- Electric Circuit

## Nodal Analysis

What is nodal analysis.

Nodal analysis is used for solving any electrical network, and it is defined as

The mathematical method for calculating the voltage distribution between the circuit nodes.

This method is also known as the node-voltage method since the node voltages are with respect to the ground. The following are the three laws that define the equation related to the voltage that is measured between each circuit node:

- Kirchhoff’s voltage law
- Kirchhoff’s current law

## Features of Nodal Analysis

- Nodal analysis is an application of Kirchhoff’s current law.
- When there are ‘n’ nodes in a given electrical circuit, there will be ‘n-1’ simultaneous equations to be solved.
- To obtain all the node voltages, ‘n-1’ should be solved.
- The number of non-reference nodes and the number of nodal equations obtained are equal.

## Recommended Video:

## Procedure of Nodal Analysis

The following steps are to be followed while solving any electrical circuit using nodal analysis:

To identify the principal nodes and select one of them as a reference node. This reference node will be treated as the ground.

All the node voltages with respect to the ground from all the principal nodes should be labelled except the reference node.

The nodal equations at all the principal nodes except the reference node should have a nodal equation. The nodal equation is obtained from Kirchhoff’s current law and then from Ohm’s law.

To obtain the node voltages, the nodal equations can be determined by following Step 3.

Hence, for a given electrical circuit, the current flowing through any element and the voltage across any element can be determined using the node voltages.

## Types of Nodes in Nodal Analysis

There are two types of nodes in nodal analysis:

- Non-reference node
- Reference node

## Non-Reference Node

The node with a definite node voltage is a non-reference node.

## Reference Node

The node that acts as a reference point to all the other nodes is known as the reference node, which is also known as the datum node.

There are two types of reference nodes, and they are:

- Chassis ground

A reference node that acts as a common node for more than one circuit is called chassis ground.

- Earth ground

In any circuit where the earth’s potential is used as a reference, it is known as the earth’s ground.

## What is a Super Node?

The super node is defined as a voltage source connected between the two non-reference nodes such that these two nodes form a generalized node.

## Properties of Super Node

The following are the properties of super nodes:

- The difference between the voltage of two non-reference nodes can be determined at the super node.
- A super node does not have its own voltage.
- Kirchhoff’s current law and Kirchhoff’s voltage law are both applied for solving the super node.

## Nodal Analysis Examples

For the following circuit, find V a by nodal analysis.

Step 1: To select a reference node and label the node voltages

Step 2: Apply Kirchhoff’s current law to all the nodes

From eq.1,2,3 we know that

Step 3: To find the required quantities

Applying Kirchhoff’s voltage law in the above loop, we get

-V 1 – V a + V 2 = 0

Determine the voltage at each node of the given circuit using nodal analysis.

The number of nodes that are present in the given circuit is 3

The nodes that are present in the circuit are numbered as shown in the figure

Let node 2 be the reference node, and this node’s voltage will be zero.

Using Kirchhoff’s current law at each node, we get

This is a result of KCL at node 1

This is a result of KCL at node 3

Solving the above equations we get

V 1 = 68.2v

V 3 = 27.3v

## Frequently Asked Questions – FAQs

If there are 12 nodes in a circuit, how many equations do we get.

We know that one node is always considered a reference node, so the number of equations we can get will always be less than the given number of nodes. As a result, the number of equations that may be derived here is 11.

## Which law is used in nodal analysis?

Kirchhoff’s current law is used in nodal analysis.

## What does Kirchhoff’s voltage law state?

According to Kirchhoff’s voltage law, the algebraic sum of the potential difference in any loop should equal zero.

## What does Kirchhoff’s current law state?

According to Kirchhoff’s current law, the current flowing into a node should equal the current flowing out of it.

## What is the advantage of nodal analysis?

The benefits of nodal analysis include the reduced number of equations necessary to determine unknown values

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## What is Nodal Analysis in Circuit Design?

Key takeaways.

Nodal analysis is a mathematical method that is used to analyze circuits in terms of voltage drops between nodes in a circuit diagram.

Nodal analysis can be performed in the time domain or frequency domain, but it is limited to linear time-invariant (LTI) systems.

Although the solution algorithm used in nodal analysis is only defined for linear systems, nonlinear systems can be approximated around an operating point and used in nodal analysis.

Circuit simulation results need to be compared with in-situ measurements during prototyping and production.

Analyzing electrical circuits is vital to ensuring modern technology works as expected. Most modern electronics go through some sort of simulation and evaluation process to ensure designs work as intended, and to provide a set of reference calculations for comparison with in-circuit tests. SPICE simulations are the workhorses for circuit design and analysis, and today’s commercial solutions provide many built-in simulations.

Among the simulations found in SPICE simulators, nodal analysis is a fundamental technique used to examine voltage and current distribution in a circuit. This technique effectively combines both of Kirchhoff’s laws and Ohm’s law into a single matrix equation. Keep reading to learn more about what nodal analysis is and how it’s used in circuit design and analysis.

## What is Nodal Analysis?

Nodal analysis is a mathematical method for calculating the distribution of voltages between nodes in a circuit. Also referred to as the node-voltage method, this strategy uses Ohm’s law , Kirchhoff’s voltage law, and Kirchhoff’s current law to define an equation relating the voltage measured between each circuit node and some reference (usually ground). The voltage drops measured between neighboring nodes are taken as the variables in a set of linear equations and the system can be solved using a standard algorithm (e.g., Gauss-Jordan elimination).

## Constructing a Matrix Equation in Nodal Analysis

The general process for constructing a matrix equation for nodal analysis is to use Kirchoff’s laws between each node in a circuit diagram to write out a set of equations relating the voltage drop across different components to the currents flowing into each node. Use the following process:

Draw out the circuit diagram with currents defined at each node.

Pick a reference node (usually ground) and write a variable for the voltage at each node with respect to the reference node.

Write out Kirchoff’s current law for each node in terms of the circuit impedances and the voltages at neighboring nodes.

Rewrite the system of equations for the node voltages in matrix form.

Solve the matrix equation using an inverse matrix.

To see how this works, it helps to look at a simple example.

## An Example of Nodal Analysis

The example circuit diagram below shows 4 impedances with 1 voltage source and 4 distinct nodes. Node A is just the input voltage, leaving the voltages at the 3 other nodes to be determined. By applying steps 1-3, we arrive at the following set of equations for the voltages at each node:

The circuit diagram and the system of equations used to determine the voltages at each node.

We can then rewrite the above 3 equations as a matrix equation:

Rewriting the circuit diagram’s system of equations as a matrix equation.

This matrix equation is then easily solved by multiplying by the inverse of the coefficient matrix. In the above circuit diagram, we’ve defined a “reference node” at the common ground connection. The voltage variables are all measured with respect to this point. This is important for more than just creating a closed circuit; it sets the measurement reference for all voltage variables that appear in the above matrix equation.

## Reference Nodes

One key to doing nodal analysis successfully is to properly define reference nodes, also called datum nodes. In nodal analysis, a reference node is the point that is used for comparison when measuring voltages, because a voltage is only defined between two points in space. Imagine you were using a voltmeter on your circuit diagram; the reference node would be the point where the negative lead was connected, and the positive lead would connect to any other node. This is why voltage measurement probes in SPICE simulations are normally shown as a single probe; the measured voltage is taken with respect to a reference node in the circuit.

In this PSpice simulation result, the voltage measured at this probe is just the voltage measured between R4 and GND; this is determined using nodal analysis.

In SPICE simulations , the reference node is normally taken as ground. If there are two grounds, the reference node is taken as the nearest connected ground potential. In nodal analysis, any node can be taken as a reference point. This gives you an advantage in that you can define a potential difference between two different ground points, such as you might find in a galvanically isolated PCB with two different ground planes.

## How Does Nodal Analysis Work on Nonlinear Circuits?

Because the current and voltage in a nonlinear component is a nonlinear function (meaning the relationship does not obey Ohm’s law ), the matrix equation developed in nodal analysis will have terms that are functions of the voltages at different nodes.

The solved voltage column vector in this equation is defined in terms of a nonlinear impedance matrix and the current in each portion of the circuit. This transcendental matrix equation may include coupled terms, which require simultaneous iterative numerical techniques to determine solutions.

In this general formulation, applying nodal analysis to nonlinear circuits requires solving a transcendental matrix equation for multiple unknowns. This can be a very difficult numerical problem to solve, depending on the form of the derived matrix equation.

Therefore, in order to examine nonlinear circuits using nodal analysis, the current vs. voltage relationship measured between adjacent nodes in the circuit needs to be approximated around some operating point. The typical method to do this is to use one of the following general approximation techniques for a nonlinear time-invariant system:

Taylor series : For a system of N components, define 2N new variables that are shifted away from the intended operating point with a linear transformation. Now, expand the current-voltage function for each nonlinear component as a Taylor series in terms of these new variables.

Maclaurin series : Expand the current-voltage function for each nonlinear component directly around the operating point without defining new variables as a Maclaurin series.

By keeping only the first-order terms (i.e., linear terms), you now have a linearized circuit that includes nonlinear components. The results from this procedure are only valid within a small deviation from the operating point, depending on the impedance function for each nonlinear component in the system. Circuit simulators that use iterative solution techniques will apply these approximations for you and can solve operating point matrix equations in nodal analysis for nonlinear circuits.

Vital node voltage and nodal voltage analysis requires a strong understanding of current source in your electronic circuit but when tasked with unknown node voltages or independent voltage source , your nodal equation can get difficult. Looking at principal nodes for your current source and branch current references can encourage a better understanding of nodal voltage . But generally for electric circuit performance, circuit analysis through SPICE simulation will unlock all types of circuit variables from transient analysis to parallel circuits to various matrices and beyond. Use your node equations wisely in your next electrical circuit design .

If you’re still unclear as to what nodal analysis is, you can use an industry-standard SPICE simulator to calculate voltage and current throughout your circuits. Once you’ve got your results and you’re ready to create your physical design, use the best set of PCB design and analysis software to capture your schematics as a blank circuit board. Allegro PCB Editor from Cadence gives you the layout and routing features you need to create your circuit boards and prepare them for production. You can then use nodal analysis results as a comparison for electrical tests.

If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts .

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## What is Nodal Analysis?

Nodal Analysis is a method for determining the branch currents in a circuit. In this method, one of the nodes is taken as the reference node. The potentials of all the nodes in the circuit are measured with respect to this reference node.

The nodal analysis is based on the Kirchhoff’s Current Law , which states that "the algebraic sum of incoming currents and outgoing currents at a node is equal to zero".

$$\mathrm{\sum\:\mathit{I}_{incoming}\:+\:\sum\:\mathit{I}_{outgoing}=0}$$

Node – A node is a point in a network where two or more circuit elements meet.

Junction – A junction is point where three or more circuit elements meet.

In the nodal analysis, we have to find the potentials at the junction points instead of nodes. The number of independent node-pair equations needed is one less than the number of junctions in the network. That is, if “n” is the number of independent node equations and “j” is the number of junctions, then,

$$\mathrm{n\:=\:\mathit{j}\:-\:1}$$

## Explanation

Consider the circuit shown below. In the circuit, A, B, C and D are four nodes and node B and D are the junction points, where node D is taken as reference node. Hence,

Assuming the voltage V B at the node B higher than all the other nodes of the circuit.

Applying KCl at the node B, we obtain,

$$\mathrm{\mathit{I}_{1}\:+\:\mathit{I}_{2}\:+\:\mathit{I}_{3}=0}\:\:\:…(1)$$

Now, by Ohm’s law, the branch currents are,

$$\mathrm{\mathit{I}_{1}=\frac{\mathit{V}_{B}\:-\mathit{E}_{1}\:-0}{\mathit{R}_{1}}=\frac{\mathit{V}_{B}\:-\:E_{1}}{\mathit{R}_{1}}}$$

$$\mathrm{\mathit{I}_{2}=\frac{\mathit{V}_{B}\:-\mathit{E}_{2}\:-0}{\mathit{R}_{2}}=\frac{\mathit{V}_{B}\:-\:E_{2}}{\mathit{R}_{2}}}$$

$$\mathrm{\mathit{I}_{3}=\frac{\mathit{V}_{B}\:-0}{\mathit{R}_{3}}=\frac{\mathit{V}_{B}}{\mathit{R}_{3}}}$$

Substituting the values of I 1 , I 2 and I 3 in the equation (1), we get,

$$\mathrm{\frac{\mathit{V}_{B}\:-\mathit{E}_{1}}{\mathit{R}_{1}}+\frac{\mathit{V}_{B}\:-E_{2}}{\mathit{R}_{2}}+\frac{\mathit{V}_{B}}{\mathit{R}_{3}}=0}\:\:\:…(2)$$

The voltage VB at node B can be obtained by solving the equation (2). As, the emfs, E 1 , E 2 and resistance values are known. Thus, we can determine the values branch currents.

## Numerical Example

Find the currents in the various branches of the circuit in figure given below by nodal analysis.

Mark the currents in various branches as shown in the circuit diagram. If the value of any current comes out to be negative in the solution, it means that actual direction of the current is opposite to that of assumed. Take point B as the reference node. Thus,

Applying KCL at node A,

$$\mathrm{\mathit{I}_{1}\:+\:\mathit{I}_{2}\:+\:\mathit{I}_{3}=0}\:\:\:…(3)$$

By Ohm’s law,

$$\mathrm{\mathit{I}_{1}=\frac{\mathit{V}_{A}-100}{20}}$$

$$\mathrm{\mathit{I}_{2}=\frac{\mathit{V}_{A}}{10}}$$

$$\mathrm{\mathit{I}_{3}=\frac{\mathit{V}_{A}-50}{30}}$$

Substituting the values of I 1 , I 2 and I 3 in the equation (3), we obtain,

$$\mathrm{\frac{\mathit{V}_{A}-100}{20}+\frac{\mathit{V}_{A}}{10}+\frac{\mathit{V}_{A}-50}{30}=0}$$

$$\mathrm{\Rightarrow\frac{3(\mathit{V}_{A}-100)+6\mathit{V}_{A}+2(\mathit{V}_{A}-50)}{60}=0}$$

$$\mathrm{\Rightarrow\:3\mathit{V}_{A}-300+6\mathit{V}_{A}+2\mathit{V}_{A}-100=0}$$

$$\mathrm{\Rightarrow\:11\mathit{V}_{A}=400}$$

$$\mathrm{\mathit{V}_{A}=\frac{400}{11}=36.36\mathit{V}}$$

Hence, the voltage at the node A is equal to 36.36 V, therefore the branch current are,

$$\mathrm{Current\:\mathit{I}_{1}=\frac{\mathit{V}_{A}-100}{20}=\frac{36.36-100}{20}=-3.182\:A}$$

$$\mathrm{Current\:\mathit{I}_{2}=\frac{\mathit{V}_{A}}{20}=\frac{36.36}{20}=3.636\:A}$$

$$\mathrm{Current\:\mathit{I}_{3}=\frac{\mathit{V}_{A}-50}{30}=\frac{36.36-50}{30}=0.455\:A}$$

The negative sign for currents I 1 and I 3 3 shows that actual direction of currents is opposite to that of assumed.

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## Nodal Voltage Analysis

Nodal Voltage Analysis finds the unknown voltage drops around a circuit between different nodes that provide a common connection for two or more circuit components

Nodal Voltage Analysis complements the previous mesh analysis in that it is equally powerful and based on the same concepts of matrix analysis. As its name implies, Nodal Voltage Analysis uses the “Nodal” equations of Kirchhoff’s first law to find the voltage potentials around the circuit.

So by adding together all these nodal voltages the net result will be equal to zero. Then, if there are “n” nodes in the circuit there will be “n-1” independent nodal equations and these alone are sufficient to describe and hence solve the circuit.

At each node point write down Kirchhoff’s first law equation, that is: “ the currents entering a node are exactly equal in value to the currents leaving the node ” then express each current in terms of the voltage across the branch. For “n” nodes, one node will be used as the reference node and all the other voltages will be referenced or measured with respect to this common node.

For example, consider the circuit from the previous section.

## Nodal Voltage Analysis Circuit

In the above circuit, node D is chosen as the reference node and the other three nodes are assumed to have voltages, Va, Vb and Vc with respect to node D . For example;

As Va = 10v and Vc = 20v , Vb can be easily found by:

again is the same value of 0.286 amps , we found using Kirchhoff’s Circuit Law in the previous tutorial.

From both Mesh and Nodal Analysis methods we have looked at so far, this is the simplest method of solving this particular circuit. Generally, nodal voltage analysis is more appropriate when there are a larger number of current sources around. The network is then defined as: [ I ] = [ Y ] [ V ] where [ I ] are the driving current sources, [ V ] are the nodal voltages to be found and [ Y ] is the admittance matrix of the network which operates on [ V ] to give [ I ].

## Tutorial Summary

The basic procedure for solving Nodal Analysis equations is as follows:

- 1. Write down the current vectors, assuming currents into a node are positive. ie, a ( N x 1 ) matrices for “N” independent nodes.
- Y 11 = the total admittance of the first node.
- Y 22 = the total admittance of the second node.
- R JK = the total admittance joining node J to node K .
- 3. For a network with “N” independent nodes, [ Y ] will be an ( N x N ) matrix and that Ynn will be positive and Yjk will be negative or zero value.
- 4. The voltage vector will be ( N x L ) and will list the “N” voltages to be found.

We have now seen that a number of theorems exist that simplify the analysis of linear circuits. In the next tutorial we will look at Thevenins Theorem which allows a network consisting of linear resistors and sources to be represented by an equivalent circuit with a single voltage source and a series resistance.

## Read more Tutorials inDC Circuits

- 1. DC Circuit Theory
- 2. Ohms Law and Power
- 3. Electrical Units of Measure
- 4. Kirchhoffs Circuit Law
- 5. Mesh Current Analysis
- 6. Nodal Voltage Analysis
- 7. Thevenin’s Theorem
- 8. Nortons Theorem
- 9. Maximum Power Transfer
- 10. Star Delta Transformation
- 11. Voltage Sources
- 12. Current Sources
- 13. Kirchhoff’s Current Law
- 14. Kirchhoff’s Voltage Law
- 15. Voltage Dividers
- 16. Current Dividers
- 17. Electrical Energy and Power

## 129 Comments

Notify me of follow-up comments by email.

the lay out is not that interesting, it could be better

Ok its well understood. But can someone please help me on this? =>given a three bus system with all the parameters using nodal analysis generate the Y bus.

So whenever using nodal analysis, the reference node should have no voltage?

Find the voltage across the 3 A current source using node analysis.

powerful presentation, with out a doubt the presenter is the best tutor on the internet

Outstanding explanation of node,s formula my hats off to author

Me is the student study electrical and electronics engineering at must University at mbeya I am join in that app to get the material

Very impressed learning a lot

I am an electronics engineering

Is there any app for finding solutions of node and mesh analysis

I need some solution

Nodal analysis

Nice lecture notes and example here posted

You never fail to please.

I want to understand electronic more

I still don’t understand ,but helpfull

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- Nodal Analysis: A Detailed Study

## Nodal Analysis: An Introduction

Node voltages are used as the circuit variables in the nodal analysis, offering a general procedure for studying circuits. The Node-Voltag e Method is an alternative name for nodal analysis. The characteristics of nodal analysis incl ude:

On the use of Kirchhoff's Current Law (KCL), nodal analysis is utilised

There will be "n-1" simultaneous equations to solve with "n" nodes

All of the node voltages can be found by solving 'n-1' equations

The number of possible nodal equations is equal to the number of non-reference nodes.

Keep reading through the following content to learn more about nodal analysis, types of nodes, examples, solving circuits using nodal analysis and more.

## Kirchhoff’s Current Law

A technique for figuring out the branch currents in a circuit is termed nodal analysis . One of the nodes is chosen to serve as the reference node in this method. All the circuit's node potentials are calculated in relation to this reference node.

Kirchhoff's Current Law (KCL) asserts that " the algebraic sum of incoming currents and outgoing currents at a node is equal to zero ," and serves as the foundation for the nodal analysis.

\[\begin{array}{l}\sum {\mathop I\nolimits_{incoming} + } \mathop I\nolimits_{outgoing} = 0\\\end{array}\]

Node : In a network, a node is a location where two or more circuit elements converge.

Junction: Junction is the place where three or more circuit elements come together. We must locate the potentials at the junction points rather than the nodes in the nodal analysis. There are fewer independent node-pair equations required than there are junctions in the network. In other words, if there are "n" independent node equations and "j" junctions, then:

\[n = j - 1\]

## Types of Nodes

Non-Reference Node: It has a fixed node voltage.

Reference Node: It serves as a point of reference for all other nodes in the network.

## Reference Node Types

Chassis Ground: This kind of reference node serves as a common node for multiple circuits.

Earth Ground: This type of reference node is known as Earth Ground when earth potential is utilised as a reference in any circuit.

## Step-by-step Procedures

To resolve nodal analysis in an electrical circuit or network, follow these nodal analysis steps below:

Recognise the primary nodes in Step 1 and choose one of them to serve as the reference node. Thus, this node is handled the same as the ground.

Mark all the primary node’s voltages with respect to the ground terminal in Step 2 , excluding the reference node.

Write nodal equations in Step 3 at each of the primary nodes aside from the reference node. You can arrive at this equation by first applying KCL and then Ohm's law.

We must resolve the nodal equations we obtained in Step 3 to obtain the node voltages in Step 4 .

## Difference Between Mesh and Nodal Analysis

In contrast to mesh analysis, which uses current values in a specific branch of a circuit, nodal analysis of an electrical network allows for the observation of voltages at a given branch. Mesh analysis is one type of technique used in the electrical network to solve planar circuits, specifically for the currents at any branch. Without putting any wires in contact with one another, these circuits can be sketched on a flat surface. Any circuit employs a more all-encompassing technique called loop analysis.

The KCL (Kirchhoff's Current Law) application is necessary for nodal analysis . The "n-1" instantaneous equations are required to solve n nodes. These equations can be solved to obtain all the node voltages. The number of acquirable nodal non-reference is equal to the number of reference nodes.

## Solving of Circuit Using Nodal Analysis

Look at the circuit below named Figure 1. A B, C, and D are the four nodes in the circuit, and nodes B and D are the junction points, with node D serving as the reference node.

Nodal analysis problems for an explanation of the mathematical solution.

Hence, If node B has a higher voltage \[\mathop V\nolimits_B \] than any of the other nodes in the circuit.

By putting KCL on node B, we get,

\[\mathop I\nolimits_1 + \mathop I\nolimits_2 + \mathop I\nolimits_3 = 0............(1)\]

The branch currents are now determined by Ohm's law to be,

\[\mathop I\nolimits_1 = \dfrac{{\mathop V\nolimits_B - \mathop E\nolimits_1 - 0}}{{\mathop R\nolimits_1 }} = \dfrac{{\mathop V\nolimits_B - \mathop E\nolimits_1 }}{{\mathop R\nolimits_1 }}\]

\[\mathop I\nolimits_3 = \dfrac{{\mathop V\nolimits_B - \mathop E\nolimits_2 - 0}}{{\mathop R\nolimits_2 }} = \dfrac{{\mathop V\nolimits_B - \mathop E\nolimits_2 }}{{\mathop R\nolimits_2 }}\]

${I_2} = \dfrac{V_B}{R_3}$

In equation (1), we obtain by substituting the values of \[\mathop I\nolimits_1 \] , \[\mathop I\nolimits_2 \] , and \[\mathop I\nolimits_3 \] .

\[\dfrac{{\mathop V\nolimits_B - \mathop E\nolimits_1 }}{{\mathop R\nolimits_1 }} + \dfrac{{\mathop V\nolimits_B - \mathop E\nolimits_2 }}{{\mathop R\nolimits_2 }} + \dfrac{{\mathop V\nolimits_B }}{{\mathop R\nolimits_3 }} = 0...........(2)\]

The equation can be solved to determine the voltage \[\mathop V\nolimits_B \] at node B. (2). Since the resistance values, \[\mathop E\nolimits_1 \] , and \[\mathop E\nolimits_2 \] values are known. As a result, we can calculate the branch current values.

Nodal Analysis Examples

By using nodal analysis, determine the currents in the various circuit branches shown in the accompanying Figure 2.

As illustrated in the circuit schematic, indicate the currents in the various branches. The true direction of the current is the opposite of the assumed direction if the value of any current in the solution turns out to be negative. As the reference node, use point B. Thus,

Using Ohm’s law,

\[\mathop I\nolimits_1 = \dfrac{{\mathop V\nolimits_A - 100}}{{20}}\]

\[\mathop I\nolimits_2 = \dfrac{{\mathop V\nolimits_A }}{{10}}\]

\[\mathop I\nolimits_3 = \dfrac{{\mathop V\nolimits_A - 50}}{{30}}\]

\[\dfrac{{\mathop V\nolimits_A - 100}}{{20}} + \dfrac{{\mathop V\nolimits_A - 50}}{{30}} + \dfrac{{\mathop V\nolimits_A }}{{10}} = 0\]

\[\dfrac{{3(\mathop V\nolimits_A - 100)}}{{60}} + \dfrac{{2(\mathop V\nolimits_A - 50)}}{{60}} + \dfrac{{6\mathop V\nolimits_A }}{{60}} = 0\]

\[\mathop {3V}\nolimits_A - 300 + \mathop {6V}\nolimits_A + \mathop {2V}\nolimits_A - 100 = 0\]

\[\mathop {11V}\nolimits_A = 400\]

\[\mathop V\nolimits_A = \dfrac{{400}}{{11}} = 36.36V\]

Current \[\mathop I\nolimits_1 = \dfrac{{\mathop V\nolimits_A - 100}}{{20}} = \dfrac{{36.36 - 100}}{{20}} = - 3.182V\]

Current \[\mathop I\nolimits_2 = \dfrac{{\mathop V\nolimits_A }}{{20}} = \dfrac{{36.36}}{{20}} = 3.636V\]

Current \[\mathop I\nolimits_3 = \dfrac{{\mathop V\nolimits_A - 50}}{{30}} = \dfrac{{36.36 - 50}}{{30}} = - 0.455V\]

The fact that currents have a negative sign indicates that their real directions are different from what was previously supposed.

This was all about a general overview of nodal analysis. When the circuit diagram doesn't show any conductor lane crossings, nodal analysis can be applied for circuit analysis prominently and uses Kirchhoff's Current Law & node equations to solve voltage values within the circuit. When the voltage at the ground terminal is equal to 0 volts, this nodal analysis can be used to determine the voltage at each node with respect to a reference node that is typically known as ground.

## FAQs on Nodal Analysis: A Detailed Study

1. How many equations are there in a circuit with 12 nodes?

The number of equations we can obtain will always be smaller than the specified number of nodes because we are aware that one node is always regarded as a reference node. There are 11 equations that may be obtained because of this.

2. How does nodal analysis determine node voltage?

With nodal analysis, anyone can determine the voltage at each node of the given circuit. Assume that node 2 serves as the reference node, with zero voltage.

3. What is the Kirchhoff current law?

Kirchhoff's current law states that the current entering a node should be equal to the current leaving it.

## What is Nodal Analysis : Circuit with Example

In an electric or electronic circuit , a node is used to connect different components to each other. According to KCL ( Kirchoff’s Current Law ), the current that enters into a node should leave from a node. In the circuit, each point on the node has the same voltage, so this voltage is known as the node voltage. It is the voltage disparity between the node at an arbitrary location & the ground point. The nodes within real circuits can be made up of wires, but the voltage is not the same all over the node. To solve any electrical circuit, there are two techniques are used mesh analysis and nodal analysis. In nodal analysis, we use the node voltages along with the ground and is also known as the node voltage technique. This article discusses an overview of nodal analysis with an example.

## What is Nodal Analysis?

Nodal analysis is one kind of method used in electrical networks to analyze circuits through node voltages like the circuit variables. An alternate name of this method is Node-Voltage Method. The main features of nodal analysis include the following.

- Nodal Analysis depends on the KCL which is known as Kirchhoff’s Current Law.
- If the nodes in the circuit are ‘n’ then ‘n-1’ simultaneous equations will be there to solve.
- The voltage on all nodes in the circuit can be obtained by solving ‘n-1’ equations.
- The total nodal equations are equivalent to total nonreference nodes that can be obtained.

In nodal analysis, there are two kinds of nodes available like reference and nonreference.

- A node in the network which performs like a reference point to remaining nodes is known as the reference node or Datum Node.
- A node in the network which includes an exact node voltage is known as a non-reference node. For instance, node 1 & 2 are the non-reference nodes

Again, reference nodes are classified into two types like chassis ground and earth ground.

- Chassis Ground is a kind of reference node that works like an ordinary node for above one circuit.
- In any network, once earth potential is utilized like a reference then this kind of reference node is known as Earth Ground.

## Step by Step Procedure

The step-by-step procedure of nodal analysis is used to solve nodal analysis in an electrical circuit or network.

- In Step-1, recognize the main nodes & select one of them as a reference node. So, this node is treated like the Ground.
- In Step-2, mark the node voltages with respect to the ground terminal from all the main nodes apart from the reference node.
- In Step-3, at all the main nodes excluding the reference node, write nodal equations. This equation can be attained by applying KCL first & after that Ohm’s law .
- In Step-4, to get the node voltages, we need to solve the nodal equations which are achieved in Step-3.

So by using node voltages, we can discover the current flowing throughout any part & the voltage across any part that is there in the given circuit through node voltages.

## Electrical Sources

Electrical sources are classified into two types namely independent & dependent.

An independent electrical source gives a set value of current or voltage that is connected to the circuit. These sources are nothing but batteries & power supplies . Here, the power supply gives a stable set value while batteries provide a stable fixed value eventually without recharging them.

A dependent source is nothing but a current or voltage source whose value mainly depends on a current or voltage value anywhere within the circuit. These sources are used to analyze amplifiers and the main characteristics of amplifiers are current gain (Ai) & voltage gain (AV).

## Linear Dependent Sources

Linear-dependent sources are classified into four types like following.

- Voltage-controlled voltage source
- Current-controlled voltage source
- Current-controlled current source
- Voltage-controlled current source

In a voltage-controlled voltage source, the o/p is ‘V’, AV is the voltage gain & VCD is the parameter being detected. The equation given below is allied with this linear dependent source.

In a current-controlled voltage source, the o/p is V; the resistance is RM & IC is the parameter being detected. The below equation can be allied through a current-controlled voltage source.

In the current-controlled current source, the o/p is ‘I’ & ‘Ai’ is the current gain & IC is the parameter being detected. The following mentioned equation can be allied through a current-controlled current source.

In voltage-controlled current source, the o/p is ‘I’, GM is the conductance & VCD is the parameter being detected. The below equation can be connected through a voltage-controlled current source.

I = GM * VCD

## Difference between Mesh and Nodal Analysis

The difference between mesh and nodal analysis includes the following.

In a nodal analysis of an electrical network, the voltages can be observed at a certain branch whereas, in mesh analysis, current values are used in a certain branch of a circuit.

In the electrical network, mesh analysis is one kind of technique, used to solve planar circuits particularly for the currents at any branch. These circuits can be drawn over a flat surface without crossing wires with each other. A more general method like loop analysis is used in any circuit.

Nodal Analysis depends on the KCL (Kirchhoff’s Current Law) application. If the circuit has n-nodes then there will be ‘n-1’ instantaneous equations to resolve. By solving these equations, we can get all the node voltages. The amount of nonreference nodes is equivalent to the number of Nodal equations that can be acquired.

## Nodal Analysis Circuit

In the following circuit, the nodal analysis method is generally used to determine the voltage through KCL at a node. In this circuit, extraordinary nodes are V1, V2, V3 & V3 is in reference to GND.

In the following circuit, voltages like V1, V2 & V3 are extraordinary nodes whereas V3 is ground.

Apply Kirchoff current law at nodes 1 & 2 to the above circuit

i1+i2+i3 = 0 ……(1)

i1= (V1-Vbs)/R1+R2

i3 = V1-V2/R4

Thus, substitute 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

i4+i5+i6 = 0 ……(2)

i4=V2-V1/R4

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

## Example Problem

Find the node voltage ‘V1’ by applying nodal analysis in the following circuit.

i1+i2+i3 = 0

(V1-10/100)+(V1/100)+(V1-5/200+100)

(V1/100 -10/100) + V1/100+ (V1-5/200+100)

(V1/100+V1/100+V1/300) –(10/100)-(5/300)

V1(1/100+1/100+1/300) –(10/100)-(5/300)

(3+3+1/300)V1 = (30/300+5/300)

(7/300)V1 = 30+5/300

V1 = (35/300)x(300/7)

V1 = (35/7)

## Nodal Analysis with Current Source

In the following example circuit, the nodal analysis using current sources is discussed.

Ex: For the following circuit, calculate the nodal voltage

There are three nodes in the following circuit where one is a reference node and the remaining two are nonreference nodes like node 1 & node2.

In step 1, node voltages are denoted with v1 & v2, and also branch currents directions can be marked with respect to the reference nodes.

In this step, apply Kirchoff current law to two nodes like node1 and node2 When Kirchoff current law is applied to node1 in the above circuit

i1 =i2+i3……(1)

Similarly, at node 2

i2+i4 = i1+i5….(2)

When ohms law is applied to KCL equations

At node1, apply ohms law to KCL equations

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

Once the above equation is simplified then

20 = 3V1-V2……(3)

At node2, apply ohms law to KCL equation2

i2+i4 = i1+i5

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

Simplifying the above equation we get

60 = -3V1+5V2 ….(4)

Solve the equations like 3 & 4 to obtain the v1 & v2 values

Using elimination method

20 = 3V1-V2

60 = -3V1+5V2

When V2 = 20 is substituted in equation (3) then we can get

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

Therefore, node voltages like v1 = 13.33 Volts & v2 = 20 Volts.

Similarly, nodal analysis with a voltage source can also be calculated

Thus, this is all about an overview of Nodal analysis , nodal analysis examples, etc. It is one kind of circuit analysis that works with Kirchhoff’s Current Law & node equations to solve voltage values within the circuit wherever the circuit diagram does not include any conductor lanes crossing. This nodal analysis can be applied for concluding the voltage at every node with respect to a reference node that is normally known as ground wherever the voltage at the ground terminal is equivalent to 0 Volts. Here is a question for you, what is mesh analysis?

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## NODAL ANALYSIS

The term "nodal analysis" refers to the process of solving any electrical network. The process of determining the voltage distribution between nodes in a circuit using mathematics.

## Table of Content

When the branch constitutive relations of all circuit elements have an admittance representation, nodal analysis is possible. Nodal analysis generates a small set of network equations that can be solved by hand if the network is small or quickly solved by computer using linear algebra. The branch current method, also known as nodal analysis, node-voltage analysis or the branch current method, is a method of calculating the voltage (potential difference) between “nodes” (points where elements or branches link) in an electrical circuit using branch currents. Nodal analysis is a way of analysing circuits in terms of voltage dips between nodes in a circuit diagram.

Nodal analysis can be done in the time or frequency domains, but it is only applicable to LTI systems. Nonlinear systems can be approximated around an operational point and employed in nodal analysis, despite the fact that the solution algorithm used in nodal analysis is only defined for linear systems.

## What is Nodal analysis?

Nodal analysis is a method for estimating the voltage distribution between nodes in a circuit using mathematics. This procedure, also known as the node-voltage method, employs Ohm’s law, Kirchhoff’s voltage law and Kirchhoff’s current law to create an equation that relates the voltage measured between each circuit node and a reference voltage (usually ground). The voltage drops between nearby nodes are used as variables in a set of linear equations, which may be solved with a typical technique (e.g., Gauss-Jordan elimination).

## Features of Nodal analysis

- Kirchhoff’s current law is used to do nodal analysis.
- There will be n-1 simultaneous equations to solve when there are n nodes in a particular electrical circuit.
- ‘n-1’ must be solved to obtain all of the node voltages.
- The number of non-reference nodes and nodal equations to be obtained are both the same.
- Nodal Voltage Analysis is a powerful complement to the preceding mesh analysis because it uses the same matrix analysis ideas.
- As a result, the net outcome of summing all of these nodal voltages is zero. Then, if the circuit has “n” nodes, there will be “n-1” independent nodal equations, which are enough to describe and solve the circuit.

## Procedure of Nodal Analysis

The general method for developing a matrix equation for nodal analysis is to write out a set of equations linking the voltage drop across different components to the currents flowing into each node using Kirchoff’s laws between each node in a circuit diagram. The procedure is as follows:

- Create a circuit diagram with each node’s currents defined.
- Choose a reference node (typically ground) and create a variable to represent the voltage at each node in relation to the reference node.
- In terms of the circuit impedances and voltages at surrounding nodes, write down Kirchoff’s current law for each node.
- In matrix form, rewrite the node voltages system of equations.
- Using an inverse matrix, solve the matrix equation.

## Types of nodes in Nodal Analysis

- A non-reference node is one that has a fixed Node Voltage. Non Reference nodes are, for example, Node 1 and Node 2.
- A reference node is a node that serves as a point of reference for all other nodes. The Datum Node is another name for it.

There are several different types of reference nodes: –

- Chassis Ground — This type of reference node serves as a shared ground for multiple circuits.
- Earth Ground — This type of reference node is utilised when the earth potential is employed as a reference in any circuit.

When two non-reference nodes are joined by a voltage source (independent or dependent), a generalised node called the Super node is formed. As a result, a Super node can be thought of as a surface that encloses the voltage source as well as its two nodes.

- At the Super node, the voltage differential between two non-reference nodes is always known.
- A super node does not produce its own voltage.
- To solve a super node, you’ll need to use both KCL and KVL.
- The super node can be formed by connecting any element in parallel with the voltage source.
- A Super node, like a simple node, satisfies the KCL.

Analysing electrical circuits is essential for ensuring that current equipment functions properly. Most modern electronics go through some type of simulation and evaluation procedure to ensure that designs work as intended and to offer a set of reference calculations for in-circuit tests to compare to. SPICE simulations are the workhorses for circuit design and analysis, with numerous built-in simulations in today’s commercial solutions.

Nodal analysis is a fundamental approach for examining voltage and current distribution in a circuit and it is one of the simulations available in SPICE simulators. Kirchhoff’s laws and Ohm’s law are efficiently combined in a single matrix equation using this technique. Continue reading to discover more about nodal analysis and its applications in circuit design and analysis.

## Frequently asked questions

Get answers to the most common queries related to the NEET UG Examination Preparation.

## What is nodal analysis and mesh analysis?

What is nodal theorem, why do we use nodal analysis, where is nodal analysis applied, differentiate between loop and nodal analysis.

Mesh and nodal analysis are different in that nodal analysis uses Kirchhoff’s current law to calculate the voltages at each node in an equation. Mesh analysis, on the other hand, is a method of computing current that employs Kirchhoff’s voltage law.

To find the voltage potentials around the circuit, Nodal Voltage Analysis uses the “Nodal” equations of Kirchhoff’s first law. So, when all of these nodal voltages are added together, the net result is zero.

On circuits, we apply nodal analysis to get numerous KCL equations, which are then used to solve for voltage and current in the circuit. A circuit’s number of nodes is one less than the number of KCL equations necessary.

Since each node, whether planar or non-planar, may be assigned a voltage, nodal analysis can be used to analyse both planar and non-planar networks.

Our calculations are mostly based on Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL), both of which we are already familiar with (KVL). KCL is used to determine node voltages in a nodal analysis, whereas KVL is used to determine loop currents in a loop analysis.

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Nodal analysis is one kind of method used in electrical networks to analyze circuits through node voltages like the circuit variables.

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What is Nodal analysis? Nodal analysis is a method for estimating the voltage distribution between nodes in a circuit using mathematics. This procedure, also