Low Overhead Constant Current LED Driver

I came up with this circuit in trying to come up with a cheap LED flashlight for poorer people. The inexpensive flashlights tend to just use a resistor only, and would waste too much power (money) for those who only make a dollar or two a week. I finally settled on 2.6mA as the minimum usable light level, which I've been using for a few years. I've changed the batteries once, and they seem to last forever. I do believe that the overhead of this circuit is about 200mV, which is what you wanted. Replace R9 with a trimpot or a potentiometer to make it adjustable, or to find the actual value of the resistor for your particular LED. Remains fairly stable in the temperature range that I expect a flashlight to operate in, but it might not work well in Alaska or Death Valley, and need the trim-pot instead of a resistor if your product goes world-wide.

A Linear Regulated Discrete Constant Current LED Driver delivering about 10mA:

You could use something like the below, which requires less than 60mV of headroom. Sense voltage is only 50mV so the offset voltage of the op-amp will tend to greatly affect the current. The LED could be controlled by connecting R3 to a port pin.

(R4 and C1 may not be necessary- it may be possible to omit C1 and short R4 but that would have to be determined- that phase margin is enough for good stability)

^{simulate this circuit – Schematic created using CircuitLab}

Note that if your supply is nominal 3.3V and your LED nominal 3.1V Vf you still may not have enough headroom if the LED happens to be on the high side and your 3.3 turns out to be on the low side.