ee541F10HWSolutions05

ee541F10HWSolutions05 - U niversity of S outhern C...

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U niversity of S outhern C alifornia USC Viterbi School of Engineering Ming Hsieh Department of Electrical Engineering EE 541: Solutions, Homework #05 Fall, 2010 Due: 10/07/2010 Choma Solutions Problem #20: A linear feedback amplifier, whose equivalent circuit is the topology embedded within the dashed box in Figure (P20), is terminated at its output port in a resistance, R L , and is driven by a signal source whose Thévenin equivalent resistance is R S . When meas- ured with respect to a characteristic impedance of R o , the reflection coefficients of the load re- sistance, R L , the source resistance, R S , and the feedback resistance, R F , are ρ L = 9/11 , 0 , and ρ F = 19/21 , respectively. If g m R L = 10 , find the input port reflection coefficient and all scattering pa- rameters, where it is understood that the scattering parameters are measured with respect to the reference resistance, R o . + R F R S g V m1 R L V 1 I 1 V 2 V S Figure (P20) The diagrams in Figure (P20.1) display the configurations pertinent to the evaluation of the driving point input resistant, R in (R L ) , and the driving point output resistance, R out (R S ) . A straightforward analysis of each of these circuits reveals that gV R L R F V 2 I 1 V 1 (a). R S R F V 1 (b). I x V x Figure (P20.1) () 1F L in L 1m VR R L R R I1 g R + == + , (P20-1) and x FS out S x mS R R R. (P20-2) g R + + With ρ L =9/11 , S = 0 , and F = 19/21 ,
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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #05 55 Fall Semester, 2010 LL oL SS oS FF oF o so s R1 ρ 10 ρ ρ 1 ρ . ρ 20 ρ R RR 20 R + == + + = (P20-3) Furthermore, since g m R L = 10 , () L mL mo o S mS o R gR gR 1 0 1 R . R g Rg R g R 1 1g R = = = (P20-4) These results combine with (P20-1), (P20-2), and (P20-3) to give in L o out S o 20 10 30 R 1 10 11 . 20 1 21 1 2 + + + + (P20-5) It follows that the reflection coefficients, ρ in and out , of the input and output ports, respectively, are in L o in in L o out S o out out S o RR R 1 30 11 1 19 ρ R R R 1 30 11 1 41 . R 1 21 2 1 19 ρ R R R 1 21 2 1 23 = ++ = (P20-6) The scattering parameter, S 11 , is simply the reflection coefficient of the input port under the condi- tion of a load termination matched to R o . Returning to (P20-1), in o Fo om o RR 1 21 . R 1gR 2 + + (P20-7) Thus, in o o 11 in o o RRR 1 21 2 1 19 S. R R R 12 1 2 3 = (P20-8) Similarly, S 22 is the reflection coefficient of the output port under the condition of a source termi- nation matched to R o . Appealing to (P20-2), it is clear that R out (R o ) R in (R o ) , which implies that 22 S 11 = 19/23 . The scattering parameters, S 21 and S 12 , are intimate with the forward and reverse network voltage gains when both the input and output ports are terminated in the reference resistance, R o . Figure (P20.2a) is the model pertinent to the computation of the forward voltage gain, A vf , while the dia- gram in Figure (P20.2b) pertains to the evaluation of the reverse voltage gain, A vr . An analysis of these circuits reveals quickly that
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EE 541 University of Southern California Viterbi School of Engineering Choma () mF 2 vf sm o F o 1 vr xm o F o gR 1 V A Vg R R R 2 .
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ee541F10HWSolutions05 - U niversity of S outhern C...

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