Nodal

1. NODAL ANALYSIS

2. What is Nodal Analysis?
Nodal analysis is used for solving any electrical network, and it is
defined as
The mathematical method for calculating the distribution of
voltage between the nodes in a circuit.
 This method is also known as the node-voltage method since
the node voltages are with respect to ground. The following are
the three laws that define the equation related to the voltage that
is measured between each circuit node:
 Ohm’s law
 Kirchhoff’s voltage law
 Kirchhoff’s current law

3. Procedure of Nodal Analysis
The following steps are to be followed while solving any
electrical circuit using nodal analysis:
Step 1:
◦ To identify the principal nodes and select one of them as
a reference node. This reference node will be treated as
the ground.
Step 2:
oAll the node voltages with respect to the ground from all
the principal nodes should be labelled except the
reference node.

4. Step 3: The nodal equations at all the principal nodes
except the reference node should have a nodal
equation.
The nodal equation is obtained from Kirchhoff’s
current law and then from Ohm’s law.
Step 4: To obtain the node voltages, the nodal
equations can be determined by following Step 3.
Hence, for a given electrical circuit the current flowing
through any element and the voltage across any
element can be determined using the node voltages.

5. Types of Reference Nodes
Chassis Ground – This type of
reference node acts a common node for
more than one circuits is called chassis
ground
Earth Ground – When earth potential
is used as a reference in any circuit then
this type of reference node is called Earth
Ground.

6. TYPES OF NODAL ANALYSIS
Non-Reference
Node:
The node that has a
definite node
voltage is known as
a non-reference
node.
Reference
Node:
The node that acts
as a reference point
to all the other
nodes is known as
the reference node,
which is also known
as the datum node.

7. Features of Nodal Analysis
Nodal analysis is an application of Kirchhoff’s current
law.
 When there are ‘n’ nodes in a given electrical circuit,
there will be ‘n-1’ simultaneous equations to be solved.
 To obtain all the node voltages, ‘n-1’ should be
solved.
The number of non-reference nodes and the number
of nodal equations to be obtained are equal.

8. Steps of Nodal Analysis
5Ω
4Ω
-3A.
7Ω
-25A.
V1
V2 V3
1Ω
-8A.
3Ω
constant
,
,
,
,
constant
,
,
33
13
12
11
3
2
1
3
3
33
2
32
1
31
2
3
23
2
22
1
21
1
3
13
2
12
1
11












a
a
a
a
b
b
b
b
x
a
x
a
x
a
b
x
a
x
a
x
a
b
x
a
x
a
x
a

B
AX
1. Assign a reference node => a node having
the most electric elements connecting to
it or a ground node if it is given
2. Apply KCL to each node except the
reference node.
3. Solving the simultaneous equations.

9. KCL@Node 1
V1 V2
3Ω
-3A.
4Ω
V3
-8A.
i1
i2
)
2
(
11
3
4
)
1
(
0
8
3
3
4
3
4
4
0
8
3
2
1
3
1
2
1
3
1
2
1
2
3
1
13
1
2
1























v
v
v
v
v
v
v
v
v
v
i
v
v
v
i
i
i

10. KCL@Node 2
-3A.
3Ω 7Ω
1Ω
V1
V2
V3
i2
i3
i4
)
3
(
3
7
3
3
1
7
3
0
3
2
3
2
1
2
2
3
2
2
1
4
3
2













 






 






v
v
v
v
v
v
v
v
v
v
i
i
i

11. KCL@Node 3
4Ω
7Ω
5Ω
-25A.
V1
V2 V3
)
4
(
25
5
4
7
3
1
3
2
3





 v
v
v
v
v
From (2), (3) and (4) we get
v1 = 5.412 V , v2 = 7.736 V , v3 = 46.32 V

12. Example 1
4Ω
2Ω 8Ω
4Ω
2ix
3A
.
(1)
(2)
(3)
ix
KCL @ node 1
)
1
(
3
2
4
2
1
3
1




 v
v
v
v
KCL @ node 2
)
2
(
0
4
8
2
2
3
2
1
2





 v
v
v
v
v
KCL @ node 3
)
3
(
0
4
2
8
4
2
2
3
1
3











 v
v
v
v
v
Determine v1, v2 and v3
From (1), (2) and (3) we get
v1 = 4.8 V , v2 = 2.4 V , v3 = -2.4 V

13. Example 2
4Ω
3Ω
1Ω
-8A.
-25A.
-3A.
22V.
V3
V1
V2
5Ω
Supernode
Supernode internal relationship
)
1
(
22
2
3 

 v
v
KCL @ node 1
)
2
(
11
3
4
2
1
3
1





 v
v
v
v
KCL @ supernode
)
3
(
0
25
5
4
3
1
3
3
1
3
2
1
2








 v
v
v
v
v
v
Determine v1, v2 and v3
From (1), (2) and (3) we get
v1 , v2 , and v3

14. Example 3
0.5Ω 2Ω
0.2vy
2.5Ω
12V.
14A.
0.5vx
1Ω
vy
vx
V2
V3
V4
Ref.
+
-
-
+
Supernode
(2)
2
.
0
)
1
(
V
12
4
3
1






y
v
v
v
v
)
3
(
0
2
14
5
.
0
3
2
1
2





 v
v
v
v
Determine the voltage of the unknown
node to reference voltages.
We know that
KCL @ node 1
 Because it connects to reference node,
the node equation has not to determine.
KCL @ node 2

15. Example 4
0.5Ω 2Ω
0.2vy
2.5Ω
12V.
14A.
0.5vx
1Ω
vy
vx
V2
V3
V4
Ref.
+
-
-
+
Supernode
)
6
(
)
5
(
)
4
(
0
5
.
2
1
5
.
0
2
1
4
1
2
1
4
4
2
3













v
v
v
v
v
v
v
v
v
v
v
v
y
x
x
Determine the voltage of the unknown
node to reference voltages.
KCL @ supernode (3,4)
From (1) - (6) we get
v1 = -12 V, v2 = -4 V,
v3 = 0 V and v4 = -2 V

16. Summarize of Nodal Analysis
1. Select the node to which the highest number of branches is connected
as the referenced node.
2. Set up KCL equations for other nodes by expressing the unknown
currents as a function of the node voltages measured with respect to
the referenced node.
3. If the given circuit contains voltage sources, KCL equations of those two
nodes connected by a voltage source are combined to eliminate the
redundancy of KCL equations since the additional information is
available through the node voltage.
4. Solve KCL equations to determine the node voltages.

17. THANK YOU
PRESENTED BY-
N.DHARANISH
K.S.DHANUSH KUMAR
S.ASWANTH VISHNU