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Lecture 15_16 - Lecture 15 16 RLC finding the natural...

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Spring 2011 ECE 2100, Cornell Univ. Prof. A. Molnar 1 Lecture 15 + 16 RLC: finding the natural response different kinds of damping
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Spring 2011 ECE 2100, Cornell Univ. Prof. A. Molnar 2 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 ( ) 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 ω α α ω α ω α ω 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
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Spring 2011 ECE 2100, Cornell Univ. Prof. A. Molnar 3 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 ω α α τ ω α α = = = + = s s 2 0 2 ω α α ± = s + V - L C R I L I C I R ( ) t s t s Be Ae t V 2
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