How do capacitors improve power factor without destroying the capacitor?
I'm familiar with capacitors in DC circuits as tanks for storing energy


Don't think about storing energy when talking about capacitors in AC circuits!


  wouldn't the constant reversal of polarity destroy the capacitor?


As far as you don't have a polarized capacitor: Normally no.

It depends on the waveform (e.g. square or sine) you apply to the capacitor. We are talking about sine voltages here.

Unlike batteries capacitors can be charged and discharged without wearing.


  And wouldn't it be a huge waste of energy


Indeed you have some energy loss because of current flow in the capacitor. However due to the better power factor you have less current flow in the power supply system.

This means you also have less energy loss in the power supply system.

You pay a minimum of more energy loss (in the capacitor) and get rid of a much higher energy loss (in the supply system).


  how adding capacitors to ... AC power circuit helps improve power factor


Many AC engines behave like they have a coil in parallel.

Just like capacitors coils store energy and emit that energy later. When operating at 50Hz for example (sine wave voltages) both capacitors and coils store energy 100 times per second and in between they emit the energy stored before 100 times per second.

For the power supply system this is not good:

The energy stored in the coils must be transported from the power station to the engine over transmission lines and when the energy is emitted it must be transported back. Transmission lines have a resistance (of course) so whenever energy is transported over the transmission line there is also a loss of energy.

Therefore it is not wanted that electrical devices behave like they have a coil in parallel but unfortunately for many devices this effect cannot be avoided.

The more detailed explaination how the capacitor works would be looking at the differential equations of the capacitor and the coil.

The more simple explaination is looking on the way the two parts are storing the energy:

The higher the absolute voltage is the more energy is stored in a capacitor. When the voltage is zero the capacitor has no energy stored. This is true for DC, AC and any other voltages.

When a coil is connected to a sine voltage (note: this is only true for sine voltages) it is exactly the opposite: When the voltage is in the maximum the coil does not have any energy stored; when the voltage is zero the coil has the maximum of energy stored.

This means: Whenever the coil emits energy the capacitor will store energy and whenever the coil stores energy the capacitor will emit energy.

The trick is to choose a capacitor whose capacitance has a value so that the energy stored by the capacitor and the energy stored by the coil is the same.

In this case no more (additional) energy must be transported over the transmission lines because the energy is exchanged between the coil and the capacitor.