On the Closed-Form Expression of Carson’s Integral

Abstract

Carson’s integral is used in power systems analysis for evaluating the earth-return impedance of overhead conductors above homogeneous earth. Contrary to the widespread belief that it does not evaluate to a closed-form expression, a scrutiny of Carson’s historical (1926) paper reveals that such an expression exists and was actually presented therein. Instead of using the original expression that involves the evaluation of just one Struve and one Bessel function, modern approaches wander between simplified but crude approximations of the integral and numerical evaluations by adaptive quadrature methods. In this paper we present the facts and show that neither of these approaches are necessary, since we can readily and rapidly evaluate the integral (to computer machine accuracy) by using the closed-form expression.