Re: Fuel level sending unit test
The fuel gauge and a variable resistor (rheostat) in the fuel tank make up a series circuit operated by 12v usually from the ignition switch ( to keep it from draining the battery when not in use). Resistance in the tank varies from somewhere around 35 ohms to 135 ohms and system OEM's caution against short testing as the meter can be damaged by the slug of unresisted current, they say even for a short "short" test.
I don't remember which is which, i.e. 35 ohms full or vice versa. Doesn't matter as you can figure it out while testing.
The only series testing would be with power off using your ohmmeter and moving the float from one limit to the other while observing the resistance reading.
Problem with fuel gauges is the inability to accurately position the float as a function of how much fuel is in the tank and how much reserve fuel you want in your tank when the gauge reads empty.
The center screw is the 12v lead and the disc is the - or ground. If you get them reversed, the current reverses hence the needle reverses and could be damaged.
If you are getting a different reading with the key on vs the engine running you are getting some interference or your alternator is putting out it's recharge voltage which is roughly 14.5v rather than 12.75 so the current will be more and for a given position (fuel level) , you will read differently. The position of the needle is strictly how much current runs through it.
HTH,
Mark