Arlec BC58112 v battery charger

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The board has DZ1 printed under it, the resistors are marked R1 etc, and I have just found, using a magnifier, that the glass tube has fuse printed under it.

That is the thermal switch and it will work as I described to "regulate" the charge current.

elements in the circuit would you expect to be destroyed?

The zener, if it is one, would be associated with the led circuit. For what it's worth, using LED's to indicate charge status on such a cheap charger is almost worthless anyway. You could simply disconnect anything associated with the LED's and as long as the transformer, the rectifier diodes and the thermal switch are ok the charger will work.

You can check to see if the zener is ok by using a DMM (diode range) to measure it in both directions to see if it short circuit. If it measures around 0.6 in the forward direction and open in the other it might still be ok. To test it fully you need to use an adjustable bench power supply and a series resistor (say 470 ohm 1/2W). Connect the resistor in series with the zener and hitch it up to the psu output - zener cathode to +ve. Now raise the voltage output while measuring the voltage across the zener. You should see the voltage reading plateau at its rated voltage even when the psu voltage keeps increasing. If your PSU has both voltage and current meters you can perform this test without an external meter across the zener. In this case you will see the current meter jump up from near zero while the output voltage is increased slowly. With a current of around 20mA stop increasing the voltage and read the output voltage on the psu and that will be the nominal zener voltage. If you dont' get this result the zener is faulty and needs replacing.