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# chapt12 - CHAPTER 12 PLANE WAVES AT BOUNDARIES AND IN...

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CHAPTER 12 PLANE WAVES AT BOUNDARIES AND IN DISPERSIVE MEDIA In Chapter 11, we considered basic electromagnetic wave principles. We learned how to mathematically represent waves as functions of frequency, medium prop- erties, and electric field orientation. We also learned how to calculate the wave velocity, attenuation, and power. In this chapter, we consider wave reflection and transmission at planar boundaries between media having different properties. Our study will allow any orientation between the wave and boundary, and will also include the important cases of multiple boundaries. We will also study the practical case of waves that carry power over a finite band of frequencies, as would occur, for example, in a modulated carrier. We will consider such waves in dispersive media, in which some parameter that affects propagation (permittivity for example) varies with frequency. The effect of a dispersive medium on a signal is of great importance, since the signal envelope will change its shape as it propagates. This can occur to an extent that at the receiving end, detection and faithful representation of the original signal become problematic. Dispersion thus becomes an important limiting factor in allowable propagation distances, in a similar way that we found to be true for attenuation. 387

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12.1 REFLECTION OF UNIFORM PLANE WAVES AT NORMAL INCIDENCE In this section we will consider the phenomenon of reflection which occurs when a uniform plane wave is incident on the boundary between regions composed of two different materials. The treatment is specialized to the case of normal inci- dence , in which the wave propagation direction is perpendicular to the boundary. In later sections we remove this restriction. We shall establish expressions for the wave that is reflected from the interface and for that which is transmitted from one region into the other. These results will be directly applicable to impedance- matching problems in ordinary transmission lines as well as to waveguides and other more exotic transmission systems. We again assume that we have only a single vector component of the electric field intensity. Referring to Fig. 12.1, we define region l ± 1 ; ² 1 as the half-space for which z < 0; region 2 ± 2 ; ² 2 is the half-space for which z > 0. Initially we establish the wave traveling in the z direction in region l, E x 1 z ; t E x 10 e ± ³ 1 z cos ! t ± ´ 1 z or E xs 1 E x 10 e ± jk 1 z 1 where we take E x 10 as real. The subscript 1 identifies the region and the super- script + indicates a positively traveling wave. Associated with E x 1 z ; t is a magnetic field H ys 1 1 µ 1 E x 10 e ± jk 1 z 2 where k 1 and µ 1 are complex unless ± 00 1 (or 1 ) is zero. This uniform plane wave in region l which is traveling toward the boundary surface at z 0 is called the incident wave. Since the direction of propagation of the incident wave is perpen- dicular to the boundary plane, we describe it as normal incidence.
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chapt12 - CHAPTER 12 PLANE WAVES AT BOUNDARIES AND IN...

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