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Let's start with this:

enter image description here

I know the top image is of a flyback diode. I also get that its purpose is to protect the power supply from a voltage spike when the electromagnet releases. So I've gotten that far.

The second diagram is what gets me, I guess. In this diagram, I am assuming the diode is being used in a more conventional setting and it is wired inline (in series?)" The cathode stripe in this picture, though, is toward the negative end of the circuit, which is exactly the opposite of the way it is being used in the former picture. So, that's confusing to me. Why is the polarity different between the two? Also, why do I need to wire a flyback diode in parallel? Shouldn't I just be able to put it inline in the positive wire to protect the voltage source? (And at that point, would I put the cathode end toward the solenoid?

I'm just trying to wrap my head around orientation and application here. I see lots of videos explaining how they act as "valves" and can rectify AC current, but very little about how flyback diodes work (and why they are wired in parallel).

Still getting my sea legs, here. Thanks for any helpful input.

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Looking at figure two, the polarity has been marked such that, if you hooked it up to a battery like that, current would flow.

In figure one, the diode is reverse biased by the power supply, and current will not flow.

If you unplug the power supply, the solenoid drops out because the current stops flowing through the coil. However, the coil has inductance, which has the property of resisting any change in current flow - that is, the power supply will stop producing current, and the coil will start producing current in the same direction, until its magnetic field is totally gone.

Think of the stripe as an arrow showing you which way current is allowed to flow through it, and imagine you just unplugged the power supply. Understanding that current will no longer go through the power supply, you should be able to see that it will now flow out of the coil and through the diode.

If that diode were not installed, the coil can produce something called inductive kick, which can and will destroy the power supply by pushing voltage higher until it gets current flow - possibly through something that shouldn't be passing current.

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Why is the polarity different between the two?

The diode is there so that if the solenoid creates a voltage that's negative, current can flow through the diode instead of backwards through your power supply. To do this, the diode has to be backwards - if it was the same polarity as the power supply, then you'd be instead stopping the power supply from producing the correct voltage.

Also, why do I need to wire a flyback diode in parallel? Shouldn't I just be able to put it inline in the positive wire to protect the voltage source?

It's in parallel so that this current (out of the solenoid) has a path different from the power supply to flow. I'm not sure about wiring it in series and why it's better to use a parallel branch.

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