Non-invasive measurement of the distribution of current with a set of sinusoidally-wound toroidal coils

An analytical solution is presented for the voltage which a time-dependent current induces on a toroidal coil in which the number of turns per unit length is proportional to the sine or cosine of the product of an integer and the azimuthal angle. It is shown that when a wire carrying a time-dependent current passes through the aperture of such a coil, the induced voltage is related to the current by functions of the azimuthal and radial coordinate of the wire which are unique for coils wound with each value of the integer. Thus, the voltages measured on a set of coils having different values for the integer can serve as a basis to determine the location of the wire and the current. The azimuthal functions are orthogonal but the radial functions are not. Thus, this procedure is limited to determining the locations and currents for several wires, or forming a low-resolution picture of a continuous distribution of current that is within the aperture.