R₁ 0.75 X R3 W 0.52 i(t) R₂ · 0,5 Ω www 2H Hii v(t) 1 C 0.5 F
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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Question
![**Circuit Analysis Problem**
**Problem Statement:**
The switch in the circuit shown in the figure below has been closed for a long time before it is opened at \( t = 0 \). Determine the current \( i(t) \) for \( t \ge 0 \).
**Circuit Diagram:**
- The circuit consists of a voltage source \( V_s = 4 \text{ V} \), a series resistor \( R_1 = 0.75 \Omega \), two parallel resistors \( R_2 = 0.5 \Omega \) and \( R_3 = 0.5 \Omega \), an inductor \( L = 2 \text{ H} \), and a capacitor \( C = 0.5 \text{ F} \).
- Before \( t = 0 \), the switch is closed, allowing the circuit to reach a steady state.
- At \( t = 0 \), the switch is opened, and the current \( i(t) \) through the inductor and the voltage \( v(t) \) across the capacitor are analyzed for \( t \ge 0 \).
**Figure:**
```
___ ___
+4V ___|R1|_____|R3|___ v(t)
0.75Ω 0.5Ω |____ ____
| | |
_|_ R2 C
L 0.5Ω 0.5F
2H | |
_|_ ______
___ ___
```
**Analysis:**
To find the current through the inductor \( i(t) \), the characteristic equation of the differential equation describing the circuit dynamics can be expressed as:
\[ s^2 + a_1s + a_0 = 0 \]
**Question:**
What is the value of \( a_1 \) in the characteristic equation?
**Explanation of Graphs/Diagrams:**
The circuit diagram shown above is a representation of the electrical setup under analysis. This circuit includes a voltage source \( V_s = 4 \text{ V} \) that powers the circuit. The resistor \( R_1 = 0.75 \Omega \) is in series with the parallel combination of resistors \( R_2 = 0.5 \Omega \) and \( R_3 =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdaad9e7e-1f14-44d5-98ac-51623b651f9b%2F249de073-a449-4d02-841b-bb9b6fabed95%2Fhl47f9g_processed.png&w=3840&q=75)
Transcribed Image Text:**Circuit Analysis Problem**
**Problem Statement:**
The switch in the circuit shown in the figure below has been closed for a long time before it is opened at \( t = 0 \). Determine the current \( i(t) \) for \( t \ge 0 \).
**Circuit Diagram:**
- The circuit consists of a voltage source \( V_s = 4 \text{ V} \), a series resistor \( R_1 = 0.75 \Omega \), two parallel resistors \( R_2 = 0.5 \Omega \) and \( R_3 = 0.5 \Omega \), an inductor \( L = 2 \text{ H} \), and a capacitor \( C = 0.5 \text{ F} \).
- Before \( t = 0 \), the switch is closed, allowing the circuit to reach a steady state.
- At \( t = 0 \), the switch is opened, and the current \( i(t) \) through the inductor and the voltage \( v(t) \) across the capacitor are analyzed for \( t \ge 0 \).
**Figure:**
```
___ ___
+4V ___|R1|_____|R3|___ v(t)
0.75Ω 0.5Ω |____ ____
| | |
_|_ R2 C
L 0.5Ω 0.5F
2H | |
_|_ ______
___ ___
```
**Analysis:**
To find the current through the inductor \( i(t) \), the characteristic equation of the differential equation describing the circuit dynamics can be expressed as:
\[ s^2 + a_1s + a_0 = 0 \]
**Question:**
What is the value of \( a_1 \) in the characteristic equation?
**Explanation of Graphs/Diagrams:**
The circuit diagram shown above is a representation of the electrical setup under analysis. This circuit includes a voltage source \( V_s = 4 \text{ V} \) that powers the circuit. The resistor \( R_1 = 0.75 \Omega \) is in series with the parallel combination of resistors \( R_2 = 0.5 \Omega \) and \( R_3 =
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