Nodal Analysis
Nodal analysis is the general
technique for analysing an electric circuit and it is based on Krichhof’s
Current law (KCL) which states that the sum of all current entering into a node
is equal to the total current leaving from a node. In nodal analysis we are
interested in finding node voltages. Choosing node voltage rather than voltage
across elements as a circuit variable is convenient and it reduces the number
of equations to be solved. In nodal analysis we have to write KCL equations for
each non reference node. The number of equations derived for a circuit having n
number of nodes is (n – 1).
Nodal Analysis with Current
Source
The process of a nodal analysis
of an electric circuit having current source involves the following steps to be
taken
Step 1 Identify the total number of
nodes in the given electric circuit. A node is point in electric circuit where
two or more than two branches are connected. For example refer the given
figure.
Step 2 Out of all the nodes, designate
a node as a reference node. A reference node a common node for all the other
node against which node voltages of all the nodes measured. Generally, the
ground is taken as reference node.
Step 3 Assign unknown voltage
variables to all nonreference nodes.
Step 4 Develop KCL equations for each
non reference node ( i.e. the sum of current coming towards the node is equal
to the sum of current going outwards from the node).
Step 5 Express the driven equation in
terms of nodal voltages by the help of Ohm's Law i.e. (I = V/R) and
also express any other unknown voltage and current other than the node voltage
in terms of nodal voltage.
Step 6 Arrange the driven
equation and solve them for finding the node voltage.
Let us understand these steps with the help of an example
Consider the circuit shown in the given figure.
According to second step,
designate a reference node. In the given circuit we have clearly seen that node
3 is common for all other node (node 1 and node2) and its potential is at
ground potential that's why it is taken as reference node.
Once the reference node is selected then as per step 3 assign the voltage designation to all other nonreference nodes. In the given example we have assign V_{1} variable of node 1 and V_{2} for node 2
At node 1 applying KCL
I_{1} = i_{1} + i_{3} ………………… (1)
At
node 2 applying KCL
i_{1} = i_{2} + I_{2
} ……………………(2)
According to step 5 there is
no any unknown voltage or current present in the given circuit rather than node
voltage. Actually, this happens in the circuit having dependent sources.
Write the derived equation in
the form of voltage by the help of Ohm's Law (I = V/R).
Put these Values in equation (1) and (2)
After writing these equations
in the form of node voltages we have got two equations to be solved for the
given circuit having three nodes. As in the introduction part we discuss that
in Nodal analysis n  1 equation to be solved for the circuit analysis.
Nodal Analysis with Voltage Source
Nodal analysis of an electric
circuit having voltage source connected between two nonreference nodes
introduces the concept of supernode. In an electric circuit a voltage
source is connected in two ways either it is connected between a nonreference
node and a reference node or it is connected between two nonreference nodes.
If voltage source is connected
between a nonreference node and a reference node than the node voltage of
nonreference node is equal to the voltage of the voltage source.
If voltage is connected between
two nonreference nodes than it is difficult to find the current in that branch.
Hence, KCL equation cannot be developed for such nodes. That’s why the branch
having voltage source connected between two nonreference node is considered as
supernode. Let us understand the concept of supernode the help of an example.
Consider the circuit as shown in
given figure
In the given figure we saw
that there are two voltage sources present in the circuit. In which one voltage
source is connected between a reference (node 4) and a nonreference (node 1)
and other voltage source is connected between two nonreference node (node 2
and node 3).
So, if we calculate the
voltage at node 1 it is simply equals to the voltage of the voltage source V1.
But if we drive the KCL
equations for node 2 and node 3 for calculating the node voltages we will run
into some difficulty because we do not know the current in the branch having
voltage source or there is no way by which we can express the current in
the form of voltage. So, for the convenience node 2, node 3 and voltage of
source together treated as supernode, the supernode is indicated by the region
enclosed by the broken line shown in figure. By considering it as supernode we
can apply KCL to the both node at same time.
Let us develop the nodal
equations for this circuit.
In the given circuit there are four nodes (node 1,2,3,4) in
which node 4 is the reference node and node 1, 2, 3 are the nonreference
nodes. As per step second assigning voltage variables for each non reference
node va vb and vc are the voltage variables for node 1, 2, 3 respectively. Now
develop the KCL equations for each noreference node keeping the supernode
concept in mind that we have discussed previously.
At node 1
v_{a} = V1
discussed earlier
I =
i_{1} + i_{2} ……………….(1)
Put these values in equation 1
At node 2 and 3
Remember that node 2 and node 3 combined treated as supernode.
i_{1}  i_{3}

i_{4} + I_{1} =
0
I_{1} = i_{3} + i_{4}  i_{1}
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