When the pushbutton switch is closed, capacitors CREF and CX are charged through their respective PNP transistors. When the switch is opened, these capacitors begin to discharge. CREF (assuming it is in good condition) has additional external discharge resistors. The capacitor under test, CX, discharges through its own internal resistance. If the leakage of CX is greater than the leakage of CREF through RREF, the voltage on CX drops faster. As a result, the voltage at the non-inverting input of the op amp will be lower than the inverting input, making the op amp output low and the red LED illuminated. This LED is used to indicate that the capacitor under test has leakage. Testing of the circuit has shown that even 1nF ceramic capacitors can be detected. Check the rating of the test capacitor to make sure it is higher than the voltage it will be charged to, in this case VSUPPLY is -1.8V.
The minimum supply voltage of the LF357 is 10V, but the test only requires 6V, so a low upper voltage limit is allowed for the test capacitor. Make sure the capacitor has a FET or MOSFET input stage.
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