To start with, I'm not an Electrical Engineer, so bear with me. I will try to explain how a tesla coil works in laymen terms
( in english ;)  ).
 
 

    Tesla Coils can work off of purely transformer action ( normally a poorly tuned coil will do this). With transformation of voltages, the number of windings of the primary coil compared to the number of windings in the secondary coil can give a voltage increase. Lets say you have 10 "windings" in the primary and 1000 "windings" in the secondary, you will have a 1 to 100 ratio. If you are putting 15000 volts into the primary, you will end up with about 1,500,000 volts in the secondary ( this is in a theoretical sence). This will put out a spark but not much of one since air core transformers are "lossy" ( not truely 100% effective) the voltages at the top will be most likely 1/20 of the expected values.

       A truely well preforming Tesla coil works by something called resonate rise. Resonate rise is the buildup of energy through the action of adding power a little at a time to the existing power that was there from the original "pulse" ( energy wave).

    Resonate rise transformers ( tesla coils) behave differently from standard transformers. What happens in this senerio is that the voltages get stronger by "bouncing" the electrical wave back and forth from the bottom of the secondary coil to the top of it many times adding a bit more energy each time it bounces.

    My best way to explain this is to do it with a Slinky toy. If you take a slinky and attach it to the wall near the floor and then streach it out and hold onto the end of it. Best place to do this is on a smooth floor. This will simulate the resistive losses ( friction ) of the pulsed wave applied to the slinky.

    Ok the slinky is streached out on a smooth floor and the next step is to "push" the slinky a bit. As you push the slinky it creates a compression wave that will travel down the slinky and eventually hits the wall. That wave has energy in it and as it travels across the floor, the resistive (frictional) forces will reduce the original amount of energy applied to the "push". Sooo.. we have a energy loss already. The remaining energy will then come in contact with the wall and we loose a bit more energy when the compression wave hits the wall. But since there is some energy left over, it gets bounced back towards you. That remaining energy which is left over will "echo" back towards you.

    If you were to push the slinky at the exact moment the compression wave returned to your hand, you will resend that wave back towards the wall with the new energy you just added. If you did this long enuf, and timed it just right, the slinky would eventually have enuf power to break the wall. This is what resonate rise is.

    Sooo...  how does a Slinky and a Tesla Coil have its relationship you ask?  Its all a matter of timing. ( tuning is the word used in coiling).

    Here comes some technical stuff so hold on...   the secondary coil in combination with the toriod is comparable to the amount of length you have streached out the Slinky from the wall. The longer you streach the Slinky, the longer it takes for the wave to hit the wall and return to you. If you make the Slinky really short and push it, the wave will come back at you very fast, and you will have to be very quick to resend the wave out again. ( this is like having a secondary with no toroid). If you add a toriod to the secondary you change the time needed to "echo" the energy wave back. By adding a larger toroid you in effect streach the slinky longer.

    Once you have placed a toroid on the secondary coil ( and if you keep that same toriod on it ) you have basically set the "length" of the slinky. Now you need to pulse the slinky ( the secondary coil ) to get the energy to start building up in it. If you were to pulse the slinky with no idea of the timing needed to make the energy build up, you are most likely out of tune with the slinky and building up the energy in the slinky will be minimal. But with careful timing, you get the energy wave to bounce back and forth just right at the same time you pulse the slinky, and then you get an energy build up.

    Okay...  now you ask how do we pulse a secondary coil just right to make the energy build up??   We use a primary coil and a capacitor along with some hi voltage and a spark gap.

    A little explanation first about capacitors and coils...

A capacitor is capable of holding a charge ( kinda like a battery ) but the difference is that a capacitor can be charged both ways unlike a battery that can be charged only one way. It can let all of the energy be used up very very quickly. It basically "dumps" its full charge into the circiut all at one time. This is why Tesla coiling can be dangerous, alot of energy is dumped at one time from the capacitors.

 A coil can be seen by electricity as another way to hold a charge. By using alternating current ( AC ) electricity, a magnetic field is created around the coil. This field has energy in it and if you reverse or stop the electricity flowing thru the coil, the magnetic field will colapse and and recharge the coil.

    If you had a capacitor hooked up to the coil, the colapsing field would recharge the capacitor to about 70% of the original charge. Once the capacitor was recharged by the energy from the colapsing magnetic field, it would redump the power back into the coil and produce another magnetic field which builds up untill the capacitor is empty again. Once the capacitor is empty, the coil colapses again, recharging the capacitor. This continues to happen untill there is not enough energy left to either charge the capacitor or make a magnetic field strong enough to charge the capacitor. This is what is known as a resonate circuit too.

    This combination of capacitor and coil pulses at a specific timing. The timing can be changed by either changing the coil ( more turns/less turns) or by changing the value of the capacitor.

    This is where a spark gap comes into play... the spark gap is the switch that allows the capacitor to recharge the coil and the coil to recharge the capacitor. Once the voltage drops low enough, the spark gap no longer stays "connected" and this loss of connection allows the transformer to recharge the capacitor. When the capacitor is recharged enough from the transformer, the spark gap reconnects ( begins to arc) and the charge/discharge cycle is started all over again between the capacitor and coil ( primary).

    Okay ...  great, we have a capacitor that gets charged by a transformer, the capacitor builds up electricity and makes a spark gap fire. The gap fires allowing the energy to cycle back and forth ( oscilates) between the capacitor and the primary coil. The resulting magnetic field produced by the primary coil is transfered into the secondary coil also.  Sounds simple huh...   then why doesn't the coil work well??? Timing...  tuning ...   we need to get the pulse rate of the primary coil and capacitor to match the pulse rate of the secondary and toriod.

    Remeber that we set the "timing" of the secondary by putting a toriod on it. ( actually, the secondary has its own timing by design, width of tube, number of windings, length of windings)  a small coil design has a fast timing, a bigger coil has a slower timing. (( timing = frequency))  a little note here...  the timing happens extremely fast ... 100's of thousands of times each second   in other words ... a secondary coil of lets say 8" diameter with about 35" of winding length with a #22 wire is "tuned" to about 180,000 pulses per second. This = 180 kHz.

    How does one time/tune the 2 coil sets together to make the big sparks? Since we have "set" the timing of the secondary/toroid by simply making them, we need to be able to adjust the timing somewhere else. This other place to do the adjustments would be easiest at the primary coil. Changing the capacitors value is more difficult than changing the position of the attachment wire on the primary coil. Soooo...   we make the adjustments at the primary.

    By changing the attachment point of the wire leading to the primary coil, we can change the timing of the pulses ( frequency) of the primary/capacitor circuit. Since the secondary/toroid is at a set value ( set frequency by design) we want to adjust the frequency of the primary/capacitor to match that of the secondary/toroid frequency. A rough estimate of where the primary tap point should be found using either math or some of the tesla programs available on the internet. Once this estimated point is determined, a bit of testing is needed to find the exact point which will produce the best spark length.