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The Formula for the Equivalent Resistance of Complex Resistance Network(Circuits)

Discussion in 'General Electronics Discussion' started by Deli Zhang, Apr 10, 2015.

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  1. Deli Zhang

    Deli Zhang

    Apr 8, 2015
    Hi All,
    I am back!
    In 2008 I posted the thread and just provided a example. Now I have time to complete the English paper The Formula for the Equivalent Resistance of Complex Resistance Network.pdf as below attachment.
    In this paper I given the formula of equivalent resistance for arbitrary circuits and proof. To understand the paper you may have to learn Graph Theory and Metrix Theory, the wording statement of the paper is professional to a mathematics student, even so I will help you understand it.

    The example that year took is still the best.
    Assuming a four vertices graph (means a circuit with 4 nodes), the conductance between two vertices(it is
    symmetric. ):
    g(1,2)=1; g(1,3)=2; g(1,4)=3; g(2,3)=4; g(2,4)=5; g(3,4)=6;

    The corresponding metrix for the graph:
    then got the matrix of matrix-tree theory,
    M4 =

    Attention on the main diagonal elements, which are the sum of all
    conductance on the same row and then multiplied by -1

    Then, remove the last row and the last column, we get determinant T3.
    T3 =

    Next, let the conductance g(1,2) between vertex 1 and vertex 2
    replaced by number '1', we got T3((g1,2)=1). which is same to T3, because g(1,2)=1 originally.
    And let the conductance g(1,2) replaced by number '0', we got
    T3(g(1,2)=0) =

    Finally, we get the Equivalent Resistance G(1,2) between vertex 1 and
    vertex 2
    = |T3| / (|T3(g(1,2)=1)| - |T3((g(1,2)=0)|)
    = (-556) / ((-556)-(-424))
    = 139/33
    = 4.2121
    Equivalent Resistance R(1,2) = 1 / G(1,2) = 0.2374

    Same method, you can get any others,
    e.g. G(2,4)

    the raw value of g(2,4) is 5, now replaced by 1, so
    T3(g(2,4)=1) =
    and replaced by 0, got
    T3(g(2,4)=0) =

    G(2,4) = |T3| / (|T3(g(2,4)=1)| - |T3((g(2,4)=0)|)
    = (-556) / ((-284)-(-216))
    = 139/17
    = 8.1765
    Equivalent Resistance R(2,4) = 1 / G(2,4) = 0.1223
    ... ...

    Any question, please raise.

  2. Deli Zhang

    Deli Zhang

    Apr 8, 2015
    Another example. You will feel how magical the formula is!


    What's the equivalent resistance R(A,B) between A and B?

    Use conductance to describe the circuit.

    So the equivalent resistance R(A,B)=5/6
  3. Laplace


    Apr 4, 2010
    Here is an alternate procedure of solving for the lattice resistance using KCL with node equations. Not sure which method should be regarded as the most magical.

    Attached Files:

  4. pebe


    Sep 3, 2013
    I make it 1ohm.

    Edit: Oops! My mistake.
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