I've played with fairy lights like this for several years, using them mostly for small displays. What I wanted to do was to power them with a "wall wart" controlled by my garage light timer rather than batteries. The displays would come on at dusk and go off about midnight.There are a number of fairy light LED strings similar to this item. Many just use two CR2032, 3 V coin cells in series (to get 6V). At first, years ago, I simply removed the coin cells and connected in a 6V wall wart. But, the fairy LEDs burned out quickly. After some head scratching and online research, I learned that coin cells have a fairly high internal resistance, maybe up to 20 ohms. Power supplies, in general, have very low internal resistance so that the voltage delivered to the "load" doesn't drop much.The coin cell's internal resistance was, in effect, limiting the current actually supplied to the fairy LEDS and preventing burn out. So, I connected a 5V wall wart (not 6V, because that's what I had) in series with a 39 ohm resistor (a standard value that I had) and then to the fairy LED string. No burn out and the LED light emitted seemed fine with the 5V wall wart.I didn't do any other testing or analysis. 39 ohms is about 40 ohms which is 2 times 20 ohms. I suspect that a lower resistor value would work as well. If you want to use a 12V wart or other voltage, a higher resistance value would probably be needed.
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