Anand K. Verma - Introduction To Modern Planar Transmission Lines

(4.5.6)

where is the propagation constant, i.e. the wavenumber (k 0) 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 ris 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 E 0e −jkx= E 0e −j(β − jα)x= (E 0e −αx) e −jβx.

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/ωε 0ε r) ≪ 1, can be used. In such a medium σ ≪ ωε 0ε 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 (4.5.11a):

(4.5.16)

The above equation computes the dielectric loss of a low‐loss dielectric medium. The approximation is used to get an approximate value of β for such medium from equation (4.5.11b):

(4.5.17)

For a low‐loss dielectric medium the dielectric loss, due to tan δ , increases linearly with frequency ω. However, the propagation constant β is dispersionless, giving the frequency‐independent phase velocity. The above approximation can also be carried out in a little different way:

(4.5.18)

In the above equation, is the characteristic ( intrinsic ) impedance of free space. A low‐loss medium is a mildly dispersive medium, with the frequency‐dependent phase velocity: