Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 173

Maxwell’s equations in the unbounded medium could be also written in the vector algebraic form. The del operator can be replaced as follows: . Using equation (4.4.9), for the charge‐free lossless medium ρ = σ = 0, two sets of Maxwell equations, for the isotropic and anisotropic media, are written in the following algebraic forms:

Set #I for the isotropic medium:

Set #II for the anisotropic medium:

(4.5.31)

Equations (4.5.31a) and (4.5.31b) show that for the positive values of μ and ε, the triplet follows the right‐handed orthogonal coordinate system. Equations (4.5.31c) and (4.5.31d) show that in an isotropic medium, the field vectors and are orthogonal to the wavevector . Equations (4.5.31c) and (4.5.31d) directly follow from the first two Maxwell equations by taking their dot product with the wavevector . Equation (4.5.31a) further shows that is normal to both vectors, and equation (4.5.31b) shows that is normal to both vectors. In brief, the vectors are orthogonal to each other, and there is no field component along the wavevector _, i.e. the wave is a transverse electromagnetic (TEM) type. Also, in an isotropic medium, is parallel to vector and is parallel to vector . This statement does not hold for the anisotropic medium. Maxwell equations (4.5.31e)–(4.5.31h) apply to an anisotropic medium. In an anisotropic medium, equations (4.5.31e) and (4.5.31f) show that is normal to vectors and is normal to vectors . However, is not parallel to . Also, is not parallel to . It is discussed in subsection (4.2.3).

Equations (4.5.31a) and (4.5.31b) are solved to get the vector algebraic form of wave equation as follows:

(4.5.32)

Likewise, the wave equation for could be written that is useful for the EM‐wave propagation in an anisotropic medium. Equation (4.5.32b) is an eigenvalue equation, and the nontrivial solution for E ≠ 0 provides the eigenvalue of k belonging to the propagating waves in the unbounded isotropic medium. The medium supports two numbers of linearly y‐polarized waves known as normal modes propagating in ±x‐directions with same phase velocity (v_p) given below:

(4.5.33)

In the above equation, is the wavenumber in free space. Using the intrinsic impedance with equation (4.5.31a), the magnetic field vector and Poynting vector are obtained below:

(4.5.34)

In the case of the propagation of waves in an isotropic medium, the wavevector and Poynting vector both are in the same direction. It provides the phase and group velocities in the same direction.