Nodal analysis is a method we can apply virtually on any electric circuit. In general, the technique is considered as a universal solution. As it almost handles all the practical electric circuits.

Nodal analysis is based on the applied “Kirchhoff’s Current Law.” Which creates a series of nodal equations to solve nodal voltages. It is also based on Ohm’s law and will be in the form of:

I=V/RI=V/R,

Or more generally, I=(1/RX)⋅VA+(1/RY)⋅VB

Once we find the node voltage, it’s simpler to find branch currents or connected components’ power in the electrical network.

This educational piece will discuss the node voltage method, nodal circuit analysis, mesh analysis examples, the difference between nodal and mesh analysis, and more.

## Kirchoff’s Current Law

In the electrical circuit, a node is a point where different electrical components are connected. Kirchoff’s current law states that the current entry into a node must leave from a node. In the circuits, each nodal point has the same voltage.

That’s why this voltage is known as the node voltage. It’s the voltage disparity at the arbitrary location and the ground point. In real circuits, the node is made up of wires. However, the nodal voltage is not the same all over the nodes.

Two methods are used to solve any electrical circuit or do supernodes nodal analysis. Which are mesh analysis and nodal analysis. In the nodal analysis method nodal voltage, we also use the ground node. This method is also known as the node voltage method.

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## What Is Nodal Analysis?

Nodal analysis is one of the methods used in electric circuits for circuit analysis through node voltages like circuit variables. Another name for this method is the node voltage method. The main features of this method are as follows:

■ Nodal analyses are based on KCL equations or law. That is Kirchoff’s Current Law.

■ If there are “n” nodes in the network or electric circuit, then we have “n-1” simultaneous equations to solve.

■ The voltage of all the nodes in the circuit can be calculated by solving “n-1” equations.

■ The total nodal equations are equal to non-referenced nodes that can be obtained.

In the nodal analysis method, two kinds of nodes are available. Which are reference nodes and non-reference nodes.

■ A node in the network which acts as a reference point for the remaining nodes is called a reference node or Datum node.

■ A node in the network that includes the exact node voltage is known as the non reference node. For example, node 1 and node 2 are non reference nodes.

Moreover, reference nodes are further classified into two node types. These are chassis ground and earth ground.

■ Chassis ground is the type of reference node that acts like an ordinary node in the given circuit.

■ In a network, if earth potential is used as a reference, then this type of reference node is known as the earth ground.

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## Nodal Analysis In Electric Circuit Or Network Here are the complete steps for nodal analysis in the electric circuit or network.

■ Step 1: Recognize the main nodes and select one node among them as a reference node. That means this node will be treated like ground.

■ Step 2: Analyse the node voltages as per the ground terminal from all the main nodes except the reference node.

■ Step 3: Write nodal equations at all the main nodes except the reference node. These equations can be obtained by applying KCL, followed by Ohm’s law.

■ Step 4: Determine the node voltages. For that, solve nodal equations, which are obtained in step 3.

By utilizing node voltages, we can determine the current flowing and the voltages throughout any part of the given circuit through node voltages.

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## Electric Sources

Electric sources can be broadly classified into two types. Which are independent and dependent sources.

The independent electric source delivers a set value of current or voltage connected to the circuit. These electric sources are batteries and power supplies. The power supplies deliver the stable set values, while batteries eventually offer the stable set values without recharging them.

A dependent electric source delivers current or voltage from any circuit part. These sources are used to analyze amplifiers. The main characteristics of the amplifiers are current gain (Ai) and voltage gain (AV).

## Difference Between Nodal And Mesh Analysis There is a specific difference between the nodal and mesh analysis.

The voltages are observed in certain branches using nodal analysis in the electric circuit. However, in mesh analysis, current values are utilized in the specific circuit branch.

In the electric network, mesh analysis is used to calculate the current flows through the planar circuit. Planar circuits are those which can be drawn on the plane surface, and now wires are crossing each other. That’s why mesh analysis is also known as the loop analysis or mesh-current method. And understand how to write a Spatial Order essay here.

Nodal analysis is completely based on the application of Kirchoff’s Current Law. Suppose a circuit has “n” nodes. Then there will be “n-1” simultaneous equations to solve. By solving these equations, we can evaluate the node voltages. The count of non reference nodes is equal to the nodal equations which can be obtained. Understand the difference between Fellowship Vs Internship here.

## Nodal Analysis

Nodal analysis is the technique to determine voltages by applying KCL at the node. In the above circuit, extraordinary nodes are V1, V2 & V3 in reference to the ground. The ground symbol is three diagonal lines, and the decreasing horizontal bars represent chassis and earth ground, respectively.

In the above figure, let’s apply KCL at nodes 1 & 2.

At node 1:

i1+i2+i3 = 0 ……(1)

i1= (V1-Vbs)/R1+R2

i2= V1/R3

i3 = V1-V2/R4

So, substituting these current values in the above equation (1).

((V1-Vbs)/R1+R2) + (V1/R3) + (V1-V2/R4) = 0

(V1/ R1+R2)-(Vbs/R1+R2)+ (V1/R3)+(V1/R4)- (V2/R4)

V1((1/ R1+R2)+1/R3+1 /R4) – (V2/ R4) = Vb/R1+R2

At node 2:

i4+i5+i6 = 0 ……(2)

i4=V2-V1/R4

i5=V2/R5

i6=I0

Now, let’s substitute these current values in the above equation (2).

((V2-V1)/R4)+ (V2/R5)+ I0

(-1/R4)V1+ (1/R4 + 1/R4)V2= I0

Simultaneous equation solution.

a11V1+a12V2 = b1

a21V1+a22V2 = b2

## How To Do Nodal Analysis With Current Sources

In the example below, let’s perform nodal analysis with a current source. We will calculate the nodal voltage in the following circuit. You will find the three nodes in the above circuit. One is the reference node, and the remaining two nodes are the non reference nodes. As node 1 & node 2.

Step 1:

In the first step, the node voltages are represented with V1 and V2. Also, the branch’s current directions can be marked with respect to reference nodes.

Step 2:

In the second step, we will apply KCL to two nodes which are node 1 and node 2.

Applying KCL to node1 in the above circuit

i1 =i2+i3……(1)

In the same way, at node 2

i2+i4 = i1+i5….(2)

Step 3:

Here we will apply Ohm’s law to KCL equations.

Let’s apply Ohm’s law to the KCL equations.

i1 =i2+i3

5 = (V1-V2/4) + (V1-0/2)

Let’s simplify the equation, and we will get the following:

20 = 3V1-V2……(3)

Now apply Ohms law to KCL equation (2) at node2

i2+i4 = i1+i5

(V1-V2/4)+10 = 5 + V2-0/6

Simplifying this, we will get the following:

60 = -3V1+5V2 ….(4)

Solving equations (3) and (4), we will get V1 & V2 values.

Let’s use the elimination method:

20 = 3V1-V2

60 = -3V1+5V2

4V2 = 80

V2= 20

Substituting V2 = 20 in equation (3) then, we can get

20 = 3V1-V2

20 = 3V1-20 => V1 = 40/3 = 13.3 V

Hence, node voltages like V1 = 13.33 Volts & V2 = 20 Volts.

In the same way, we can have the nodal analysis with a voltage source.

## How to Determine Current Direction In Nodal Analysis

■ Step 1: Identify every node in the circuit and select a reference node.

■ Step 2: Now proceed with Kirchhoff’s Current Law equation for any unknown nodal voltages.

■ Step 3: Use Ohm’s law to rewrite currents in terms of nodal voltages.

■ Step 4: Now substitute currents from Step 3 into the KCL equations from step 2.

■ Step 5: Solve and get the values of the unknown voltages.

## How To Choose Nodes In Nodal Analysis

As we know, nodal analysis is the general method based on KCL or current equations. Here are the steps to select nodes:

■ Identify all the nodes in the given circuit.

■ Select the reference node. You can identify it with the reference ground symbol. The best selection is the node with the highest branches.

■ You can also select the node that can immediately give the other node voltage. This can be found below the voltage source.

■ Assign voltage variables to the other nodes. These are the voltage across nodes.

■ Using a system of equations, write a KCL equation for each node. Here you need to sum the currents leaving the nodes and set them equal to zero. Now rearrange these equations as A*V1+B*V2 = C.

■ Now solve a system of equations obtained from step 4. You can use various methods like – simple substitution, adjoint matric method, etc.

## Complications In Nodal Analysis

There are some complications in the nodal analysis. Which are:

### 1. Dependent Current Source

The solution is to write KCL equations for each node followed by expressing the additional variable in terms of node voltages. Rearrange them as shown in step 4 above. Solve the remaining nodes in step 5.

### 2. Independent Voltage Source

Problem: We don’t have an idea about the current through the voltage source. So, we can’t write KCL equations for the nodes connected to the unknown voltage source.

Solution: In case the voltage source is between the reference node and any other node. Then we can have the free node voltage.

Note that node voltage must be equal to the voltage value from the source. If not, you can use the “super node,” which consists of the source and nodes connected to it.

Now write the KCL equation for all current entering and leaving the super node. We can have one equation with the two unknowns: node voltages.

Another equation relating to these voltages is the equation given by the voltage source. Which is (V2-V1=source value). This new system of equations can be solved as in Step 5 above.

### 3. Dependent Voltage Source

Solution: It is almost similar to the independent voltage source with an additional step. Start writing the super node KCL equation. Then write the quality controlling the source in terms of the node voltages.

Now, rearrange the equation in the form A*V1+B*V2=C and solve the system as we have seen above.

## Nodal Analysis With Voltage Sources

Case 1: In case the voltage source is connected between the reference node and the non reference node. Then we can simply put the value at the non reference node, which is equal to the voltage of the voltage source. Its analysis can be performed with current sourced V1 = 10 Volts.

Case 2: If the voltage source is situated between the two non reference nodes, it creates a super node. Its analysis is done as given below.

Super Node Analysis

Before doing the super node analysis, it’s imperative to understand it. When an independent or dependent voltage source is connected between the two non reference nodes. Then these two nodes create a general node known as a super node.

That means the super node is the place enclosing the voltage source and related two nodes. The properties of a super node are given below:

■ It is always the difference between the voltages of two non reference nodes. Which is known as a super node.

■ The super node doesn’t have its own voltage.

■ The super node needs the application of both KCL and KVL equations to solve it.

■ We can connect any element parallel to the voltage source, forming the super node.

■ The super node satisfies the KCL similar to the simple node.

## Solving Circuit Containing Super Node

Let’s see how to solve the circuit with the super node. In the circuit above 2V voltage source is connected between Node-1 and Node-2. That means it forms a super node. It also has a 10Ω resistor in parallel.

Any component connected in parallel with the voltage source forming a super node doesn’t make any difference. Because V2– V1 = 2V always whatever the value of the resistor may be.

So, we can remove 10 Ω from the circuit. It can be redrawn and applying KCL to the super node; we get the following circuit: Let’s express it in terms of the node voltages:

2 =  = 2V1 + V2 + 28 = V2 = -2V1 – 20 …… (5)

V1 + 2 – V2 = 0 = V2 = V1 + 2 ……. (6)

Using equations 5 and 6, we can write as:

V2 = V1 + 2 = -2V1 – 20 = 3V1 = -22

Hence,

V1 = -7.333 V &

V2 = -5.333 V

So we have got the required answers.

## Nodal Analysis By Inspection

The system of equations can be easily solved if there is no voltage source. Let’s summarize the inspection method below:

1. Verify the circuit uses only a current source with resistors and no voltage sources. In case there are voltage sources, they must be converted to current sources.
2. Now get all the current summing nodes and number them. Also, decide on the reference node ground.
3. Locate the first node to get the equations. This is the node of interest, and the next few steps are associated with it.
4. Now add the current sources feeding the node of interest. Entering the deemed positive while exiting is considered negative. The calculation is placed on one side of the equals sign.
5. Proceed with finding all the resistors connected to the node of interest and write them as a sum of conductance on the other side of the equals sign.
6. Now find all resistors connected to the node of interest and other nodes except the ground reference.
7. For each of these other nodes, multiply the sum of the conductance between the node of interest and this other node by this other node’s voltage and then subtract that product from the equation built so far. Once all other nodes are considered, this equation is finished.
8. Find the next node and consider this as the new node of interest.
9. Repeat steps 4 through 7 until all nodes have been treated as the node of interest.

Each iteration creates a new equation. There will be the same equations as the number of nodes excluding the reference node.

## Conclusion

This is all about nodal analysis. It is a kind of circuit analysis that works with the Kirchhoff Current Law and node equations. It is used to solve the voltage values in the given circuit. We hope you have gone through every aspect of the nodal analysis given here.

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#### What is the nodal formula?

It is the number of radial nodes = n – l – 1. A number of angular nodes = l. Then we have a total number of nodes = n – 1.

#### When to use nodal analysis and mesh analysis?

Mesh analysis is used to determine currents in a circuit loop. It requires solving KVL equations for the voltages across each component in the loop. It is generally used in planar circuits.

#### What is the definition of nodal analysis?

It is the simple calculation used to get the voltage values and distribution across the network or electric circuit. This method is also known as the node voltage method.

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