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RLC Locus Diagrams

  1. Apr 14, 2010 #1
    1. The problem statement, all variables and given/known data


    In a series circuit V = V<0 @50hz with an R and C, show that the graph of the loci or the port impedance, Z, and port admittance, Y, as the resistance R is varied from 0 to inf ohms are as shown.
    [URL]http://ivila.net/prob.png[/URL]


    [URL]http://ivila.net/loci.png[/URL]

    2. Relevant equations



    3. The attempt at a solution

    Z = R + j*X;
    Y = Z-1;

    Map to Y = G + jB (Separate Real and Complex)
    G = R/(R2 + X2);
    B = -X/(R2 + X2);

    G2 + B2 = 1/(R2+X2) = B/-X

    Then I get lost,

    I know I need to arrange into the format below with the condition that the R term (varying from 0 to inf) remains in the LHS, ie, out of the radius. however I am stuck .
    (G-x)2 + (B-y)2 = radius2

    Any help would be much appreciated.
     
    Last edited by a moderator: Apr 25, 2017
  2. jcsd
  3. Apr 15, 2010 #2
    I found the solution!

    Firstly.

    Z = R - jX (incorrectly stated in my first post)
    Y = G + jB
    G = R/(R^2 + X^2)
    B = X/(R^2 + X^2)
    G^2 + B^2 = 1/(R^2 + X^2) this is very close to B, infact B/X = G^2 + B^2

    using this we can say.

    G^2 + B^2 = 1/X * B
    G^2 + (B^2 - B/X) = 0

    Now Complete the Square

    G^2 + (B^2 - B/X + a^2) = a^2
    2a = -1/x, a = -1/2x

    Therefore

    (G^2) + (B + -1/2x)^2 = (-1/2x)^2

    or, a circle, origin at 0,-1/2x radius -1/2x

    Thanks!
    Alex
     
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