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 niversity of S outhern 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 CR 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 12 o io i io 23 2 io io RR 1s RC 1s RC V Z( s ) I sL R , L 1s2 s L Cs L R+R      (P19-1) where the zero frequency value of the forward transimpedance is io io 1 2 R 0 ) RR. (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  do do do 21 D ( s ) 1 Ts Ts Ts . 51 5  (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 L T2 R C , R +R (P19-4) 2 do 5 TL C , 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 T1 5 L R 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 T6 R 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 12 L 6R C 2R C , R +R  (P19-8) whence L4 R 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. A sixth order, Butterworth, lowpass filter whose load and source resistances are matched has a sig- nal source -to- load port transfer function, say H s (p) , which abides by 2 o ss s 61 2 12 s V 0.25 0.25 H (p) H (p)H (-p) , V 1p 11 p  (P20-1) where the normalized complex frequency, p , relates to the Laplace operator, s , as α ps ω , (P20-2)
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ee541F11HWSolutions04 - EE 541 University of Southern...

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