Nodal Voltage Analysis Circuit and Example

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Nodal voltage analysis finds the voltage drops around a circuit between different nodes. These nodes provide a common connection for two or more circuit components.

KCL and KVL are almost enough for analyzing an electric circuit with less complexity. We will get more equations if we use KVL or KCL for circuits with more complex branches, nodes, and elements.

You will burden yourself with a lot of equations to solve.

What will we do then? It will be a lot of work if we have to use elimination and substitution for more than 4 equations.

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

We will leave the mesh analysis for a later post. Right now, let’s focus on Nodal Voltage Analysis or Nodal Analysis Circuit.

As the name implies, we will use the node voltages method in respect to the Ground, we call it Node Voltage Analysis.

What is Nodal Voltage Analysis

Nodal analysis circuits complement each other with mesh analysis circuits. The nodal analysis circuit uses the first Kirchhoff’s law, the Kirchhoff’s current law (KCL). Like we mentioned above, the name implies that we use node voltages and use it along with the KCL.

Nodal analysis requires us to calculate nodal voltages in each node with respect to the ground voltage (reference node), hence we call it the node-voltage method.

Nodal analysis is based on a 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? Keep in mind that we will get ‘n-1’ equations, where n is the number of nodes including the reference node. Using this circuit analysis method means we will focus on node voltages in the circuit.

Nodal analysis circuit properties:

Nodal analysis circuit uses the Kirchhoff’s current law (KCL)
For the ‘n’ nodes (including reference node) there will be ‘n-1’ independent nodal voltage equations
Solving all the equations will grant us the nodal voltages value
The number of nodes (except non-reference nodes) is equal to the number of the nodal voltage equation we can get.

What is Node Voltage

Before moving on, let us define ‘what is nodal voltage‘.

Notice at the circuit above, where v₁, v₂, and v₃ are the node voltages, connecting the corresponding node with element/s and another node.

Not only that, but we also need to define a reference node (ground), hence this node is always called a ground node. Thus, this node voltage is 0 V.

We have read a lot about node voltage. But what node voltage actually is?

Node voltage means the potential difference (voltage) between two nodes where the element or branch is present. The nodal analysis provides us with a mathematical equation for every non-reference node where the sum of the currents in a node is zero.

There are two types of node:

Reference nodes: reference nodes are the ground node
Non-reference nodes: the node voltages used for solving the circuit (v₁, v₂, v₃,… , v_n)

Nodal analysis with resistor

This one is the most basic because almost every circuit will contain at least one resistor. Assume that we have a resistor between two nodes and the current flows from node V₁ to V₂:

And then we get the equation:

That is the equation for a resistor between a node.

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

The equation will be the same as above, but we will set the V2 to 0 since it is a ground node.

Nodal analysis with voltage source

It is often that a branch consists of a voltage source in a resistor just as shown below:

We need to take care of the voltage source polarity. From the figure above, the voltage source positive polarity is facing against V₁ and I. It means the current from the voltage source is flowing against I and V₁. The equation will be:

If the voltage source is facing right, it means the current I will be summed with the current from Vs.

Hence,

If the V₂ is a reference node, you just need to set the V2 to 0 like before.

Nodal analysis with current source

We use nodal analysis to get work with KCL in which, acquiring the current equation using known nodal voltages. What will happen if there is a current source? This will make our equation simpler. First, look at the figure below.

We set both I₁ and I₂ leaving the node V₁ while the current source Is is entering node V₁. From the KCL, the currents leaving a node are equal to the currents entering that node.

The equation will be:

If the current source Is leaving the node V₁, the equation will be:

Nodal Analysis Circuit Procedure

Observe the circuit below for our first practice. The circuit has three resistors and two current sources all connected in parallel. Why do we only use current sources? At the end of this topic you will get the answer why we only use current sources at this moment.

Moving on,

The first step is to define a node as the reference or datum node or the ground node. The reference node is usually called the ground since it is assumed to have zero potential.

The second step, we assign voltage designations to reference nodes.

Take a look below where node 0 is the reference node (v = 0), while nodes 1 and 2 are assigned with voltages v₁ and v₂ respectively.

Remember, node voltages are defined with respect to the reference node. Each node voltage is the voltage rise from the reference node to the non-reference node or simply from the voltage node to the reference node.

The third step is to apply KCL to each non-reference node in the circuit. We use i₁, i₂, and i₃ as the currents flowing through resistors R₁, R₂, and R₃ respectively.

At node 1 we apply KCL and gives

At node 2 gives

We apply Ohm’s law to express the unknown value of i₁, i₂, and i₃ in terms of node voltages.

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

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

We can use this principle as,

We obtain the currents from,

Substituting the equations for i₁, i₂, and i₃ to KCL equations at node 1 and 2 results

And

Substituting with conductances, two equations above become

And

The fourth step is solving with node voltages. Applying KCL to n – 1 non reference node, we obtain n – 1 simultaneous equations.

For the circuit in example above, we solve all the equations to get the node voltages v₁ and v₂ using any standard method such as substitution method, elimination method, Cramer’s rule, or matrix inversion.

For now we will use matrix form as

which can be solved to get v₁ and v₂.

Please take note that we will find the resistor, voltage source, and current source in the circuit. There will be special treatment to a voltage source and current source.

Nodal Voltage Analysis Examples

For better understanding, let us review some examples below :

1. Calculate the node voltages in the circuit below.

Solution:

Consider the circuit below where the circuit above has been prepared for nodal analysis. The currents have been selected for KCL except for the branches with current sources.

The labeling of the current is arbitrary but consistent. (Consistent means if, for example, i₂ enters the 4 Ω 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, applying KCL and Ohm’s law gives