ECE 6345Spring 2011Homework 21) A certain probe-fed microstrip antenna has a resistance of 50 [] at resonance. Thebandwidth (defined by SWR < 2) is 2%. The resonance frequency is 2.0 GHz. Assume thatthe probe inductance is negligible.a) Plot the input resistance and reactance versus frequency for this antenna, from 1.9 to2.1 GHz.b) Plot |S11| versus frequency over the same range, assuming that the antenna is fed by a50 [] feed line. c) Plot the normalized input impedance for this antenna on the Smith chart, over thesame frequency range. 2) Redo the previous problem, assuming now that the probe reactance is j10 []. (The probereactance may be assumed to be constant over the frequency range of interest.) Comment onhow the probe reactance affects the various plots. 3) A probe-fed microstrip antenna has an input resistance of 50 [] at resonance. The probereactance may be neglected. What is the input impedance of this antenna at the frequencies1f and2f? These are the two frequencies (lower and upper) at which the SWR is 2.0, whenthe antenna is fed by a 50 [] feed line. 4) Show that when the probe inductance is accounted for, the resonance frequency fres (where inthe input impedance is purely real) is given by21 1 42prespXxX- -=where01/ .resres rresrresresrp px Q fffffX X R� �= -� �� �==5) As a continuation of the previous problem, denote f0 as the resonance frequency of a probe-fed microstrip antenna when the probe inductance is neglected. Let fres = f0 +f be theresonance frequency when the probe inductance is included. That is, fres is the frequency for1which the input impedance of the microstrip antenna (including the probe inductance) is areal number. Derive the following approximate formula for the shift in resonance frequencydue to the probe inductance:( )01BW2pXff R� �D� �=� �� �� �� �,where BW is the bandwidth of the antenna (SWR < 2 definition), Xp is the probe reactance(assumed to be constant over the frequency range of interest), and R is the input resistance atfrequency f0.6) Consider a microstrip patch antenna that is gap coupled (with a small gap) to a microstripline (i.e, there is a capacitive gap between the line and the patch). The gap is represented as aseries capacitor Cg. What will the input resistance seen by the feeding line be (at the gaplocation) when the patch is operating at the resonance frequency where the input impedanceis purely real? Assume that the patch is modeled as an RLC circuit with known values of (R,L, C) when the patch is fed by a direct contact at the edge. From your result, explain how thegap coupling can be used to lower the input resistance seen by the feed line, from R to alower value resinR.7) As we saw in the short-course notes, using a double-tuned resonator effect is a good way toincrease the bandwidth of a microstrip antenna. Consider one microstrip antenna that iscapacitively coupled to another one. An approximate model for this is shown below. Thecapacitor 0C is the coupling capacitance between the two antennas. Show that the input admittance of this circuit is approximately  2121 /incYY Yj Y B ,where 1Y is the input admittance of antenna tank circuit one, 2Yis the input admittance ofantenna tank circuit two, and 0 0cB Cw=, where 0 is the center frequency of the double-tuned circuit. The input admittances of the antenna tank circuits can be written as  1 1 1 1111 1/r rY jQ f fR        2 2 2 2211 1/r rY jQ f fR       where 2101rfff=and202rfff=.Note that we can write121rrff  ,where 02 0101f ff-D= is the normalized shift in resonance frequencies between the two antenna tank circuits (youshould show this).8) Make a plot of SWR vs. fr1 for the following case: R1 = 50 [], Q1 = 25, R2 = 100 [], Q2 = 25,Bc = 0.0001,  = 0.009, and Z0 = 50 []. Because of the very small value of Bc, Thiscorresponds to the response of the single resonator 1. Determine the bandwidth from yourplot. Compare with the bandwidth from the formula 112BWQ=.3R1R2L1L2C1C2C0Z09) Make a plot of SWR vs. fr1 for the following case: R1 = 75 [], Q1 = 25, R2 = 100 [], Q2 = 25,Bc = 0.016,  = 0.009, and Z0 = 50 []. Verify that a double-tuned response is obtained.Determine the bandwidth from your plot. How much larger is the bandwidth compared tothat of the single resonator in the previous problem (i.e., what is the ratio of the twobandwidths)? 10) Try varying the values R1, R2, Bc, and , to see what the maximum bandwidth is that you canobtain, assuming that bandwidth is defined from SWR < 2. Keep both Q values at 25, and thefeed-line impedance Z0 at 50 []. (There is not necessarily a unique best solution to this problem.)