A Concise Analytical Formula of Mutual Inductance for Circular Rogowski Coil Revealing Spatial Distribution Characteristics of Mutual Inductance Deviation

Mutual inductance between primary wire and circular Rogowski coil varies with the position of primary conductor, the deviation of mutual inductance referred to center of circle causes the measurement error. Usually, the mutual inductance of Rogowski coil can be calculated by the summation of each turn of coil, however the spatial distribution characteristics of mutual inductance deviation has not been revealed. A concise analytical formula of mutual inductance is derived based on the summation form and verified by experiment in this paper, which leads to three characteristics of mutual inductance deviation. The first is the mapping relation of mutual inductance deviation between inside and outside Rogowski coil which established connection between positional accuracy and anti-interference ability. The second is the zero deviation line of mutual inductance deviation given by simple equation, which describes the position of primary wire where the measurement error vanishes. The third is approximate function on turns number, inner and outer radius of Rogowski coil and measurement error, which is easy to calculate and simplifies the design progress of Rogowski coil. Furthermore the formula and characteristics of mutual inductance can be generalized from uniform Rogowski coil and perpendicular primary wire to discontinuous PCB RC, primary conductor with cross section, and inclined primary wire. The mapping relation, zero deviation line and improved formula of mutual inductance deviation are discussed.