Current in a loop (Kirchhoff's Laws)

[maxsthekat]

Homework Statement

Given the following circuit:

Calculate the currents I₁, I₂, and I₃
The given resistances are:
r: 1.05 ohm
R1: 11.7 Ohm
R2: 10.5 Ohm
R3: 7.20 Ohm
R4: 14.3 Ohm
R5: 18.8 Ohm

The given emf's are:
E₁ = E₂: 9.0 V
E₃: 3.0 V

Homework Equations

We know Kirchhoff's laws for current and voltage. Namely, I_in = I_out for any junction, and voltage around any loop must = 0.

The Attempt at a Solution

I attempted to use Kirchhoff's voltage laws around the big loop (all the way around the outside), the upper (top) loop, and the lower (bottom) loop. Then, I put these into a matrix, and tried to use the reduced row echelon form (rref) to solve for the currents.

Big loop:
E₃ - I₃r - I₃R5 - I₁R1 + E₁ - I₁r - I₁R3 - I₃R4 = 0

This is equivalent to: I₁(-R1 - r - R3) + I₂(0) + I₃(-r - R5 - R4) = -E₃ - E₁

Top loop:
E₂ - I₂r - I₂R2 - I₁R1 + E₁ - I₁r - I₁R3 = 0

Equivalent to: I₁(-R1 - r - R3) + I₂(-r - R2) + I₃(0) = E₂ - E₁

Bottom loop:
E₃ - I₃r - I₃R5 + I₂R2 + I₂r - E₂ - I₃R4 = 0

Equivalent to: I₁(0) + I₂(R2 + r) + I₃(-r - R5 - R4) = -E₃ + E₂

Resultant matrix

[(-R1 - r - R3)       0      (-r - R5 - R4)  ]  [ I1 ]      [ -E3 - E1  ]
[(-R1 - r - R3)  (-R2 - r)       0            ]  [ I2 ]  =  [  E2 - E1  ]
[       0         (R2 + r)  (-r - R5 - R4)   ]  [ I3 ]      [ -E3 + E2 ]

However, the rref of this matrix ends up with a row of all 0's (implying infinite solutions). Therefore, I'm assuming there is a mistake in how I went around the loops and setup the equations. Can anyone help?

 

[maxsthekat]

Aha! Solved it. I needed one more equation, which I neglected.

1 -1 -1 = 0 (I1 = I2 + I3)

with this, the matrix can be solved :)

 

[nlightner]

It seems like you have correctly applied Kirchhoff's voltage law to the three loops in the circuit. However, there may be an error in your calculations or in the values given for the resistances and emfs. I suggest checking your calculations and double checking the given values to ensure they are accurate. If the rref of the matrix still results in a row of all 0's, it is possible that the circuit is not solvable using Kirchhoff's laws alone and may require additional information or a different approach.