Anand K. Verma - Introduction To Modern Planar Transmission Lines

The relative permittivity matrix for gyroelectric medium, under the applied magnetic field in the z‐direction, is expressed as follows:

(4.22)

In the above expressions, ω pand ω care the plasma frequency and cyclotron frequency . The cyclotron frequency is also called the gyrofrequency. The jκ is a cross‐coupling factor responsible for the gyration property of the relative permittivity of a medium. Other parameters are N: electron density, e: electron charge, m: electron mass, and B 0: DC magnetic field. In the case of the extremely high magnetic field, ω c→ ∞ and ε r= 1, κ = 0. Also, in the absence of magnetic field, B 0= 0, κ = 0 ,and the gyrotropic medium is reduced to a uniaxial dielectric medium. In some cases, there is no plasma medium in the z‐direction and ε r, zz= 1.

The relative permeability matrix of a gyromagnetic ferrite medium is given below:

(4.2.12)

In the above equations, H 0is the biasing magnetic field in the z‐direction, M sis the saturation magnetization, and the parameter γ is the gyromagnetic ratio . Again the cross‐coupling factor jκ is responsible for the gyration property of the relative permeability of a medium. The permeability function shows a singularity at the frequency ω = ω 0. It is suppressed in the presence of losses. In the case of unbiased and demagnetized ferrite, H 0= 0 and M s= 0 leading to k = 0 and μ r= 1. The ferrite, in this case, acts as a dielectric medium with permeability μ = μ 0.

The magneto‐dielectric (MD) materials have both electric and magnetic properties. These materials are characterized by the simultaneous presence of permittivity and permeability tensors. These tensors for uniaxial MD materials are expressed as follows:

(4.2.13)

The above‐given expressions for the uncoupled independent relative permittivity and permeability tensors show that there is no coupling between the electric and magnetic fields. Ferrites have such property. However, ferrites are very lossy at microwave frequency. The low‐loss MD materials are synthesized by mixing ferrites/hexaferrite and their composites with a polymer as a host medium [J.3]. The periodic structures are embedded in the host medium to engineer MD materials for antenna applications in VHF and UHF bands [J.4]. The metamaterial composites are also MD materials with simultaneously negative permittivity and permeability in a certain frequency band.

The magnetoelectric materials are general bianisotropic electromagnetic materials with cross‐coupling of electric and magnetic fields . These materials have anisotropy for both the permittivity and permeability with additional cross‐coupling of the electric and magnetic field. In such materials, the electric flux density vector and also magnetic flux density depends on both applied and . It shows that in the bianisotropic materials, the fields not only generate an electric polarization but also create the magnetic polarization, i.e. magnetization. Similarly, fields applied to such materials create both magnetization and electric polarization. The constitutive relation relating four flux and field vectors for a linear magneto‐electric medium is expressed as follows [B.21–B.23]:

(4.2.14)

The above given four material parameter tensors , , , and describe the bianisotropic magnetoelectric materials. The tensors and are usual permittivity and permeability tensors, whereas and are magneto‐electric coupling tensors . In general, 36 complex material parameters are required to characterize bianisotropic materials. However, the material parameter matrix could be diagonalized. So, for the uniaxial bianisotropic , i.e. the magneto‐electric , materials, constitutive relations are written as follows [B.15, B.21]:

(4.2.15)

In the absence of cross‐coupling, i.e. for , the bianisotropic medium is reduced to an MD medium. Figure (4.5a)shows groups of general bianisotropic medium ‐ isotropic , bi‐isotropic , biaxial anisotropic , and bianisotropic media [B.25]. Two cases have already been discussed.

The bi‐isotropic materials are isotropic materials also showing cross‐coupling of electric and magnetic fields. However, Fig. (4.5b)shows that the general bi‐isotropic medium has special forms – isotropic, Pasture, Tellegen , and bi‐isotropic . For general bi‐isotropic medium, the medium tensors are reduced to scalars, and the constitutive relations given by equation (4.2.14)are reduced to the following simpler form [B.23]:

(4.2.16)