ee541F11FinalSolution

ee541F11FinalSolution - EE 541 University of Southern...

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EE 541 University of Southern California Viterbi School of Engineering J. Choma. Final Examination Solutions 6 Fall Semester, 2011 U niversity of S outhern C alifornia USC Viterbi School of Engineering Ming Hsieh Department of Electrical Engineering EE 541: FINAL EXAMINATION 13 December 2011 (SOLUTIONS) 11:00 AM -to- 1:00 PM Problem #1: Solution (20%) The tapped inductor resonator, whose schematic diagram appears in Figure (E1), is capable of transforming the indicated load resistance, R l to a strictly resistive input resistance, R in , at a tuned frequency of ω o ; that is, Z in (j ω o ) = R in . For simplicity, assume that both induc- tances have infinitely large quality factor (infinite Qs ). L 1 L 2 R l C Z (j ) in V out V in Z (j ) a Figure (E1) (a). If we define Q 2 as the quality factor, Q 2 Δ R l / ω o L 2 , associated with the shunt interconnec- tion of inductance L 2 and resistance R l , reduce the network to an equivalent shunt RLC fil- ter. Express pertinent elements of this reduced network in terms of network branch para- meters and the quality factor, Q 2 . The impedance, Z l (j ω ) , of the shunt interconnection of resistance R l and inductance L 2 is   l2 ll l l o 2 2 Rj ω L RR Z( j ω ). R ω ω L 1 1j Q j ω L ω     (E1-1) This impedance function can be rationalized to produce o l l 2 o o 2 2 ω R1 j Q ω R j ω ), ω ω Q 1Q ω ω (E1-2) or 2 o 2 l 22 oo ω Q R ω j ω )j ω L. ωω   (E1-3) Clearly, the impedance herewith is of the form,
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EE 541 University of Southern California Viterbi School of Engineering J. Choma. Final Examination Solutions 7 Fall Semester, 2011 2 o 2 l l2 l l l l 22 oo ω Q R ω Z( j ω )j ω LR j ω L, ωω 1Q         (E1-4) with the understanding that o o ll ll ll o 2 2 2 o 2 2 2 2 ll 2 ll o 2 2 RR R; R ω ω . ω Q QL ω LL ; L ω ω  (E1-5) Consequently, the filter in Figure (E1) collapses to the electrically equivalent structure given in Figure (E1.1a). L 1 L ll R ll C Z (j ) in V out V in Z (j ) a (a). L a R a C Z (j ) in V in Z (j ) a (b). Figure (E1.1) Now, the impedance, Z a (j ω ) , is   al l 1 l l j ω )R j ω , (E1-6) and its corresponding admittance, Y a (j ω ) , is    ll 1 ll a 2 2 2 a ll 1 ll 1l l ll Rj ω 11 Y( j ω ). j ω ) ω R ω  (E1-7) This admittance implies the shunt interconnection of a resistance, R a , and an inductance, L a , as de- picted in Figure (E1.1b), where   ll 1 ll a 2 2 2 aa l ll ω 1 j ω ), ω L R ω (E1-8) with
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EE 541 University of Southern California Viterbi School of Engineering J. Choma. Final Examination Solutions 8 Fall Semester, 2011  2 1l l al l ll 2 l ll a2 2 l ll ω LL RR 1 R ω .
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ee541F11FinalSolution - EE 541 University of Southern...

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