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I have several Christmas lights with different AC adapters.

First case

In the label of one of them, I read:

Primary: 220-240 V ~ 50/60 Hz
Secondary: 31 V DC 3.6 W

In the label of the cable connecting this AC adapter to the LED array:

Lamp: 3.2 V
0.064 W
80 LED

I can not understand if this is referred to the single LEDs, but the total amount of LED seems to be 80.

How are they connected, then?

In the case of a series connection, each of them would have a 0.3875 V voltage drop, which maybe is too small for a LED. If they were parallel, 31 V seems too much for each of them, and the total available current 3.6 W / 31 V = 0.1161 would be 0.00145 per LED. So, neither of these solutions seems correct.

Second case

In another LED string, I have a 4.5 V battery power supply, with no labels and therefore no indication about the current and the number of LEDs. Do they have a parallel connection?

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    \$\begingroup\$ Current to light up each LED is around 20mA and the voltage on each is approx 3v. The battery one is probably less current per LED but the voltage will be around 3V also (this is a fundamental physical characteristic of LEDs... There are no such thing as 0.3V LEDs) Your LEDs are likely wired in a series/parallel configuration. I.e.Maybe 4 parallel strings of 20 series wired LEDs. Look closer at the wires \$\endgroup\$ – Kyle B Jan 3 at 19:38
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    \$\begingroup\$ Add: The battery one is gonna all be wired all parallel... There isn't enough voltage (4.5v) to light up a series of even just 2 LEDs (6v). There probably aren't even series resistors added to limit individual LED current. They are disposable so proper design isn't critical. \$\endgroup\$ – Kyle B Jan 3 at 20:44
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The LEDs must be connected in series strings of LEDs. A string of 8 LEDs would work, with a single small resistor to linit the current. Connect ten strings in parallel, and you have 80 LEDs.

Alternatively, connect eight strings of ten LEDs, and rely on current limiting in the power supply. This will work if the LEDs in the strings are all matched.

For the 4.5V version, connect all the LEDs in parallel, with a single current limiting resistor. Again, all the LEDs need to be reasonably well matched.

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