Introduction To Modern Planar Transmission Lines. Anand K. Verma
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In the above equation,
4.5.3 Uniform Plane Waves in Linear Lossless Homogeneous Isotropic Medium
Figure (4.9a) shows the propagation of the TEM wave in the x‐direction of an unbounded medium. Figure (4.9b) further shows that for the TEM wave, electric and magnetic field components, i.e. the pairs (Ey, Hz), or (Ez, Hy), are transverse to the direction of propagation, i.e. ± x ‐ direction. For the field pair (Ey, Hz), the wave is y‐polarized; and for the field pair(Ez, Hy), it is the z‐polarized. The polarization of an EM‐wave is determined by the direction of the
The field components of a uniform plane wave do not change with y and z coordinates, i.e. ∂/∂y(field E or H) = ∂/∂z(field E or H) = 0. The field components are a function of x only. So, the field components of the EM‐wave propagating in the x‐direction can be written as follows:
In the above expressions, the (−) sign shows the wave propagation in the positive x‐direction, whereas the (+) sign is for the wave propagation in the negative x‐direction. The wave propagation in the positive x‐direction is discussed below.
Figure 4.9 TEM mode wave in an unbounded medium.
The above expressions related to a uniform plane wave, in an external source‐free
(4.5.21)
On separating each component of the fields, the following expressions are obtained:
(4.5.22)
Likewise, the following expressions are obtained from Maxwell's second curl expression (4.4.1b):
(4.5.23)
It is seen from the above equations that the Ex and Hx components, in the direction of propagation, are time‐independent, i.e. constant. They do not play any role in the wave propagation and can be assumed to be zero, without affecting the wave propagation [B.3]. Only transverse field components play a role in wave propagation. The time‐varying Hy component generates the Ez, whereas the time‐varying Ey component generates the Hz. It is also true for another time‐varying pair (Hz, Ez). Maxwell divergence relations also show ∂Ex/∂x = ∂Hx/∂x = 0. Again, Ex and Hx components do not show any variation along the direction of propagation that is essential for wave propagation. So, in the TEM mode propagation, the longitudinal field components are zero, i.e. Ex = Hx = 0.
By using equation (4.5.20) with the above equations, the time‐harmonic fields are rewritten as follows:
On using equation (4.5.20) with the above equations, the following algebraic expressions are obtained for the EM‐wave propagating in the positive x‐direction:
The wave impedance in free space, or in homogeneous unbounded medium, is defined in a plane normal to the direction of propagation [B.2]. For instance, Fig. (4.9a) shows the direction of propagation is along the x‐axis, and wave impedance is defined in the (y‐z) plane. It is also called the intrinsic impedance η0 of free space and intrinsic impedance η of material filled homogeneous space. The following expression is obtained for the wave impedance η, for y‐polarized waves, from equation (4.5.25a):
(4.5.26)
Equation (4.5.25b) provides the following wave impedance for the z‐polarized waves:
(4.5.27)
The positive wave impedance of the (Ey, Hz) or (−Ez, Hy) fields corresponds to a wave traveling in the +x direction. However, for the (Ez, Hy) fields, the wave impedance is negative showing the wave propagation in the negative x‐direction. Under certain conditions, a medium can have an imaginary value of propagation constant, i.e. βx = − jp. In this case, the wave impedance becomes reactive, and there is no wave propagation. Again, the wave equation (4.5.20) is reduced to Ei = E0ie−px ejωt and Hi = H0i e−px ejωt for the wave propagation in the positive x‐direction. These are