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

Question

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.)?

Answer 1

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

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.