V (o)=0 VG) If B=0 = • Asin Bix VG-C)=√₁₂ = V₂ - Asin BL Asin BL A = -√₂ Sinßl Ta=jV₂ Prove -j Vs Ces Bx Z₂ Sinße
V (o)=0 VG) If B=0 = • Asin Bix VG-C)=√₁₂ = V₂ - Asin BL Asin BL A = -√₂ Sinßl Ta=jV₂ Prove -j Vs Ces Bx Z₂ Sinße
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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Transcribed Image Text:**Transcription of Handwritten Notes for Educational Use**
### Context
These notes pertain to the analysis of a voltage function \( V(x) \) and its implications in a given scenario. The mathematical expressions suggest a focus on wave or signal analysis, possibly within the field of electrical engineering or physics.
### Detailed Transcription
- **Condition and Implications**
- If \( V(0) = 0 \), then it implies that \( B = 0 \).
- **Expression for Voltage**
- \( V(x) = A \sin \beta x \)
- At x = L:
- \( V(L) = V_s = -A \sin \beta L \)
- From this equation, it follows that:
- \( A = \frac{V_s}{\sin \beta L} \)
- **Proof**
- The goal is to prove the expression:
- \( I(x) = -i \frac{V_s \cos \beta x}{Z_0 \sin \beta L} \)
### Explanation
- **Variables and Parameters**
- \( V(x) \) represents the voltage as a function of position \( x \).
- \( A \) and \( B \) are constants related to the amplitude of the wave.
- \( \beta \) is likely the propagation constant.
- \( V_s \) denotes the voltage at the position \( x = L \).
- \( Z_0 \) represents the characteristic impedance.
- \( I(x) \) is the current related to \( x \).
### Note
These equations are often encountered in the study of transmission lines or in the analysis of sinusoidal steady-state responses in circuits. Understanding the boundary conditions and the resultant expressions is crucial for validating the behavior of electrical signals over a medium.
![The image contains a mathematical derivation related to electrical circuits, specifically discussing current \( I(x) \). Below is the transcription and explanation of the content:
### Transcription:
\[ I(x) = - \frac{1}{Z} \frac{dV}{dx} = -\frac{1}{j\omega L} [A \beta \cos \beta x - B \beta \sin \beta x] \]
\[ -\frac{1}{j} = j \]
\[ \omega^2 LC = 1 \]
\[ I(x) = \frac{j}{Z_0} (A \cos \beta x - B \sin \beta x) \]
\[ \sqrt{\frac{V_C}{V_C}} \]
### Explanation of Diagram:
The diagram illustrates a transmission line with various parameters:
- It is labeled with the voltage \( V(x) \) and distance \( x \).
- The transmission line is represented by a rectangular shape, aligned horizontally, with \( x = 0 \) on one end and \( x = -l \) on the other. The arrow indicates the direction of increasing \( x \).
- The diagram and equations suggest that this is part of an analysis of wave propagation or signal transmission in a linear medium, incorporating parameters such as impedance \( Z \) and characteristics of the medium defined by \( \omega, L, \) and \( C \).
This content appears to be part of an advanced lesson on transmission lines in the context of electrical engineering or physics, focusing on wave equations and impedance matching.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3af11e6a-2a61-4302-a4ee-f3c4efdfe331%2F5d1ad1e1-70f1-464d-a6a9-80e11d4955f7%2F7tbjx3t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical derivation related to electrical circuits, specifically discussing current \( I(x) \). Below is the transcription and explanation of the content:
### Transcription:
\[ I(x) = - \frac{1}{Z} \frac{dV}{dx} = -\frac{1}{j\omega L} [A \beta \cos \beta x - B \beta \sin \beta x] \]
\[ -\frac{1}{j} = j \]
\[ \omega^2 LC = 1 \]
\[ I(x) = \frac{j}{Z_0} (A \cos \beta x - B \sin \beta x) \]
\[ \sqrt{\frac{V_C}{V_C}} \]
### Explanation of Diagram:
The diagram illustrates a transmission line with various parameters:
- It is labeled with the voltage \( V(x) \) and distance \( x \).
- The transmission line is represented by a rectangular shape, aligned horizontally, with \( x = 0 \) on one end and \( x = -l \) on the other. The arrow indicates the direction of increasing \( x \).
- The diagram and equations suggest that this is part of an analysis of wave propagation or signal transmission in a linear medium, incorporating parameters such as impedance \( Z \) and characteristics of the medium defined by \( \omega, L, \) and \( C \).
This content appears to be part of an advanced lesson on transmission lines in the context of electrical engineering or physics, focusing on wave equations and impedance matching.
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