Coupling of Differential and Common Modes of Two-line Circuits in the Multi-conductor Transmission-line Theory Including Radiation

We study the coupling of the differential mode with the common (sum) mode in two-line circuits, where the two lines have different radii including the radiation effect through the retardation terms. We found that the two modes couple each other when the two lines have different radii. This feature is not described in the standard circuit theory using the Heaviside telegraphic equation. We encountered, however, a difficulty of numerical stability when the retardation effect was included in the numerical calculations as the number of mesh points are increased. We then study the fundamental features of the delay differential equations so that we will find suitable methods for various cases of interest with numerical stability.