### Example Problem on Electrical Network Analysis#### Problem Statement (E7.6):Consider the network in Fig. E7.6. If the switch opens at \( t = 0 \), find the output voltage \( v_o(t) \) for \( t \geq 0 \).#### Solution:**Given that the switch opens at \( t = 0 \), the output voltage \( v_o(t) \) is determined as follows:**\[ v_o(t) = 6 - \frac{10}{3}e^{-3t} \text{ V} \]#### Circuit Description:The figure (Figure E7.6) depicts an electrical circuit containing the following components:- A 12V voltage source.- A 4V voltage source.- Three resistors of value 2Ω each.- An inductor with an inductance of 2H.- A switch that opens at \( t = 0 \).#### Detailed Analysis:1. **Initial Setup:** - At \( t < 0 \), with the switch closed, the circuit reaches a steady-state condition. - At \( t = 0 \), the switch opens, changing the circuit dynamics. 2. **Solving for \( v_o(t) \):** - Utilizing Kirchhoff's laws and the natural response for RL circuits, the transient response occurs due to the inductor and resistors.3. **Mathematical Formulation:** - The voltage across the inductor is described by an exponential decay \( e^{-3t} \), where the constant derived is influenced by the resistor and inductor values.### Graph / Diagram Explanation:- **Circuit Diagram:** - **Top Branch:** - Begins with a 12V source and a 2Ω resistor in series. - Encountering the switch (which opens at \( t=0 \)). - Followed by a 2H inductor. - **Middle Branch:** - In parallel with the top branch. - Contains a 4V source and 2Ω resistor in series. - **Bottom Branch:** - Contains only a 2Ω resistor and connects back to the negative terminals of both the 12V and 4V sources. - **Output Voltage \( v_o(t) \):** - Measured across the rightmost 2Ω resistor in the circuit's

Answered: Image /qna-images/answer/8d1ff3a6-2c3b-4b8c-87f7-7954f369c8f6.jpg

Answered: E7.6 Consider the network in Fig. E7.6.…

E7.6 Consider the network in Fig. E7.6. If the switch opens at t = 0), find the output voltage v (1) fort > 0. Figure E7.6 12 V ww 20 2025 2H t=0 4 V 2023 vo(t) ANSWER: v (1) =6- 10 3 e² V.

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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Current Division Method

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Question

### Example Problem on Electrical Network Analysis

#### Problem Statement (E7.6):
Consider the network in Fig. E7.6. If the switch opens at \( t = 0 \), find the output voltage \( v_o(t) \) for \( t \geq 0 \).

#### Solution:
**Given that the switch opens at \( t = 0 \), the output voltage \( v_o(t) \) is determined as follows:**

\[ v_o(t) = 6 - \frac{10}{3}e^{-3t} \text{ V} \]

#### Circuit Description:
The figure (Figure E7.6) depicts an electrical circuit containing the following components:
- A 12V voltage source.
- A 4V voltage source.
- Three resistors of value 2Ω each.
- An inductor with an inductance of 2H.
- A switch that opens at \( t = 0 \).

#### Detailed Analysis:
1. **Initial Setup:**
- At \( t < 0 \), with the switch closed, the circuit reaches a steady-state condition.
- At \( t = 0 \), the switch opens, changing the circuit dynamics.

2. **Solving for \( v_o(t) \):**
- Utilizing Kirchhoff's laws and the natural response for RL circuits, the transient response occurs due to the inductor and resistors.

3. **Mathematical Formulation:**
- The voltage across the inductor is described by an exponential decay \( e^{-3t} \), where the constant derived is influenced by the resistor and inductor values.

### Graph / Diagram Explanation:
- **Circuit Diagram:**
- **Top Branch:**
- Begins with a 12V source and a 2Ω resistor in series.
- Encountering the switch (which opens at \( t=0 \)).
- Followed by a 2H inductor.
- **Middle Branch:**
- In parallel with the top branch.
- Contains a 4V source and 2Ω resistor in series.
- **Bottom Branch:**
- Contains only a 2Ω resistor and connects back to the negative terminals of both the 12V and 4V sources.
- **Output Voltage \( v_o(t) \):**
- Measured across the rightmost 2Ω resistor in the circuit's

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