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Circuit Terminology Chapter Contents 11
12
13
14
15 CellPhone Circuit Architecture
Historical Timeline
Units, Dimensions, and Notation
Electric Charge and Current
Voltage and Power
Circuit Elements
Chapter Highlights Objectives
Upon learning the material presented in this chapter, you should
be able to:
1. Differentiate between active and passive devices; analysis
and synthesis; device, circuit, and system; and dc and ac.
2. Point to important milestones in the history of electrical
and computer engineering.
3. Use multiple and submultiple preﬁxes.
4. Relate electric charge to current; voltage to energy;
power to current and voltage; and apply the passive sign
convention.
5. Describe the properties of dependent and independent
sources.
6. Deﬁne the i v relationship for: a voltage source; a current
source; a resistor; a capacitor; and an inductor.
7. Describe the operation of SPST and SPDT switches. 2 CHAPTER 1 CIRCUIT TERMINOLOGY CellPhone Circuit Architecture
Electronic circuits are contained in just about every gadget we
use in daily living. In fact, electronic sensors, computers, and
displays are at the operational heart of most major industries,
from agricultural production and transportation to healthcare
and entertainment. The ubiquitous cell phone (Fig. 11), which
has become practically indispensable, is a perfect example
of an integrated electronic architecture made up of a large
number of interconnected circuits. It includes ampliﬁer circuits,
oscillators, frequency up and downconverters, and circuits
with many other types of functions (Fig. 12). Factors such as
compatibility among the various circuits and proper electrical
connections between them are critically important to the overall
operation and integrity of the cell phone.
Usually, we approach electronic analysis and design through
a hierarchical arrangement where we refer to the overall entity
as a system, its subsystems as circuits, and the individual
circuit elements as devices or components. Thus, we may
regard the cell phone as a system (which is part of a much
larger communication system); its audiofrequency ampliﬁer,
for example, as a circuit, and the resistors, integrated circuits
(ICs), and other constituents of the ampliﬁer as devices. In
actuality, an IC is a fairly complex circuit in its own right,
but its input/output functionality is such that usually it can be RF = Radio Frequency
IF = Intermediate Frequency
LO = Local Oscillator
RF Power
Mixer = Frequency Up or
Amp
DownConverter
RF Filter
Transmit Path Figure 11: Cell phone. represented by a relatively simple equivalent circuit, thereby
allowing us to treat it like a device. Generally, we refer to Human Interface,
Dialing, Memory
Battery Power Control
Mixer (Speech,
video, data)
In Out Microprocessor
Control IF Amp
Modulator Antenna LO Received Signal ~ ~
IF Amp LO D/A and A/D
Converters
and
Filters Demodulator
Transmitted Signal Diplexer/Filter RF Low Mixer
Noise Amp IF Filter Receive Path
Antenna and
Propagation RF FrontEnd IF Block Figure 12: Cellphone block diagram. BackEnd Baseband 11 HISTORICAL TIMELINE devices that do not require an external power source in order to
operate as passive devices; these include resistors, capacitors,
and inductors. In contrast, an active device (such as a transistor
or an IC) cannot function without a power source.
This book is about electric circuits. A student once asked:
“What is the difference between an electric circuit and an
electronic circuit? Are they the same or different?” Strictly
speaking, both refer to the ﬂow of electric charge carried
by electrons, but historically, the term “electric” preceded
“electronic,” and over time the two terms have come to signify
different things:
An electric circuit is one composed of passive devices,
in addition to voltage and current sources, and possibly
some types of switches. In contrast, the term “electronic”
has become synonymous with transistors and other active
devices.
The study of electric circuits usually precedes and sets the
stage for the study of electronic circuits, and even though a
course on electric circuits usually does not deal with the internal
operation of an active device, it does incorporate active devices
in the circuit examples considered for analysis, but it does so by
representing the active devices in terms of equivalent circuits.
An electric circuit, as deﬁned by Webster’s English
Dictionary, is a “complete or partial path over which current
may ﬂow.” The path may be conﬁned to a physical structure
(such as a metal wire connecting two components), or it may
be an unbounded channel carrying electrons through it. An
example of the latter is when a lightning bolt strikes the
ground, creating an electric current between a highly charged
atmospheric cloud and the earth’s surface.
The study of electric circuits consists of two complementary
parts: analysis and synthesis (Fig. 13). Through analysis, we
develop an understanding of “how” a given circuit works. If
we think of a circuit as having an input—a stimulus—and an
output—a response, the tools we use in circuit analysis allow
us to relate mathematically the output response to the input
stimulus, enabling us to analytically and graphically “observe”
the behavior of the output as we vary the relevant parameters of
the input. An example might be a speciﬁc ampliﬁer circuit,
in which case the objective of circuit analysis might be to
establish how the output voltage varies as a function of the
input voltage over the full operational range of the ampliﬁer
parameters. By analyzing the operation of each circuit in a
system containing multiple circuits, we can characterize the
operation of the overall system.
As a process, synthesis is the reverse of analysis. In
engineering, we tend to use the term design as a synonym for
synthesis. The design process usually starts by deﬁning the
operational speciﬁcations that a gadget or system should meet, 3
and then we work backwards (relative to the analysis process)
to develop circuits that will satisfy those speciﬁcations. In
analysis, we are dealing with a single circuit with a speciﬁc
set of operational characteristics. We may employ different
analysis tools and techniques, but the circuit is unique, and
so are its operational characteristics. That is not necessarily
the case for synthesis; the design process may lead to multiple
circuit realizations—each one of which exhibits or satisﬁes the
desired speciﬁcations.
Given the complementary natures of analysis and synthesis,
it stands to reason that developing proﬁciency with the tools
of circuit analysis is a necessary prerequisite to becoming a
successful design engineer. This textbook is intended to provide
the student with a solid foundation of the primary set of tools
and mathematical techniques commonly used to analyze both
direct current (dc) and alternating current (ac) circuits, as
well as circuits driven by pulses and other types of waveforms.
A dc circuit is one in which voltage and current sources are
constant as a function of time, whereas in ac circuits, sources
vary sinusoidally with time. Even though this is not a book
on circuit design, design problems occasionally are introduced
into the discussion as a way to illustrate how the analysis and
synthesis processes complement each other. Review Question 11: What are the differences between a device, a circuit, and a system?
Review Question 12: What is the difference between analysis and synthesis? 11 Historical Timeline We live today in the age of electronics. No ﬁeld of science
or technology has had as profound an inﬂuence in shaping the Analysis vs. Synthesis
Analysis
Circuit Functionality Synthesis
Circuit Specs
(Design) Figure 13: The functionality of a circuit is discerned by
applying the tools of circuit analysis. The reverse process, namely
the realization of a circuit whose functionality meets a set of
speciﬁcations, is called circuit synthesis or design. 4
operational infrastructure of modern society as has the ﬁeld of
electronics. Our computers and communication systems are at
the nexus of every major industry, from food production and
transportation to health care and entertainment. Even though
no single event marks the beginning of a discipline, electrical
engineering became a recognized profession sometime in
the late 1800s (see chronology). Alexander Graham Bell
invented the telephone (1876); Thomas Edison perfected
his incandescent light bulb (1880) and built an electrical
distribution system in a small area in New York City;
Heinrich Hertz generated radio waves (1887); and Guglielmo
Marconi demonstrated radio telegraphy (1901). The next
50 years witnessed numerous developments, including radio
communication, TV broadcasting, and radar for civilian and
military applications—all supported by electronic circuitry that
relied entirely on vacuum tubes. The invention of the transistor
in 1947 and the development of the integrated circuit (IC)
shortly thereafter (1958) transformed the ﬁeld of electronics by
setting it on an exponentially changing course towards “smaller,
faster, and cheaper.”
Computer engineering is a relatively young discipline.
The ﬁrst allelectronic computer, the ENIAC, was built and
demonstrated in 1945, but computers did not become available
for business applications until the late 1960s and for personal
use until the introduction of Apple I in 1976. Over the past 20
years, not only have computer and communication technologies
expanded at a truly impressive rate (see Technology Brief 2),
but more importantly, it is the seamless integration of the two
technologies that has made so many business and personal
applications possible.
Generating a comprehensive chronology of the events and
discoveries that have led to today’s technologies is beyond the
scope of this book, but ignoring the subject altogether would
be a disservice to both the reader and the subject of electric
circuits. The abbreviated chronology presented on the next few
pages represents our compromise solution. CHAPTER 1 CIRCUIT TERMINOLOGY ca. 900 BC According to legend, a shepherd in northern Greece, Magnus,
experiences a pull on the iron nails in his sandals by the black rock he
was standing on. The rock later became known as magnetite [a form of
iron with permanent magnetism].
ca. 600 BC Greek philosopher Thales describes how amber, after being rubbed
with cat fur, can pick up feathers [static electricity]. 1600 William Gilbert (English) coins the term electric after the Greek word for
amber (elektron ) and observes that a compass needle points north to
south because the Earth acts as a bar magnet. 1614 John Napier (Scottish) develops the logarithm system. 1642 Blaise Pascal (French) builds the ﬁrst adding machine using multiple
dials. 1733 Charles Francois du Fay (French) discovers that electric charges are
¸
of two forms and that like charges repel and unlike charges attract. 1745 Pieter van Musschenbroek (Dutch) invents the Leyden jar, the ﬁrst
electrical capacitor. 1800 Alessandro Volta (Italian) develops the ﬁrst electric battery. 1827 Georg Simon Ohm (German) formulates Ohm’s law relating electric
potential to current and resistance. 1827 Joseph Henry (American) introduces the concept of inductance and
builds one of the earliest electric motors. He also assisted Samuel Morse
in the development of the telegraph. Chronology: Major Discoveries, Inventions, and
Developments in Electrical and Computer
Engineering
ca. 1100 BC Abacus is the earliest known calculating device. 11
1837 HISTORICAL TIMELINE
1888 Nikola Tesla (CroatianAmerican) invents the ac motor. 1893 Valdemar Poulsen (Danish) invents the ﬁrst magnetic sound recorder
using steel wire as recording medium. 1895 1876 Samuel Morse (American) patents the electromagnetic telegraph using
a code of dots and dashes to represent letters and numbers. 5 Wilhelm Rontgen (German) discovers Xrays. One of his ﬁrst Xray
¨
images was of the bones in his wife’s hands. [1901 Nobel prize in
physics.] 1896 Guglielmo Marconi (Italian) ﬁles his ﬁrst of many patents on wireless
transmission by radio. In 1901, he demonstrates radio telegraphy across
the Atlantic Ocean. [1909 Nobel prize in physics, shared with Karl Braun
(German).] Alexander Graham Bell (ScottishAmerican) invents the telephone: the
rotary dial becomes available in 1890, and by 1900, telephone systems
are installed in many communities. 1879 Thomas Edison (American) demonstrates the operation of the
incandescent light bulb, and in 1880, his power distribution system
provided dc power to 59 customers in New York City. 1887 Heinrich Hertz (German) builds a system that can generate electromagnetic waves (at radio frequencies) and detect them. 6 CHAPTER 1 1897 Karl Braun (German) invents the cathode ray tube (CRT). [1909 Nobel
prize, shared with Marconi.] 1897 Joseph John Thomson (English) discovers the electron and measures
its chargetomass ratio. [1906 Nobel prize in physics.] 1902 Reginald Fessenden (American) invents amplitude modulation for
telephone transmission. In 1906, he introduces AM radio broadcasting
of speech and music on Christmas Eve. 1904 John Fleming (British) patents the diode vacuum tube. 1907 CIRCUIT TERMINOLOGY Lee De Forest (American) develops the triode tube ampliﬁer for wireless
telegraphy, setting the stage for longdistance phone service, radio, and
television. Robert Watson Watt (Scottish) invents radar.
John Mauchly and J. Presper Eckert (both American) develop the
ENIAC, the ﬁrst allelectronic computer. 1947 William Schockley, Walter Brattain, and John Bardeen (all Americans)
invent the junction transistor at Bell Labs. [1956 Nobel prize in physics.] Birth of commercial radio broadcasting; Westinghouse Corporation
establishes radio station KDKA in Pittsburgh, Pennsylvania. Vladimir Zworykin (RussianAmerican) invents television. In 1926,
John Baird (Scottish) transmits TV images over telephone wires from
London to Glasgow. Regular TV broadcasting began in Germany (1935),
England (1936), and the United States (1939). Vannevar Bush (American) develops the differential analyzer, an analog
computer for solving differential equations. Edwin Howard Armstrong (American) invents the superheterodyne
radio receiver, dramatically improving signal reception. In 1933, he
develops frequency modulation (FM), providing superior sound quality
of radio transmissions over AM radio. 1923 1930 1945 1920 Transatlantic telephone service established between London and New
York. 1935 1917 1926 11 HISTORICAL TIMELINE 1948 Claude Shannon (American) publishes his Mathematical Theory of
Communication, which formed the foundation of information theory,
coding, cryptography, and other related ﬁelds. 1950 Yoshiro Nakama (Japanese) patents the ﬂoppy disk as a magnetic
medium for storing data. 1954 Texas Instruments introduces the ﬁrst AM transistor radio. 1955 The pager is introduced as a radio communication product in hospitals
and factories. 1955 Navender Kapany (IndianAmerican) demonstrates optical ﬁber as a
lowloss, lighttransmission medium. 1956 John Backus (American) develops FORTRAN, the ﬁrst major programming language. 1958 Charles Townes and Arthur Schawlow (both Americans) develop the
conceptual framework for the laser. [Townes shared 1964 Nobel prize in
physics with Aleksandr Prokhorov and Nicolay Bazov (both Soviets).] In
1960 Theodore Maiman (American) builds the ﬁrst working model of a
laser. 1958 Bell Labs develops the modem. 1958 Jack Kilby (American) builds the ﬁrst integrated circuit (IC) on
germanium, and independently, Robert Noyce (American) builds the
ﬁrst IC on silicon. 7
1959 Ian Donald (Scottish) develops an ultrasound diagnostic system. 1960 Echo, the ﬁrst passive communication satellite is launched and
successfully reﬂects radio signals back to Earth. In 1962, the ﬁrst
communication satellite, Telstar, is placed in geosynchronous orbit. 1960 Digital Equipment Corporation introduces the ﬁrst minicomputer, the
PDP1, which was followed with the PDP8 in 1965. 1962 Steven Hofstein and Frederic Heiman (both American) invent the
MOSFET, which became the workhorse of computer microprocessors. 1964 IBM’s 360 mainframe becomes the standard computer for major
businesses. 1965 John Kemeny and Thomas Kurtz (both American) develop the BASIC
computer language. 8 CHAPTER 1 1965 Konrad Zuse (German) develops the ﬁrst programmable digital
computer using binary arithmetic and electric relays. 1968 Douglas Engelbart (American) demonstrates a wordprocessor system,
the mouse pointing device, and the use of a Windowslike operating
system. 1969 ARPANET is established by the U.S. Department of Defense, which is to
evolve later into the Internet. 1970 James Russell (American) patents the CDROM, as the ﬁrst system
capable of digitaltooptical recording and playback. 1971 Texas Instruments introduces the pocket calculator. CIRCUIT TERMINOLOGY Palm Pilot becomes widely available. 1997 The 17,500mile ﬁberoptic cable extending from England to Japan is
operational. 2002 Cell phones support video and the Internet. 2007 The powerefﬁcient White LED invented by Shuji Nakamura (Japanese)
in the 1990s promises to replace Edison’s lightbulb in most lighting
applications. IBM introduces the laser printer. Japan builds the ﬁrst cellular telephone network: Sabeer Bhatia (IndianAmerican) and Jack Smith (American) launch
Hotmail as the ﬁrst webmail service. Godfrey Hounsﬁeld (British) and Alan Cormack (South African–
American) develop the computerized axial tomography scanner (CAT
scan) as a diagnostic tool. [1979 Nobel Prize in physiology or medicine.] 1979 Tim BernersLee (British) invents the World Wide Web by introducing a
networking hypertext system. Intel introduces the 4004 fourbit microprocessor, which is capable of
executing 60,000 operations per second. Apple Computer sells Apple I in kit form, followed by the fully assembled
Apple II in 1977, and the Macintosh in 1984. First transatlantic optical ﬁber cable between the U.S. and Europe is
operational. 1997 1976 Worldwide Internet becomes operational. 1996 1976 1984 1989 1972 IBM introduces the PC. 1988 1971 1981 • 1983 cellular phone networks start in the United States.
• 1990 electronic beepers become common.
• 1995 cell phones become widely available.
1980 Microsoft introduces the MSDOS computer disk operating system.
Microsoft Windows is marketed in 1985. 12 12 UNITS, DIMENSIONS, AND NOTATION Units, Dimensions, and Notation The standard system used in today’s scientiﬁc literature to
express the units of physical quantities is the International
System of Units, (SI) abbreviated after its French name Syst` me
e
Internationale. Time is a fundamental dimension, and the
second is the unit by which it is expressed relative to a speciﬁc
reference standard. The SI conﬁguration is based on the six
fundamental dimensions listed in Table 11, and their units are
called Fundamental SI units. All other dimensions, such as
velocity, force, and energy, are regarded as secondary because
their units are based on and can be expressed in terms of the six
fundamental units. Appendix A provides a list of the quantities
used in this book, together with their symbols and units.
In science and engineering, a set of preﬁxes commonly are
used to denote multiples and submultiples of units. These
preﬁxes, ranging in value between 10−18 and 1018 , are listed in
Table 12. An electric current of 3 × 10−6 A, for example, may
be written as 3 μA. 9
The physical quantities we will discuss in this book (such as
voltage and current) may be constant in time or may vary with
time. As a general rule, we shall use:
• A lowercase letter, such as i for current, to represent
the general case:
i • A lowercase letter followed with (t) to emphasize
time:
i(t) Dimension Unit Length
Mass
Time
Electric Current
Temperature
Amount of substance meter
kilogram
second
ampere
kelvin
mole Symbol
m
kg
s
A
K
mol Table 12: Multiple and submultiple preﬁxes.
Preﬁx Symbol Magnitude exa
peta
tera
giga
mega
kilo E
P
T
G
M
k 1018
1015
1012
109
106
103 milli
micro
nano
pico
femto
atto m
μ
n
p
f
a 10−3
10−6
10−9
10−12
10−15
10−18 is a timevarying quantity • An uppercase letter if the quantity is not time varying;
thus:
I Table 11: Fundamental SI units. may or may not be time varying is of constant value (dc quantity) • A letter printed in boldface to denote that:
I
has a speciﬁc meaning, such as a vector, a
matrix, the phasor counterpart of i(t), or the Laplace
or Fourier transform of i(t) Convert the following quantities to
scientiﬁc notation: (a) 52 mV, (b) 0.3 MV, (c) 136 nA,
and (d) 0.05 Gbits/s. Exercise 11: Answer: (a) 5.2 × 10−2 V, (b) 3 × 105
(c) 1.36 × 10−7 A, and (d) 5 × 107 bits/s. (See ) V, Exercise 12: Convert the following quantities to a preﬁx format such that the number preceding the preﬁx is
between 1 and 999: (a) 8.32 × 107 Hz, (b) 1.67 × 10−8 m,
(c) 9.79 × 10−16 g, (d) 4.48 × 1013V, and (e) 762 bits/s.
Answer: (a) 83.2 MHz, (b) 16.7 nm, (c) 979 ag,
(d) 44.8 TV, and (e) 762 bits/s. (See )
Simplify the following operations
into a single number, expressed in preﬁx format:
(a) A = 10 μV + 2.3 mV, (b) B = 4THz − 230 GHz,
(c) C = 3 mm/60 μm.
Exercise 13: Answer: (a) A = 2.31 mV, (b) B = 3.77 THz,
(c) C = 50. (See ) 10 TECHNOLOGY BRIEF 1: MICRO AND NANOTECHNOLOGY Technology Brief 1: Micro and Nanotechnology
History and Scale
As humans and our civilizations developed, our ability to control the environment around us improved dramatically. The
use and construction of tools was essential to this increased control. A quick glance at the scale (or size) of manmade
and natural is very illustrative (Fig. TF11). Early tools (such as ﬂint, stone, and metal hunting gear) were on the order
of tens of centimeters. Over time, we began to build eversmaller and everlarger tools. The pyramids of Giza (ca.,
2600 BCE) are 100m tall; the largest modern construction crane is the K10,000 Kroll Giant Crane at 100m long and
82m tall; and the current (2007) tallest manmade structure is the KVLYTV antenna mast in Blanchard, North Dakota
at 0.63 km! Miniaturization also proceeded apace; for example, the ﬁrst hydraulic valves may have been Sinhalese
valve pits of Sri Lanka (ca., 400 BCE), which were a few meters in length; the ﬁrst toilet valve (ca., 1596) was tens
of centimeters in size; and by comparison, the largest dimension in a modern microﬂuidic valve used in biomedical
analysis chips is less than 100 μm! FigureTF11: The scale of natural and manmade objects, sized from nanometers to centimeters. (Courtesy of U.S. Department
of Energy.) TECHNOLOGY BRIEF 1: MICRO AND NANOTECHNOLOGY 11 In electronic devices, miniaturization has been a key enabler in almost all of the technologies that shape the world
around us. Consider computation and radiofrequency communications, two foundations of 21stcentury civilization.
The ﬁrst true automated computer was arguably the ﬁrst Babbage Difference Engine, proposed by Charles Babbage
to the Royal Astronomical Society (1822). The complete engine would have had 25,000 moving parts and measured
approximately 2.4 m × 2.3 m × 1 m. Only a segment with 2000 parts was completed and today is considered the
ﬁrst modern calculator. The ﬁrst generalpurpose electronic computer was the Electronic Numerical Integrator and
Computer (ENIAC), which was constructed at the University of Pennsylvania between 1943 and 1946. The ENIAC
was 10ft tall, occupied 1,000 square feet, weighed 30 tons, used ∼100,000 components and required 150 kilowatts of
power! What could it do? It could perform simple mathematical operations on 10digit numbers at approximately 2,000
cycles per second (addition took 1 cycle, multiplication 14 cycles, and division and square roots 143 cycles). With the
invention of the semiconductor transistor in 1947 and the development of the integrated circuit in 1959 (see Technology
Brief 7 on IC Fabrication Process), it became possible to build thousands (now trillions) of electronic components onto
a single substrate or chip. The 4004 microprocessor chip (ca., 1971) had 2250 transistors and could execute 60,000
instructions per second; each transistor had a “gate” on the order of 10 μm (10−5 m). In comparison, the 2006 Intel
Core has 151 million transistors with each transistor gate measuring 65 nm (6.5 × 10−8 m), and it can perform 27 billion
instructions per second when running on a 2.93 GHz clock!
Similar miniaturization trends are obvious in the technology used to manipulate the electromagnetic spectrum.
The ability of a circuit component to interact with electromagnetic waves depends on how its size compares with
the wavelength (λ) of the signal it is trying to manipulate. For example, to efﬁciently transmit or receive signals, a
wire antenna must be comparable to λ in length. Some of the ﬁrst electromagnetic waves used for communication
were in the 1MHz range (corresponding to λ = 300 m) which today is allocated primarily to AM radio broadcasting.
[The frequency f (in Hz) is related to the wavelength λ (in meters) by λf = c, where c = 3 × 108 m/s is the velocity
of light in vacuum.] With the advent of portable radio and television, the usable spectrum was extended into the
megahertz range 102 to 103 MHz or (λ = 3 m to 30 cm). Modern cell phones operate in the low gigahertz (GHz) range
(1 GHz = 109 Hz). Each of these shifts has necessitated technological revolutions as components and devices continue
to shrink. The future of electronics looks bright (and tiny) as the processing and communication of signals approaches
the terahertz (THz) range (1012 Hz)! 64 Gbits ∗ Number of bits per chip 1010
Human memory
Human DNA 109 4 Gbits
1 Gbits 256 Mbits
108 64 Mbits
16 Mbits Book 107 Encyclopedia
2 hrs CD Audio
30 sec HDTV 4 Mbits
106 1 Mbits
256 Kbits 105 64 Kbits Doubling every
2 years
Page 104
1970 1980 1990 2000 2010 Year
Figure TF12: Chip capacity has increased at a logarithmic rate for the past 30 years. (Courtesy of Jan Rabaey.) 12 TECHNOLOGY BRIEF 1: MICRO AND NANOTECHNOLOGY Scaling Trends and Nanotechnology
It is an observable fact that each generation of tools enables the construction of a new, smaller, more powerful generation
of tools. This is true not just of mechanical devices, but electronic ones as well. Today’s highpower processors could
not have been designed, much less tested, without the use of previous processors that were employed to draw and
simulate the next generation. Two observations can be made in this regard. First, we now have the technology to build
tools to manipulate the environment at atomic resolution. At least one generation of microscale techniques (ranging
from microelectromechanical systems—or MEMS—to microchemical methods) has been developed which, useful
onto themselves, are also enabling the construction of newer, nanoscale devices. These newer devices range from
5 nm (1 nm = 10−9 m) transistors to femtoliter (10−15 ) microﬂuidic devices that can manipulate single protein molecules.
At these scales, the lines between mechanics, electronics and chemistry begin to blur! It is to these everincreasing
interdisciplinary innovations that the term nanotechnology rightfully belongs.
Second, the rate at which these innovations are occurring seems to be increasing exponentially! Consider Fig.TB12
and TB13 and note that the y axis is logarithmic and the plots are very close to straight lines. This phenomenon, which
was observed to hold true for the number of transistors that can be fabricated into a single processor, was noted by
Gordon Moore in 1965 and was quickly coined “Moore’s Law” (see Technology Brief 2: Moore’s Law). Figure TF13: Time plot of computer processing power in MIPS per $1000 (From “When will computer hardware match the
human brain?” by Hans Moravec, Journal of Transhumanism, Vol. 1, 1998.) 13 ELECTRIC CHARGE AND CURRENT 13 Electric Charge and Current 13.1 13 Charge The current ﬂows from the positive (+) terminal of the
battery to the negative (−) terminal, along the path external
to the battery. At the atomic scale, all matter contains a mixture of neutrons,
positively charged protons, and negatively charged electrons.
The nature of the force induced by electric charge was
established by the French scientist Charles Augustin de
Coulomb (1736–1806) during the latter part of the 18th century.
This was followed by a series of experiments on electricity
and magnetism over the next 100 years, culminating in J. J.
Thompson’s discovery of the electron in 1897. Through these
and more recent investigations, we can ascribe to electric charge
the following fundamental properties:
1. Charge can be either positive or negative.
2. The fundamental (smallest) quantity of charge is that
of a single electron or proton. Its magnitude usually
is denoted by the letter e.
3. According to the law of conservation of charge, the
(net) charge in a closed region can neither be created
nor destroyed.
4. Two like charges repel one another, whereas two
charges of opposite polarity attract.
The unit for charge is the coulomb (C) and the magnitude of e
is
e = 1.6 × 10−19
(C).
(1.1)
The symbol commonly used to represent charge is q . The
charge of a single proton is qp = e, and that of an electron,
which is equal in magnitude but opposite in polarity, is qe = −e.
It is important to note that the term charge implies “net charge,”
which is equal to the combined charge of all protons present in
any given region of space minus the combined charge of all
electrons in that region. Hence, charge is always an integral
multiple of e.
The last of the preceding properties is responsible for the
movement of charge from one location to another, thereby
constituting an electric current. Consider the simple circuit
in Fig. 14 depicting a battery of voltage V connected across a
resistor R using metal wires. The arrangement gives rise to an
electric current given by Ohm’s law (which will be discussed
in some detail in Chapter 2):
I=
As shown in Fig. 14: V
.
R Through chemical or other means, the battery generates a supply
of electrons at its negatively labeled terminal by ionizing some
of the molecules of its constituent material. A convenient model
for characterizing the functionality of a battery is to regard the
internal path between its terminals as unavailable for the ﬂow
of charge, forcing the electrons to ﬂow from the (−) terminal,
through the external path, and towards the (+) terminal to
achieve neutrality. It is important to note that:
The direction of electric current is deﬁned to be the same
as the direction of ﬂow that positive charges would follow,
which is opposite to the direction of ﬂow of electrons.
Even though we talk about electrons ﬂowing through the wires
and the resistor, in reality the process is a drift movement
rather than freeﬂow. The wire material consists of atoms
with loosely attached electrons. The positive polarity of the
(+) terminal exerts an attractive force on the electrons of the
hitherto neutral atoms adjacent to that terminal, causing some
of the loosely attached electrons to detach and jump to the (+)
terminal. The atoms that have lost those electrons now become
positively charged (ionized), thereby attracting electrons from
their neighbors and compelling them to detach from their hosts
and to attach themselves to the ionized atoms instead. This
process continues throughout the wire segment (between the Expanded view of wire
e− e−
Atom V +
_ Electron I
R
e− (1.2)
Figure 14: The current ﬂowing in the wire is due to electron transport
through a drift process, as illustrated by the magniﬁed structure of the
wire. 14 CHAPTER 1 (+) battery terminal and the resistor), into the longitudinal path
of the resistor, and ﬁnally through the wire segment between
the resistor and the (−) terminal. The net result is that the
(−) terminal loses an electron and the (+) terminal gains one,
making it appear as if the very same electron that left the (−)
terminal actually ﬂowed through the wires and the resistor and
ﬁnally appeared at the (+) terminal. It is as if the path itself
were not involved in the electron transfer, which is not the case.
The process of sequential migration of electrons from one
atom to the next is called electron drift, and it is this process
that gives rise to the ﬂow of conduction current through a
circuit. To illustrate how important this process is in terms of
the electronic transmission of information, let us examine the
elementary transmission experiment represented by the circuit
shown in Fig. 15. The circuit consists of an 8volt battery and a
switch on one end, a resistor on the other end, and a 60mlong
twowire transmission line in between. The wires are made
of copper, and they have a circular cross section with a 2mm
diameter. After closing the switch, a current will start to ﬂow
through the circuit. It is instructive to compare two velocities
associated with the consequence of closing the switch, namely
the electron drift velocity inside the copper wires and the
transmission velocity (of the information announcing that the
switch has been closed) between the battery and the resistor.
For the speciﬁed parameters of the circuit shown in Fig. 15,
the electron drift velocity—which is the actual physical velocity
of the electrons along the wire—can be calculated readily and
shown to be on the order of only 10−4 m/s. Hence, it would
take about 1 million seconds (∼ 10 days) for an electron to
physically travel over a distance of 120 m. In contrast, the
time delay between closing the switch at the sending end and
observing a response at the receiving end (in the form of current
ﬂow through the resistor) is extremely short (≈ 0.2 μs). This is
because the transmission velocity is on the order of the velocity
of light c = 3 × 108 m/s. Thus:
The rate at which information can be transmitted
electronically using conducting wires is about 12 orders
of magnitude faster than the actual transport velocity of
the electrons ﬂowing through those wires! This fact is at the heart of what makes electronic communication
systems possible. 13.2 Current Moving charge gives rise to current.
Electric current is deﬁned as the time rate of transfer of
electric charge across a speciﬁed boundary.
For the wire segment depicted in Fig. 16, the current i ﬂowing
through it is equal to the amount of charge dq that crosses the
wire’s cross section over an inﬁnitesimal time duration dt , given
as
i= +
8V _ dq
dt (1.3) (A), and the unit for current is the ampere (A). In general, both
positive and negative charges may ﬂow across the hypothetical
interface, and the ﬂow may occur in both directions. By
convention, the direction of i is deﬁned to be the direction
of the net ﬂow of (net) charge (positive minus negative). The
circuit segment denoted with an arrow in Fig. 17(a) signiﬁes
that a current of 5 A is ﬂowing through that wire segment in the
direction of the arrow. The same information about the current
magnitude and direction may be displayed as in Fig. 17(b),
where the arrow points in the opposite direction and the current
is expressed as −5 A.
When a battery is connected to a circuit, the resultant current
that ﬂows through it usually is constant in time (Fig. 18(a))—
at least over the time duration of interest—in which case we
refer to it as a direct current or dc for short. In contrast,
the currents ﬂowing in household systems (as well as in many
electrical systems) are called alternating currents or simply ac,
because they vary sinusoidally with time (Fig. 18(b)). Other
time variations also may occur in circuits, such as exponential Wire t=0 CIRCUIT TERMINOLOGY Direction of
electron flow Cross section −
Wire Switch 100 Ω
60 m Figure 15: After closing the switch, it takes only 0.2 μs to observe a
current in the resistor. − − − i −
− − − Electron i Current direction
Figure 16: Direction of (positive) current ﬂow through a conductor
is opposite that of electrons. 13 ELECTRIC CHARGE AND CURRENT 5A Circuit
(a) −5 A Circuit
(b) 15
rises and decays (Fig. 18(c) and (d)), exponentially damped
oscillations (Fig. 18(e)), and many others.
Even though in the overwhelming majority of cases the
current ﬂowing through a material is dominated by the
movement of electrons (as opposed to positively charged ions),
it is advisable to start thinking of the current in terms of
positive charge, primarily to avoid having to keep track of the
fact that current direction is deﬁned to be in opposition to the
direction of ﬂow of negative charges.
Example 11: Charge Transfer Figure 17: A current of 5 A ﬂowing “downward” is the same as −5 A
ﬂowing “upward” through the wire. I In terms of the current i(t) ﬂowing past a reference cross section
in a wire:
(a) Develop an expression for the cumulative charge q(t) that
has been transferred past that cross section up to time t . Apply
the result to the exponential current displayed in Fig. 19(a),
which is given by dc
i(t) = t 0
6e−0.2t A for t < 0,
for t ≥ 0. (1.4) (b) Develop an expression for the net charge Q(t1 , t2 ) that
ﬂowed through the cross section between times t1 and t2 , and
then compute Q for t1 = 1 s and t2 = 2 s. (a)
i(t)
ac
t
i(t) i(t) (b)
6A Decaying Current t t
i(t) (c)
(a) Rising
t q(t)
30 C (d)
i(t) Charge Damped oscillatory t
t
(e) (b) Figure 18: Graphical illustrations of various types of current
variations with time. Figure 19: The current i(t) displayed in (a) generates the cumulative
charge q(t) displayed in (b). 16 CHAPTER 1 Solution:
(a) We start by rewriting Eq. (1.3) in the form: Example 12: Current The charge ﬂowing past a certain location in a wire is given by dq = i dt.
Then by integrating both sides over the limits −∞ to t , we have
t t dq =
−∞ q(t) = Solution:
(a) Application of Eq. (1.3) yields which yields
t q(t) − q(−∞) = dq
dt
d
= (5te−0.1t )
dt i= (1.5) i dt,
−∞ where q(−∞) represents the charge that was transferred
through the wire “at the beginning of time.” We choose −∞
as a reference limit in the integration, because it allows us to
set q(−∞) = 0, implying that no charge had been transferred
prior to that point in time. Hence, Eq. (1.5) becomes
t (C). i dt (1.6) −∞ = 5e−0.1t − 0.5te−0.1t
= (5 − 0.5t)e−0.1t A.
Setting t = 0 in the expression gives i(0) = 5 A.
Note that i = 0, even though q(t) = 0 at t = 0.
(b) To determine the value of t at which q(t) is a maximum,
we ﬁnd dq/dt and then set it equal to zero:
dq
= (5 − 0.5t)e−0.1t
dt
= 0, For i(t) as given by Eq. (1.4), i(t) = 0 for t < 0. Hence,
t 6e−0.2t dt = q(t) = −6 −0.2t
e
0.2 t
0 = 30[1 − e−0.2t ] C. A plot of q(t) versus t is displayed in Fig. 19(b). The
cumulative charge that would transfer after a long period
of time is obtained by setting t = +∞, which would yield
q(+∞) = 30 C.
(b) The cumulative charge that has ﬂowed through the cross
section up to time t1 is q(t1 ), and a similar deﬁnition applies
to q(t2 ). Hence, the net charge that ﬂowed through the cross
section over the time interval between t1 and t2 is
Q(t1 , t2 ) = q(t2 ) − q(t1 ) −∞ −∞ Q(1, 2) = 6e
1 −0.2t t = 10 s, or t = ∞. as well as when
e−0.1t = 0 The ﬁrst value (t = 10 s) corresponds to a maximum and t = ∞
corresponds to a minimum (which can be veriﬁed either by
graphing q(t) or by taking the second derivative of q(t) and
evaluating it at t = 10 s and t = ∞).
At t = 10 s, t2 i dt = i dt.
Review Question 13: What are the four fundamental t1 For t1 = 1 s, t2 = 2 s, and i(t) as given by Eq. (1.4),
2 or q(10) = 5 × 10e−0.1×10 = 50e−1 = 18.4 C. t1 i dt − = which is satisﬁed when
5 − 0.5t = 0 0 t2 0
for t < 0,
5te−0.1t C for t ≥ 0. Determine (a) the current at t = 0 and (b) the instant at which
q(t) is a maximum and the corresponding value of q . i dt,
−∞ q(t) = CIRCUIT TERMINOLOGY 6e−0.2t
dt =
−0.2 2
1 = −30(e−0.4 − e−0.2 ) = 4.45 C. properties of electric charge?
Review Question 14: Is the direction of electric current in a wire deﬁned to be the same as or opposite to the
direction of ﬂow of electrons?
Review Question 15: How does electron drift lead to the conduction of electric current? 14 VOLTAGE AND POWER 17 Exercise 14: If the current ﬂowing through a given resistor in a circuit is given by i(t) = 5[1 − e−2t ] A for
t ≥ 0, determine the total amount of charge that passed
through the resistor between t = 0 and t = 0.2 s.
Q(0, 0.2) = 0.18 C. (See Answer: ) Exercise 15: If q(t) has the waveform shown in Fig. E1.5, determine the corresponding current waveform. electrically neutral structure, assume that we are able to detach
an electron from one of the atoms at point a and move it to
point b. Moving a negative charge from the positively charged
atom against the attraction force between them requires the
expenditure of a certain amount of energy. Voltage is a measure
of this expenditure of energy relative to the amount of charge
involved, and it always involves two spatial locations:
Voltage often is denoted vab to emphasize the fact that it is
the voltage difference between points a and b. q(t)
2C
1 2 3 4 5 6 7 8 t (s) Figure E1.5 The two points may be two locations in a circuit or any two
points in space.
Against this background, we now offer the following formal
deﬁnition for voltage:
The voltage between location a and location b is the ratio
of dw to dq, where dw is the energy in joules (J) required to
move (positive) charge dq from b to a (or negative charge
from a to b). Answer: i(t)
2A −2 A 1 2 3 4 5 6 7 8 t (s) 14 Voltage and Power
14.1 Voltage
The two primary quantities used in circuit analysis are current
and voltage. Current is associated with the movement of
electric charge and voltage is associated with the polarity of
charge. Before we offer a formal deﬁnition for voltage, let us
examine the energy implications of polarizing a hitherto neutral
material, thereby establishing opposite electrical polarities on
its two ends. Suppose we have a piece of material (such as a
resistor) to which we connect two short wires and label their
end points a and b, as shown in Fig. 110. Starting out with an vab = (1.7) and the unit for voltage is the volt (V), named after the inventor
of the ﬁrst battery, Alessandro Volta (1745–1827). Voltage also
is called potential difference. In terms of that terminology, if
vab has a positive value, it means that point a is at a potential
higher than that of point b. Accordingly, points a and b in
Fig. 110 are denoted with (+) and (−) signs, respectively. If
vab = 5 V, we often use the terminology: “The voltage rise
from b to a is 5 V”, or “The voltage drop from a to b is 5 V”.
Just as 5 A of current ﬂowing from a to b in a circuit
conveys the same information as −5 A ﬂowing in the opposite
direction, a similar analogy applies to voltage. Thus, the two
representations in Fig. 111 convey the same information with
regard to the voltage between terminals a and b. Also, the a 12 V Circuit vab
e− dw
,
dq a e−
e− That is, a
−12 V Circuit b
b (a) b
(b) Any material
Figure 110: The voltage vab is equal to the amount of energy required
to move one unit of negative charge from a to b through the material. Figure 111: In (a), with the (+) designation at node a , Vab = 12 V.
In (b), with the (+) designation at node b, Vba = −12 V, which is
equivalent to Vab = 12 V. [That is, Vab = −Vba .] 18 CHAPTER 1 R1 V1 R2 Node 1
V0 +
_ R3 V2
Node 2 CIRCUIT TERMINOLOGY Volts Voltmeter R4 Amps V12 V 1 + R 2 I Ammeter − Voltage reference (ground)
Figure 112: Ground is any point in the circuit selected to serve as a
reference point for all points in the circuit. terms dc and ac deﬁned earlier for current apply to voltage as
well; a constant voltage is called a dc voltage and a sinusoidally
timevarying voltage is called an ac voltage. Figure 113: An ideal voltmeter measures the voltage difference
between two points (such as nodes 1 and 2) without interfering with the
circuit. Similarly, an ideal ammeter measures the current magnitude
and direction without extracting a voltage drop across itself. To measure the current ﬂowing through a wire, it is necessary
to insert an ammeter in that path, as illustrated by Fig. 113.
The voltage drop across an ideal ammeter is zero. Ground
Open and Short Circuits
Since by deﬁnition voltage is not an absolute quantity but rather
the difference in electric potential between two locations, it is
sometimes convenient to select a reference point in the circuit,
label it ground, and then deﬁne the voltage at any point in the
circuit with respect to that ground point. Thus, when we say
that the voltage V1 at node 1 in Fig. 112 is 6 V, we mean that the
potential difference between node 1 and the ground reference
point is 6 V, which is equivalent to having assigned the ground
point a voltage of zero.
When a circuit is constructed in a laboratory, the chassis
often is used as the common ground point—in which case it
is called chassis ground. As discussed later in Section 86,
in a household electrical network, outlets are connected to
three wires—one of which is called Earth ground because it is
connected to the physical ground next to the house.
Voltmeter and Ammeter
The voltmeter is the standard instrument used to measure
the voltage difference between two points in a circuit. To
measure V12 in the circuit of Fig. 113, we connect the (+)
terminal of the voltmeter to terminal 1 and the (−) terminal
to terminal 2. Connecting the voltmeter to the circuit does
not require any changes to the circuit, and in the ideal case,
the voltmeter will have no effect on any of the voltages and
currents associated with the circuit. In reality, the voltmeter
has to extract some current from the circuit in order to perform
the voltage measurement, but the voltmeter is designed such
that the amount of extracted current is so small as to have a
negligible effect on the circuit. An open circuit refers to the condition of path discontinuity
(inﬁnite resistance) between two points. No current can ﬂow
through an open circuit, regardless of the voltage across it. The
path between terminals 1 and 2 in Fig. 114 is an open circuit. In
contrast, a short circuit constitutes the condition of complete
path continuity (with zero electrical resistance) between two
points, such as between terminals 3 and 4 in Fig. 114. No
voltage drop occurs across a short circuit, regardless of the
magnitude of the current ﬂowing through it.
Switches come in many varieties, depending on the intended
function. The simple ON/OFF switch depicted in Fig. 115(a)
is known as a singlepole singlethrow (SPST) switch. The
ON (closed) position acts like a short circuit, allowing current
to ﬂow while extracting no voltage drop across the switch’s
terminals; the OFF (open) position acts like an open circuit.
The speciﬁc time t = t0 denoted below or above the switch
(Fig. 115(a)) refers to the time t0 at which it opens or closes. Open circuit
1 V +
_ 2
R1 Short circuit
3 4
R2 Figure 114: Open circuit between terminals 1 and 2, and short circuit
between terminals 3 and 4. 14 VOLTAGE AND POWER t = t0 19 SPST switches
t = t0 Switch initially open,
then closes at t = t0 Switch initially closed,
then opens at t = t0 Switch open
Vbat (a)
1
SPDT switch (a)
I t = t0 2
(b) Switch initially connected to terminal 1,
then moved to terminal 2 at t = t0
Figure 115: (a) Singlepole singlethrow (SPST) and (b) singlepole
doublethrow (SPDT) switches. If the purpose of the switch is to combine two switching
functions so as to connect a common terminal to either of two
other terminals, then we need to use the singlepole doublethrow (SPDT) switch illustrated in Fig. 115(b). Before t = t0 ,
the common terminal is connected to terminal 1; then at t = t0 ,
that connection ceases (becomes open), and it is replaced with
a connection between the common terminal and terminal 2. 14.2 +
− Power The circuit shown in Fig. 116(a) consists of a battery and a
light bulb connected by an SPST switch in the open position.
No current ﬂows through the open circuit, but the battery has
a voltage Vbat across it, due to the excess positive and negative
charges it has at its two terminals. After the switch is closed
at t = 5 s, as indicated in Fig. 116(b), a current I will ﬂow
through the circuit along the indicated direction. The battery’s
excess positive charges will ﬂow from its positive terminal
downward through the light bulb towards the battery’s negative
terminal, and (since current direction is deﬁned to coincide with
the direction of ﬂow of positive charge) the current direction will
be as indicated in the ﬁgure.
The consequences of current ﬂow through the circuit are: (1)
The battery acts as a supplier of power and (2) The light bulb
acts as a recipient of power, which gets absorbed by its ﬁlament,
causing it to heat up and glow, resulting in the conversion of
electrical power into light and heat. A power supply, such
as a battery, offers a voltage rise across it as we follow the
current from the terminal at which it enters (denoted with a
(−) sign) to the terminal from which it leaves (denoted with
a (+) sign). In contrast, a power recipient (such as a light
bulb) exhibits a voltage drop across its corresponding terminals.
This set of assignments of voltage polarities relative to the Switch closes at t = 5 s +
Vbat
− + Vbulb − (b) Figure 116: Current ﬂow through a resistor (lightbulb ﬁlament) after
closing the switch. direction of current ﬂow for devices generating power versus
those consuming power is known as the passive sign convention
(Fig. 117). We will adhere to it throughout the book.
Our next task is to establish an expression for the power p
delivered to or received by an electrical device. By deﬁnition,
power is the time rate of change of energy,
p= dw
dt (W), (1.8) and its unit is the watt (W), named after the Scottish engineer
and inventor James Watt (1736–1819) who is credited with the
development of the steam engine from an embryonic stage into Passive Sign Convention
i p>0
p<0 v
Device
p = vi
power delivered to device
power supplied by device *Note that i direction is defined as entering
(+) side of v.
Figure 117: Passive sign convention. 20 TECHNOLOGY BRIEF 2: MOORE’S LAW AND SCALING Technology Brief 2: Moore’s Law and Scaling
In 1965, Gordon Moore, cofounder of Intel, predicted that the number of transistors in the minimumcost processor
would double every two years (initially, he had guessed they would double every year). Amazingly, this prediction has
proven true of semiconductor processors for 40 years, as demonstrated by Fig. TF21.
In order to understand Moore’s Law, we have to understand the basics about how transistors work. As we will see
later in Section 37, the basic switching element in semiconductor microprocessors is the transistor: All of the complex
components in the microprocessor (including logic gates, arithmetic logic units, and counters) are constructed from
combinations of transistors. Within a processor, transistors have different dimensions depending on the component’s
function; larger transistors can handle more current, so the subcircuit in the processor that distributes power may be
built from larger transistors than, say, the subcircuit that adds two bits together. In general, the smaller the transistor,
the less power it consumes and the faster it can switch between binary states (0 and 1). Hence, an important goal of a
circuit designer is to use the smallest transistors possible in a given circuit. We can quantify transistor size according
to the smallest drawn dimension of the transistor, sometimes called the feature size. In the Intel 4004, for example, the
feature size was approximately 10 μm, which means that it was not possible to make transistors reliably with less than
10μm features drawn in the CAD program. In modern processors, the feature size is 0.065 μm or 65 nm. (Remember
that 1 nm = 10−9 m.)
The questions then arise: How small can we go? What is the fundamental limit to shrinking down the size of
a transistor? As we ponder this, we immediately observe that we likely cannot make a transistor smaller than the
diameter of one silicon or metal atom (i.e., ∼ 0.2 to 0.8 nm). But is there a limit prior to this? Well, as we shrink
transistors such that they are made of just one or a few atomic layers (∼ 1 to 5 nm), we run into issues related to the Transistors 10,000,000,000
DualCore Itanium 2
Itanium 2
Itanium 1,000,000,000
100,000,000 Pentium 4
Pentium III 10,000,000 Pentium II
Pentium II
386 1,000,000 286
8086 100,000 6000
8008
4004 1965 1970 Intel CPUs
10,000 8000 1975 1980 1985 1990 1995 2000 2005 1,000
2010 Figure TF21: Moore’s Law predicts that the number of transistors per processor doubles every two years. TECHNOLOGY BRIEF 2: MOORE’S LAW AND SCALING 21 stochastic nature of quantum physics. At these scales, the random motion of electrons between both physical space
and energy levels becomes signiﬁcant with respect to the size of the transistor, and we start to get spurious or random
signals in the circuit. There are even more subtle problems related to the statistics of yield. If a certain piece of a
transistor contained only 10 atoms, a deviation of just one atom in the device (to a 9atom or an 11atom transistor)
represents a huge change in the device properties! (Can you imagine your local car dealer telling you your sedan will
vary in length by ±10 percent when it comes from the factory!?) This would make it increasingly difﬁcult to economically
fabricate chips with hundreds of millions of transistors. Additionally, there is an interesting issue of heat generation:
Like any dissipative device, each transistor gives off a small amount of heat. But when you add up the heat produced by
100 million transistors, you get a very large number! Figure TF21 compares the power density (due to heat) produced
by different processors with the heat produced by rocket engines and nuclear reactors.
None of these issues are insurmountable. Challenges simply spur driven people to come up with innovative solutions.
Many of these problems will be solved, and in the process, provide engineers (like you) with jobs and opportunities.
But, more importantly, the minimum feature size of a processor is not the end goal of innovation: It is the means to it.
Innovation seeks simply to make increasingly powerful processors, not smaller feature sizes. In recent years, processor
companies have lessened their attempts at smaller, faster processors and started lumping more of them together to
distribute the work among them. This is the idea behind the dual and quad processor cores that power the computers
of the last few years. By sharing the workload among various processors (called distributed computing) we increase
processor performance while using less energy, generating less heat, and without needing to run at warp speed. So
it seems, as we approach eversmaller features, we simply will transition into new physical technologies and also new
computational techniques. As Gordon Moore himself said, “It will not be like we hit a brick wall and stop.” Power Density (W/cm2) 10000 Rocket
Nozzle 1000 Nuclear
Reactor 100 8086
10 4004
Hot Plate
P6
8008 8085
Pentium® proc
386
286
486
8080
1
1970 1980 1990
Year 2000 2010 Light Bulb
Power dissipation
Surface area
Heat flux Integrated Circuit 100 W 50 W 106 cm2
(bulb surface area) 1.5 cm2 (die area) 0.9 W/cm2 33.3 W/cm2 Figure TF22: The power density generated by an IC in the form of heat is approaching the densities produced by a nuclear reactor.
(Courtesy of Jan Rabaey.) 22 CHAPTER 1 0.2 A a viable and efﬁcient source of power. Using Eqs. (1.3) and
(1.7), we can rewrite Eq. (1.8) as
p= dw
dw dq
=
·
dt
dq dt CIRCUIT TERMINOLOGY 12 V +
_ 12 V Device or simply
p = vi (W). (1.9) (a)
Device 1 Consistent with the passive sign convention:
The power delivered to a device is equal to the voltage
across it multiplied by the current entering through its (+)
voltage terminal.
If the algebraic value of p is negative, then the device is a
supplier of energy. For an isolated electric circuit composed of
multiple elements, the law of conservation of power requires
that the algebraic sum of power for the entire circuit be always
zero. That is, for a circuit with n elements,
n pk = 0 , (1.10) k =1 which means that the total power supplied by the circuit always
must equal the total power absorbed by it.
Power supplies are sometimes assigned ratings to describe
their capacities to deliver energy. A battery may be rated as
having an output capacity of 200 amperehours (Ah) at 9 volts,
which means that it can deliver a current i over a period of
time t (measured in hours) such that i t = 200 Ah, and it
can do so while maintaining a voltage of 9 V. Alternatively, its
output capacity may be expressed as 1.8 kilowatthours (kWh),
which represents the total amount of energy it can supply,
namely, W = vi t (with t in hours).
Example 13: Conservation of Power For each of the two circuits shown in Fig. 118, determine how
much power is being delivered to each device and whether it is
a power supplier or recipient.
Solution:
(a) For the circuit in Fig. 118(a), the current entering the (+)
terminal of the device is 0.2 A. Hence, the power P (where we
use an uppercase letter because both the current and voltage are
dc) is:
P = V I = 12 × 0.2 = 2.4 W, +
12 V _ 3A 18 V
6V Device 2 (b)
Figure 118: Circuits for Example 13. and since P > 0, the device is a recipient of power. As we
know, the law of conservation of power requires that if the
device receives 2.4 W of power then the battery has to deliver
exactly that same amount of power. For the battery, the current
entering its (+) terminal is −0.2 A (because 0.2 A of current is
shown leaving that terminal), so according to the passive sign
convention, the power that would be absorbed by the battery
(had it been a passive device) is
Pbat = 12(−0.2) = −2.4 W.
The fact that Pbat is negative is conﬁrmation that the battery
is indeed a supplier of power.
(b) For device 1 in Fig. 118(b), the current entering its (+)
terminal is 3 A. Hence,
P1 = V1 I1 = 18 × 3 = 54 W,
and the device is a power recipient.
For device 2,
P2 = V2 I2 = (−6) × 3 = −18 W,
and the device is a supplier of power (because P2 is negative).
By way of conﬁrmation, the power associated with the battery
is
Pbat = 12(−3) = −36 W,
thereby satisfying the law of conservation of power, which
requires the net power of the overall circuit to be exactly zero. 15 CIRCUIT ELEMENTS 23 Example 14: Energy Consumption A resistor connected to a 100V dc power supply was consuming
20 W of power until the switch was turned off, after which the
voltage decayed exponentially to zero. If t = 0 is deﬁned as
the time at which the switch was turned to the off position and
if the subsequent voltage variation was given by
v(t) = 100e−2t V for t ≥ 0 Solution:
Before t = 0, the current ﬂowing through the resistor was
I = P /V = 20/100 = 0.2 A. Using this value as the initial
amplitude of the current at t = 0 and assuming that the current
will exhibit the same time variation as the voltage, i(t) can be
expressed as
for t ≥ 0. The instantaneous power is
p(t) = v(t) · i(t) = (100e−2t )(0.2e−2t ) = 20e−4t W.
We note that the power decays at a rate (e−4t ) much faster
than the rate for current and voltage (e−2t ). The total energy
dissipated in the resistor after engaging the switch is obtained
by integrating p(t) from t = 0 to inﬁnity, namely
∞ 20e−4t dt = − 0 20 −4t
e
4 ∞
0 = 5 J. 0 Exercise 16: If a positive current is ﬂowing through a resistor from its terminal a to its terminal b, is vab positive
or negative?
Answer: vab > 0. (See ) Exercise 17: A certain device has a voltage difference of
5 V across it. If 2 A of current is ﬂowing through it from
its (−) voltage terminal to its (+) terminal, is the device a
power supplier or a power recipient, and how much energy
does it supply or receive in 1 hour? Answer: P = V I = 5(−2) = −10 W. Hence, the
device is a power supplier. W  = P  t = 36 kJ.
(See ) 4 minutes. (See ) Circuit Elements Electronic circuits used in functional systems employ a wide
range of circuit elements, including transistors and integrated
circuits. The operation of most electronic circuits and devices—
no matter how complex—can be modeled (represented) in
terms of an equivalent circuit composed of basic elements
with idealized characteristics. The equivalent circuit offers a
circuit behavior that closely resembles the behavior of the actual
electronic circuit or device over a certain range of speciﬁed
conditions, such as the range of input signal level or output
load resistance. The set of basic elements commonly used in
circuit analysis include voltage and current sources; passive
elements (which include resistors), capacitors, and inductors;
and various types of switches. The basic attributes of switches
were covered in Section 14.1. The nomenclature and current–
voltage relationships associated with the other two groups are
the subject of this section. 15.1 ∞ p(t) dt = W= Answer: 15 (where t is in seconds), determine the total amount of energy
consumed by the resistor after the switch was turned off. i(t) = 0.2e−2t A Exercise 18: A car radio draws 0.5 A of dc current when
connected to a 12V battery. How long does it take for the
radio to consume 1.44 kJ? i –v Relationship The relationship between the current ﬂowing through a device
and the voltage across it deﬁnes the fundamental operation of
that device. As was stated earlier, Ohm’s law states that the
current i entering into the (+) terminal of the voltage v across
a resistor is given by
v
i= .
R
This is called the i –v relationship for the resistor. We note
that the resistor exhibits a linear i –v relationship, meaning
that i and v always vary in a proportional manner, as shown
in Fig. 119(a), so long as R remains constant. A circuit
composed exclusively of elements with linear i –v responses
is called a linear circuit. The linearity property of a circuit
is an underlying requirement for the various circuit analysis
techniques presented in this and future chapters. Diodes and
transistors exhibit nonlinear i –v relationships, but we still
can apply the analysis techniques speciﬁc to linear circuits
to circuits containing nonlinear devices by representing those
devices in terms of linear subcircuits that contain dependent
sources. The concept of a dependent source and how it is used
is introduced in Section 15.3. 24 CHAPTER 1 i v
i= R Resistor
1
Slope = R
v (a)
i v = Vs
Ideal voltage source Is i = Is
Ideal current source Vs v (b)
vs
vs = αvx VCVS
Slope = α
vx (c)
Figure 119: i –v relationships for (a) an ideal resistor, (b) ideal,
independent current and voltage sources, and (c) a dependent, voltagecontrolled voltage source (VCVS). 15.2 Independent Sources An ideal, independent voltage source provides a speciﬁed
voltage across its terminals, regardless of the type of load or
circuit connected to it. Hence, for a voltage source with a
speciﬁed voltage Vs , its i –v relationship is given by
v = Vs for any i, so long as it is not connected to a short circuit. Similarly, an
ideal, independent current source provides a speciﬁed current
ﬂowing through it, regardless of the voltage across it (but it
cannot do so if connected to an open circuit). Its i –v relationship
is
for any v.
i = Is
The i –v proﬁle of an ideal voltage source is a vertical line,
as illustrated in Fig. 119(b), whereas the proﬁle for the ideal
current source is a horizontal line.
The circuit symbol used for independent sources is a
circle, as shown in Table 13, although for dc voltage
sources the traditional “battery” symbol is used as well. A
household electrical outlet connected through an electrical CIRCUIT TERMINOLOGY powerdistribution network to a hydroelectric or nuclearpower generating station provides continuous power at an
approximately constant voltage level. Hence, it may be
classiﬁed appropriately as an independent voltage source. On a
shorter time scale, a ﬂashlight’s 9volt battery may be regarded
as a voltage source, but only until its stored charge has been
used up by the light bulb. Thus, strictly speaking, a battery is
a storage device (not a generator), but we tend to treat it as a
generator so long as it acts like a constant voltage source.
In reality, no sources can provide the performance
speciﬁcations ascribed to ideal sources. If a 5V voltage source
is connected across a short circuit, for example, we run into
a serious problem of ambiguity. From the standpoint of the
source, the voltage is 5 V, but by deﬁnition, the voltage is
zero across the short circuit. How can it be both zero and
5 V simultaneously? The answer resides in the fact that our
description of the ideal voltage source breaks down in this
situation. More realistic models for voltage and current
sources include a series resistor in the case of the voltage source,
and a shunt (parallel) resistor in the case of the current source,
as shown in Table 13. The real voltage source (which may have
an elaborate circuit conﬁguration) behaves like a combination
of an equivalent, ideal voltage source vs in series with an
equivalent resistance Rs . Usually, Rs has a very small value
for the voltage source and a very large value for the current
source. 15.3 Dependent Sources As alluded to in the opening paragraph of Section 15, we often
use equivalent circuits to model the behavior of transistors and
other electronic devices. The ability to represent complicated
devices by equivalent circuits composed of basic elements
greatly facilitates not only the circuit analysis process but
the design process as well. Such circuit models incorporate
the relationships between various parts of the device through
the use of a set of artiﬁcial sources known as dependent
sources. The voltage level of a dependent voltage source is
deﬁned in terms of a speciﬁc voltage or current elsewhere in
the circuit. An example of circuit equivalence is illustrated
in Fig. 120. In part (a) of the ﬁgure, we have a Model
741 operational ampliﬁer (op amp), denoted by the triangular
circuit symbol, used in a simple ampliﬁer circuit intended to
provide a voltage ampliﬁcation factor of −2; that is, the output
voltage v0 = −2vs , where vs is the input signal voltage. The op
amp, which we will examine later in Chapter 4, is an electronic
device with a complex architecture composed of transistors,
resistors, capacitors, and diodes, but in practice, its circuit
behavior can be represented by a rather simple circuit consisting
of two resistors (input resistor Ri and output resistor Ro ) and
a dependent voltage source, as shown in Fig. 120(b). The
voltage v2 on the righthand side of the circuit in Fig. 120(b) 15 CIRCUIT ELEMENTS 25 Table 13: Voltage and current sources. Independent Sources
Ideal Voltage Source Realistic Voltage Source Rs Battery +
_
−
+ Vs or vs dc source +
_
−
+ +
− Realistic Current Source Rs is is dc source +
_
−
Any source Any source* Ideal Current Source Is vs + Vs Any source Any source Dependent Sources
VoltageControlled Voltage Source (VCVS) +
− vs = αvx CurrentControlled Voltage Source (CCVS) +
− vs = rix VoltageControlled Current Source (VCCS) is = gvx CurrentControlled Current Source (CCCS) is = βix Note: α , g , r , and β are constants; vx and ix are a speciﬁc voltage and a speciﬁc current elsewhere in the
circuit. ∗ Lowercase v and i represent voltage and current sources that may or may not be time varying,
whereas uppercase V and I denote dc sources. is given by v2 = Avi , where A is a constant and vi is the
voltage across the resistor Ri located on the lefthand side of
the equivalent circuit. In this case, the magnitude of v2 always
depends on the magnitude of vi , which depends in turn on the
input signal voltage vs and on the values chosen for some of
the resistors in the circuit. Since the controlling quantity vi
is a voltage, v2 is called a voltagecontrolled voltage source
(VCVS). Had the controlling quantity been a current source, the dependent source would have been called a currentcontrolled
voltage source (CCVS) instead. A parallel analogy exists for
voltagecontrolled and currentcontrolled current sources. The
characteristic symbol for a dependent source is the diamond
(Table 13). Proportionality constant α in Table 13 relates
voltage to voltage. Hence, it is dimensionless, as is β , since
it relates current to current. Constants g and r have units of
(A/V) and (V/A), respectively. Because dependent sources are 26 CHAPTER 1 30 kΩ 30 kΩ
15 kΩ vs vo (a) Ro = 75 Ω 15 kΩ _
741
+ +
_ CIRCUIT TERMINOLOGY Opamp circuit vs +
_ vi (b) Ri = 3 MΩ +
_ v2 = Avi vo Equivalent circuit with dependent source Figure 120: An operational ampliﬁer is a complex device, but its circuit behavior can be represented in terms of a simple equivalent circuit
that includes a dependent voltage source. characterized by linear relationships, so are their i –v proﬁles.
An example is shown in Fig. 119(c) for the VCVS.
Example 15: Dependent Source Find the magnitude of the voltage V1 of the dependent source
in Fig. 122. What type of source is it? 5Ω 10 V 2Ω +
− I1 +
_ V1 = 4I1 Solution:
Since V1 depends on current I1 , it is a currentcontrolled voltage
source with a coefﬁcient of 4 V/A.
The 10V dc voltage is connected across the 2 resistor.
Hence, the current I along the designated direction is
10
= 5 A.
2 Consequently,
V1 = 4I1 = 4 × 5 = 20 V. Passive Elements Table 14 lists three passive elements. For the resistor, capacitor, and inductor, their current–voltage (i –v ) relationships are
given by
vR = RiR , (1.11a) iC = C Figure 121: Circuit for Example 15. I1 = 15.4 dvC
,
dt (1.11b) vL = L diL
,
dt (1.11c) where R , C , and L are the resistance in ohms ( ), capacitance
in farads (F), and inductance in henrys (H), respectively; vR , vC ,
and vL are the voltages across the resistor, capacitor, and
inductor, respectively; and iR , iC , and iL are the corresponding
currents ﬂowing through them. It should be noted that:
The i –v relationships are deﬁned such that for any of the
three devices, the current direction is into the positive
terminal of the voltage across it and out of the negative
terminal.
We observe that the resistor exhibits a linear i–v relationship,
meaning that i and v always vary in a proportional manner. The
capacitor and inductor are characterized by i –v relationships
that involve the time derivative d/dt . In fact, if the voltage
across the capacitor (vC ) is constant with time (dc), then
dvC /dt = 0, and consequently, the current iC = 0 (no matter 15 CIRCUIT ELEMENTS 27 Table 14: Passive circuit elements and their symbols.
Element Symbol i –v Relationship Solution:
See Fig. 123. iR
vR Resistor SPST switch that closes at t = 5 s. Generate circuit diagrams
that include only those elements that have current ﬂowing
through them for (a) t < 0, (b) 0 ≤ t < 5 s, and (c) t ≥ 5 s. R R1 vR = RiR V0
iC
vC Capacitor iC = C C dvC
dt vL R6 R2 R7
(a) t < 0 iL
Inductor +
− R1 L vL = L diL
dt V0 +
− R3 R4 R7
how large or small vC might be). Similarly, if the current iL
ﬂowing through the inductor is dc, then the voltage across it (vL )
is zero. Because the time variations of voltage and current are at
the core of what makes capacitors and inductors useful devices,
they are used primarily in timevarying circuits. The resistor
is used in both dc and timevarying circuits. Our examination
of circuits containing capacitors and inductors—both of which
are energy storage devices—begins in Chapter 5. (b) 0 < t < 5 s
R1
V0 +
− R3
R6 R5
R4 R7
Example 16: Switches The circuit in Fig. 122 contains one SPDT switch that changes
position at t = 0, one SPST switch that opens at t = 0, and one (c) t > 5 s
Figure 123: Solutions for circuit in Fig. 122. R1 V0 t=0 + Review Question 16: What is the difference between an SPDT − R6
R7 R2 SPST SPST
t=5s
t=0 R5 R3 SPST switch and an SPDT switch?
Review Question 17: What is the difference between R4 an independent voltage source and a dependent voltage
source? Is a dependent voltage source a real source of
power?
Review Question 18: What is an “equivalentcircuit” Figure 122: Circuit for Example 16. model? How is it used? 28 CHAPTER 1 Exercise 19: Find Ix from the diagram in Fig. E1.9. CIRCUIT TERMINOLOGY Exercise 110: In the circuit of Fig. E1.10, ﬁnd I at (a) t < 0 and (b) t > 0. 2Ω I + V1 −
5Ω 5A Ix = t=0 V1
4
12 V +
− 3Ω Figure E1.9 Answer: Ix = 2.5 A. (See SPDT ) 4Ω Figure E1.10 Answer: (a) I = 4 A, (b) I = 3 A. (See ) Chapter 1 Relationships
Ohm’s law i = v/R Current
i = dq/dt
Direction of i = direction of ﬂow of (+) charge
t Charge transfer q(t) =
t2 Q = q(t2 ) − q(t1 ) = Direction of i is into +v
terminal of device
p = vi
device absorbs power
device delivers power Passive sign convention i dt
−∞ i dt
t1 Power
If p > 0
If p < 0
i –v relationships
Resistor
Capacitor
Inductor vR = RiR
iC = C dvC /dt
vL = L diL /dt Voltage = potential energy difference CHAPTER HIGHLIGHTS
• Active devices (such as transistors and ICs) require an
external power source to operate; in contrast, passive
devices (resistors, capacitors, and inductors) do not. the (+) side of v ; if p > 0, the device is recipient
(consumer) of power, and if p < 0, it is a supplier of
power. • Analysis and synthesis (design) are complementary
processes. • Independent voltage and current sources are real
sources of energy; dependent sources are artiﬁcial
representations used in modeling the nonlinear behavior
of a device in terms of an equivalent linear circuit. • Current is related to charge by i = dq/dt ; voltage
between locations a and b is vab = dw/dq , where dw
is the work (energy) required to move dq from b to a ;
and power p = vi .
• Passive sign convention assigns i direction as entering • A resistor exhibits a linear i –v relationship, while
diodes and transistors do not; the i –v relationships for
a capacitor and an inductor involve d/dt . PROBLEMS 29 GLOSSARY OF IMPORTANT TERMS
Provide deﬁnitions or explain the meaning of the following terms:
ac
active device
amperehours
analysis
conduction current
cumulative charge
dc
dependent source
design
electric charge
electric circuit electric current
electron drift
equivalent circuit
independent source
i –v characteristic
kilowatthours
linear response
open circuit
passive device
passive sign convention
polarization PROBLEMS
Sections 12 and 13: Dimensions, Charge, and Current
1.1 Use appropriate multiple and submultiple preﬁxes to
express the following quantities:
(a) 3,620 watts (W)
(b) 0.000004 amps (A)
(c) 5.2 × 10−6 ohms ( )
(d) 3.9 × 1011 volts (V)
(e) 0.02 meters (m)
(f) 32 × 105 volts (V)
1.2 Use appropriate multiple and submultiple preﬁxes to
express the following quantities:
(a) 4.71 × 10−8 seconds (s)
(b) 10.3 × 108 watts (W)
(c) 0.00000000321 amps (A)
(d) 0.1 meters (m)
(e) 8,760,000 volts (V)
(f) 3.16 × 10−16 hertz (Hz)
1.3 Convert: (a) 16.3 m to mm
(b) 16.3 m to km
(c) 4 × 10−6 μF (microfarad) to pF (picofarad)
(d) 2.3 ns to μs 1.4 potential difference
power
preﬁx
short circuit
SI units
SPST
SPDT
synthesis
voltage Convert: (a) 4.2 m to μm
(b) 3 hours to μseconds
(c) 4.2 m to km
(d) 173 nm to m
(e) 173 nm to μm
(f) 12 pF (picofarad) to F (farad)
1.5 The total charge contained in a certain region of space is
−1 C. If that region contains only electrons, how many does it
contain?
1.6 A certain cross section lies in the x –y plane. If 3 × 1020
electrons go through the cross section in the zdirection in 4
seconds, and simultaneously 1.5 × 1020 protons go through
the same cross section in the negative zdirection, what is the
magnitude and direction of the current ﬂowing through the cross
section?
1.7 Determine the current i(t) ﬂowing through a resistor if
the cumulative charge that has ﬂowed through it up to time t is
given by
(a) q(t) = 3.6t mC
(b) q(t) = 5 sin(377t) μC (e) 3.6 × 107 V to MV (c) q(t) = 0.3[1 − e−0.4t ] pC (f) 0.03 mA (milliamp) to μA (d) q(t) = 0.2t sin(120π t) nC 30 CHAPTER 1 1.8 Determine the current i(t) ﬂowing through a certain
device if the cumulative charge that has ﬂowed through it up to
time t is given by
(a) q(t) = −0.45t 3 μC
(b) q(t) = 12 sin2 (800πt) mC
(c) q(t) = −3.2 sin(377t) cos(377t) pC CIRCUIT TERMINOLOGY 1.14 Given that the current in (mA) ﬂowing through a wire is
given by:
⎧
⎪0
for t < 0
⎨
i(t) = 6t
for 0 ≤ t ≤ 5 s
⎪ −0.6(t −5)
⎩
for t ≥ 5 s,
30e
(a) Sketch i(t) versus t . (d) q(t) = 1.7t [1 − e−1.2t ] nC (b) Sketch q(t) versus t . 1.9 Determine the net charge Q that ﬂowed through a
resistor over the speciﬁed time interval for each of the following
currents: 1.15 The plot in Fig. P1.15 displays the cumulative amount
of charge q(t) that has entered a certain device up to time t .
What is the current at
(a) t = 1 s (a) i(t) = 0.36 A, from t = 0 to t = 3 s (b) t = 3 s (b) i(t) = [40t + 8] mA, from t = 1 s to t = 12 s (c) t = 6 s (c) i(t) = 5 sin(4π t) nA, from t = 0 to t = 0.05 s q(t) (d) i(t) = 12e−0.3t mA, from t = 0 to t = ∞
1.10 Determine the net charge Q that ﬂowed through a
certain device over the speciﬁed time intervals for each of the
following currents:
(a) i(t) = [3t + 6t 3 ] 4C 0 mA, from t = 0 to t = 4 s (b) i(t) = 4 sin(40π t) cos(40πt)
t = 0.05 s μA, from t =0 to (d) i(t) = 4s t
6s 8s −4 C (c) i(t) = [4e−t − 3e−2t ] A, from t = 0 to t = ∞
12e−3t 2s Figure P1.15: q(t) for Problem 1.15. cos(40πt) nA, from t = 0 to t = 0.05 s 1.11 If the current ﬂowing through a wire is given by
i(t) = 3e−0.1t mA, how many electrons pass through the wire’s
cross section over the time interval from t = 0 to t = 0.3 ms?
1.12 The cumulative charge in mC that entered a certain
device is given by
⎧
⎪0
⎨
q(t) = 5t
⎪
⎩
60 − t for t < 0,
for 0 ≤ t ≤ 10 s,
for 10 s ≤ t ≤ 60 s 1.16 The plot in Fig. P1.16 displays the cumulative amount of
charge q(t) that has exited a certain device up to time t . What
is the current at
(a) t = 2 s
(b) t = 6 s
(c) t = 12 s q(t)
4C 4e−0.2(t−8) 2C
(a) Plot q(t) versus t from t = 0 to t = 60 s.
(b) Plot the corresponding current i(t) entering the device. 0
4s 1.13 A steady ﬂow resulted in 3 × 1015 electrons entering a
device in 0.1 ms. What is the current? 8s Figure P1.16: q(t) for Problem 1.16. t PROBLEMS 31 Sections 14 and 15: Voltage, Power, and Circuit Elements
1.17 For each of the eight devices in the circuit of Fig. P1.17,
determine whether the device is a supplier or a recipient of
power and how much power it is supplying or receiving. 1.20 A 9V ﬂashlight battery has a rating of 1.8 kWh. If
the bulb draws a current of 100 mA when lit; determine the
following:
(a) For how long will the ﬂashlight provide illumination?
(b) How much energy in joules is contained in the battery?
(c) What is the battery’s rating in amperehours? + 8 V_
4 + 6 V_
+
1 16 V
_ 5 v(t) = 5 cos(4π t) V, 2A 1A 2 4A 1.21 The voltage across and current through a certain device
are given by + 4 V_ +
10 V 3
_ 3A _ 7V+ 1A 6
+ 12 V _ 9V 7 _ + 8 i(t) = 0.1 cos(4π t) A. Determine:
(a) The instantaneous power p(t) at t = 0 and t = 0.25 s.
(b) The average power pav , deﬁned as the average value of
p(t) over a full time period of the cosine function (0 to
0.5 s). Figure P1.17: Circuit for Problem 1.17. 1.22 The voltage across and current through a certain device
are given by
v(t) = 100(1 − e−0.2t ) V,
1.18 For each of the seven devices in the circuit of Fig. P1.18,
determine whether the device is a supplier or a recipient of
power and how much power it is supplying or receiving. i(t) = 30e−0.2t mA. Determine:
(a) The instantaneous power p(t) at t = 0 and t = 3 s.
(b) The cumulative energy delivered to the device from t = 0
to t = ∞.
1.23 The voltage across a device and the current through it
are shown graphically in Fig. P1.23. Sketch the corresponding
power delivered to the device and calculate the energy absorbed
by it.
i(t) + 6 V_ 24 V 1 4 4A
10 A 5 5A 2A 3A 6 7 +V
6_ + _ + 4 V_ _V V
8_ 3 1A 12 + 5A + + 2 t 0 _V 10 Figure P1.18: Circuit for Problem 1.18. v(t) 2s 1s 2A 1s 2s 5V
t 0
1.19 An electric oven operates at 120 V. If its power rating
is 0.6 kW, what amount of current does it draw, and how much
energy does it consume in 12 minutes of operation? Figure P1.23: i(t) and v(t) of the device in Problem 1.23. 32 CHAPTER 1 1.24 The voltage across a device and the current through it
are shown graphically in Fig. P1.24. Sketch the corresponding
power delivered to the device and calculate the energy absorbed
by it. i(t) 1.26 For the circuit in Fig. P1.26, generate circuit diagrams
that include only those elements that have current ﬂowing
through them for
(a) t < 0
(b) 0 < t < 2 s
(c) t > 2 s R1 10 A
t 0
v(t) 1s 2s V1 +
_ +
V2 _ 5V CIRCUIT TERMINOLOGY SPST
t=0
SPDT R2 t=2s R3 R5
t=0 R6
SPST t 0
1s 2s Figure P1.26: Circuit for Problem 1.26. Figure P1.24: i(t) and v(t) of the device in Problem 1.24. 1.25 For the circuit in Fig. P1.25, generate circuit diagrams
that include only those elements that have current ﬂowing
through them for
(a) t < 0
(b) 0 < t < 2 s
(c) t > 2 s R1 V0 t=0
R2 +
_ R3
R4
t=2s R5 Figure P1.25: Circuit for Problem 1.25. R6 R4 ...
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