DeCarlo Ch17b Solutions

Linear Circuit Analysis: The Time Domain and Phasor Approach

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5/15/01 P17-1 R.A. DeCarlo & P.M. Lin C HAPTER 17 P ROBLEM S OLUTIONS S OLUTION P ROBLEM 17.11. Case 1: suppose R 1 > R 2 . From example 17.3, page 696, if L and C are connected as indicated in part (a), then Z 1 can be made real and larger than R 2 . This means we can solve the problem at least for Z 1 . Specifically, consider the figure below From example 17.3, at a specified frequency, ϖ r , for which Z 1 is real, then L, C, and R 2 must satisfy, ϖ r = 1 LC - R 2 2 L 2 (1) Further from example 17,3, at ϖ r , Z 1 ( j ϖ r ) = L R 2 C We require that Z 1 ( j ϖ r ) = R 1 in which case R 1 R 2 = L C (2) It is necessary to solve equations (1) and (2) simultaneously for L and C. From (2), L = R 1 R 2 C . Substituting into the square of (1) yields ϖ r 2 = 1 R 1 R 2 C 2 - R 2 2 R 1 2 R 2 2 C 2 Hence C = 1 ϖ r R 1 R 1 - R 2 R 2 It follows that
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5/15/01 P17-2 R.A. DeCarlo & P.M. Lin L = R 1 R 2 1 ϖ r R 1 R 1 - R 2 R 2 = 1 ϖ r R 2 R 1 - R 2 ( 29 Observe that since R 1 > R 2 , both C and L are real, i.e., exist. Please note that this connection would not result in real values of C and L had R 1 < R 2 . If we can now show that Z 2 = R 2 , then parts (a) and (b) are valid for this case. By direct computation Z 2 ( j ϖ r ) = j ϖ r L + 1 j ϖ r C + 1 R 1 = j ϖ r L + R 1 j ϖ r CR 1 + 1 = j R 2 R 1 - R 2 ( 29 + R 1 j R 1 - R 2 R 2 + 1 = j R 2 R 1 - R 2 ( 29 + R 1 - jR 1 R 1 - R 2 R 2 1 + R 1 - R 2 R 2 = j R 2 R 1 - R 2 ( 29 + R 2 - jR 2 R 1 - R 2 R 2 = R 2 Thus, (a) and (b) are true for the case R 1 > R 2 . We can also arrive at the conclusion that Z 2 = R 2 using maximum power transfer concepts. Since Z 1 is constructed so that Z 1 = R 1 , we have set up the conditions for maximum power transfer of a V-source in series with R 1 to the "load" Z 1 . Since the LC coupling network is lossless, whatever average power is received by the network to the right of R 1 , will be dissipated by R 2 . Therefore maximum power is transferred to the load R 2 . Looking back from R 2 , it must be that R 2 sees a Thevenin resistance Z 2 = R 2 since it is known that there is a non-zero R 1 . Case 2, R 1 < R 2 . Now consider the configuration
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5/15/01 P17-3 R.A. DeCarlo & P.M. Lin Interchanging the subscripts of 1 and 2 in case 1 produces the derivation for this case.
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