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 :)