Fields and Waves I Sping 005 Homewok 8 Tansmission Lines Due: 3 May 005. Multiple Choice (6) a) The SWR (standing wave atio): a) is a measue of the match between the souce impedance and line impedance b) may be equal to c) is a measue of the match between the load impedance and line impedance d) may be equal to 0 b) The impedance of a lossless tansmission line depends on a) mateial popeties b) voltage c) geomety d) cuent c) If the total voltage as a function of position on a tansmission line is: L = + e -jβz + + e jβz a) The load is an open cicuit b) The tansmission line is lossless c) Γ L = - d) SWR = (infinity) d) The chaacteistic impedance of typical cable T tansmission line is a) 93 Ω b) 50 Ω c) 75 Ω d) 300 Ω. Fo the following expessions state if the wave is a standing o taveling wave, explain. If it is a taveling wave give the diection. Fo eithe case give the wavelength. C o is an abitay constant. (0 Pts). a) C o sin(000t+π/)cos(3000x+4) Standing because the aguments depend only on t o x, not both. b) C o cos(4000t+375y+) Taveling because the agument depends on both t and y. The wave tavels in the negative y diection because the sign of the y tem is the same as the sign of the t tem. K. A. Conno 6 Apil 005 Rensselae Polytechnic Institute Page Toy,New Yok, USA
Fields and Waves I Sping 005 Homewok 8 3. u = = 8. x0 8 lc l. e 7 Zo = = = 40 c 375. e 0 A component on a pinted cicuit boad sends a 5 [], 0.5 [ns] pulse down a 0 [cm] long tansmission line. The cicuit is shown in the figue. Assume that the line is lossless. a) Daw a eflection (bouncing) diagam illustating the cicuit behavio. Include all appopiate numbes. Model the behavio until afte the signal gets to the load fo the thid time (i.e. 3 eflections off the load and off the geneato). ( pts) 0 40 0 40 Γ S = = 06. Γ L = = 0 + 40 0 + 40 oltage divide: 40 IN = 5 = 4 40 + 0 t +0.36 05. T +0.8-0.6 00. = = 55. x0 8 8. x0 t 0 3T -. 3T T T Path of the pulse T 4 T z=0 z=d b) Plot the voltages at the load and the input as a function of time. Be sue to include all numeical values on the axis. ( pts) At the output end, T T 3T 4T 5T 6T 7T K. A. Conno 6 Apil 005 Rensselae Polytechnic Institute Page Toy,New Yok, USA
Fields and Waves I Sping 005 Homewok 8 6.0 4.0.0 0 -.0 0s 0.5ns.0ns.5ns.0ns.5ns 3.0ns 3.5ns 4.0ns 4.5ns 5.0ns (T:B+) Time PSpice is used to plot this out just fo neatness. It can always be used to check esults fo homewok. At the input and output ends 4.0.0 0 SEL>> -.0 8.0 (R:) 4.0 0-4.0 0s 0.5ns.0ns.5ns.0ns.5ns 3.0ns 3.5ns 4.0ns 4.5ns 5.0ns (T:B+) Time K. A. Conno 6 Apil 005 Rensselae Polytechnic Institute Page 3 Toy,New Yok, USA
Fields and Waves I Sping 005 Homewok 8 Electostatics 4. Multiple Choice (6) a) The capacitance of two fixed conductos with fee space between them: a) inceases if the voltage diffeence between the conductos deceases b) inceases if the magnitude of the chage on the conductos inceases c) inceases if a dielectic mateial is placed between the conductos d) inceases if the stoed electic enegy W E inceases b) Which of the following ae always continuous acoss a mateial bounday? a) Tangential electic field, E t b) Nomal electic field, E n c) Tangential electic flux, D t d) oltage, c) Which of the following diffeential opeations esult in a scala ( A is an abitay vecto, f is an abitay scala) a) the divegence A b) the gadient f c) the cul A d) the l e) the Laplacian, f d) The field inside a cube with sides of length a is E = E a$ + E a$. The stoed enegy in this egion is: a) 3 ε( Ex + Ey) a b) 3 ε( Ex + Ey) a c) ε( Ex + Ey ) a d) 0 3 x x y y K. A. Conno 6 Apil 005 Rensselae Polytechnic Institute Page 4 Toy,New Yok, USA
Fields and Waves I Sping 005 Homewok 8 5. Use the finite diffeence appoximation to estimate the voltages at the fou points shown in the figue. (3 pts) 8. A. B. C. D -8 The voltage at each point the fist time though is equal to: A B C D = ( + 8+ 0+ 0)/ 4 = 5 = ( + 8 + 5 + 0)/ 4 = 6. 5 = ( 8 + 5 + 0)/ 4 =. 5 = ( 8 +. 5 + 6. 5)/ 4 =. 5 / 4 = 35. The voltage at each point the second time though is equal to: A B C D = ( + 8 + 6. 5 +. 5)/ 4 = 75. = ( + 8 + 7. 5 + 35. )/ 4 = 7.565 = ( 8 + 7. 5 + 35. )/ 4 = 3.565 = ( 8 + 7. 565 + 3565. )/ 4 =. 5 / 4 = 3. 783 The voltage at each point the nth time though is equal to: A B C D = 8 = 8 = 4 = 4 The voltage can also be guessed and checked. This same esult came fom a speadsheet. K. A. Conno 6 Apil 005 Rensselae Polytechnic Institute Page 5 Toy,New Yok, USA
Fields and Waves I Sping 005 Homewok 8 6. A coaxial cable is constucted using two insulating mateials. Thee is a voltage 0 on the inne conducto and the oute conducto is gounded. The chage pe unit length on the inne conducto is ρ l [C/m]. ε ε 0 (side view) a (end view) b c a) Detemine the electic field E in the egion a c (7pts) The electic flux density is D = ρ l π so the electic field is E whee ε = ε o ε = ε fo a<<b o b<<c, espectively. D ρl = = ε πε b) Detemine the voltage (potential), at all points in the same egion, a c (7 pts) a b ρ The voltage is a E d d a l ρl d ρl b c ( ) = = = ln + ln c c πε b πε π ε a ε b b ρ E d d l ρl d ρl b c ( ) = = = ln + ln c c fo a<<b πε b πε π ε ε b ρl E d d ρl c () = = = ln c fo b<<c c πε π ε c) Detemine the capacitance, C pe unit length of this cable? (7 pts) ρl b c ( a) = o = ln + ln so that the capacitance pe unit length is π ε a ε b C ρl b c π = ln + ln = which is exactly the same esult one l π ε a ε b b c ln + ln ε a ε b would get by taking the seies combination of the two cylindical capacitos that epesent each egion of insulato. K. A. Conno 6 Apil 005 Rensselae Polytechnic Institute Page 6 Toy,New Yok, USA
Fields and Waves I Sping 005 Homewok 8 Magnetic Fields 7. Multiple Choice (6) a) Which of the following is tue egading the magnetic popeties of mateials: a) elative pemeability can be less than b) elative pemeability can be geate than 0000 c) plastic makes an excellent coe fo a tansfome d) it is possible to make fogs levitate b) Fo Beakman s Moto: a) The field due to the cuent in the coil may be appoximated as a solenoid type of field b) The battey intenal esistance wasn t negligible in the cicuit analysis c) The movie bought teas to my eyes; I can t wait fo the sequel d) The oscilloscope impedance is appoximately the same magnitude as the esistive components in the cicuit (wie, pape clips, etc.) c) Fo a cuent caying wie with a unifom cuent density : a) B = 0 inside the wie b) B = 0 outside the wie c) H = 0 inside the wie d) H = 0 outside the wie d) Hysteesis: a) descibes the nonlinea elationship between H and B b) descibes the nonlinea elationship between E and D c) applies to feomagnetic mateials d) applies to dielectic mateials K. A. Conno 6 Apil 005 Rensselae Polytechnic Institute Page 7 Toy,New Yok, USA
Fields and Waves I Sping 005 Homewok 8 8. A cicula conducto of adius 0 = [cm] has an intenal field H = 04 ( a sina a cosa)a$ [A/m] whee a = φ π. Find the total cuent in the 0 conducto. (0 pts) The total cuent can be detemined fom the line integal of H aound the bounday of the conducto Iencl = H dl Fist evaluate H at the bounday o =. H 04 ( a sina a cosa)a 0 4 = $ = ( sin π cos π φ )a$ = 0 4 φ a $ φ Iencl = H dl = π 0 4 9. A tooidal magnet is constucted using a ectangula coss section tooidal coe aound which N tuns of wie caying a cuent I ae wound. The pemeability of the coe is µ>>µ 0. a) Find the magnetic field, B eveywhee? (6 pts) The magnetic field outside of the coe is zeo. Inside the coe, we can use Ampee s Law µ NI B dl = π Bφ = µ NI so that Bφ =. π b) Detemine the magnetic flux, ψ linked by the wies? (6 pts) µ NI b d µ NI b The magnetic flux ψ = B ds = h = π ln π a h a c) Detemine the total magnetic field enegy, W m stoed in the coe mateial (6 pts) b NI NI The magnetic field enegy is W B Hdv h d N I b a h m = = = µ µ π a π π π ln d) Detemine the inductance of the tooid, L fom you answe to (c) (6 pts) N I b The inductance detemined fom the enegy W = LI = µ a h m ln is π N b L = µ ln π a h K. A. Conno 6 Apil 005 Rensselae Polytechnic Institute Page 8 Toy,New Yok, USA
Fields and Waves I Sping 005 Homewok 8 EM Waves 0. Multiple Choice (6) a) A tansvese EM wave in fee space a) is taveling with the speed of light (c=3x0 8 m/s) b) has a magnetic field which is paallel to the diection of popagation c) has an intinsic impedance of 377 Ohms d) has electic and magnetic fields pependicula to each othe b) Complex pemittivity a) should be consideed when conductivity is finite in a dielectic b) is independent of fequency c) has only a eal pat if the medium is lossless d) is a function of skin depth c) A plane wave in fee space is nomally incident on a lossy dielectic mateial a) at the location of the bounday, the electic field in both egions must be the same b) at the location of the bounday, the magnetic field in both egions must be the same c) the magnitude of the eflection coefficient will be geate than d) a cuent density will be induced in the bulk of the mateial d) Polaization a) descibes the state (e.g. cicula, linea, etc) of the wave b) descibes the locus of the tip of E c) depends on the phase diffeence between two othogonal components of the E d) depends on the atio of the magnitudes of two othogonal components of the E. A plane wave is nomally incident on distilled wate (σ 0, ε =80). Find the eflection and tansmission coefficients. (6 pts) η η 0π 0π Γ= η = ηo = 0π η = ηo = = = 34. π = 4 η + η 80 80 894. η τ = η + η Γ= 80 =. 8 τ = = + 80 + 80 0. K. A. Conno 6 Apil 005 Rensselae Polytechnic Institute Page 9 Toy,New Yok, USA
Fields and Waves I Sping 005 Homewok 8. The electic field phaso of a unifom plane wave taveling in a lossless medium with an intinsic impedance of 88.5 [Ω] is given by ~ E e j 4π = 0 y a $ z [m/m]. Detemine a) The associated magnetic field phaso H ~ (6 pts) The magnitude of the magnetic field phaso ~ ~ H E e j 4π 0 y = = η 885. The wave popagates in the y-diection, which must be the diection of the coss poduct of E and H. E is in the z-diection which means that H is in the x-diection. Thus, ~ j4πy 0e H = a$ x 885. b) The instantaneous expession fo E if the medium is non-magnetic (µ=µ 0 ) (6 pts) E( y, t) = a$ 0cos( ωt βy) Fom the given expession, β = 4π = ω µ oεo ε. We z µ also know that η = o ε ε = 885. which gives us ε = 377 = 4. We then have 885. o ω 8 8 that 4π = ω µ oεo ε = so that ω = ( π) 3x0 = 6πx 0 30 x 8 c) The instantaneous expession fo H if the medium is non-magnetic (µ=µ 0 )(6 pts) 0 H( y, t) = a$ x cos( ωt βy) = a$ x 053. cos( ωt βy) 885. EXTRA CREDIT (0 pts) Fo a single unifom plane wave in a lossless medium, show that the aveage Poynting vecto S, equals the poduct of the aveage enegy density times the phase velocity. jβz + jβz * Eme Eme Em Save = Re( E H ) = = η η u = µε and the enegy density is * * wm = Re( D E ) + Re( B H ) = 4 4 poduct is equal to uwm = εe m = Em = µε µ ε Em η εe m so the K. A. Conno 6 Apil 005 Rensselae Polytechnic Institute Page 0 Toy,New Yok, USA