ECSE 352: Electromagnetic Waves

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Marvin Watts
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1 December 2008 Final Examination ECSE 352: Electromagnetic Waves 09:00 12:00, December 15, 2008 Examiner: Zetian Mi Associate Examiner: Andrew Kirk Student Name: McGill ID: Instructions: This is a CLOSED BOOK examination. Only the EQUATION SHEET provided may be used. STANDARD CALCULATOR permitted only. SPACE IS PROVIDED on the examination to answer all the questions, and SHOW ALL WORK to receive full credits The first section of this test consists of ten short questions, and the second section consists of four questions requiring calculation. You are required to answer all the questions in both sections I and II. Do not try to fully answer each question if it means that you will have insufficient time to attempt the others. The total number of marks for the test is 100. Questions in section 1 are worth a total of 30 points. Section 2 is worth a total of 70 points. The test consists of 8 pages (including this page), plus a formula sheet. Please ensure that you have a COMPLETE examination paper before starting. 1 of 8

2 Section I 1. The resistance of a transmission line typically increases as the operating frequency increases, please explain why. [3 points] When frequency increases, the sin depth decreases. This leads to a reduced effective area for transmission lines. Therefore, resistance increases. 2. What are the lowest TM and TE modes in a rectangular waveguide? [3 points] TM11, TE01 or TE10 (depending on the geometry) 3. What are the essential differences in the propagation characteristics of dielectric and conducting waveguides? [3 points] Dielectric: evanescent wave in the cladding region; The lowest mode has no cutoff; For a given design, the number of operating modes is limited. 4. Is it possible that lossy transmission lines, i.e., 0, can be distortionless? Please support your statement. [3 points] Yes. Heavyside s condition G/C=R/L no dispersion even for lossy lines. 5. Explain the concept of radiation resistance for an antenna. [3 points] The equivalent resistance that would transmit (or dissipate) the same amount of power. 2 of 8

3 6. Why does a rectangular waveguide not support the TEM mode? [3 points] Due to the two-dimensional confinement, E or H fields will always have a component along the propagation direction and therefore a rectangular waveguide can not support TEM modes. 7. A uniform plane wave is incident from air onto glass at an angle θ. The incident wave is left circularly polarized. It is determined that the reflection coefficients for s-polarized and p-polarized waves are and at this incident angle, respectively. What is the fraction of power that is reflected? [3 points] (0.222^ ^2)/2=3.5%. 8. Do signals become distorted as a result of transmission through a waveguide? Briefly explain why. [3 points] Yes. The group velocity is a function of frequency in waveguides. 9. A circularly polarized wave is incident on a dielectric medium at the Brewster angle. What are the polarization states of the reflected and transmitted waves? Please support your statement. [3 points]. Reflected: only s-wave linearly polarized. Transmitted: both s- and p-waves elliptically polarized. 10. Why is double stub matching superior to single stub matching [3 points]? No longer require precise positioning. 3 of 8

4 Section II 1. You are required to design an antireflection coating for a microscope objective lens which is made of glass with a relative dielectric constant of The lens is designed for use in air at normal incidence and you are told to design the coating for a center wavelength of 600 nm in air. a) What is the thickness and refractive index of the coating that you should use? [6] b) For the coating design in (a), please calculate the reflection coefficient for light which has a wavelength of 500 nm in air. [6] c) Why do antireflection coatings not work as well for light which is incident at an angle? [3] a) ; refractive index = 1.265; thickness: b) Γ Note that 500 nm is the wavelength in air not in the coating. c) s and p-polarization for oblique incidence have different reflection coefficients. Therefore, a coating that can give zero reflections for both waves may not be achieved. 4 of 8

5 2. The E-field of an electromagnetic wave traveling in an isotropic medium with an intrinsic impedance of Ω is represented by the equation below.,,, V/m. a) Calculate the frequency and wavelength in vacuum [5 points] b) What is the polarization state of the wave? [2 points] c) Calculate the average power density of the wave [5 points] d) Assume at certain position, the medium terminates with another medium whose intrinsic impedance is 200 Ω, and the wave is normally incident on the boundary. This leads to a partial standing wave in the incident medium. Please determine positions of the first maxima and minima that are closest to the boundary. [5 points] e) Assume at certain position, the medium terminates with air. It is also known the wave is incident on the boundary at an angle of 45, please determine the fraction of power that is reflected back to the medium. [3 points] a) b) Linear polarization c) / d) Since, 2 1,, 1, 0; 1 2 2,, 0, 0.205μ e) This incidence angle is above the critical angle. Therefore all the wave will be reflected. 5 of 8

6 3. An air-filled rectangular waveguide has dimensions 6 and 4. (a) Determine the frequency range in which the guide operates single mode. [4 points] (b) If the operating frequency is 3 GHz, calculate the phase and group velocities. [8 points] (c) If the operating frequency is 6 GHz, estimate the total number of modes in the waveguide. [3 points] a) First TE and TM modes have the same cutoff frequency b) Frequency range for single mode operation: 2.5 < f < 3.75 GHz It may be noted that the lowest TM mode is TM11 which has a higher cutoff frequency. c) First calculate the cutoff frequencies: 4.5 ; 6.25 ; 7.91 ; 5.0 The possible modes are TE01, 10, 11, 20, and TM11. There is no TM20 mode. 6 of 8

7 4. A 100 MHz generator with 10 and internal resistance 50 Ω is connected to a lossless 50 Ω air line that is 3.6 m long and terminated in a 25 + j25 Ω load. (a) Calculate the voltage standing wave ratio on the line. [5 points] (b) Calculate the current and voltage phasor at the input terminal. [8 points] (c) Calculate the average power delivered to the load. [4 points] In the case of a matched load (Z L = 50 Ω), what is the average power delivered to the load? [3 points] a) First calculate the reflection coefficient: Γ Then the standing wave ratio 2.62 b) First calculate the input impedance c) Average power: 0.2 For matched load, of 8

Waveguides. Metal Waveguides. Dielectric Waveguides

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