Nodal Analysis Nodal Analysis is a method of circuit analysis that is based on systematic application of Kirchhoff’s Current Law insofar as as possible. The first step in this process is to identify and label all of the nodes in the circuit, being careful in the process to merge all junctions and terminals that are connected directly together together without an intervening component component into a single node. For example, each different-colored group of connections in the circuit below is a node:
One of these nodes must be chosen as the reference node. The method yields a voltage for each of the other nodes, measured relative to the reference node. The goal is to write a KCL equation for a closed surface encompassing each node except the reference node. Each of the resistor currents in these equations equations is expressed in terms of the node voltages, thus yielding a system of simultaneous equations which can be solved for those node voltages.
Now, suppose I choose the blue node node as the reference. In this case, I have no particular reason reason for choosing the blue node. node. I have to choose one, one, and it’s the one I chose. Just to remind remind me of
my choice, I’ll label the blue node with the number “0” and the other two nodes as “1” and “2”. Again, all three choices are arbitrary. My choices are shown in the diagram below.
The node voltages V 1 and V 2 , are conventionally defined, respectively, at nodes “1” and “2” with respect to the reference node, as shown below.
Given the presence of the node labels, the node voltages are usually not shown on the circuit diagram because the convention is well known. Now, the idea is to write a KCL equation for each closed surface, except the one chosen as the reference node. Note that there are 5 wires crossing the green surface, so there will be 5 currents involved in the KCL equation for node 1. Similarly, since 4 wires cross the red surface, there will be 4 currents in the KCL equation for node 2. In this case, those equations are: V2
4
V V V1 1A 2 1 0 3 2
V2
and V1
5
V V V2 1A 1 2 6A 0 3 2
V1
Why not include an equation for the reference node? Because the KCL equation corresponding to that node is not independent of the others. In this case, it’s a linear combination of the other two, as shown below. V V V V V V V V V V V V 2 2 1 1A 2 1 1 1 2 1A 1 2 6A 2 1 6A 0 3 2 5 3 2 4 5 4
The right-hand portion of this equation is the KCL equation for the reference node. Thus, the KCL equation for the reference node would provide no additional information, and can be ignored. (Actually, we could choose to leave out any node. It’s convention, though, to leave out the reference node.) In fact, keeping that equation in the system would eventually cause problems, because the fact that it is not independent of the others would result in a singular coefficient matrix whose inverse would be impossible to compute. Writing the two useful KCL equations in matrix form, we have a system of simultaneous linear equations, called the Node Equations: 1 1 3 2 1 1 1 5 3 2
1A 4 3 2 V 1 1 1 V 2 1A 6A 3 2 1
1
1
or
5 13 6 12 V 1 1 31 5 V 2 7 30 6 These equations can be solved for the node voltages using any desired matrix solution technique. Here, for the results, we have: 1
5 13 270 V 1 1 17 15.8824 6 12 V 31 5 7 192 11.2941 2 30 6 17
Thus, V 1 15.8824V and
V 2
11.2941V . These results are sufficient to determine any
voltage or current in the circuit. Now, you may have noticed that the example circuit contained only independent current sources. What do we do if the circuit contains independent voltage sources? It is often inconvenient, if not outright difficult, to express the current through an independent voltage source in terms of the circuit’s node voltages as we would expect to need to do here. Actually, in a sense, the presence of independent voltage sources makes writing the equations a bit easier, because each voltage source gives a constraint relationship between two node voltages. For each equation we can get this way, we reduce the number of KCL equations we must write by one. This also turns out to be convenient in another way, because the fact that we have the constraint equation means we don’t have to find an expression for the current through that source, as we’ll see shortly. As an almost trivial example, consider the circuit shown below. The voltage source connected between node 1 and the reference node determines the node voltage at node 1. Thus, there are only two unknown node voltages, and two equations (together with the known value of the third node voltage) are adequate to solve for them. The two necessary equations can be derived by applying KCL at each of these nodes.
V 1
5
V2
V1 4
V3
V1
10
V 2
20 V3
V2
V 3 8
0
V2 1 0 8
A voltage source connected between two nodes, neither of which is the reference node, yields a constraint relationship between the corresponding two node voltages. In either case, the conventional approach, then, is to enclose the voltage source and its two nodes inside a closed surface, and call it a supernode. Then we write a KCL equation for the supernode, hence avoiding any necessity for determining how to express the current through that voltage source in terms of the node voltages.
V3
V 1 5
V2
V1 4
V1
10
V1
V 2
20
V 2 4
V2
V 3 8
V3
V 2 8
0
1 0
Multiple voltage sources means multiple constraint equations and multiple supernodes. Rules of thumb to remember are: 1. Any two supernodes that involve a common node are merged into a single supernode after their respective constraint equations are written. 2. A KCL equation is not written for any supernode involving the reference node, just as we did not write a KCL equation for the reference node itself in the earlier discussion. A few examples follow:
V 1
10
V3
V 2 5
V2
V1 4
V3
V 1 8
1
V 2
20
0
V 1
10
V2
V 1 5
V3
V1 8
V3
V2
V 1 10
V3
V 2 5
V1
4
V 2
20
V2 1 0 4
1 0
In the last example, I did not include the current through the 8 resistor in the third equation. Why? Because both ends of that resistor are connected to the supernode, meaning that the current through it enters the supernode at one end and leaves it at the other. Thus, the two terms cancel. What if the circuit contains controlled sources? That topic will be discussed in a future revision of this note.
