repairing the bmw e34 fuel sender unit by fileoffset
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The fuel sender unit is a fairly simple unit. It sits on the top of the fuel tank and is held in place with a rubber o-ring (#2) and a large plastic circular seal that screws in (#3) and holds it all in place:

(#1 is the sender unit itself, the other assembly shown is the fuel pump)
This repair guide is for the plastic unit only, though the more common (in america at least) metal tank/sender unit combination should be very similar.
Common problems with the sender unit include:
1. Misrepresentation of the amount of fuel in the tank
2. Inconsistent readings, guage drops in and out
3. The fuel light is intermittant or does not work at all
4. The guage doesn't work at all
If you have any of problems 1 to 3 this guide should help you fix them. If you have 4 then it may or may not work, more diagnosis may be required and is not covered here.
Removal
Follow the instructions in your manual to remove the sender unit. If you dont have a manual i've written up quick instructions, its a fairly simple procedure:
1. Open the boot
2. Take up the boot lining
3. Undo the 5 or so screws holding the access plate in place
4. Undo the two hose clamps going to the sender unit, pull them off and block them up with a bolt or something similar
5. Undo the large outer screw (#3 above)
6. Lift the unit out of the tank carefully
7. There will be two wires and two hoses going to the fuel pump, fixed inside the tank. you can either take the fuel pump out aswell, or do as i did and just undo the hose clamps at the fuel sender side. the wires you need to undo on the pump itself, which can be a bit fiddly if you don't take the pump out first.
Opening the sender unit
Once out of the tank you will need to open the unit itself. This involves pulling off the large cylindrical cover on the bottom of the unit, its held in place by two tabs on either side. This is probably the trickiest part of the whole operation. I found it best to use two small screwdrivers, push up over both tabs and twist the cover clockwise.

As you pull it out, unthread the wires and you will have something like this:
Pull the very top off slowly and carefull. Once off you will see the real guts of the sender unit:

The black thing in the middle is the float, it floats on top of the fuel as the fuel rises and lowers. There are three wires that thread from the top to the bottom, as well as the main guiding bar in the centre. Two of the wires are 'filament' style wires and are extremely thin, the third wire is slightly thicker and seems to act mainly as a guide for the float, so it doesn't twist around and snap the filaments.
The two filament wires, and the float are the most important parts of the whole system:

As the float moves up and down, it should be in contact with the two filament wires at all times. test that the two filament wires are taut. if they are not, then slackness will develop and the float will lose contact, which in turn will cause the guage to show no fuel!
If the wires aren't taut (one of mine wasn't) then try to pull it tight. you can do this by using a pair of needle nose pliers and slowly and very carefully crimping one of the little hook ends which the wire threads under (shown above) down a little bit. if the wire is very very loose, you can try and re-solder it, though this could be pretty tricky and i wouldn't recommend it unless you really know what you are doing (or have four hands :P).
While you are at it, i also recommend getting some very fine sand or emery paper and giving the filament wires a quick rub, being careful not to use too much force and snapping them.
On each side of the float where it contacts the filament wires, it has two tiny contacts which are sprung to grip the wire on each side, kind of like tiny little hands clasped together over the wire. get a sliver of emery paper and slide it in, carefully cleaning both 'hands' where they contact the wire.

Lastly, if you have inconsistant or non working fuel warning light, then clean and adjust the pads on the bottom of the unit:
Clean the contacts and make sure they are level. when the float is resting at the bottom, both pads should come in contact. with mine, only one was, so i used needle nose pliers to bend it up into place.
Now that that is all done, its time to test the resistance values. the bentley manual gave me the following table:
Simply connect your multimeter to G and 31 and move the float up and down the rail slowly, it helps to have an extra set of hands here. the resistance should be in the range shown in the table. With my sender unit, the readings were continuous now, and the Empty continuity was reading properly aswell, but the resistance readings were still off by about 20 or 30 ohms. this means that if i was to put it back in the car it would work but it would always show around 15 to 20 litres under what was in the tank, this is better than nothing but its hardly ideal. This turns out to be a fairly common problem. after a long time, and face it these cars are getting on in years now, the solder in the joints start to deteriorate and no longer conduct as they used to. the easiest way to fix this is to de-solder the main joints and re-solder them. Using my multimeter i kept reading the ohms in different parts of the circuit and i finally found the culprit:
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The wires from the top of the float assembly join to the sender unit output plug as above. also attached to the plug are the two wires that attach to the fuel pump, giving it power.
All you have to do is de-solder the three coloured wires, grey, green and brown, leaving the two thicker wires in place. i used solder wick to de-solder, and then freshly soldered them all back in place. cautious note! after re- soldering the wires, wait until the contacts have cooled down completely before re-measuring the resistance.
After re soldering the wires i found that my resistance was only about 3 or 4 ohms off the correct values, good enough for me!
Note: if you still have too high a resistance and you have tried resoldering the joints, then you can actually solder a resister in parallel to the circuit. there are equations on the internet for working out what resistance resister you need to solder in parallel to bring the total overall resistance down to the correct level. this will work but it should be a last resort.
Thats it! your sender unit should be good as new! or close enough anyway :)