analysingNodal Analysis

Nodal Analysis is a foundational circuit analysis technique based on Kirchhoff's Current Law (KCL). The method's primary objective is to determine the voltage potential at each principal node within an electrical network. The process involves identifying all principal nodes, selecting one as a "reference node" with a potential of zero volts, and then developing a KCL equation for every other "non-reference" node. This results in a system of n-1 simultaneous equations, where n is the total number of principal nodes. Solving this system yields the unknown node voltages, which can then be used to calculate other circuit parameters like branch currents. A significant advantage of Nodal Analysis over Mesh Analysis is its universal applicability to both planar and non-planar networks.

1.0Core Principles of Nodal Analysis

Nodal Analysis provides a systematic approach to analysing complex electrical circuits. Its methodology is built upon a few key definitions and a foundational law of circuit theory.

Foundation in Kirchhoff's Current Law (KCL)

The entire framework of Nodal Analysis is derived from Kirchhoff's Current Law. This contrasts directly with Mesh Analysis, which is based on Kirchhoff's Voltage Law (KVL). As stated in the source, "nodal analysis is based on KCL," which dictates that the algebraic sum of currents entering a node is equal to the sum of currents leaving it.

2.0Node Definitions

• A "node" is defined as a common point where two or more circuit elements are connected. For the purpose of Nodal Analysis, a distinction is made between two types of nodes:
• Simple Node: A point where only two elements are connected. At a simple node, "current division does not take place."

• Principal Node: A point where more than two elements are connected. At a principal node, "current division takes place," which allows for the application of KCL.

Note: Crucially, within the context of Nodal Analysis, the term "node" exclusively refers to a principal node.

Super Node: A supernode is formed when a dependent or independent voltage source is connected between two non-reference nodes, along with any elements that may be in parallel with that source, and is enclosed as a single extended node.

Properties of a Super Node

•  A supernode does not have a single, well-defined voltage by itself.
•  To analyze a supernode, you must apply both Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL) to obtain the necessary equations for solving node voltages.

3.0Reference and Non-Reference Nodes

Once all principal nodes are identified, one is designated as the reference node, also known as the datum.

• The potential of the reference node is defined as zero volts.
• It is often advisable to select the node where the maximum number of branches connect, although a common convention is to always "take the bottom node as the reference node."
• All other principal nodes in the circuit are classified as non-reference nodes. KCL equations are developed for each of these non-reference nodes.

4.0Systematic Procedure for Nodal Analysis

The application of Nodal Analysis follows a structured, four-step process to solve for unknown node voltages.

5.0Key Characteristics and Applications

Applicability

• One of the most significant advantages of Nodal Analysis is its broad applicability. The source notes: "It is applicable for both planar and non planar networks." This makes it a more versatile tool than Mesh Analysis, which "was applicable only for planar networks."

Number of Equations

• The number of KCL equations required to solve a network using Nodal Analysis is directly related to the number of principal nodes. The formula is given as:
• Number of Equations = n - 1 , Where n is the total number of principal nodes.

Illustration-1.Using nodal analysis determine the current in the  20 Ω resistor.

Solution: Transform all the voltage sources into their corresponding current sources by converting each voltage source and its series resistor into a current source in parallel with the same resistor.

$I_1=\frac{10}{10}=1A$

$I_2=\frac{20}{10}=2A$

Applying KCL at node 1

$1+2=\frac{V_1}{10}+\frac{V_1}{20}+\frac{V_1}{10}$

$3=V_1\left(\frac{1}{10}+\frac{1}{20}+\frac{1}{10}\right)$

$3=0.25V_1\Rightarrow V_1=\frac{3}{0.25}=12V$

The current through 20 Ω is $I_{20}$

$I_{20}=\frac{V_1}{20}=\frac{12}{20}=0.6A$

Illustration-2.Write the system of nodal equations for the circuit presented and then calculate the branch currents throughout the network shown in the diagram

Solution:

Assigning voltage to each node and applying KCL at node 1  

$5=i_{10}+i_3$

$5=\frac{V_1}{10}+\frac{V_1-V_2}{3}$

$5=\frac{V_1}{10}+\frac{V_1}{3}-\frac{V_2}{3}$

$5=V_1\left(\frac{1}{10}+\frac{1}{3}\right)-\frac{V_2}{3}............(1)$

applying KCL at node 2 

$0=i_3+i_5+i_1$

$0=\frac{V_2-V_1}{3}+\frac{V_2}{5}+\frac{V_2-10}{1}$

$0=V_2\left(\frac{1}{3}+\frac{1}{5}+1\right)-\frac{V_1}{3}..................(2)$

Solving equation (1) and (2) we get,

$V_1=19.85V,V_2=10.84V$

$i_{10}=\frac{V_1}{10}=\frac{19.87}{10}=1.987A$

$i_3=\frac{V_2-V_1}{3}=\frac{10.84-19.87}{3}=-3.01A\Rightarrow i_3=3.01A$

$i_5=\frac{V_2}{5}=\frac{10.84}{5}=2.17A$

$i_1=V_2-10=10.84-10=0.84A$