The topic in this problem is nodal analysis, and we're going to solve circuits with independent sources in them. In this problem, we're looking for the current I sub zero in the circuit shown below. We see that we have two independent sources. We have a current source, 4 milliamps, and we have a voltage source, which is a 12 volt source at the top of the circuit. Additionally, we have a number of resistors. There's five resistors in this problem as well. So let's solve it using a nodal analysis. We notice that the first thing we have to do in nodal analysis is we have to identify all of the nodes. So let's do that first. We know that there's a node at the upper-left hand corner of the circuit going along the top of the circuit, another node, a third node, and a fourth node at the top of the circuit. We also notice that we have a ground node at zero volts at the bottom of the circuit as well. So, we're going to solve this problem using nodal analysis, which means that we're going to use the Kirchhoff's current law to find the solution. And in this case, we're going to sum the currents into the nodes. So, if we look at node one, we notice that there's two currents flowing into node one. The current through the 3 Kilo ohm resistor, and the current through the 12 volt source. But we don't know what the current is through the 12 volt source, and we can't identify it in terms of nodal voltages. So, we would ultimately, if we tried to solve this using Kirchhoff's current law, and summing currents into node 1, we'd be introducing an additional variable, I12 volts, for the current flowing through the 12 volt source. So we'd have the four unknowns from the nodal voltages, V1, V2, V3, and V4. And we'd have this additional unknown for the current flowing through the 12V source. We can only come up with four equations for this circuit, the four nodal equations associated with the nodes. So, we would end up with five unknowns, and four equations, and would not be able to solve the problem, because we have too many unknowns for the number of equations that we have. So, in order to this problem, we have to utilize the concept of a supernode to solve it. And we use this supernode concept when we're doing nodal analysis and we have circuits where voltage sources are floating between two nodal points, where 1 ohm is not a reference node. So, in this case, the 12 volt source is between V1, or between node one and node two, and neither one of those nodal voltages are known. They're unknowns at this point. They're variables at this point. So, we're going to identify our supernode first. And our supernode is the node that encompasses our node 1, it encompasses node 2, and the 12 volt source as shown in the dash line. So that is our supernode. So, we're going to write Kirchhoff's Current Laws about the supernode and the other nodes in the circuit. So, we see that the supernode has three currents flowing into it. The current up through the 3 kilo ohm resistor, the current flowing up through the 2 kilo ohm resistor, and the current flowing through the 3 kilo ohm resistor at the top of the circuit from right to left. So, adding those currents up, starting with the 3 kilo ohm on the left-hand side of the circuit, it starts at the zero volt, minus the V sub 1, divided by 3K. That's our first current into the supernode. We also have the current flowing up through the 2 Kilo ohm resistor, which is zero for the ground node, reference node, minus V2 divided by 2K. So, it's zero minus V2 divided by 2K. And we have a current flowing from node 3 to node 2 through the 3 kilo ohm resistor. And that's all of our currents flowing into the supernode. So, the sum of those is equal to zero. We can also write our nodal equation about node 3. In this case, we again have three currents flowing in to node 3. We have, first of all, the current flowing left to right through the 3 kilo ohm resistor. V2 minus V3 over 3K. We have the current flowing right to left from node 4 to node 3. That's going to be V4 minus V3 divided by 2K. Plus, we have the current flowing up through the source, the current source, which is minus 4 milliamps, since the 4 milliamp source is flowing out of node 3. That's all of our currents, and the sum is equal to zero. Now, if we look at node 4, the right-hand side of the circuit, it has two currents flowing into it. First, through the 2 kilo ohm resistor at the top flowing into node 4, V3 minus V4 over 2K. And then, the current flowing up through the 1 kilo ohm resistor is going to be zero volts minus 4 volts divided by 1K. So, it's zero minus V4 over 1K is equal to zero. So now, we have three equations, and we have four unknowns. We need one additional equation. And our one addition equation is our constraining equation for the supernode, which tells us that V2 minus V1 is equal to 12 volts. So now, we have all those. We have V1, V2, V3, and V4 identified as variables, and four independent equations. So we have a set of four equations, four unknowns, and we could solve for all the nodal voltages if we wanted to. But in this problem, we're looking for I sub zero. And I sub zero, as you can see from the circuit, is the current down through the 2 kilo ohm resistor on the center left part of the circuit. And so, that current is V2 minus the reference voltage, zero volts, divided by 2K. So what we really need, is we need to find the nodal voltage for node 2. That's really the one that's of interest to us for solving this problem. So if we can find that, then we can solve for I sub zero, using our equation that we have stated here. So if you solve for V2 using our set of four equations and four unknowns, V2 is equal to 2 volts. And using our equation on the left hand side of our page, I sub zero will be 1 milliamp.