operational amplifiers are used as comparators for detecting voltage. The two 10K variable resistors are used to adjust the green and red LEDs to goes OFF and ON on desired voltages.

For adjusting the circuit you have to use an adjustable power supply and set its voltage on 12.3V, and connect it with this battery monitor circuit in the place of battery. Now slowly adjust the 10K variable resistor connected with the pin 5 of the IC Until the green LED become switch off. Now again change the power supply voltage to 12V and adjust the 10K variable resistor connected with pin 2 of the IC until the red LED goes off. After making these settings make a final test with the circuit and again increase the power supply voltage to 12.6V on 12.6V both the LEDs should be activated, now slightly lower the power supply voltage to 12.3V, on this voltage the Green LED should become deactivated and red LED will still stay ON. Now slightly lower the voltage till 12V, on 12V the red LED should be deactivated. After these adjustments your circuit will be ready to use with your 12V batteries. Now on fully charged battery or on 12.6V both LEDs will glow showing the battery is full. When the battery voltage falls on 12.3V the Green LED will goes off showing the battery is half, and red LED will stay on showing the battery is low of half. On 12V the red LED will also goes off showing the battery is very low and require charging.
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This simple two stage battery high low monitor circuit can be used with variety of batteries from 6V to 12V. The circuit is quite simple to build and using a low cost LM358 opamp IC. The LM358 is a low power dual op amp IC contains two separate operational amplifiers in one chip. In this circuit both of its operational amplifiers are used as comparators for detecting voltage.
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Battery High Low Monitor Circuit Diagram