Hertzian Dipole Radiation over Isotropic Magnetodielectric Substrates

: This report investigates dipole antennas printed on grounded magnetodielectric substrates. The interest in magnetodielectric substrates stems from the expectation that new magnetic materials may revolutionize the engineering of thin planar antennas. Since these materials often require complicated constitutive equations to describe their properties rigorously, the solution of intricate boundary value problems has become important to understanding the associated radiating structures. In this report, the magnetodielectric is modeled as a simple material with scalar constitutive parameters, an approximation that is likely not satisfactory. However, with such an approximation, our problem reduces to a very well-known boundary value problem in applied electromagnetics work. Despite the extensive amount of existing literature, no completely self-contained treatment of the mathematical boundary value problem itself was found. Consequently, the missing mathematical details are provided in this report with the intent to be useful to future research of magnetic materials described by more realistic constitutive equations. Furthermore, some novel formulas are given as they relate to the square roots appearing in the theory of complex integration. The directivity, radiation resistance, and efficiency are derived, and some results are plotted, from which we observe complicated dependencies on material properties and substrate thickness.