ee541F11MidtermSolution

ee541F11MidtermSolution - EE 541 University of Southern...

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EE 541 University of Southern California Viterbi School of Engineering J. Choma. Midterm Examination Solutions 1 Fall Semester, 2011 U niversity of S outhern C alifornia USC Viterbi School of Engineering Ming Hsieh Department of Electrical Engineering EE 541: MIDTERM EXAMINATION 21 October 2010 (SOLUTIONS) 12:30 -to- 1:55 Problem #1: Solution (25%) The lowpass network shown in Figure (E1) is to be designed so that the frequency response implied by its transfer function, H(s) = V o /V i , closely approximates that of a second order Butterworth filter. Moreover, the network is to be designed to deliver a 3-dB bandwidth of B (in radians/sec ) while sustaining a zero frequency transfer function value that closely ap- proaches one. R 1 L 1 C 2 R 2 V o V i Figure (E1) (a). What general constraint must resistance R 2 satisfy to ensure a zero frequency transfer func- tion value that is very close to one? An inspection of the network at hand confirms a low frequency transfer function value that equals the resistive divider, R 2 /(R 2 + R 1 ) . If this divider is to approach one, it is clearly necessary to in- voke the constraint, R 2 >> R 1 ; that is, resistance R 1 must be very small in comparison to resis- tance R 2 . (b). In terms of resistance R 2 and bandwidth B , how must inductance L 1 and capacitance C 2 be selected? The transfer function of the network in Figure (E1) is 2 o 22 2 i 11 2 21 2 2 2 2 12 R V 1s R C H(s) R V sL R R C R RR . LR R C R s L C      (E1-1) Since R 1 must be very small to satisfy the zero frequency transmission requirement, 2 1 2 1 , L s L C R (E1-2) whose generic second order form is
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EE 541 University of Southern California Viterbi School of Engineering J. Choma. Midterm Examination Solutions 2 Fall Semester, 2011 2 2 1 12 2 2 n n 11 H(s) . L s s 1s s L C 1 R Q ω ω      (E1-3) If the network is to deliver a Butterworth frequency response, the quality factor, Q , must be the in- verse of the square root of two , wherein the self-resonant frequency, ω n , becomes the network 3-dB bandwidth. We note that n 1 B ω , LC  (E1-4) and n1 1 22 2 ω L L 2. QR R R (E1-5) It follows that 2 2 L2 R C . (E1-6) Moreover,  2 2 1 B , 2R C 2RC (E1-7) which stipulates, 2 2 1 C. 2R B (E1-8) Using this result, (E1-6) becomes 2 1 2R L. B (E1-9) Problem #2: Solution (30%) Consider the normalized impedance function, 2 1a p Z(p) .
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ee541F11MidtermSolution - EE 541 University of Southern...

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