covered in class only!) a) Find the differential (loop) equation relating output "y(t)" to input "x(t)". b) Find the zero input component of the loop current y (t). ( y.(0)-0,y (0)-2) c) Find the zero-state input component of the loop current y(t). The system impulse response is h(t) = (−2e¹²¹ + 3e™¹³¹)u(t) (I.E, find the system response y(t) to the external input x(t)). X(t) alt)
covered in class only!) a) Find the differential (loop) equation relating output "y(t)" to input "x(t)". b) Find the zero input component of the loop current y (t). ( y.(0)-0,y (0)-2) c) Find the zero-state input component of the loop current y(t). The system impulse response is h(t) = (−2e¹²¹ + 3e™¹³¹)u(t) (I.E, find the system response y(t) to the external input x(t)). X(t) alt)
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
![### RLC Circuit Analysis
#### Problem Statement
5. A voltage \( x(t) = e^{-2t}u(t) \) is applied at the input of the RLC circuit illustrated below.
- Given:
- \( L = 1 \, \text{H} \)
- \( R = 5 \, \Omega \)
- \( C = \frac{1}{6} \, \text{F} \)
(*Use methods that we covered in class only!*)
##### Tasks:
a) **Find the differential (loop) equation relating output \( y(t) \) to input \( x(t) \).**
b) **Find the zero input component of the loop current \( y_o(t) \).**
Given conditions: \( y_o(0^-)=0,\,\dot{y}_o(0^-)=2 \)
c) **Find the zero-state input component of the loop current \( y_I(t) \).**
The system impulse response is:
\[ h(t) = (-2e^{-2t} + 3e^{-3t})u(t) \]
(**I.E., find the system response \( y_I(t) \) to the external input \( x(t) \).**)
#### Circuit Diagram
The circuit diagram consists of a series combination of an inductor \( L \), resistor \( R \), and capacitor \( C \). The voltage source \( x(t) \) is applied across the series combination, and the loop current \( y(t) \) flows through the elements.
- **L**: Inductor
- **R**: Resistor
- **C**: Capacitor
- **\(x(t)\)**: Input voltage
- **\(y(t)\)**: Loop current through the circuit
The problem requires the differential equation relating the input and output, as well as the zero input and zero-state responses of the loop current.
#### Detailed Steps:
1. **Formulate the differential equation** for \( y(t) \).
2. **Solve for the zero input response \( y_o(t) \)** using initial conditions.
3. **Determine the zero-state response \( y_I(t) \)** using the given impulse response \( h(t) \).
#### Important Note
- Focus on methods covered in](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fecd9d238-92a1-4a47-af9e-c758e887edd1%2Fb7c43bb7-724e-4312-9be8-e449e9370a96%2Fbvumwso_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### RLC Circuit Analysis
#### Problem Statement
5. A voltage \( x(t) = e^{-2t}u(t) \) is applied at the input of the RLC circuit illustrated below.
- Given:
- \( L = 1 \, \text{H} \)
- \( R = 5 \, \Omega \)
- \( C = \frac{1}{6} \, \text{F} \)
(*Use methods that we covered in class only!*)
##### Tasks:
a) **Find the differential (loop) equation relating output \( y(t) \) to input \( x(t) \).**
b) **Find the zero input component of the loop current \( y_o(t) \).**
Given conditions: \( y_o(0^-)=0,\,\dot{y}_o(0^-)=2 \)
c) **Find the zero-state input component of the loop current \( y_I(t) \).**
The system impulse response is:
\[ h(t) = (-2e^{-2t} + 3e^{-3t})u(t) \]
(**I.E., find the system response \( y_I(t) \) to the external input \( x(t) \).**)
#### Circuit Diagram
The circuit diagram consists of a series combination of an inductor \( L \), resistor \( R \), and capacitor \( C \). The voltage source \( x(t) \) is applied across the series combination, and the loop current \( y(t) \) flows through the elements.
- **L**: Inductor
- **R**: Resistor
- **C**: Capacitor
- **\(x(t)\)**: Input voltage
- **\(y(t)\)**: Loop current through the circuit
The problem requires the differential equation relating the input and output, as well as the zero input and zero-state responses of the loop current.
#### Detailed Steps:
1. **Formulate the differential equation** for \( y(t) \).
2. **Solve for the zero input response \( y_o(t) \)** using initial conditions.
3. **Determine the zero-state response \( y_I(t) \)** using the given impulse response \( h(t) \).
#### Important Note
- Focus on methods covered in
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