1 Find V.(t) for t>0 in the circuit below and plot the response including the time internal just prior to closing the switch 202 t = 0 1 R₂ 12 V (+ 10 ο Συρι Ω 1 400 TI F 1 H +
1 Find V.(t) for t>0 in the circuit below and plot the response including the time internal just prior to closing the switch 202 t = 0 1 R₂ 12 V (+ 10 ο Συρι Ω 1 400 TI F 1 H +
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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![**Problem Statement**
Find \( V_o(t) \) for \( t > 0 \) in the circuit below and plot the response including the time interval just prior to closing the switch.
**Circuit Description:**
The given circuit comprises the following components:
- A voltage source rated at 12V
- A resistor \( R_1 \) with a resistance value of \( 2 \Omega \)
- A switch that closes at \( t = 0 \)
- A capacitor with a capacitance value of \( \frac{1}{400} \text{ F} \)
- An inductor with an inductance value of \( 1 \text{ H} \)
- A resistor \( R_2 \) with a resistance value of \( 10 \Omega \)
**Circuit Diagram:**
Below is the description of the provided circuit diagram:
- The 12V voltage source is connected in series with resistor \( R_1 \) (2Ω).
- The switch, when open, breaks the circuit connecting \( R_1 \) to the rest of the circuit to the right.
- The right half of the circuit includes, in parallel combination:
- A branch with a capacitor of value \( \frac{1}{400} \text{ F} \)
- A branch with a series combination of an inductor (1H) and resistor \( R_2 \) (10Ω), across which the output voltage \( V_o(t) \) is measured
**Steps to Solve:**
1. **Initial Conditions:**
- Analyze the circuit just before the switch is closed to understand the voltage across the capacitor and the current through the inductor.
2. **Circuit Analysis Post Switch Closure:**
- For \( t = 0^+ \), apply Kirchhoff’s laws (KVL/KCL), initially considering the capacitor and inductor voltage/current relationships.
3. **Solve the Differential Equation:**
- Develop the differential equations governing the circuit for \( t > 0 \).
- Solve for the complementary and particular solutions to the differential equation.
4. **Determine \( V_o(t) \):**
- Find \( V_o(t) \) by combining both solutions and applying boundary conditions derived from initial conditions.
5. **Plotting Response:**
- Plot the time response of \( V_o(t) \) from just before switching \( t = 0 \) to a sufficiently](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F735da1a4-726a-480c-8ef4-9ca4d85d4b7e%2F864717a5-6bda-4d06-a320-8a3ecc16c22f%2F6114dah_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find \( V_o(t) \) for \( t > 0 \) in the circuit below and plot the response including the time interval just prior to closing the switch.
**Circuit Description:**
The given circuit comprises the following components:
- A voltage source rated at 12V
- A resistor \( R_1 \) with a resistance value of \( 2 \Omega \)
- A switch that closes at \( t = 0 \)
- A capacitor with a capacitance value of \( \frac{1}{400} \text{ F} \)
- An inductor with an inductance value of \( 1 \text{ H} \)
- A resistor \( R_2 \) with a resistance value of \( 10 \Omega \)
**Circuit Diagram:**
Below is the description of the provided circuit diagram:
- The 12V voltage source is connected in series with resistor \( R_1 \) (2Ω).
- The switch, when open, breaks the circuit connecting \( R_1 \) to the rest of the circuit to the right.
- The right half of the circuit includes, in parallel combination:
- A branch with a capacitor of value \( \frac{1}{400} \text{ F} \)
- A branch with a series combination of an inductor (1H) and resistor \( R_2 \) (10Ω), across which the output voltage \( V_o(t) \) is measured
**Steps to Solve:**
1. **Initial Conditions:**
- Analyze the circuit just before the switch is closed to understand the voltage across the capacitor and the current through the inductor.
2. **Circuit Analysis Post Switch Closure:**
- For \( t = 0^+ \), apply Kirchhoff’s laws (KVL/KCL), initially considering the capacitor and inductor voltage/current relationships.
3. **Solve the Differential Equation:**
- Develop the differential equations governing the circuit for \( t > 0 \).
- Solve for the complementary and particular solutions to the differential equation.
4. **Determine \( V_o(t) \):**
- Find \( V_o(t) \) by combining both solutions and applying boundary conditions derived from initial conditions.
5. **Plotting Response:**
- Plot the time response of \( V_o(t) \) from just before switching \( t = 0 \) to a sufficiently
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