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Kirchoff's Laws Giving Singular System

  1. Nov 3, 2014 #1
    Hi, I'm trying to use Kirchoff's voltage law to find a system of equations which I can use to find I1, I2, I3, I4, and I5; which are the respective currents going through each resistor in the image.


    I have 5 unknowns so I need to find 5 non redundant equations. The problem is, every system of equations I come up with is singular. In other words... I am trying to set up a matrix equation ##A x = b## but my matrix A keeps having determinant zero!

    Is there any general rule to follow in order to find a non redundant set of equations? This is really time consuming!

    If you are curious, here is an example system which I came up with, but in some way it's redundant.
    Code (Text):
    V = R1 I1 + R4 I4
    V = R2 I2 + R5 I5
    V = R1 I1 + R3 I3 + R5 I5
    V = R2 I2 - R3 I3 + R4 I4
    0 = R1 I1 - R2 I2 + R4 I4 - R5 I5
    V is the voltage of the battery.
  2. jcsd
  3. Nov 3, 2014 #2


    Staff: Mentor

    I only see two unknowns. Maybe three.

    Have you been introduced to node voltage or mesh current analysis?
  4. Nov 3, 2014 #3


    Staff: Mentor

    My favorite analysis technique is node voltage analysis. Basically, you look at the circuit and you identify any unknown voltages, each unknown voltage is a node. In this circuit, there are only two. Then, at each node you use KCL to write that the total current leaving the node is 0. Then you solve those two equations in two unknowns to get the unknown voltages. All of the currents are then uniquely determined.
  5. Nov 3, 2014 #4
    I'm confused, what are the unknowns you are seeing? I'm trying to find the current going through each resistor, and I0, the current going through the wires near the EMF. I need to know the answer of each of these six values.

    I'm also not exactly sure what you mean by node voltage and current analysis. I'm familiar with both of Kirchoff's laws though.

    1. The total potential difference around any closed loop is zero.
    2. At any node there is as much current leaving as is entering.
    Last edited: Nov 3, 2014
  6. Nov 3, 2014 #5


    Staff: Mentor

    The only unknown voltages are the voltages at the two nodes touching R3. All other voltages are known. Call those two voltages Va and Vb. Then at Va and Vb write KCL.

    Once you have Va and Vb then all of the currents are obtained through Ohms law.
  7. Nov 3, 2014 #6
    Oh I see! Thank you!

    It's like magic...
  8. Nov 3, 2014 #7


    Staff: Mentor

    Glad I could help!

    There is another similar technique, called mesh current analysis. It is based on KVL, but I don't like it as much.
  9. Nov 3, 2014 #8
    Thank you, I will surely look into it in due time :)
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