The mutual inductance of a linear current source and a non-coplanar, parallel, rectangular loop

An integral expression for the low-frequency mutual inductance between a short linear track (exhibiting a rectangular conductor cross-section) and a noncoplanar, parallel, rectangular loop (exhibiting negligible conductor cross-section) is derived in terms of printed-circuit board (PCB) parameters. These parameters are: the wire-to-loop planar separation, the loop dimensions, the wire-to-loop lateral and longitudinal offsets, and the linear track cross-section. This expression yields numerical results which compare favorably to mutual inductance values calculated by other researchers using finite-element methods. Families of curves, computed from the integral expression, are presented as a PCB design tool.<>