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09 - Second Order Circuits

09 - Second Order Circuits - EECS 215 Second Order Circuits...

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EECS 215 Second Order Circuits J. Phillips EECS 215 Second Order Circuits A second order circuit is characterized by a second order differential equation Resistors and two energy storage elements Determine voltage/current as a function of time Initial and final values of voltage/current, and their derivatives are needed
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J. Phillips EECS 215 Source-Free RLC Circuit Find Natural Response Of RLC Circuit From KVL, 0 1 = + + dt i C dt di L Ri t Differentiate with respect to time 0 2 2 = + + LC i dt di L R dt i d Need two initial conditions ( ) 0 0 I i = ( ) ( ) 0 0 1 0 V RI L dt di + = Assume solution is exponential of form st Ae i = J. Phillips EECS 215 Source-Free Series RLC Circuit Find Natural Response Of RLC Circuit Insert Ae st into differential equations 0 2 = + + st st st e LC A se L AR e As 0 1 2 = + + LC s L R s Ae st 0 1 2 = + + LC s L R s LC L R L R s 1 2 2 2 2 , 1 ± = 2 0 2 2 , 1 ω α α ± = s L R 2 = α LC 1 0 = ω
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J. Phillips EECS 215 Underdamped Response Underdamped, α < ω 0 Damping: loss of stored energy ( ) ( ) t B t B e t i d d t ω ω α sin cos 2 1 + = factor damping = α frequency resonant 0 = ω 2 0 2 2 , 1 ω α α ± = s L R 2 = α LC 1 0 = ω 2 2 0 α ω ω = d J. Phillips EECS 215
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