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Now say the current is increasing. Discharge the capacitor to an output capacitor while reversing the connection so the positive terminal is connected to the negative and vice versa.

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The inductor resists change in current.

Inductor negative voltage. In this case energy from the magnetic field is being returned to the circuit. The tps60400 family is an example of such a device. So it generates a negative voltage across it.
In this scenario the inductor will be acting as a source with the negative side of the induced voltage on the end where electrons are exiting and the positive side of the induced voltage on the end where electrons are entering. Energy stored in an inductor. If you start with thinking about current through an inductor what you have to think about is the rate of change of current through the inductor resulting in certain amplitude of voltage across it that opposes the rate of change.
This approach generates a negative voltage equal to the input voltage for example 5v from a 5v supply. The inductor will try to oppose the change by creating a current the opposite direction from point b to a. With a steady state dc current flowing through the inductor and therefore zero induced voltage across it the inductor acts as a short circuit equal to a piece of wire or at the very least a very low value resistance.
If the current is decreasing the induced voltage will be negative at the currents entrance point and positive at the exit point tending to maintain the current. So say you measure the voltage across the inductor from point a to point b the current is flowing in from a towards b. The more rapidly current is decreased the more voltage will be produced by the inductor in its release of stored energy to try to keep current constant.
In other words the opposition to the flow of current offered by an inductor is very different between ac and dc circuits.
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