ee541F11HWSolutions08

ee541F11HWSolutions08 - EE 541 University of Southern...

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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #08 119 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 #08 Fall, 2011 Due: 11/22/2011 Choma Solutions Problem #42: The distributed line in Figure (P42) is lossless and is characterized by a characte- ristic impedance of Z o . It operates at a frequency for which its physical length is N- wavelengths. Z o Z in Figure (P42) (a). In terms of Z o and the sine and cosine of appropriate functions of N , derive an expression for the indicated impedance, Z in . The schematic diagram of the network at hand is redrawn as Figure (P42.1) to highlight the I/O port currents, I 1 and I 2 , as well as the I/O port voltages, V 1 and V 2 . The interrelationships among these electrical port variables are arguably best modeled through use of the y-parameters for the distributed network. In particular, Z o Z in I 1 I 2 I 2 I V 1 V 2 V Figure (P42.1) ir 11 ri 22 YY IV V , V     (P42-1) where use has been made of the fact that the port voltages, V 1 and V 2 , in Figure (P42.1) are identi- cally equal to the indicated voltage, V . In (P42-1), a lossless line gives  i oo Y, Zj 2 π Nj Z 2 π N tanh tan (P42-2) and r Y. 2 π Z 2 π N sinh sin  (P42-3) Since current I is the sum of the port currents, I 1 and I 2 , 12 o 21 1 III 2 Y Y V V . jZ 2 π N2 π N tan sin     (P42-4)
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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #08 120 Fall Semester, 2011 In view of the fact that Z in is simply V/I ,  o o in Z Z j2 π N j V 2 2 Z . 11 I2 π N1 2 π N2 π N sin cos tan sin     (P42-5) (b). Evaluate Z in for line lengths of ¼ and ½ wavelengths. For N = ¼ , Z in = –jZ o /2 , while for N = ½ , Z in = 0 . Solutions Problem #43: The quarter wavelength transmission line in the schematic diagram of Figure (P43) has a real characteristic impedance of Z o . The circuit is to be designed as a resonant structure having a radial tuned frequency of o . R l C l C s Z o ¼ V o V i V s R s Z( j ) in V x x Figure (P43) (a). In terms of o , load resistance R l , signal source resistance R s , and load capacitance C l , give an expression for capacitance C s , such that the indicated input impedance, Z in (j ω ) , matches the source resistance at frequency o . From the Class Lecture Aids, the impedance, Z x (j ω ) , seen looking into the input port of a quarter wavelength line terminated in an impedance, Z l (j ω ) , is 22 2 2 oo o x ll o l l ZZ Z j ω )1 j ω RC j ω ZC. j ω )R R (P43-1) It follows that the given network can be supplanted by the lumped equivalent model offered in Fig- ure (P43.1).
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ee541F11HWSolutions08 - EE 541 University of Southern...

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