I would like to know the conductancee,\$C\$, and capacitance,\$G\$, of my lossy transmission line which is configured as an open circuit.

In such a case, I know that the expression that defines the input impedance of the line, \$ Z_{in}\$ is given by the following expression

$$\ Z_L = \infty \Rightarrow Z_{in} = Z_{o}*coth \left( \gamma*l \right) = $$ $$\ = Z_{o}*coth \left( \ \left( \alpha+j\beta \right)*l \right) \tag 1$$


$$\ \gamma = \sqrt{ \left( R+j\omega L \right)* \left( G+j\omega C \right) } = $$ $$\ = \sqrt{ \left( RG-\omega^2LC \right)+ j\omega* \left(RC+LG \right) } = \alpha + j\beta $$ $$\ Z_o = \sqrt{ \frac{R+j\omega L}{G+j\omega C} } = \frac{\sqrt{ \gamma}}{G+j\omega C}$$

I can use a non-linear method to solve \$(1)\$ since I know \$ Z_{in}, L, R\$ and \$ \omega\$

My question is related to the units of \$C\$ and \$G\$ obtained when solving, are these units per length, \$\left[C\right]= \frac{F}{m}\$ and \$\left[G \right]=\frac{S}{m}\$, or the total of the line \$\left[C\right]= F\$ and \$\left[G\right]= S \$?


In the context of your transmission line formulas, C and G are "distributed" elements, which mean they are "per unit". So you have to deal with [C]=F/m and [G]=S/m.


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