because I have no idea what a constant current power supply is
Edit: Is a constant current supply one that adjusts the voltage depending on the resistance of the load in order to keep the current across that load constant?
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You say that, yet you showed that you can figure it out for yourself. In fact I think you grasped to concept very quickly.
Added to that, there are well known circuits using op-amps (and others using discrete components) which can give you the constant current you need without having to understand a great deal.
As for the Op-Amp I need, what you said also went beyond me. I think I'll just leave it and send it off to a company for design, there's literally hundreds of different Op-Amps out there and I have no clue what I'm looking for, I don't even understand the terminology yet.
There are a lot of concepts to grasp. However the ones you need for this are not that exhaustive.
Conceptually, an op-amp is a perfect device having certain unachievable characteristics which allows it to flawlessly perform a number of tasks.
Practically, op-amps are imperfect and different designs have different limitations and deviations from the theoretically perfect device.
The ones that are important to you are that:
a) it can operate from a single 5V supply.
b) the input common mode range includes ground (don't worry too much about this other than it means the op-amp will work with both inputs grounded)
c) the output pulls very close to ground.
An LM358 might do the trick. I think the main limitation will be the fact that it can only pull the output down to within 20mV of the ground rail (maybe a little better).
If you are going for a count of 0 to 800 (corresponding to 0 to 3.906V) then 20 mV corresponds to a count of 4, which is (assuming full scale is 10 ohms) is 0.1 ohm.
That's probably not good enough.
I need to ask. Why are you doing this? If it's to measure low resistances then there are devices you can purchase cheaper than getting someone to design and build them.