Tesla Coils are simple devices. You don't need a Ph.D. in Electrical Engineering to understand how they work.
The basic setup incorporates to major parts: the primary coil and the secondary coil. The secondary coil has MANY more windings than the primary coil and is placed inside of the of the windings of the primary coil. This concentric configuration forms a transformer. A transformer of this type will TRANSFORM a small voltage across the primary coil into a voltage on the secondary coil. The voltage on the secondary coil is directly proportional to the ratio of windings on the secondary to the windings on the primary. Wow that was a mouthful. In short , the voltage on the secondary coil will be much larger than that on the primary coil. This is because the primary coil usually has only 2-10 windings while the secondary has as many as 1000. It should also be noted that no electrical connection exists between the primary and secondary coils, they are MAGNETICALLY COUPLED. Magnetically coupled means that energy is transferred between the primary and secondary coils via a common magnetic field.
On the primary coil side the circuit consists only of the primary coil itself, a capacitor, and a high voltage source. By properly connecting the capacitor and primary coil, a resonant circuit is obtained. The term resonant circuit implies that the circuit will generate a voltage that will oscillate with time. Furthermore the term resonant implies there is some frequency at which the amplitude of the oscillating voltage will be maximum, the resonant frequency. The idea of resonance is not constrained to electricity. You can think of the primary circuit behavior to be very much like pushing a child on a swing. When you put some energy into the child (by pushing) the child and swing will oscillate. If you continue to add energy to the child/swing at the right times you can make the child can swing higher; you would then be adding energy to the child/swing system at its resonant frequency. The electrical analogue of this child/swing behavior is employed in the primary circuit. By charging capacitor (like pushing the child) at the resonant frequency of the coil, a large output voltage appears across the primary coil. As a result an even larger voltage is present on the secondary coil owing to the aforementioned transformer action between the two.
The secondary side consists of the same to circuit elements as the primary side. The secondary coil and a capacitor. There is one distinct difference in how they are connected however. The capacitor and secondary coil are connected together at top of the secondary. The bottom of the secondary is connected to 'ground'; a steak is literally driven into the earth. The capacitor used is unique. All most all capacitors have two 'terminals', the capacitor in the secondary circuit uses only one terminal. Now by selecting the proper values for the capacitor and the secondary coil, the resonant frequency of the secondary circuit can be made identical to that of the primary circuit. This is done so the maximum amount of voltage can be developed on the top of the secondary circuit.
Now when the primary circuit is operated, it causes a very large voltage to appear on the capacitor of the secondary circuit. This voltage is high enough to make the surrounding air conduct an electrical current and you get the desired lightning bolts.
Pictured at the right is a Tesla Coil in action. The long cylinder in the middle is the secondary coil. Atop the secondary coil is a doughnut-shaped piece of metal that acts as the single terminal capacitor. As you can see this is where the voltage is developed and the arcs eminate from. If you follow the large bolt in the middle of the picture you'll see that it is striking the top winding of the primary coil. If you just to the left of the primary coil you can see the primary capacitors (they look like two gray boxes.
You might be asking yourself how long the arc from a coil can be. The answer is that it depends on a number of factors. First the ratio of the primary and seconday coil windings has a strong influence on the voltage (and therefore arcs) appearing at the top of the secondary coil. Second, the more energy you have available to put into the primary coil, the longer you can make the arcs. Finally it depends on how well the resonant frequencies of the primary and seconday circuits are matched. The better the matching, the longer the arc. The latter of the two factors have a direct analogue with the child/swing system discussed earlier. Assuming you push at the right time, the stronger you are the higher the child can swing. In short, you need to push the child at the right time or it will not matter how strong you are.
Finally I thought you might be interested in some typical arc lengths
from coils. The average modern 'garage' coil typically ranges from 1 foot
to 10 foot long arcs. Some of the exhibition coils are capable of arcs
upto 30'. The longest arc length that I have ever heard is 135 feet. The
135 foot bolts came from Nikola Tesla's Colorado Springs coil. To put the
power involved in generating such an arc in perspective, at one point Tesla
drew so much power from the power power plant that it melted the generator.
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