Superposition Principle to Solve Circuit

[Gwozdzilla]

Homework Statement

Homework Equations

KVL: ΣV = 0
KCL: ΣI = 0

v = iR

The Attempt at a Solution

I started by removing the independent current source to leave an open circuit. Then I attempted nodal analysis at nodes v₁, v₂, and v₃:

KCL @ v₁:
(v₁ - v₂)/10 = v₁/40

KCL @ v₂:
(v₂ - v₁)/10 - 4i_o + (v₂ - (-30))/20 = 0

KCL @ v₃:
v₃ = -30 V

i_o = (v₁ - v₂)/10

Plugging equation 4 into euation 2 and simplifying gives...

.1v₂ - .1v₁ -.4v₁ + .4v₂ + (v₂ - (-30))/20 = 0

.55v₂ - .5v₁ + 1.5 = 0

Simplifying equation 1:

.1v₁ - .1v₂ = .025v₁

v₂ = .75v₁

Plugging this in...

.55(.75v₁) - .5v₁ + 1.5 = 0

...

v₁ = 17.1V

v₂ = 12.9V

This makes i_o = .42A which is wrong.

Where did I mess up?

 

[gneill]

In your first node equation you've fallen into the trap of giving a wrong sign to one of the terms. This happens most often when one tries to write the equation on both sides of the equal sign rather than as a sum of terms all on one side. It's natural to think that the two current terms are equal and so write them both as positive values on either side, but consider what happens if you move a term from one side to the other.