Anand K. Verma - Introduction To Modern Planar Transmission Lines

Maxwell's equation ( 4.4.11a,b)of chapter 4could be written for a lossless (σ = 0) DNG medium by using (−|ε r|) and (−|μ r|) in place of their usual positive values:

(5.5.2)

Equations ( 4.5.31a)and ( 4.2.3b)of chapter 4show Maxwell's equations for a DPS medium in terms of the wavevector and the field vectors . They form the wavevector triplet ( ) relations in the right‐hand (RH) coordinate system, shown in Fig (5.8a)for a DPS medium. Normally, the direction of the wavevector determines the direction of the wave propagation, i.e. the direction of the phase velocity v p. However, the Poynting vector ( ) given by equation ( 4.4.20) provides another power‐vector triplet ( ) that determines the true direction of wave propagation. It is the direction of the energy flow from the source to a load. So the corresponding group velocity v g defines the direction of the wave propagation .

Figure (5.8c)shows both the sets of the vector triplets of a DPS medium. The field vectors are rotated in cyclic order so that the power flow is maintained from left to right‐hand side, i.e. from a source located at the origin O, power flows outwardly in a positive direction. For both the phase and group velocities, the DPS medium follows the right‐hand (RH) coordinate system, so a normal DPS medium is called the right‐handed, i.e. the RH‐medium . As the RH‐system holds for both triplets, they are combined into one diagram as shown in Fig (5.8c). For a DPS medium, the directions of the vectors and are identical, i.e. they are parallel vectors giving the relation . Therefore, in a DPS medium, both the phase and group velocities are in the same direction giving . It is shown in the first quadrant of Fig (5.7). The phase of the propagating EM‐wave in the DPS medium lags while traveling in the ‐direction .

In the case of a DNG medium, both permittivity and permeability are negative. Using Maxwell's equations (5.5.2c,d), and replacing ∇ → − jk the wavevector triplet‐ relations for the DNG medium are written as follows:

(5.5.3)

However, in the above expression reversal of the direction of the magnetic field involves the reversal of the direction of the power flow toward the source. Physically, it is not possible, so the above equations are rearranged as follows by associating the negative sign with the wavevector :

(5.5.4)

The above wavevector triplet‐ is shown in Fig (5.8b), i.e. in the left‐hand (LH) coordinate system, so the DNG medium is also called the left‐handed, LH‐medium . Both the power‐vector and wavevector triplets and their combination are further shown in Fig (5.8d). The field vectors are rotated to maintain the power flow in a positive direction. Figure (5.8d)shows that the phase and group velocities are opposite to each other ( ) as for a DNG medium the vectors and are antiparallel, i.e. . The DNG occupies the third quadrant in the (μ r, ε r)‐plane as shown in Fig (5.7).

In conclusion, a DNG medium supports the backward wave propagation, whereas a forward wave is supported by the DPS medium. As the phase velocity travels toward the source, while energy is traveling from the source to a load, a propagating EM‐wave in a DNG medium, in the direction of the vector , has a leading phase . This is a unique property of the DNG medium. It significantly influences the EM‐wave characteristics of the DNG medium [J.8, B.6, B.10].

Figure 5.8 RH and LH‐coordinate systems for the DPS and DNG media.

The above discussion shows that Maxwell's equations in the DNG medium are written in the LH‐coordinate system. However, the wave equation ( 4.5.32) of chapter 4for the DPS medium remains valid for a lossless (σ = 0) DNG medium. It provides the following expressions for the propagation constant β = k DPSand refraction index of a DPS medium:

(5.5.5)