Historical and Technical Overview of Electromagnetic Fields in Stratified Media

Method of Calculating the Inductance Value of MEMS Suspended Inductors with Silicon Substrates

“…Only the magnetic flux density along the direction that is perpendicular to the plane of the substrate (along the z axis in Figure 2) expressed as $B_{0z}$, contributes to the magnetic flux. In the cylindrical coordinates, $B_{0z}$ due to a horizontal electric dipole at ($0, 0,d$) can be expressed as [18]:$\begin{array}{ll}{B}_{0z}\left(\rho ,\phi ,z\right)=& \frac{i{\mu }_{0}Idl}{4\pi }\sin \phi \{-\left(\frac{\rho }{r_1}\right)\left(\frac{k_0}{r_1}+\frac{i}{r_1{}^2}\right)e^{i{k}_0{r}_1}+\left(\frac{\rho }{r_2}\right)\left(\frac{k_0}{r_2}+\frac{i}{r_2{}^2}\right)e^{i{k}_0{r}_2}\\ & +2\mathsf{\pi }{\displaystyle {\displaystyle \sum }_j}\frac{{\lambda }_{jB}{{}^{*}}^2\tan {\gamma }_{1B}{}^{*}l}{p^{\prime }\left({\lambda }_{jB}{}^{*}\right)}·e^{i{\gamma }_{0B}{}^{*}\left(z+d\right)}·H_1{}^{\left(1\right)}\left({\lambda }_{jB}{}^{*}\rho \right)-2k_0{}^2\sqrt{\frac{1}{\mathsf{\pi }{k}_0\rho }}·e^{i{k}_0{r}_2}\\ & ·[\sqrt{\frac{\mathsf{\pi }}{k_0\rho }}-\frac{\mathsf{\pi }}{\sqrt{2}}\end{array}$…”

“…They presented approximate formulas for the electromagnetic fields of vertical electric dipoles and horizontal electric dipoles, in the presence of a three-layered region [13,14,15]. Then several studies improved King’s theory by considering the trapped surface wave that was previously ignored [16,17,18,19]. …”

Impulsive Radiation From a Horizontal Electric Dipole Above an Imperfectly Conducting Surface

“…Studies of perfect dielectric media can be found in e.g. [22], [23], and transient problems are found in e.g. [24], [25], [26].…”