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

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 األسئلة اآلتية. قليل من األسئلة خارج نطاق ما تم تدريسه فال 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. Find the four lowest cut-off frequencies. What is the frequency range for single mode operation? If the guide is used at a frequency 1.25 the lowest cut-off frequency, determine the phase shift constant, the phase velocity, the guide wavelength, and the wave impedance in terms of free space TEM values. 3- A lossless rectangular wave guide has internal dimensions a = 0.02 m, and b = 0.01 m. The frequency is 10 GHz. What effective load impedance should it have to avoid reflections? If the load impedance is 250 , what is the VSWR?

4- A rectangular waveguide of inside dimensions 40x20 mm is to propagate TE10 mode of frequency 5 GHz. A dielectric of constant r = 3.0 fills the guide for z>0. with an air dielectric for z<0. Assuming the dielectric-filled part to be matched, find the reflection coefficient at z = 0 and the standing wave ratio in the air-filled part. Find the length and dielectric constant of a quarter wave matching section to be placed between the air and the given dielectric. 5-(a) Consider a WR-284 rectangular air waveguide with inner dimensions 7.214 cm X 3.404 cm to operate in the dominant mode at 2.45 GHz. Assuming the breakdown electric field of air to be 15 kV/cm (with a safety factor of 2 at sea level), calculate the maximum time-average power that can be carried by this guide. (b) Another waveguide (WR-430 air-filled, with inner dimensions 10.922 cm X 5.461 cm is proposed to increase the power-handling capability at 2.45 GHz. Calculate the maximum time- average power that can be carried by this guide and compare it with the previous one. 6- The inside dimensions of the WR-42 air-filled K-band waveguide are 10.7x4.3 mm. Determine the cut-off frequencies of the first two propagating modes. The recommended operating range for this guide is from 18.0 to 26.5 GHz. Determine the percentage reduction in bandwidth that this operating range represents relative to the theoretical band width for single mode operation. 7-a The electric field of the TE10 mode in a rectangular waveguide of dimensions a x b m is E(x,t) = Em sin(x/a) exp(jt – z) ay, write expressions for the magnetic field components 2 and show that the power transmitted by the guide is PT = (Em) ab/ 4 ZTE. -b Find the maximum power that can be carried by a 6-GHZ TE10 wave in an air-filled waveguide 40 mm wide and 20 mm high, taking the breakdown field in air at that frequency as 2000 kV/m. -c Assume the inner walls have a surface resistance Rs and use the magnetic field components to estimate the electric surface current density on the walls from J = n x H, where n is unit normal to the surface. Why can a narrow slot be cut along the centerline of the broad wall of the guide without perturbing the operation of the guide? (Such a slot is often used in a slotted line to sample the standing wave field inside the line.) 8-a Derive an approximate expression for the attenuation constant due to dielectric losses in a dielectric-filled waveguide from d = PLd/ 2 PT, where PLd is the power loss in the dielectric per unit length. -b Compute the TE10 mode attenuation, in dB/m, for K-band waveguide operating at 20.0GHz. The waveguide is made of brass ( = 2.564 x 107 S/m), and is filled with a dielectric material having r = 2.2 and tan = 0.002. 2 2 1/2 1/2 Note: c = PLc/ 2 PT = Rs ( 1 + (2b/a) (fc/f) ) / b ZTE (1 – (fc/f) ) Np/m, Rs = (/2) 9- For f = 3 GHz, design a rectangular waveguide with copper conductor and air dielectric so that the TE10 wave will propagate with a 30% safety factor (f = 1.3 fc) but also so that the wave type with next higher cut-off frequency will be 20% below its cut-off frequency. Calculate the attenuation due to copper losses in decibels per meter. Note: cu = 5.813x107 S/m, 2 2 1/2 1/2 c = PLc/ 2 PT = Rs ( 1 + (2b/a) (fc/f) ) / b ZTE (1 – (fc/f) ) Np/m, and Rs = (/2) 10- A rectangular waveguide with internal dimensions 2.286 x 1.016 cm supporting TE10 mode at 5 GHz is filled with a dielectric of relative permittivity r . What are the limits of r if only the dominant TE10 mode is to propagate? If r = 2.25 and the loss tangent tan = -3 10 , calculate g, vph, and the attenuation factor due to dielectric losses d. 11- Show that the power transmitted by a rectangular waveguide in the TE10 mode is guide is 2 PT = (Em) ab/ 4 ZTE A rectangular guide with inner dimensions 2.286 x 1.012 cm is operating at 9.4 GHz in the dominant TE10 mode with incident power 15 mW. Calculate the real components of the electromagnetic fields.

12-Starting from Ey = Em sin(x/a) exp(-j z) for the y component of the electric field inside a rectangular waveguide operating in the dominant TE10 mode and of cross-sectional dimensions a x b m, derive expressions for the other electric and magnetic field components and the power transmitted through the guide. The WR-90 X-band single mode air-dielectric rectangular copper waveguide has inside dimensions 22.86  10.16 mm. The guide is transmitting 15 mW at a frequency 9.4 GHz. Determine the phase shift constant, the wavelength, the wave impedance, and the phase and group velocites inside the guide. Determine also the electric surface current density on each of the inner walls of the waveguide. (2005) 13- The WR-90 X-band single mode air-dielectric rectangular copper waveguide has inside dimensions 22.86  10.16 mm. Determine the frequency band for single mode operation. If the guide is transmitting 15 mW at a frequency 9.4 GHz, determine the phase shift constant, the wavelength, the wave impedance, and the phase and group velocities inside the guide. Determine also the peak electric field inside the guide. (2006)

14- Why are microwave waveguides usually designed to support only a single mode? The inside dimensions of the WR-42 air-filled waveguide are a = 10.7mm and b = 4.3 mm. a-Determine the cut-off frequencies of the first two propagating modes, hence suggest, giving reasons, which range of frequencies would be most appropriately covered by this guide. b- For the TE10 mode at 20 GHz, determine the phase shift constant, the wavelength, the phase and group velocities, and the wave impedance inside the guide. (2007)

 jz 15- Starting from the expression Hz = A cos(x / a)e for the transverse axial magnetic field of the dominant mode inside a rectangular guide, derive expressions for the power transmitted by the mode and the attenuation due to dielectric losses. The WR-90 X-band single mode air-dielectric rectangular waveguide has inside dimensions 22.86  10.16 mm. If the guide is used to transport a 2 mW CW at a frequency 1.25 the cutoff frequency of the dominant mode, determine the maximum electric field inside the guide. Determine also the wavelength, the phase and group velocities, and the phase shift constant inside the guide. (2003)

 jz 16- Starting from the expression E = Em sin(x / a)e yˆ for the electric field of the dominant mode inside a rectangular guide, derive expressions for the magnetic field components and the power transmitted by the mode. The WR-90 X-band single mode air-dielectric rectangular waveguide has inside dimensions 22.86  10.16 mm. If the guide is used to transmit a CW at a frequency 1.25 the cutoff frequency of the dominant mode, determine the wavelength, the phase and group velocities, and the phase shift constant inside the guide. Show that the group velocity increases as the operating frequency increases. Could you explain why? What is the maximum power that can be carried by this guide if the breakdown field in air at the given frequency is 1500 kV/m? (2004)

IV- RESONATORS and MATCHING SECTIONS

IV-1 Questions with short answers

Q.1 Define Q of a resonator. How is it related to the stored energy and the dissipated power? Q.2 What is the loaded Q of a resonator connected to a load R? Q.3 What is the reactance of a short length (d<<) of a lossless short-circuited transmission line? Q.4 What is the reactance of a short length (d<<) of a lossless open-circuited transmission line? Q.5 A short-circuited transmission line section of length /4 is equivalent to parallel or series resonant circuit? What is the answer if the length is /2? Q.6 An open-circuited transmission line section of length /4 is equivalent to parallel or series resonant circuit? What is the answer if the length is /2? Q.7 Which type of coaxial transmission line sections would you choose to use as a resonator or as a stub line: short circuited section or open circuited section? Q.8 Which type of microstrip transmission line sections would you choose to use as a resonator or as a stub line: short circuited section or open circuited section? Q.9 What are cavity resonators? What applications do they have? Q.10 What two variables determine the primary frequency of a resonant cavity? Q.11 Energy can be inserted or removed from a cavity by what three methods? Q.12 Where should a small loop be placed to couple to the rectangular cavity TE101 mode? The loop is formed by extending the inner conductor of a coaxial cable feed, bending it into a loop, and connecting to the inside of the cavity. Q.13 Inductive tuning of a resonant cavity is accomplished by placing a nonmagnetic slug in what area? Q.14 Which one of the following cavity tuning components is not easily adjustable: screw, iris, or plunger? Q.15 What is a /4 transformer? What applications does it have? Q.16 What is a /2 transformer? What applications does it have? Q.17 What is a stepped transformer? What are its advantages over ordinary /4 transformers? Q.18 When stubs are used to match a reactive load, would there be no reflections on all line sections? Q.19 Why is it better to place stub matching sections close to the load? Q20. A microstrip cavity is made of a conducting patch over a dielectric substrate with a ground plate under the substrate. What are the most likely causes of losses in this structure?

IV-2 Problems 1- A transmission line resonator is fabricated from a quarter wavelength short-circuited lossy line. Find the Q of this resonator in terms of the constants and of the line. 2- A transmission line resonator is fabricated from a quarter wavelength open-circuited lossy line. Find the Q of this resonator in terms of the constants and of the line. 3- A transmission line resonator is fabricated from a half wavelength short-circuited lossy line. Find the Q of this resonator in terms of the constants and of the line. 4- A transmission line resonator is fabricated from a half wavelength open-circuited lossy line. Find the Q of this resonator in terms of the constants and of the line. 5- A resonator is constructed from a 3.0 cm length of a 100 air-filled coaxial line, shorted at one end and terminated with a capacitor at the other end. Determine the capacitor value to achieve the lowest order resonance at 6.0 GHz. If a 10,000 resistor is connected in parallel with the capacitor, find Q of the resonator. 6- A transmission line has the following unit length parameters: L = 0.2 H/m, C = 300 pF/m, R = 5 /m, G = 0.01 S/m. Calculate the propagation constant ( = + j) and the characteristic impedance of the line at 500 MHz. If a half wavelength short-circuited section of this line is used as a UHF resonator find Q of this resonator. 7-A rectangular waveguide cavity is made from a piece of copper WR-187 H-band waveguide, with a = 47.55 mm and b = 22.15 mm. The cavity is filled with polyethylene (r = 2.25, tan = 0.0004). If resonance is to occur at f = 5 GHz, find the required length, d, and the resulting Q for the  =1 and the  = 2 resonant modes. 8- Consider matching a load impedance ZL = 60 – j80 to a 50 coaxial line using a shunt 50 short circuited stub. Determine the length and the location, nearest to the load, of the stub. What will be the reflection coefficients: on the stub, on the line section near the load, and on the line section on the other side of the stub?

9- A 50 transmission line is matched to a 10V source and feeds a load ZL = 150 – j 50 . The line is 0.7 long. Find the magnitude and phase of the voltage reflection coefficient at the load, the VSWR on the line, and the values of the minimum and maximum impedances on the line. Consider matching the load to the line using: i) series 50 short circuited stub, ii) parallel 50 short circuited stub. Determine the length and the location, nearest to the load, of the stub in each case. Find the reflection coefficient on each line section. 10- Discuss the types of losses and dispersion that may occur in RF transmission lines and microwave waveguides. Derive expressions for and for a low-loss transmission line in terms of the line parameters L, C, R, and G. A transmission line resonator is fabricated from a quarter wavelength open-circuited line. Find Q of this resonator in terms of the line parameters. (2004) 11- Derive expressions for the attenuation constant and the phase shift constant of a low-loss transmission line in terms of the line parameters R, L, G, C. The coaxial line RG58B/U has a copper inner conductor of diameter 0.813 mm. The dielectric filling the space between the inner and outer conductor is polyethylene of outer diameter 2.95 mm. If a section of this line is used as a half-wavelength open-circuited resonator, find the length in mm, Q, and the bandwidth of this resonator. [For a coaxial line: L = (/2) ln(b/a), C = (2) / ln(b/a), R = (Rs/2) (1/a + 1/b), 1/2 G = (2 ”) / ln(b/a), and Rs= (f /) .] (2005,2006) 12- If the TE101 mode is to be excited in a rectangular cavity by a small monopole antenna that is the extension of the center conductor of a coaxial line, where should the antenna be placed? Find the resonant frequency for the TE101 cavity mode if a = 2b = d = 0.04 m and the cavity is air filled. What would be the resonant frequency if the cavity is filled with Teflon (r =2.08)? Comment on the result. (2001) 13- Where should a small loop be placed to couple to the rectangular cavity TE101 mode? The loop is formed by extending the inner conductor of a coaxial cable feed, bending it into a loop, and connecting to the inside of the cavity. Show that for a cubical cavity (a=b=d) in the TE101 mode, the unloaded Q is given by: Q = 2V/S, where V = a3 is the volume of the cavity, S = 6a2 is the surface area, and V is the skin depth. (2000) 14- Consider matching a load impedance ZL = 60 – j80 to a 50 coaxial line using a shunt 50 short circuited stub. Determine the length and the location, nearest to the load, of the stub. What will be the reflection coefficients: on the stub, on the line section near the load, and on the line section on the other side of the stub?.

15- A 50 transmission line is connected between a 10 V source of internal impedance 50 and a load ZL = 150 – j 50 . The line is 0.7 long. a) Find the magnitude and phase of the voltage reflection coefficient at the load, the VSWR on the line, and the values of the minimum and maximum impedances on the line. b) Consider matching the load to the line using: i) series 50 short circuited stub, ii) parallel 50 short circuited stub. Determine the length and the location, nearest to the load, of the stub in each case. What will be the reflection coefficient on each line section in each case? (2004) 16-A transmission line of characteristic impedance 50 is terminated into a load 75 + j 50 . The line is of length 0.60 wavelength at the operating frequency. The voltage across the input terminals is 10 V. Find the voltage standing wave ratio on the line, the input admittance, the input current, and the power through the load. If a 50 short cicuited stub is used to match the load to the line, find the length of the stub and the distance between the load and the stub. (2006) 17-What is a quarter-wavelength transformer? When is it used? Consider matching a load impedance ZL = 60 – j80 to a 50 coaxial line using a shunt 50 short circuited stub. Determine the length and the location, nearest to the load, of the stub. What will be the reflection coefficients: on the stub, on the line section near the load, and on the line section on the other side of the stub? (2007)

V- MICROWAVE PASSIVE COMPONENTS

V-1 Questions with short answers Q.1 What are the two basic types of T junctions? Q.2 Why is the H-type T junction so named? Q.3 Why is the E-type T junction so named? Q.4 The magic-T is composed of what two basic types of T junctions? Q.5 What are the primary disadvantages of the magic-T? Q.6 What is the primary purpose of a directional coupler? Q.7 How far apart are the two holes in a simple directional coupler? Q.8 What is the purpose of the absorbent material in a directional coupler? Q.9 For a directional coupler, define the terms: directivity, coupling, isolation, and throughput. Q.10 Hybrid rings are used primarily for what purpose?