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Taking e to a complex power (telecommunication formula)

  1. Feb 28, 2012 #1
    Hello, I am have this telecommunications formula which calculates the magnitude of a wave on a transmission line:
    VO = VI e^(-yi)

    where:
    VO = voltage out.
    VI = voltage in (5)
    y = propagation coefficient (0.745 + j1.279)
    i = length of line (2500)

    I am trying to use it to calculate the magnitude of a wave on a transmission line that is:
    - 2.5km from the source
    - whose input signal is 5V

    My attempt is to write it out as below:
    VO = 5*e^((-0.745 + j1.279)2500)

    But I am unsure if I am have written this out correctly and if so, how to take 'e' to the power of a complex number.

    Any help regarding how to finish this would be greatly appreciated.
     
  2. jcsd
  3. Feb 28, 2012 #2

    jbunniii

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    Here's the definition of the exponential of a complex number:

    e^(a + bi) = (e^a)(cos b + i*sin b)

    where a,b are the real and imaginary parts of the complex number.
     
  4. Feb 28, 2012 #3
    thanks for that.
    My attempt is below but I doubt I have implemented the "negative" y exponent correctly. Could someone please tell me how I have gone wrong.

    Original formula:
    VO = VI e^(-yi)
    = VO = 5*e^((-0.745 + j1.279)2500)

    My rewrite using:
    e^(a + bi) = (e^a)(cos b + i*sin b)

    is:

    5 * e^(-0.745)(-cos 1.279 - sin 1.279)(2500)

    = 0.4747(-1.022)(2500)

    = -1212.94




     
  5. Feb 28, 2012 #4

    jbunniii

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    No, you dropped the "i" in the term i*sin b. Your result should be a complex number.

    Also, isn't the 2500 part of the exponent?
     
  6. Feb 28, 2012 #5
    Thanks for your patience. I have tried again. Is this an improvement ?

    Original formula:
    VO = VI e^(-yi)
    = VO = 5*e^((-0.745 + j1.279)2500)

    My rewrite using:
    e^(a + bi) = (e^a)(cos b + i*sin b)

    is:

    5 * e^(-0.745*2500)(-cos 1.279 - isin 1.279)

    = -1862.5(-.9998 - j.0223)


     
  7. Feb 28, 2012 #6

    jbunniii

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    No, this still isn't right. Multiply out the complex number first:

    (-0.745 + j1.279)2500 = -1862.5 + j*3197.5

    Then a = -1862.5, b = 3197.5
     
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