This very simple circuit uses a transformer ,two diodes , a capacitor and an ammeter.

To charge a battery just connect the + and - terminals of the circuit to the corresponding terminals of the battery. When the battery is not charged, the ammeter reading shows 1-3 amps.

When the battery is fully charged the ammeter reads Zero or nearly zero, after which the battery should be removed from the charger.

The circuit is a full wave rectifier using 2 diodes for dc power supplyrectification. The capacitor is used for smoothing.

I think the circuit works fine without the capacitor since the battery itself acts a BIG capacitor. But when you are using the circuit to supply 12V (as a battery eliminator) the capacitor needs to be present.

Care should be taken NOT to reverse the + and - terminals while connecting it to the battery.