I have a circuit that normally runs at 3-5v and then when triggered, it gets pulled to ground (around .1-2 mv leakage). I want to use this to trigger a relay to close a different low voltage circuit. Can I use the "Sainsmart DC 5V Relay Module for Arduino PIC ARM DSP AVR MSP430 TTL Logic" board to achieve this?
Yes. A low level input WILL operate this OK as long as it can provide enough current to operate the optocoupler. See below
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The circuit diagram is shown below.