ECE201_25_2nd_order_const_input

ECE201_25_2nd_order_const_input - ECE 201 Lecture 25 2nd...

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ECE 201 Lecture 25 2 nd order circuits: RLC with constant inputs
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Serial RLC Circuit are given KCL KVL In terms of v C In terms of i L differentiate v C (t)=?, i L (t)=? Memorize equations with circuit!
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Further Analysis of Differential Equation 2 2 1 0 C C C d v dv R v dt L dt LC + + = 2 2 1 0 L L L d i di R i dt L dt LC + + = 2 2 1 0 C L d x R dx x dt L dt LC x v or i + + = = for serial RLC circuits
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Initial Conditions are given Initial conditions for the differential equations? with initial conditions: with initial conditions: KVL Ohm’s Law Notice how to make the bridge from 0 + to 0 - ! i L continuous
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Parallel RLC Circuit are given v C (t)=?, i L (t)=? Differential equations: 2 2 1 1 0 C L d x dx x dt RC dt LC x v or i + + = = for parallel RLC circuits
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Review of Source Free 2 nd Order Circuit and given with characteristic equation Problem is reduced to solving the linear differential equation: Case 1 ( two distinct real roots s 1 , s 2 ): Case 2 ( two identical real roots s 1 =s 2 ): Case 3 ( two conjugate complex roots s 1 =- σ +j ϖ d , s 2 =- σ -j ϖ d ): if both s 1 and s 2 are on the left half of the complex plane (not including the imaginary axis) stable stable unstable unstable Will be given in exams
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A Relatively Complicated Example Differential equation Characteristic equation has two complex roots Initial condition
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A Relatively Complicated Example (cont.) Differential equation Initial condition Solution: To satisfy the initial condition: At time t=1
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Characteristic equation
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This note was uploaded on 01/12/2012 for the course ECE 201 taught by Professor All during the Fall '08 term at Purdue University-West Lafayette.

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ECE201_25_2nd_order_const_input - ECE 201 Lecture 25 2nd...

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