Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
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Question
**Title: Calculating the Time Constant in RC Circuits** **Introduction:** In this section, we aim to determine the time constant, \( \tau \), of the output current, \( i_o(t) \), in a given electrical circuit configuration. The circuit is depicted in Figure P7.7, and specific component values are provided for calculations. **Circuit Description (Figure P7.7):** The circuit consists of the following components: - A voltage source with a potential difference of 6V. - Four resistors with the following values: - \( R1 = 12 \, \text{k}\Omega \) - \( R2 = 8 \, \text{k}\Omega \) - \( R3 = 9 \, \text{k}\Omega \) - \( R4 = 6 \, \text{k}\Omega \) - A capacitor with a capacitance \( C = 187 \, \mu \text{F} \). **Circuit Analysis:** - The resistors \( R1 \), \( R2 \), \( R3 \), and \( R4 \) are connected in a combination that affects how the capacitor charges and discharges. - The output current \( i_o(t) \) is influenced by the time constant \( \tau \) of the RC circuit, which is crucial for understanding transient response. - The switch in the circuit is initially open and closes at \( t = 0 \) to initiate the circuit's transient response. **Objective:** Determine the time constant \( \tau \), which characterizes how quickly the circuit responds to the switching event, primarily influenced by \( R \) and \( C \) values. **Conclusion:** The time constant \( \tau = R_{\text{eq}} \cdot C \), where \( R_{\text{eq}} \) is the equivalent resistance affecting the capacitor. Identifying \( R_{\text{eq}} \) requires combining resistances based on the configuration and simplified using series-parallel resistance calculations. Understanding these parameters assists in predicting the circuit's behavior over time, a critical aspect of electronic design and analysis.
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Transcribed Image Text:**Title: Calculating the Time Constant in RC Circuits** **Introduction:** In this section, we aim to determine the time constant, \( \tau \), of the output current, \( i_o(t) \), in a given electrical circuit configuration. The circuit is depicted in Figure P7.7, and specific component values are provided for calculations. **Circuit Description (Figure P7.7):** The circuit consists of the following components: - A voltage source with a potential difference of 6V. - Four resistors with the following values: - \( R1 = 12 \, \text{k}\Omega \) - \( R2 = 8 \, \text{k}\Omega \) - \( R3 = 9 \, \text{k}\Omega \) - \( R4 = 6 \, \text{k}\Omega \) - A capacitor with a capacitance \( C = 187 \, \mu \text{F} \). **Circuit Analysis:** - The resistors \( R1 \), \( R2 \), \( R3 \), and \( R4 \) are connected in a combination that affects how the capacitor charges and discharges. - The output current \( i_o(t) \) is influenced by the time constant \( \tau \) of the RC circuit, which is crucial for understanding transient response. - The switch in the circuit is initially open and closes at \( t = 0 \) to initiate the circuit's transient response. **Objective:** Determine the time constant \( \tau \), which characterizes how quickly the circuit responds to the switching event, primarily influenced by \( R \) and \( C \) values. **Conclusion:** The time constant \( \tau = R_{\text{eq}} \cdot C \), where \( R_{\text{eq}} \) is the equivalent resistance affecting the capacitor. Identifying \( R_{\text{eq}} \) requires combining resistances based on the configuration and simplified using series-parallel resistance calculations. Understanding these parameters assists in predicting the circuit's behavior over time, a critical aspect of electronic design and analysis.
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