You can check the current page or check for previous versions at the Internet Archive. Yahoo! is not affiliated with the authors of this page or responsible for its content.
RESISTIVE NETWORK ANALYSIS
September 23, 2005 17:13
riz63473_ch03
Sheet number 1 Page number 81
magenta black
81
C
H
A
P
T
E
R
3
RESISTIVE NETWORK ANALYSIS
hapter 3 illustrates the fundamental techniques for the analysis of resistive
circuits. The chapter begins with the denition of network variables and of
network analysis problems. Next, the two most widely applied methods
node analysis and mesh analysisare introduced. These are the most gener-
ally applicable circuit solution techniques used to derive the equations of all electric
circuits; their application to resistive circuits in this chapter is intended to acquaint
you with these methods, which are used throughout the book. The second solution
method presented is based on the principle of superposition, which is applicable only
to linear circuits. Next, the concept of Thévenin and Norton equivalent circuits is
explored, which leads to a discussion of maximum power transfer in electric circuits
and facilitates the ensuing discussion of nonlinear loads and load-line analysis. At
the conclusion of the chapter, you should have developed condence in your ability
to compute numerical solutions for a wide range of resistive circuits. The following
box outlines the principal learning objectives of the chapter.
*(866) 487-8889*
CONFIRMING PROOFS
MASTER SET
Please mark
all alterations
on this set only September 23, 2005 17:13
riz63473_ch03
Sheet number 2 Page number 82
magenta black
82
Chapter 3
Resistive Network Analysis Learning Objectives
1.
Compute the solution of circuits containing linear resistors and independent and
dependent sources by using node analysis. Sections 3.2 and 3.4.
2.
Compute the solution of circuits containing linear resistors and independent and
dependent sources by using mesh analysis. Sections 3.3 and 3.4.
3.
Apply the principle of superposition to linear circuits containing independent sources.
Section 3.5.
4.
Compute Thévenin and Norton equivalent circuits for networks containing linear
resistors and independent and dependent sources. Section 3.6.
5.
Use equivalent-circuit ideas to compute the maximum power transfer between a
source and a load. Section 3.7.
6.
Use the concept of equivalent circuit to determine voltage, current, and power for
nonlinear loads by using load-line analysis and analytical methods. Section 3.8.
3.1
Network Analysis
The analysis of an electric network consists of determining each of the unknown
branch currents and node voltages. It is therefore important to dene all the rele-
vant variables as clearly as possible, and in systematic fashion. Once the known and
unknown variables have been identied, a set of equations relating these variables
is constructed, and these equations are solved by means of suitable techniques. The
analysis of electric circuits consists of writing the smallest set of equations sufcient
to solve for all the unknown variables. The procedures required to write these equa-
tions are the subject of Chapter 3 and are very well documented and codied in the
form of simple rules. The analysis of electric circuits is greatly simplied if some
standard conventions are followed.
Example 3.1 denes all the voltages and currents that are associated with a
specic circuit.
EXAMPLE 3.1
Problem
Identify the branch and node voltages and the loop and mesh currents in the circuit of Figure 3.1.
Solution
The following node voltages may be identied:
Node voltages
Branch voltages
v
a
= v
S
(source voltage)
v
S
= v
a
v
d
= v
a
v
b
= v
R
2
v
R
1
= v
a
v
b
v
c
= v
R
4
v
R
2
= v
b
v
d
= v
b
v
d
= 0 (ground)
v
R
3
= v
b
v
c
v
R
4
= v
c
v
d
= v
c
v
R
4
+
_
i
a
i
b
d
a
b
c
R
1
R
4
v
R
1
R
2
+
_
+
_
+
_
+
_
v
S
i
c
+
_
v
R
3
v
R
2
Figure 3.1
Comments:
Currents i
a
, i
b
, and i
c
are loop currents, but only i
a
and i
b
are mesh currents.
*(866) 487-8889*
CONFIRMING PROOFS
MASTER SET
Please mark
all alterations
on this set only September 23, 2005 17:13
riz63473_ch03
Sheet number 3 Page number 83
magenta black
Part I
Circuits
83
MAKE THE
CONNECTION
Thermal Systems
A useful analogy can be
found between electric cir-
cuits and thermal systems.
The table below illustrates
the correspondence be-
tween electric circuit var-
iables and thermal system
variables, showing that the
difference in electrical po-
tential is analogous to the
temperature difference be-
tween two bodies. When-
ever there is a temperature
difference between two bod-
ies, Newtons law of cooling
requires that heat ow from
the warmer body to the
cooler one. The ow of heat
is therefore analogous to the
ow of current. Heat ow
can take place based on
one of three mechanisms:
(1) conduction, (2) convec-
tion, and (3) radiation. In this
sidebar we only consider
the rst two, for simplicity.
Electrical
Thermal
variable
variable
Voltage
Temperature
difference
difference
v, [V]
T
, [ C]
Current
Heat ux
i
, A
q
, [W]
Resistance
Thermal
R
, [ /m]
resistance
R
t
[ C
/W]
Resistivity
Conduction
, [ /m]
heat-transfer
coefcient
k
,
W
m
C
(No exact
Convection
electrical
heat-transfer
analogy)
coefcient, or
lm coefcient
of heat-transfer
h
,
W
m
2
C
In the example, we have identied a total of 9 variables! It should be clear that
some method is needed to organize the wealth of information that can be generated
simply by applying Ohms law at each branch in a circuit. What would be desirable at
this point is a means of reducing the number of equations needed to solve a circuit to the
minimum necessary, that is, a method for obtaining N equations in N unknowns. The
remainder of the chapter is devoted to the development of systematic circuit analysis
methods that will greatly simplify the solution of electrical network problems.
3.2
THE NODE VOLTAGE METHOD
Node voltage analysis is the most general method for the analysis of electric circuits.
In this section, its application to linear resistive circuits is illustrated. The node voltage
method is based on dening the voltage at each node as an independent variable. One
of the nodes is selected as a reference node (usuallybut not necessarilyground),
and each of the other node voltages is referenced to this node. Once each node voltage
is dened, Ohms law may be applied between any two adjacent nodes to determine
the current owing in each branch. In the node voltage method, each branch current
is expressed in terms of one or more node voltages; thus, currents do not explicitly
enter into the equations. Figure 3.2 illustrates how to dene branch currents in this
method. You may recall a similar description given in Chapter 2.
Once each branch current is dened in terms of the node voltages, Kirchhoffs
current law is applied at each node:
i
= 0
(3.1)
Figure 3.3 illustrates this procedure.
i
R
v
a
v
b
i = v
a

v
b
R
In the node voltage method, we
assign the node voltages v
a
and v
b
;
the branch current flowing from a
to b is then expressed in terms of
these node voltages.
Figure 3.2 Branch current
formulation in node analysis
i
1
R
1
v
a
v
b
i
3
v
c
v
d
R
3
R
2
i
2
By KCL: i
1
i
2
i
3
= 0. In the node
voltage method, we express KCL by
v
a
v
b
R
1
v
b
v
c
R
2
v
b
v
d
R
3
= 0 + + +
Figure 3.3 Use of KCL in
node analysis
The systematic application of this method to a circuit with n nodes leads to
writing n linear equations. However, one of the node voltages is the reference voltage
and is therefore already known, since it is usually assumed to be zero (recall that
the choice of reference voltage is dictated mostly by convenience, as explained in
Chapter 2). Thus, we can write n
1 independent linear equations in the n 1 inde-
pendent variables (the node voltages). Node analysis provides the minimum number
of equations required to solve the circuit, since any branch voltage or current may be
determined from knowledge of node voltages.
*(866) 487-8889*
CONFIRMING PROOFS
MASTER SET
Please mark
all alterations
on this set only September 23, 2005 17:13
riz63473_ch03
Sheet number 4 Page number 84
magenta black
84
Chapter 3
Resistive Network Analysis
MAKE THE
CONNECTION
Thermal
Resistance
To explain thermal resis-
tance, consider a heat treat-
ed engine crankshaft that
has just completed some
thermal treatment. Assume
that the shaft is to be
quenched in a water bath at
ambient temperature (see
the gure below). Heat ows
from within the shaft to the
surface of the shaft, and
then from the shaft surface
to the water. This process
continues until the tempera-
ture of the shaft is equal to
that of the water.
The rst mode of heat
transfer in the above de-
scription is called
conduc-
tion
, and it occurs because
the thermal conductivity of
steel causes heat to ow
from the higher temperature
inner core to the lower-
temperature surface. The
heat transfer conduction
coefcient k is analogous to
the resistivity
of an electric
conductor.
The second mode of
heat transfer,
convection,
takes place at the boundary
of two dissimilar materials
(steel and water here). Heat
transfer between the shaf