# Reasons and assumptions behind the use of per unit length quantities in the telegraph equations

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.)?

• 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.) – Transistor Aug 15 '18 at 13:34
• 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. – user103380 Aug 15 '18 at 14:28
• @KingDuken Thank you. This actually answers some of the questions I have had. – Mike Aug 16 '18 at 15:04

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