The circuit given is wrong. The whole point of a 4-wire resistor is that no current flows along the sense wires (so no IR error voltage is developed across the wire's resistance.
So connect high-side sense wire to pin 3, low side sense wire to pin2 (but NOT to ground). The current-carrying low-side wire is connected to ground of the opamp circuit. The reference pin and output go to Arduino gnd and analog input pin respectively (don't common up the reference wire with supply ground, use a separate wire - it has to accurately sense the Arduino ground voltage.
For measuring low voltages you always have to separate current-carrying wires from those used to sense the small voltages.
Once you have the circuit set up correctly see if its still misbehaving (sounds like an oscillation, probably due to the bad circuit topology you have).
Shouldn't be any problem with saturation till about 4.8V - sounds like its oscillating, turn down the gain to 100 and see if its doing better. A ground-plane is pretty much essential with a gain of 60dB (x 1000). You should have good supply decoupling, 100nF, 1uF and 1000uF caps ought to do it.
If oscillation is a problem and you can't use a ground plane use two stages of amplification with x30 each. and keep inputs away from outputs.
ABSOLUTELY NOT SO. Ohms law does apply. The point is that reactance and resistance are losses in circuit and work perfectly as long as you take into consideration the frequency of the AC current or volatge, a 1 ohm reactance will drop the same voltage as a 1 ohm resistor providing that you measure the drop at the frequency where the reactive component = 1 ohm. A resistance of 1 ohm will drop the same voltage with ac or dc. A reactance is C or L taken singly or combinations of both Vs frequency i.e. XL = 2Pi * F * L so a 1Hy coil has a reactance of 6.28 ohms at 1 Hz and a capacitor has an XC of 1/6.28 * F * C, so a 1 F capacitor has a reactance of of .15915 ohms at 1 Hz. Resonance is the point where XL = XC. The difference between XL and XC is the phase angle as a capacitor's instantaneous output current leads its voltage output and an inductor's current lags it's output voltage. At resonance the impedance is a very low value in a series configuration and a very high value in a parallel circuit. Attempting to explain more Basic AC Theory is I think beyond the scope of this forum. IMO
Do the math, It might be strange but it is eqasy. It's worked for me for 45 Years. Basic electricity is something everyone should know... especially If IT's more than blinking lights... I might point out at this junction that Allegro makes a lone of Hall effect ammeters with the sense element built in, range from a few amps to hundreds of amps. LEM makes a very wonderful current transformer that and be "wound" by wound I mean multiple windings through the hole for lower current measurements. I had a 25A lem current transformer that worked very for lower currents, 0 - 5A. THe Allegro device is a lot easier to use than the OP_AMP device mentioned. The Allegro has only one little issue and that is the output voltage is 1/2 Vcc so both positive and negative... Charge and Discharge...
Is it necessary to limit voltage drop to 1-3 mV? Why not set 10x or 30x resistor and decrease gain proportionally? On a link, they measure with gain about 10, less trouble for them.
That is the best bet, to increase the sense voltage and decrease the gain. From what you describe the circuit is almost certainly oscillating... This current sense wouldn't have anything to do with output voltage control??? feedback for a current or voltage regulator?
Hmm, yes, it does make sense to simply increase the sense resistor. Boosted that and it's helped a lot. Not sure why I chose such a small value to start. I'm going to grab all those capacitors anyway, doing an order now.
The current is regulated separately, there is no feedback.
(Oh yeah, and it's still saturating at 1.25 volts. Do I have a bad IC?)