Nodal Analysis
Nodal analysis provides a procedure for analyzing circuits using node
voltages as the circuit variables.
Nodal Analysis
Problem: Calculate the node voltages in the circuit.
Nodal Analysis
Solution: Assign node voltages
Labeling
arbitrary
of
the
currents
is
At node 1, applying KCL
i1  i2  i3
v1  v2 v1  0
5

4
2
v1  v2  2v1
35v1 v2  20.....( i )
4
Nodal Analysis
At node 2, applying KCL
i2  i4  i1  i5
v1  v2
v2  0
 10  5 
4
6
v1  v2  40 30  v2

4
6
 3v1  5v2  60.....( ii )
Nodal Analysis
After solving equation (i) and (ii), we get,
v1  13.333 V
v2  20 V
Nodal Analysis
Problem: Determine the voltages at the nodes.
Nodal Analysis
Solution: Assign node voltages and labeling the current
At node 1, applying KCL
3  i1  ix
v1  v3 v1  v2
3

4
2
v1  v3  2v1  2v2
3
4
3v1  2v2  v3  12.....( i )
Nodal Analysis
At node 2, applying KCL
ix  i2  i3
v1  v2 v2  v3 v2  0


2
8
4
 4v1  7v2  v3  0.....( ii )
Nodal Analysis
At node 3, applying KCL
i1  i2  2ix
v1  v3 v2  v3
v1  v2

2
4
8
2
2v1  2v3  v2  v3
 v1  v2
8
6v1  7v2  v3  0.....( iii )
Nodal Analysis
After solving equation (i), (ii) and (iii), we get,
v1  4.8 V
v 2  2 .4 V
v3   2 . 4 V
Nodal Analysis with Voltage Sources
If a voltage source is connected between the reference node and a
nonreference node, we simply set the voltage at the nonreference node
equal to the voltage of the voltage source.
v1  10 V
Nodal Analysis with Voltage Sources
(Super node)
If the voltage source (dependent or independent) is connected between
two nonreference nodes, the two nonreference nodes form a
generalized node or supernode; we apply both KCL and KVL to
determine the node voltages.
Nodal Analysis with Voltage Sources
A supernode is formed by enclosing a (dependent or independent)
voltage source connected between two nonreference nodes and any
elements connected in parallel with it.
A supernode has no voltage of its own.
A supernode requires the application of both KCL and KVL
Nodal Analysis with Voltage Sources
Determine the voltages at the nodes.
Nodal Analysis with Voltage Sources
Solution: The supernode contains the 2V source, nodes 1 and 2, and
the 10Ω resistor.
Labeling
arbitrary
of
the
currents
Applying KCL at supernode,
2  i1  i2  7
v1  0 v2  0
2

7
2
4
is
2v1  v2  28
2
4
2v1  v2  20.......( i )
Nodal Analysis with Voltage Sources
To get the relationship between v1 and v2 we apply KVL
 v1  2  v2  0
 v1  v2  2......( ii )
After solving equation (i) and (ii), we get,
v1  7.333 V
v2  5.333 V
Nodal Analysis with Voltage Sources
Problem: Find the node voltages
Nodal Analysis with Voltage Sources
Solution: Nodes 1 and 2 form a supernode
Nodes 3 and 4 form another supernode
Labeling the direction of currents
Nodal Analysis with Voltage Sources
Applying KCL at supernode 1-2,
i3  10  i1  i2
5v1  v2  v3  2v4  60.......( i )
v3  v 2
v1  v4 v1  0
 10 

6
3
2
Nodal Analysis with Voltage Sources
Applying KCL at supernode 3-4,
i1  i3  i4  i5
4v1  2v2  5v3  16v4  0.......( ii )
v1  v4 v3  v2 v4  0 v3  0



3
6
1
4
Nodal Analysis with Voltage Sources
Apply KVL to loop 1,
 v1  20  v2  0
v1  v2  20....( iii )
Apply KVL to loop 2,
 v3  3v x  v4  0
3v1  v3  2v4  0.......( iv )
v x  v1  v4
Nodal Analysis with Voltage Sources
After solving equation (i), (ii), (iii) and (iv), we get,
v1  26.67 V
v3  173 .33 V
v2  6.667 V
v 4  46.67 V