The image shows an electrical circuit used to determine the output voltage \( v_o(t) \) using the differential equation method. Here is the description and analysis of the circuit:### Circuit Description:1. **Voltage Source (\( V_S \))**: - \( V_S = 18V \)2. **Resistors**: - \( R_1 = 3k\Omega \) - \( R_2 = 6k\Omega \) - \( R_3 = 12k\Omega \) - \( R_4 = 6k\Omega \)3. **Capacitor (\( C \))**: - \( C = 100\mu F \)4. **Switch**: - Initially open, it closes at \( t = 0 \).### Circuit Configuration:- The circuit is powered by the voltage source \( V_S \).- \( R_1 \) is in series with the switch and \( R_2 \) is in parallel to the path containing \( R_3 \) and the capacitor \( C \).- \( R_3 \) is in series with the parallel arrangement of the capacitor \( C \) and resistor \( R_4 \).- The output voltage \( v_o(t) \) is measured across \( R_4 \).### Objective:- To determine the output voltage \( v_o(t) \) for \( t > 0 \) using the differential equation method.### Explanation:When the switch closes at \( t = 0 \), the circuit becomes a first-order RC circuit. A differential equation describing the voltage across the capacitor, and hence \( v_o(t) \), can be derived based on the circuit’s time constant and the resistive and capacitive components involved. This allows for calculating the transient response of the circuit, helping in understanding how \( v_o(t) \) evolves over time.

The given circuit isWe need to determine the output voltage, vo(t) using differential equation…

n the following circuit below, at t = 0 the switch closes: R₁ ww R3 ww + Vs Vs = 18V Xt=0 m R₂ R₁ = 3kN R4 = 6kN C R₂ = 6kN C = 100μF vo(t). R3 = 12kN RA Use the differential equation method to determine the output voltage

n the following circuit below, at t = 0 the switch closes: R₁ ww R3 ww + Vs Vs = 18V Xt=0 m R₂ R₁ = 3kN R4 = 6kN C R₂ = 6kN C = 100μF vo(t). R3 = 12kN RA Use the differential equation method to determine the output voltage

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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Question

The image shows an electrical circuit used to determine the output voltage \( v_o(t) \) using the differential equation method. Here is the description and analysis of the circuit:

### Circuit Description:

1. **Voltage Source (\( V_S \))**:
- \( V_S = 18V \)

2. **Resistors**:
- \( R_1 = 3k\Omega \)
- \( R_2 = 6k\Omega \)
- \( R_3 = 12k\Omega \)
- \( R_4 = 6k\Omega \)

3. **Capacitor (\( C \))**:
- \( C = 100\mu F \)

4. **Switch**:
- Initially open, it closes at \( t = 0 \).

### Circuit Configuration:

- The circuit is powered by the voltage source \( V_S \).
- \( R_1 \) is in series with the switch and \( R_2 \) is in parallel to the path containing \( R_3 \) and the capacitor \( C \).
- \( R_3 \) is in series with the parallel arrangement of the capacitor \( C \) and resistor \( R_4 \).
- The output voltage \( v_o(t) \) is measured across \( R_4 \).

### Objective:

- To determine the output voltage \( v_o(t) \) for \( t > 0 \) using the differential equation method.

### Explanation:

When the switch closes at \( t = 0 \), the circuit becomes a first-order RC circuit. A differential equation describing the voltage across the capacitor, and hence \( v_o(t) \), can be derived based on the circuit’s time constant and the resistive and capacitive components involved. This allows for calculating the transient response of the circuit, helping in understanding how \( v_o(t) \) evolves over time.

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