# wave guide

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## Wave Guide

a dielectric channel, or guiding system, for the propagation of electromagnetic waves. The surface of the channel is an interface between two media. At this interface the dielectric constant ∊, the magnetic permeability *μ*, and the electrical conductivity σ change abruptly. The surface may be of any shape. Cylindrical wage guides, however, are used most often—especially hollow metallic cylindrical wave guides filled with air or some other gas. The cross section of a metallic wave guide may be, for example, rectangular, circular, U-shaped, or H-shaped (Figure 1). We usually classify as wave guides only channels whose cross sections mathematically are simply connected regions. The propagation of electromagnetic waves in transmission lines with multiply connected cross sections is treated in the theory of long lines. An example of such a transmission line is the two-conductor coaxial line shown in Figure 1,e

It can be shown that the wave field propagates inside the wave guide along the guide’s axis; the field is a result of the multiple reflection of the waves from the guide’s inner walls and of the interference of the reflected waves. That propagation occurs in this way accounts for the principal characteristic of wave propagation in wave guides: the propagation of waves in a wave guide is possible only if the transverse dimension of the wave guide is comparable to or greater than the wavelength λ. For example, if λ = 30 cm, the longer dimension *a* of the cross section of a rectangular wave guide will be 20–25 cm. For this reason, wave guides are used primarily with microwaves.

Wave guides are used as transmission lines in, for example, radar sets for the transmission of energy from the transmitter to the transmitting antenna and from the receiving antenna to the receiver. A microwave transmission line consists of wave-guide sections that differ in shape and in cross-sectional dimensions and of a number of other structures, such as bends and rotary joints (Figure 2). Tapered wave-guide sections, such as the taper in Figure 2, are used to join wave guides of different cross section.

The chief advantage of a metallic wave guide over two-conductor balanced and coaxial lines is that the losses are low at microwave frequencies. There are two reasons for this. First, almost no energy is radiated into the surrounding space. Second, if the outer dimensions of the wave guide and of the, for example, two-conductor line are the same, the wave guide’s surface, on which electric currents flow when waves are propagating, is always greater than the surface of the conductors in the two-conductor line. Since the depth of penetration of the currents is determined by the skine effect, the current density in the wave guide and, consequently, the Joule heat losses are lower than in the line. Four disadvantages of a wave guide can be mentioned. First, there is a lower limit to the frequencies it can pass. Second, the structure required for decimeter or longer waves is unwieldy. Third, a high precision of manufacture and special finishing of the inner surfaces of the walls are required. Fourth, assembly is complicated.

Since the transverse dimensions of a wave guide are comparable to λ, the problems of the propagation and excitation of electromagnetic fields in the wave guide can be solved by integrating Maxwell’s equations for given boundary conditions and field sources. The theory of wave guides deals with the methods for solving these problems. In the case of a rectangular wave guide (Figure 3), for any component *f* of the electric field **E** and magnetic field **H** the theory yields the wave equation

where *k = 2π/λ = ω/c* is the wave number, *ω* is the oscillation frequency, and c is the speed of light. The solution of this equation for an infinitely long rectangular wave guide reduces to the following expressions for the complex amplitudes of the components of the vectors E and H:

Here, *a* and *b* are the dimensions of the cross section of the rectangular wave guide, *m* and *n* are any positive integers, and *A _{x}, A_{y}, A_{z}, B_{x}, B_{y}*, and

*B*are constants determined by the excitation conditions of the wave guide. The propagation constant γ determined by (2) and (1) is given by

_{z}The presence of trigonometric factors in (2) indicates the formation of standing waves in directions perpendicular to the walls of the wave guide. The tangential components of the electric

field have nodes at the walls, and the normal components have antinodes. The numbers *m* and *n* define the number of half-wavelengths that fit into the dimensions *a* and *b*, respectively. The greater the *m* and *n*, the more complex the field in the cross section of the wave guide.

The wave field in a wave guide is the sum of the fields of an infinite number of wave modes. Three types of wave modes are possible: *TE*, or *H*, waves; *TM*, or *E*, waves; and *TEM* waves. The *T* here stands for “transverse.” Each wave mode has its own field structure. In *TE* waves, the electric field has only transverse components, but the magnetic field has both longitudinal and transverse components. In *TM* waves, the magnetic field has only transverse components, and only the electric field has a longitudinal component. *TEM* waves have no longitudinal field components at all and can exist only in multiply connected wave guides.

In the standard notation for wave modes, the *m* and *n* appear as subscripts: *TM _{mn}* and

*TE*, or

_{mn}*E*and

_{mn}*H*. The waves with the lowest possible

_{mn}*m*and

*n*are called the simplest, or lowest-order, waves. In the case of

*TM*waves, where

*H*

_{z}= 0, the simplest wave is the wave TM11 (Figure 4).

*TM*

_{10}and

*TM*

_{01}waves are not possible because the magnetic field lines must be closed.

More complicated waves arise if we increase the transverse dimensions of the wave guide or the frequency so that more than one half-wavelength fits into the dimensions *a* and *b*. In this case, the cross section of the wave guide, like a vibrating membrane, is broken up into cells identical in structure to the cross section of a TM_{11} wave (Figure 5).

*TE* waves, where E_{z} = 0, can exist when *m =* 0 and n 0 or when *η* ≠ 0 and *m* ≠ 0. The reason for this is that the electric field lines can be straight lines that originate and terminate on opposite walls of the wave guide (Figures 6 and 7). Higher-order *TE* waves are composed of “cells” of TE_{10} and TE_{11} waves (Figure 8).

The factor *e-*^{γz}determines the changes in amplitude and phase of a wave as the wave propagates along the axis of the wave guide. In the absence of losses, γ must be a purely imaginary quantity: γ = iα. Thus,

This corresponds to the condition for frequency:

This means that a wave guide transmits without attenuation only oscillations of a frequency that is higher than some limiting frequency ωc, which is called the cutoff frequency. To ωc there corresponds the critical, or cutoff, wavelength λc. The smaller the wave-guide dimensions *a* and *b*, the higher the cutoff frequency ωc. For a given operating frequency ω, higher-order waves—that is, modes with larger *m* and *n—* require larger *a* and *b*.

The wavelength inside the wave guide ∧ is greater than the wavelength in free space:

The phase velocity of propagation inside the wave guide is equal to

that is, the phase velocity is always greater than the speed of light and depends on the frequency. Thus, a wave guide is dispersive. The wider the frequency range of the transmitted signals, the greater the distortion caused by dispersion in the signals.

The attenuation of a wave in a wave guide is described by the real part of the complex propagation constant γ = *β + i* α. In actual wave guides this attenuation is accounted for by losses in the walls and in the dielectric that fills the wave guide. If ω < ωc in an ideal, or loss-free, wave guide, the electromagnetic field is attenuated without any energy losses, owing to total reflection. The wave-guide dimensions can be chosen so as to permit the wave guide to operate at one of the *TE* modes. For example, for a rectangular wave guide and the TE_{10} mode, the choice of a is governed by the condition *α > λ > 2a*. The dimension *a* is usually taken as *a* = 0.72λ cm. Thus, *a* = 72 mm for λ = 10 cm, and a = 23 mm for λ = 3.2 cm (see Table 1).

The sum of the *TE* and *TM* waves in each wave guide forms the complete wave system. This means that propagation in a wave guide is possible only for electromagnetic fields with structures that can be represented as the superposition of *TE* and *TM* waves.

For wave guides of circular cross section, the basic equation is not (1) but the Bessel equation, which has solutions in the form of cylindrical functions. In a circular wave guide, it is also possible to choose the diameter so that the wave guide can transmit only one of the first modes (see Table 1). The first mode, however, is not always the most convenient one. For example, because of the axial symmetry of the fields of the TM_{0I}*a* and TE_{01} waves in a circular wave guide (Figures 9 and 10), these waves are used in rotary joints. Figures 11 and 12 show field structures for TM_{11} and TE_{11} waves in a circular wave guide. The use of waves with a comparatively small X_{c} is difficult

because if the propagation conditions for these waves are satisfied, unwanted modes of all other types will also be propagated at the same time.

The TE_{01} wave in a circular wave guide has an unusual property: the losses in the walls of the wave guide decrease continuously as λ decreases. This property can be made use of to construct wave-guide communications lines in the millimeter wavelength range with repeater stations spaced 50–60 km apart. Such lines are capable of carrying up to 1,500 telephone channels and 100 television channels. The principal difficulty here consists in providing the necessary field purity of the TE_{01} wave along the entire line through the elimination of the other wave modes that arise as a result of various discontinuities. In lossy wave guides the concept of a sharp transmission cutoff at ω_{c}

Table 1. Cutoff wavelengths λ_{c} for rectangular and circular wave guides | ||||||||
---|---|---|---|---|---|---|---|---|

Rectangular wave guide | Circular wave guide | |||||||

Wave mode ........ | TE_{10} | TE_{20} | TE_{10} | TE_{11} | TM_{01} | TE_{21} | TM_{11} | TE01 |

λc.............. | 2a | a | 2b | 3.41p | 2.61p | 2.06p | 1.64p | 1.64p |

loses its simple meaning. Lossy wave guides transmit some waves, although weakly, that are beyond the cutoff wavelength (λ > λ_{c}) computed for a loss-free wave guide.

Dielectric wave guides can be used for the transmission of centimeter and millimeter waves. In such wave guides the inner surface of a dielectric rod serves as the boundary that guides the wave. Since dielectric wave guides are sensitive to external influences and have additional losses associated with energy leakage beyond the boundaries of the wave guide, practical application is difficult.

In surface-wave wave guides the surface of a metallic strip or a cylindrical conductor is ribbed or is coated with a dielectric (Figure 13). Waves of various modes, for example, TM_{10}, can propagate along such wave guides. The energy of the field is concentrated in the surrounding space. The field radius—that is, the distance at which the field is still appreciable—depends on the width and conductivity of the strip; this radius decreases rapidly as λ decreases. Surface-wave guides exhibit less attenuation than metallic wave guides, are simpler in design, and are capable of transmitting greater power in a wide range of frequencies. The field of the surface wave surrounds the wave guide on the outside, a circumstance responsible for the disadvantages of such wave guides: various discontinuities—such as deformations of the wave guide, supports, joints, and shields—result in radiation, that is, losses of energy. Despite these disadvantages, surface-wave wave guides are used as guiding systems and as radiating elements in decimeter-, centimeter-, and millimeter-wave antennas.

Figure 14. Methods of excitation of a TE_{10} wave: (a) by a probe, (b) by a loop, (c) by an aperture

Three methods of field excitation are used in wave guides: by means of a probe, or linear conductor; by means of a loop; and by means of an aperture, or slot, in the side wall or end face of the wave guide. The probe is located parallel to the lines of electric force. The plane of the loop is perpendicular to the lines of magnetic force. The aperture is cut through the metallic surface in the direction of the lines of magnetic force on this surface. For better coupling, the exciting elements are located in the an-tinodes of the electric or magnetic field (Figure 14).

The impedance matching of individual wave-guide sections to each other or to the load can be achieved by means of matching devices (Figure 15) in the form of combinations of probes and inductive or capacitive irises. Tapers are also often used as matching devices. A disadvantage of most matching devices is the small frequency range for which the devices can be used. Matching can generally be achieved in a range of frequencies that is 1–2 percent of ω; 10–20 percent of ω is possible in only a few cases.

The problem of constructing high-power wave guides is of practical importance. The power that a wave guide with cross-sectional dimensions appropriate for the propagation of only first-mode waves can transmit is merely of the order of 3–4 megawatts. If larger dimensions are used for a given wavelength, higher-order modes will propagate through the wave guide.

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