HW1_solution - MAE107 Homework#1 Solution Prof M’Closkey...

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MAE107 Homework #1 Solution Prof. M’Closkey Problem 1 The circuit to analyze is R 1 C 1 C 2 R 2 + V in + V out V i 1 i 2 Let i 1 be the current through capacitor C 1 and let i 2 be the current through C 2 and R 2 (since no current flows into the V out measurement system). The relations for each component are: V in V = R 1 ( i 1 + i 2 ) (1) C 1 ˙ V = i 1 (2) C 2 ( ˙ V ˙ V out ) = i 2 (3) V out = R 2 i 2 (4) Manipulate these relations to eliminate V , i 1 and i 2 . Substitute (2) and (4) into (3) and (1): V in V = R 1 ( C 1 ˙ V + 1 R 2 V out ) (5) C 2 ˙ V C 2 ˙ V out = 1 R 2 V out . (6) Differentiate (5), ˙ V in ˙ V = R 1 C 1 ¨ V + R 1 R 2 ˙ V out . (7) Differentiate (6), ¨ V = ¨ V out + 1 R 2 C 2 ˙ V out (8) Substitute (6) and (8) into (7) to eliminate V and its derivatives, ˙ V in ˙ V out + 1 R 2 C 2 V out = R 1 C 1 ¨ V out + 1 R 2 C 2 ˙ V out + R 1 R 2 ˙ V out . Rearraging into “standard” form, R 1 C 1 ¨ V out + 1 + R 1 C 1 R 2 C 2 + R 1 C 1 ˙ V out + 1 R 2 C 2 V out = ˙ V in , (9) so, a = R 1 C 1 , b = 1 + R 1 C 1 R 2 C 2 + R 1 C 1 , c = 1 R 2 C 2 , d = 1 . 1
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Problem 2 Simplify these equations into a relation between V out, 2 and V in, 1 : ˙ V out, 1 + 1 R 1 C 1 V out, 1 = 1 R 1 C 1 V in, 1 (10) R 2 C 2 ˙ V out, 2 + V out, 2 = R 2 C 2 ˙ V in, 2 = R 2 C 2 ˙ V out, 1 (11) Differentiate (10), ¨ V out, 1 + 1 R 1 C 1 ˙ V out, 1 = 1 R 1 C 1 ˙ V in, 1 . (12) From (11), ˙ V out, 1 = ˙ V out, 2 + 1 R 2 C 2 V out, 2 , (13) which is differentiated, ¨ V out, 1 = ¨ V out, 2 + 1 R 2 C 2 ˙ V out, 2 . (14) Substitute (13) and (14) into (12), R 1 C 1 ¨ V out, 2 + 1 R 2 C 2 ˙ V out, 2 + ˙ V out, 2 + 1 R 2 C 2 V out, 2 = ˙ V in, 1 which is rearranged to, R 1 C 1 ¨ V out, 2 + 1 + R 1 C 1 R 2 C 2 ˙ V out, 2 + 1 R
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