Biomedical Engineering Reference
In-Depth Information
resistor and then divides into three equal currents of
3
A through each
I
2
flows through the 1
O
12
O
resistor. The current
I
1
can also be found by applying the current divider rule as
0
@
1
A ¼
1
12
1
12
1
3
A
I
1
¼
I
2
12
¼
1
12
þ
1
12
þ
1
12
1
12
þ
1
12
þ
1
9.5 LINEAR NETWORK ANALYSIS
Our methods for solving circuit problems up to this point have included applying Ohm's
law and Kirchhoff's laws, resistive circuit simplification, and the voltage and current
divider rules. This approach works for all circuit problems, but as the circuit complexity
increases, it becomes more difficult to solve problems. In this section, we introduce the
node-voltage method to provide a systematic and easy solution of circuit problems. The
application of the node-voltage method involves expressing the branch currents in terms
of one or more node voltages and applying KCL at each of the nodes. This method provides
a systematic approach that leads to a solution that is efficient and robust, resulting in a min-
imum number of simultaneous equations that save time and effort.
The use of node equations provides a systematic method for solving circuit analysis
problems by the application of KCL at each essential node. The node-voltage method involves
the following two steps:
1.
Assign each node a voltage with respect to a reference node (ground). The reference
node is usually the one with the most branches connected to it and is denoted with the
symbol
. All voltages are written with respect to the reference node.
2.
Except for the reference node, we write KCL at each of the
N-1
nodes.
The current through a resistor is written using Ohm's law, with the voltage expressed as the
difference between the potential on either end of the resistor with respect to the reference
node, as shown in Figure 9.20. We express node-voltage equations as the currents leaving
the node. Two adjacent nodes give rise to the current moving to the right (like
Figure 9.20a) for one node and the current moving to the left (like Figure 9.20b) for
I
A
¼
V
R
¼
V
1
V
2
the other node. The current
is written for (a) as
and for (b) as
R
I
B
¼
V
R
¼
V
V
2
1
:
It is easy to verify in (a) that
V
¼
V
1
V
2
by applying KVL.
R
I
A
I
B
+V
-
-
V
+
+
V
1
-
+
V
2
-
+
V
1
-
+
V
2
-
R
R
(a)
(b)
FIGURE 9.20
Ohm's law written in terms of node voltages.
