ECE201_Lecture18

ECE201_Lecture18 - E CE201 Linear Cir cuit Analysis I L...

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1 ECE201 Linear Circuit Analysis I Lecture 18 Topic: First-order Circuits: zero input response 1. 1 st -order circuits (1) Circuits that consist of sources, resistors and ONE capacitor or ONE inductor (2) v L = di dt & i c = C dv (3) KVL & KCL & Ohm’s Law (for R) still work (4) Circuit equations are 1 st -order linear differential equations 2. A general solution to 1 st -order differential equations (1) A general circuit equation dx ( t ) = λ x ( ) + f ( ) Initial condition: x(t o ) = x o where: x(t) can be i or v λ is a constant f(t) is a function of t. The equation is valid for t ≥ t o e.g. (from last lecture) di L (t) dt = − 25i L + 50 i L (0) = 1A = i L , = –25, f(t) = 50, t o = 0, x o = 1 A ( ) (2) derivation of solution Step 1. multiply both sides by e - λ t e −λ t dx(t) dt = λ e t x(t) + e t f(t) Step 2. integrate both sides over [t o , t] left side = t o t e - λτ dx( τ ) d τ − λ e - x( τ ) d τ i L (t) 5A L=1H 15 10
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2 Note that d e −λ t x(t) [ ] dt = e − λ t dx(t) dt + x(t)( −λ )e t = e − λ t dt − λ e t x(t) Then: left side = e - λτ x( τ ) t t o
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ECE201_Lecture18 - E CE201 Linear Cir cuit Analysis I L...

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