1. Parallel-plate waveguide modes. A parallel-plate waveguide
consists of two perfectly conducting infinite plates spaced 2 cm apart
in air. Find the propagation constant γ for the TM0, TE1, TM1, TE2, and
TM2 modes at an operating frequency of (a) 5 GHz, (b) 10 GHz, and (c) 20
GHz. Get solution
2. Parallel-plate waveguide modes. A parallel-plate air waveguide has a plate separation of 6 mm and width of 10 cm. (a) List the cutoff frequencies of the seven lowest-order modes (TEm and TMm) that can propagate in this guide. (b) Find all the propagating modes (TEm and TMm) at 40 GHz. (c) Find all the propagating modes at 60 GHz. (d) Repeat part (c) if the waveguide is filled with polyethylene (assume it is lossless, with ∈r ≃ 2.25, μr = 1). Get solution
3. Parallel-plate waveguide modes. An air-filled parallel-plate waveguide with a plate separation of 1 cm is to be used to connect a 25-GHz microwave transmitter to an antenna. (a) Find all the propagating modes. (b) Repeat part (a) if the waveguide is filled with polyethylene (assume it is lossless, with ∈r ≃ 2.25, μr = 1). Get solution
4. VLF propagation in the Earth–ionosphere waveguide. The height of the terrestrial earth– ionosphere waveguide considered in Example varies for VLF (3–30 kHz) from 70 km during the day to about 90 km during the night.39 (a) Find the total number of propagating modes during the day at 12 kHz. (Assume both the ionosphere and the earth to be flat and perfect conductors.) (b) Repeat part (a) for during the night. (c) Find the total number of propagation modes during the day at 24 kHz. (d) Repeat part (c) for during the night. (e) Does the TM17 mode propagate during the day at 30 kHz? (f) Does the TM17 mode propagate during the night at 30 kHz?Example: ELF propagation in the Earth–ionosphere waveguide. Extremely low frequencies (ELF) are ideal for communicating with deeply submerged submarines, because below 1 kHz, electromagnetic waves penetrate into seawater.14 Propagation at these frequencies takes place in the “waveguide” formed between the earth and the ionosphere (Figure); low propagation losses allow nearly worldwide communication from a single ELF transmitter.In J. R. Wait’s simple model,15 the surface of the earth and the bottom of the ionosphere form the boundaries of a terrestrial “parallel”-plate waveguide with lossy walls. The ionosphere is approximated by an isotropic layer beginning at a given altitude and extending to infinity with no horizontal variations allowed. Energy is lost through the “walls” either into the finitely conducting ionosphere or into the ground, with the former loss being dominant. The important feature of propagation below 1 kHz is that there is a single propagating mode, a so-called quasi-TEM mode. All the other modes are evanescent and are almost undetectable at distances in excess of 1000 km. In the far field, the wave consists of a vertical electric field and a horizontal magnetic field transverse to the direction of propagation (Figure). The leakage of energy from this wave into the ocean (see Section 9.8) gives rise to a plane wave propagating vertically downward, and it is this signal that the submarine receiver detects.Consider an idealized earth–ionosphere waveguide where both the ionosphere and the earth are assumed to be perfect conductors. In addition, neglect the curvature of the waveguide and assume it to be flat. The height of the terrestrial waveguide can vary anywhere from 70 km to 90 km depending on conditions; for our purposes, assume it to be 80 km. Find all the propagating modes at an operating frequency of (a) 100 Hz, (b) 1 kHz, and (c) 10 kHz....Figure ELF propagation and submarine reception. Get solution
5. Waveguide in the earth’s crust. It has been proposed40 that radio waves may propagate in a waveguide deep in the earth’s crust, where the basement rock has very low conductivity and is sandwiched between the conductive layers near the surface and the high-temperature conductive layer far below the surface. The upper boundary of this waveguide is on the order of 1 km to several kilometers below the earth’s surface. The depth of the dielectric layer of the waveguide (the basement rock) can vary anywhere from 2 to 20 km, with a conductivity of 10−6 to 10−11 S-m−1 and a relative dielectric constant of ~6. This waveguide may be used for communication from a shore sending station to an underwater receiving station. Consider such a waveguide and assume it to be an ideal nonmagnetic parallel-plate waveguide. (a) If the depth of the dielectric layer of the guide is 2 km, find all the propagating modes of this guide below an operating frequency of 2 kHz. (b) Repeat part (a) (same depth) below 5 kHz. (c) Repeat parts (a) and (b) for a dielectric layer depth of 20 km. Get solution
6. Single-mode propagation. Consider a parallel-plate air waveguide with plate separation a. (a) Find the maximum plate separation amax that results in single-mode propagation along the guide at 10 GHz. (b) Repeat part (a) for the waveguide filled with a dielectric with∈r ≃ 2.54, μr = 1. Get solution
7. Evanescent wave attenuator design. A section of a parallel-plate air waveguide with a plate separation of 7.11 mm is constructed to be used at 15 GHz as an evanescent wave attenuator to attenuate all the modes except the TEM mode along the guide. Find the minimum length of this attenuator needed to attenuate each mode by at least 100 dB. Assume perfect conductor plates. Get solution
8. Evanescent wave filter design. Consider a parallel-plate air waveguide to be designed such that no other mode but the TEM mode will propagate along the guide at 6 GHz. If it is required by the design constraints that the lowest-order nonpropagating mode should face a minimum attenuation of 20 dB-(cm)−1 along the guide, (a) find the maximum plate separation amax of the guide needed to satisfy this criterion. (b) Using the maximum value of a found in part (a), find the dB-(cm)−1 attenuation experienced by the TE2 and the TM2 modes. Get solution
9. Cutoff and waveguide wavelengths. Consider a parallel-plate waveguide in air with a plate separation of 3 cm to be used at 8 GHz. (a) Determine the cutoff wavelengths of the three lowest-order nonpropagating TMm modes. (b) Determine the guide wavelengths of all the propagating TMm modes. Get solution
10. Guide wavelength of an unknown mode. The waveguide wavelength of a propagating mode along an air-filled parallel-plate waveguide at 15 GHz is found to be λ = 2.5 cm. (a) Find the cutoff frequency of this mode. (b) Recalculate the guide wavelength of this same mode for a waveguide filled with polyethylene (lossless, with ∈r ≃ 2.25, μr = 1). Get solution
11. Guide wavelength, phase velocity, and wave impedance. Consider a parallel-plate air waveguide having a plate separation of 5 cm. Find the following: (a) The cutoff frequencies of the TM0, TM1, and TM2 modes. (b) The phase velocity ῡpm , waveguide wavelength λm and wave impedance ZTMm of the above modes at 8 GHz. (c) The highest-order TMm mode that can propagate in this guide at 20 GHz. Get solution
12. Parallel-plate waveguide design. Design a parallel-plate air waveguide to operate at 5 GHz such that the cutoff frequency of the TE1 mode is at least 25% less than 5 GHz, the cutoff frequency of the TE2 mode is at least 25% greater than 5 GHz, and the power-carrying capability of the guide is maximized. Get solution
13. TEM decomposition of TEm modes. For an air-filled parallel-plate waveguide with 4 cm plate separation, find the oblique incidence angle θi and sketch TE1, TE2, TE3 , and TE4 modes in terms of two TEM waves at an operating frequency of 20 GHz. Get solution
14. Unknown waveguide mode. The electric field of a particular mode in a parallel-plate air waveguide with a plate separation of 2 cm is given by...(a) What is this mode? Is it a propagating or a nonpropagating mode? (b) What is the operating frequency? (c) What is the similar highest-order mode (TEm or TMm) that can propagate in this waveguide? Get solution
15. Unknown waveguide mode. The magnetic field of a particular mode in a parallel-plate air waveguide with a plate separation of 2.5 cm is given by...where x and y are both in meters. (a) What is this mode? Is it a propagating or a nonpropagating mode? (b) What is the operating frequency? (c) Find the corresponding electric field E(x, y). (d) Find the lowest-order similar mode (TEm or TMm) that does not propagate in this waveguide at the same operating frequency. Get solution
16. Maximum power capacity. (a) In Problem. determine the value of constant C1 (assumed to be real) which maximizes the power carried by all the propagating modes without causing any dielectric breakdown (use 15 kV-m−1 for maximum allowable electric field in air, which is half of the breakdown electric field in air at sea level). (b) Using the value of C1 found in part (a), find the maximum time-average power per unit width carried by the mode found in part (a) of Problem.ProblemUnknown waveguide mode. The magnetic field of a particular mode in a parallel-plate air waveguide with a plate separation of 2.5 cm is given by...where x and y are both in meters. (a) What is this mode? Is it a propagating or a nonpropagating mode? (b) What is the operating frequency? (c) Find the corresponding electric field E(x, y). (d) Find the lowest-order similar mode (TEm or TMm) that does not propagate in this waveguide at the same operating frequency Get solution
17. Power-handling capacity of a parallel-plate waveguide. A parallel-plate air waveguide with a plate separation of 1.5 cm is operated at a frequency of 15 GHz. Determine the maximum time-average power per unit guide width in units of kW-(cm)−1 that can be carried by the TE1 mode in this guide, using a breakdown strength of air of 15 kV-(cm)−1 (safety factor of approximately 2 to 1) at sea level. Get solution
18. Power capacity of a parallel-plate waveguide. Show that the maximum power-handling capability of a TMm mode propagating in a parallel-plate waveguide without dielectric breakdown is determined only by the longitudinal component of the electric field for fcm f √ 2fcm and by the transverse component of the electric field for f >√ 2fcm. Get solution
19. Power capacity of a parallel-plate waveguide. For a parallel-plate waveguide formed of two perfectly conducting plates separated by air at an operating frequency of f = 1.5fcm , find the maximum time-average power per unit area of the waveguide that can be carried without dielectric breakdown [use 15 kV-(cm)−1 for maximum allowable electric field in air, which is half of the breakdown electric field in air at sea level] for the following modes: (a) TEM, (b) TE1, and (c) TM1. Get solution
20. Attenuation in a parallel-plate waveguide. Consider a parallel-plate waveguide with plate separation a having a lossless dielectric medium with properties ∈ and μ. (a) Find the frequency in terms of the cutoff frequency (i.e., find f /fcm ) such that the attenuation constants αc due to conductor losses of the TEM and the TEm modes are equal. (b) Find the attenuation αc for the TEM, TEm, and TMm modes at that frequency. Get solution
21. Attenuation in a parallel-plate waveguide. For a TMm mode propagating in a parallel-plate waveguide, do the following: (a) Show that the attenuation constant αc due to conductor losses for the propagating TMm mode is given by...(b) Find the frequency in terms of fcm (i.e., find f /fcm ) such that the attenuation constant αc found in part (a) is minimum. (c) Find the minimum for αc . (d) For an air-filled waveguide made of copper plates 2.5 cm apart, find αc for TEM, TE1, and TM1 modes at the frequency found in part (b). Get solution
22. Parallel-plate waveguide: phase velocity, wavelength, and attenuation. A parallel-plate waveguide is formed by two parallel brass plates (σ = 2.56 × 107 S-m−1, ∈r = 1, μr = 1) separated by a 1.6-cm thick polyethylene slab (∈′r ≃ 2.25, tan δ≃ 4 × 10−4, μr = 1) to operate at a frequency of 10 GHz. For TEM, TE1, and TM1 modes, find (a) the phase velocity ῡp and the waveguide wavelength λ and (b) the attenuation constants αc due to conductor losses and αd due to dielectric losses. Get solution
23. Losses in a parallel-plate waveguide. Consider TEM wave propagation in a parallel-plate waveguide. Although ideally the only nonzero wave components for this mode are Ex and Hy (as given by equation (10.17)), it is clear that there should be a small Ez component due to the finite (Js = n × H) current that flows in the conductors, as mentioned in Footnote 21 on page 834. Since Jz is equal and oppositely directed at the top (x = a) and the bottom (x = 0) conductors, we expect Ez to also have the same magnitude and the opposite sign at x = a and x = 0. Thus, it is reasonable to assume a linear variation between the two values of Ez as:...where K is a constant. Since we must have ∇ • E = 0 within the waveguide, we expect that the fields Ex and Hy may have to be modified to be consistent with this nonzero Ez . Find the modified expressions for Ex and Hy and determine all components of the time-average Poynting flux Sav at x = a/4 inside the waveguide. Get solution
24. Semi-infinite parallel-plate waveguide. Two perfectly conducting and infinitesimally thin sheets in air form a semi-infinite parallel-plate waveguide, with mouth in the plane z = 0 and sides parallel to the y-z plane as shown in Figure. Two perpendicularly polarized (i.e., electric field in the y direction) uniform plane waves 1 and 2 of equal strength are incident upon the mouth of the guide at angles θi as shown. The two waves are in phase, so their separate surfaces of constant phase intersect in lines lying in the y-z plane. The peak electric field strength for each wave is known to be 1 V-m−1. (a) Find an expression (in terms of θi) for the time-average power flowing down the inside of the guide, per unit meter of the guide in the y direction. (b) If the wavelength λ of the incident waves is such that sin θi = λ/(2a) = √ 3/2 and a = 1 cm, find the numerical value of the time-average power transmitted, per unit guide width in the y direction....Figure Semi-infinite parallel-plate waveguide. Problem. Get solution
25. Reflection and transmission in a parallel plate waveguide. The region z >0 in a parallelplate waveguide is filled with nonmagnetic dielectric material with permittivity ∈2 = 3∈0, as shown in Figure. (a) Assuming that the TM1 wave is incident from the left with an incident magnetic field intensity given by...determine the magnitude of the magnetic fields of the reflected and transmitted TM1 waves. (b) A fellow engineer claims that the TM1 wave can be completely transmitted (i.e., without any reflection) across the interface, as long as ∈2 has a specific value. Comment on whether the engineer is correct and if so, determine the value of ∈2 for which there is no reflection....Figure Parallel-plate waveguide with a dielectric. Problem. Get solution
26. Propagating modes in a dielectric slab waveguide. Consider a dielectric slab waveguide with thickness d and refractive indices of 1.5 (for the guide) and 1.48 (for the cladding). (a) Find all the propagating modes at air wavelength λ = 2μm if d = 5μm. (b) Repeat part (a) if d = 15μm. (c) Repeat part (a) if the refractive index of the cladding is 1.49. Get solution
27. Single TE1 mode dielectric slab design. Consider a dielectric slab waveguide with guide thickness d and refractive index 1.55 sandwiched between two cladding regions each with refractive index 1.5. (a) Design the guide such that only the TE1 mode is guided at air wavelength λ = 1μm. (b) Repeat part (a) for a cladding region with a refractive index of 1.53. Get solution
28. Millimeter-wave dielectric slab waveguide. Consider a dielectric slab waveguide with ∈d = 4∈0 and μd = μ0 surrounded by air to be used to guide millimeter waves. Find the guide thickness d such that only the TE1 mode can propagate at frequencies up to 300 GHz. Get solution
29. TE2 mode in a dielectric slab waveguide. Find the cutoff frequency for the TE2 mode in a dielectric slab waveguide with ∈ = 7∈0 and thickness 2 cm, which is embedded in another dielectric with ∈ = 3∈0. Assume nonmagnetic case. Get solution
30. Dielectric slab waveguide modes. A dielectric slab waveguide in air is used to guide electromagnetic energy along its axis. Assume that the slab is 2 cm in thickness, with ∈ = 5∈0 and μ = μ0. (a) Find all of the propagating modes for an operating frequency of 4 GHz and specify their cutoff frequencies. (b) Find αx (in np-m−1) and βx (in rad-m−1) at 8 GHz for each of the propagating modes. (c) Considering a ray theory analysis of the propagating modes, find the incidence angles θi of the component TEM waves within the slab for all of the propagating modes at 8 GHz. Get solution
31. Dielectric slab waveguide modes. Consider a dielectric slab waveguide surrounded by air made of a dielectric core material of d = 10 μm thickness with a refractive index of nd = 1.5 covered with a cladding material of refractive index nc = 1.45, which is assumed to be of infinite extent. (a) Find all the propagating modes at 1 μm air wavelength. (b) Determine the shortest wavelength allowed in the single-mode transmission. Get solution
32. Dielectric waveguide. Consider a nonmagnetic dielectric slab waveguide consisting of a slab with relative permittivity ∈1r = 2.19 surrounded by another dielectric with relative permittivity ∈2r = 2.13. The frequency of operation is 50 GHz. Determine the thickness of the dielectric slab if the lowest order TE mode propagates at a ray angle of θi = 85°. Get solution
33. Dielectric slab waveguide thickness. A dielectric slab waveguide is made of a dielectric core and cladding materials with refractive indices of nd = 1.5 and nc = 1, respectively. If the number of propagating modes is 100 at a free-space wavelength of 500 nm, calculate the thickness of the core material. Get solution
34. TEM decomposition of TMm modes. A dielectric slab waveguide is designed using a core dielectric material of refractive index nd= 3 and thickness 5 μm covered by a cladding dielectric material of refractive index nc = 2.5. Find all the propagating TMm modes and corresponding angles of incidence with which they are bouncing back and forth between the two boundaries at 3 μm air wavelength. Get solution
35. Dielectric above a ground plane. A planar perfect conductor of infinite dimensions is coated with a dielectric material (∈r = 5, μr = 1) of thickness 5.625 cm. (a) Find the cutoff frequencies of the first four TE and/or TM modes, and specify whether they are odd or even. (b) For an operating frequency of 1 GHz, find all of the propagating TE modes. (c) For each of the TE modes found in part (b), find the corresponding propagation constant β. Assume the medium above the coating to be free space. Get solution
36. Group velocity. Derive the expression (equation (10.73)) for the group velocity for Tem or TMm modes in a parallel-plate waveguide. Get solution
37. Group velocity. Consider the propagation in seawater (σ = 4 S/m, ∈ = 81∈0, μ = μ0) of a uniform plane wave signal consisting of the superposition two frequency components at 19 and 21 kHz. (a) Stating all assumptions, determine the phase and group velocities at 20 kHz. (b) Assuming that the two signals are in phase at position z0 = 0 and at time t0 = 0, determine the minimum distance z1 (and the time t1) at which the two signals are once again in phase, that is, have a phase difference of a multiple of 2π. Is the group velocity vg at 20 kHz equal to z1/t1? If not, why not? Get solution
38. Dielectric waveguide. Consider the TEm mode in a nonmagnetic dielectric slab waveguide consisting of a slab with thickness d and permittivity ∈d surrounded by air. (a) Show that the dispersion relation (i.e., β-ω relation) is given by...(b) For ∈d = 4∈0 and for the TE1 mode, find the value of d such that the waveguide phase velocity ῡp = ω/β is equal to the geometric mean of c = (μ0∈0)−1/2 and vpd= (μd∈d ) −1/2. Get solution
39. Dispersion in seawater. Reconsider Problem in view of your knowledge of group velocity. Assuming that the frequency dependence of the phase velocity is as given in Problem. derive an expression for the group velocity in seawater, and plot both vp and vg as a function of frequency between 0.5 and 2.5 kHz.ProblemDispersion in sea water. A uniform plane electromagnetic wave in free space propagates with the speed of light, namely, c ≃ 3 × 108 m-s−1. In a conducting medium, however, the velocity of propagation of a uniform plane wave depends on the signal frequency, leading to the “dispersion” of a signal consisting of a band of frequencies. (a) For sea water (σ = 4 S-m−1, ∈r = 81, and μr = 1), show that for frequencies much less than ~890 MHz, the velocity of propagation is approximately given by vp ≃ k1 √ f, where k1 is a constant. What is the value of k1? (b) Consider two different frequency components of a signal, one at 1 kHz, the other at 2 kHz. If these two signals propagate in the same direction in seawater and are in phase at z = 0, what is the phase delay (in degrees) between them (e.g., between their peak values) at a position 100 m away? Get solution
40. Group velocity in a plasma. A cold ionized gas consisting of equal numbers of electrons and protons behaves (see Section 11.1) as a medium with an effective permittivity ∈eff = ∈0(1 − ω2 p/ω2), where ωp = ... is known as the plasma frequency, with N being the volume density of free electrons, qe ≃ −1.6 × 10−19 C the electronic charge, and me ≃ 9.11 × 10−31 kg the electronic mass. (a) Derive the expression for the group velocity vg in this medium. (b) Evaluate the group velocity (and express it as a fraction of the speed of light in free space) for a 1 MHz radio signal propagating through the earth’s ionosphere, where N ≃ 1011 m−3. (c) Repeat part (b) for 100 kHz. Get solution
41. Group velocity in a dielectric slab waveguide. Derive an expression for the group velocity vg for the odd TMm modes in a dielectric slab waveguide of slab thickness d and permittivity ∈d. Get solution
2. Parallel-plate waveguide modes. A parallel-plate air waveguide has a plate separation of 6 mm and width of 10 cm. (a) List the cutoff frequencies of the seven lowest-order modes (TEm and TMm) that can propagate in this guide. (b) Find all the propagating modes (TEm and TMm) at 40 GHz. (c) Find all the propagating modes at 60 GHz. (d) Repeat part (c) if the waveguide is filled with polyethylene (assume it is lossless, with ∈r ≃ 2.25, μr = 1). Get solution
3. Parallel-plate waveguide modes. An air-filled parallel-plate waveguide with a plate separation of 1 cm is to be used to connect a 25-GHz microwave transmitter to an antenna. (a) Find all the propagating modes. (b) Repeat part (a) if the waveguide is filled with polyethylene (assume it is lossless, with ∈r ≃ 2.25, μr = 1). Get solution
4. VLF propagation in the Earth–ionosphere waveguide. The height of the terrestrial earth– ionosphere waveguide considered in Example varies for VLF (3–30 kHz) from 70 km during the day to about 90 km during the night.39 (a) Find the total number of propagating modes during the day at 12 kHz. (Assume both the ionosphere and the earth to be flat and perfect conductors.) (b) Repeat part (a) for during the night. (c) Find the total number of propagation modes during the day at 24 kHz. (d) Repeat part (c) for during the night. (e) Does the TM17 mode propagate during the day at 30 kHz? (f) Does the TM17 mode propagate during the night at 30 kHz?Example: ELF propagation in the Earth–ionosphere waveguide. Extremely low frequencies (ELF) are ideal for communicating with deeply submerged submarines, because below 1 kHz, electromagnetic waves penetrate into seawater.14 Propagation at these frequencies takes place in the “waveguide” formed between the earth and the ionosphere (Figure); low propagation losses allow nearly worldwide communication from a single ELF transmitter.In J. R. Wait’s simple model,15 the surface of the earth and the bottom of the ionosphere form the boundaries of a terrestrial “parallel”-plate waveguide with lossy walls. The ionosphere is approximated by an isotropic layer beginning at a given altitude and extending to infinity with no horizontal variations allowed. Energy is lost through the “walls” either into the finitely conducting ionosphere or into the ground, with the former loss being dominant. The important feature of propagation below 1 kHz is that there is a single propagating mode, a so-called quasi-TEM mode. All the other modes are evanescent and are almost undetectable at distances in excess of 1000 km. In the far field, the wave consists of a vertical electric field and a horizontal magnetic field transverse to the direction of propagation (Figure). The leakage of energy from this wave into the ocean (see Section 9.8) gives rise to a plane wave propagating vertically downward, and it is this signal that the submarine receiver detects.Consider an idealized earth–ionosphere waveguide where both the ionosphere and the earth are assumed to be perfect conductors. In addition, neglect the curvature of the waveguide and assume it to be flat. The height of the terrestrial waveguide can vary anywhere from 70 km to 90 km depending on conditions; for our purposes, assume it to be 80 km. Find all the propagating modes at an operating frequency of (a) 100 Hz, (b) 1 kHz, and (c) 10 kHz....Figure ELF propagation and submarine reception. Get solution
5. Waveguide in the earth’s crust. It has been proposed40 that radio waves may propagate in a waveguide deep in the earth’s crust, where the basement rock has very low conductivity and is sandwiched between the conductive layers near the surface and the high-temperature conductive layer far below the surface. The upper boundary of this waveguide is on the order of 1 km to several kilometers below the earth’s surface. The depth of the dielectric layer of the waveguide (the basement rock) can vary anywhere from 2 to 20 km, with a conductivity of 10−6 to 10−11 S-m−1 and a relative dielectric constant of ~6. This waveguide may be used for communication from a shore sending station to an underwater receiving station. Consider such a waveguide and assume it to be an ideal nonmagnetic parallel-plate waveguide. (a) If the depth of the dielectric layer of the guide is 2 km, find all the propagating modes of this guide below an operating frequency of 2 kHz. (b) Repeat part (a) (same depth) below 5 kHz. (c) Repeat parts (a) and (b) for a dielectric layer depth of 20 km. Get solution
6. Single-mode propagation. Consider a parallel-plate air waveguide with plate separation a. (a) Find the maximum plate separation amax that results in single-mode propagation along the guide at 10 GHz. (b) Repeat part (a) for the waveguide filled with a dielectric with∈r ≃ 2.54, μr = 1. Get solution
7. Evanescent wave attenuator design. A section of a parallel-plate air waveguide with a plate separation of 7.11 mm is constructed to be used at 15 GHz as an evanescent wave attenuator to attenuate all the modes except the TEM mode along the guide. Find the minimum length of this attenuator needed to attenuate each mode by at least 100 dB. Assume perfect conductor plates. Get solution
8. Evanescent wave filter design. Consider a parallel-plate air waveguide to be designed such that no other mode but the TEM mode will propagate along the guide at 6 GHz. If it is required by the design constraints that the lowest-order nonpropagating mode should face a minimum attenuation of 20 dB-(cm)−1 along the guide, (a) find the maximum plate separation amax of the guide needed to satisfy this criterion. (b) Using the maximum value of a found in part (a), find the dB-(cm)−1 attenuation experienced by the TE2 and the TM2 modes. Get solution
9. Cutoff and waveguide wavelengths. Consider a parallel-plate waveguide in air with a plate separation of 3 cm to be used at 8 GHz. (a) Determine the cutoff wavelengths of the three lowest-order nonpropagating TMm modes. (b) Determine the guide wavelengths of all the propagating TMm modes. Get solution
10. Guide wavelength of an unknown mode. The waveguide wavelength of a propagating mode along an air-filled parallel-plate waveguide at 15 GHz is found to be λ = 2.5 cm. (a) Find the cutoff frequency of this mode. (b) Recalculate the guide wavelength of this same mode for a waveguide filled with polyethylene (lossless, with ∈r ≃ 2.25, μr = 1). Get solution
11. Guide wavelength, phase velocity, and wave impedance. Consider a parallel-plate air waveguide having a plate separation of 5 cm. Find the following: (a) The cutoff frequencies of the TM0, TM1, and TM2 modes. (b) The phase velocity ῡpm , waveguide wavelength λm and wave impedance ZTMm of the above modes at 8 GHz. (c) The highest-order TMm mode that can propagate in this guide at 20 GHz. Get solution
12. Parallel-plate waveguide design. Design a parallel-plate air waveguide to operate at 5 GHz such that the cutoff frequency of the TE1 mode is at least 25% less than 5 GHz, the cutoff frequency of the TE2 mode is at least 25% greater than 5 GHz, and the power-carrying capability of the guide is maximized. Get solution
13. TEM decomposition of TEm modes. For an air-filled parallel-plate waveguide with 4 cm plate separation, find the oblique incidence angle θi and sketch TE1, TE2, TE3 , and TE4 modes in terms of two TEM waves at an operating frequency of 20 GHz. Get solution
14. Unknown waveguide mode. The electric field of a particular mode in a parallel-plate air waveguide with a plate separation of 2 cm is given by...(a) What is this mode? Is it a propagating or a nonpropagating mode? (b) What is the operating frequency? (c) What is the similar highest-order mode (TEm or TMm) that can propagate in this waveguide? Get solution
15. Unknown waveguide mode. The magnetic field of a particular mode in a parallel-plate air waveguide with a plate separation of 2.5 cm is given by...where x and y are both in meters. (a) What is this mode? Is it a propagating or a nonpropagating mode? (b) What is the operating frequency? (c) Find the corresponding electric field E(x, y). (d) Find the lowest-order similar mode (TEm or TMm) that does not propagate in this waveguide at the same operating frequency. Get solution
16. Maximum power capacity. (a) In Problem. determine the value of constant C1 (assumed to be real) which maximizes the power carried by all the propagating modes without causing any dielectric breakdown (use 15 kV-m−1 for maximum allowable electric field in air, which is half of the breakdown electric field in air at sea level). (b) Using the value of C1 found in part (a), find the maximum time-average power per unit width carried by the mode found in part (a) of Problem.ProblemUnknown waveguide mode. The magnetic field of a particular mode in a parallel-plate air waveguide with a plate separation of 2.5 cm is given by...where x and y are both in meters. (a) What is this mode? Is it a propagating or a nonpropagating mode? (b) What is the operating frequency? (c) Find the corresponding electric field E(x, y). (d) Find the lowest-order similar mode (TEm or TMm) that does not propagate in this waveguide at the same operating frequency Get solution
17. Power-handling capacity of a parallel-plate waveguide. A parallel-plate air waveguide with a plate separation of 1.5 cm is operated at a frequency of 15 GHz. Determine the maximum time-average power per unit guide width in units of kW-(cm)−1 that can be carried by the TE1 mode in this guide, using a breakdown strength of air of 15 kV-(cm)−1 (safety factor of approximately 2 to 1) at sea level. Get solution
18. Power capacity of a parallel-plate waveguide. Show that the maximum power-handling capability of a TMm mode propagating in a parallel-plate waveguide without dielectric breakdown is determined only by the longitudinal component of the electric field for fcm f √ 2fcm and by the transverse component of the electric field for f >√ 2fcm. Get solution
19. Power capacity of a parallel-plate waveguide. For a parallel-plate waveguide formed of two perfectly conducting plates separated by air at an operating frequency of f = 1.5fcm , find the maximum time-average power per unit area of the waveguide that can be carried without dielectric breakdown [use 15 kV-(cm)−1 for maximum allowable electric field in air, which is half of the breakdown electric field in air at sea level] for the following modes: (a) TEM, (b) TE1, and (c) TM1. Get solution
20. Attenuation in a parallel-plate waveguide. Consider a parallel-plate waveguide with plate separation a having a lossless dielectric medium with properties ∈ and μ. (a) Find the frequency in terms of the cutoff frequency (i.e., find f /fcm ) such that the attenuation constants αc due to conductor losses of the TEM and the TEm modes are equal. (b) Find the attenuation αc for the TEM, TEm, and TMm modes at that frequency. Get solution
21. Attenuation in a parallel-plate waveguide. For a TMm mode propagating in a parallel-plate waveguide, do the following: (a) Show that the attenuation constant αc due to conductor losses for the propagating TMm mode is given by...(b) Find the frequency in terms of fcm (i.e., find f /fcm ) such that the attenuation constant αc found in part (a) is minimum. (c) Find the minimum for αc . (d) For an air-filled waveguide made of copper plates 2.5 cm apart, find αc for TEM, TE1, and TM1 modes at the frequency found in part (b). Get solution
22. Parallel-plate waveguide: phase velocity, wavelength, and attenuation. A parallel-plate waveguide is formed by two parallel brass plates (σ = 2.56 × 107 S-m−1, ∈r = 1, μr = 1) separated by a 1.6-cm thick polyethylene slab (∈′r ≃ 2.25, tan δ≃ 4 × 10−4, μr = 1) to operate at a frequency of 10 GHz. For TEM, TE1, and TM1 modes, find (a) the phase velocity ῡp and the waveguide wavelength λ and (b) the attenuation constants αc due to conductor losses and αd due to dielectric losses. Get solution
23. Losses in a parallel-plate waveguide. Consider TEM wave propagation in a parallel-plate waveguide. Although ideally the only nonzero wave components for this mode are Ex and Hy (as given by equation (10.17)), it is clear that there should be a small Ez component due to the finite (Js = n × H) current that flows in the conductors, as mentioned in Footnote 21 on page 834. Since Jz is equal and oppositely directed at the top (x = a) and the bottom (x = 0) conductors, we expect Ez to also have the same magnitude and the opposite sign at x = a and x = 0. Thus, it is reasonable to assume a linear variation between the two values of Ez as:...where K is a constant. Since we must have ∇ • E = 0 within the waveguide, we expect that the fields Ex and Hy may have to be modified to be consistent with this nonzero Ez . Find the modified expressions for Ex and Hy and determine all components of the time-average Poynting flux Sav at x = a/4 inside the waveguide. Get solution
24. Semi-infinite parallel-plate waveguide. Two perfectly conducting and infinitesimally thin sheets in air form a semi-infinite parallel-plate waveguide, with mouth in the plane z = 0 and sides parallel to the y-z plane as shown in Figure. Two perpendicularly polarized (i.e., electric field in the y direction) uniform plane waves 1 and 2 of equal strength are incident upon the mouth of the guide at angles θi as shown. The two waves are in phase, so their separate surfaces of constant phase intersect in lines lying in the y-z plane. The peak electric field strength for each wave is known to be 1 V-m−1. (a) Find an expression (in terms of θi) for the time-average power flowing down the inside of the guide, per unit meter of the guide in the y direction. (b) If the wavelength λ of the incident waves is such that sin θi = λ/(2a) = √ 3/2 and a = 1 cm, find the numerical value of the time-average power transmitted, per unit guide width in the y direction....Figure Semi-infinite parallel-plate waveguide. Problem. Get solution
25. Reflection and transmission in a parallel plate waveguide. The region z >0 in a parallelplate waveguide is filled with nonmagnetic dielectric material with permittivity ∈2 = 3∈0, as shown in Figure. (a) Assuming that the TM1 wave is incident from the left with an incident magnetic field intensity given by...determine the magnitude of the magnetic fields of the reflected and transmitted TM1 waves. (b) A fellow engineer claims that the TM1 wave can be completely transmitted (i.e., without any reflection) across the interface, as long as ∈2 has a specific value. Comment on whether the engineer is correct and if so, determine the value of ∈2 for which there is no reflection....Figure Parallel-plate waveguide with a dielectric. Problem. Get solution
26. Propagating modes in a dielectric slab waveguide. Consider a dielectric slab waveguide with thickness d and refractive indices of 1.5 (for the guide) and 1.48 (for the cladding). (a) Find all the propagating modes at air wavelength λ = 2μm if d = 5μm. (b) Repeat part (a) if d = 15μm. (c) Repeat part (a) if the refractive index of the cladding is 1.49. Get solution
27. Single TE1 mode dielectric slab design. Consider a dielectric slab waveguide with guide thickness d and refractive index 1.55 sandwiched between two cladding regions each with refractive index 1.5. (a) Design the guide such that only the TE1 mode is guided at air wavelength λ = 1μm. (b) Repeat part (a) for a cladding region with a refractive index of 1.53. Get solution
28. Millimeter-wave dielectric slab waveguide. Consider a dielectric slab waveguide with ∈d = 4∈0 and μd = μ0 surrounded by air to be used to guide millimeter waves. Find the guide thickness d such that only the TE1 mode can propagate at frequencies up to 300 GHz. Get solution
29. TE2 mode in a dielectric slab waveguide. Find the cutoff frequency for the TE2 mode in a dielectric slab waveguide with ∈ = 7∈0 and thickness 2 cm, which is embedded in another dielectric with ∈ = 3∈0. Assume nonmagnetic case. Get solution
30. Dielectric slab waveguide modes. A dielectric slab waveguide in air is used to guide electromagnetic energy along its axis. Assume that the slab is 2 cm in thickness, with ∈ = 5∈0 and μ = μ0. (a) Find all of the propagating modes for an operating frequency of 4 GHz and specify their cutoff frequencies. (b) Find αx (in np-m−1) and βx (in rad-m−1) at 8 GHz for each of the propagating modes. (c) Considering a ray theory analysis of the propagating modes, find the incidence angles θi of the component TEM waves within the slab for all of the propagating modes at 8 GHz. Get solution
31. Dielectric slab waveguide modes. Consider a dielectric slab waveguide surrounded by air made of a dielectric core material of d = 10 μm thickness with a refractive index of nd = 1.5 covered with a cladding material of refractive index nc = 1.45, which is assumed to be of infinite extent. (a) Find all the propagating modes at 1 μm air wavelength. (b) Determine the shortest wavelength allowed in the single-mode transmission. Get solution
32. Dielectric waveguide. Consider a nonmagnetic dielectric slab waveguide consisting of a slab with relative permittivity ∈1r = 2.19 surrounded by another dielectric with relative permittivity ∈2r = 2.13. The frequency of operation is 50 GHz. Determine the thickness of the dielectric slab if the lowest order TE mode propagates at a ray angle of θi = 85°. Get solution
33. Dielectric slab waveguide thickness. A dielectric slab waveguide is made of a dielectric core and cladding materials with refractive indices of nd = 1.5 and nc = 1, respectively. If the number of propagating modes is 100 at a free-space wavelength of 500 nm, calculate the thickness of the core material. Get solution
34. TEM decomposition of TMm modes. A dielectric slab waveguide is designed using a core dielectric material of refractive index nd= 3 and thickness 5 μm covered by a cladding dielectric material of refractive index nc = 2.5. Find all the propagating TMm modes and corresponding angles of incidence with which they are bouncing back and forth between the two boundaries at 3 μm air wavelength. Get solution
35. Dielectric above a ground plane. A planar perfect conductor of infinite dimensions is coated with a dielectric material (∈r = 5, μr = 1) of thickness 5.625 cm. (a) Find the cutoff frequencies of the first four TE and/or TM modes, and specify whether they are odd or even. (b) For an operating frequency of 1 GHz, find all of the propagating TE modes. (c) For each of the TE modes found in part (b), find the corresponding propagation constant β. Assume the medium above the coating to be free space. Get solution
36. Group velocity. Derive the expression (equation (10.73)) for the group velocity for Tem or TMm modes in a parallel-plate waveguide. Get solution
37. Group velocity. Consider the propagation in seawater (σ = 4 S/m, ∈ = 81∈0, μ = μ0) of a uniform plane wave signal consisting of the superposition two frequency components at 19 and 21 kHz. (a) Stating all assumptions, determine the phase and group velocities at 20 kHz. (b) Assuming that the two signals are in phase at position z0 = 0 and at time t0 = 0, determine the minimum distance z1 (and the time t1) at which the two signals are once again in phase, that is, have a phase difference of a multiple of 2π. Is the group velocity vg at 20 kHz equal to z1/t1? If not, why not? Get solution
38. Dielectric waveguide. Consider the TEm mode in a nonmagnetic dielectric slab waveguide consisting of a slab with thickness d and permittivity ∈d surrounded by air. (a) Show that the dispersion relation (i.e., β-ω relation) is given by...(b) For ∈d = 4∈0 and for the TE1 mode, find the value of d such that the waveguide phase velocity ῡp = ω/β is equal to the geometric mean of c = (μ0∈0)−1/2 and vpd= (μd∈d ) −1/2. Get solution
39. Dispersion in seawater. Reconsider Problem in view of your knowledge of group velocity. Assuming that the frequency dependence of the phase velocity is as given in Problem. derive an expression for the group velocity in seawater, and plot both vp and vg as a function of frequency between 0.5 and 2.5 kHz.ProblemDispersion in sea water. A uniform plane electromagnetic wave in free space propagates with the speed of light, namely, c ≃ 3 × 108 m-s−1. In a conducting medium, however, the velocity of propagation of a uniform plane wave depends on the signal frequency, leading to the “dispersion” of a signal consisting of a band of frequencies. (a) For sea water (σ = 4 S-m−1, ∈r = 81, and μr = 1), show that for frequencies much less than ~890 MHz, the velocity of propagation is approximately given by vp ≃ k1 √ f, where k1 is a constant. What is the value of k1? (b) Consider two different frequency components of a signal, one at 1 kHz, the other at 2 kHz. If these two signals propagate in the same direction in seawater and are in phase at z = 0, what is the phase delay (in degrees) between them (e.g., between their peak values) at a position 100 m away? Get solution
40. Group velocity in a plasma. A cold ionized gas consisting of equal numbers of electrons and protons behaves (see Section 11.1) as a medium with an effective permittivity ∈eff = ∈0(1 − ω2 p/ω2), where ωp = ... is known as the plasma frequency, with N being the volume density of free electrons, qe ≃ −1.6 × 10−19 C the electronic charge, and me ≃ 9.11 × 10−31 kg the electronic mass. (a) Derive the expression for the group velocity vg in this medium. (b) Evaluate the group velocity (and express it as a fraction of the speed of light in free space) for a 1 MHz radio signal propagating through the earth’s ionosphere, where N ≃ 1011 m−3. (c) Repeat part (b) for 100 kHz. Get solution
41. Group velocity in a dielectric slab waveguide. Derive an expression for the group velocity vg for the odd TMm modes in a dielectric slab waveguide of slab thickness d and permittivity ∈d. Get solution
