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Math Help - [SOLVED] Setting Up Linear Equations Using Kirshoff's Laws

  1. #1
    Len
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    [SOLVED] Setting Up Linear Equations Using Kirshoff's Laws

    I am having a real difficult time with the first part of my assignment. I need to set up 9 linear equations from this diagram.



    Using Kirshoff's laws which are:

    1.) The potential drop around each closed loop in the circuit board adds up to zero.
    2.) (Current conservation) The current at each node is conserved: The amount of current flowing into
    the node is equal to the amount of current flowing out of the node.

    More info can be found here with an example for each law but I still do not understand. Maybe someone can help
    http://www.math.yorku.ca/Who/Faculty...ss2041-409.pdf

    Thanks, Len

    Edit: The equations I am setting up are to solve i and i_n (n=1..8)
    Last edited by Len; November 17th 2009 at 06:47 PM. Reason: More info:
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  2. #2
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    example vertex and loop equations.

    For each vertex in your circuit, the incoming current must be equal to the outgoing current. So for instance,
    i4 = i2+i3 means the incoming currents, i2 and i3, must sum up to the outgoing current i4. Imagine each vertex as a door - all the people entering the door must leave the door (nobody stays there). You can do that for each vertex - there are 6 of them. Another set of equations you can write down are the loop equations. Going around a loop can't raise your potential ( think of hiking around the top of a mountain - you wind up where you started. The Escher pictures try to fool you into thinking you've gone higher or lower after 1 revolution). In your circuit, you would write down: i1*r + i3*r -i2*r = 0. (I'm assuming you didn't draw in the resistors). The negative on the i2 is because the direction of the current is backwards. One of your loop equations involves the battery. Without the battery nothing will happen. The bottom loop would then give you
    V - i7*r + i5*r + i1*r = 0. I count 4 loop equations in your circuit. Hope this gets you started.
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  3. #3
    Len
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    Thanks so much for the reply, following your advice this is what I have set up:

    Using the first law and assuming r and R=1
    1) Loop 1) i_1+i_3-i_2=0
    2) Loop 2) i_6-i_4-i_3+i_5=0
    3) Loop 3) i_8-i_6-i_7=0
    4) Loop 4)(Given to me in the hand out) -V+i+i+i_7-i_5-i_1+i=0

    I'm kind of stuck at the next part. Is it dependent on the way the arrows are facing? Or does it work from right to left?

    For example the top right point has all the arrows pointing to it.( i_4,i_6,i_8) If it is a door and everyone goes in it, where do they come out?
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  4. #4
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    signs on currents

    Just use the negative of that current to switch direction. So the upper right vertex could be written as:
    i4+i6=-i8 [enter via i4 or i6 and leave on i8] or
    i4=-i6-i8 [enter via i4 and leave via i6 or i8] or
    i6+i8=-i4 [enter via i6 or i8 and leave via i4], etc.
    which are all mathematically the same.
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  5. #5
    Len
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    Thanks again. I now have all 9 equations.

    It has been so long since I have done any linear algebra. I will be using the program Maple to solve these equations. What is the best way to solve them?

    Edit: NVM solved it, thanks again for the help.
    Last edited by Len; November 17th 2009 at 10:03 PM.
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