An RLC circuit with L = 1 h of inductance, R = 40 of resistance, and C = 0.25 f of capacitance has an initial charge of 1 c on the capacitor and an initial current 0 Amps. 5 V DC is impressed on the system. The system can be modeled by the following initial value problem: q" +4q' + 4q = 5,q (0) = 1, i (0) = 0 a. Solve the initial value problem. You may use any method you would like. (I recommend undetermined coefficients.) b. Describe the circuit as overdamped, underdamped, or critically damped. Explain how you know. c. Describe the long-term, steady-state solution for this system. Write a brief explanation. d. Identify the plot that best illustrates this system.
An RLC circuit with L = 1 h of inductance, R = 40 of resistance, and C = 0.25 f of capacitance has an initial charge of 1 c on the capacitor and an initial current 0 Amps. 5 V DC is impressed on the system. The system can be modeled by the following initial value problem: q" +4q' + 4q = 5,q (0) = 1, i (0) = 0 a. Solve the initial value problem. You may use any method you would like. (I recommend undetermined coefficients.) b. Describe the circuit as overdamped, underdamped, or critically damped. Explain how you know. c. Describe the long-term, steady-state solution for this system. Write a brief explanation. d. Identify the plot that best illustrates this system.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![## Problem 2: RLC Circuit Analysis
An RLC circuit with the following parameters is considered:
- Inductance (\(L\)) = 1 henry
- Resistance (\(R\)) = 4 ohms
- Capacitance (\(C\)) = 0.25 farads
The circuit has:
- An initial charge (\(q\)) of 1 coulomb on the capacitor
- An initial current of 0 amperes
- A 5 V DC source impressed on the system
The system can be modeled by the following initial value problem:
\[ q'' + 4q' + 4q = 5, \quad q(0) = 1, \quad i(0) = 0 \]
### Tasks:
a. Solve the initial value problem. You may use any method you would like. (I recommend undetermined coefficients.)
b. Describe the circuit as overdamped, underdamped, or critically damped. Explain how you know.
c. Describe the long-term, steady-state solution for this system. Write a brief explanation.
d. Identify the plot that best illustrates this system.
### Graphs:
**Plot A**:
- The plot features a curve starting at roughly 1.2, slightly increasing before stabilizing at 1.5.
- The graph has labeled axis markings, with increments of 0.5 on the vertical, and 0.5 on the horizontal scale, up to 2 on the x-axis.
**Plot B**:
- This plot displays a damped oscillation pattern.
- The red and black curves show oscillation, gradually level off, suggesting a settling over time.
- Axis markings show increments similar to Plot A, with the x-axis extending to 3.5. Curves cross the x-axis at various points, indicating oscillation behavior.
Select the graph (A or B) that best illustrates the system based on your solution and explanations from parts (a) to (c).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9454a43d-48d8-460c-9970-8c1a520f5a48%2F88f245a9-e9b8-4a25-800c-9355a598b5aa%2Fhbtcl5h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Problem 2: RLC Circuit Analysis
An RLC circuit with the following parameters is considered:
- Inductance (\(L\)) = 1 henry
- Resistance (\(R\)) = 4 ohms
- Capacitance (\(C\)) = 0.25 farads
The circuit has:
- An initial charge (\(q\)) of 1 coulomb on the capacitor
- An initial current of 0 amperes
- A 5 V DC source impressed on the system
The system can be modeled by the following initial value problem:
\[ q'' + 4q' + 4q = 5, \quad q(0) = 1, \quad i(0) = 0 \]
### Tasks:
a. Solve the initial value problem. You may use any method you would like. (I recommend undetermined coefficients.)
b. Describe the circuit as overdamped, underdamped, or critically damped. Explain how you know.
c. Describe the long-term, steady-state solution for this system. Write a brief explanation.
d. Identify the plot that best illustrates this system.
### Graphs:
**Plot A**:
- The plot features a curve starting at roughly 1.2, slightly increasing before stabilizing at 1.5.
- The graph has labeled axis markings, with increments of 0.5 on the vertical, and 0.5 on the horizontal scale, up to 2 on the x-axis.
**Plot B**:
- This plot displays a damped oscillation pattern.
- The red and black curves show oscillation, gradually level off, suggesting a settling over time.
- Axis markings show increments similar to Plot A, with the x-axis extending to 3.5. Curves cross the x-axis at various points, indicating oscillation behavior.
Select the graph (A or B) that best illustrates the system based on your solution and explanations from parts (a) to (c).
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