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I've been reading that it is possible to connect a LED to the motherboards CHA fan controller. The LED I had in mind was this

Online people have been saying you would need a resistor as the voltage may burn out the LED otherwise. What I'm not understanding is how I determine the necessary resistance. The LEDs website says it runs at 9~14.5 VDC, it also says the Current Draw is 50mA

This answer says $$Resistance = \frac{Source\,Voltage - LED\,Forward\,Voltage\,Drop}{Desired\,Current}$$

But what is my Forward Voltage Drop? Would my desired current be 50mA and the source voltage 12V.

Also how would I go about connecting the two wires from the LED to the pins on the board. Would something like this be appropriate.

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Forward drops of red LEDs are about 1.7-2.5V, with the high output ones being towards the higher end, but there's some information you're lacking here. The page says that there are 3 LEDs on the assembly, but doesn't indicate whether they're connected in series or parallel. A 50mA draw for a high brightness LED would be reasonable to indicate that they are in series, which would be logical for a supply of sufficiently high voltage to allow that. Fortunately, these assemblies have a series resistor built in (or at least the ones I have used have), to allow them to run on the stated voltage range, so you don't need to add any external resistor.

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The LED you refer to in your link is really an LED assembly, not a single LED. From the description, it is designed to be operated from 9 - 14.5 volts, so will include appropriate current limiting. There is no need to add a series resistor with it.

Individual "bare" LEDs do require current-limiting resistors. Their forward voltages will vary from ~ 1.7 - 3.5 volts, depending on colour.

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But what is my Forward Voltage Drop? Would my desired current be 50mA and the source voltage 12V.

There is an old recipe for cooking rabbit which starts, "First, catch a rabbit." In your case, the first thing you need to do is decide which LED you are going to use. Part of this process is to find a data sheet for that LED. Second, you need to decide what current you want to drive it with. The fan drive circuit will provide at least 50 mA (probably more like 1 A), but a great many LEDs will be destroyed by 50 mA. That is why you need a data sheet.

In the case of the LED you've linked, read ALL of the web page. The LED housing contains 3 LEDs (almost certainly in series) and a limiting resistor, so you don't need to worry about that part. You can just hook it up to 12 volts and you're good.

Also how would I go about connecting the two wires from the LED to the pins on the >board. Would something like this be appropriate.

There is no way to know. You have not shown the pins on the board. The connector is intended for .025 inch square pins spaced 0.1 inch apart. If that describes your drive pins, you're in good shape. If not, not so much. You need to measure your drive pins before you can tell what they require, and the measurement must be fairly precise - a ruler probably won't work. However, at 75 cents apiece, you could consider just buying one or two and trying it, since 0.1 spacing is pretty standard. If it doesn't work, just use it as the foundation for your junk box.

With this said, please look closely at the CHA FAN connector. If it has more than 2 pins, your idea may not work. You need to look up the different varieties of fan which you might find.

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