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N A
I 3
V 1
I 1
R 3
10 �
6 A R1 4 � R2 2 � 4 A
I 2
V 2
I 3
Supernode
FIG. 8.52
Defining the supernode for the network of Fig. 8.51.
network as shown in Fig. 8.52. The result is a single supernode for which
Kirchhoff’s current law must be applied. Be sure to leave the other defined
nodes in place and use them to define the currents from that region of the
network. In particular, note that the current I3 will leave the supernode at
V1 and then enter the same supernode at V2.Itmust therefore appear twice
when applying Kirchhoff’s current law, as shown below:
Σ Ii � Σ Io 6 A � I 3 � I 1 � I 2 � 4 A � I 3
or I 1 � I 2 � 6 A � 4 A � 2 A
V1 �
R1
Then � �2 A
V 1
V2 �
R2
V 2
and � ���2 A
4 � 2 �
Relating the defined nodal voltages to the independent voltage source,
we have
V1 � V2 � E � 12 V
which results in two equations and two unknowns:
0.25V1 � 0.5V2 � 2
V1 � 1V2 � 12
Substituting:
V1 � V2 � 12
0.25(V2 � 12) � 0.5V2 � 2
and 0.75V2 � 2 � 3 ��1
�1
so that V2 ����1.333 V
0.75
and V1 � V2 � 12 V ��1.333 V � 12 V � �10.667 V
The current of the network can then be determined as follows:
V 10.667 V
I1 �����2.667 A
R1 4 �
V2 1.333 V
I2 �����0.667 A
R2 2 �
V1 � V2 10.667 V � (�1.333 V) 12 V
I3 ������ ���1.2 A
10 �
10 �
10 �
NODAL ANALYSIS (GENERAL APPROACH) ⏐⏐⏐ 285