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:

4 Parallel RLC Circuit There is an initial voltage across the capacitor. There is an initial current thorough the inductor.

5 Applying Kirchhoff’s Current Law at the top node:

6 Differentiating with respect to t gives :

7 Or:

8 This is the second-order, linear differential equation that describes the behavior of the circuit.

9 Based on experience with first-order circuits, we assume a solution of the form:

10 Substitute the assumed solution into the differential equation:

11 Substitute the assumed solution into the differential equation:

12 This may be factored:

13 This requires that:

14 This is called the Characteristic Equation of the differential equation The Characteristic equation determines the behavior of the circuit.

15 This equation has roots given by the quadratic equation:

16 The two roots are given by:

17 These are the Characteristic Roots of the differential equation.

18 To aid in characterizing the equation we define:

19 So the Characteristic Roots may be written:

20 The solution to the differential equation is:

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.

22 Over-damped Condition

23 Over-damped Condition

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30 Critically Damped Condition

31 Critically Damped Condition

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37 Under-damped Condition

38 Under-damped Condition

39 Under-damped Condition

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