# i 0e t with t 0 l r dvc d constant i c c 0 dt dt v

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Unformatted text preview: • • dvC d (constant) i =C =C =0 dt dt • v (0 ) = V0 • C dv v + = 0 for t dt R 0 • v (0 ) = V0 = v (0+) C dv v + = 0 for t dt R v (t ) = v (0+)e t RC = V0e t RC 0 for t 0 v (t ) = v (0+)e t RC = V0e v (t ) V i (t ) = = 0e R R t RC t RC for t for t i (0 ) = 0; i (0+) = V0 R 0 0 • V02 p(t ) = i (t ) v (t ) = e R 2t RC for t 1 w (t ) = p (t ) dt = CV02 1 e 0 2 t ( 2t RC ) 0 for t 0 • = RC dy (t ) + y (t ) = 0 dt dy (t ) + y (t ) = 0 dt dy (t ) = dt y (t ) y (t ) = A e st • y (t ) = Ae st dy (t ) + y (t ) = 0 dt st ( s + 1) Ae = 0 s +1 = 0 • y (t ) = Ae st t 0 Ae st y (0) = A s= 1 0 s +1 = 0 • y (t ) = y (0)e t/ 1 y ( ) = y (0)e = y (0) e 1 dy ( t ) = dt t =o 1 y (0)e t/ = t =0 y (0) 0.37 y (0) • • • • • • • • dy (t ) + y (t ) = x (t ) dt • 1 et / et / dy (t ) e t / et / + y (t ) = x (t ) dt t/ te d t/ (e y (t ))dt = 0 x (t )dt 0 dt t d t/ et / (e y (t )) = x (t ) dt et / d t/ (e y (t )) = x (t ) dt t 0 d (e y (t )) = y (t )e t/ t/ 1 y (0) = t t/ x (t )e dt • • t/ 0 e y (t ) e y (0) = 1 1 t t/ x (t )e dt t x (t )e t / dt 0 0 0 • t/ te d t/ (e y (t ))dt = 0 x (t )dt 0 dt t y (t ) = y (0)e t/ + 1 e t/ t 0 x (t )e t / dt y complete = y natural + y forced y complete = y (t ) y natural = y (0)e t / 1 t/ t y forced = e x (t )e t / 0 • • • x (t ) = X S (with X s being a constant) • 0 for t < 0 = X S u (t ) x (t ) = X S for t 0 x(t) Xs • • dy (t ) + y (t ) = X S dt 1 y forced = e t/ t 0 t/ t/ X S e dt = y forced = X S (1 e y natural = y (0)e for t t/ ) XS e t/ t 0 for t for t 0 0 0 e t / dt = XS e t/ [ et / t 0 for t 0 • y (t ) = y natural + y forced = y (0)e = y (0)e t/ + XS XSe t/ t/ + X S (1 e for t t/ )= 0 • y ( ) = y (0)e + X S XSe = XS • y (t ) = y (0)e t/ XSe t/ + X S = [ y (0) y ( )]e t/ + y ( ) for...
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## This document was uploaded on 01/24/2014.

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