ECE357H1S ELECTROMAGNETIC FIELDS TERM TEST 1. 8 February 2016, 19:00 20:00. Examiner: Prof. Sean V. Hum

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1 UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING The Edward S. Rogers Sr. Department of Electrical and Computer Engineering ECE57HS ELECTROMAGNETIC FIELDS TERM TEST 8 February 6, 9:00 :00 Examiner: Prof. Sean V. Hum NAME: STUDENT NUMBER: TOTAL POINTS: Include units in your answers. You can use your Cheng textbook. No other reference materials allowed. All programmable and non-programmable electronic calculators are permitted. You cannot use devices with wireless communications capabilities (e.g. smartphones) as your calculator. Only answers that are fully justified will be given full credit. Unless otherwise stated, assume that all materials are non-magnetic (µ r = ). GOOD LUCK!

2 USEFUL FORMULAE Refractive index of non-magnetic dielectrics: n = ɛ r Characteristic impedance Z 0 = R+jωL G+jωC Complex propagation constant γ = α + jβ = (R + jωl)(g + jωc) Speed of light in vacuum c = 8 m/s Input reflection coefficient Γ in (l) = Γ L e jβl e αl Load reflection coefficient Γ L = Z L Z 0 Z L +Z 0 VSWR S = + Γ L Γ L Quarter-wavelength transformer Z in Z L = Z 0 Admittance of an open stub Y in = jy 0 tan(βl) Impedance of a shorted stub Z in = jz 0 tan(βl) +Γ Impedance transformation Z in = Z in (l) 0 = Z Γ in (l) 0 Z L+Z 0 tanh γl Z 0 +Z L tanh γl ( Line voltage V (z ) = Z 0V g [ ] Z 0 +Z g e γz +Γ L e γz = I L (ZL + Z 0 )e γz + (Z L Z 0 )e γz Line current I(z ) = Γ gγ L e γl ) ( ) Vg Z 0 +Z g e γz Γ L e γz Γ gγ L e γl = I L Z 0 [ (ZL + Z 0 )e γz (Z L Z 0 )e γz ]

3 ECE57 Midterm Page PROBLEM #. [ POINTS] A load Z L = j0 Ω terminated a lossless transmission line with a characteristic impedance of Z 0 = Ω, as shown in Figure. A time-harmonic source, with a voltage of V g = 0 and a frequency of f = 800 MHz, excites the line via a source resistance R g = 70 Ω as shown in the figure. The line has a length of l =.75 cm, and the speed of light along the line is v p = 8 m/s. R g l V g Z 0, γ Z L z z Figure : Transmission line circuit You may solve this problem using analytical or graphical means. a) Determine the reflection coefficient of the load Γ L. [ point] b) Determine the input impedance seen looking into the input of the line Z in and the associated reflection coefficient, Γ in. [ points]

4 ECE57 Midterm Page c) Determine the location(s) of all the maxima and minima of V (z) along the line, as measured as distances from the load. [ points] d) Sketch the standing wave pattern V (z) as a function of z along the line. Indicate on the graph the numerical values for max( V (z) ) and min( V (z) ). Also indicate the locations of the maxima and minima of V (z). [ points]

5 ECE57 Midterm Page e) The load Z L is replaced with a new load Z L = + jx L Ω, where X L > 0. What is the value of X L which would produce the same VSWR as the original load? [ point]

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7 ECE57 Midterm Page PROBLEM #. [ POINTS] A series RL load with R L = 0 Ω and L L =.8 nh is to be matched to Z 0 = 80 Ω at f = GHz using a shunt open-circuit stub tuner as shown shown in Figure. Ignore the resistor R s except in part (e). Treat all transmission lines as lossless for this problem. Z 0 d L L Z 0, γ R L l R s Figure : Single-stub tuner a) If the speed of light along the line is v p = 8 m/s, determine the per-unit length parameters L and C of the underlying transmission lines. [ points] b) Determine the load admittance using graphical techniques. [ points]

8 ECE57 Midterm Page 5 c) Determine the lengths of d and l, in centimetres, to achieve the required impedance match looking into the tuner. Choose the solution that yields the smallest value of d. [ points] d) What is the VSWR observed along the section of transmission line between the stub and the load? [ point] e) If a 00 Ω resistor was placed at the end of the stub in place of the open circuit, but the dimensions of tuner remain unchanged, determine the input impedance and VSWR seen looking into the tuner. [ points]

9 7 The Complete Smith Chart Black Magic Design ± > WAVELENGTHS TOWARD GENERATOR > < WAVELENGTHS TOWARD LOAD < INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES 8-9 ANGLE OF REFLECTION COEFFICIENT IN DEGREES SWR dbs RTN. LOSS [db] RFL. COEFF, P RFL. COEFF, E or I RADIALLY SCALED PARAMETERS TOWARD LOAD > < TOWARD GENERATOR CENTER ATTEN. [db] S.W. LOSS COEFF RFL. LOSS [db] S.W. PEAK (CONST. P) TRANSM. COEFF, P TRANSM. COEFF, E or I ORIGIN

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11 7 The Complete Smith Chart Black Magic Design ± > WAVELENGTHS TOWARD GENERATOR > < WAVELENGTHS TOWARD LOAD < INDUCTIVE REACTANCE COMPONENT (+jx/zo), OR CAPACITIVE SUSCEPTANCE (+jb/yo) RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) CAPACITIVE REACTANCE COMPONENT (-jx/zo), OR INDUCTIVE SUSCEPTANCE (-jb/yo) ANGLE OF TRANSMISSION COEFFICIENT IN DEGREES 8-9 ANGLE OF REFLECTION COEFFICIENT IN DEGREES SWR dbs RTN. LOSS [db] RFL. COEFF, P RFL. COEFF, E or I RADIALLY SCALED PARAMETERS TOWARD LOAD > < TOWARD GENERATOR CENTER ATTEN. [db] S.W. LOSS COEFF RFL. LOSS [db] S.W. PEAK (CONST. P) TRANSM. COEFF, P TRANSM. COEFF, E or I ORIGIN