Is conductivity equivalent to conductance per unit length?

Question

Conductivity is measured in units \$S/m\$. Conductance is units \$S\$, and length is units \$m\$, so conductance per length would also be \$S/m\$. Does this mean conductivity and conductance per length are equivalent? The units match, but I have some doubts.

What got me thinking about this was the typical model of characteristic impedance of a transmission line:

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

where \$G\$ is conductance per unit length of the dielectric. I'm unable to find actual values of \$G\$ for common coax cable, probably because \$G\$ is almost universally insignificant in this calculation, so no one bothers to provide an actual value. PET has a conductivity on the order of \$10^{-21}S/m\$; is this a valid value to use in this calculation? Or, is \$G\$ a function of both the dielectric's conductance and the cable's geometry? If geometry is relevant, how?

Answer 1

I'd rather think of the units of conductivity as \$\dfrac{\mathrm{S}\cdot\mathrm{m}}{\mathrm{m}^2}\$. Which of course simplifies to S / m.

But what's really going on is that if the length of a conductor increases, the conductance decreases, and if the cross sectional area increases the conductance increases. \$ G = \sigma A / l \$. Writing out the cancelled units tells more of the full story.

PET has a conductivity on the order of 10⁻²¹ S/m; is this a valid value to use in this calculation? Or, is G a function of both the dielectric's conductance and the cable's geometry? If geometry is relevant, how?

This value is \$\sigma\$, the material conductivity, and not G from the characteristic impedance equation. But they are related.

In the characteristic impedance equation, you're interested in current conducted through the dielectric from the inner to the outer conductor. So the "length" dimension in the transmission line is one of the two "area" dimensions in the conductance equation. The other area dimension in a coaxial line would be some average of circumference of the line (between the inner and outer conductor). The lenght dimension in the conductance equation would be the distance between the two conductors in the coax.

So

\$ G_z = \dfrac{2\pi{}\bar{r}}{r_2 - r_1}\sigma\$

where G_z is the conductance per unit length needed for the characteristic impedance equation; r₁ and r₂ are the outer and inner conductor radii, and \$\bar{r}\$ is an "average" radius of the dielectric. I haven't thought through the correct way to take the average, but it would be whatever gives the correct overall conductance for a fixed length of line.

Answer 2

Yes, Resistance = Resistivity * m / m2 = (1/S) = (m/G*m*m). The unit of Area is m*m, the unit of length is m, and G*m = S.

If you have a non-Euclidean area, in non-Euclidian space, then the non-Euclidian length unit might not cancel out with part of the area unit, but at normal speeds in almost-Euclidian space it works out ok.

In your equation for Z0, G is the leakage conductance between the two plates of the capacitor made by the core and the shield. The DC leakage conductance is a good value to use if you want Z0 for DC conditions. For a better model, replace G+jwC with a capacitor equivilant circuit: both G and C vary with frequency, and there is an effective series resistance as well. Yes, the geometry will affect the field strength in the dielectric, but of course temperature and humidity will also have an effect on C and G. I think you normally measure things like this rather than trying to predict them.