In electrical and electronics engineering, the analysis represents the process of
determining the response of the circuit such as current, voltage, power,
energy, etc. for a given excitation. In the case of analysis, the circuit is given
and we have to find the excitation or response of the circuit.
For the electric
circuit analysis, a number of analysis tools are present such as Ohm’s law,
KCL, KVL, nodal analysis, mesh analysis, and various network theorems.
What is Nodal Analysis?
Nodal analysis
is a very powerful and efficient tool for analyzing an electric circuit. The
nodal analysis is based on the application of KCL (Kirchhoff’s Current Law) on the different nodes of the
circuit.
In the nodal analysis, we assume voltages at the different junctions
of the circuit with respect to a reference node and write KCL equations for
all the junctions.
Therefore, if the given circuit has N junctions, then there
will be 1 reference node and we write KCL equations for the remaining (N-1) junctions.
Then, in order to get the results, we solve these (N-1) KCL equations.
Basic Terminology related to Nodal Analysis
Node – A node is a
point in the electric circuit where two or more circuit elements meet.
Junction – A
point in the electric circuit where three or more branches meet, is known as a junction.
Branch – A branch
is the part of an electric circuit connected between two junctions.
Which node is to be taken as the reference node?
In nodal analysis, it is a usual practice to select a node
as the reference node which has the largest number of branches connected to it.
Basically, the reference node is a ground node (node to which negative
terminals of all the sources are connected) that has zero potential.
Steps to Solve the Circuit Using Nodal Analysis
The following steps are required to follow to solve a
circuit using nodal analysis:
Step 1 – Identify
the reference node and other junction points and assume the voltages for all
the junctions with respect to the reference node as shown in figure-1.
Step 2 – Assume the branch currents and their directions in the circuit as shown in the figure-2.
Step 3 – Write the KCL equations for all the junctions.
Explanation:
Applying KCL at node a, we get,
`\I_1+I_2+I_3=0" "…(1)`
By Ohm’s law, we have,
`\I_1=(V_a-V_1)/R_1 ;" "I_2=V_a/R_2 ;" "I_3=(V_a-V_2)/R_3`
On substituting these values in equation
(1), we get,
`\(V_a-V_1)/R_1 +V_a/R_2 +(V_a-V_2)/R_3 =0" "…(2)`
Step 4 – Solve
all the KCL equations to obtain the voltages at the junctions.
Step 5 – By using
these voltage values, we can obtain the value of branch currents and other
electrical quantities in the circuit.
Numerical Example
– By using nodal analysis, determine the branch current in the circuit shown in
the figure-3.
Solution –
Step 1 – Mark the branch currents as shown in figure-4:
Step 2 – Write the KCL equations for each junction point:
`\I_1+I_2+I_3=0`
By Ohm’s law, we can get,
`\(V_1-6)/2+V_1/4+(V_1-12)/6=0`
Step 3 – Solve
the KCL equations, to obtain the junction voltages:
`\(6(V_1-6)+3V_1+2(V_1-12))/12=0`
`\⟹6V_1-36+3V_1+2V_1-24=0`
`\11V_1=60`
`\∴V_1=60/11" volts"=5.45" V"`
Step 4 –
Calculate the branch current by using the values of junction voltage obtained
in step 4:
`\I_1=(V_1-6)/2=(5.45-6)/2=-0.275" A"`
Here, the negative sign shows that the
current direction is opposite to the assumed direction.
Similarly,
`\I_2=V_1/4=5.45/4=1.36" A"`
And
`\I_3=(V_1-12)/6=(5.45-12)/6=-1.092" A"`
Hence, using the nodal analysis method, we
have successfully obtained the branch currents in the given circuit.
Conclusion
Nodal analysis is an easy and efficient powerful tool for analyzing an electric circuit with the help of KCL. In this article, we have discussed the basics of nodal analysis, the steps involved in solving the circuit, and a solved numerical example to elaborate on the concept.
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