I constantly read that capacitors provide reactive power. So how are they used instead of a battery? I mean in a battery the voltage and current are in the same phase. How is the phase change between the voltage and current in a capacitor eliminated without using an inductor, to use capacitor as a power source?
A battery stores charge in a different manner than does a battery, but charge is charge. In terms of comparing it to a battery of the same physical size, a capacitor holds less charge, can be charged to a high voltage, and has a very steep discharge curve. The good news is that it can be recharged almost instantly.
I mean in a battery the voltage and current are in the same phase.
This is an incorrect expansion of a narrow condition. The phase of the voltage and current are set by whatever the load is, not by the power source. If the load is a simple resistor, the voltage and current are in phase no matter what the source is; battery, capacitor, generator, whatever. If the load is a capacitor, then while it is charging from a DC source the current and voltage are not in phase no matter what the source is.
How is the phase change between the voltage and current in a capacitor eliminated without using an inductor, to use capacitor as a power source?