After understood the fundamental laws of circuit theory (Ohm's law and Kirchhoff's laws), we will now proceed to use these two to develop two powerful techniques for circuit analysis :

- Nodal analysis
- Mesh analysis

Nodal analysis is based on a systematic application of Kirchhoff's current law (KCL). With these technique we will be able to analyze any linear circuit.

## Nodal Analysis

Nodal analysis provides a technique to analyze circuits using node voltages as the circuit variables. Using node voltages instead of element voltages as circuit variables is more convenient and reduces the equations number to solve simultaneously.

To make it simpler, let us assume circuits in this part do not have voltage sources. The circuits with voltage sources will be covered in the next post. In

*nodal analysis*, we are supposed to find node voltages. Given a circuit with*n*nodes without voltage sources, the nodal analysis uses following three steps.Steps to Determine Node Voltages :

- Select a node as the reference node. Assign voltages
vto the remaining_{1}, v_{2}, ...., v_{n-1}n -1 nodes. The voltages are referenced with respect to the reference node.- Apply KCL for each
n - 1nonreference nodes. Use Ohm's law to find the current flowing in the branches.- Solve the resulting simultaneous equations to obtain the unknown node voltages.

The full explanation below.

**First step**is selecting a node as the

*reference*or

*datum node*. The reference node is usually called the ground since it is assumed to have zero potential. The symbol of reference node is shown in Figure.(1).

*Earth ground*is shown in Figures.(1a) and (1b) and

*chasis ground*is shown in Figure.(1c).

Figure 1. Common symbols of ground |

*v*= 0), while nodes 1 and 2 are assigned with voltages

*v*and

_{1}*v*respectively.

_{2}__Remember,__node voltages are defined with the respect to the reference node. As shown in Figure.(2a), each node voltage is the voltage rise from the reference node to the nonreference node or simply of the voltage node to reference node.

Figure 2. Typical circuit of nodal analysis |

**Second step**is apply KCL to each nonreference node in the circuit. To reduce the complexity of variables, Figure.(2a) is redrawn in Figure.(2b), where we use

*i*and

_{1}, i_{2},*i*as the currents flowing through resistors

_{3}*R*and

_{1}, R_{2},*R*respectively.

_{3}
At node 1 we apply KCL and gives

(1) |

(2) |

*i*and

_{1}, i_{2},*i*in terms of node voltages. Since the resistance is passive element, using the passive sign convention, current must always flow from higher potential to a lower potential.

_{3}Current flows froma higherpotential toa lowerpotential in a resistor.

We can use this principle as,

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

**third step**is solving with node voltages. Applying KCL to

*n -*1 nonreference nodes, we obtain

*n -*1 simultaneous equation such as Equations.(5) and (6) or (7) and (8). For circuit in Figure.(2) we solve Equations.(5) and (6) or (7) and (8) to get the node voltages

*v*and

_{1}*v*using any standard method such as substitution method, elimination method, Cramer's rule, or matrix inversion.

_{2}For example Equations.(7) and (8) will use matrix form as

(9) |

*v*and

_{1}*v*.

_{2}This method can be modified into supernode analysis.

Nodal analysis also works for ac circuit. Make sure to read it to advance your study.

## Nodal Analysis Examples

For better understanding, let us review some examples below :**1.Calculate the node voltages in the circuit in Figure.(3a)**

Figure 3 |

__Solution :__
Consider Figure.(3b) where the circuit in Figure.(3a) has been prepared for nodal analysis. The currents have been selected for KCL except the branches with current sources. The labeling of the current is arbitrary but consistent. (Consistent means if , for example,

*i*enters the 4 Ω resistor from left hand side,_{2}*i*must leave the resistor from the right hand side)._{2}
The reference node is selected and the node voltages

*v*and_{1}*v*are now determined._{2}
At node 1, applying KCL and Ohm's law gives

Multiplying each term in the last equation by 4, we obtainor

(1.1) |

Multiplying each term by 12 results

or

(1.2) |

*v*and

_{1}*v*.

_{2}**Method 1**

Using elimination method we add (1.1) and (1.2) gives

Subsituting the result above with (1.1) gives

**Method 2**

Use Cramer's rule, we put (1.1) and (1.2) to matrix form as

The determinant is

We now obtain the voltages as

**2.Determine the voltages at the nodes in Figure.(4a)**

__Solution :__
In this example we will need three nonreference nodes instead of only two. We assign three nodes as can be seen in Figure.(4b)

At node

**1**,

Multiplying by 4 and rearranging terms, we get

(2.1) |

**2**,

Multiplying by 8 and rearranging terms,we get

(2.2) |

**3**,

Multiplying by 8, rearranging terms, and dividing by 3, we get

(2.3) |

(2.4) |

(2.5) |

From (2.3), we get

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