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Lecture 16_17_rev2

# Lecture 16_17_rev2 - Lecture 16 17 RLC finding the natural...

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Lecture 16 + 17 RLC: finding the natural response different kinds of damping

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Parallel RLC circuit KCL gave us a 2 nd order differential equation Can reformulate in a general form Subbing in an exponential gives a quadratic equation Solve for roots Question: how does behavior change as a function of ω 0 , α ? 3 cases of interest ( 29 2 0 2 2 2 0 2 2 0 2 0 2 2 2 2 1 2 1 2 1 2 1 0 exp 1 , 2 1 2 1 0 ϖ α α ϖ α ϖ α ϖ α - ± - = - ± - = + + = + + = = = = + + = + + = LC RC RC s V s s LC RC s s st V LC RC V dt dV dt V d LC V dt dV RC dt V d + V - L C R I L I C I R
Case 1: over damped If α > ω 0, then Has two real roots, specifically: Note that τ 1> τ 2 Solution takes the form: Where initial conditions dictate that 2 2 0 2 2 1 2 0 2 1 1 1 τ ϖ α α τ ϖ α α - = - - - = - = - + - = s s 2 0 2 ϖ α α - ± - = s + V - L C R I L I C I R ( 29 t s t s Be Ae t V 2 1 + = ( 29 ( 29 ( 29 ( 29 2 1 0 0 0 0 Bs As C I RC V dt dV B A V L + = - - = + =

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2 nd order RC also over damped Have two 1 st order differential equations Can substitute to get one 2 nd order differential
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Lecture 16_17_rev2 - Lecture 16 17 RLC finding the natural...

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