Try this one and then ask away...
This has exactly the same problem in my simulation.
I set V2 to go from 1-5V, and got the attached results.
I told you, this type of schematic CANNOT go down to zero on the output.. Compliance voltage on the good ones with a very small shunt are 1-2V ..
The diagram shows that load starts working nicely at around 2 Volts, which is great with a 0.1 Ohm shunt that drops a 1V on that current.. Together with 0.1 Ohm of the mosfet that is just right...
By using mosfet with very small RdsOn and very small shunt you could get down to 1 Volt maybe...
If you want to test something with this architecture down to 0 V, you have to connect a voltage source in series with a load to create few volts bias to compensate for dropout voltage of the load...
Why 0? I'm talking about 1+ volt.
This thing is designed as a battery tester. There are dozens of schematics that show this circuit as a load for battery testing. For example MJLorton (used code because otherwise it would embed the video, don't know how to prevent this) https://www.youtube.com/watch?v=xLI3yqZvHbs&t=10m40s
@ 10:40, you can see it being stable down to 0.5V. His circuit is pretty much a copy of Dave and can be seen in this video @ ~1:35. So it works in real live in his video.
So the circuit works down to that voltage just fine. So either the simulation is wrong, or the mosfet is too different. Dave used an MTP3055, MJLorton used an BUZ31LH - cannot find a ltspice model for either. I've also seen an IRLZ44N being used, simulation this one yields the same results.
So how can I find a similar mosfet to the two above and can they be so different?
IRF530: http://www.vishay.com/docs/91019/91019.pdf (http://www.vishay.com/docs/91019/91019.pdf)
MTP3055: http://pdf1.alldatasheet.com/datasheet-pdf/view/23382/STMICROELECTRONICS/MTP3055/+Q3_84UORlHDyRHOIpa/1XXyxeobifKHxO+/datasheet.pdf (http://pdf1.alldatasheet.com/datasheet-pdf/view/23382/STMICROELECTRONICS/MTP3055/+Q3_84UORlHDyRHOIpa/1XXyxeobifKHxO+/datasheet.pdf)
BUZ31LH : http://pdf1.alldatasheet.com/datasheet-pdf/view/397044/INFINEON/BUZ31LH/+2J7Q57VGL.pOhSSeHbFwzUy+/datasheet.pdf (http://pdf1.alldatasheet.com/datasheet-pdf/view/397044/INFINEON/BUZ31LH/+2J7Q57VGL.pOhSSeHbFwzUy+/datasheet.pdf)
When I modify the design to the one shown in the attachment (both files are the same circuit), I get it mostly stable in all voltage ranges.