There are two major types or RGB lighting systems dominating the market now. BUT before them was a third.
The earliest lighted fans are called LED Fans. Typically they have only ONE colour of LED mounted in the fan frame, and they have only ONE standard cable from the motor with a 3- or 4-pin female connector that plugs into a mobo fan header. The LED's are simply wired in parallel with the motor so they light up whenever the fan has power. You have NO control over those lights. Their brightness may dim when the fan is running slowly. THIS type might be connected directly to a PSU power output (rather than to a mobo header) and still light up. This type is NOT called an RGB or ARGB lighting system.
The two actual RGB or ARGB lighted fan systems usually have TWO cables from the fan motor. One is for the motor only and ends in a standard female fan connector that goes to a mobo fan header. Most often now that will be one with 4 holes ("4-pin fan"), but some 3-pin fans still come on the market. The second cable is for the LIGHTS only - really, you have two devices, a fan motor and a lighting system, in one unit. This second cable ends in a wider connector with three or four holes in it in a straight line, and it plugs into the correct mobo lighting header.
The plain RGB lighting system (light strips or lighted fan) uses a FOUR-pin connection system that supplies to the lights a common +12 VDC power line and three separate Ground lines - one each for the Red, Green and Blue LED's. Along the strip, all LED's of one colour (say, Red) are connected to the +12 VDC line and one of the Ground lines so they all do the same thing. Same for the Greens, and for the Blues. The mobo header manipulates the three Ground lines to create thousands of colours that can change over time. At any one moment, though, the entire strip is ONE colour.
The more complex Addressable RGB (ADDR RGB, ARGB, Digital RGB) system uses a THREE-pin connector that looks just like the 4-pin one, but with one pin missing (hole blocked off on the female connector). This supplies to the strip common +5 VDC and Ground lines, plus a digital Control Line. The Control Line carries data packets that are each addressed to specific places and contain signals for the display. The strip has those same three colours of LED's. But along the strip, they are grouped into NODES. Each Node contains one LED of each of the three colours, plus a Control chip. That Control chip listens to the Control Line and responds only to a data packet addressed to it, then uses that to control its own set of three LED's. Each Node will be changed every time a new data packet for it arrives. Thus at any one moment, every Node along the strip can have a different colour and brightness, and this system can generate much more complex displays including rainbows - either stationary or moving.
Because the power supply voltage and the mechanism of display control is so different between these two systems you can NOT mix the two on one circuit. So when buying, you must match the type of header available on your mobo to the type of lighting devices (either strips or in fan frames) you buy.
Although power and control of RGB or ARGB lights is easiest to do from mobo headers, some mobos have none, or the wrong type for the lights you got. And some makers actually intend you NOT to use mobo headers. So there are third-party makers who sell RGB or ARGB Controllers that can do the job without any mobo header. HOW they do the control comes in three variations. The simplest are small control boxes with a few buttons (often three) that connect to a PSU output for power and then to all your lighting devices. You use the three buttons to manually select the display characteristics, so you need access to that box. Next are ones that come with a box to go inside your case with all the connections, but also have a hand-held battery-powered remote control for you to use to make your selections. SOME of these also include ways to control the fan motor speeds, too. The third type has an inside-the-case Controller box that also has a cable to plug into a mobo USB2 header. Then you download and use the maker's software utility that uses that USB2 connection to communicate its instructions to the Controller box. So this way your control system is by the keyboard and that utility software. In all of these cases, you do NOT use the software utility supplied by your mobo for lighting control, because control is NOT being done by the mobo header.
Note this source of confusion, too. Mobo makers sell some mobos with NO RGB headers, some with only plain RGB, some with only ARGB, and some with both types of headers. Each maker has their own software utility they supply for use with their board to control those headers and the the lights. They have their own proprietary names for this, like MSI Mystic Light, ASUS Aura Sync, Gigabyte RGB Fusion, ASRock RGB Sync, etc., and you will see lighting devices sold as "compatible with ..." those names. BUT those tools are designed to work with BOTH types of lighting systems on their maker's mobos. So the NAME of the software utility does NOT tell you the TYPE of lights that match your mobo. You must match the HARDWARE type of header your mobo has.
And one more source of confusion to alert. There are "3-pin" and "4-pin" fans now, each using a different means of controlling their motor speeds. The SAME names - "3-pin" and "4-pin" - are used to describe the types of LIGHTING systems, even though there is NO link between them. For example, there are lots of fans with 4-pin fan motors (the newer PWM style) and 3-pin ARGB lights in their frames.
So, with that background, to your specific questions. Several of them are rooted in the early LED Fan design. As I said, these should NOT be called RGB or ARGB fans. These are fans with lights in the frame but only ONE cable that plugs into a mobo fan header (or maybe into a PSU 4-pin Molex power output). Their light display cannot be controlled at all. Their motor speed cannot be controlled IF they are connected to the PSU directly, because that source has no way to vary the electrical signals to the fan motor.
Plain RGB or ARGB fans almost always have two cables from them - one for the motor only, and the other for the lights only - and these plug into different mobo headers. Note that there are some systems that are designed NOT to use mobo resources, and may NOT have two cables from each fan. Instead they have a single cable, often with more than 4 wires in it, ending in a non-standard connector that only fits into the custom Controller box supplied with the whole system. Generally these cannot be connected to your mobo. However, some provide the option to connect a cable from their Controller box to a mobo RGB or ARGB header, and an option to let the mobo take control, rather than using the system in the supplied Controller box.
As I said above, the older LED Fans will light up whenever the fan gets power, but there is no control. When you have real plain RGB or ARGB lighting units (either strips or in fan frames), BOTH power for them and control of the displays is provided by the Controller, be it by separate box or from a mobo header. So you cannot get the lights to work without that Controller. In some cases the control system may not be working and you might get a single fixed set of lights, or no lights at all. If you do not connect the LIGHTING cable from such a fan to a Controller or mobo header, generally you will get no lights, although you might get a "frozen" display of one bit of light.
You may be familiar with fan SPLITTERS that allow you to connect two or more fan motors to a single fan header. Similarly, there are 4-pin plain RGB Splitters and 3-pin ARGB Splitters you can use to connect two more more lighting unit cables to a single mobo header or separate Controller. And similar to fan headers, the mobo lighting headers have a limit to how much max current they can supply to many connected lighting devices. The mobo manuals tell you this limit PER HEADER. Unfortunately, suppliers of RGB or ARGB Fans often do NOT tell you the current the LIGHTS ONLY in their fans will consume. If you need help with that little detail, post a query back here. But most often if you need to do this, the header can handle three or four fans' LIGHTS from a single header with a Splitter. For someone with more lighting devices, there also are fan lighting HUBS similar to the concept of fan Hubs - they get power from the PSU rather than from the mobo header, and only take from the header the control signals to relay to the lights.
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