Let's keep this simple, so anyone can understand. 

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A tesla coil has two main parts: the primary and secondary.  The primary is the low voltage side, typically seen at the bottom of the "tower", or in my case, the PCB itself! The secondary is the "tower" and the top torus. 

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How does it work? Well, it's easy to think of the tesla coil as a swing set. If you push someone at just the right intervals, the swinger will have a better and better time (that is, until their mv^2/r > mg and they flip right over. I digress. The big idea is that if you hit that swing at the right time, you are pumping that swinger with more and more energy. This is the exact same deal with the tesla coil. 

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The "secondary" is kind of an electronic swingset, and for the pros out there, this is called an LC circuit. While we could go through the differential equations and stuff, it's more important to have an intuition. We still haven't answered the obvious question: how do we "push" the secondary?

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Well, a part of Maxwell's equations (Faraday's Law) states that any changing magnetic flux (or replace "flux" with "field", but don't' tell EE's I said that) causes an electric field. So, to drive the secondary, aka "push" it, we need to change the magnetic field inside the "tower". How do we do this? The "primary" of course! The primary coil is something we control, and we use special spark gap timers to "fire" this primary at the perfect frequency. Recall that this frequency is the resonance frequency of the secondary. Going back to the swing analogy, the primary coil is the guy pushing the swing, and the secondary coil is the guy on the swing. 

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Just as how pushing at the right intervals builds up energy, by hitting that resonance frequency of the secondary, the primary transfers energy to the secondary. If it were off-frequency, this energy wouldn't stay--it would get tossed between the primary and secondary like a game of catch. However, at resonance, the energy transfer is one-way. Obviously the secondary isn't going to rise in altitude or move faster as the swing would. Instead, this energy is stored inside a changing magnetic field, with the voltage rising higher and higher. When the electric field caused by this rising voltage hits above the dielectric breakdown strength of air, we get an arc! Now, this causes the voltage to rapidly drop, but that's no problem! The primary will keep on "driving" the secondary until this breakdown happens again.

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Now, for the pros again: the coil of the secondary is the inductor, but where is the "C"? Where is the capacitor? That's where the top torus comes in. This is actually a very small capacitor (typically picofarads) WRT the earth.

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In the "old" days, as mentioned before, the "driving" frequency of the primary was determined by spark gaps.  They would use giant capacitors and wait for the electric field around a small gap to reach dielectric breakdown and then discharge it through the primary. This would happen in very predictable intervals, but it was loud and most importantly, incredibly dangerous. Large, high-voltage capacitor banks are a no-no for amateurs. So, some genius came up with an amazing idea. What if we use another LC circuit to drive the primary? Now, if this LC circuit had the same frequency as the secondary, the timing would be solved! This is as if we had two swingers next to each other, and they linked arms. However, who does the pushing? Well, primary voltage levels are much lower, so we can actually use a timer chip and IGBTs (a type of transistor) to pulse mains voltage across the primary. We couldn't do this for the secondary because the voltage is way too high, normally at millions of volts. Tesla coils with this modern construction are called Dual Resonance Solid-State Tesla Coils, or DRSSTC. This type of coil is what you see in the video.

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Now, how do we play music? Contrary to popular belief, we can't just adjust this resonance frequency. Even if we could, this swingset is yeeting around at millions of times per second--definitely not in the audible frequency. Instead, we actually take a more macro perspective. We just turn on and off the entire system at audible frequencies. This will make audible sound, and this is the basic idea behind my musical tesla coil.

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If you thought this was interesting, I invite you to learn more about LC resonance in Young's College Physics, where they do go over the differential equations and derivations. (I'm not sponsored).