Electronic – using TTL to drive LED

TL;DR version:

Use two alkaline AA cells, a 2.7 kOhm resistor and the LED in series, such that the LED is provided the bare minimum current that lights it up sufficiently.

Summarizing from chat discussion around this question.

• Determine empirically what minimum current through your specific LED provides sufficient light to be acceptable as "staying on". Done: 0.25 mA
• Measure precisely the forward voltage of the LED at around twice that current. Done: 1.86 Volts at 0.5 mA
  • ^{Why the higher current? So that the "staying on" condition remains valid even when the current drops to half the starting value due to battery depletion.}
• Determine nominal voltage for your power source: Alkaline AA = ~ 1.5 x 2 = 3 Volts
• Simply add a series resistor to the LED such that ~ 0.5 mA is supplied to the LED. Calculation: R = (V_batt - V_led) / I_led = (3 - 1.86) / 0.0005 = 2.28 kOhms
• Next higher E12 value = 2.7 kOhms, gives 0.42 mA, well within our desired current range, so we are done.

Now, calculating power consumed and longevity of the solution

• Circuit power consumption: P = V x I = 3 x 0.00042 = 1.27 milliwatts
• Capacity of batteries:
  • Assuming Energizer E91 Alkaline AA.
  • From Milliamp Hours Capacity chart, extrapolating to discharge rate less than 1 mA, but discharging only to 1.27 Volts gives approximately 3000 mAh. This assumption is such that the voltage does not drop below 2.535 Volts for 2 batteries in series, so that current across LED does not drop below 0.25 mA.
  • Keeping that as a worst-case number, 2 cells = 6000 mAh (Approximation!)
  • At 0.42 mA, minimum longevity = 6000 / 0.42 = 14285.7 hours = ~ 595 days, close to 2 years.
• Even if we discount the calculations above significantly, this solution will still give around a year of battery life with a good estimation buffer.

_{As per experiments conducted and mentioned in chat, using a resistor of 2.4K, it consumes 0.25mA at 2.5V, 0.42mA at 3.0V. This proves that the calculations above are conservative, a longer duration is not inconceivable.}

From what I can see this little box is designed to take a 12V low-current signal and amplify it to a 12V high-current signal.

It is not suitable for directly connecting to an Arduino.

You will need three NPN transistors to connect between V+ and each of the R, G and B inputs, and then drive the bases of the transistors through a resistor from the Arduino.

But that's only a wild stab in the dark since there seems to be absolutely no documentation available anywhere.

After much hunting I managed to find this image:

so it seems it is designed to go midway through a chain of LEDs to power more of them, not be driven by a low voltage device like an Arduino.