The topic of this problem is nodal analysis. And we're going to solve circuits with current sources in them. In particular, we have a problem that has both dependent and independent current sources. The problem itself is to determine V sub 0. That's the output voltage across the 1 kilohm resistor on the right hand side of the circuit. This problem, again, has two current sources. It has a dependent source, which is a current of voltage controlled current source. It's 2 V sub X divided by 1000. Where the controlling parameter is a voltage. V sub X which is dropped across the 1 kilohm resistor on the left hand side of the circuit. We also have a two milliamp source, independent source in the circuit. So you know when we're doing the analysis, the first step in nodal analysis is to find the equations for the nodes using Kirchhoff's Current Laws. So we're going to use Kirchhoff's Current Law to sum the currents either into or out of the nodes of the circuit. In this circuit we have four nodes we have the node at the upper left hand side of the circuit which is node 1, we have node 2 kind of the center of the circuit. And the upper side we have node 3, on the right hand side of the circuit. Node 3 is the node which encompasses all the different elements flowing into this particular point in the circuit. We also have a reference node which we know is at 0 volts at the bottom of the circuit. So we have four nodes in the circuit. We're going to use Kirchhoff's Current Law to sum the currents either into or out of node 1, node 2 and node 3. So we're going to choose to solve the problem using Kirchhoff's Current Law and summing the currents into the different nodes. So we're going to start with node 1. If we look at node 1 and we want to sum the currents into node 1, we have current through the 1 kilohm resistor on the left hand side of the circuit, and we have current flowing through the 1 kilohm resistor on the top of the circuit. So the current flowing through the 1 kilohm resistor on the left hand side of the circuit, if it's flowing into node 1, it originates at the ground node, and flows up into node 1 through the 1 kilohm resistor, so it's 0- V sub 1, which is the nodal voltage for node 1, divided by 1 kilohm. We also have the current through the one kilohm resistor at the top of the circuit, flowing from node 2 to node 1, since we're summing currents into the nodes. Starting at 2 flowing into 1 is going to be nodal voltage of V2- nodal voltage V1 divided by 1K. Those are the only two currents flowing into node 1. So the sum of those at any estimate time is equal to 0. It's important to note that with nodal analysis, what we're solving for are the nodal voltages, V1, V2, and V3. If we can find those nodal voltages then we can find any of the associated currents and any of the branches of the circuit and we can find the voltage drops as well. So there's our first independent equation. Our second equation is for node 2 in the center of the circuit. Again, assuming the currents into node 2 looking at the 1 kilohm resistor at the top of the circuit first. The current flowing into node 2 originates at node 1 flows to node 2 through the 1 kilohm resistor. So it's V1- V2 divided by 1 kilohm. We also have a current following up through the 2 milliamp source into node 2. It's a positive current so it's + 2 milliamps and we have the current through the 2 kilohm resistor at the top of the circuit as well. Originating at node 3 to node 2 through the 2 kilohm resistor so it's + V3- V2 divided by 2k and that's = 0 that's all the current flowing into node 2. Now looking at node 3. Node 3 has again three currents flowing into it. We have a current flowing through the 2 kilohm from left to right. It's V2- V3 over 2k, Plus the current flowing up through the dependent source. Which is 2V sub x divided by 1000. And then we have the current flowing through the 1K resistor on the right hand side of the circuit starting at the ground node, the reference node 0 volts, flowing up to node 3. So its + 0- V3 divided by 1K = 0. So if we look at these three equations we have three equations and three unknowns. The unknowns are the nodal voltages of V1, V2 and V3. And everything else is given to us so we can solve for each one of those voltages. [COUGH] We also need to find an equation that relates V sub x to the unknown voltages V1, and V2, and V3. because in fact, the independent current source adds yet another a invariable V sub x. So we need to find the relationship between V sub x, our controlling parameter and our nodal voltages. And if we look at the circuit, it's quite easy to find that constraint equation. And we know that V1 = V sub x. So now we have our four equations and our four unknowns if we include our unknown from the dependent current source. So we can solve these simultaneously. Openly, we see the V sub 0 = V sub 3. It's the same voltage. And if we solve these four equations and four unknowns, then we get a V sub 0 = 4 v.
