Basic Equations for Electromagnetics and ApplicationsMIT 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 Quiz 2, November 17, 2005 6.013 Formula Sheets attached. Problem 1 11meter2λ= 0250Z =Ω 01100Z=Ω ()()8100cos 10vt tπ=× 50sR =Ω ()it sjX + _ ()50 1LZj=+Ω 20.44λ= meter A transmission line system incorporates two transmission lines with characteristic impedances of as illustrated above. A voltage source is applied at the left end, . At this frequency, line 1 has length of 01 02100 and 50ZZ=Ω =Ω)(8( ) 100cos 10vt tπ=×11meter2λ= and line 2 has length of 20.4 meter4λ= , where 1and2λλ are the wavelengths along each respective transmission line. The two transmission lines are connected by a series reactance sjX and the end of line 2 is loaded by impedance ()50 1LZj=+Ω. The voltage source is connected to line 1 through a source resistance . 50sR =Ω a) What are the speeds and of electromagnetic waves on each line? 1c2c b) It is desired that sX be chosen so that the source current ()()80cos 10it I tπ=× is in phase with the voltage source. What is sX? c) For the value of sXin part (b), what is the peak amplitude0I of the source current ()?it Note that the value of sX itself is not needed to answer this question or part (. )dProblem 2 A parallel plate waveguide is to be designed so that only TEM modes can propagate in the frequency range 0 . The dielectric between the plates has a relative dielectric constant of 2 GHzf<<9rε= and a magnetic permeability of free space 0μ. a) What is the maximum allowed spacing between the parallel plate waveguide plates? maxd b) If the plate spacing is 2.1 cm, and f= 10 GHz, what modes will propagate? TE and TMnnProblem 3 0 + 100 Volts _ 0100 ,ZTc=Ω= 100sR =Ω 300LZ=Ω zSwitch opens at 0t = A transmission line of length , characteristic impedance 0100Z=Ω , and one-way time of flight Tc= is connected at to a 100 volt DC battery through a series source resistance and a switch . The end is loaded by a 3000z =100sR =Ω z = Ω resistor. a) The switch at the end has been closed for a very long time so that the system is in the DC steady state. What are the values of the positive and negative traveling wave voltage amplitudes0z =()()andVzct Vzct+−−+? Part b, on the next page, to be handed in with your exam. Put your name at the top of the next page.Name: b) With the system in the DC steady state, the switch is suddenly opened at time t . 0=)ct+−) i) Plot the positive and negative traveling wave voltage amplitudes, Vz and , as a function of ((Vzct−+zat time 2Tt . = ()/2VzcT+− ()/2VzcT−+ -100 -50 50 100 z -50 50 100 -50 50 100 -100 z ii) Plot the transmission line voltage (),vz as a function of tzat time 2.Tt = (),2Tvzt= -50 50100-100z Please tear out this page and hand in with your exam. Don’t forget to put your name at the top of this page.6.013 Quiz 2 Formula Sheet November 17, 2005 Cartesian Coordinates (x,y,z): ˆˆˆxy∂Ψ ∂Ψ ∂Ψ∇Ψ = + +∂∂∂xyzz yxzAAAAxy∂∂∂∇= + +∂∂∂iz yyzxzAAAAAAˆˆˆyz zx xy∂∂⎛⎞ ⎛∂∂∂⎛⎞∇× = − + − + −⎜⎟ ⎜⎜⎟∂∂ ∂∂ ∂∂⎝⎠⎝⎠ ⎝xyzxA⎞∂⎟⎠ 222222xyz∂Ψ ∂Ψ ∂Ψ∇Ψ= + +∂∂∂2 Cylindrical coordinates (r,φ,z): 1ˆrˆˆrrzφ∂Ψ ∂Ψ ∂Ψ∇Ψ = + +∂∂φz∂ ()rzArAA11Arr r zφ∂∂∂∇= + +∂∂φi∂ ()zrz rrzˆˆˆrAAAAA A111ˆAdˆˆrzzrrr rArAAφφφφ∂⎛⎞∂⎛⎞∂∂∂ ∂⎛⎞∇× = − + − + − = ∂ ∂ ∂ ∂φ ∂ ∂⎜⎟⎜⎟⎜⎟∂φ ∂ ∂ ∂ ∂ ∂φ⎝⎠⎝⎠ ⎝ ⎠rr zrzφetrz ()22222 211rrr rrz∂∂Ψ ∂Ψ∂Ψ∇Ψ= + +∂∂∂φ∂ Spherical coordinates (r,θ,φ): 11ˆˆˆrr rsin∂Ψ ∂Ψ ∂Ψ∇Ψ = + +∂∂θ θrθφ∂φ ()()2r2ArAsin A11 1Arrsin rsinrφθ∂∂∂θ∇= + +∂θ∂θθi∂φ ()()()rrsin A rArAAAA1111ˆˆAˆrsin rsin r r r rrθφφφθθ∂θ ∂⎛⎞⎛ ⎞⎛⎞∂∂∂∂∇× = − + − + −⎜⎟⎜⎟⎜ ⎟θ ∂θ ∂φ θ ∂φ ∂ ∂ ∂θ⎝⎠⎝⎠⎝ ⎠2rˆˆrsin ˆ1det rrsinArArsinAθφθθφ=∂∂∂∂θ∂∂φθθrr ()()22222 2211 1rsinrrrrsin rsin∂∂Ψ ∂ ∂Ψ ∂∇Ψ= + θ +∂∂ ∂θ ∂θ2Ψθθ∂φ Gauss’ Divergence Theorem: VAG dv G daˆ∇=∫∫iin Stokes’ Theorem: ()ACG da Gdˆ∇× =∫∫iinVector Algebra: xxyyzzˆˆˆ∇=∂∂+∂∂+∂∂ AB AB AB ABxx yy zz•=++ ()A0∇•∇× = ()()2AA∇× ∇× =∇ ∇• −∇ABasic Equations for Electromagnetics and ApplicationsFundamentals 1//E2//E0−= ()[]ofqEv HN=+×μ 1// 2//sHHJnˆ−=× EB∇× =−∂ ∂t0⊥⊥ BB12−= cAdEds Bdadt•=− •∫∫ 12sDD⊥⊥−=ρ HJ Dt∇× = +∂ ∂ 0 = if σ = ∞ cA AdHds Jda Ddadt•= •+ •∫∫ ∫ Electromagnetic Waves AVDDda∇• =ρ→ • = ρ∫∫ˆn 1 2 dv ()222t E 0 [Wave Eqn.]∇−με∂ ∂ = AB0 Bda0∇• = → • =∫ ()22 jkokE0, EEe−∇+ = =ir Jt∇• =−∂ρ ∂ k = ω(με)0.5 = ω/c = 2π/λ E = electric field (Vm-1) kx2 + ky2 + kz2 = ko2 = ω2με H = magnetic field (Am-1) vp = ω/k, vg = (∂k/∂ω)-1 D = electric displacement (Cm-2) θr = θi B = magnetic flux density (T) t i it itsin sin kk nnθθ= = Tesla (T) = Weber m-2 = 10,000 gauss ()1ctsin n n−θ=i ρ = charge density (Cm-3) ()0.51Btitan for TM−θ= εε J = current density (Am-2) xjkzzticEETe+α −θ>θ ⇒ = σ = conductivity (Siemens m-1) kk'jk'' =− sJ = surface current density (Am-1) T1Γ=− ρs = surface charge density (Cm-2) []()TEittiT21cos cos=+η θ η θ εo 8.854 × 10≈-12 Fm-1 []()TMttiiT21coscos=+η θη θ μo = 4π × 10-7 Hm-1Transmission Lines c
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