I take I1 to the Right in R1, I2 to the left in R2 and I3 up in R3. My Kirchoff’s Rules equations
are:
Junction:
Upper Loop
Lower Loop
I2 = I1 + I3
Ɛ1 – I1 R1 – I2 R2 = 0
Ɛ2 – I3R3 – I2 R2 = 0
I1
– I2 + I 3 = 0
R1 I1 + R2 I2
= Ɛ1 →
R2 I2 + R3 I3 = Ɛ2
I’ll arrange these in the form
| I1| |k1|
for Cramer’s Rule
|A| | I2 | = |k2|
| I3| |k3|
| 1 -1 1 | | I1|
|0|
| 1 -1 1 | | I1|
|0 |
| R1 R2 0 | | I2 | = | Ɛ1| → | 25 48 0 | | I2 | = |90 |
| 0 R2 R3 | | I3 | | Ɛ2|
| 0 48 35 | | I3 | |120|
Now to Evaluate:
| 1 -1 1 |
Det |A| = Det | 25 48 0| = 1(48*35 – 0*48) – (-1)(25*35- 0*0) + 1(25*48 – 48*0) = 3755
| 0 48 35|
| 0 -1 1 |
Det |Ax| = Det| 9 48 0| = 171
|12 48 35 |
I1 = Det |Ax| / Det |A| = 0.04 Amps to the Right
|1 0 1 |
Det |Ay| = Det |25 9 0|
|0 12 35|
I2 = Det |Ax| / Det |A| = 0.16 Amps to the Left
= 615
| 1 -1 0 |
Det |Az| = Det | 25 48 9| = 444
| 0 48 12|
I3 = Det |Ax| / Det |A| = 0.12 Amps “Up”
Extra Notes:
Microsoft Excel has a function mdeterm to evaluate determinants. Just remember that you must
define (name) each Array.