Leakage Conductance of Cylindrical Coax Cable

[polarmystery]

Homework Statement

Find the leakage conductance per meter of a cylindrical coax cable whose inner conductor (r1) is 0.125" (3.175*10^-3 m) and whose outer conductor (r2) = 0.5" (1.27*10^-2 m) if the space between them is filled with a material whose: $$\varepsilon_r = 2$$ and $$\sigma_c = 6*10^{-8} \mho /m.$$

Homework Equations

I used this formula for calculating the resistance of the coax wire:

$$R = \dfrac{ln (\dfrac{r2}{r1})}{l2\pi\sigma_c}$$

The Attempt at a Solution

After solving for R, I get


$$R = \dfrac{ln (\dfrac{1.27*10^{-2}}{3.175*10^{-3}})}{(1)2\pi(6*10^{-8})} = 3.667 M\Omega$$

My question is, do I just invert R in order to get the line leakage conductance? It seems like I should be using the relative permittivity somewhere in the calculation, but not sure where. Thanks in advance.

 

[NascentOxygen]

I can't comment on the formula, not without looking it up. But I'll assume you used it correctly. Yes, its reciprocal gives conductance.

Permittivity is a property concerned with capacitance, etc., whereas resistance (i.e., conductance^-1 ) does not involve capacitance. They are independent properties of a dielectric, or of any material.

 

[polarmystery]

Something seems to be amiss here. I was thinking about it and I remember that typical coax cable resistance is somewhere like 50 ohms. I got this formula from our slides in class. It doesn't make sense that the answer is so high so I think there is something wrong, but honestly I don't know. Any help/suggestions? I think I might have missed something.

I understand that typical coax lines have a characteristic impedance of being 50 ohms, but it still doesn't make sense that the actual wire resistance I calculated is so high. Can someone point me in the right direction? Am I attacking this problem correctly?

 

[NascentOxygen]

The 50 ohms is like the wire resistance, sort of, but at RF. We like it to be predictable so RF loads and sources can be matched to it. The 3.6 Mohms is the insulation resistance; we'd like it to be very high. A few megohms seems low, but this is just an exercise.

 

[rwisz]

Yes, you are correct. In order to find the leakage conductance per meter, you need to invert the resistance value you calculated. This is because conductance is the inverse of resistance. Therefore, the leakage conductance per meter would be 1/3.667 MΩ or 0.272 mΩ.

As for the relative permittivity, it is not needed in this calculation as it is only used to determine the capacitance of the cable, which is not relevant to finding the leakage conductance. However, if you were trying to find the overall conductance of the cable, you would need to incorporate the relative permittivity into the calculation.

I hope this helps clarify your understanding. Keep up the good work in your studies!