1 ECE 6345 Spring 2011 Homework 2 1) A certain probe-fed microstrip antenna has a resistance of 50 [] at resonance. The bandwidth (defined by SWR < 2) is 2%. The resonance frequency is 2.0 GHz. Assume that the probe inductance is negligible. a) Plot the input resistance and reactance versus frequency for this antenna, from 1.9 to 2.1 GHz. b) Plot |S11| versus frequency over the same range, assuming that the antenna is fed by a 50 [] feed line. c) Plot the normalized input impedance for this antenna on the Smith chart, over the same frequency range. 2) Redo the previous problem, assuming now that the probe reactance is j10 []. (The probe reactance may be assumed to be constant over the frequency range of interest.) Comment on how the probe reactance affects the various plots. 3) A probe-fed microstrip antenna has an input resistance of 50 [] at resonance. The probe reactance may be neglected. What is the input impedance of this antenna at the frequencies 1f and2f? These are the two frequencies (lower and upper) at which the SWR is 2.0, when the antenna is fed by a 50 [] feed line. 4) Show that when the probe inductance is accounted for, the resonance frequency fres (where in the input impedance is purely real) is given by 21 1 42prespXxX where 01/.resres rresrresresrppx 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 the resonance frequency when the probe inductance is included. That is, fres is the frequency for2 which the input impedance of the microstrip antenna (including the probe inductance) is a real number. Derive the following approximate formula for the shift in resonance frequency due to the probe inductance:  01BW2pXffR, 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 at frequency f0. 6) Consider a microstrip patch antenna that is gap coupled (with a small gap) to a microstrip line (i.e, there is a capacitive gap between the line and the patch). The gap is represented as a series capacitor Cg. What will the input resistance seen by the feeding line be (at the gap location) when the patch is operating at the resonance frequency where the input impedance is 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 the gap coupling can be used to lower the input resistance seen by the feed line, from R to a lower value resinR. 7) As we saw in the short-course notes, using a double-tuned resonator effect is a good way to increase the bandwidth of a microstrip antenna. Consider one microstrip antenna that is capacitively coupled to another one. An approximate model for this is shown below. The capacitor 0C is the coupling capacitance between the two antennas. Show that the input admittance of this circuit is approximately  2121/incYYYj Y B, where 1Y is the input admittance of antenna tank circuit one, 2Yis the input admittance of antenna tank circuit two, and 00cBC, 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/rrY jQ f fR    2 2 2 2211 1/rrY jQ f fR   where3 101rfff and 202rfff. Note that we can write 121rrff , where 02 0101fff is the normalized shift in resonance frequencies between the two antenna tank circuits (you should 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, This corresponds to the response of the single resonator 1. Determine the bandwidth from your plot. Compare with the bandwidth from the formula 112BWQ. R1 R2 L1 L2 C1 C2 C0 Z04 9) 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 to that of the single resonator in the previous problem (i.e., what is the ratio of the two bandwidths)? 10) Try varying the values R1, R2, Bc, and , to see what the maximum bandwidth is that you can obtain, assuming that bandwidth is defined from SWR < 2. Keep both Q values at 25, and the feed-line impedance Z0 at 50 []. (There is not necessarily a unique best solution to this