Sheet resistivity measurements on rectangular surfaces–general solution for four point probe conversion factors

Voltage-current ratios, measured using probes on the surface of a homogeneous conducting sheet, are converted to resistance per square by “conversion factors”. For rectangles, a closed form solution for these factors is obtained by using the complex Jacobian sine-amplitude function
$x + iy\ =\ {\rm sn}[(u+iv),k]$
as a transformation. After transformation, an insulated edge rectangle becomes a semi-infinite sheet. Two conjugate current point images establish the boundary condition. A double-sided rectangle becomes an infinite sheet directly, needing no images. New tables have been prepared for a pattern of probe center locations uniformly distributed over the surface. The probe chosen for these tables is a common arrangement having four equally spaced points on a line parallel to the longer edge of the rectangle.