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Unformatted text preview: C H A PT E R 1 Circuit Terminology Chapter Contents 1-1 1-2 1-3 1-4 1-5 Cell-Phone 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 Cell-Phone 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. 1-1), 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 down-converters, and circuits with many other types of functions (Fig. 1-2). 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 audio-frequency 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 Down-Converter RF Filter Transmit Path Figure 1-1: 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 Front-End IF Block Figure 1-2: Cell-phone block diagram. Back-End Baseband 1-1 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. 1-3). 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 1-1: What are the differences between a device, a circuit, and a system? Review Question 1-2: What is the difference between analysis and synthesis? 1-1 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 1-3: 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 all-electronic 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. 1-1 1837 HISTORICAL TIMELINE 1888 Nikola Tesla (Croatian-American) 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 X-rays. One of his first X-ray ¨ 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 (Scottish-American) 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 charge-to-mass 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 long-distance phone service, radio, and television. Robert Watson Watt (Scottish) invents radar. John Mauchly and J. Presper Eckert (both American) develop the ENIAC, the first all-electronic 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 (Russian-American) 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 1-1 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 (Indian-American) demonstrates optical fiber as a low-loss, light-transmission 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 PDP-1, which was followed with the PDP-8 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 word-processor system, the mouse pointing device, and the use of a Windows-like 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 CD-ROM, as the first system capable of digital-to-optical recording and playback. 1971 Texas Instruments introduces the pocket calculator. CIRCUIT TERMINOLOGY Palm Pilot becomes widely available. 1997 The 17,500-mile fiber-optic cable extending from England to Japan is operational. 2002 Cell phones support video and the Internet. 2007 The power-efficient 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 (Indian-American) 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 Berners-Lee (British) invents the World Wide Web by introducing a networking hypertext system. Intel introduces the 4004 four-bit 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 MS-DOS computer disk operating system. Microsoft Windows is marketed in 1985. 1-2 1-2 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 1-1, 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 1-2. 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 1-2: 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 time-varying quantity • An uppercase letter if the quantity is not time varying; thus: I Table 1-1: 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 1-1: 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 1-2: 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 1-3: 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. TF1-1). Early tools (such as flint, stone, and metal hunting gear) were on the order of tens of centimeters. Over time, we began to build ever-smaller and ever-larger tools. The pyramids of Giza (ca., 2600 BCE) are 100-m tall; the largest modern construction crane is the K10,000 Kroll Giant Crane at 100-m long and 82-m tall; and the current (2007) tallest man-made structure is the KVLY-TV 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! FigureTF1-1: The scale of natural and man-made 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 radio-frequency communications, two foundations of 21st-century 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 general-purpose 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 10-ft 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 10-digit 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 1-MHz 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 TF1-2: 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 high-power 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 micro-scale techniques (ranging from microelectromechanical systems—or MEMS—to micro-chemical methods) has been developed which, useful onto themselves, are also enabling the construction of newer, nano-scale 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 ever-increasing interdisciplinary innovations that the term nanotechnology rightfully belongs. Second, the rate at which these innovations are occurring seems to be increasing exponentially! Consider Fig.TB1-2 and TB1-3 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 TF1-3: 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.) 1-3 ELECTRIC CHARGE AND CURRENT 1-3 Electric Charge and Current 1-3.1 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. 1-4 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. 1-4: 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 free-flow. 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 1-4: 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. 1-5. The circuit consists of an 8-volt battery and a switch on one end, a resistor on the other end, and a 60-m-long two-wire transmission line in between. The wires are made of copper, and they have a circular cross section with a 2-mm 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. 1-5, 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. 1-3.2 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. 1-6, 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. 1-7(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. 1-7(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. 1-8(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. 1-8(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 1-5: After closing the switch, it takes only 0.2 μs to observe a current in the resistor. − − − i − − − − Electron i Current direction Figure 1-6: Direction of (positive) current flow through a conductor is opposite that of electrons. 1-3 ELECTRIC CHARGE AND CURRENT 5A Circuit (a) −5 A Circuit (b) 15 rises and decays (Fig. 1-8(c) and (d)), exponentially damped oscillations (Fig. 1-8(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 1-1: Charge Transfer Figure 1-7: 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. 1-9(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 1-8: Graphical illustrations of various types of current variations with time. Figure 1-9: 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 1-2: 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. 1-9(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 1-3: 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 1-4: 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 1-5: How does electron drift lead to the conduction of electric current? 1-4 VOLTAGE AND POWER 17 Exercise 1-4: 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 1-5: 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) 1-4 Voltage and Power 1-4.1 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. 1-10. 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. 1-10 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. 1-11 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 1-10: 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 1-11: 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 1-12: 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 time-varying voltage is called an ac voltage. Figure 1-13: 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. 1-13. 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. 1-12 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 8-6, 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. 1-13, 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. 1-14 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. 1-14. 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. 1-15(a) is known as a single-pole single-throw (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. 1-15(a)) refers to the time t0 at which it opens or closes. Open circuit 1 V + _ 2 R1 Short circuit 3 4 R2 Figure 1-14: Open circuit between terminals 1 and 2, and short circuit between terminals 3 and 4. 1-4 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 1-15: (a) Single-pole single-throw (SPST) and (b) single-pole double-throw (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 single-pole doublethrow (SPDT) switch illustrated in Fig. 1-15(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. 1-4.2 + − Power The circuit shown in Fig. 1-16(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. 1-16(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 1-16: Current flow through a resistor (light-bulb 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. 1-17). 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 1-17: Passive sign convention. 20 TECHNOLOGY BRIEF 2: MOORE’S LAW AND SCALING Technology Brief 2: Moore’s Law and Scaling In 1965, Gordon Moore, co-founder of Intel, predicted that the number of transistors in the minimum-cost 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. TF2-1. In order to understand Moore’s Law, we have to understand the basics about how transistors work. As we will see later in Section 3-7, 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 sub-circuit in the processor that distributes power may be built from larger transistors than, say, the sub-circuit 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 10-μm 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 Dual-Core 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 TF2-1: 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 9-atom or an 11-atom 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 TF2-1 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 ever-smaller 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 TF2-2: 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 ampere-hours (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 kilowatt-hours (kWh), which represents the total amount of energy it can supply, namely, W = vi t (with t in hours). Example 1-3: Conservation of Power For each of the two circuits shown in Fig. 1-18, 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. 1-18(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 1-18: Circuits for Example 1-3. 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. 1-18(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. 1-5 CIRCUIT ELEMENTS 23 Example 1-4: Energy Consumption A resistor connected to a 100-V 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 1-6: 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 1-7: 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 1-4.1. The nomenclature and current– voltage relationships associated with the other two groups are the subject of this section. 1-5.1 ∞ p(t) dt = W= Answer: 1-5 (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 1-8: A car radio draws 0.5 A of dc current when connected to a 12-V 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. 1-19(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 1-5.3. 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 1-19: i –v relationships for (a) an ideal resistor, (b) ideal, independent current and voltage sources, and (c) a dependent, voltagecontrolled voltage source (VCVS). 1-5.2 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. 1-19(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 1-3, although for dc voltage sources the traditional “battery” symbol is used as well. A household electrical outlet connected through an electrical CIRCUIT TERMINOLOGY power-distribution 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 9-volt 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 5-V 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 1-3. 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. 1-5.3 Dependent Sources As alluded to in the opening paragraph of Section 1-5, 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. 1-20. 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. 1-20(b). The voltage v2 on the right-hand side of the circuit in Fig. 1-20(b) 1-5 CIRCUIT ELEMENTS 25 Table 1-3: 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 Voltage-Controlled Voltage Source (VCVS) + − vs = αvx Current-Controlled Voltage Source (CCVS) + − vs = rix Voltage-Controlled Current Source (VCCS) is = gvx Current-Controlled 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 left-hand 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 voltage-controlled voltage source (VCVS). Had the controlling quantity been a current source, the dependent source would have been called a current-controlled voltage source (CCVS) instead. A parallel analogy exists for voltage-controlled and current-controlled current sources. The characteristic symbol for a dependent source is the diamond (Table 1-3). Proportionality constant α in Table 1-3 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 Op-amp circuit vs + _ vi (b) Ri = 3 MΩ + _ v2 = Avi vo Equivalent circuit with dependent source Figure 1-20: 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. 1-19(c) for the VCVS. Example 1-5: Dependent Source Find the magnitude of the voltage V1 of the dependent source in Fig. 1-22. What type of source is it? 5Ω 10 V 2Ω + − I1 + _ V1 = 4I1 Solution: Since V1 depends on current I1 , it is a current-controlled voltage source with a coefficient of 4 V/A. The 10-V 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 1-4 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 1-21: Circuit for Example 1-5. I1 = 1-5.4 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 1-5 CIRCUIT ELEMENTS 27 Table 1-4: Passive circuit elements and their symbols. Element Symbol i –v Relationship Solution: See Fig. 1-23. 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 time-varying circuits. The resistor is used in both dc and time-varying 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 1-6: Switches The circuit in Fig. 1-22 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 1-23: Solutions for circuit in Fig. 1-22. R1 V0 t=0 + Review Question 1-6: 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 1-7: 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 1-8: What is an “equivalent-circuit” Figure 1-22: Circuit for Example 1-6. model? How is it used? 28 CHAPTER 1 Exercise 1-9: Find Ix from the diagram in Fig. E1.9. CIRCUIT TERMINOLOGY Exercise 1-10: 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 ampere-hours analysis conduction current cumulative charge dc dependent source design electric charge electric circuit electric current electron drift equivalent circuit independent source i –v characteristic kilowatt-hours linear response open circuit passive device passive sign convention polarization PROBLEMS Sections 1-2 and 1-3: 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 z-direction in 4 seconds, and simultaneously 1.5 × 1020 protons go through the same cross section in the negative z-direction, 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 1-4 and 1-5: 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 9-V 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 ampere-hours? + 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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