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exp6 - Tam Wilson Chi Hang ENGR 261 SP08 Experiment#6...

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Tam, Wilson Chi Hang ENGR 261, SP08 28/04/2008 Experiment #6: Second Order Transient Response of LRC Circuits Objective: This lab investigates the complete response of series RLC circuits and how numerical values of the circuit elements determine whether the response will be under-damped, over-damped, or critically damped. Theory: Second-Order circuits are named after the second-order differential equations that describe their responses. The general form of the governing second-order differential equation: ? 2 𝑥 ?? 2 + 2 𝑥 ?𝑥 ?? + 𝜔 ? 2 𝑥 = ? ? Where 𝑥 is the current or voltage response, α is the damping factor, 𝜔 ? is the undamped natural frequency, and ? ( ? ) is the forcing function For the series LRC circuit shown below: 𝛼 = ? 2 𝐿 , 𝑎?? 𝜔 ? = 1 𝐿𝐶 Figure 1 In this experiment, we will consider cases when the forcing function is a step function, and the response is a combination of a natural response (the solution with no forcing), and a forced response that is also a step function. The natural part of the solution can be one of the three forms depending on the numerical values of the circuit elements. First Case: The Underdamped Case If 𝛼 < 𝜔 ? , the response of the circuit is of the form: 𝑥 ? = ? −𝛼? A 1 cos 𝜔? + 𝐴 2 sin ( 𝜔? ) Where the constants A 1 and A 2 , are determined using initial conditions. The response of the circuit is a decaying oscillation a sinusoidal function with exponentially decaying amplitude.
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Second Case: The Overdamped Case If 𝛼 > 𝜔 ? , the response of the circuit is of the form: 𝑥 ? = 𝐴 1 ? −? 1 ? + 𝐴 2 ? −? 2 ? Where ? 1 = −𝛼 + 𝛼 2 − 𝜔 ? 2 𝑎?? ? 2 = −𝛼 − 𝛼 2 − 𝜔 ? 2 The response of the circuit is the sum of two exponentially decaying functions. Third Case: The Critically Damped Case If 𝛼 = 𝜔 ? , the response of the circuit is of the form: 𝑥 = ? −𝛼? 𝐴 1 + 𝐴 2 ? The response of the circuit is the product of an exponentially decaying function and a linear function. For times close to zero, the exponential function is close to being unity so that the response is nearly linear. For large values of time, the exponential decay dominates and the response is asymptotic to zero. Procedures: Part A : The LC Tank Figure 2 The simplest second- order circuit is the “LC Tank” shown above, consisting of an ideal inductor and an ideal capacitor. The energy in the circuit comes from an initial current in the inductor, or an initial voltage in the capacitor. The response of the circuit is a sinusoidal function at a natural frequency of 𝜔 ? = 1/ 𝐿𝐶 . 1.) Compute the natural frequency of the LC tank for L=10mH, and C=0.1μF. 2.) Setup and run a MultiSIM simulation of a circuit in Figure 2. Initial Voltage =2V. Set an appropriate time step and time range to plot about five to ten oscillations of the voltage across the capacitor.
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