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ee541 - EE 541 University of Southern California Viterbi...

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EE 541 University of Southern California Viterbi School of Engineering J. Choma. Midterm Examination Solutions 1 Fall Semester, 2010 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 passive pi-architecture filter shown in Figure (E1) is formed of an ideal induc- tance, L , two identical capacitances, C , and two resistances, R 1 and R 2 . The filter is to be de- signed 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 CR 2 R 1 I i V o Figure (E1) (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 12 oi io i 23 io io RR 1s RC 1s RC VR o 2 Z (s) , I L sL 2 s L Cs L R+R ⎛⎞ ⎜⎟ ++ ⎝⎠ == = + + (E1-1) where the zero frequency value of the forward transimpedance is io io 1 2 R Z( 0 ) RR. = ± (E1-2) which is obvious by 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 do do do 21 3 D ( s ) 1 Ts Ts Ts . 51 5 =+ + + (E1-3) Upon equating like coefficients of complex frequency s in D(s) and the denominator polynomial on the far right hand side of (E1-1), we deduce the constraints, do io L T2 R C , R +R (E1-4) 2 do 5 TL C 2 = , . (E1-5) and 3 2 io do T1 5 L R C = (E1-6) If (E1-6) is divided by (E1-5), the zero frequency group delay, T do , is found to satisfy do io 1 2 T6 R C 6 R R C . (E1-7)
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EE 541 University of Southern California Viterbi School of Engineering J. Choma. Midterm Examination Solutions 2 Fall Semester, 2010 (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 (E1-7) is substituted into (E1-4), io io 12 L 6R C 2R C , R +R =+ (E1-8) whence L4 R R C = . (E1-9) Problem #2: Solution (35%) In the filter of Figure (E2), the inductance, L , is chosen in accordance with the con- straint, 2 o L RC, = + L C R o R o R o V s V o Z in Figure (E2) where R o is the resistance terminating the output port of the filter, as well as representing the Thévenin resistance of the signal source applied to the filter input port. In addition, note that a resistance of value R o shunts inductance L in the filter. (a). Determine the input port scattering parameter, S 11 , referred to a characteristic impedance of R o . The input impedance, Z in , of the filter is () 3 oa oo in 2 o o 2 o o Rs L R C R Z o R s L 1s R C 1s R C R C R C R.
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ee541 - EE 541 University of Southern California Viterbi...

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