2 added 91 characters in body
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Here is an alternative method that is almost as simple and is more repeatable (and, probably at least as important, has a lot of compliance):

schematic

simulate this circuit – Schematic created using CircuitLab

Use a MOSFET rated to work on 2.5V drive if your logic input is 3.3V. It needs to turn on fully with about 2.7V to get the full compliance range. You can also use a BJT in place of M1.

The current is about 0.6V/R2.

It works by Q1 reducing the gate voltage to M1 (or base current if a BJT) as the voltage across R2 approaches 0.6V or so and Q1 starts to turn on.

It will drop out around 0.6-0.7V or so if the parts are well chosen, so a 3V LED can be maintained at fully brightness/current to about 3.7V input.

Here is an alternative method that is almost as simple and is more repeatable (and, probably at least as important, has a lot of compliance):

schematic

simulate this circuit – Schematic created using CircuitLab

Use a MOSFET rated to work on 2.5V drive if your logic input is 3.3V. You can also use a BJT in place of M1.

The current is about 0.6V/R2.

It works by Q1 reducing the gate voltage (or base current) as the voltage across R2 approaches 0.6V or so and Q1 starts to turn on.

It will drop out around 0.6-0.7V or so if the parts are well chosen, so a 3V LED can be maintained at fully brightness/current to about 3.7V input.

Here is an alternative method that is almost as simple and is more repeatable (and, probably at least as important, has a lot of compliance):

schematic

simulate this circuit – Schematic created using CircuitLab

Use a MOSFET rated to work on 2.5V drive if your logic input is 3.3V. It needs to turn on fully with about 2.7V to get the full compliance range. You can also use a BJT in place of M1.

The current is about 0.6V/R2.

It works by Q1 reducing the gate voltage to M1 (or base current if a BJT) as the voltage across R2 approaches 0.6V or so and Q1 starts to turn on.

It will drop out around 0.6-0.7V or so if the parts are well chosen, so a 3V LED can be maintained at fully brightness/current to about 3.7V input.

1
source | link

Here is an alternative method that is almost as simple and is more repeatable (and, probably at least as important, has a lot of compliance):

schematic

simulate this circuit – Schematic created using CircuitLab

Use a MOSFET rated to work on 2.5V drive if your logic input is 3.3V. You can also use a BJT in place of M1.

The current is about 0.6V/R2.

It works by Q1 reducing the gate voltage (or base current) as the voltage across R2 approaches 0.6V or so and Q1 starts to turn on.

It will drop out around 0.6-0.7V or so if the parts are well chosen, so a 3V LED can be maintained at fully brightness/current to about 3.7V input.