Nodal Analysis


            Nodal analysis provides a general procedure for analyzing circuits using node voltages as the circuit variables. Choosing node voltages instead of element voltages as circuit variables is convenient and reduces the number of equations one must solve simultaneously. To simplify matters, we shall assume in this section that circuits do not contain voltage sources. Circuits that contain voltage sources will be analyzed in the next section.

              In nodal analysis, we are interested in finding the node voltages. Given a circuit with “n” nodes without voltage sources, the nodal analysis of the circuit involves taking the following three steps.


Steps to Determine Node Voltages:
Select a node as the reference node. Assign voltages v1,v2,…,vn−1 to the remaining n−1 nodes. The voltages are referenced with respect to the reference node.

Apply KCL to each of the n−1 nonreference nodes. Use Ohm’s law to express the branch currents in terms of node voltages.

Solve the resulting simultaneous equations to obtain the unknown node voltages.

Take Note:

  • Current flows from a higher potential to a lower potential in a resistor.

We can express this principle as:

i = (Vhigher−Vlower) / R


       We now consider how voltage sources affect nodal analysis. Consider the following two possibilities.

If a voltage source is connected between the reference node and a nonreference node, we simply set the voltage at the nonreference node equal to the voltage of the voltage source.

Thus our analysis is somewhat simplified by this knowledge of the voltage at this node.

If the voltage source (dependent or independent) is connected between two nonreference nodes, the two nonreference nodes form a generalized node or supernode; we apply both KCL and KVL to determine the node voltages.

  • A supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it.

Note the following properties of a supernode:

  1. The voltage source inside the supernode provides a constraint equation needed to solve for the node voltages.
  2. A supernode has no voltage of its own.
  3. A supernode requires the application of both KCL & KVL.


First, select a reference node and label the other nodes. Since each node has the same number of connected branches (4), we’ll simply choose the bottom node as the reference. There are only two other nodes, which we will label V1 and V2.

Now write KCL at each node (except the reference):

KCL at V1:

-5A + V1/5 + (V1-V2)/10 + [V1-(V2+4)]/10 = 0

Note that there are four terms in the equation, one for each branch leaving the node. The terms list the current leaving right, down, left, and up.

KCL at V2:

(V2-V1)/10 + V2/2 – 2A + [V2-(V1-4)]/10 = 0

Note that there are four terms in the equation, one for each branch leaving the node. The terms list the current leaving right, down, left, and up.

Now gather terms (multiplying through by 10 to clear up the fractions):

4V1 – 2V2 = 54
-2V1 + 7V2 = 16

Now solve the set of 2 equations with 2 unknowns.

V1 = 17.08V
V2 = 7.17V

We can now determine the current through the 5 ohm by Ohm’s law:

I = V1/5 = 3.41A

The current through the 4V source can be found as:

I = [V1-(V2+4)]/10 = 0.59A


Studying Nodal Analysis and the knowledge in KCL and KVL are very important

  1. First we discuss what is Nodal analysis and How important is the application of Nodal analysis in every circuit.
  2. We also discuss what is the process in getting the Voltage using nodal analysis with Voltage and without voltage source or in the other term for With voltage source is Supernode. Supernode occurs when there is a voltage source between to corresponding nodes.


The aim of nodal analysis is to determine the voltage at each node relative to the reference node (or ground). A node is all the points in a circuit that are directly interconnected. We assume the interconnections have zero resistance so all points within a node have the same voltage.

  1. The knowledge in KCL & KVL are very important to undergo with the Nodal Analysis.
  2. We can assign one reference and non-reference nodes anywhere we want in a given number of nodes.
  3. Shortcut method in getting the equations can executed in the circuit with or without supernodes.


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