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Capacity (of a derailer)
The "capacity" of a particular derailer model is the largest range of sprocket sizes it can handle:
For front derailers, when the derailer is mounted high enough to clear the largest chainwheel, there is a certain minimum size that you need for the smallest ring so that the chain won't be dragging over the bottom of the front cage. Different front derailers have different capacities depending on how tall their cages are. It is expressed in a number of teeth, which is the difference between the largest and smallest chainwheel. For instance, a 52/42/30 crank set would call for a front derailer with a minimum 22 tooth (52-30) capacity.
Front derailers are also designed to be used with a certain size for the largest chain ring. The curvature of the outer cage plate is matched to this size. If you use a different-sized big ring, capacity may be reduced. If the big ring is substantially larger than the derailer is designed for, shifting precision will suffer. If the big ring is much smaller than the derailer is designed for, it may shift OK, but you are likely to have to "trim" the front derailer as you shift the rear derailer to the extremes.
A front derailer for triple chainrings also may need to have an inner cage plate adapted to the size difference between the outer and middle chainrings. If this difference is more than a few teeth, the inner cage plate must be deep from top to bottom, so it can push the chain from small chainwheel to the middle one without overshifting it onto the large one. The cage plate also may have a special shape, intended to improve this shift.
For rear derailers, the capacity relates to the amount of chain slack the derailer can take up, and is equal to the front range (22 in the example above) plus the rear range. Thus, if you have a 52/42/30 crank set, and a 12-28 (16 tooth difference) cluster, the total capacity required would theoretically be 38 teeth (22 front difference + 16 rear difference).
Manufacturers specify this fairly conservatively. They must do so, because they have to assume that some of their derailers will be sold to incompetent cyclists, who will abuse their drive trains by using the smallest chainwheel with the smaller rear sprockets.
Competent riders can considerably exceed the official rated capacity, since they will not misuse the granny ring by running it with the smaller rear sprockets, so it doesn't matter if the chain hangs slack in those gears.
Rear derailers are also commonly designed for a particular maximum size of rear sprocket. If you exceed this size by too much, the jockey pulley may rub against the sprocket when using the lowest gear.
Derailers will also commonly work with larger sprockets than manufacturers' ratings suggest. For instance, Shimano's models designated as "road" derailers are generally listed for a "maximum" sprocket of 27 teeth...because 27 teeth is the largest size that bShimano makes in a designated "road" cassette. However, in almost all cases, these derailers, even the short-cage models, will handle rear sprockets as large as 30 teeth in practice. (This somewhat depends on the design of the frame's derailer hanger, so once in a while you will find a particular installation where you can't use a 30, but I've never seen one where a 28 wouldn't work.)
I'll also mention that many folks seem to be a bit mystified and intimidated by derailers. They're actually quite inexpensive and easy to replace, so it is foolish to allow the limitations of a particular derailer to keep you from having the gearing appropriate to your riding style/needs.
