I'm trying to put together a constant current sink circuit for discharging small Lithium-Ion coin cells. I need to be able to sink from about 200 uA to 12 mA. My first thought was to use some voltage regulators like the LM317 or the LP2951. Unfortunately for the LM317 as the battery discharges from 4.2V to 3V the sink current doesn't remain constant at low currents (see attached picture2.png). 

I've attached the circuit (picture1.png)- I'm measuring voltage across the 10 Ohm resistor to measure current. R2 is a 256 step digital potentiometer to set sink current. In Multisim it seems like I shouldn't have a problem getting down to ~200 uA, but in practice, I don't get the same currents when the voltage source is at 3V or 4.2V (I'm using a power supply to simulate a battery). In the attached graph of current vs. pot value, I can only maintain stable current between 3 and 4.2V at currents > 1mA. 

I tried the LP2951 in the same circuit but got even worse results (picture3.png). What gives? I thought the 2951 was a low dropout regulator, and so should have no problem with the low output voltage?

Does anyone have a suggestion for a better voltage regulator or circuit design for sinking such low currents?

Thanks!

[![picture1.png](https://i.stack.imgur.com/34Reo.png)](https://i.stack.imgur.com/34Reo.png)

[![picture2.png](https://i.stack.imgur.com/CqMqs.png)](https://i.stack.imgur.com/CqMqs.png)

[![picture3.png](https://i.stack.imgur.com/R8Rb1.png)](https://i.stack.imgur.com/R8Rb1.png)