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I have a friend who has asked me to help them with tuning their tesla coil. Every schematic I can find for building one simplifies down to essentially the circuit below. I recognize the primary circuit as an LC circuit. And I see that essentially what we have is a resonant transformer. My question is, how do the primary LC circuit, and the secondary coil interact to create the resonant frequency. I can solve the primary LC circuit easily enough, but what role does the secondary play on the frequency emitted?

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The secondary circuit is also an LC because the topload ("torus") in your circuit acts as one plate of a capacitor, with ground forming the other plate. The windings in the secondary also have some parasitic capacitance. Generally speaking, it is safe to say that the primary tank circuit is made up of a high capacitance and low inductance with a certain resonant frequency, and the secondary tank circuit is made up of a low capacitance and high inductance with a similar frequency. The most practical way to tune the secondary is to change the capacitance (i.e. change the size of the topload), though this is more of a fine adjustment. For most cases you'll want to try the coarse adjustment first, which usually involves changing the primary coil "tap point". Changing the inductance of the primary coil will also change the resonant frequency, thus bringing it closer (or further away) from the resonant frequency of the secondary. Usually streamer loading will reduce the resonant frequency of the secondary by about 10%, so your primary circuit should be tuned about 10% lower than the resonant frequency of the secondary.

JavaTC is a tool created by Bart Anderson to help design and tune Tesla coils. I recommend putting in your specs and see what JavaTC has to say. It even has an "Auto-Tune" button!

http://www.classictesla.com/java/javatc/javatc.html

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The actual resonant frequency of the Tesla coil would be lower than that measured on the primary, on account of the loading effect of the secondary.

Likewise, the actual resonant frequency of the secondary would be lower than that measured, on account of streamer loading.

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