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1 ECE 3301 General Electrical Engineering Section 30 Natural Response of a Parallel RLC Circuit
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3 Parallel RLC Circuit Consider the parallel RLC circuit:
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4 Parallel RLC Circuit There is an initial voltage across the capacitor. There is an initial current thorough the inductor.
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5 Applying Kirchhoff’s Current Law at the top node:
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6 Differentiating with respect to t gives :
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7 Or:
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8 This is the second-order, linear differential equation that describes the behavior of the circuit.
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9 Based on experience with first-order circuits, we assume a solution of the form:
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10 Substitute the assumed solution into the differential equation:
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11 Substitute the assumed solution into the differential equation:
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12 This may be factored:
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13 This requires that:
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14 This is called the Characteristic Equation of the differential equation The Characteristic equation determines the behavior of the circuit.
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15 This equation has roots given by the quadratic equation:
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16 The two roots are given by:
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17 These are the Characteristic Roots of the differential equation.
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18 To aid in characterizing the equation we define:
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19 So the Characteristic Roots may be written:
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20 The solution to the differential equation is:
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21 The solution will be different depending on the roots of the characteristic equation. There are three cases 1.Real, un-equal roots – The Over-damped Condition. 2.Real, equal roots – The Critically Damped Condition. 3.Complex Conjugate Roots – The Under-damped Condition.
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22 Over-damped Condition
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23 Over-damped Condition
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30 Critically Damped Condition
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31 Critically Damped Condition
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37 Under-damped Condition
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38 Under-damped Condition
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39 Under-damped Condition
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