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Wave Guides Summary and Problems ECE 144 Electromagnetic Fields and Waves Bob York General Waveguide Theory Basic Equations (ωt γz) Consider wave propagation along the z-axis, with fields varying in time and distance according to e − . The propagation constant γ gives us much information about the character of the waves. We will assume that the fields propagating in a waveguide along the z-axis have no other variation with z,thatis,the transverse fields do not change shape (other than in magnitude and phase) as the wave propagates. Maxwell’s curl equations in a source-free region (ρ =0andJ = 0) can be combined to give the wave equations, or in terms of phasors, the Helmholtz equations: 2E + k2E =0 2H + k2H =0 ∇ ∇ where k = ω√µ. In rectangular or cylindrical coordinates, the vector Laplacian can be broken into two parts ∂2E 2E = 2E + ∇ ∇t ∂z2 γz so that with the assumed e− dependence we get the wave equations 2E +(γ2 + k2)E =0 2H +(γ2 + k2)H =0 ∇t ∇t (ωt γz) Substituting the e − into Maxwell’s curl equations separately gives (for rectangular coordinates) E = ωµH H = ωE ∇ × − ∇ × ∂Ez ∂Hz + γEy = ωµHx + γHy = ωEx ∂y − ∂y ∂Ez ∂Hz γEx = ωµHy γHx = ωEy − − ∂x − − − ∂x ∂Ey ∂Ex ∂Hy ∂Hx = ωµHz = ωEz ∂x − ∂y − ∂x − ∂y These can be rearranged to express all of the transverse fieldcomponentsintermsofEz and Hz,giving 1 ∂Ez ∂Hz 1 ∂Ez ∂Hz Ex = γ + ωµ Hx = ω γ −γ2 + k2 ∂x ∂y γ2 + k2 ∂y − ∂x w W w W 1 ∂Ez ∂Hz 1 ∂Ez ∂Hz Ey = γ + ωµ Hy = ω + γ γ2 + k2 − ∂y ∂x −γ2 + k2 ∂x ∂y w W w W For propagating waves, γ = β,whereβ is a real number provided there is no loss. Rewriting the above for propagating waves, and using the substitution k2 γ2 + k2,gives c ≡ ∂Ez ∂Hz ∂Ez ∂Hz Ex = β + ωµ Hx = ω β −k2 ∂x ∂y k2 ∂y − ∂x c w W c w W ∂Ez ∂Hz ∂Ez ∂Hz Ey = β + ωµ Hy = ω + β k2 − ∂y ∂x −k2 ∂x ∂y c w W c w W Theanalyticprocedureforfinding waveguide fields and propagation constants is to solve the wave equations for the z-components of the fields, subject to the boundary conditions for the waveguide, and then find the transverse field components from the above. Mode Classification In uniform waveguides it is common to classify the various wave solutions found from the previous analysis into the following types: TEM waves: waves with no electric or magnetic field in the direction of propagation (Hz = Ez =0). • Plane waves and transmission-line waves are common examples. TM waves: waves with an electric field but no magnetic field in the direction of propagation (Hz = • 0,Ez = 0). These are sometimes referred to as E waves. TE waves: waves with a magnetic field but no electric field in the direction of propagation (Hz = • 0,Ez = 0). These are sometimes referred to as H waves. Hybrid waves: Sometimes the boundary conditions require all field components. These waves can be • considered as a coupling of TE and TM modes by the boundary. Note that these are not the only way to categorize the different wave solutions, but have been standardized by long usage. Mode Impedances, Propagation Constants, and Cutoff Frequencies The various types of wave solutions have many common features, regardless of the shape of the waveguiding structure. Examining the propagation constant γ = k2 k2 we see that for some frequencies it is imaginary c − (γ = β), corresponding to propagating waves, and for others it is real (γ = α), corresponding to exponen- 0 tially decaying, or evanescent fields. The dividing line is at a frequency known as the cutoff frequency,given by kc fc = cutoff frequency 2π√µ we can write the propagation constant in terms of fc as follows β = k 1 (f /f)2 f>f γ = c c − 2 α = kc 1 (f/fc) f<fc F 0 − Thus above the cutoff frequency, waves can propagate.0 Note from the field expression derived previously 2 2 that TEM waves can only have non-zero fields if γ + k =0,orγ = k = ω√µ.Inthiscase,kc = 0, hence TEM waves have no cutoff frequency. From our expressions for the transverse field components we can also define wave impedances for the various modes. The wave impedance concept is important because it allows us to relate the E and H fields through thesimplerelationshipH =ˆz E/Z, thus unifying the presentation of electromagnetic waves. And from × a practical standpoint the wave impedance allows us to use transmission-line theory to describe non-TEM waveguide circuits. Mode Propagation Constant, β Guide Wavelength, λg Wave Impedance, Z TEM k = ω√µ λ =1/f√µ η = µ/ 2 λ γ 0 2 TM k 1 (fc/f) = η 1 (fc/f) 2 − 1 (fc/f) ω − 0 − 0 2 λ ωµ η TE k 1 (fc/f) 0 = 2 2 − 1 (fc/f) γ 1 (fc/f) 0 − − 0 0 Below cutoff the wave impedance is imaginary, indicating that there is no net transfer of power. Since the propagation constants and wave impedances for non-TEM modes are nonlinear functions of frequency, such modes are dispersive. Try the following questions. Some few questions are beyond the scope of the course حاول إجابة اﻷسئلة اﻵتية. قليل من اﻷسئلة خارج نطاق ما تم تدريسه فﻻ يلتف إليه. III-WAVEGUIDES III-1 Questions with short answers Q.1 What is the region of the frequency spectrum from 1000 MHz to 100,000 MHz called? Q.2Why are coaxial lines more efficient at microwave frequencies than two-wire transmission lines? Q.3 What kind of material must be used in the construction of waveguides? Q.4 The large surface area of a waveguide greatly reduces what type of loss that is common in two-wire and coaxial lines? Q.5 What causes the current-carrying area at the center conductor of a coaxial line to be restricted to a small layer at the surface? Q.6 Why are square wave guides not in use? Q.7 What is the primary lower-frequency limitation of waveguides? Q.8 The frequency range of a waveguide is determined by what dimension? Q.9 When the frequency is decreased so that two quarter-wavelengths are longer than the "a" (wide) dimension of the waveguide, what will happen? Q.10 What primary condition must magnetic lines of force meet in order to exist? Q.11 For an electric field to exist at the surface of a conductor, the field must have what angular relationship to the conductor? Q.12 Compared to the velocity of propagation of waves in air, what is the velocity of propagation of waves in waveguides? Q.13 What term is used to identify the forward progress velocity of wavefronts in a waveguide? Q.14 What term is used to identify each of the many field configurations that can exist in waveguides? Q.15 What field configuration is easiest to produce in a given waveguide? Q.16 The field arrangements in waveguides are divided into what two categories to describe the various modes of operation? Q.17 The electric field is perpendicular to the "a" dimension of a waveguide in what mode? Q.18 The number of half-wave patterns in the "b" dimension of rectangular waveguides is indicated by which of the two descriptive subscripts? Q.19 Which subscript, in circular waveguide classification, indicates the number of full- wave patterns around the circumference? Q.20 What determines the frequency, bandwidth, and power-handling capability of a waveguide probe? Q.21 Loose or inefficient coupling of energy into or out of a waveguide can be accomplished by the use of what method? Q.22 What is the result of an impedance mismatch in a waveguide? Q.23 A diaphragm placed along the "a" dimension wall produces what kind of reactance? Q.24 What is a capacitive window? Q.25 A diaphragm placed along the "b" dimension wall produces what kind of reactance? Q.26 What is an inductive window? Q.27 A diaphragm that has portions along both the "a" and "b" dimension walls act at the resonant frequency? Q.28 What is a resonant window? Q.29 What device is used to produce a gradual change in impedance at the end of a waveguide? Q.30 When a waveguide is terminated in a resistive load, the load must be matched to what property of the waveguide? Q.31 What is the result of an abrupt change in the size, shape, or dielectric of a waveguide? Q.32 A waveguide bend must have what minimum radius? Q.33 What is the most likely cause of losses in waveguide systems? Q.34 Which transmission medium is most suitable for handling high powers: rectangular wave guides, coaxial lines, or microstrip lines? Q.35 For which applications are circular guides preferred to rectangular guides? Q.36 Give reasons, why the dominant TE10 mode is preferred with rectangular waveguides. Q.37 Which one of the following waveguide tuning components is not easily adjustable: screw, stub, iris, or plunger? III-2 Problems 1- Calculate cutoff frequency for TE1, TE2, TE3, TM1, TM2, and TM3 waves between perfectly conducting planes 15 mm apart with air dielectric between them. Repeat for a glass dielectric of dielectric constant 4. Suppose excitation at 8 GHz is provided at a cross section of the air-filled guide and all the six waves are excited, which of them will propagate without attenuation? At what distance from the excitation plane will each of the non- propagating waves be attenuated to 1/e of its value at the excitation plane? 2- An air-filled waveguide has inside dimensions 22.9x10.2 mm.
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