EE310 Lecture 5.pptx - EE310 Lecture 5 8.7 General...

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EE310 Lecture 5
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8.7 General Second-Order Circuits Now we are prepared to apply the ideas to any second-order circuit having one or more independent sources with constant values. Although the series and parallel RLC circuits are the second-order circuits of greatest interest. Given a second-order circuit, we determine its step response (which may be voltage or current) by taking the following four steps: 1. We first determine the initial conditions and and the final value as discussed in Section 8.2. 2. We turn off the independent sources and find the form of the transient response by applying KCL and KVL. Once a second-order differential equation ( homogeneous ) is obtained, we determine its characteristic roots. Depending on whether the response is overdamped, critically damped, or underdamped , we obtain with two unknown constants as we did in the previous sections. t x 0 x dt dx / 0 x t x 1 t x 1
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3. We obtain the steady-state response as Where is the final value of x , obtained in step 1. 4. The total response ( solution of the nonhomogenous differential equation ) is now found as the sum of the transient response and steady-state response We finally determine the constants associated with the transient response by imposing the initial conditions and determined in step 1. We can apply this general procedure to find the step response of any second-order circuit. The following examples illustrate the four steps. 51 . 8 x t x ss x 52 . 8 1 t x t x t x ss 0 x dt t dx /
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Example 8.9 Find the complete response v and then i for t 0 ˃ in the circuit of Fig. 8.25 Solution: We first find the initial and final values. At the circuit is at steady state. The switch is open; the equivalent circuit is shown in Fig. 8.26(a).
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