In researching how to design a planar rectangular PCB coil, I've come across numerous papers describing many topologies, except for a rectangular one. Square, hexagonal, circular are common ones, but a description of a rectangular does not seem to exist.
For a square one, there is a relationship between the outer "diameter" and the inner "diameter" at the end of the spiral. But, I think this assumes the lenght of each side of the outer loop is the same, which is not the case for a rectangular coil. I'm wondering why the rectangular geometry is not covered in any of these papers. Is it an obvious extension of the square one (but apparently not obvious to me), or is there something that makes it a much more complex analytical problem than the square one?
Edited- per request, references include in question. Also added comment.
Also, wanted to add that after giving this some thought, it occurred to me that maybe I can just approximate the inductance of the rectangular coil by scaling the square-coil's inductance by the ratio of the areas of the coils. For example, if the square-coil's area is 4 times the area of the rectangular one, could I just divide the inductance of the square coil by 4 to approximate the inductance of the rectangular one?.