METHODS OF
ANALYSIS AND
SELECTED
TOPICS (dc)
det EXAMPLE
10
Apply
branch-current analysis to
det
(a)
FIG. 24
TI-89 solution for the current Il of Fig. 22.
After you select the last ENTER key, the screen shown in Fig.
24(b) appears.
network in Fig. 26.
Solution: Again, the
current
directions were chosen to match the "pressure"
of each battery. The polarities are then added,
and Kirchhoff's voltage law is applied around
each closed loop in the clockwise direction. The
result is as follows:
loop 1: +15 v - (40)11 + 20V=O loop2: +20v - (10 - (5
0)12 + 40v=o
-I.OOEO
=
12
(b)
13 = 11 + 12
It is now important that the impact of the results obtained be
understood. The currents Il, 12, and 13 are the actual currents in the
branches in which they were defined. A negative sign in the solution
means that the actual current has the opposite direction than initially
defined—the magnitude is correct. Once the actual current directions
and their magnitudes are inserted in the original network, the various
voltages and power levels can be determined. For this example, the
actual current directions and their magnitudes have been entered on
the original network in Fig. 25. Note that the current through the
series elements RI and El is I A; the current through R3, is I A; and
the current through the series elements R2 and E2 is 2 A. Due to the
minus sign in the solution, the direction of Il is opposite to that shown
in Fig. 22. The voltage across any resistor can now be found using
Ohm's law, and the power delivered by either source
or to any one of the three resistors can be found using the appropriate
power equation.
Applying Kirchhoff's voltage law around the loop indicated in Fig.
25 gives
FIG. 25
or
Reviewing the results of the analysis of the
network in Fig. 22.
(4 Q)13 + (1 Q)12 = 6V
and
4V+2V=6V
6V = 6V
the
(checks)
a
RI
50
FIG. 26
Example 10.
299
El 40 V
M
ETHODS OF ANALYSIS AND SELECTED TOPICS (dc)
Applying Kirchhoff's current law at node a gives
11 13 = 12
Substituting the third equation into the other two yields (with units removed
for clarity)
15 - 411 + 1013 20 0
Substituting for 12 (since it
20 — 1013 — 5 (Il + 13) + 40 = 0 only once in the two equations)
—411 + 1013 = 5
or
511 - 1513 = -60
Multiplying the lower equation by — 1, we have
—411 + 1013 — 5 511
+ 1513 = 60
5 10
60 15
75 — 600
-525
-60 - 50
—110
4.77 A
13
-240 — 25 -265
=
-110
= 2.41 A
-110
—110
12 = Il + 13 = 4.77 A + 2.41 A = 7.18 A
revealing that the assumed directions were the actual directions, with 12
equal to the sum of Il and 13.
7 MESH ANALYSIS (GENERAL APPROACH)
The next method to be described—mesh analysis—is actually an
extension of the branch-current analysis approach just introduced. By
defining a unique array of currents to the network, the information
provided by the application of Kirchhoff's current law is already
included when we apply Kirchhoff's voltage law. In other words, there is
no need to apply step 4 of the branch-current method.
The currents to be defined are called mesh or loop currents. The two
terms are used interchangeably. In Fig. 27(a), a network with two
"windows" has had two mesh currents defined. Note that each forms a
closed "loop" around the inside of each window; these loops are similar
to the loops defined in the wire mesh fence in Fig. 27(b)—hence the use
Of the term mesh for the loop currents. We will find that the number of
mesh currents required to analyze a network will equal the number of "windows" of the configuration.
The defined mesh currents can initially be a little confusing because it appears that two currents have been
defined for resistor R3. There is no problem with El and RI, which have only current Il, or with E2 and R2' which
have only current 12. However, defining the current through R3 may seem a little troublesome. Actually, it is quite
straightforward. The current
(b) through R3 is simply the difference between Il and 12, with the direction FIG. 27 being that
of the larger. This is demonstrated in the examples to follow.
Defining the mesh (loop) current: (a) "two-window" Because mesh currents can result in more than one current through
an network; (b) wire mesh fence analogy. element, branch-current analysis was introduced first. Branch-current
300