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Power Transmitted in a Coaxial Cable

  1. Dec 2, 2017 #1
    1. The problem statement, all variables and given/known data
    A coaxial transmission line consists of an inner cylindrical conductor of radius a = 1 mm and a
    cylindrical outer conductor chosen to make the characteristic impedance 75 ohm. The space
    between the conductors is lled with a gas which can stand a maximum eld of 105 V/m
    without dielectric breakdown. Estimate the maximum mean radio-frequency power that
    can be transmitted along this line into a matching load.

    2. Relevant equations
    On transmission line
    ## Z = \sqrt{L/C} ##
    For coaxial with radius a and b:
    ##C = \frac{2\pi \epsilon \epsilon_0}{ln(b/a)} ##
    ##L = \frac{\mu_0 ln(b/a)}{2\pi} ##
    3. The attempt at a solution
    The voltage between the two conductors is by Gauss's law:
    ## V = \int \boldsymbol{E} \cdot d\boldsymbol{r} = E_0 ln(b/a) ##
    hence:
    ## V_{max} = E_{max} ln(b/a) ##
    also:
    ##Z = \sqrt{\mu / \epsilon \epsilon_0} \frac{ln(b/a)}{2\pi} ##
    Using ##P = VI = V^2/Z ## and asuuming ##epsilon \approx 1## gives:
    ##P_{max} = 0.21 GW##, but the answer is supposed to be 104 W.

    What am I doing wrong? Many thanks!
     
  2. jcsd
  3. Dec 2, 2017 #2

    kuruman

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    This equation doesn't make sense. The left side is in volts, the right side is in volts/meter. Check your derivation.
     
  4. Dec 3, 2017 #3
    Thank you, I got it!
     
  5. Dec 3, 2017 #4

    Delta²

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    I had my doubts about this step but for another reason. There is vector potential ##A## inside coaxial cable, so it is
    ##E=\nabla V+\frac{\partial A}{\partial t}##

    so that
    ##V=\int Edr## doesn't hold (because it is equivalent to ##E=\nabla V##.
    Do we make the approximation ##A\approx 0 ##?
     
  6. Dec 4, 2017 #5

    rude man

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    It's not necessary to invoke the magnetic potential.

    You want to use Gauss's theorem to relate E field to potential difference between the conductors. So first you want to determine the geometry of the cable, i.e. the inner and outer radii. Is there a relation between these radii and Z0?

    Then: use Gauss's theorem relating the E field to the potential difference V and the geometry and compute the max. allowable V for the given max. E.

    Don't forget they want the rms power figure (I think - wording is not clear to me). Also assume an ideal matched line.
     
  7. Dec 4, 2017 #6

    Delta²

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    Can you show how you find V using Gauss's theorem(have to solve Poisson's equation?) without using that ##V=\int Edr## because that last integral becomes path dependent when there is vector potential.
     
  8. Dec 5, 2017 at 12:24 AM #7

    rude man

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    We are dealing solely with electric, not magnetic, fields. I have never run into a vector potential such that E = ∇ x Ae the way we write H = ∇ x A (or sometimes B = ∇ x A). The former is in fact impossible if there is charge present since ∇⋅E = ρ, not zero, so no vector potential can exist. In any case, forget all vector potentials.

    You can relate V to E by solving Laplace's equation for the space between the conductors but you'd still need to do step 1 below. It would in fact be good to do so as a check if you went with my suggestion which is:

    (1) Find the outer radius of said space; (2) relate E(r) to V, the potential between the conductors using Gauss; (3) find where E is the largest, then you can determine the max. allowable V given the max allowable E.

    To do step 1 you will need one item of data given to you.
     
    Last edited: Dec 5, 2017 at 11:58 AM
  9. Dec 5, 2017 at 11:58 AM #8

    rude man

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    Actually, solving Laplace's equation is the more direct route. Use the cylindrical coordinate expression ∇2V = (1/r) d/dr (r dV/dr) = 0 since there are no derivatives with respect to θ or z.

    But again, your first task is to find the outer radius of the space between the conductors.
     
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