How to charge a dc battery with ac output?

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The best way to charge a battery from an AC source is to use a high-quality commercial battery charger that is of the correct voltage and charging profile for your battery. By "charging profile," I mean a charger that is matched to the battery's chemistry and amp-hour capacity. WARNING: Not every type of battery is rechargeable, and even rechargeable batteries can do nasty things if they are not charged properly. If you are not sure you have the right charger, please ask someone for help.

If you want to make your own charger, the answer to "how" depends upon the AC source, the kind of battery, and how "good" a charger you want to make.

At a minimum, you need some kind of rectification of your AC power. Your battery will tend to discharge during the part of the cycle where it is not reverse-biased, so you will need something like a diode to make sure the battery can't drain itself back into the charging circuit. Cheap chargers for car batteries, for example, are sometimes nothing more than a transformer (to convert the mains voltage down to something suitable for the battery) and a rectifier (two or four diodes).

More sophisticated charger circuits are used to maximize battery life (heat from recharging can damage batteries, so charging circuits use various strategies to control the heat) and to provide the appropriate charging profile for the battery chemistry. You'll see a pretty dramatic shortening of the life of your batteries if you don't use a specialized circuit to charge some kinds of batteries. For this reason, you may want to use a commercial charger. If you really want to build a charger, try to get a kit or build from a known-good circuit.
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How a Seminar on Why can not you store AC power in battery like DC power?

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