Lecture 15

# Lecture 15 - Lecture 15 LC circuits RLC natural response...

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Lecture 15 LC circuits, RLC: natural response

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Parallel LC circuit ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 1 cos cos cos 2 2 2 2 2 V A LC t A LC dt V d LC V t A t A t V dt V d LC V dt dV C I dt dI L V = = - - = - = = = - = - = = ϖ + V - L C By governing equations: This is a 2 nd order differential equation. Guess solution to be: V 0 V t This assumes I(0)=0. more generally, I V 0 C ω ( 29 ( 29 ( 29 ( 29 0 sin cos I C L B t B t A t V - = + =
Parallel LC circuit: energy ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] 2 sin cos 2 sin 2 sin 2 2 cos 2 2 sin ) ( sin ) ( cos 2 0 2 2 2 0 2 2 0 2 2 0 2 2 2 0 2 0 0 0 C V W t t C V W W W t C V t L C L V L I W t C V C V W t L C V t I t C V dt dV C t I t V t V tot L C tot L C = + = + = = = = = = = = - = = ϖ + V - L C For V(0)=V 0 , I(0)=0 Now compute energy stored in C, L V 0 V t Total energy is constant: no dissipation! I V 0 C ω =1

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Parallel RLC circuit Start with KCL – Sub in governing equations – Take derivative of everything, divide by R Assume soln is exponential
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Lecture 15 - Lecture 15 LC circuits RLC natural response...

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