Advanced microwave effective medium theory for two-component nonmagnetic metamaterials: fundamentals and antenna substrate application

Abstract

A modification of effective medium theory for two-component nonmagnetic metal–dielectric metamaterials is developed for use in the microwave frequency range. The metamaterial is represented as an unbounded isotropic dielectric host material with periodically embedded nonmagnetic metallic inclusions of cylindrical or spherical shape. The effective electromagnetic response of the metamaterial is represented by the tensor of the effective relative permittivity and tensor of the effective relative permeability. The losses of the metamaterial are also evaluated in this study. A physical interpretation for the nature of the effective properties of such metamaterials is given. Analytical models of the proposed effective medium theory are benchmarked against numerical simulations using commercial electromagnetic software. Two compact microwave rectangular dual-band patch antennas on such metamaterial substrates are designed in this study.

Appendix

If the external magnetic field has vector $\mathbf {H}_{0}$ parallel to the axes of the cylinders, the magnetic component inside the cylinders is given by [43]

$$\begin{aligned} H=H_0+\left( 1-F_{\mathrm{cyl}}\right) j, \end{aligned}$$

(29)

where j is the current flowing per unit length.

The net electromotive force (EMF) around the contour of the cylinder of a unit cell

$$\begin{aligned} \hbox {EMF}=-\mu _\mathrm{m}\mu _0\frac{\partial }{\partial t} \int _s H\,\mathrm {d}s-\oint _L j\delta \,\mathrm {d}l \end{aligned}$$

(30)

is balanced ($\hbox {EMF}=0$), where s is the cylinder cross-sectional area, which is equal to some S, and L is the contour of the cylinder cross-section, which is equal to some l. Then, integrating the right part of (30) gives

$$\begin{aligned} \mathrm{i}\omega \mu _\mathrm{m}\mu _0\left( H_0+\left[ 1-F_{\mathrm{cyl}}\right] j\right) =0. \end{aligned}$$

(31)

Solving Eq. (31) for the current j gives

$$\begin{aligned} j=-\frac{H_0}{1-F_{\mathrm{cyl}}+\mathrm{i}\delta \cdot l/\omega S \mu _\mathrm{m}\mu _0} . \end{aligned}$$

(32)

The average B-component over the entire unit cell is

$$\begin{aligned} B_{\mathrm{ave}}=\mu _\mathrm{m}\mu _0 H_0, \end{aligned}$$

(33)

while the average H-field along a line lying entirely outside the cylinders is given by

$$\begin{aligned} H_{\mathrm{ave}}=H_0-F_{\mathrm{cyl}}j. \end{aligned}$$

(34)

Substituting Eq. (32) and Eq. (6) into the relation between $B_{\mathrm{ave}}$ and $H_{\mathrm{ave}}$ in the form

$$\begin{aligned} B_{\mathrm{ave}}=\mu _{\mathrm{eff}} \mu _0 H_{\mathrm{ave}}, \end{aligned}$$

(35)

after some rather routine manipulations, finally gives

$$\begin{aligned} \mu _{\mathrm{eff}}=\mu _\mathrm{m}\left( 1-\frac{F_{\mathrm{cyl}}}{1+i\delta \cdot l/\mu _\mathrm{m} \mu _0 S\omega }\right) . \end{aligned}$$

(36)