ee541F11HWSolutions07

ee541F11HWSolutions07 - EE 541 University of Southern...

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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #07 103 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 #07 Fall, 2011 Due: 11/09/2011 Choma Solutions Problem #37: When terminated at its output port in a resistance, R , the passive network abstracted in Figure (P37a) is to produce the voltage transfer function,  2 o 2 s ss 1 V 1 aQ a H(s) , V2 a1s s 11 a a 1 aQ a         where parameters “a” and “Q” are positive constants and the transfer function is written in terms of a normalized impedance of 1 ohm and a normalized frequency of 1 radian per second . In general, a normalized one-ohm impedance corresponds to an actual R-ohm impedance, while 1 radian per second reflects an actual frequency of o radians -per-second . V o V s V s Passive RLC Network R R R Z(s ) a Z(s) b R R R V o (a). (b). Figure (P37) (a). If the network is realized as the constant resistance tee structure offered in Figure (P37b), give normalized mathematical expressions for the requisite impedances, Z a (s) and Z b (s) . The given transfer function can be written in the form, 2 2 2 2 2 1+ aQ a 1 , sa 1 s a 1 s s Qa aQ a Q a 1 aQ a (P37-1) which implies that, with R = 1 ,

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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #07 104 Fall Semester, 2011 2 a 2 b sa 1 s Qa 1 Z (s) . Z( s ) ss 1+ aQ a     (P37-2) (b). Determine the relationship between parameters “Q” and “a,” such that Z a (s) is realizable as an interconnection of a minimal number of branch elements. By continued fraction expansion,  2 a 22 2 1 s 1 Z (s) . s 1 s a 1 Q aQ a a Q a Q 1 s s     (P37-3) Observe that the coefficient of the (s/a)-term in this expansion is potentially negative, which would preclude the realization of Z a (s) with positive resistances, capacitances, and inductances. On the other hand, if this term were to be constrained to zero, the (s/a)-term vanishes, thereby giving rise to the possibility of realizing Z a (s) with a minimum number of branch elements. Accordingly, set 1 Q. a1 (P37-4) (c). Given the relationship between “Q” and “a” found in the preceding part of this problem, draw the normalized realizations of impedances Z a (s) and Z b (s) . Express all normalized element values exclusively in terms of “Q . With Q constrained in accordance with the solution to the preceding part of this problem, the ex- pansion for Z a (s) becomes a 2 2 2 11 Z (s) , Q 1 s s a Q a aa 1 Qs 1 s s  (P37-5) which suggests a normalized impedance realization consisting of an inductance of value 1/Q placed in shunt with the series combination of a capacitance, Q/a 2 , and a resistance, a(a+1) . Since 5 2 2 2 QQ
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ee541F11HWSolutions07 - EE 541 University of Southern...

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