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Equivalent Resistance

  1. Oct 16, 2012 #1
    1. The problem statement, all variables and given/known data
    A circuit extends forever to the right, and all the resistors have the same value ##R##. Calculate the equivalent resistance measured across the two terminals at left.
    circuit.jpg


    2. Relevant equations
    The series and parallel equivalent resistance equation.


    3. The attempt at a solution
    I have no idea to begin, could someone give some suggestions for me.
    Sincerely.
     
  2. jcsd
  3. Oct 16, 2012 #2

    CWatters

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    There is a paper on this on the web but if I post a link to it that will give you too much help.

    If the chain is infinite then adding another "stage" to the input won't change the resistance because it's equivalent to shifting the whole chain physically one place to the left.

    So call the existing resistance Z and write an equation for the resistance with an additional stage added...

    Z = Z//R + 2R

    Over to you..
     
    Last edited: Oct 16, 2012
  4. Oct 16, 2012 #3
    Dear CWatters:

    Could you post the link to me?
    I want to understand it more clearly.

    Regards
     
  5. Oct 16, 2012 #4

    CWatters

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    The forum rules discourage us giving away the whole solution so before I do that which bit are you struggling with?

    Here is a diagram explaning of why adding a stage doesn't change the resistance. It works because ∞ + 1 = ∞.

    Although you are adding a stage to the left the result is indistinguisable from adding a stage to the right hand side which is at infinity.
     

    Attached Files:

  6. Oct 16, 2012 #5

    CWatters

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    Here is how you derive the equation for Z that you need to solve...
     

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  7. Oct 16, 2012 #6
    I think I understand the argument, so the equations are as follows:
    ##Z=Z//R+2R=(\frac{1}{Z}+\frac{1}{R})^{-1}+2R=(\frac{R+Z}{RZ})^{-1}+2R=\frac{ZR}{Z+R}+2R##
    ##ZR+Z^2=ZR+2R^2+2ZR##
    ##Z^2-2RZ-2R^2=0##
    ##Z=\frac{2R±√(4R^2+8R^2)}{2}=(1±√3)R##
    There is a problem for choosing the sign.
    Is there any argument that one of which is right?
    If my calculations are correct, could you please let me see the paper?
    Thank you for kind reply!
     
  8. Oct 16, 2012 #7
    I think that Z should be positive, therefore, ##Z=(1+\sqrt 3)R##.
     
  9. Oct 17, 2012 #8

    CWatters

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    Last edited: Oct 17, 2012
  10. Oct 17, 2012 #9
    Dear CWatters:

    Thank you for kind help!

    Regards.
     
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