A battery charger like
this
costs £4. It is unlikely you can build one for that price. To build items for the fun of building yes but by time you buy a box to mount it all in, cable, rectifier, voltage reg, and LED to try and build for less than you can buy is not really going to work.
I found this out a long time ago. I lived on a caravan park full of contractors caravans most touring vans and we all wanted to charge batteries to run our lights and water pump. The cheap battery charger has no control it is designed that way as in the main they are designed to put on a car battery every 6 months to give it an equalling charge. Mainly from dynamo days.
This would result in one of two problems. Either the battery would go flat or run short of water due to being over charged. Either way batteries did not last long. Today you can buy regulated battery chargers without a problem but back then they were rare.
But if one tried to put a transistor in the supply the 0.6 volt drop was too much for a standard battery charger to then give full output. So I made a unit which would turn excess into light and heat. An Opamp compared the voltage output to a zener controlled standard and when the voltage raised it would turn on a transistor (2N3055) which powered a 36W lamp. This stopped the overcharging and also gave visual indication that the battery was charged. First one made with bits out of junk box. It worked well so I decided to make more. However this is where it all failed. It would cost around £10 and by time I added my time it just was not worth making.
The battery charger typically has a very poor quality transformer and it uses this to produce a natural steep cut off at the battery becomes charged so for a 8 amp car battery charger at 12v it will supply 8A but as the volts rise by time it reaches 14.5v the charge rate is around 0.5A.
So there is the unregulated. The float voltage regulated. The 3 stage step battery charger, and the pulse battery charger each getting more expensive. The last two are designed to quick charge. In the old days of open cells the hydrometer allowed us to monitor the battery. Today however with sealed batteries not so easy. So we have to decide good charger or short battery life.
Last Radio Rally I went to regulated battery chargers for small 12AH batteries were going for around £10.