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ee541F11HWSolutions04

ee541F11HWSolutions04 - EE 541 University of Southern...

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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #04 53 Fall Semester, 2011 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 #04 Fall, 2011 Due: 10/04/2011 Choma Solutions Problem #19: The passive pi-architecture filter shown in Figure (P19) is formed of an ideal inductance, L , two identical capacitances, C , and two resistances, R 1 and R 2 . The filter is to be designed so that its transimpedance function, Z io (s) = V o /I i , provides a Bessel frequency response featuring a steady state (or group) delay at zero frequency of T do . L C C R 2 R 1 I i V o Figure (P19) (a). How is the zero frequency group delay related to resistances R 1 and R 2 and capacitance C ? The transimpedance, Z io (s) = V o /I i , of the subject filter evaluates as the function 1 2 1 2 o io 1 2 i 1 2 io 2 3 2 io io 1 2 R R 1 sR C 1 sR C V Z (s) R R I sL 1 sR C 1 sR C R , L 1 s 2R C s LC s LR C R + R    (P19-1) where the zero frequency value of the forward transimpedance is io io 1 2 R Z (0) R R . (P19-2) which is obvious through inspection of the given schematic diagram. Clearly, the filter at hand is a third order architecture. For a third order Bessel filter delivering a zero frequency group delay of T do , the applicable characteristic polynomial, say D(s) , is 2 3 do do do 2 1 D(s) 1 T s T s T s . 5 15 (P19-3) Upon equating like coefficients of complex frequency s in D(s) with the denominator polynomial on the far right hand side of (P19-1), we deduce the constraints, do io 1 2 L T 2R C , R + R (P19-4) 2 do 5 T LC , 2 (P19-5)
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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #04 54 Fall Semester, 2011 and 3 2 io do T 15LR C . (P19-6) If (P19-6) is divided by (P19-5), the zero frequency group delay, T do , is found to satisfy do io 1 2 T 6R C 6 R R C . (P19-7) (b). In terms of resistances R 1 and R 2 and capacitance C , how must inductance L be selected to satisfy the design requirements? If the delay, T do , from (P19-7) is substituted into (P19-4), io io 1 2 L 6R C 2R C , R + R (P19-8) whence 1 2 L 4R R C . (P19-9) Solutions Problem #20: A certain communication system application requires a sixth order , lowpass, lossless Butterworth filter whose input port is driven by a 75 signal source and whose output port is terminated in a purely resistive 75 load. The 3-dB bandwidth of the filter is to be 900 MHz . (a). Design the normalized form of this filter and show the schematic diagram of the norma- lized filter architecture. Choose the normalizing resistance as the source/load resistance of the filter and the normalizing frequency as the 3-dB bandwidth of the filter.
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