Introduction To Modern Planar Transmission Lines. Anand K. Verma

product of , known as the Poynting vector:

(4.4.20)

      The divergence of the above equation provides the power entering, or leaving, a location in the space:

(4.4.21)

      The energy contained per unit volume, i.e. the energy density, in a dispersive and a nondispersive medium, in the form of the electric and magnetic energy, is given by the following expressions:

(4.4.22)

A physical medium is dispersive. For a dispersive medium, the modified equation (4.4.22b) is valid [B.2, B.8, B.11–B.15]. The energy density W is a positive quantity, i.e. the following relations must be satisfied:

      (4.4.23)

      The medium having finite conductivity σ dissipates the EM‐energy in the form of heat given by the Joule's law:

      (4.4.24)

      The total power carried in a medium, in the form of the EM‐wave, is

      (4.4.25)

      The integration is carried over the cross‐section of the medium carrying the EM‐power. The total energy stored in volume V is

      (4.4.26)

      The total power dissipated in volume V of the medium is

      (4.4.27)

      Thus, power (P_ext) supplied by the external source is balanced by the following equation:

(4.4.28)

Using Maxwell equations, the power balance equation (4.4.28) is evaluated in terms of the field quantities and external sources. The resulting power balance equation identifies the above‐mentioned expressions for the power in a wave, energy dissipated in the medium, energy stored in the medium, and also the energy supplied by the external electric and magnetic currents.

      The dot product of equation (4.4.1a) is taken with , and the dot product of equation (4.4.1b) with , and subtract one from another to get the following expression:

      (4.4.29)

In the above equation, is used. In the case of a time‐invariant medium, the relative permittivity and relative permeability of a medium are constants. By using an equation (4.4.21), the above equation is rewritten as follows:

      (4.4.30)

On taking volume integral of the above equation and further using Gauss divergence theorem (4.4.13), the following expression is obtained:

      (4.4.31)

      The power supplied by the external sources is positive. So, the total power in a medium is negative. Out of the total power supplied to a medium, P_wave is power carried away by the EM‐wave, P_dis is power loss in the medium due to the finite conductivity of the medium, and is the oscillating electric and magnetic energy in the medium.

4.5 EM‐waves in Unbounded Isotropic Medium

      Maxwell’s coupled vector differential equations (4.4.1a) and (4.4.1b) are solved in an external source‐free medium with to obtain separate wave equations for the electric and magnetic fields. It is like getting the voltage and current wave equations on a transmission line. Maxwell equations are further presented in the vector algebraic form. The present section is concerned with the uniform 1D wave propagation in an unbounded isotropic dielectric medium, also in a conducting medium. The EM‐wave propagation in the anisotropic medium is discussed in section (4.7).

      4.5.1 EM‐wave Equation

      Maxwell’s coupled equations (4.4.1), in the external source‐free medium , are solved, using the rule of vector algebra, for the electric field intensity by substituting from equation (4.4.1b) to equation (4.4.1a):

      (4.5.1)

      In the above equation, the identity is used. The above wave equation for the electric field is valid in an isotropic, homogeneous, and lossy medium. In a homogeneous medium, the (μ_r, ε_r) are not a function of position. Further, the medium is taken as charge‐free, i.e. ρ = 0, . Likewise, the following wave equation is obtained for the magnetic field:

      (4.5.2)

      To get the time‐harmonic field, i.e. , the time differential variable is replaced as follows: ∂/∂t → jω and ∂²/∂t² → − ω².