Problem 8.1MIT OpenCourseWare http://ocw.mit.edu 6.013/ESD.013J Electromagnetics and Applications, Fall 2005 Please use the following citation format: Markus Zahn, Erich Ippen, and David Staelin, 6.013/ESD.013J Electromagnetics and Applications, Fall 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/termsMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.013 Electromagnetics and Applications Problem Set #8 Issued: 11/1/05 Fall Term 2005 Due: 11/9/05 _____________________________________________________________________________________ Suggested Reading Assignment: 5.2.1, 5.2.2, 9.2 Quiz 2 will be on Thursday, November 17 at 10-11 a.m. It will cover material through P. S. #8,with a focus on sinusoidal steady state and transient waves on transmission lines; parallel plate,rectangular, and dielectric waveguides. Quiz 2 Formula Sheets (as attached to this problem set)will be provided. Problem 8.1 Switched transmission line systems with an initial dc voltage can be used to generate high voltage pulses of short time duration. The line shown is charged up to a dc voltage when at t=0 the load switch is closed and the source switch is opened. 0V Problem 8.10 in Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 1987. Used with permission.(a) What are the initial line voltage and current at t=0? What are V+ and V? −(b) Sketch the time dependence of the load voltage. Problem 8.2 A transmission line of length and characteristic impedance l0Z has a matched load at 0z=and is short circuited at z= l. The speed of electromagnetic waves on the transmission line is c so that the one-way transit time is T= l c. The transmission line is unexcited until at it is hit by lightning so that the line voltage and current at 0t =0t= are ()()0,0 0,00 0vzt V zizt z== <<== <<ll 0,ZTc=l 0Z z 0z = l 12a) What are Vt+()− zcand Vt−()+ zcat t = 0 ? b) Plot the voltage and current at z = 0 as a function of time. c) Plot the current at z = as a function of time. d) Plot the voltage and current as a function of zfor 0 < z < at timetT= 2 . Problem 8.3 An air-filled metal rectangular waveguide has cross-sectional area dimensions a =2 cm and b = 1 cm. a) What TEmn and TMmn mode have the lowest cut-off frequencies and what are these frequencies? b) Over what frequency range will this waveguide operate at a single mode? What is the mode? Problem 8.4 z d ε,μix y A parallel plate waveguide with spacingd , dielectric permittivity ε, and magnetic permeability μ supports TEnnand TM modes given by: TEnEs=−Ek0inzxzsin(ωtkx)iyEH =−0⎣⎡kηkzcos kzzcos(ωωt−kxixx) +kxsin kzzsin( t−kxixz)⎤⎦TMn EE =−0⎡sin sin(ωωt kxi) −⎣k kkzkzzxx xcos kzzcos( t−kxixz)EHc=−0oskcoωkxiηzxzs(t)ynπkkωε22zx==,μ−kdz⎤⎦a) What are the surface charge densities on the z=0 and z=d surfaces for modes? nnTE and TM b) What are the surface current densities on the z=0 and z=d surfaces for modes? nTE and TMn) c) Find the equation for the magnetic field lines that go through coordinates (00,xz for the modes at t=0 defined as: nTEzxHdzdx H= Hint: cot ln(sin ) Constantudu u=+∫ d) Find the equation for the electric field lines that go through coordinates()00,xz for the modes at t=0 defined as: nTMzxEdzdx E= Hint: tan ln(cos ) Constantudu u=− +∫ e) Extra Credit: Using your favorite drawing program, draw some field lines of parts (c) and (d) that illustrate the fundamental shape of the TE1 and TM1 modes.
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