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me375_sp11_hwk08_soln

me375_sp11_hwk08_soln - Homework No 8 Solution Spring 2011...

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Homework No. 8 Solution – Spring 2011 Problem No. 1 Elemental laws v R 1 = R 1 i 1 (1) v L 1 = L 1 di 1 dt (2) v L 2 = L 2 di 2 dt (3) i 3 = C dv C dt (4) Kirchhoff’s node law (node A): i 1 = i 2 + i 3 (5) Kirchhoff’s loop law: loop 1: v in ! v R 1 ! v L 1 ! v C = 0 (6) loop 2: v L 2 ! v C = 0 (7) Combining above: 7 ( ) ! v L 2 = v C ! v C = L 2 di 2 dt ! dv C dt = L 2 d 2 i 2 dt 2 (8) 4 ( ) , 8 ( ) ! dv C dt = i 3 C = L 2 d 2 i 2 dt 2 ! i 3 = CL 2 d 2 i 2 dt 2 (9) 5 ( ) , 9 ( ) ! i 1 = i 2 + CL 2 d 2 i 2 dt 2 (10) 6 ( ) , 1 ( ) ! 3 ( ) , 8 ( ) , 10 ( ) " v in = R 1 i 1 + L 1 di 1 dt + L 2 di 2 dt = R 1 i 2 + CL 2 d 2 i 2 dt 2 # $ % & ' ( + L 1 di 2 dt + CL 2 d 3 i 2 dt 3 # $ % & ' ( + L 2 di 2 dt " v in = CL 1 L 2 d 3 i 2 dt 3 + CR 1 L 2 d 2 i 2 dt 2 + L 1 + L 2 ( ) di 2 dt + R 1 i 2 + + + + ! ! ! ! A loop 1 loop 2
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Z 1 Z 2 Z 3 V in Z 1 = R 1 + L 1 s = V 1 I 1 ! V 1 = Z 1 I 1 = Z 1 I 2 + I 3 ( ) Z 2 = L 2 s = V 2 I 2 ! V 2 = Z 2 I 2 Z 3 = 1 Cs = V 3 I 3 ! V 3 = Z 3 I 3 V 2 = V 3 ! Z 2 I 2 = Z 3 I 3 ! I 3 = Z 2 / Z 3 ( ) I 2 V in = V 1 + V 2 = Z 1 I 2 + I 3 ( ) + Z 2 I 2 = Z 1 1 + Z 2 / Z 3 ( ) + Z 2 ! " # $ I 2 = Z 1 Z 2 + Z 3 ( ) + Z 2 Z 3 Z 3 I 2 = Cs R 1 + L 1 s ( ) L 2 s + 1/ Cs ( ) + L 2 / C ! " # $ I 2 = R 1 + L 1 s ( ) CL 2 s 2 + 1 ( ) + L 2 s !
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