Introduction To Modern Planar Transmission Lines. Anand K. Verma

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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. The above wave equations for the time‐harmonic fields are written as,

(4.5.3)

The complex propagation constant γ = α + jβ is defined as follows:

(4.5.4)

      For a lossless medium σ = 0, and the propagation constant is a real quantity:

      (4.5.5)

      In a homogeneous medium, propagation constant β is also expressed as the wavenumber k. In free space, μ_r = ε_r = 1. The velocity of the EM‐wave is equal to the velocity of light (c) in free space:

      (4.5.6)

      where is the propagation constant, i.e. the wavenumber (k₀) in free space. A lossless material medium is electrically characterized by (ε_r, μ_r). However, it is also characterized by the refractive index . In the case of a dielectric medium, it is . The velocity of the EM‐wave propagation in a medium is

      (4.5.7)

      For a lossy medium, the complex propagation constant can be further written as:

      (4.5.8)

      For a lossy dielectric medium, ε_r is defined as a complex quantity:

      (4.5.9)

      It is like the previous discussion on the complex relative permittivity in a lossy dielectric medium, with the following expressions for the loss‐tangent and propagation constant:

(4.5.10)

      In the above equation, the real part of the complex relative permittivity is .

      On separating the real and imaginary parts, the attenuation constant (α) and propagation constant (β) are obtained:

(4.5.11)

The wave equation (4.5.3a) and (4.5.3b) for the () fields in a lossy and lossless (α = 0) media are rewritten below:

(4.5.12)

      The propagation constant β is also expressed as the wavenumber k of the wavevector . Sometimes in place of the complex propagation constant γ, the complex wavevector k is used as a complex propagation constant, i.e. k^* = β − jα. On using the complex k, the field is written as E₀ e^−jkx = E₀ e^{−j(β − jα)x} = (E₀e^−αx) e^−jβx.

      4.5.2 1D Wave Equation

For the wave propagating only in the x‐direction, equations (4.5.12a) and (4.5.12b) are reduced to the 1D wave equations:

(4.5.13)

Equation (4.5.13a) has the solution, . The time‐harmonic wave propagating in the x‐direction is

      (4.5.14)

The field equations in the time‐domain are also written as follows:

      (4.5.15)

In the case of a lossy medium, equation (4.5.11) shows that both α and β depend on the loss‐tangent of a medium. For a lossless medium, tan δ = 0, leading to α = 0, and . In the case of low conductivity, i.e. the low‐loss medium, the approximation tan δ ≪ 1, or (σ/ωε₀ε_r) ≪ 1, can be used. In such a medium σ ≪ ωε₀ε_r, the contribution of the conduction current is small as compared to that of the displacement current. Such a medium is a dielectric medium with a small loss. On approximating, the following expression is obtained from equation

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