Analytical design of magnetic loop induction systems

The method of orthogonal fields, presented in the companion article, permits the design of a loop system with little spreading out of the resultant magnetic field. This article describes the mathematical basis of the design. Good approximation and ease of handling are obtained by the use of complex function theory and conformal mapping. Two solutions for the problem emerge, one based on the properties of hyperbolic functions, the other on those of elliptic functions. In the first solution, the field decays monotonically to zero outside the primary area of communication. In the second solution, the field exhibits equiripple character in the region where communication is to be prevented. The ultimate design was chosen from the second category, after elaborate computation with the help of an electronic computer.