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#13: The switch has been open a long time (no current) and then closes at t = 0. Find the voltages v1 and v2, currents il and i2 each at the following three times: just before the switch closes, just after the switch closes and a long time after the switch closes.

#13: The switch has been open a long time (no current) and then closes at t = 0. Find the voltages v1 and v2, currents il and i2 each at the following three times: just before the switch closes, just after the switch closes and a long time after the switch closes.

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
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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Subject: Fundamentals of circuit analysis

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**Transcription for Educational Website:** --- ### Transient Analysis of Capacitor Circuits #### Initial Conditions: \( t < 0 \) - **Capacitors' Behavior:** In DC conditions, capacitors act like an open circuit with no current flow. - **Voltages and Currents:** - \( i_1(0^-) = 0 \) - \( i_2(0^-) = 0 \) - \( v_1(0^-) = 60 \, \text{Volt} \) - \( v_2(0^-) = 60 \, \text{Volt} \) #### At the Moment After Switching: \( t = 0^+ \) - **Continuity of Capacitor Voltage:** The capacitor voltage remains continuous; hence, there is no change. - **Voltages:** - \( v_1(0^+) = 60 \, \text{Volt} \) - \( v_2(0^+) = 60 \, \text{Volt} \) - **Current Calculations Using Ohm’s Law:** - \( i_1(0^+) = \frac{-60}{30} = -2 \, \text{Amp} \) - **Kirchhoff's Current Law (KCL) at Top Middle Node:** - \( i_2(0^+) = 0 \) #### Behavior Over Time: \( t \rightarrow \infty \) - **Capacitors' Behavior:** In DC conditions, capacitors act like an open circuit with no current flow. - **Voltages and Currents:** - \( i_1(\infty) = 0 \) - \( i_2(\infty) = 0 \) - **Voltage Divider Principle:** - \( V_1(\infty) = 30 \, \text{Volt} \) - **Using Kirchhoff's Voltage Law (KVL):** - \( V_2(\infty) = 40 \, \text{Volt} \) --- This analysis outlines the foundational behavior of capacitors in DC circuits, emphasizing initial and steady-state conditions. It is critical for understanding how capacitors impact current and voltage over time in electrical circuits.
Transcribed Image Text:**Transcription for Educational Website:** --- ### Transient Analysis of Capacitor Circuits #### Initial Conditions: \( t < 0 \) - **Capacitors' Behavior:** In DC conditions, capacitors act like an open circuit with no current flow. - **Voltages and Currents:** - \( i_1(0^-) = 0 \) - \( i_2(0^-) = 0 \) - \( v_1(0^-) = 60 \, \text{Volt} \) - \( v_2(0^-) = 60 \, \text{Volt} \) #### At the Moment After Switching: \( t = 0^+ \) - **Continuity of Capacitor Voltage:** The capacitor voltage remains continuous; hence, there is no change. - **Voltages:** - \( v_1(0^+) = 60 \, \text{Volt} \) - \( v_2(0^+) = 60 \, \text{Volt} \) - **Current Calculations Using Ohm’s Law:** - \( i_1(0^+) = \frac{-60}{30} = -2 \, \text{Amp} \) - **Kirchhoff's Current Law (KCL) at Top Middle Node:** - \( i_2(0^+) = 0 \) #### Behavior Over Time: \( t \rightarrow \infty \) - **Capacitors' Behavior:** In DC conditions, capacitors act like an open circuit with no current flow. - **Voltages and Currents:** - \( i_1(\infty) = 0 \) - \( i_2(\infty) = 0 \) - **Voltage Divider Principle:** - \( V_1(\infty) = 30 \, \text{Volt} \) - **Using Kirchhoff's Voltage Law (KVL):** - \( V_2(\infty) = 40 \, \text{Volt} \) --- This analysis outlines the foundational behavior of capacitors in DC circuits, emphasizing initial and steady-state conditions. It is critical for understanding how capacitors impact current and voltage over time in electrical circuits.
**Question #13:** The switch in the circuit has been open for a long time (resulting in no current flow) and then closes at time \( t = 0 \). Determine the voltages \( v_1 \) and \( v_2 \), and the currents \( i_1 \) and \( i_2 \) at three specific instances: right before the switch closes, immediately after it closes, and a long duration after the switch has closed. **Circuit Diagram Details:** - **Components:** - One 30 Ω resistor in series with a capacitor \( C_1 \). - A 10 Ω resistor. - A 50 Ω resistor. - A 20 Ω resistor. - A 60 V voltage source. - Two capacitors are marked with voltages \( v_1 \) and \( v_2 \). - The currents flowing through the capacitors are labeled as \( i_1 \) and \( i_2 \), respectively. - **Configuration:** - The switch is initially open, interrupting the path and preventing current flow through the circuit. - Upon closing the switch (at \( t = 0 \)), the circuit becomes a closed loop allowing current to flow. **Analysis:** - Analyze the circuit in three timeframes: 1. **Just before the switch closes:** Assuming initial conditions with capacitors fully discharged or in a steady state without current flow. 2. **Just after the switch closes:** Consider transient analysis as the circuit begins to conduct current, affecting voltages across capacitors and current through resistors. 3. **A long time after switch closure:** Likely reaching a steady state where capacitors are fully charged according to the DC steady state. The analysis helps in understanding transient and steady-state behavior in RLC circuits.
Transcribed Image Text:**Question #13:** The switch in the circuit has been open for a long time (resulting in no current flow) and then closes at time \( t = 0 \). Determine the voltages \( v_1 \) and \( v_2 \), and the currents \( i_1 \) and \( i_2 \) at three specific instances: right before the switch closes, immediately after it closes, and a long duration after the switch has closed. **Circuit Diagram Details:** - **Components:** - One 30 Ω resistor in series with a capacitor \( C_1 \). - A 10 Ω resistor. - A 50 Ω resistor. - A 20 Ω resistor. - A 60 V voltage source. - Two capacitors are marked with voltages \( v_1 \) and \( v_2 \). - The currents flowing through the capacitors are labeled as \( i_1 \) and \( i_2 \), respectively. - **Configuration:** - The switch is initially open, interrupting the path and preventing current flow through the circuit. - Upon closing the switch (at \( t = 0 \)), the circuit becomes a closed loop allowing current to flow. **Analysis:** - Analyze the circuit in three timeframes: 1. **Just before the switch closes:** Assuming initial conditions with capacitors fully discharged or in a steady state without current flow. 2. **Just after the switch closes:** Consider transient analysis as the circuit begins to conduct current, affecting voltages across capacitors and current through resistors. 3. **A long time after switch closure:** Likely reaching a steady state where capacitors are fully charged according to the DC steady state. The analysis helps in understanding transient and steady-state behavior in RLC circuits.
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