Transmission line: leakage current differential equation

  • #1
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Homework Statement


I have a coaxial cable with internal conductor of radius r1 and external conductor of radii r2 and r3. The material of the conductors has a conductivity ##\sigma_1##. Between the conductors there is a imperfect dielectric of conductivity ##\sigma_2##.

Consider the approximation that cable conductors are perfect (that is, cable voltage U is constant along the longitudinal coordinate z). Determine the evolution of the cable current intensity along z, a consequence of the leakage currents crossing the imperfect dielectric.

Homework Equations


3. The Attempt at a Solution [/B]
So, my attempt was to use the fundamental equation div J = 0. By applying the divergence in cylindrical coordinates, you obtain ##\frac{dJ_y}{dy}=0##. I thought about now integrating over a section of the cable I obtain ##\frac{dI}{dy}=K##. My question now is how do I prove that this constant should be equal to GU, where G is the per-unit length transverse conductance of the dielectric medium.

I'm only having trouble in that last step of the derivation. Can someone help me?
<mentor edit: fix latex, add ##>
 
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Answers and Replies

  • #2
158
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Really can't anyone help me? I've been around this problem for an entire day I'm losing my mind with this, it must be a simple math trick I have no freaking idea about...
 
  • #3
epenguin
Homework Helper
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, you obtain ##\frac{dJ_y}{dy}=0##. I thought about now integrating over a section of the cable I obtain ##\frac{dI}{dy}=K##. My question now is how do I prove that this constant should be equal to GU, where G is the per-unit length transverse conductance of the dielectric medium.

I'm only having trouble in that last step of the derivation. Can someone help me?
I can't except except to recommend that you make imore readable by using ## (or maybe $$) in place of $ everywhere so your text will look like above
 

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