chapter 12

# chapter 12 - Chapter WAVEGUIDES If a man writes a better...

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Chapter WAVEGUIDES If a man writes a better book, preaches a better sermon, or makes a better mouse- trap than his neighbor, the world will make a beaten path to his door. —RALPH WALDO EMERSON 12.1 INTRODUCTION As mentioned in the preceding chapter, a transmission line can be used to guide EM energy from one point (generator) to another (load). A waveguide is another means of achieving the same goal. However, a waveguide differs from a transmission line in some respects, although we may regard the latter as a special case of the former. In the first place, a transmission line can support only a transverse electromagnetic (TEM) wave, whereas a waveguide can support many possible field configurations. Second, at mi- crowave frequencies (roughly 3-300 GHz), transmission lines become inefficient due to skin effect and dielectric losses; waveguides are used at that range of frequencies to obtain larger bandwidth and lower signal attenuation. Moreover, a transmission line may operate from dc (/ = 0) to a very high frequency; a waveguide can operate only above a certain frequency called the cutofffrequency and therefore acts as a high-pass filter. Thus, wave- guides cannot transmit dc, and they become excessively large at frequencies below mi- crowave frequencies. Although a waveguide may assume any arbitrary but uniform cross section, common waveguides are either rectangular or circular. Typical waveguides 1 are shown in Figure 12.1. Analysis of circular waveguides is involved and requires familiarity with Bessel functions, which are beyond our scope. 2 We will consider only rectangular waveguides. By assuming lossless waveguides (a c — °°, a ~ 0), we shall apply Maxwell's equations with the appropriate boundary conditions to obtain different modes of wave propagation and the corresponding E and H fields. _ ; 542 For other t\pes of waveguides, see J. A. Seeger, Microwave Theory, Components and Devices. E n glewood Cliffs, NJ: Prentice-Hall, 1986, pp. 128-133. 2 Analysis of circular waveguides can be found in advanced EM or EM-related texts, e.g., S. Y. Liao. Microwave Devices and Circuits, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1990, pp. 119-141.

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12.2 RECTANGULAR WAVEGUIDES 543 Figure 12.1 Typical waveguides. Circular Rectangular Twist 90° elbow 12.2 RECTANGULAR WAVEGUIDES Consider the rectangular waveguide shown in Figure 12.2. We shall assume that the wave- guide is filled with a source-free (p v = 0, J = 0) lossless dielectric material (a — 0) and its walls are perfectly conducting (a c — °°). From eqs. (10.17) and (10.19), we recall that for a lossless medium, Maxwell's equations in phasor form become k z E s = 0 = 0 (12.1) (12.2) Figure 12.2 A rectangular waveguide with perfectly conducting walls, filled with a lossless material. / («, jX, <T = 0 )
544 Waveguides where k = OJVUB (12.3) and the time factor e J01t is assumed. If we let - (E xs , E ys , E zs ) and - (H xs , H ys , H zs ) each of eqs. (12.1) and (12.2) is comprised of three scalar Helmholtz equations. In other words, to obtain E and H fields, we have to solve six scalar equations. For the z-compo- nent, for example, eq. (12.1) becomes d 2 E zs dx 2 dy 2 dz (12.4) which is a partial differential equation. From Example 6.5, we know that eq. (12.4) can be

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chapter 12 - Chapter WAVEGUIDES If a man writes a better...

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