I've tried to make the descriptions as simple as possible and I've tried not to use electronic or technical terms.
Some electronic or technical terms are of course un-avoidable.
1) Voltage is applied , the primary capacitor (C) charges , the voltage across the primary capacitor increases.
2) Spark gap (SG) fires, when the voltage across C reaches the spark gap break down voltage.
3) Capacitor (C) is discharged through the spark gap (SG) and primary coil (P).
4) The magnetic field in the primary (P) increases as the voltage across the capacitance (C) decreases.
5) When the voltage across the capacitor (C) is zero, the field in the primary coil (P) starts to collapse, creating a voltage which re-charges the capacitor (C).
This creates an alternating current in the primary coil (P).
6) Because the secondary coil (S) is connected (by transformer action) to the primary coil (P). The alternating current in the primary coil is transferred to the secondary coil (S).
It takes a few of the above cycles for all of the power to be transferred into the secondary coil (S).
7) The topload (T) and the secondary coil (S) are designed to swap power between them in the same way and at the same rate as the primary capacitor (C) and the primary coil (P). This matching of frequencies multiplies the induced voltage in the secondary coil (S).
8) When there is more voltage than the topload can hold on its surface. The voltage breaks out into an arc.
9) This process continues until there is no more power in the primary capacitor (C) and primary coil (P). As there is now not enough voltage to maintain the arc across the spark gap (SG), The arc extinguishes.
This alows the capacitor(C) to charge again and the sequence repeats.
This is a video that describes how a Tesla coil works in basic terms. (https://www.youtube.com/watch?v=M-vz82dx6gQ)
See more Tesla Coil videos
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