Chapter 2
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
An ideal voltage source provides a prescribed
voltage across its terminals irrespective of the
current flowing through it. The amount of
current supplied by the source is determined
by the circuit connected to it.
Ideal voltage sources
Various representations of an electrical system
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 to it.
Symbol for ideal current source
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 (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 a diamond, is used
to represent dependent sources and to distinguish
them from independent sources.
The symbols typically used to represent
dependent sources are depicted below. The
table illustrates the relationship between the
source voltage or current and the voltage or
current it depends on—vx or ix, respectively—
which can be any voltage or current in the
circuit.
Symbols for dependent sources
Branch
A branch is any portion of a circuit with two
terminals connected to it. A branch may
consist of one or more circuit elements. In
practice, any circuit element with two
terminals connected to it is a branch.
Definition of a branch
Node
A node is the junction of two or more branches
(one often refers to the junction of only two
branches as a trivial node). The figure below
illustrates the concept.
Definitions of node and supernode
Loop
A loop is any closed connection of branches.
Various loop configurations are illustrated below.
Definition of a loop
Mesh
A mesh is a loop that does not contain other
loops.
Definition of a mesh
Network Analysis
The analysis of an electrical network consists
of determining each of the unknown branch
currents and node voltages.
Electric current is defined as the time rate
of change of charge passing through a
predetermined area.
Current flow in an electric conductor
In order for the current to flow, there must exist a closed circuit.
The figure below depicts a simple circuit, composed of a battery (e.g., a
dry-cell or alkaline 1.5-V battery) and a lightbulb.
Note that in the circuit of this figure, the current i flowing from the battery
to the lightbulb is equal to the current flowing from the lightbulb to the
battery. In other works, no current (and therefore no charge) is “lost”
around the closed circuit. This principle was observed by the German
scientist G.R. Kirchhoff and is now known as Kirchhoff’s current law
(KCL). Kirchhoff’s current law states that because charge cannot be
created but must be conserved, the sum of the currents at a node must
equal zero. Formally,
A simple electric circuit
The significance of Kirchhoff’s current law is
illustrated below.
Illustration of Kirchhoff’s current law
The total work per unit charge associated with the
motion of charge between two points is called
voltage. Thus, the units of voltage are those of
energy per unit charge; they have been called
volts in honor of Alessandro Volta.
Experimental observations led Kirchhoff to the formulation of
the second of his laws, Kirchhoff’s voltage law, or KVL.
The principle underlying KVL is that no energy is lost or
created in an electric circuit; in circuit terms, the sum of all
voltages associated with source must equal the sum of the
load voltages, so that the net voltage around a closed circuit
is zero.
Voltages around a circuit
In general, refer to elements that provide energy
as sources and to elements that dissipate energy
as loads. Standard symbols for a generalized
source-and-load circuit are shown below.
Sources and loads in an electric circuit
The concept of reference voltage finds a practical use in the
ground voltage of a circuit. Ground represents a specific
reference voltage that is usually a clearly identified point in a
circuit.
Analogy between electrical and earth ground
The electric power generated by an active
element, or that dissipated or stored
by a passive element, is equal to the product of
the voltage across the element and the current
flowing through it.
The passive sign convention
FOCUS ON METHODOLOGY
THE PASSIVE SIGN CONVENTION
1. Choose an arbitrary direction of current flow.
2. Label polarities of all active elements (voltage and current sources).
3. Assign polarities to all passive elements (resistors and other loads); for
passive elements, current always flows into the positive terminal.
4. Compute the power dissipated by each element according to the
following rule: If positive current flows into the positive terminal of an
element, then the power dissipated is positive (i.e., the element
absorbs power); if the current leaves the positive terminal of an
element, then the power dissipated is negative (i.e.,the element
delivers power).
The relationship between current and voltage at the
terminals of a circuit element defines the behavior of
that element within the circuit. The figure below
depicts the representation that is employed to denote
a generalized circuit element.
Generalized representation of circuit elements
An ideal resistor is a device that exhibits linear resistance
properties according to Ohm’s law, which states that
The resistance of a material depends on a property called
resistivity. The inverse of resistivity is called conductivity. For
a cylindrical resistence element (shown below), the resistance is
proportional to the length of the sample l and inversely
proportional to its cross sectional area A and conductivity σ .
The resistance element
In addition to the resistance in ohms, the maximum
allowable power dissipation (or power rating) is
typically specified for commercial resistors.
Exceeding this power rating leads to overheating
and can cause the resistor to literally burn up. For
a resistor R, the power dissipated can be
expressed by
A circuit element with resistance approaching
zero is called a short circuit.
The short circuit
A circuit element whose resistance approaches
infinity is called an open circuit.
The open circuit
Definition
Two or more circuit elements are said to be in
series if the current from one element
exclusively flows into the next element. From
KCL, it then follows that all series elements
have the same current.
The voltage across each resistor in a series
circuit divides in direct proportion to the
individual series resistances.
Two or more circuit elements are said to be
in parallel if the elements share the same
terminals. From KVL, it follows that the
elements will have the same voltage.
Current Divider
The ohmmeter is a device that, when connected
across a circuit element, can measure the resistance
of the element.
Ohmmeter and measurement of resistance
The resistance of an element can be measured
only when the element is disconnected from
any other circuit.
The ammeter is a device that, when connected
in series with a circuit element, can measure the
current flowing through the element.
1. The ammeter must be placed in series with
the element whose current is to be measured
(e.g., resistor R2).
2. The ammeter should not restrict the flow of
current (i.e., cause a voltage drop), or else it will
not be measuring the true current flowing in the
circuit. An ideal ammeter has zero internal
resistance.
Measurement of current
The voltmeter is a device that can measure the
voltage across a circuit element.
1. The voltmeter must be placed in parallel with
the element whose voltage it is measuring.
2. The voltmeter should draw no current away
from the element whose voltage it is measuring,
or else it will not be measuring the true voltage
across that element. Thus, an ideal voltmeter
has infinite internal resistance.
Measurement of voltage