How much current does it take to switch a diode?

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Ohm's Law for a diode, relates the current through the diode to the voltage across it, by an exponential relationship, the 0.7 volts is just an approximation, and it really depends on where on that curve you are.

With a reverse voltage then the current should be zero, but really it is just a minimal value so the circuit should be "off"

The next question is how and when do you reverse bias that diode in the circuit, at what point in the pulse train? It is reverse biased when the voltage on the secondary is lower than the voltage on the choke at the end that connects to the diode.

But since voltage is relative it becomes more unclear, you have to choose a reference for the circuit, which may or may not be grounded on the secondary side.

I think a differential probe across the diode would be a useful step.

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yea current can only pass in the forward bias direction. If you try to pass current in reverse bias, you will cause an avalanche effect and damage the diode. This is why in my video where I explain how the the LC circuit operates, I saw that due to the diode you will have only the positive phase current flowing through the circuit. The diode also keeps the positive phase of the L2 choke from reaching the secondary. Another thing to remember is that positive voltage is lacking electrons and a negative voltage has an abundance of electrons. While keeping this in mind, think of how a a transistor operates. You have a P material sandwiched between two N materials to form a NPN transistor. In Stephens radio interviews he says the cell acts much like a transistor, you have the two plate (N material) and the water in the middle (P material). That is something for you guys to think about