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+ switch 240 V 4 Ω ΖΩ 0.5 F 2 H The switch has been closed for a long time. It is opened at t = 0. Use Laplace transform method to find the voltage across the capacitor v (t), for t≥ 0.

+ switch 240 V 4 Ω ΖΩ 0.5 F 2 H The switch has been closed for a long time. It is opened at t = 0. Use Laplace transform method to find the voltage across the capacitor v (t), for t≥ 0.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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**Example of RLC Circuit Analysis Using Laplace Transforms** **Circuit Explanation:** The provided RLC circuit consists of: - A DC voltage source of 240 V. - A switch that has been closed for a long time and is opened at \( t = 0 \). - A series combination of a 4 Ω resistor and a parallel combination of a 0.5 F capacitor with a series combination of a 2 Ω resistor and a 2 H inductor. **Objective:** To determine the voltage across the capacitor, denoted as \( v_c(t) \), for \( t \geq 0 \), using the Laplace transform method. **Steps for Solution:** 1. **Initial Conditions**: Since the switch has been closed for a long time, the capacitor would be fully charged, and the inductor would behave as a short circuit in the steady state. 2. **Laplace Transform Application**: - Apply the Laplace transform to the circuit, converting it into the s-domain. - The initial voltage across the capacitor, \( V_{c}(0^-) \), can be found based on the steady state of the circuit before \( t = 0 \). 3. **RLC Circuit in s-domain**: - Transform the capacitor and inductor to their s-domain equivalents. - The inductor's s-domain representation will include its initial current as a dependent source. - Similarly, the capacitor's s-domain representation will include its initial voltage as a dependent source. 4. **Solve for \( V_c(s) \)**: - Using Kirchoff's laws, or network analysis techniques like Mesh Analysis or Nodal Analysis, formulate the equations in the s-domain. - Solve for the Laplace transformed voltage \( V_c(s) \). 5. **Inverse Laplace Transform**: - Find the Inverse Laplace Transform of \( V_c(s) \) to obtain \( v_c(t) \). **Instructions:** - For detailed understanding, please refer to textbooks or course materials on Laplace Transforms and RLC Circuit Analysis. This problem demonstrates the application of Laplace Transforms in analyzing RLC circuits, particularly in transitioning from steady-state to transient conditions.
Transcribed Image Text:**Example of RLC Circuit Analysis Using Laplace Transforms** **Circuit Explanation:** The provided RLC circuit consists of: - A DC voltage source of 240 V. - A switch that has been closed for a long time and is opened at \( t = 0 \). - A series combination of a 4 Ω resistor and a parallel combination of a 0.5 F capacitor with a series combination of a 2 Ω resistor and a 2 H inductor. **Objective:** To determine the voltage across the capacitor, denoted as \( v_c(t) \), for \( t \geq 0 \), using the Laplace transform method. **Steps for Solution:** 1. **Initial Conditions**: Since the switch has been closed for a long time, the capacitor would be fully charged, and the inductor would behave as a short circuit in the steady state. 2. **Laplace Transform Application**: - Apply the Laplace transform to the circuit, converting it into the s-domain. - The initial voltage across the capacitor, \( V_{c}(0^-) \), can be found based on the steady state of the circuit before \( t = 0 \). 3. **RLC Circuit in s-domain**: - Transform the capacitor and inductor to their s-domain equivalents. - The inductor's s-domain representation will include its initial current as a dependent source. - Similarly, the capacitor's s-domain representation will include its initial voltage as a dependent source. 4. **Solve for \( V_c(s) \)**: - Using Kirchoff's laws, or network analysis techniques like Mesh Analysis or Nodal Analysis, formulate the equations in the s-domain. - Solve for the Laplace transformed voltage \( V_c(s) \). 5. **Inverse Laplace Transform**: - Find the Inverse Laplace Transform of \( V_c(s) \) to obtain \( v_c(t) \). **Instructions:** - For detailed understanding, please refer to textbooks or course materials on Laplace Transforms and RLC Circuit Analysis. This problem demonstrates the application of Laplace Transforms in analyzing RLC circuits, particularly in transitioning from steady-state to transient conditions.
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