ECE201_24_Jung

# ECE201_24_Jung - ECE 201 Lecture 24 2 nd order RLC circuits...

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Unformatted text preview: ECE 201 Lecture 24 2 nd order RLC circuits, Source free or constant input (Undamped) LC Circuit Initial conditions: Goal: KCL and KVL: V-I Relations: Differential equation for v C (t) Differential equation for i L (t) Differential Equations for v C (t) and i L (t) and are both of the general form with solution Check: amplitude phase angular velocity Conclusions: for some K and θ for some K ’ and θ ’ “ undamped oscillation ” natural (resonant) frequency Total Energy = ( 29 ( 29 2 2 1 1 . 2 2 Tot C L E C t Li t Cons υ = + = How to determine K and θ ? for some K and θ with Use the initial conditions: Take the quotient: Use the fact that How to determine K and θ ? for some K and θ with Use the initial conditions: Take the quotient: Use the fact that ( 29 ( 29 sin L i t K t ϖ θ = + ( 29 ( 29 cos L L di t L LK t dt υ ϖ ϖ θ = = + ( 29 ( 29 L L i i i +- = = ( 29 sin K i θ = ( 29 ( 29 ( 29 L C C υ υ υ υ + +- = = = ( 29 cos LK ϖ θ υ = ( 29 tan L i ϖ θ υ = 1 tan L i ϖ θ υ- = 2 2 2 2 2 2 1 i K L K υ ϖ + = 2 2 2 2 K i L υ ϖ = ± + sin and cos combination! with How to find A & B? ( 29 ( 29 sin L i t K t ϖ θ = + ( 29 ( 29 L L i i i B +- = = = ( 29 ( 29 ( 29 L C C LA υ υ υ υ ϖ + +- = = = = ( 29 L i i- = ( 29 C υ υ- = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 sin sin cos cos sin L i t K t K t t ϖ θ ϖ θ ϖ θ = + = + ( 29 ( 29 ( 29 ( 29 cos sin sin cos K t K t θ ϖ θ ϖ = + ( 29 ( 29 sin cos A t B t ϖ ϖ = + ( 29 ( 29 ( 29 cos sin L L di t L L A t B t dt υ ϖ ϖ ϖ ϖ = =- A L υ ϖ = Serial RLC Circuit are given KCL KVL In terms of v C In terms of i L differentiate v C (t)=?, i L (t)=?...
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ECE201_24_Jung - ECE 201 Lecture 24 2 nd order RLC circuits...

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