Closed-Form Expressions for the Total Power Radiated by an Electrically Long Multiconductor Line

Two analytical solutions based on transmission-line theory for the total power radiated by a multiconductor line above a ground plane are proposed. The line is not assumed to be electrically short or close to the ground plane, thus making the proposed model suitable for assessing the emission/immunity of actual transmission lines employed in industrial contexts such as in the automotive domain, railway lines, and power-distribution lines. The model allows an imperfect ground plane to be considered through the complex-image approximation together with propagation losses. Numerical and experimental results are provided as a validation, while an empirical rule to assess the accuracy of the results is proposed. The two expressions aim at allowing fast parametric analysis of radiation during the design phase of the electrical and geometrical configuration of an unshielded multiconductor transmission line.