In addition to Dan's very good post, the main reason that there is an aperture limit on Phase Detect AF systems is not about image brightness. After all the Contrast Detect AF system used during Live View operation will still work at any aperture, as long as there is enough light to form some contrast. Phase Detect AF systems operate by splitting the image, so that light rays from either side of the image (which sides depends on AF spot type, Line/Cross/Double cross and orientation. When the image is out of focus the phase (relative amplitude of the wave signal) is out of alignment on either side. The amount of phase difference actually tells the AF system how far out of focus the image is, and in what direction. This means that the AF system can simply drive the lens x far and know that the image will be in focus. This speeds up the AF a lot. All the AF system has to do is Look Drive and you are ready to take the picture. Sometimes it is possible to configure the system to do a look drive look which adds accuracy but at the cost of a slightly slower system. In order for there to be a useable amount of phase difference for the AF to work with there is a limit on how far apart the two sides of the image are. F numbers are useful here as they scale nicely with focal length and so angle of view. A wider angle of view can use a smaller physical aperture, which is exactly what the f number system gives us. So if you have an aperture of f/2.8 or wider then you can measure the phase difference with more accuracy as it will be a bigger value, as phase difference becomes more the further apart physically the signals are. The actual positioning of the AF sensor in relation to the image also matters. If you position the AF sensor in the center of the image it will again receive the maximum phase difference. This is why the center spot is always the most accurate. So for PDAF systems you have two limitations, they need light that is sufficient to actually form a signal, but they also need a wide enough aperture that the two sides are far enough apart so that a phase difference is actually measurable.
Now like any system the limits of accuracy and even the ability to work are not really that hard and fast, the performance drops off slowly. The manufacturers though place limits on the AF system that means that the system will work reliably for pretty much all of the time that there is a possibility of using the camera to take photos. This seems to be f/5.6 for most sensors, although it is possible to get the center spot to work at f/8 in some cases. This is why some non reporting teleconverters, or pin taping on reporting converters will sometimes works. There may just be enough measurable phase difference left for the system to actually operate. usually though this is pretty hit and miss. It also explains why for Canon cameras other brand lenses will still AF at f/6.3, as long as the lens actually reports the right value for max aperture to the camera.
So that's Phase Detect AF, so why don't we just use contrast detect more if there are no aperture limits, it seems like it would be a much better solution. The problem is speed. Because PDAF usually knows how far and in what direction it is out of focus it only needs to look and drive to obtain focus. This is really fast. Contrast Detect on the other hand looks a line of pixels, although you could look at a 2 dimensional group of pixels, and looks for a difference in brightness across them, it also looks at how quickly that change takes place. So what the system does is have a look and then as it doesn't know which way you will have to move it takes a guess and moves the lens by a certain amount. It then has to look again. If the contrast has improved, so that the change in tone is across less pixels that is great and it will then move in the same direction again. If the contrast has got worse it will then start moving in the opposite direction. These initial movements will be quite large, and eventually the signal will stop gaining contrast, and get worse again. At this point the system will then move back in the opposite direction, but by a smaller jump. It will keep doing this moving back wards and forwards in smaller and smaller jumps until it has reached the point where it can't jump any smaller and has reached the maximum level of contrast. Even if they system is in perfect focus at the start of this process it will always have to move the lens to check on the level of contrast. Remember that PDAF knows that ti is in focus just by looking the once. This makes contrast detect a very slow process, that is really very unsuited to moving subjects. The big advantage that CDAF has over PSAF is accuracy on stationary targets, as it uses the actual imaging sensor to do the looking.
Alan