## What is Nodal Matrix?

The nodal matrix or simply known as admittance matrix, generally in engineering term it is called Y Matrix or Y bus, since it involve matrices so it is also referred as a n into n order matrix that represents a power system with n number of buses. It shows the buses' nodal admittance in a power system. The Y matrix is rather sparse in actual systems with thousands of buses. In the power system the transmission cables connect each bus to only a few other buses. Also the important data that one needs for have a power flow study is the Y Matrix.

Admittance Matrix illustrates the nodal admittance of several buses in a power system. Each bus is linked to the others through a transmission line. The admittance matrix is used to analyse the data required in the load or power flow studies of the buses. It describes network admission and topology.

With the help of the Admittance matrix, the amount of current existing in the bus can be estimated. It is written as indicated below.

$\left[\begin{array}{c}{I}_{1}\\ {I}_{2}\\ {I}_{n}\end{array}\right]=\left[\begin{array}{ccc}{Y}_{11}& {Y}_{12}& {Y}_{1n}\\ {Y}_{21}& {Y}_{22}& {Y}_{2n}\\ {Y}_{n1}& {Y}_{n2}& {Y}_{nn}\end{array}\right]\left[\begin{array}{c}{V}_{1}\\ {V}_{2}\\ {V}_{n}\end{array}\right]$

The above matrix is written in the simplest form as shown below.

$I=\left[Y\right]V$

Here,

• The vector form of the bus current is I.
• The admittance matrix here is Y.
• The bus voltage vector here is V.

The (3 into 3) admittance matrix is generated as indicated below using the diagram for the bus admittance.

$Y=\left[\begin{array}{ccc}{Y}_{1}+{Y}_{12}+{Y}_{13}& -{Y}_{12}& -{Y}_{13}\\ -{Y}_{12}& {Y}_{2}+{Y}_{12}+{Y}_{13}& -{Y}_{23}\\ -{Y}_{13}& -{Y}_{23}& {Y}_{3}+{Y}_{13}+{Y}_{23}\end{array}\right]$

The bus admittance matrix have some diagonal element and these elements are known as the self admittance, while the elements that are other than the diagonal elements are called the off diagonal matrix.

#### Observation of Nodal Admittance Matrix

• The nodal admittance matrix is sparse.
• The use of diagonal elements is prevalent.
• The elements that are not on the diagonal are symmetric.
• In matrices, each node's diagonal element is the sum of the admittance that are connected to it.
• Negated admittance is the off diagonal element.

### Steps to determine Nodal Admittance Matrix

The steps to determine the admittance matrix is described below.

• To begin, create the bus admittance matrix.
• To solve the network, choose the reference bus.
• For all other types of buses, define the known variables.
• Assign the initial voltage and angle values to all of the buses.
• Calculate the power injection current and the power mismatch vector.
• Apply iteration methods such as Newton-Raphson, Gauss-Siedel, and others.
• Examine the mismatching vector to see if it falls inside the 0.001 per unit restriction. If yes, end the procedure; if no, continue with the iteration steps to retrieve the new values.
• Check the data once more to see if they are inside the limit.

## What is Nodal Analysis ?

The nodal analysis is used to solve for electrical current, voltage and other parameters. Generally the values of the branch current are determine the nodal method. The nodes are fixed to determine the node voltages and the node currents in the circuit, that is voltage source and current source.

A node is a terminal or connection that connects more than two elements. Nodal analysis is widely employed in networks with multiple parallel circuits that share a common terminal ground. This method uses a smaller number of equations to solve the circuit.

The algebraic sum of all incoming currents at a node must equal the algebraic sum of all outgoing currents at that node, according to Kirchhoff's Current Law (KCL), which is used in Nodal Voltage Analysis.

It is a method for determining the potential difference between the elements or branches of an electric circuit. This approach specifies the voltage at each node of the circuit. There are two types of nodes in this method. These are the non-reference node and the reference node.The non-reference nodes have a constant voltage, while the reference node serves as the reference point for all other nodes.

The number of independent node pair equations required in the nodal technique is one fewer than the number of network junctions. If n is the number of independent node equations and j is the number of junctions, then this is the case.

$n=j-1$

When calculating the current expression, the node potentials are assumed to always be higher than the other voltages in the equations.

For a circuit that have current sources, the direction of the current at the node can either be negative or positive, but the convention must be kept same for all such current sources for the circuit, the final direction of the current is determined by the values obtained after the calculation. The same concept is also applied for other dependent source of the circuit.

### Steps for solving Nodal Analysis

The procedures that follow are based on the circuit schematic shown above. There are two Voltage source and 5 resistor; R1, R2, R3, R4, and R5 resistors.

1. Identify and label several nodes in the present circuit, labelled the nodes A and B.
2. Select one of the nodes as the reference, or zero potential nodes at which the greatest number of elements are connected. Node D is used as the reference node in the diagram above. Let VA and VB be the voltages at nodes A and B, respectively.
3. Apply KCL to the various nodes now.

Using KCL at node A. The equation is,

${I}_{1}+{I}_{2}+{I}_{3}=0$

Write the current in terms of the node voltages.

$\frac{\left({V}_{A}-{V}_{1}\right)}{{R}_{1}}+\frac{{V}_{A}}{{R}_{2}}+\frac{\left({V}_{A}-{V}_{B}\right)}{{R}_{3}}=0\phantom{\rule{0ex}{0ex}}{V}_{A}\left[\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}+\frac{1}{{R}_{3}}\right]-\frac{{V}_{B}}{{R}_{3}}=\frac{{V}_{1}}{{R}_{1}}.............\left(1\right)$

Using KCL at node B,

$-{I}_{3}+{I}_{4}-{I}_{5}=0$

$\frac{-\left({V}_{A}-{V}_{B}\right)}{{R}_{3}}+\frac{{V}_{B}}{{R}_{4}}-\frac{\left({V}_{B}-{V}_{2}\right)}{{R}_{5}}=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{V}_{B}\left[\frac{1}{{R}_{3}}+\frac{1}{{R}_{4}}-\frac{1}{{R}_{5}}\right]-\frac{{V}_{A}}{{R}_{3}}=-\frac{{V}_{2}}{{R}_{5}}...........\left(2\right)$

Find the values of VA and VB by solving equations (1) and (2).

• The bus admittance matrix data preparation is relatively straightforward.
• The creation and adjustment of the bus admittance matrix is simple.
• The bus admittance matrix is a sparse matrix, requiring less computer memory.

## Context and Applications

Nodal matrices are used in many simulations and in different applications such as

• Application in solving power system equations
• Application in load flow analysis
• To obtain values of current source and voltage sources in complex circuits.

In each of the expert exams for undergraduate and graduate publications, this topic is huge and is mainly taught in

• Bachelor of Technology in Electrical Engineering.
• Master of Technology in Electrical Engineering.
• Master of Technology in Power Engineering
• Impedance matrix
• Voltage Nodal analysis
• Current source and voltage source nodal analysis
• KCL Nodal Analysis
• Gauss Seidel method
• Newton Raphson method

## Common Mistakes

• Simplifying circuits incorrectly.
• Make sure voltage polarities and current directions are labelled properly.
• Making frequent calculation mistakes while designing circuit.
• Making assumptions about open and short circuits can lead to erroneous.
• Misidentifying series and parallel device connections.
• Formulating node voltage equations incorrectly.

## Practice Problems

Q1. The nodal admittance matrix is also known as .......

A. Y Bus

B. Z Bus

C. Both A and B

D. Nodal analysis

Explanation: The nodal admittance matrix is also called as Y Bus matrix.

Q2. The bus admittance matrix have some diagonal element and these elements are known as the ............

A. Off Diagonal Matrix

C. Matrice

D. None

Explanation: The bus admittance matrix have some diagonal element and these elements are known as the self admittance.

Q3. The algebraic sum of all incoming currents at a node must equal the algebraic sum of all outgoing currents at that node. This statement is known as ..................

A. KCL

B. KVL

C. Thereon

D. None

Explanation: The algebraic sum of all incoming currents at a node must equal the algebraic sum of all outgoing currents at that node, is known as Kirchhoff's Current Law (KCL) .

Q4. The elements that are not on the diagonal are ........... .

A. Unsymmetrical

B. Semi-Symmetric

C. Symmetric

D. Nodal Analysis

Explanation: The elements that are not on the diagonal are symmetric.

Q5. In Nodal Analysis which law is used to fine current and voltage values?

A. KCL

B. KVL

C. Thereon

D. None

Explanation: Kirchhoff's Current Law (KCL) is used in Nodal Analysis.

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