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The resistor in question is R2.

The below circuit provides roughly constant current above a certain Vcc.

For R2, I arrived at the value by trial and error by checking simulation outputs.

I would like to make a more precise decision on the value of this resistor. How can I start to to think about doing this?

Here is the current through the LED:

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    \$\begingroup\$ Here you find the analysis of the same circuit but with NPN transistors instead of a PNP electronics.stackexchange.com/questions/353904/… \$\endgroup\$ – G36 Apr 17 at 13:25
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    \$\begingroup\$ JC123, And for still more detail, if desired, please see here. \$\endgroup\$ – jonk Apr 17 at 17:52
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R2 provides base current for pass transistor Q1. If it is too high then LED current will be less than VBE/R1 and vary depending on the HFE of Q1. HFE is quite temperature sensitive and depends on the transistor. If R2 is too low then the circuit wastes more power than it needs to. R2 is 2K on your simulated circuit which gives more than 10mA drive at 24 V. This means that LED current would be fine at low temps with low gain transistors. If your minimum gain was about 50 you could increase R2 to say 4K.

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    \$\begingroup\$ There is one more consideration: R2 plays a role in determining how low the voltage can go while remaining in current regulation (compliance range). \$\endgroup\$ – Dave Tweed Apr 17 at 13:32
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    \$\begingroup\$ @Dave Tweed would you kindly explain how that works? \$\endgroup\$ – Eon Apr 17 at 14:04
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    \$\begingroup\$ @Eon: Vcc gets divided across three components: R1, D1 and Q1. When in regulation, the voltage across R1 and D1 is fixed, so any variation must occur across Q1. Since Q1's V_be is also fixed, this variation also appears across R2. If R2's value is too large, when Vcc is low, it won't supply enough base current to Q1 to keep it in regulation. But on the other hand, if R2's value is too small, it will dissipate too much power when Vcc is high. \$\endgroup\$ – Dave Tweed Apr 17 at 17:56

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