Nodal Analysis w/ Controlled Sources

[bob29]

Homework Statement

Find V1, V2 and V3.

Homework Equations

Nodal Analysis
Supernode Equation
Capacitance = 1/R
KCL & KVL

The Attempt at a Solution

KCL @ node 0: Ic = v2(-G1) +v3(-G3)
Node 1: Ic + I1 +v3(G2+G3) +V2(-G3) = 0 (subbing in Ic)
v2(G1+G3) + v3(g-3) = I1....Equation (1)

Node 2: v2(G1+G3) +v3(-G3) - I1 = 0...Equation (2)

Node 3: Same as equation 1

KVL @ supernode: -Hm*IC +V3-V2=0

I'm confused on how the equations were setup in the image.

 

[The Electrician]

The first equation in the image is the same as what you've called equation 2. It's just the application of KCL to node 2.

The second equation in the image is a constraint equation; it's setting V1 equal to zero.

The third equation in the image is found like this:

The current in the Hm*Ic source is equal to the sum of the currents in R3 and R2, and is equal to G3*(v3-V2) + G2(V3).

Now, over at node 1, we have Ic + I1 + G3*(v3-V2) + G2(V3) = 0 or:

-Ic = G3*(v3-V2) + G2(V3) + I1

Equating the voltage across the Hm*Ic source we have V3 - V1 = Hm*Ic

giving V3 - V1 = -Hm*(G3*(v3-V2) + G2(V3) + I1)

or V3 - V1 + Hm*(G3*(v3-V2) + G2(V3) = -I1*Hm

Rearranging gives equation 3 in the image.