C H A P T E R
2
FUNDAMENTALS OF ELECTRIC
CIRCUITS
hapter 2 presents the fundamental laws that govern the behavior of electric
circuits, and it serves as the foundation to the remainder of this book. The chap-
ter begins with a series of definitions to acquaint the reader with electric circuits;
next, the two fundamental laws of circuit analysis are introduced: Kirchhoff’s
current and voltage laws. With the aid of these tools, the concepts of electric power
and the sign convention and methods for describing circuit elements—resistors in
particular—are presented. Following these preliminary topics, the emphasis moves
to basic analysis techniques—voltage and current dividers, and to some applica-
tion examples related to the engineering use of these concepts. Examples include a
description of strain gauges, circuits for the measurements of force and other related
mechanical variables, and of the study of an automotive throttle position sensor. The
chapter closes with a brief discussion of electric measuring instruments. The following
box outlines the principal learning objectives of the chapter.
9
10 Chapter 2 Fundamentals of Electric Circuits
➲ Learning Objectives
1. Identify the principal elements of electric circuits: nodes, loops, meshes, branches,
and voltage and current sources. Section 2.1.
MAKE THE 2. Apply Kirchhoff’s laws to simple electric circuits and derive the basic circuit
CONNECTION equations. Sections 2.2 and 2.3.
3. Apply the passive sign convention and compute the power dissipated by circuit
elements. Calculate the power dissipated by a resistor. Section 2.4.
Mechanical 4. Apply the voltage and current divider laws to calculate unknown variables in simple
series, parallel, and series-parallel circuits. Sections 2.5 and 2.6.
(Gravitational)
5. Understand the rules for connecting electric measuring instruments to electric
Analog of Voltage circuits for the measurement of voltage, current, and power. Sections 2.7 and 2.8.
Sources
The role played by a voltage
source in an electric circuit is
equivalent to that played by 2.1 DEFINITIONS
the force of gravity. Raising
a mass with respect to a In this section, we formally define some variables and concepts that are used in the
reference surface increases
remainder of the chapter. First, we define voltage and current sources; next, we define
its potential energy. This
potential energy can be the concepts of branch, node, loop, and mesh, which form the basis of circuit analysis.
converted to kinetic energy Intuitively, an ideal source is a source that can provide an arbitrary amount of
when the object moves to a energy. Ideal sources are divided into two types: voltage sources and current sources.
lower position relative to the Of these, you are probably more familiar with the first, since dry-cell, alkaline, and
reference surface. The
lead-acid batteries are all voltage sources (they are not ideal, of course). You might
voltage, or potential
difference across a voltage have to think harder to come up with a physical example that approximates the
source plays an analogous behavior of an ideal current source; however, reasonably good approximations of
role, raising the electrical ideal current sources also exist. For instance, a voltage source connected in series
potential of the circuit, so that with a circuit element that has a large resistance to the flow of current from the source
current can flow, converting
provides a nearly constant—though small—current and therefore acts very nearly as
the potential energy within
the voltage source to electric an ideal current source. A battery charger is another example of a device that can
power. operate as a current source.
Ideal Voltage Sources
An ideal voltage source is an electric device that generates a prescribed voltage at
its terminals. The ability of an ideal voltage source to generate its output voltage is
not affected by the current it must supply to the other circuit elements. Another way
to phrase the same idea is as follows:
An ideal voltage source provides a prescribed voltage across its terminals
➲
LO1 irrespective of the current flowing through it. The amount of current supplied
by the source is determined by the circuit connected to it.
Figure 2.1 depicts various symbols for voltage sources that are employed
throughout this book. Note that the output voltage of an ideal source can be a function
of time. In general, the following notation is employed in this book, unless otherwise
noted. A generic voltage source is denoted by a lowercase v. If it is necessary to
emphasize that the source produces a time-varying voltage, then the notation v(t) is
Part I Circuits 11
+ + + +
vs (t) + vs (t) Vs vs (t) vs (t) + vs (t)
_ _~
– – – –
Circuit Circuit Circuit
– –
General symbol A special case: A special case:
for ideal voltage DC voltage sinusoidal
source. vs (t) source (ideal voltage source,
may be constant battery) vs (t) = V cos ωt
(DC source).
Figure 2.1 Ideal voltage sources
employed. Finally, a constant, or direct current, or DC, voltage source is denoted by
the uppercase character V . Note that by convention the direction of positive current
flow out of a voltage source is out of the positive terminal.
The notion of an ideal voltage source is best appreciated within the context of the
source-load representation of electric circuits. Figure 2.2 depicts the connection of an
energy source with a passive circuit (i.e., a circuit that can absorb and dissipate energy).
Three different representations are shown to illustrate the conceptual, symbolic, and
physical significance of this source-load idea.
i RS i
+ +
Source v Load
– VS +
_ v R + –
Car battery Headlight
i –
Power flow
(a) Conceptual (b) Symbolic (circuit) (c) Physical
representation representation representation
Figure 2.2 Various representations of an electrical system
In the analysis of electric circuits, we choose to represent the physical reality
of Figure 2.2(c) by means of the approximation provided by ideal circuit elements,
as depicted in Figure 2.2(b).
iS, IS
Ideal Current Sources
An ideal current source is a device that can generate a prescribed current independent iS, IS
of the circuit to which it is connected. To do so, it must be able to generate an arbitrary Circuit
voltage across its terminals. Figure 2.3 depicts the symbol used to represent ideal
current sources. By analogy with the definition of the ideal voltage source just stated,
we write that Figure 2.3 Symbol for
ideal current source
An ideal current source provides a prescribed current to any circuit connected
to it. The voltage generated by the source is determined by the circuit connected
➲ LO1
to it.
12 Chapter 2 Fundamentals of Electric Circuits
The same uppercase and lowercase convention used for voltage sources is employed
in denoting current sources.
Dependent (Controlled) Sources
MAKE THE
CONNECTION The sources described so far have the capability of generating a prescribed voltage
or current independent of any other element within the circuit. Thus, they are termed
independent sources. There exists another category of sources, however, whose output
Hydraulic Analog (current or voltage) is a function of some other voltage or current in a circuit. These
are called dependent (or controlled) sources. A different symbol, in the shape of
➲
LO1
of Current a diamond, is used to represent dependent sources and to distinguish them from
Sources independent sources. The symbols typically used to represent dependent sources are
The role played by a current depicted in Figure 2.4; the table illustrates the relationship between the source voltage
source in an electric circuit is or current and the voltage or current it depends on—vx or ix , respectively—which can
very similar to that of a pump be any voltage or current in the circuit.
in a hydraulic circuit. In a
pump, an internal mechanism
(pistons, vanes, or impellers)
forces fluid to be pumped Source type Relationship
from a reservoir to a hydraulic Voltage controlled voltage source (VCVS) vS = !vx
circuit. The volume flow rate
+ Current controlled voltage source (CCVS) vS = rix
of the fluid q, in cubic meters vS _ iS
per second, in the hydraulic Voltage controlled current source (VCCS) iS = gvx
circuit, is analogous to the Current controlled current source (CCCS) iS = "ix
electrical current in the circuit.
Figure 2.4 Symbols for dependent sources
Positive Displacement Pump
slip
Dependent sources are very useful in describing certain types of electronic
circuits. You will encounter dependent sources again in Chapters 8, 10, and 11, when
flow flow electronic amplifiers are discussed.
Suction Discharge An electrical network is a collection of elements through which current flows.
low high The following definitions introduce some important elements of a network.
pressure pressure
A hydraulic pump Branch
A branch is any portion of a circuit with two terminals connected to it. A branch may
➲
LO1
Pump symbols
Left: Fixed consist of one or more circuit elements (Figure 2.5). In practice, any circuit element
capacity pump.
Right: Fixed with two terminals connected to it is a branch.
capacity pump
with two directions
of flow.
Left: Variable
capacity pump.
Right: Variable
Node
A node is the junction of two or more branches (one often refers to the junction of
➲
LO1
capacity pump only two branches as a trivial node). Figure 2.6 illustrates the concept. In effect,
with two directions
of flow. any connection that can be accomplished by soldering various terminals together is
Courtesy: Department of a node. It is very important to identify nodes properly in the analysis of electrical
Energy networks.
It is sometimes convenient to use the concept of a supernode. A supernode
is obtained by defining a region that encloses more than one node, as shown in the
rightmost circuit of Figure 2.6. Supernodes can be treated in exactly the same way as
nodes.
Part I Circuits 13
+ i
A
Branch v Branch R
voltage current
rm
–
b
A branch Ideal A battery Practical
resistor ammeter
Examples of circuit branches
Figure 2.5 Definition of a branch
Supernode
... ... Node a R1
Node c Node a +
R2 + V
+ − S2
vS iS R4 −
VS1 +
_
Node
−
... Node b R3 R5
Node b
...
Examples of nodes in practical circuits
Figure 2.6 Definitions of node and supernode
Loop
Aloop is any closed connection of branches. Various loop configurations are illustrated
➲ LO1
in Figure 2.7.
Note how two different loops
in the same circuit may R
include some of the same
elements or branches.
Loop 1 Loop 2 vS iS R1 R2
Loop 3 1-loop circuit 3-loop circuit
(How many nodes in
this circuit?)
Figure 2.7 Definition of a loop
Mesh
A mesh is a loop that does not contain other loops. Meshes are an important aid to
➲ LO1
certain analysis methods. In Figure 2.7, the circuit with loops 1, 2, and 3 consists of two
meshes: Loops 1 and 2 are meshes, but loop 3 is not a mesh, because it encircles both
loops 1 and 2. The one-loop circuit of Figure 2.7 is also a one-mesh circuit. Figure 2.8
illustrates how meshes are simpler to visualize in complex networks than loops are.
