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A 5V LED should run for a predefined time period once a button is pressed. The LED should be illuminated for the same duration as the button is pressed, but only once the button is released and illumination should last at least 5 seconds. The time deviation should not be more than 2 seconds. Since the controller needs to be positioned on the backside of a wood panel, only limited physical space is available (the smaller the better). A 5V power source is provided for. The project is part of an art project that is to be displayed on a music festival in the summer of 2020. At least 200 of these panels+controllers are needed, so cost should be reduced to a minimum.

Previously (prototype), an Arduino Nano was used. Now, cost and size need to be reduced further. What would be the minimal components required to complete this task?

cheesus
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There are many low-cost approaches for solving this problem. I will list the two that I would consider:

1) CD4060 (or 74HC4060) timer / oscillator. These chips cost several pennies from Asian suppliers. A few resistors and a couple of capacitors are required. Probably may also need an output buffer for the LED, depending upon LED current.

2) 6-pin Microchip PIC. These are also available for very little money, although my favourite part from that family (PIC10F321 or 322) does cost a bit more. The chip can directly supply up to 20mA or so to the LED, so a buffer may not be needed. Again - depends on the LED. You can also feed the LED with multiple pins for more current.

Still need a couple of resistors and a couple of capacitors but otherwise very simple.

The PIC-based solution is probably the most flexible and would also have the smallest footprint. You could literally fit the whole circuit on a PCB the size of a postage stamp.

Dwayne Reid
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As I read your question, the LED is to stay on for the same amount of time as the switch is on, but for at least five seconds. ( I don't know why others are saying five minutes.) I have designed analog circuits that could do this, but by far the least expensive solution is the PIC. One six or eight pin PIC, a bypass cap, the switch, a resistor or two, and a FET if the LED requires more than 20 miliamps. All would fit on a PCB about half an inch on a side. ( Assuming the switch is located off board (and possibly the LED.))

EinarA
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When timing doesn't need to be that exact, the schematic below is another option: it is also very simple, no programming and it doesn't require more components than shown, so it satisfies the condition least components the best.
Maybe its dimensions are a bit larger due to the capacitor.

EDIT
Timing requirements have been changed by OP, which makes this solution better/smaller:
For the capacitor now a way smaller capacitor like a SMD 0603 4.7uF 10V can be used, costing hardly nothing.

For a low power LED, a low power PMOS can be choosen which may be compatative in privce to the PIC solution. And this solution may also have a smaller total footprint.
For a higher power LED, the PIC solution as well as this solution require a power mosfet, which makes this solution definitely cheaper.

end of edit

The PMOS and R2 depend on your choice for D1. Also R1 and C1 values can be tweaked a bit to get closer to the 5 minutes when desired (current values should give about 335 seconds).

schematic

simulate this circuit – Schematic created using CircuitLab

Huisman
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    The problem with this circuit is that the LED takes a long time to turn OFF completely. That may not be a problem - only the original poster can say. – Dwayne Reid Apr 03 '19 at 16:54
  • @DwayneReid Ah, haha, I read 5 minutes instead of 5 seconds... makes the circuit way smaller... thanks for addressing it. Will change it soon. – Huisman Apr 03 '19 at 17:20