ee541F11HWSolutions02

# ee541F11HWSolutions02 - EE 541 University of Southern...

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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #02 17 Fall Semester, 2011 U niversity of S outhern C alifornia USC Viterbi School of Engineering Ming Hsieh Department of Electrical Engineering EE 541: Solutions, Homework #02 Fall, 2011 Due: 09/08/2011 Choma Solutions Problem #06: Consider the linear, bilateral, and symmetric two-port network abstracted in Figure (P6a). Observe that the load (R l ) terminating the output port of the network is purely resistive, as is the intrinsic impedance (R s ) associated with the applied signal source. Linear, Bilateral, Symmetric Two-Port Network I 2 V 2 V 1 R l R s V s I 1 Z in Figure (P6a) (a). In terms of the open circuit impedance parameters, z ij , of the subject network, show that maximum power transfer between the applied signal source and the network input port re- quires  2 11 s 11 l 12 zR z .  If a general, linear two-port network is modeled by its open circuit impedance parameters, z ij , the terminal volt-ampere characteristics of the network abide by 11 1 1 1 2 2 Vz I z I ,  (P6-1) and 22 1 1 2 2 2 I z I . (P6-2) Since the terminating load resistance, R l , in the subject linear network subjugates the output port current, I 2 , and the output port voltage, V 2 , to the Ohm’s constraint, V 2 = R l I 2 , (P6-2) delivers 1 1 2 2 2 l 2 I z I R I ,  (P6-3) which, in turn, produces 21 1 2 22 l zI I.  (P6-4) Note then that z 21 is a measure of the degree to which input port current, I 1 is transferred to the out- put port as a current, I 2 , since the input -to- output (I/O) current ratio I 2 /I 1 , is clearly proportional to the transimpedance parameter, z 21 . If we now insert (P6-4) into (P6-1), we arrive at a driving point input impedance, Z in , of 2 2 1 in 11 12 2 l z Z z. Iz R  (P6-5) But in a bilateral network, z 12 = z 21 , while a symmetric two-port exhibits z 22 = z 11 . Thus, (P6-5) for

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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #02 18 Fall Semester, 2011 a symmetric, bilateral, linear two-port network is 2 12 in 11 11 l z Z z. zR  (P6-6) Now, maximum signal power transfer occurs at the network input port when the driving point input impedance of the subject network is a conjugate match to the source impedance. In this case, the source impedance is a purely real resistance, R s , which means that Z in = R s is required to effect maximum power transfer between the applied signal source and the network input port. Setting Z in = R s in (P6-6), we therefore conclude that the applicable design requirement is  2 11 s 11 l 12 z .  (P6-7) (b). What network condition must be satisfied if the indicated input impedance, Z in is designed to be KR l , where K is a positive, frequency invariant constant?
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ee541F11HWSolutions02 - EE 541 University of Southern...

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