Node Analysis for Electrical Circuits and Example Problems

The node circuit analysis finds an unknown voltage drop in a circuit between different nodes that have a common connection between two or more circuit components.

Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL) are almost sufficient to analyze electrical circuits of lower complexity.

What would we gain from KVL and KCL for more complex circuits with more complex branches, nodes, and elements?

What are we going to do next? Of course it will take a lot of time if we have to use elimination and substitution for equations greater than 4.

This is where we will use three powerful techniques for circuit analysis:

  • Wye delta transformation
  • Node analysis
  • Mesh analysis

We will ignore mesh analysis for the next post. For now, let’s focus on node voltage analysis or node circuit analysis.

As the name suggests, we will use the node-to-ground voltage method, so it is known as Node Voltage Analysis.

What is Node Circuit Analysis

Network node analysis is the opposite of mesh network analysis. The node analysis circuit uses Kirchhoff’s first law, Kirchhoff’s current law (KCL).

As we discussed earlier, the name itself reflects that we are using node voltages and using them with KCL.

Node analysis requires us to calculate the node voltage at each node with respect to the ground voltage (reference node), so this is known as the node voltage method.

The node analysis is based on the systematic application of Kirchhoff’s current law (KCL). With this technique, we will be able to analyze any linear circuit.

What do you need to prepare before using this method? Remember that you will get the ‘n-1’ equation, where n is the number of nodes including the reference node.

Using this method of circuit analysis means we will focus on the voltages of the nodes in the circuit.

Network node analysis properties:

  1. Analysis of the node circuit using Kirchhoff’s current law (KCL).
  2. For ‘n’ nodes (including reference nodes) there will be ‘n-1’ independent node voltage equations.
  3. Solving all the equations will give us the node voltage values.
  4. The number of nodes (except non-reference nodes) is equivalent to the number of node voltage equations we obtain

What is Node Voltage

Before proceeding, let’s define ‘what is node voltage’. Node voltage is the potential difference between two nodes in a circuit.

Notice in the circuit above, where v1, v2 and v3 are the node voltages, connecting nodes with respect to other elements and nodes.

Not only that, we also need to define a reference node (ground), so this node is always called the ground node. So, the voltage of this node is 0 V.

Node Types in Node Sequence Analysis

We’ve read a lot about node voltage. But what is node voltage actually? Node voltage means the difference in polarity (voltage) between the two nodes where the element or branch is located.

The node analysis provides a mathematical equation for each non-referenced node where the sum of the currents at a node is zero.

There are two types of nodes:

  1. Reference node: the reference node is the ground node
  2. Non-reference node: node voltage used to complete the circuit (v1, v2, v3,… , vn).

Reference Node Type

There are two types of reference nodes:

  1. Earth ground: Fig.(1a) and (1b).
  2. Chassis ground: Fig.(1c).

You can choose one of them when completing the node series analysis.

Node Circuit Analysis Step

The first step is to define a node as a reference or ground node. The reference node is usually called ground because it has zero potential.

The symbol of the reference node can be seen in Fig.(1). The earth ground is shown in Fig. (1a) and (1b) and the chassis ground is shown in Fig. (1c).

After we get the reference node, we give the voltage pointer at the reference node.

Consider Fig.(2a) where node 0 is the reference node (v=0), while nodes 1 and 2 represent the voltages v1 and v2 respectively.

Remember, the node voltage is defined with respect to the reference node. As depicted in Fig.(2a), each node voltage is the increase in voltage from the reference node to the non-reference node or simply the node-to-reference voltage.

The second step is to apply KCL to each non-reference node in the chain. To reduce the complexity of the variables, Fig.(2a) is redrawn in Fig.(2b), where we use i1, i2, and i3 as the currents flowing through R1, R2, and R3 respectively.

We use Ohm’s law to express the unknown values of i1, i2, and i3 with node voltages.

Since resistance is a passive element, using the passive sign convention, current must always flow from higher potential to lower potential.

The third step is to solve the node voltage. Using KCL to n-1 reference nodes, we obtain n-1 sequential equations such as Eq.(5) and (6) or (7) and (8).

For the circuit in Fig.(2) we solve Equations.(5) and (6) or (7) and (8) to obtain the node voltages v1 and v2 using any basic method such as substitution method, elimination method, Cramer’s rule, or inversion matrix.

Which can be solved to get v1 and v2.

Please note that we will find resistors, voltage sources, and current sources in the circuit.

There will be special treatment for voltage sources and current sources. If the steps above are quite complex, you may want to read the brief explanation below to address each element:

Node analysis with resistor

This one is the most basic as almost all circuits will have at least one resistor. Assume we have a resistor between two nodes and current flows from node V1 to V2:

And we will get the equation:

And that’s the equation for the resistors between the nodes.

What if node 2 is ground (reference node) as shown below?

The equation will remain the same as above, but we will set V2 to 0.

Node analysis with voltage source

Quite often a branch has a voltage source across a resistor as shown below:

We need to pay attention to the polarity of the voltage source. From the figure above, the positive polarity of the voltage source is against V1 and I. This means that the current from the voltage source flows against I and V1. The equation will be:

If the voltage source is facing right, this means that the current I will be added to the current from VS.

If V2 is a reference node, you just need to write V2 to 0 as before.

Example of Node Stress Analysis Problems

For a better understanding, let’s look at the example questions below:

1. Calculate the node voltages in the circuit in Fig.(3a)

Consider Fig.(3b) where the circuit in Fig.(3a) has been prepared for node analysis. Current selected for KCL except branch with current source.

Free-flow naming but consistent. (Consistent means that if, for example, i2 enters the 4 resistor from the left-hand side, i2 must leave the resistor from the right-hand side).

The reference node is selected and the node voltages v1 and v2 are now determined.

At node 1, using KCL and Ohm’s law gives

That’s a review about Node Analysis for Electrical Circuits and Example Problems What the team can describe. Hopefully the articles that we provide can be useful.