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For the circuit shown above, determine the following:

For the circuit shown above, determine the following:

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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### Circuit Analysis Problem **Circuit Diagram:** - The diagram shows a series-parallel circuit with a 12 V voltage source, a switch, a capacitor \( C = 100 \, \mu F \), and resistors. - Resistor values: - \( 3 \, k\Omega \) - \( 2 \, k\Omega \) - \( 4 \, k\Omega \) The switch is initially closed at \( t = 0 \). **Tasks:** For the circuit shown above, determine the following: i. **Thevenin Resistance**: What is the Thevenin resistance of the circuit represented to the capacitor when the switch is open? - Answer: \( \_\_\_ \, k\Omega \) ii. **Time Constant (\(\tau\))**: What is the time constant for the circuit? - Answer: \( \_\_\_ \, s \) iii. **Initial Capacitor Voltage**: What is the voltage across the capacitor \( V_C(t) \) at \( t = 0^- \) (just before the switch is opened)? - Answer: \( \_\_\_ \, V \) iv. **Final Capacitor Voltage**: After the switch is opened, what is the value of \( V_C(\infty) \)? - Answer: \( \_\_\_ \, V \) v. **Capacitor Voltage Equation**: Write the equation for \( V_C(t) \) in terms of time \( t \). - Equation: \( V_C(t) = \_\_\_ \, V \) #### Explanation: - **Thevenin Resistance Calculation**: Determine the equivalent resistance seen by the capacitor when the switch is open. - **Time Constant Calculation**: Given by \(\tau = R_{\text{th}} \times C\), where \( R_{\text{th}} \) is the Thevenin resistance. - **Initial and Final Voltages**: Calculate the initial and steady-state voltages across the capacitor based on circuit conditions. - **Voltage Equation**: Derive \( V_C(t) \) using the exponential decay formula \( V_C(t) = V_C(0) \times e^{-t/\tau} \) for a discharging capacitor.
Transcribed Image Text:### Circuit Analysis Problem **Circuit Diagram:** - The diagram shows a series-parallel circuit with a 12 V voltage source, a switch, a capacitor \( C = 100 \, \mu F \), and resistors. - Resistor values: - \( 3 \, k\Omega \) - \( 2 \, k\Omega \) - \( 4 \, k\Omega \) The switch is initially closed at \( t = 0 \). **Tasks:** For the circuit shown above, determine the following: i. **Thevenin Resistance**: What is the Thevenin resistance of the circuit represented to the capacitor when the switch is open? - Answer: \( \_\_\_ \, k\Omega \) ii. **Time Constant (\(\tau\))**: What is the time constant for the circuit? - Answer: \( \_\_\_ \, s \) iii. **Initial Capacitor Voltage**: What is the voltage across the capacitor \( V_C(t) \) at \( t = 0^- \) (just before the switch is opened)? - Answer: \( \_\_\_ \, V \) iv. **Final Capacitor Voltage**: After the switch is opened, what is the value of \( V_C(\infty) \)? - Answer: \( \_\_\_ \, V \) v. **Capacitor Voltage Equation**: Write the equation for \( V_C(t) \) in terms of time \( t \). - Equation: \( V_C(t) = \_\_\_ \, V \) #### Explanation: - **Thevenin Resistance Calculation**: Determine the equivalent resistance seen by the capacitor when the switch is open. - **Time Constant Calculation**: Given by \(\tau = R_{\text{th}} \times C\), where \( R_{\text{th}} \) is the Thevenin resistance. - **Initial and Final Voltages**: Calculate the initial and steady-state voltages across the capacitor based on circuit conditions. - **Voltage Equation**: Derive \( V_C(t) \) using the exponential decay formula \( V_C(t) = V_C(0) \times e^{-t/\tau} \) for a discharging capacitor.
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