I am working out the derivations of the telegraph equations of a transmission line

$$\begin{gathered} \frac{{\partial V}}{{\partial x}} = - L\frac{{\partial I}}{{\partial t}} - RI \hfill \\ \frac{{\partial I}}{{\partial x}} = - C\frac{{\partial V}}{{\partial t}} - GV \hfill \\ \end{gathered}$$ where I is the current, 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 G is the conductance per unit length.

Working into the derivations, two questions arose: Why do we express the parameters on a per unit length basis? What are the assumptions underlying the transformation from total values to per unit length (e.g., linearity, isotropy, etc.)?

  • \$\begingroup\$ I don't understand your question, "Why do we express the parameters on a per unit length basis?" How else would you do it when there is no standard length for a cable? (I probably can't answer the question anyway.) \$\endgroup\$ – Transistor Aug 15 '18 at 13:34
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    \$\begingroup\$ We represent per unit length because you're talking about a cross section of resistance at point \$\Delta x\$. You can practically multiply the length per cross section of the resistive value to your length to get a total resistance. It's just a way to represent the material you're using as a conductor. I.E. Oh with this wire diameter, at this temperature, per unit distance, and we're reading X number of Ohms, you must be talking about Conductor A. \$\endgroup\$ – user103380 Aug 15 '18 at 14:28
  • \$\begingroup\$ @KingDuken Thank you. This actually answers some of the questions I have had. \$\endgroup\$ – Mike Aug 16 '18 at 15:04

In a real cable of non-zero length you can't model it very accurately with a single lumped parameter model like this: -

enter image description here

So, to start the process you conceive of the idea that for a very short piece of cable you can model it with lumped parameters and those parameters have a "per length" aspect about them.

Then when you begin manipulating the formulas and taking "actual" length down to zero you find that all the per length parts cancel out and you are left with the formula for characteristic impedance being: -

$$Z_0 = \sqrt{\dfrac{R+j\omega L}{G+j\omega C}}$$

Now you should be able to realize thatif you have lumped parameters for 1 metre or "X" metres, \$Z_0\$ will always be the same value.


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