The Telegraph Equation

  • #1
durandal
5
2
Homework Statement
Find the general solution of the system of telegraphers equations using the eigenvalues and -vectors.
Relevant Equations
See image in description
Screenshot 2024-04-25 at 16.24.05.png

I have tried row reduction to solve for the first eigenvector but I don't feel like I get any closer to the solution:

Screenshot 2024-04-25 at 19.07.34.png
 
Physics news on Phys.org
  • #2
Looks great so far, you just need to keep going.

An important thing to remember about eigenvectors is that there isn't a single vector solution. Eigenvectors are the vectors that satisfy ##Av=\lambda v## so any vector in the same direction as ##v## is also an eigenvector. This means that there is an unspecified degree of freedom that corresponds to the length of the eigenvectors. So, you get to arbitrarily choose one value. You're not solving for a vector, you're solving for a set of vectors of arbitrary magnitude. In practice we only pick and show one, which can confuse people.

ex. if your system has ##V_1=[1, -1]^T## as an eigenvector, then ##V_1=[-13, 13]^T## is also an eigenvector. We would normally say it's the same eigenvector, but that is sloppy shorthand for it's a member of that set of vectors (the eigenspace).

https://www.khanacademy.org/math/li...-finding-eigenvectors-and-eigenspaces-example

PS: Thank you for clearly stating the problem and your attempt. It makes it easy for us to understand and comment. If it's not easy to understand, people might not respond. This is a great example of how to ask a technical question.
 
Last edited:
  • Like
Likes durandal
  • #3
DaveE said:
Looks great so far, you just need to keep going.

An important thing to remember about eigenvectors is that there isn't a single vector solution. Eigenvectors are the vectors that satisfy ##Av=\lambda v## so any vector in the same direction as ##v## is also an eigenvector. This means that there is an unspecified degree of freedom that corresponds to the length of the eigenvectors. So, you get to arbitrarily choose one value. You're not solving for a vector, you're solving for a set of vectors of arbitrary magnitude. In practice we only pick and show one, which can confuse people.

ex. if your system has ##V_1=[1, -1]^T## as an eigenvector, then ##V_1=[-13, 13]^T## is also an eigenvector. We would normally say it's the same eigenvector, but that is sloppy shorthand for it's a member of that set of vectors (the eigenspace).

https://www.khanacademy.org/math/li...-finding-eigenvectors-and-eigenspaces-example

PS: Thank you for clearly stating the problem and your attempt. It makes it easy for us to understand and comment. If it's not easy to understand, people might not respond. This is a great example of how to ask a technical question.
Thank you very much. I found the eigenvectors to be v1 = [1 , -Zc^-1] and v2 = [1 , Zc^-1]. My professor verified it, but he also said that the eigenvectors could be found just by "looking" at the matrix itself. I have asked him about this, but he hasn't answered me. Do you have any idea which "technique" he was referring to?
 
  • #4
durandal said:
Thank you very much. I found the eigenvectors to be v1 = [1 , -Zc^-1] and v2 = [1 , Zc^-1]. My professor verified it, but he also said that the eigenvectors could be found just by "looking" at the matrix itself. I have asked him about this, but he hasn't answered me. Do you have any idea which "technique" he was referring to?
About 30 years ago I could have. It's really just proficiency with linear algebra and recognizing "simple" problems. I think the answer lies in understanding the "geometry" associated with linear transforms.

There are some great Linear Algebra lectures online from MIT. Particularly this one which sort of relates to your question.
https://ocw.mit.edu/courses/18-06-l...rces/lecture-21-eigenvalues-and-eigenvectors/
 
  • Like
Likes durandal
  • #5
##
\begin{bmatrix}
\gamma & -Z \\
-Y & \gamma
\end{bmatrix}

= \gamma
\begin{bmatrix}
1 & \frac{-Z}{\gamma} \\
\frac{-Y}{\gamma} & 1
\end{bmatrix}

= \gamma
\begin{bmatrix}
1 & \frac{-Z}{\sqrt{YZ}} \\
\frac{-Y}{\sqrt{YZ}} & 1
\end{bmatrix}

= \gamma
\begin{bmatrix}
1 & -\sqrt{\frac{Z}{Y}} \\
-\sqrt{\frac{Y}{Z}} & 1
\end{bmatrix}

= \gamma
\begin{bmatrix}
1 & -Z_c \\
-\frac{1}{Z_c} & 1
\end{bmatrix}

##

The e-vectors are obvious for the matrix on the left. But it's not obvious to me with the version on the right.
 

FAQ: The Telegraph Equation

What is the Telegraph Equation?

The Telegraph Equation is a second-order partial differential equation that describes the propagation of electrical signals in a transmission line. It accounts for both the resistance and capacitance of the line, and it can be expressed in the form: ∂²V/∂x² = (R + jωL)(C)(∂V/∂t), where V is the voltage, R is the resistance per unit length, L is the inductance per unit length, C is the capacitance per unit length, and ω is the angular frequency.

What are the key parameters in the Telegraph Equation?

The key parameters in the Telegraph Equation are resistance (R), inductance (L), capacitance (C), and conductance (G). These parameters characterize the transmission line and influence how signals propagate through it. Resistance and conductance relate to energy loss, while inductance and capacitance relate to the storage of energy in magnetic and electric fields, respectively.

What are the applications of the Telegraph Equation?

The Telegraph Equation is widely used in electrical engineering, particularly in the analysis of transmission lines, such as coaxial cables and twisted pairs. It helps in understanding signal distortion, attenuation, and the effects of frequency on signal propagation. Additionally, it is applicable in telecommunications, power systems, and any field that involves the transmission of electrical signals over distances.

How does the Telegraph Equation differ from the wave equation?

The Telegraph Equation differs from the wave equation in that it incorporates both resistance and capacitance, which are critical for modeling real transmission lines. While the wave equation typically describes ideal wave propagation without losses, the Telegraph Equation accounts for energy loss due to resistance and allows for the inclusion of frequency-dependent effects, making it more suitable for practical applications.

Can the Telegraph Equation be solved analytically?

Yes, the Telegraph Equation can be solved analytically under certain conditions, such as for uniform transmission lines with constant parameters. Solutions often involve techniques such as separation of variables or Fourier transforms. However, in more complex scenarios, numerical methods may be required to obtain solutions, especially when dealing with non-uniform lines or varying parameters.

Similar threads

Back
Top