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# FIELD AND WAVES I ECSE 2100

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This 9 page Class Notes was uploaded by Immanuel Brakus PhD on Monday October 19, 2015. The Class Notes belongs to ECSE 2100 at Rensselaer Polytechnic Institute taught by Kenneth Connor in Fall. Since its upload, it has received 20 views. For similar materials see /class/224761/ecse-2100-rensselaer-polytechnic-institute in ELECTRICAL AND COMPUTER ENGINEERING at Rensselaer Polytechnic Institute.

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Date Created: 10/19/15

Fields and Waves Fall 2009 Homework 1 Due 9 September 2009 at 600 pm 1 Waves and Phasor Notation Be sure that you read the following questions carefully Also when you convert an expression from phasor to time domain form or vice versa convert it back carefully following the rules to check your answer J77 a Write the following voltage phasor expression in time domain form 100e 4 Note that this is just a voltage not a voltage wave Assume f 30MH2 b Write the following time domain current wave expression in phasor at formizt a s1n a2t j a c Is the expression in part b a standing or traveling wave Ifit is a traveling wave what direction does it travel and what is its velocity 2 Plane Wave Representations The numbers given in this problem are realistic but not necessarily real That is your answers should come out in a reasonable range but the numbers are not based on a real commercially available transmission line The current on a transmission line is given byiz t cos77r106t 003712 a Is this a standing wave or a traveling wave If it is a traveling wave what direction does it travel and what is its velocity What is the period of this wave T What is the wavelength xl Plot this expression as a function of space at t 0 tT3 t2T3 using Maple or Matlab or some similar program Note that the wave propagates to the left order is blue red green 09 1 K A Connor Rensselaer Polytechnic Institute September 2009 Fields and Waves Fall 2009 Homework 1 d Write this current expression in phasor form e Assume that the transmission line has a capacitance per unit length of43p Find the characteristic impedance of the line Z0 and then the voltage on the line in phasor form 3 PSpice Simulated Experiments PSpice can show us a lot about how the voltages look at the input and output ends of transmission lines The simulation can in effect replace an experiment by giving us essentially the same results a To get some practice using PSpice set up the following simulation using the parameters of your transmission line from problem 2 That is the source line and load impedances should all be equal to the characteristic impedance of the line you determined in problem 2 The source voltage and frequency should be selected to obtain the current given in problem 2 on the line Set the offset voltage to zero Use the lossless line model and assume the length of the line is 40meters Be careful to use the zero ground since it is the only one that works with PSpice V OFF o 1 R2 VAMPL 7 1 7 REQ 7 T0 For a line that is 40m long the time delay neededfor T1 will be length 40 7 TD gammy Asmos 17143x10 Run the simulation shown below and display your results The numbers given are approximate You may want to change them some to get a better display To see the results better the time range should be reduced However this is not a critical issue Simulation Settings test E General Analysis llncludeFIlesl lelallESl Stimulus Dptmns neural an Fmberndcwl Analysis type T mwmmmamm j Hunluhms 1 second 115mm DD an Slailsavmg data altar lam mend Transientupkluns Makimumstapslze 1m 39 secnnds r amp the lhltlal tianslenx blas palnl calculation SKIPBP C as on llload Blas Pm Dulpul Fae Gullah l J g 2 K A Connor Rensselaer Polytechnic Institute September 2009 Fields and Waves Fall 2009 Homework 1 mm b Explain why your result makes sense Be sure to thoroughly label your plot The green sine wave is the input and the red sine wave is the output The line is lossless so the input and output should have the same amplitude The 40 meter line produces a time delay of 71ns which is exactly what is shown Note that to obtain the specified voltage on the line of 00V the voltage source had to be 200V as shown below R1 VOFF 0 100 T1 VAMPL 200 v1 FREQ 35e6 lt R2 quot 100 TO K A Connor 3 Rensselaer Polytechnic Institute September 2009 Fields and Waves Fall 2009 Homework 1 4 Lumped Transmission Lines A transmission line can be replaced by a series of lumped circuit elements under certain conditions In this exercise you will compare the response of such a lumped model line with the coil of coax Obtain a coil of coax a 50 Ohm terminator and one of the lumped component transmission lines from a TA Note we will be analyzing the lumped configuration using PSpice in lecture Thus your measurements should be guided by what we obtain there Provide the number of your coax coil and lumped model below Coax 39 LumpedModel s a First we will repeat an experiment from the first studio session at a different frequency Put the 50 Ohm terminator across the output of the coaxial cable and simultaneously measure input and output signals on the oscilloscope Set the input voltage at lep with a frequency of 3MHZ Measure the time delay between the signals What else is different about the input and output voltages b Replace the coaxial cable with the lumped version and repeat the experiment c You should have observed that the lumped line behaves in a qualitatively similar manner to the spool of coax Thus it must represent a similar length of line We want to determine the actual length of line Each L C combination represents some equivalent length of line Since there are 20 such combinations we only need to figure out what length each combo represents and multiply it by 20 To understand better how the lumped circuit is con gured look at the diagram below done with PSpice This diagram has a load impedance R2 of 93 Ohms which is not what we are using here Note that the inductance and capacitance for each section is indicated Given your knowledge of the actual capacitance and inductance per unit length for the RG5 8U coaxial cable approximately what length does each section represent 4 K A Connor Rensselaer Polytechnic Institute September 2009 Fields d D quot1 and Waves Homework 1 Remove the load resistor and replace it with a short circuit Measure the voltage at the 11111 node Adjust the frequency somewhere between MHz and 4MHz until the voltage at the 11111 node is a minimum It should be somewhere around 100mV Put the terminating resistor back on as a load Measure the voltage at each of the nodes for the case where the lumped line is terminated with 50 Ohms Remove the resistor and repeat for the short circuit Where are the minimas and maximas located One minimum should be at node 11 Plot your results for the measured voltages as a function of distance using the distance between nodes that you determined above Plot your results again but now in terms of wavelength rather than meters You can use the same plot if you wish and provide two sets of labels Show that the maXimas and minimas are located where they should be in terms of wavelength Run the corresponding case using the simulation at BesserNet mentioned in lecture Note that the lumped lines work slightly di Qrently if di erent capacitors are used Thus please note the color of the capacitors on the board you are using The correct answer to part d will depend on what capacitors are used on your board Before running the experiment we can first simulate it using PSpice 2 1 2 V 50 1uH 1111 1 1111 1 1w 1uH VOFF0 u C C3 C4 39 C5 VAMPL 1 390g 390131 390pF 390131 390131 FREQ 2 8M 9 0 L10 L9 L8 L7 L6 1 2 1 1 2 1uH 1uH Mi Mi 1uH C10 C9 C8 C7 C6 390131 7 390pF 390131 I 390131 7 390131 L11 L12 L13 L14 L1 1 2 1 Z 1 1 1UH w w 1uH m V c c quot c V 1 15 7 390131 390131 I 390131 7 390131 390131 L20 L19 L18 L17 L16 1 2 1 1 2 1 2 R4 IV 1111 1 1111 1 1w 1uH W b C20 C19 C18 C17 C16 1673 390131 I 390131 I 390131 T 390pF T 390pF K A Connor Rensselaer Polytechnlc Instltute Fall 2009 September 2009 Fields and Waves Fall 2009 Homework 1 For this case the signals that should be observed at the last several nodes are plotted below The thick green dashed line is for node 11 and is indeed a null The only other null observed is at the load end where there is a short circuit 5 a v tVlEH n x vVlEJX n We also should look at the BesserNet applet which shows the standing wave pattern with a null at the half wavelength distance from the shorted load The wavelength for this case should be given by the ratio of the velocity to the frequency or about 72 meters for the wavelength so the null should be located about 36 meters from the load or 9 nodes away at node 1 were we do indeedfind it BesserNetw aininy vml row 0 mil ou N511an ram m mmrr Reflectometer Calculator Type a value in one ofthe field below and mt 39enterquot Reflectlun Cuef cam l 10 0 awn i no lncidenlIRefl Wave Transmitted Wave Return Loss w m Mismatch Loss W A 6R ii i Z Standing Wave I Showtwo interfaces Resume Copyright2004E EIAl All Rights Reserved Free Applets atwwwbessernetcom 6 K A Connor Rensselaer Polytechnic Institute September 2009 Fields and Waves Homework 1 Fall 2009 Taking data from the arti cial line we see the following pattern in the voltages at each node 3 7 0 1 2 3 4 5 6 7 391011121314151617131920 The standing wave pattern is quite clear It appears that the next minimum is between nodes 1 and 2 and thus maybe the width of the half wavelength is a bit more than 36 meters These data were taken with a case where the minimum was achieved at 26MHZ the blue capacitors We can compare with an idealized PSpice calc too 8 6 7 8 Measured Psplce 91011121314151617181920 We probably cannot do much better than this but there is a systematic tilt to the data that may be due to a small resistance in the inductors and maybe even a little real conductance in the capacitors The impact of the scope is also not included K A Connor 7 Rensselaer Polytechnic Institute September 2009 Fields and Waves Fall 2009 Homework 1 llxm n E 0 V UQe V 1C 3OMH w ur C 1w 3km 7 x10 39 it 39QTrxw 7 ICe13 Reflectch E m m an x19f TE II mamp 3A Lac An ev w WK b Hana In saklwf j n Idmwtt391 A m E 39 1 I In 6 5 32 fl C bug w39u lm C TvowLKu V0 VL0LI39 1 3P 5 3 6 d 7 7 UnloMx 39 C u a shadw 7 mm age lww H m i Ev k wu k pM omsm 4 N n93 Al rad12M 8 K A Connor Rensselaer Polytechnic Institute September 2009 Fields and Waves Fall 2009 Homework 1 7 727 tilt 9m7 xlo f 1 moani i 7 TY E L39A V br vJoxm 74 i Awed w 7 7 7 7 7 7 7 76 7 7 17L LL77 77 77139Tm 2333 X106 7 if i h 7 F 7 00339quot V S 7 7 7 757 Penn 7T 7 55 Mo 7 q 39 zfa 116 H E77 12 H 7 7 P 53 5 3917 M7 7 7 C TM 3 we S NMK3M7 7 7 7 N bobbing 7 M7M 7 7 77 9 3 c0311 7 7 7 7 at L C7 7 7 quot 7 7 7 6 ED C K93 PFw 7 77 7 L7 7 77 7 7 71 W 7 7 7 1 t 7 7 7 7 7 20 cv 4Sxm WCI35Xlag i 6quot IOU JP 7 W 7 17772 7 IxL lmloiiH fth ii The current on the line is izt cos77r10 t 00372 which is phasor form is T e U 32 Thus the voltage on the line inphasor form is V ZDT 100e U 32 9 K A Connor Rensselaer Polytechnic Institute September 2009

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