Inductor Coil Design

Without knowing the details of your square coil, I would say that crudely-speaking, if the enclosed area is increased, the inductance will increase slightly. If your smoothed corner cuts into where the otherwise sharp corner would have been, without a corresponding bulge near the corner, the inductance will decrease slightly. One other thing, many formulas show that the overall perimeter length is a sensitive parameter.

About 80 to 100 years ago some folks took great care to make inductance standards fabricated from square coils. I have seen pictures of these, but I do not know how they dealt with the sharp-corner issue.

The best inductance standards were made from round coils. Here's a short book by J.G. Coffin, published by the Bureau of Standards in 1906, about such a standard coil. It makes interesting reading. As you can see from the math in the book, it's possible to exactly calculate the inductance of your approximately-square coil.

Here's a 1999 paper by Thompson, dealing with square coils. He quotes formulas from Frederick W. Grover's famous book (Dover 1946 edition), pictured at right. This book is the gold standard for calculating inductance.

Thompson also gives three references for more detailed calculations for square planar coils. These are popular for use on printed-circuit boards. If the inner distance D_i = 0 (completely-filled coil) the simplified formula is

L ~= 8.5 x 10^-10 D N^5/3

That formula is by H. Dill, from a Feb 17, 1964 Electronic Design article (whew, I suppose D is in meters and L is in Henrys). Thompson says the 5/3 exponent is thought to be due to end effects. Hah! The other formulas in the literature are not so simple.

A 1974 paper by H.M. Greenhouse shows that five different formulas for a single-turn square coil, by various experts and authorities, gives answers varying from 1.95 (Grover, expanded to include mutual-inductance effects) to 2.4 (Terman) to 3.89 (Bryan). One imagines the best course is to build one and measure it! When this is done Greenhouse says Grover gets the nod.

Here's IN3OTD's interesting web site, comparing formulas, and here's a useful paper, available from MIT. :-)