shown above. At this current level it can be seen on the oscilloscope traces below that the core has just started to saturate and so it has reached its limit.



Top to bottom:
As the current increases you can see saturation occurring as the sine wave looses its shape.
This is trace is obtained by measuring the output of a homemade current transformer that was applied to the primary's incoming live feed.

The weight of the whole assembly in the box with oil is 31 Kgs or 68 pounds. As I have some serious back issues at times this means it is too heavy for me to move now, but.......... C'est La vie.



'YouTube of Transformer."



Links I found useful for this project:

VERY Good Page with lots of info.

Homemade Transformer Projects

Hammer's Tesla Pages

The correct formulas as used in commercial designs

Luebke Lands

Amperage rating for AWG wire

Wikipedia

Transformer Measurements

Transformer Theory

Phasing of windings

UK Magnet wire seller

Ballast Coils