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ee541F10HWSolutions05

ee541F10HWSolutions05 - U niversity of S outhern C...

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U U U niversity of S S S outhern C C 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 m 1 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 g V m 1 R L R F V 2 I 1 V 1 (a). g V m 1 R S R F V 1 (b). I x V x Figure (P20.1) ( ) 1 F L in L 1 m V R R L R R I 1 g R + = = + , (P20-1) and ( ) x F S out S x m S V R R R R . (P20-2) I 1 g R + = = + With ρ L =9/11 , ρ S = 0 , and ρ 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 L L o L S S o S F F o F o F F s o s R 1 ρ 10 R 1 ρ R 1 ρ 1 R 1 ρ . R 1 ρ 20 R 1 ρ R R R 20 R R R + = = + = = + = = = × = (P20-3) Furthermore, since g m R L = 10 , ( ) ( ) L m L m o m o m o o S m S m o m o m S o R g R g R g R 10 g R 1 R . R g R g R g R 1 1 g R 1 R = = = = = = = (P20-4) These results combine with (P20-1), (P20-2), and (P20-3) to give ( ) ( ) in L o out S o R R 20 10 30 R 1 10 11 . R R 20 1 21 R 1 1 2 + = = + + = = + (P20-5) It follows that the reflection coefficients, ρ in and ρ out , of the input and output ports, respectively, are ( ) ( ) ( ) ( ) in L o in in L o out S o out out S o R R R 1 30 11 1 19 ρ R R R 1 30 11 1 41 . R R 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 F o o m o R R R R 1 21 . R 1 g R 2 + = = + (P20-7) Thus, ( ) ( ) in o o 11 in o o R R R 1 21 2 1 19 S . R R R 1 21 2 1 23 = = + + = (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 S 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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