Nodal Analysis

In electric circuits’ analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage (potential difference) between "nodes" (points where elements or branches connect) in an electrical circuit in terms of the branch currents. Nodal Analysis provides a general procedure for analyzing circuits using node voltages as the circuit variables. 

Steps to Determine Node Voltages:

1. Select a node as the reference node, Assign voltages v1, v2, . . . . . , 

vn-1 to the remaining n-1 nodes. The voltages are referenced with respect to the reference node.

2.Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law to express currents in terms of node voltages.

3. Solve the resulting simultaneous equations to obtain the unknown node voltages.

Nodal Analysis with Voltage Sources

Case 1: If the voltage source (dependent or independent) is connected between two non-reference nodes, the two non-reference nodes form a generalized node or super node, we apply both KCL and KVL to determine the node voltages.

Case 2: if a voltage source is connected between the reference node and a non-reference node, we simply set the voltage at the non-reference node equal to the voltage of the voltage source in figure 2 for example,

v₁ = 20V

What is supernode?

A supernode is formed by enclosing a (dependent or independent) voltage source connected between two non-reference nodes and any elements connected in parallel with it.

In figure 2 node 2 and node 3 form a supernode. Applying KCL at super node which are node 2 and 3 we get,

i1 + i4  = i2 + i3

To apply KVL redrawing the figure 2 circuit to figure 3 and going around the loop in the clockwise direction gives,

– v2 + 10 + v3 = 0

  Or  v2 – v3 = 10      ————————— (ii)

Overview of the Lesson :

In this lesson we tackled about the Methods of Analysis which is the Nodal Analysis. In nodal analysis, we are interested in finding the node voltages. Given a circuit with n nodes without voltage source. There are also ways in solving the node voltages; The shortcut method or the long method. The shortcut method is you can determine the equations of each node by determining the adjacent of the resistors and in the long method you can determine the equations by using Ohm’s law and Substitution method.

*Take note that; To use Crammers rule, you must derive the equation into matrix form.

This week is very stressful because we already know our scores in the 1^st prelim exam and it was very disappointing. Having low scores in exam, but I hope I can pass the 1^st prelim for bullet for the upcoming midterm and especially the finals. 

BASIC LAW: Series and Parallel Resistors

SERIES CIRCUIT

- In a series circuit, the current through each of the components is the same, and the voltage across the circuit is the sum of the voltages across each component. In a parallel circuit, the voltage across each of the components is the same, and the total current is the sum of the currents through each component.

A series circuit with a voltage source (such as a battery) and 3 resistors

Current
       

In a series circuit the current is the same for all elements.

Resistors

The total resistance of resistors in series is equal to the sum of their individual resistances:

 

PARALLEL CIRCUIT

- If two or more components are connected in parallel they have the same potential difference (voltage) across their ends. The potential differences across the components are the same in magnitude, and they also have identical polarities. The same voltage is applicable to all circuit components connected in parallel. The total current is the sum of the currents through the individual components, in accordance with Kirchhoff’s current law.

 

 Circuit containing resistors in parallel

Voltage

In a parallel circuit the voltage is the same for all elements.

Resistors

The current in each individual resistor is found by Ohm's law. Factoring out the voltage gives

.

Nodes, Branches & Loops

Node

-        A point or junction where two or more circuit’s elements (resistor, capacitor, inductor etc) meet is called Nodes

Finding Nodes in Electric Circuits

After redrawing the above circuit, it becomes as below circuit, now you can easily find the total number of Nodes as shown in fig below:

   

Branch

That part or section of circuit which locate between two junctions is called branch

In branch, one or more elements can be connected and they have two terminals.

Finding Branches in Electric Circuits:

 Loop

A closed path in circuit where more than two meshes can be occurred is called loop i.e. there may be many meshes in a loop, but a mesh does not contain on one loop.

Finding Loops in Electric Circuits:

Kirchhoff's circuit laws(KCL)

- are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by Gustav Kirchhoff. This generalized the work of Georg Ohm and preceded the work of Maxwell. Widely used in electrical engineering, they are also called Kirchhoff's rules or simply Kirchhoff's laws.

The current entering any junction is equal to the current leaving that junction. i₂ + i₃ = i₁ + i₄

Formulas:

Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be stated as:

current entering = current leaving

Kirchhoff's voltage laws(KVL)

- KVL is a fundamental law, as fundamental as Conservation of Energy in mechanics, for example, because KVL is really conservation of electrical energy. KVL and KCL are the starting point for analysis of any circuit. KCL and KVL always hold and are usually the most useful piece of information you will have about a circuit after the circuit itself.

The sum of all the voltages around the loop is equal to zero. v₁+ v₂ + v₃ - v₄ = 0

Formulas:

Similarly to KCL, it can be stated as:

voltage drop + ------> - (negative)

voltage rise - -----> +  (positive)