Calculating Current in a Kirchhoff Loop Circuit

[Faint]

Homework Statement

In the circuit below find the current I2 (in A) when R1 = 64 , R2 = 32 , R3 = 243 ; V1 = 86 V, and V2 = 200 V.
Be careful about the sign!

Homework Equations

The Attempt at a Solution

I₁ = I₂ + I₃

V1 - I1*R1 - I3*R3 + V2 = 0
V1 - I1*R1 - I2*R2 = 0
V2 - I2*R2 - I3*R3 = 0

I1*R1 + I3*R3 = V1 + V2
I1*R1 + I2*R2 = V1
-I2*R2 + I3*R3 = V2

| R1  0    R3 | | I1 |     | V1+ V2 |
| R1  R2   0  | | I2 |  =  | V1     |
| 0   -R2  R3 | | I3 |     | V2     |

Solving these using R^-1*V gives me values for I1,I2,I3 that don't follow I1 = I2 + I3, so I am doing something wrong.

Help?

 

[ehild]

This circuit consists of two independent loops. You need only two loop equations. But be careful with the directions of the currents. Show them in the picture. Writing the loop equation, the potential drops across a resistor in the direction of the current and raises in the opposite direction.ehild

 

[Faint]

this circuit consists of two independent loops. You need only two loop equations. But be careful with the directions of the currents. Show them in the picture. Writing the loop equation, the potential drops across a resistor in the direction of the current and raises in the opposite direction.


Ehild

Showing the currents (in hopefully the right way):

i₁ = i₂ + i₃

v1 - i1*r1 - i2*r2 = 0
v2 - i2*r2 - i3*r3 = 0

-i1 + i2 + i3 = 0
i1*r1 + i2*r2 = v1
-i2*r2 + i3*r3 = v2

| -1  1     1  | | i1 |     | 0   |
| r1  r2   0  | | i2 |  =  | v1 |
| 0   -r2  r3 | | i3 |     | v2 |

I get i1 = 1.1535 A, i2 = 0.3805 A, i3 = 0.7729 A which is wrong. What am I still missing?

 

[ehild]

The second loop equation is wrong. According to the current I2, which end of R2 is at more positive potential?

ehild

 

[Faint]

The second loop equation is wrong. According to the current I2, which end of R2 is at more positive potential?

ehild

Wouldn't the right end be at a more positive potential?

 

[ehild]

It is. Starting at the positive terminal of the battery, and going clockwise, the potential increases across R2 and then decreases across R3.

 

[Faint]

So v2 - i2*r2 - i3*r3 = 0 should be:
v2 + i2*r2 - i3*r3 = 0
correct? I had this written correctly in my notes, but copied it wrong.

-i2*r2 + i3*r3 = v2 should stay the same then correct?

 

[ehild]

Yes, it is correct now. So you have three equations:

I₁=I₂+I₃

I₁R₁+I₂R₂=V₁

-I₂R₂+I₃R₃=V₂

Solve.

ehild

 

[Faint]

Got it. Thank you for your help!