Physics
Rectangular Waveguide
A rectangular waveguide is a type of transmission line used to guide electromagnetic waves, typically in the microwave frequency range. It consists of a hollow metal tube with a rectangular cross-section that confines and guides the waves along its length. Rectangular waveguides are commonly used in radar systems, satellite communications, and other high-frequency applications due to their low loss and high power-handling capabilities.
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eBook - ePubElectromagnetic Fields
Theory and Applications
- Ahmad Shahid Khan, Saurabh Kumar Mukerji(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
19 Waveguides and Cavity Resonators19.1 Introduction
In Chapter 18 , it was noted that the parallel wire/plane transmission structures can only be used to propagate low frequency signals with TEM mode having zero cutoff frequency. At higher frequencies, the suitability of TEM mode diminishes due to increased attenuation. Thus, at higher frequencies the use of TE and TM modes becomes a necessity. These modes can be supported by another form of transmission structures called waveguides. These structures have a variety of forms in terms of their shapes and materials. These can be in the form of hollow rectangular or circular metallic pipes or solid dielectric rods or slabs. Except the differing modes, many of the features of these structures resemble those of wire/plane transmission lines. These include the concept of reflection, current flow in the conductor skin, properties of quarter and half wave sections, impact of discontinuities on propagation etc.Since waveguides are the key players for guiding electromagnetic energy at centimetric and millimetric ranges, this chapter is fully devoted to the study of the field behavior in these. This chapter encompasses the mathematical theory of rectangular and circular waveguides, describes different modes and their excitation, and includes the physical interpretation of various terms involved.19.2 Rectangular Waveguide
This is the simplest and most commonly used waveguide. It is a hollow rectangular pipe containing four metallic walls. Its physical structure along with its dimensional parameters and coordinate system is shown in Figure 19.1 . In view of the presence of four metallic walls E and H fields have to satisfy appropriate boundary conditions. These include the continuity of the tangential component of E and the normal component of H for appropriate values of x and y
- Constantine A. Balanis(Author)
- 2023(Publication Date)
- Wiley(Publisher)
Rectangular transmission lines (such as Rectangular Waveguides, dielectric slab lines, striplines, and microstrips) and their corresponding cavities represent a significant section of lines used in many practical radio-frequency systems. The objective in this chapter is to introduce and analyze some of them, and to present some data on their propagation characteristics. The parameters of interest include field configurations (modes) that can be supported by such structures and their corresponding cutoff frequencies, guide wavelengths, wave impedances, phase and attenuation constants, and quality factors Q. Because of their general rectilinear geometrical shapes, it is most convenient to use the rectangular coordinate system for the analyses. The field configurations that can be supported by these structures must satisfy Maxwell’s equations or the wave equation, and the corresponding boundary conditions. 8.2 Rectangular Waveguide Let us consider a Rectangular Waveguide of lateral dimensions a and b, as shown in Figure 8-3. Initially assume that the waveguide is of infinite length and is empty. It is our purpose to determine the various field configurations (modes) that can exist inside the guide. Although a TEM z field con- figuration is of the simplest structure, it cannot satisfy the boundary conditions on the waveguide walls. Therefore, it is not a valid solution. It can be shown that modes TE x , TM x , TE y , TM y , TE z , and TM z satisfy the boundary conditions and are therefore appropriate modes (field configura- tions) for the Rectangular Waveguide. We will initially consider TE z and TM z ; others will follow. Figure 8-1 Two Rectangular Waveguides (Ku-band and X-band) with flanges. Figure 8-2 X-band microwave sources: X-13 klystron and Gunn diode wafer. (a) Rear view of X-13. (b) Front view of X-13. (c) Gunn diode wafer.
- Richard G. Carter(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
4 1 2 Waveguides 2.1 Introduction Modern vacuum tubes are power amplifiers and oscillators which require the use of waveguides or coaxial lines to convey RF power into and out of them. The pur- pose of this chapter is to provide a summary of those topics which are important for the design of vacuum tubes. It also provides a foundation for the discussions of resonators in Chapter 3, and of slow-wave structures in Chapter 4. Section 2.2 sum- marises the theory of hollow metal waveguides, and two-conductor transmission lines, having uniform cross-sections. This leads to a discussion of practical coaxial lines, and rectangular, ridged and circular waveguides in Section 2.3. The properties of simple discontinuities in Rectangular Waveguides are considered in Section 2.4 followed by a discussion of matching techniques in Section 2.5. Sections 2.6 and 2.7 examine methods of coupling between waveguides of different cross-sections without, and with, changes in the mode of propagation. The final section reviews the different kinds of vacuum windows which are used in coaxial lines and rect- angular waveguides. The theory and practice of waveguides and waveguide com- ponents is covered by many books and the reader is referred to them for detailed information [1–5]. 2.2 Waveguide Theory The propagation of electromagnetic waves in a source-free region, filled with a uniform material of permittivity ε and permeability μ, is governed by the wave equations ∇ − ∂ ∂ = 2 2 2 0 E E εμ t (2.1) and ∇ − ∂ ∂ = 2 2 2 0 H H εμ t , (2.2) where E is the electric field and H is the magnetic field [1]. We shall assume that the waves are guided in the z direction by conducting boundaries whose shape Waveguides 42 4 2 does not vary with z. The vector operator can be decomposed into transverse and longitudinal parts ∇ = ∇ + ∂ ∂ T z ˆ , z (2.3) where ˆ z is the unit vector in the z direction.
eBook - PDFMicrowave Engineering
Concepts and Fundamentals
- Ahmad Shahid Khan(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
This allows a wave-guide to handle only a small range of frequencies both above and below the frequency of operation. Dimension ‘ b ’ commonly varies between 0.2 λ and 0.5 λ and is governed by the breakdown voltage of the dielectric, which is usually air. In centimetric and millimetric ranges, waveguide is the key player in a majority of applications wherein the wave is guided by the four walls. 3.3 Rectangular Waveguide To explore the guiding principles of wave propagation through a waveguide, Figure 3.4 shows a Rectangular Waveguide along with its dimensions and coordinate system. The wave is assumed to be propagating along its z -axis. As before, the time variation is accounted by e j ω t and the space variation along the z -axis by e − γ z = e − ( α + j β ) z , where parameters α , β and γ have the same mean-ing as discussed in Chapter 2. Since the waveguide configuration contains four conducting walls, boundary conditions along the x - and y -axes include the continuity of E tan and H norm at x = 0 and a , for all values of y and at y = 0 and b for all values of x . In case of waveguides the behaviour of waves is also governed by Maxwell’s equations, which are written as below. a b (a) (b) (c) m n n m m λ /4 λ /4 λ /4 λ /4 λ /4 λ /4 n and p q p q p q b a a b FIGURE 3.3 Impact of change of frequency. (a) Normal operating frequency, (b) more than operating fre-quency and (c) less than operating frequency.
- K. Kurokawa(Author)
- 1969(Publication Date)
- Academic Press(Publisher)
C H A P T E R 3 WAVEGUIDES A cylindrical pipe designed to contain one or more propagating electro-magnetic waves is called a waveguide. We shall discuss the theory of wave-guides in detail in this chapter. Since waveguides require a treatment quite different from conventional circuit theory and radically new to some of us, we shall first introduce the particular case of Rectangular Waveguides. This will enable us to become acquainted with new terminology and some methods we shall use later. Following this will be an introduction to eigen-value problems using an equation derived in the above discussion. Solutions of an eigenvalue problem are called the eigenfunctions, and these have wide applications in many branches of physics, particularly in acoustics and quantum mechanics. Suppose a function representing a linear phenomenon is to be determined, we can express it as a linear combination of eigen-functions and determine the coefficients using appropriate equations governing the process. Once this expression is obtained, we interpret the phenomenon as the superposition of simple phenomena, each corresponding to an eigenfunction; the method is called the eigenfunction approach. For this approach to be useful, it is important to select eigenfunctions whose individual behavior is simple and well understood. Therefore, depending on the problem, a set of eigenfunctions is chosen to satisfy a certain eigen-value problem closely related to the phenomenon under study. Whether or not the function to be determined can be expressed as a linear combination of eigenfunctions thus selected remains to be investigated. This, of course, depends on the eigenvalue problem used for their derivation, but when the linear combination is possible for any function of interest, the set of eigen-84 3.1. Perfect Conductors 85 functions is said to be complete.
eBook - ePub- I. S. Grant, W. R. Phillips(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
δx and to the energy in the guide at that point. The mean energy in the wave in a waveguide thus falls off exponentially with distance travelled. In contrast, when radiation is emitted from antennas (this topic is discussed in the next chapter), the energy crossing unit area per second in the radiation fields falls off inversely as the square of the distance from the antenna. Hence if one is comparing antennas and waveguides from the point of view of energy losses it is usually best to use antennas to transmit waves over long distances. This is because at large distances the exponential rate of energy decay in a waveguide always results in a smaller energy than the inverse square law fall off in free space. Waveguides, however, are much better over shorter distances and are widely used in practice for the transmission of microwaves of wavelengths in the range from a few millimetres to about one metre. For example, 3 cm radar waves are usually generated in an area near the deck of a ship where they can be controlled, and are carried to the antenna on the top of the ship’s mast with waveguides.Waveguides of rectangular cross-section are the most commonly used and we shall, for the most part, discuss these alone.12.1 THE PROPAGATION OF WAVES BETWEEN CONDUCTING PLATES
The main features of the propagation of waves in waveguides can be understood by considering the simpler problem of the propagation of waves between two parallel conducting plates.Figure 12.1 shows parts of two parallel plates separated by a distance b . We will suppose that the plates are of infinite extent and furthermore that they are perfectly conducting. This last assumption is made in order to simplify the mathematics. The problem now is to determine the characteristics of the electromagnetic fields which can be propagated down the length of the plates. The fields we are looking for have to obey Maxwell’s equations in the free space between the plates. They also have to obey certain boundary conditions at all points on the conducting walls. These conditions, as discussed in section 11.6.1, are that the tangential component of the electric field and the normal component of the magnetic field must be everywhere zero over the walls. The normal component of the electric field need not be zero, since there can be charges on the conducting surfaces, and the tangential component of the magnetic field need not be zero, since there can be surface currents in the perfectly conducting walls. The condition that the fields obey the last two of Maxwell’s equations in the space between the plates means, just as in section 11.1, that they must satisfy the wave Equations (11.5) and (11.6)
eBook - PDF- D. S. Jones, I. N. Sneddon, S. Ulam, M. Stark(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
in radar and television. As the frequency of the signal increases the dimensions of the line have to be reduced to prevent the propagation of the T E n mode. There is therefore an increased risk of breakdown if the same power is being carried. Also the losses, as indicated by (24), steadily increase with frequency and there will be additional losses in the dielectric supports holding the inner conductor. These shortcomings are absent in the circular and Rectangular Waveguides becauses, as can easily be confirmed, the attenuation is less than that of a coaxial line when the frequency is about 1.2 times the cut-off frequency. Thus at wavelengths of a few centimetres it is more efficient to use a rectangular or circular waveguide than a coaxial line. J U N C T I O N S To serve a useful purpose a waveguide must be connected to a transmitter, receiver or antenna. Such a connection will materially affect the field in the guide. The next few sections will be concerned with the theory necessary for estimating this effect. 5.6 General waveguide junction A typical junction is shown in Fig. 5.6. It consists of T and a number L of waveguide regions. The boundary S of T is closed by L terminal planes Τ λ , . . ., T L each of which is perpendicular to the axis of the guide in which it is placed. The positions of the terminal planes are, to a large extent, arbitrary but it will be assumed that they FIG. 5.6. A general waveguide'junction. THE THEORY OF WAVEGUIDES 255 are far enough from the junction for any non-propagating modes which may be present at the junction to be essentially zero. We shall also assume that only the fundamental mode can propagate in each guide; the theory can be extended in a straightforward manner to the more general case. The field on te terminal planes may then be expressed completely in terms of the fundamental modes. The region T may contain dielectrics, conducting substances and the like. It will be assumed that they are linear and passive.
eBook - PDF- Tung Tsang(Author)
- 1998(Publication Date)
- WSPC(Publisher)
CHAPTER 6. WAVE GUIDES 6.1. Metallic Boundary Conditions In the last few decades, new radar and microwave technologies have brought forth many important and useful applications. These technologies are based on the use of electro-magnetic (EM) waves with wave length in the several cm range. Many of the radar and microwave devices depend on the behavior of these waves when bounded by conducting surfaces. Typically, rectangular wave guides are used for the transmission of these waves. These guides are metal pipes with rectangular cross sections with dimensions in the several cm range (comparable to the size of household water pipes, but rectangular in cross section). More recently, fiber optics have been widely used to transmit light signals. These wave guides are glass fibers which are solid dielectric cylinders of much smaller dimensions (comparable to the visible light wave length of 10 ~ 5 cm). The general requirement for wave guide is that there will be flow of energy along the guiding structure and not perpendicular to it. The direction of energy flow will be denoted as the z-axis. We will discuss the boundary conditions at a metal-dielectric interface. We will consider a good metal with very large electrical conductivity a. Inside the metal, we have E=0. Faraday law of induction requires dB/dt=-cVxE. Any time-dependent magnetic fields will also be absent inside the metal. We will use the boundary conditions of continuous normal (subscript n) B and D, continuous tangential (subscript t) E and H. Inside the dielectric near a metal surface, the boundary conditions are: B n = 0 E t = 0 (6.1) However, D n and Ht may not be zero because of the possible presence of surface charges and surface currents on the metal surface. From the boundary conditions (6.1), we are able to calculate the dispersion relations and cutoff frequencies for the different modes of wave propagation.
eBook - PDF- Kok Yeow You(Author)
- 2018(Publication Date)
- IntechOpen(Publisher)
2003; 24 :1885-1891 [9] Sharma J. Full-wave analysis of dielectric Rectangular Waveguides. Progress in Electro-magnetics Research. 2010; 13 :121-131 [10] Sukhinin SV. Waveguide effect in a one-dimensional periodically penetrable structure. Journal of Applied Mechanics and Technical Physics. 1990; 31 :580-588 [11] Sumathy M, Vinoy KJ, Datta SK. Analysis of rectangular folded-waveguide millimeter-wave slow-wave structures using conformal transformations. Journal of Infrared, Milli-meter, and Terahertz Waves. 2009; 30 :294-301 [12] Rothwell EJ, Temme A, Crowgey B. Pulse reflection from a dielectric discontinuity in a Rectangular Waveguide. Progress in Electromagnetics Research. 2009; 97 :11-25 [13] Bulgakov AA, Kostylyova OV, Meriuts AV. Electrodynamic properties of a waveguide with layered-periodic walls. Radiophsics and Quantum Electronics. 2005; 48 :48-56 [14] Lu W, Lu YY. Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations. Journal of Computational Physics. 2012; 231 :1360-1371 [15] Eyges L, Gianino P. Modes of dielectric waveguides of arbitrary cross sectional shape. Journal of the Optical Society of America. 1979; 69 :1226-1235 Periodic Rectangular and Circular Profiles in the Cross Section of the Straight Waveguide Based on Laplace… http://dx.doi.org/10.5772/intechopen.76794 333 [16] Soekmadji H, Liao SL, Vernon RJ. Experiment and simulation on TE 10 cut-off reflection phase in gentle rectangular downtapers. Progress in Electromagnetics Research Letters. 2009; 12 :79-85 [17] Abbas Z, Pollard RD, Kelsall W. A rectangular dielectric waveguide technique for deter-mination of permittivity of materials at W-band. IEEE Transactions on Microwave Theory and Techniques. 1998; 46 :2011-2015 [18] Hewlett SJ, Ladouceur F. Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff. Journal of Lightwave Technology. 1995; 13 :375-383 [19] Binzhao C, Fuyong X.
eBook - PDFAdvances in Microwaves
Volume 4
- Leo Young(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
Recent laser developments have greatly encouraged theoretical and experimental work on the beam waveguide by stimulating research on optical resonators and transmission lines. Since Goubau gives a general description of beam waveguides in Volume 3 of this series [42], we limit our discussion to the type of continuous reflectors shown in Fig. 80. 258 T. Nakahara and N. Kurauchi Circular lenses Wing-shaped lenses between parallel plates Rectangular lenses Gas lenses Gas f low - Hot pipe Cool plpe Hot pipe Concave reflectors Continuous ref lectors J 1 FIG. 80. Types of beam waveguide. B. BEAM WAVEGUIDE WITH PARALLEL CONCAVE REFLECTORS Unlike other systems, which have a periodic structure in the direction of propagation, the system in Fig. 81 has a uniform structure. The waves are propagated by reflection between reflectors as in a closed waveguide and are converged into a beam by reflectors that are concave in the transverse cross section. Like the elementary waves in a Rectangular Waveguide mode, the modes in this system can be represented by a mixture of two elementary beam waves propagating in the direction that makes an angle 0 with the z axis in the z-x plane, as shown in Fig.,82. The angle 0 is approximately described by a mode number m (which corresponds to a field change in the x direction), a spacing d of the reflectors, and a free space wavelength h as : sin8 = mh/2d where m is an integer. MILLIMETER WAVEGUIDES AND RAILROAD COMMUNICATIONS 259 Thermal loss is approximated by the thermal loss of the TE or TM mode propagation between two parallel conductor plates of the same spacing d. The interval of the phase transformers for the elementary waves in this system is rf/tan0. The diffraction loss per reflection can be obtained by substituting d/tan Θ for the phase transformer interval or the reflector spacing in the beam wave theory [33]. This substitution shows that confocal setting of the reflections would be most favorable for low-loss transmission.
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