Economic designs for single-layer toroidal inductors

Maxwell´s problem of winding the greatest possible inductance with a given length of wire is examined for single-layer toroids. Optimum designs are obtained when the turns are square or circular or the Shafranov D-shape. In each case the optimum number of turns is proportional to the square root of the wire length, and the best obtainable inductance increases approximately as the three-halves power of the wire length. A simple graphical procedure achieves the most economical design for a required inductance and current rating.