ECE201_Lecture17

ECE201_Lecture17 - ECE201 Linear Circuit Analysis I Lecture...

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1 ECE201 Linear Circuit Analysis I Lecture 17 Topic: Inductor/Capacitor Combinations (Duality) 1. Inductor Combinations (1) Principles: #1. Inductors in series add to form L eq . L eq = L 1 + L 2 + … L n #2. For n inductors in parallel eq 1 2 n 1 1 1 1 L L L L = + + + (Reciprocal of sum of reciprocals law.) Derivation of Principle #1 :
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2 Derivation of Principle #2 eq in in in eq in L (t) v dt (t) di dt (t) di L (t) v = = 1 1 in in 1 1 2 2 in in 2 2 3 3 in in 3 3 di (t) di v v (t) L dt dt L di (t) di v v (t) L dt dt L di (t) di v v (t) L dt dt L = = = = = = in 1 2 3 i i i i (kCL) = + + 3 in 1 2 in in in in eq 1 2 3 di (t) di (t) di (t) di (t) dt dt dt dt v v v v L L L L = + + = + + eq 1 2 3 1 1 1 1 L L L L = + + Remark: Inductor combinations follow the same formula as resistors.
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3 (2) Voltage division formula in 1 1 in in eq 1 1 1 in eq 1 2 3 di v L dt di v L dt v L L v L L L L = = = = + + General: k k n in i i 1 v L v L = = (n: total # of L’s) (3) Current Division Formula General: k k n in i i 1 1 i L i 1 L = = (n: total # of parallel inductors) Derivation: t 1 in 1 t in in eq 1 i (t) v ( )d L 1 i (t) v ( )d L −∞ −∞ = = = = = 3 1 1 1 1 ) 1 ( 1 1 1 ) ( ) ( i i eq in L L L L t i t i
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4 2. Capacitor Combinations (1) Principles: #3. For n capacitors in series eq 1 2 n 1 1 1 1 C C C C = + + + #4. Capacitors in parallel add
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