Analytical calculation of the point-to-point partial inductance of a perfect ground plane

Abstract

The point-to-point partial self inductance of an infinite and perfectly conducting ground plane has been analytically calculated in closed form. The point-to-point mutual coupling between ground plane and a trace parallel to the ground plane has been expressed in terms of an integral, which should be solved numerically. The calculations are based on a new scalar potential directly related to the concept of partial inductances. Formulas for the conversion of partial inductances between the Lorenz's and Coulomb's gauges are also obtained. The definition of partial inductance in terms of the vector potential (e.g. in [1]) implicitly introduces equivalent circuits, where the voltage represents a difference of a scalar potential defined by the electric field and by the magnetic vector potential at the same time. Therefore, it should not surprise the possibility of having a partial inductance also on perfect ground planes. A part from the definition of the magnetic vector potential, the definition of ground plane partial inductance depends on the application, in particular on the considered current distribution on the ground plane. For example, in [2] the ground plane inductance is defined in terms of the induced current on the ground plane by a micro-strip conductor. In the present work, we will consider the current injected in one point and extracted from a second point on the ground plane, similarly to [3]. This type of partial inductance appears when there are connections to a ground plane, such as cables or micro-strip terminations. In a less proper way, it can be used for approximately representing the inductance related to the displacement current, when a micro-strip is decomposed in short segments carrying constant current. The numerical calculation of the point-to-point ground partial inductance is already considered in software of widespread use, such as FASTHENRY [4]. However, the segmentation of the ground plane can become a problem for configurations with high density of conductors [5]. The calculation time increases for larger ground planes, also because the image theory cannot be directly applied to the calculation of partial inductances.