Electrical Engineering Principles and Applications PDF

2 Chapter 1 Introduction 1.1 OVERVIEW OF ELECTRICAL ENGINEERING Electrical engineers design systems that have two main objectives: 1. To gather, store, process, transport, and present infor...

2 Chapter 1 Introduction 1.1 OVERVIEW OF ELECTRICAL ENGINEERING Electrical engineers design systems that have two main objectives: 1. To gather, store, process, transport, and present information. 2. To distribute, store, and convert energy between various forms. In many electrical systems, the manipulation of energy and the manipulation of information are interdependent. For example, numerous aspects of electrical engineering relating to information are applied in weather prediction. Data about cloud cover, precipitation, wind speed, and so on are gathered electronically by weather satellites, by land-based radar sta- tions, and by sensors at numerous weather stations. (Sensors are devices that convert physical measurements to electrical signals.) This information is transported by elec- tronic communication systems and processed by computers to yield forecasts that are disseminated and displayed electronically. In electrical power plants, energy is converted from various sources to electrical form. Electrical distribution systems transport the energy to virtually every factory, home, and business in the world, where it is converted to a multitude of useful forms, such as mechanical energy, heat, and light. No doubt you can list scores of electrical engineering applications in your daily life. Increasingly, electrical and electronic features are integrated into new products. Automobiles and trucks provide just one example of this trend. The electronic content of the average automobile is growing rapidly in value. Auto designers realize that electronic technology is a good way to provide increased functionality at lower cost. Table 1.1 shows some of the applications of electrical engineering in automobiles. You may find it interesting to As another example, we note that many common household appliances contain search the web for sites keypads for operator control, sensors, electronic displays, and computer chips, as related to “mechatronics.” well as more conventional switches, heating elements, and motors. Electronics have become so intimately integrated with mechanical systems that the name mechatronics is used for the combination. Subdivisions of Electrical Engineering Next, we give you an overall picture of electrical engineering by listing and briefly discussing eight of its major areas. 1. Communication systems transport information in electrical form. Cellular phone, radio, satellite television, and the Internet are examples of communication systems. It is possible for virtually any two people (or computers) on the globe to communicate almost instantaneously. A climber on a mountaintop in Nepal can call or send e-mail to friends whether they are hiking in Alaska or sitting in a New York City office. This kind of connectivity affects the way we live, the way we conduct business, and the design of everything we use. For example, communication systems will change the design of highways because traffic and road-condition information collected by roadside sensors can be transmitted to central locations and used to route traffic. When an accident occurs, an electrical signal can be emitted automatically when the airbags deploy, giving the exact location of the vehicle, summoning help, and notifying traffic-control computers. Computers that are part of 2. Computer systems process and store information in digital form. No doubt products such as appliances you have already encountered computer applications in your own field. Besides the and automobiles are called embedded computers. computers of which you are aware, there are many in unobvious places, such as house- hold appliances and automobiles. A typical modern automobile contains several  Section 1.1 Overview of Electrical Engineering 3 Table 1.1. Current and Emerging Electronic/Electrical Applications in Automobiles and Trucks Safety Antiskid brakes Inflatable restraints Collision warning and avoidance Blind-zone vehicle detection (especially for large trucks) Infrared night vision systems Heads-up displays Automatic accident notification Rear-view cameras Communications and entertainment AM/FM radio Digital audio broadcasting CD/DVD player Cellular phone Computer/e-mail Satellite radio Convenience Electronic GPS navigation Personalized seat/mirror/radio settings Electronic door locks Emissions, performance, and fuel economy Vehicle instrumentation Electronic ignition Tire inflation sensors Computerized performance evaluation and maintenance scheduling Adaptable suspension systems Alternative propulsion systems Electric vehicles Advanced batteries Hybrid vehicles dozen special-purpose computers. Chemical processes and railroad switching yards are routinely controlled through computers. 3. Control systems gather information with sensors and use electrical energy to control a physical process. A relatively simple control system is the heating/cooling system in a residence. A sensor (thermostat) compares the temperature with the desired value. Control circuits operate the furnace or air conditioner to achieve the desired temperature. In rolling sheet steel, an electrical control system is used to obtain the desired sheet thickness. If the sheet is too thick (or thin), more (or less) force is applied to the rollers. The temperatures and flow rates in chemical processes are controlled in a similar manner. Control systems have even been installed in tall buildings to reduce their movement due to wind. 4. Electromagnetics is the study and application of electric and magnetic fields. The device (known as a magnetron) used to produce microwave energy in an oven is one application. Similar devices, but with much higher power levels, are employed in manufacturing sheets of plywood. Electromagnetic fields heat the glue between 4 Chapter 1 Introduction layers of wood so that it will set quickly. Cellular phone and television antennas are also examples of electromagnetic devices. 5. Electronics is the study and application of materials, devices, and circuits used in amplifying and switching electrical signals. The most important electronic devices are transistors of various kinds. They are used in nearly all places where electrical information or energy is employed. For example, the cardiac pacemaker is an elec- tronic circuit that senses heart beats, and if a beat does not occur when it should, applies a minute electrical stimulus to the heart, forcing a beat. Electronic instru- mentation and electrical sensors are found in every field of science and engineering. Many of the aspects of electronic amplifiers studied later in this book have direct application to the instrumentation used in your field of engineering. 6. Photonics is an exciting new field of science and engineering that promises Electronic devices are based to replace conventional computing, signal-processing, sensing, and communica- on controlling electrons. tion devices based on manipulating electrons with greatly improved products Photonic devices perform similar functions by based on manipulating photons. Photonics includes light generation by lasers and controlling photons. light-emitting diodes, transmission of light through optical components, as well as switching, modulation, amplification, detection, and steering light by electrical, acoustical, and photon-based devices. Current applications include readers for DVD disks, holograms, optical signal processors, and fiber-optic communication systems. Future applications include optical computers, holographic memories, and medi- cal devices. Photonics offers tremendous opportunities for nearly all scientists and engineers. 7. Power systems convert energy to and from electrical form and transmit energy over long distances. These systems are composed of generators, transformers, distri- bution lines, motors, and other elements. Mechanical engineers often utilize electrical motors to empower their designs. The selection of a motor having the proper torque– speed characteristic for a given mechanical application is another example of how you can apply the information in this book. 8. Signal processing is concerned with information-bearing electrical signals. Often, the objective is to extract useful information from electrical signals derived from sensors. An application is machine vision for robots in manufacturing. Another application of signal processing is in controlling ignition systems of internal combus- tion engines. The timing of the ignition spark is critical in achieving good performance and low levels of pollutants. The optimum ignition point relative to crankshaft rota- tion depends on fuel quality, air temperature, throttle setting, engine speed, and other factors. If the ignition point is advanced slightly beyond the point of best performance, engine knock occurs. Knock can be heard as a sharp metallic noise that is caused by rapid pressure fluctuations during the spontaneous release of chemical energy in the combustion chamber. A combustion-chamber pressure pulse displaying knock is shown in Figure 1.1. At high levels, knock will destroy an engine in a very short time. Prior to the advent of practical signal-processing electronics for this application, engine timing needed to be adjusted for distinctly suboptimum performance to avoid knock under varying combinations of operating conditions. By connecting a sensor through a tube to the combustion chamber, an electrical signal proportional to pressure is obtained. Electronic circuits process this signal to determine whether the rapid pressure fluctuations characteristic of knock are present. Then electronic circuits continuously adjust ignition timing for optimum performance while avoiding knock.  Section 1.1 Overview of Electrical Engineering 5 Pressure (psi) 800 Knock 600 400 Figure 1.1 Pressure versus time for an internal combustion engine experiencing knock. Sensors convert 200 pressure to an electrical signal that is processed to adjust ignition timing for minimum pollution and good t (ms) performance. 1 2 3 4 5 6 7 8 Why You Need to Study Electrical Engineering As a reader of this book, you may be majoring in another field of engineering or sci- ence and taking a required course in electrical engineering. Your immediate objective is probably to meet the course requirements for a degree in your chosen field. How- ever, there are several other good reasons to learn and retain some basic knowledge of electrical engineering: 1. To pass the Fundamentals of Engineering (FE) Examination as a first step in becoming a Registered Professional Engineer. In the United States, before per- forming engineering services for the public, you will need to become registered as a Professional Engineer (PE). This book gives you the knowledge to answer questions relating to electrical engineering on the registration examinations. Save this book Save this book and course and course notes to review for the FE examination. (See Appendix C for more on notes to review for the FE exam. the FE exam.) 2. To have a broad enough knowledge base so that you can lead design projects in your own field. Increasingly, electrical engineering is interwoven with nearly all scientific experiments and design projects in other fields of engineering. Industry has repeatedly called for engineers who can see the big picture and work effectively in teams. Engineers or scientists who narrow their focus strictly to their own field are destined to be directed by others. (Electrical engineers are somewhat fortunate in this respect because the basics of structures, mechanisms, and chemical processes are familiar from everyday life. On the other hand, electrical engineering concepts are somewhat more abstract and hidden from the casual observer.) 3. To be able to operate and maintain electrical systems, such as those found in control systems for manufacturing processes. The vast majority of electrical-circuit malfunctions can be readily solved by the application of basic electrical-engineering principles. You will be a much more versatile and valuable engineer or scientist if you can apply electrical-engineering principles in practical situations. 4. To be able to communicate with electrical-engineering consultants. Very likely, you will often need to work closely with electrical engineers in your career. This book will give you the basic knowledge needed to communicate effectively. 6 Chapter 1 Introduction Content of This Book Electrical engineering is too vast to cover in one or two courses. Our objective is to introduce the underlying concepts that you are most likely to need. Circuit theory Circuit theory is the electrical is the electrical engineer’s fundamental tool. That is why the first six chapters of this engineer’s fundamental tool. book are devoted to circuits. Embedded computers, sensors, and electronic circuits will be an increasingly important part of the products you design and the instrumentation you use as an engineer or scientist. Chapters 7, 8, and 9 treat digital systems with emphasis on embedded computers and instrumentation. Chapters 10 through 14 deal with electronic devices and circuits. As a mechanical, chemical, civil, industrial, or other engineer, you will very likely need to employ energy-conversion devices. The last three chapters relate to electrical energy systems treating transformers, generators, and motors. Because this book covers many basic concepts, it is also sometimes used in intro- ductory courses for electrical engineers. Just as it is important for other engineers and scientists to see how electrical engineering can be applied to their fields, it is equally important for electrical engineers to be familiar with these applications. 1.2 CIRCUITS, CURRENTS, AND VOLTAGES Overview of an Electrical Circuit Before we carefully define the terminology of electrical circuits, let us gain some basic understanding by considering a simple example: the headlight circuit of an automobile. This circuit consists of a battery, a switch, the headlamps, and wires connecting them in a closed path, as illustrated in Figure 1.2. The battery voltage is a Chemical forces in the battery cause electrical charge (electrons) to flow through measure of the energy gained the circuit. The charge gains energy from the chemicals in the battery and delivers by a unit of charge as it moves through the battery. energy to the headlamps. The battery voltage (nominally, 12 volts) is a measure of the energy gained by a unit of charge as it moves through the battery. The wires are made of an excellent electrical conductor (copper) and are insu- lated from one another (and from the metal auto body) by electrical insulation Electrons readily move (plastic) coating the wires. Electrons readily move through copper but not through through copper but not the plastic insulation. Thus, the charge flow (electrical current) is confined to the through plastic insulation. wires until it reaches the headlamps. Air is also an insulator. The switch is used to control the flow of current. When the conducting metal- lic parts of the switch make contact, we say that the switch is closed and current flows through the circuit. On the other hand, when the conducting parts of the Electrons experience collisions switch do not make contact, we say that the switch is open and current does with the atoms of the not flow. tungsten wires, resulting in heating of the tungsten. The headlamps contain special tungsten wires that can withstand high temper- atures. Tungsten is not as good an electrical conductor as copper, and the electrons experience collisions with the atoms of the tungsten wires, resulting in heating of the tungsten. We say that the tungsten wires have electrical resistance. Thus, energy is transferred by the chemical action in the battery to the electrons and then to the Energy is transferred by the tungsten, where it appears as heat. The tungsten becomes hot enough so that copi- chemical action in the battery ous light is emitted. We will see that the power transferred is equal to the product to the electrons and then to the tungsten. of current (rate of flow of charge) and the voltage (also called electrical potential) applied by the battery.  Section 1.2 Circuits, Currents, and Voltages 7 Switch Battery Wires Headlamps (a) Physical configuration Conductors representing wires Switch + 12 V − Resistances Voltage source representing representing battery headlamps (b) Circuit diagram Figure 1.2 The headlight circuit. (a) The actual physical layout of the circuit. (b) The circuit diagram. (Actually, the simple description of the headlight circuit we have given is most appropriate for older cars. In more modern automobiles, sensors provide information to an embedded computer about the ambient light level, whether or not the ignition is energized, and whether the transmission is in park or drive. The dashboard switch merely inputs a logic level to the computer, indicating the intention of the operator with regard to the headlights. Depending on these inputs, the computer controls the state of an electronic switch in the headlight circuit. When the ignition is turned off and if it is dark, the computer keeps the lights on for a few minutes so the passengers can see to exit and then turns them off to conserve energy in the battery. This is typical of the trend to use highly sophisticated electronic and computer technology to enhance the capabilities of new designs in all fields of engineering.) Fluid-Flow Analogy Electrical circuits are analogous to fluid-flow systems. The battery is analogous to a pump, and charge is analogous to the fluid. Conductors (usually copper wires) correspond to frictionless pipes through which the fluid flows. Electrical current is The fluid-flow analogy can the counterpart of the flow rate of the fluid. Voltage corresponds to the pressure be very helpful initially in differential between points in the fluid circuit. Switches are analogous to valves. understanding electrical Finally, the electrical resistance of a tungsten headlamp is analogous to a constriction circuits. in a fluid system that results in turbulence and conversion of energy to heat. Notice that current is a measure of the flow of charge through the cross section of a circuit element, whereas voltage is measured across the ends of a circuit element or between any other two points in a circuit. Now that we have gained a basic understanding of a simple electrical circuit, we will define the concepts and terminology more carefully. 8 Chapter 1 Introduction Resistances + − Inductance Voltage source Capacitance Conductors Figure 1.3 An electrical circuit consists of circuit elements, such as voltage sources, resistances, inductances, and capacitances, connected in closed paths by conductors. Electrical Circuits An electrical circuit consists of An electrical circuit consists of various types of circuit elements connected in closed various types of circuit paths by conductors. An example is illustrated in Figure 1.3. The circuit elements elements connected in closed paths by conductors. can be resistances, inductances, capacitances, and voltage sources, among others. The symbols for some of these elements are illustrated in the figure. Eventually, we will carefully discuss the characteristics of each type of element. Charge flows easily through Charge flows easily through conductors, which are represented by lines connect- conductors. ing circuit elements. Conductors correspond to connecting wires in physical circuits. Voltage sources create forces that cause charge to flow through the conductors and other circuit elements. As a result, energy is transferred between the circuit elements, resulting in a useful function. Electrical Current Current is the time rate of flow of electrical charge. Its Electrical current is the time rate of flow of electrical charge through a conductor units are amperes (A), which or circuit element. The units are amperes (A), which are equivalent to coulombs per are equivalent to coulombs per second (C/s). second (C/s). (The charge on an electron is −1.602 × 10−19 C.) Conceptually, to find the current for a given circuit element, we first select a cross Reference direction section of the circuit element roughly perpendicular to the flow of current. Then, we select a reference direction along the direction of flow. Thus, the reference direction points from one side of the cross section to the other. This is illustrated in Figure 1.4. Next, suppose that we keep a record of the net charge flow through the cross Cross section section. Positive charge crossing in the reference direction is counted as a positive Conductor or circuit element contribution to net charge. Positive charge crossing opposite to the reference is counted as a negative contribution. Furthermore, negative charge crossing in the ref- Figure 1.4 Current is the time rate of charge flow erence direction is counted as a negative contribution, and negative charge against through a cross section of a the reference direction is a positive contribution to charge. conductor or circuit Thus, in concept, we obtain a record of the net charge in coulombs as a function element. of time in seconds denoted as q(t). The electrical current flowing through the element in the reference direction is given by Colored shading is used to indicate key equations dq(t) i(t) = (1.1) throughout this book. dt A constant current of one ampere means that one coulomb of charge passes through the cross section each second.  Section 1.2 Circuits, Currents, and Voltages 9 To find charge given current, we must integrate. Thus, we have  t q(t) = i(t) dt + q(t0 ) (1.2) t0 in which t0 is some initial time at which the charge is known. (Throughout this book, we assume that time t is in seconds unless stated otherwise.) Current flow is the same for all cross sections of a circuit element. (We reexamine this statement when we introduce the capacitor in Chapter 3.) The current that enters one end flows through the element and exits through the other end. Example 1.1 Determining Current Given Charge Suppose that charge versus time for a given circuit element is given by q(t) = 0 for t < 0 and q(t) = 2 − 2e−100t C for t > 0 Sketch q(t) and i(t) to scale versus time. Solution First we use Equation 1.1 to find an expression for the current: dq(t) i(t) = dt =0 for t < 0 = 200e−100t A for t > 0 Plots of q(t) and i(t) are shown in Figure 1.5. Reference Directions In analyzing electrical circuits, we may not initially know the actual direction of current flow in a particular circuit element. Therefore, we start by assigning current q (t) (C) i (t) (A) 2.0 200 1.0 100 0 t (ms) 0 t (ms) 0 10 20 30 40 0 10 20 30 40 Figure 1.5 Plots of charge and current versus time for Example 1.1. Note: The time scale is in milliseconds (ms). One millisecond is equivalent to 10−3 seconds. 10 Chapter 1 Introduction B D A C E Figure 1.6 In analyzing circuits, we frequently start by assigning current i2 i3 i1 variables i1 , i2 , i3 , and so forth. variables and arbitrarily selecting a reference direction for each current of interest. It is customary to use the letter i for currents and subscripts to distinguish different currents. This is illustrated by the example in Figure 1.6, in which the boxes labeled A, B, and so on represent circuit elements. After we solve for the current values, we may find that some currents have negative values. For example, suppose that i1 = −2 A in the circuit of Figure 1.6. Because i1 has a negative value, we know that current actually flows in the direction opposite to the reference initially selected for i1. Thus, the actual current is 2 A flowing downward through element A. Direct Current and Alternating Current Dc currents are constant with When a current is constant with time, we say that we have direct current, abbreviated respect to time, whereas ac as dc. On the other hand, a current that varies with time, reversing direction peri- currents vary with time. odically, is called alternating current, abbreviated as ac. Figure 1.7 shows the values of a dc current and a sinusoidal ac current versus time. When ib (t) takes a negative value, the actual current direction is opposite to the reference direction for ib (t). The designation ac is used for other types of time-varying currents, such as the triangular and square waveforms shown in Figure 1.8. ia (t) ib (t) = 2 cos 2pt (A) (A) 2 t (s) t (s) 0.5 1.0 (a) Dc current (b) Ac current Figure 1.7 Examples of dc and ac currents versus time. Double-Subscript Notation for Currents So far we have used arrows alongside circuit elements or conductors to indicate reference directions for currents. Another way to indicate the current and refer- ence direction for a circuit element is to label the ends of the element and use double subscripts to define the reference direction for the current. For example, consider the resistance of Figure 1.9. The current denoted by iab is the current through the element with its reference direction pointing from a to b. Similarly, iba is the current with its reference directed from b to a. Of course, iab and iba are the same in magnitude and  Section 1.2 Circuits, Currents, and Voltages 11 it (t) is (t) t t a (a) Triangular waveform (b) Square waveform iab iba Figure 1.8 Ac currents can have various waveforms. b Figure 1.9 Reference opposite in sign, because they denote the same current but with opposite reference directions can be indicated directions. Thus, we have by labeling the ends of iab = −iba circuit elements and using double subscripts on current variables. The reference Exercise 1.1 A constant current of 2 A flows through a circuit element. In 10 seconds direction for iab points from a to b. On the other hand, (s), how much net charge passes through the element? the reference direction for Answer 20 C.  iba points from b to a. Exercise 1.2 The charge that passes through a circuit element is given by q(t) = 0.01 sin(200t) C, in which the angle is in radians. Find the current as a function of time. Answer i(t) = 2 cos(200t) A.  Exercise 1.3 In Figure 1.6, suppose that i2 = 1 A and i3 = −3 A. Assuming that the current consists of positive charge, in which direction (upward or downward) is charge moving in element C? In element E? Answer Downward in element C and upward in element E.  Voltages When charge moves through circuit elements, energy can be transferred. In the case of automobile headlights, stored chemical energy is supplied by the battery and Voltage is a measure of the absorbed by the headlights where it appears as heat and light. The voltage associated energy transferred per unit of charge when charge moves with a circuit element is the energy transferred per unit of charge that flows through from one point in an electrical the element. The units of voltage are volts (V), which are equivalent to joules per circuit to a second point. coulomb (J/C). For example, consider the storage battery in an automobile. The voltage across its terminals is (nominally) 12 V. This means that 12 J are transferred to or from the Notice that voltage is battery for each coulomb that flows through it. When charge flows in one direction, measured across the ends of a circuit element, whereas energy is supplied by the battery, appearing elsewhere in the circuit as heat or light current is a measure of charge or perhaps as mechanical energy at the starter motor. If charge moves through the flow through the element. battery in the opposite direction, energy is absorbed by the battery, where it appears as stored chemical energy. Voltages are assigned polarities that indicate the direction of energy flow. If positive charge moves from the positive polarity through the element toward the negative polarity, the element absorbs energy that appears as heat, mechanical energy, stored chemical energy, or as some other form. On the other hand, if positive charge moves from the negative polarity toward the positive polarity, the element supplies energy. This is illustrated in Figure 1.10. For negative charge, the direction of energy transfer is reversed. 12 Chapter 1 Introduction + ⊕ Energy supplied Energy absorbed by the element by the element Figure 1.10 Energy is transferred ⊕ when charge flows through an element having a voltage across it. − + v2 − − v4 + Figure 1.11 If we do not know the voltage values and polarities in a 2 4 + + + circuit, we can start by assigning voltage variables choosing the v1 v3 v5 1 3 5 reference polarities arbitrarily. (The boxes represent unspecified circuit − − − elements.) Reference Polarities When we begin to analyze a circuit, we often do not know the actual polarities of some of the voltages of interest in the circuit. Then, we simply assign voltage variables choosing reference polarities arbitrarily. (Of course, the actual polarities are not arbitrary.) This is illustrated in Figure 1.11. Next, we apply circuit principles (discussed later), obtaining equations that are solved for the voltages. If a given voltage has an actual polarity opposite to our arbitrary choice for the reference polarity, we obtain a negative value for the voltage. For example, if we find that In circuit analysis, we v3 = −5 V in Figure 1.11, we know that the voltage across element 3 is 5 V in frequently assign reference magnitude and its actual polarity is opposite to that shown in the figure (i.e., the polarities for voltages arbitrarily. If we find at the actual polarity is positive at the bottom end of element 3 and negative at the top). end of the analysis that the We usually do not put much effort into trying to assign “correct” references value of a voltage is negative, for current directions or voltage polarities. If we have doubt about them, we make then we know that the true arbitrary choices and use circuit analysis to determine true directions and polarities polarity is opposite of the (as well as the magnitudes of the currents and voltages). polarity selected initially. Voltages can be constant with time or they can vary. Constant voltages are called dc voltages. On the other hand, voltages that change in magnitude and alternate in polarity with time are said to be ac voltages. For example, v1 (t) = 10 V is a dc voltage. It has the same magnitude and polarity for all time. On the other hand, v2 (t) = 10 cos(200π t)V + a − is an ac voltage that varies in magnitude and polarity. When v2 (t) assumes a negative value, the actual polarity is opposite the reference polarity. (We study sinusoidal ac currents and voltages in Chapter 5.) vab vba Double-Subscript Notation for Voltages − b + Another way to indicate the reference polarity of a voltage is to use double subscripts Figure 1.12 The voltage vab has a reference polarity that on the voltage variable. We use letters or numbers to label the terminals between is positive at point a and which the voltage appears, as illustrated in Figure 1.12. For the resistance shown in the negative at point b. figure, vab represents the voltage between points a and b with the positive reference  Section 1.3 Power and Energy 13 at point a. The two subscripts identify the points between which the voltage appears, and the first subscript is the positive reference. Similarly, vba is the voltage between a and b with the positive reference at point b. Thus, we can write vab = −vba (1.3) because vba has the same magnitude as vab but has opposite polarity. Still another way to indicate a voltage and its reference polarity is to use an arrow, as shown in Figure 1.13. The positive reference corresponds to the head of the arrow. v Figure 1.13 The positive Switches reference for v is at the head of the arrow. Switches control the currents in circuits. When an ideal switch is open, the current through it is zero and the voltage across it is determined by the remainder of the circuit. When an ideal switch is closed, the voltage across it is zero and the current through it is determined by the remainder of the circuit. Exercise 1.4 The voltage across a given circuit element is vab = 20 V. A positive charge of 2 C moves through the circuit element from terminal b to terminal a. How much energy is transferred? Is the energy supplied by the circuit element or absorbed by it? Answer 40 J are supplied by the circuit element.  1.3 POWER AND ENERGY Consider the circuit element shown in Figure 1.14. Because the current i is the rate i of flow of charge and the voltage v is a measure of the energy transferred per unit of + charge, the product of the current and the voltage is the rate of energy transfer. In other words, the product of current and voltage is power: v p = vi (1.4) − Figure 1.14 When current The physical units of the quantities on the right-hand side of this equation are flows through an element and voltage appears across volts × amperes = the element, energy is transferred. The rate of (joules/coulomb) × (coulombs/second) = energy transfer is p = vi. joules/second = watts Passive Reference Configuration Now we may ask whether the power calculated by Equation 1.4 represents energy supplied by or absorbed by the element. Refer to Figure 1.14 and notice that the current reference enters the positive polarity of the voltage. We call this arrange- ment the passive reference configuration. Provided that the references are picked in this manner, a positive result for the power calculation implies that energy is being absorbed by the element. On the other hand, a negative result means that the element is supplying energy to other parts of the circuit. 14 Chapter 1 Introduction If the current reference enters the negative end of the reference polarity, we compute the power as p = −vi (1.5) Then, as before, a positive value for p indicates that energy is absorbed by the element, and a negative value shows that energy is supplied by the element. If the circuit element happens to be an electrochemical battery, positive power means that the battery is being charged. In other words, the energy absorbed by the battery is being stored as chemical energy. On the other hand, negative power indicates that the battery is being discharged. Then the energy supplied by the battery is delivered to some other element in the circuit. Sometimes currents, voltages, and powers are functions of time. To emphasize this fact, we can write Equation 1.4 as p(t) = v(t)i(t) (1.6) Example 1.2 Power Calculations Consider the circuit elements shown in Figure 1.15. Calculate the power for each element. If each element is a battery, is it being charged or discharged? Solution In element A, the current reference enters the positive reference polarity. This is the passive reference configuration. Thus, power is computed as pa = va ia = 12 V × 2 A = 24 W Because the power is positive, energy is absorbed by the device. If it is a battery, it is being charged. In element B, the current reference enters the negative reference polarity. (Recall that the current that enters one end of a circuit element must exit from the other end, and vice versa.) This is opposite to the passive reference configuration. Hence, power is computed as pb = −vb ib = −(12 V) × 1 A = −12 W Since the power is negative, energy is supplied by the device. If it is a battery, it is being discharged. ia ib ic + + − va A vb B vc C − − + va = 12 V vb = 12 V vc = 12 V ia = 2 A ib = 1 A ic = −3 A (a) (b) (c) Figure 1.15 Circuit elements for Example 1.2.  Section 1.3 Power and Energy 15 In element C, the current reference enters the positive reference polarity. This is the passive reference configuration. Thus, we compute power as pc = vc ic = 12 V × (−3 A) = −36 W Since the result is negative, energy is supplied by the element. If it is a battery, it is being discharged. (Notice that since ic takes a negative value, current actually flows downward through element C.) Energy Calculations To calculate the energy w delivered to a circuit element between time instants t1 and t2 , we integrate power:  t2 w= p(t) dt (1.7) t1 Here we have explicitly indicated that power can be a function of time by using the notation p(t). Example 1.3 Energy Calculation Find an expression for the power for the voltage source shown in Figure 1.16. Compute the energy for the interval from t1 = 0 to t2 = ∞. Solution The current reference enters the positive reference polarity. Thus, we compute power as v(t) + − p(t) = v(t)i(t) i(t) = 12 × 2e−t v(t) = 12 V i(t) = 2e−t A = 24e−t W Figure 1.16 Circuit element for Example 1.3. Subsequently, the energy transferred is given by  ∞ w= p(t) dt 0  ∞ = 24e−t dt 0  ∞ = −24e−t 0 = −24e−∞ − (−24e0 ) = 24 J Because the energy is positive, it is absorbed by the source. Prefixes In electrical engineering, we encounter a tremendous range of values for currents, voltages, powers, and other quantities. We use the prefixes shown in Table 1.2 when working with very large or small quantities. For example, 1 milliampere (1 mA) is equivalent to 10−3 A, 1 kilovolt (1 kV) is equivalent to 1000 V, and so on. 16 Chapter 1 Introduction Table 1.2. Prefixes Used for Large or Small Physical Quantities Prefix Abbreviation Scale Factor........................................................................................... giga- G 109 meg- or mega- M 106 kilo- k 103 milli- m 10−3 micro- μ 10−6 nano- n 10−9 pico- p 10−12 femto- f 10−15 ia (t) ib (t) + + va (t) A vb (t) B − − ia (t) = 2t ib (t) = 10 va (t) = 10t vb (t) = 20 − 2t Figure 1.17 See Exercise 1.6. (a) (b) Exercise 1.5 The ends of a circuit element are labeled a and b, respectively. Are the references for iab and vab related by the passive reference configuration? Explain. Answer The reference direction for iab enters terminal a, which is also the posi- tive reference for vab. Therefore, the current reference direction enters the positive reference polarity, so we have the passive reference configuration.  Exercise 1.6 Compute the power as a function of time for each of the elements shown in Figure 1.17. Find the energy transferred between t1 = 0 and t2 = 10 s. In each case is energy supplied or absorbed by the element? Answer a. pa (t) = 20t 2 W, wa = 6667 J; since wa is positive, energy is absorbed by element A. b. pb (t) = 20t − 200 W, wb = −1000 J; since wb is negative, energy is supplied by element B.  1.4 KIRCHHOFF’S CURRENT LAW Kirchhoff’s current law states A node in an electrical circuit is a point at which two or more circuit elements are that the net current entering joined together. Examples of nodes are shown in Figure 1.18. a node is zero. An important principle of electrical circuits is Kirchhoff’s current law: The net current entering a node is zero. To compute the net current entering a node, we add the currents entering and subtract the currents leaving. For illustration, consider the nodes of Figure 1.18. Then, we can write: Node a : i1 + i2 − i3 = 0 Node b : i3 − i4 = 0 Node c : i5 + i6 + i7 = 0  Section 1.4 Kirchhoff’s Current Law 17 Node a Node b Node c i1 i3 i3 i5 i6 i4 i2 i7 (a) (b) (c) Figure 1.18 Partial circuits showing one node each to illustrate Kirchhoff’s current law. Notice that for node b, Kirchhoff’s current law requires that i3 = i4. In general, if only two circuit elements are connected at a node, their currents must be equal. The current flows into the node through one element and out through the other. Usually, we will recognize this fact and assign a single current variable for both circuit elements. For node c, either all of the currents are zero or some are positive while others are negative. We abbreviate Kirchhoff’s current law as KCL. There are two other equivalent ways to state KCL. One way is: The net current leaving a node is zero. To compute the net current leaving a node, we add the currents leaving and subtract the currents entering. For the nodes of Figure 1.18, this yields the following: Node a : −i1 − i2 + i3 = 0 Node b : −i3 + i4 = 0 Node c : −i5 − i6 − i7 = 0 Of course, these equations are equivalent to those obtained earlier. Another way to state KCL is: The sum of the currents entering a node equals the An alternative way to state sum of the currents leaving a node. Applying this statement to Figure 1.18, we obtain Kirchhoff’s current law is that the sum of the currents the following set of equations: entering a node is equal to the sum of the currents Node a : i1 + i2 = i3 leaving a node. Node b : i3 = i4 Node c : i5 + i6 + i7 = 0 Again, these equations are equivalent to those obtained earlier. Physical Basis for Kirchhoff’s Current Law An appreciation of why KCL is true can be obtained by considering what would happen if it were violated. Suppose that we could have the situation shown in Figure 1.18(a), with i1 = 3 A, i2 = 2 A, and i3 = 4 A. Then, the net current entering the node would be i1 + i2 − i3 = 1 A = 1 C/s In this case, 1 C of charge would accumulate at the node during each second. After 1 s, we would have +1 C of charge at the node, and −1 C of charge somewhere else in the circuit. 18 Chapter 1 Introduction B ib ia Figure 1.19 Elements A, B, C, and D A D can be considered to be connected to a common node, because all points in id a circuit that are connected directly by C ic conductors are electrically equivalent to a single point. Suppose that these charges are separated by a distance of one meter (m). Recall that unlike charges experience a force of attraction. The resulting force turns out to be approximately 8.99 × 109 newtons (N) (equivalent to 2.02 × 109 pounds). Very large forces are generated when charges of this magnitude are separated by moderate distances. In effect, KCL states that such forces prevent charge from accumulating at the nodes of a circuit. All points in a circuit that All points in a circuit that are connected directly by conductors can be considered are connected directly by to be a single node. For example, in Figure 1.19, elements A, B, C, and D are connected conductors can be considered to be a single node. to a common node. Applying KCL, we can write ia + ic = ib + id Series Circuits We make frequent use of KCL in analyzing circuits. For example, consider the ele- ments A, B, and C shown in Figure 1.20. When elements are connected end to end, we ia say that they are connected in series. In order for elements A and B to be in series, no A 1 other path for current can be connected to the node joining A and B. Thus, all elements in a series circuit have identical currents. For example, writing Kirchhoff’s current law ib at node 1 for the circuit of Figure 1.20, we have B ia = ib C 2 ic At node 2, we have Figure 1.20 Elements A, ib = ic B, and C are connected in series. Thus, we have ia = ib = ic The current that enters a series circuit must flow through each element in the circuit. Exercise 1.7 Use KCL to determine the values of the unknown currents shown in Figure 1.21. Answer ia = 4 A, ib = −2 A, ic = −8 A.  Exercise 1.8 Consider the circuit of Figure 1.22. Identify the groups of circuit elements that are connected in series. Answer Elements A and B are in series; elements E, F, and G form another series combination.   Section 1.5 Kirchhoff’s Voltage Law 19 1A 1A 3A 1A 3A 3A 2A ia ib ic 4A (a) (b) (c) Figure 1.21 See Exercise 1.7. B E A C D F G Figure 1.22 Circuit for Exercise 1.8. 1.5 KIRCHHOFF’S VOLTAGE LAW A loop in an electrical circuit is a closed path starting at a node and proceed- Kirchhoff’s voltage law (KVL) ing through circuit elements, eventually returning to the starting node. Frequently, states that the algebraic sum of the voltages equals zero for several loops can be identified for a given circuit. For example, in Figure 1.22, one any closed path (loop) in an loop consists of the path starting at the top end of element A and proceeding clock- electrical circuit. wise through elements B and C, returning through A to the starting point. Another loop starts at the top of element D and proceeds clockwise through E, F, and G, returning to the start through D. Still another loop exists through elements A, B, E, F, and G around the periphery of the circuit. Kirchhoff’s voltage law (KVL) states: The algebraic sum of the voltages equals zero for any closed path (loop) in an electrical circuit. In traveling around a loop, we Moving from + to − encounter various voltages, some of which carry a positive sign while others carry a we add va. negative sign in the algebraic sum. A convenient convention is to use the first polarity mark encountered for each voltage to decide if it should be added or subtracted in + va − the algebraic sum. If we go through the voltage from the positive polarity reference to the negative reference, it carries a plus sign. If the polarity marks are encountered in the opposite direction (minus to plus), the voltage carries a negative sign. This is illustrated in Figure 1.23. Moving from − to + For the circuit of Figure 1.24, we obtain the following equations: we subtract va. Figure 1.23 In applying Loop 1 : −va + vb + vc = 0 KVL to a loop, voltages are added or subtracted Loop 2 : −vc − vd + ve = 0 depending on their reference polarities relative to the direction of travel Loop 3 : va − vb + vd − ve = 0 around the loop. Notice that va is subtracted for loop 1, but it is added for loop 3, because the direction of travel is different for the two loops. Similarly, vc is added for loop 1 and subtracted for loop 2. 20 Chapter 1 Introduction + vb − − vd + B D + + + va A Loop C vc Loop E ve 1 2 − − − Loop 3 Figure 1.24 Circuit used for illustration of Kirchhoff’s volt- age law. Kirchhoff’s Voltage Law Related to Conservation of Energy KVL is a consequence of the law of energy conservation. Consider the circuit shown in Figure 1.25. This circuit consists of three elements connected in series. Thus, the + va − same current i flows through all three elements. The power for each of the elements is given by A + + i Element A : pa = va i B i C vb i vc Element B : pb = −vb i − − Element C : pc = vc i Figure 1.25 In this circuit, conservation of energy Notice that the current and voltage references have the passive configuration (the requires that vb = va + vc. current reference enters the plus polarity mark) for elements A and C. For element B, the relationship is opposite to the passive reference configuration. That is why we have a negative sign in the calculation of pb. At a given instant, the sum of the powers for all of the elements in a circuit must be zero. Otherwise, for an increment of time taken at that instant, more energy would be absorbed than is supplied by the circuit elements (or vice versa): pa + pb + pc = 0 Substituting for the powers, we have va i − vb i + vc i = 0 Canceling the current i, we obtain va − vb + vc = 0 This is exactly the same equation that is obtained by adding the voltages around the loop and setting the sum to zero for a clockwise loop in the circuit of Figure 1.25. One way to check our results after solving for the currents and voltages in a Two circuit elements are circuit is the check to see that the power adds to zero for all of the elements. connected in parallel if both ends of one element are Parallel Circuits connected directly (i.e., by conductors) to corresponding We say that two circuit elements are connected in parallel if both ends of one element ends of the other. are connected directly (i.e., by conductors) to corresponding ends of the other. For  Section 1.5 Kirchhoff’s Voltage Law 21 C A B D E F Figure 1.26 In this circuit, elements A and B are in parallel. Elements D, E, and F form another parallel combination. example, in Figure 1.26, elements A and B are in parallel. Similarly, we say that the three circuit elements D, E, and F are in parallel. Element B is not in parallel with D because the top end of B is not directly connected to the top end of D. The voltages across parallel elements are equal in magnitude and have the same + + − polarity. For illustration, consider the partial circuit shown in Figure 1.27. Here ele- ments A, B, and C are connected in parallel. Consider a loop from the bottom end va A vb B C vc of A upward and then down through element B back to the bottom of A. For this − − + clockwise loop, we have −va + vb = 0. Thus, KVL requires that Figure 1.27 For this circuit, va = vb we can show that va = vb = −vc. Thus, the magnitudes Next, consider a clockwise loop through elements A and C. For this loop, KVL and actual polarities of all three voltages are the same. requires that −va − vc = 0 This implies that va = −vc. In other words, va and vc have opposite algebraic signs. Furthermore, one or the other of the two voltages must be negative (unless both are zero). Therefore, one of the voltages has an actual polarity opposite to the refer- + ence polarity shown in the figure. Thus, the actual polarities of the voltages are the same (either both are positive at the top of the circuit or both are positive at the v A B C bottom). Usually, when we have a parallel circuit, we simply use the same voltage variable − for all of the elements as illustrated in Figure 1.28. Figure 1.28 Analysis is simplified by using the Exercise 1.9 Use repeated application of KVL to find the values of vc and ve for same voltage variable and reference polarity for the circuit of Figure 1.29. elements that are in parallel. Answer vc = 8 V, ve = −2 V.  Exercise 1.10 Identify elements that are in parallel in Figure 1.29. Identify elements in series. Answer Elements E and F are in parallel; elements A and B are in series.  − 5V + − −10 V + B D + + + 3V A C vc E ve F − − − Figure 1.29 Circuit for Exercises 1.9 and 1.10. 22 Chapter 1 Introduction 1.6 INTRODUCTION TO CIRCUIT ELEMENTS In this section, we carefully define several types of ideal circuit elements: Conductors Voltage sources Current sources Resistors Later in the book, we will encounter additional elements, including inductors and capacitors. Eventually, we will be able to use these idealized circuit elements to describe (model) complex real-world electrical devices. Conductors We have already encountered conductors. Ideal conductors are represented in circuit diagrams by unbroken lines between the ends of other circuit elements. We define ideal circuit elements in terms of the relationship between the voltage across the The voltage between the ends element and the current through it. of an ideal conductor is zero The voltage between the ends of an ideal conductor is zero regardless of the current regardless of the current flowing through the flowing through the conductor. When two points in a circuit are connected together conductor. by an ideal conductor, we say that the points are shorted together. Another term for an ideal conductor is short circuit. All points in a circuit that are connected by ideal All points in a circuit that are conductors can be considered as a single node. connected by ideal If no conductors or other circuit elements are connected between two parts of conductors can be considered as a single node. a circuit, we say that an open circuit exists between the two parts of the circuit. No current can flow through an ideal open circuit. Independent Voltage Sources An ideal independent voltage An ideal independent voltage source maintains a specified voltage across its ter- source maintains a specified minals. The voltage across the source is independent of other elements that are voltage across its terminals. connected to it and of the current flowing through it. We use a circle enclosing the reference polarity marks to represent independent voltage sources. The value of the voltage is indicated alongside the symbol. The voltage can be constant or it can be a function of time. Several voltage sources are shown in Figure 1.30. In Figure 1.30(a), the voltage across the source is constant. Thus, we have a dc voltage source. On the other hand, the source shown in Figure 1.30(b) is an ac voltage source having a sinusoidal variation with time. We say that these are independent sources because the voltages across their terminals are independent of all other voltages and currents in the circuit. + + 12 V 5 cos (2pt) V − − Figure 1.30 Independent voltage (a) Constant or (b) Ac voltage sources. dc voltage source source  Section 1.6 Introduction to Circuit Elements 23 Ideal Circuit Elements versus Reality Here we are giving definitions of ideal circuit elements. It is possible to draw ideal + circuits in which the definitions of various circuit elements conflict. For example, Figure 1.31 shows a 12-V voltage source with a conductor connected across its termi- + 12 V vx − nals. In this case, the definition of the voltage source requires that vx = 12 V. On the other hand, the definition of an ideal conductor requires that vx = 0. In our study of − ideal circuits, we avoid such conflicts. Figure 1.31 We avoid self- In the real world, an automobile battery is nearly an ideal 12-V voltage source, contradictory circuit and a short piece of heavy-gauge copper wire is nearly an ideal conductor. If we place diagrams such as this one. the wire across the terminals of the battery, a very large current flows through the wire, stored chemical energy is converted to heat in the wire at a very high rate, and the wire will probably melt or the battery be destroyed. When we encounter a contradictory idealized circuit model, we often have an undesirable situation (such as a fire or destroyed components) in the real-world counterpart to the model. In any case, a contradictory circuit model implies that we have not been sufficiently careful in choosing circuit models for the real circuit elements. For example, an automobile battery is not exactly modeled as an ideal voltage source. We will see that a better model (particularly if the currents are very large) is an ideal voltage source in series with a resistance. (We will discuss resistance very soon.) A short piece of copper wire is not modeled well as an ideal conductor, in this case. Instead, we will see that it is modeled better as a small resistance. If we have done a good job at picking circuit models for real-world circuits, we will not encounter contradictory circuits, and the results we calculate using the model will match reality very well. Dependent Voltage Sources A dependent or controlled voltage source is similar to an independent source except that the voltage across the source terminals is a function of other voltages or currents in the circuit. Instead of a circle, it is customary to use a diamond to represent controlled sources in circuit diagrams. Two examples of dependent sources are shown in Figure 1.32. A voltage-controlled voltage source is a voltage source having a voltage equal A voltage-controlled voltage to a constant times the voltage across a pair of terminals elsewhere in the network. source maintains a voltage across its terminals equal to a constant times a voltage elsewhere in the circuit. + + vx + 2vx − ix 3ix − − Voltage-controlled Current-controlled voltage source voltage source (a) (b) Figure 1.32 Dependent voltage sources (also known as controlled voltage sources) are represented by diamond-shaped symbols. The voltage across a controlled voltage source depends on a current or voltage that appears elsewhere in the circuit. 24 Chapter 1 Introduction An example is shown in Figure 1.32(a). The dependent voltage source is the diamond symbol. The reference polarity of the source is indicated by the marks inside the diamond. The voltage vx determines the value of the voltage produced by the source. For example, if it should turn out that vx = 3 V, the source voltage is 2vx = 6 V. If vx should equal −7 V, the source produces 2vx = −14 V (in which case, the actual positive polarity of the source is at the bottom end). A current-controlled voltage A current-controlled voltage source is a voltage source having a voltage equal to source maintains a voltage a constant times the current through some other element in the circuit. An example across its terminals equal to a constant times a current is shown in Figure 1.32(b). In this case, the source voltage is three times the value flowing through some other of the current ix. The factor multiplying the current is called the gain parameter. We element in the circuit. assume that the voltage has units of volts and the current is in amperes. Thus, the gain parameter [which is 3 in Figure 1.32(b)] has units of volts per ampere (V/A). (Shortly, we will see that the units V/A are the units of resistance and are called ohms.) Returning our attention to the voltage-controlled voltage source in Figure 1.32(a), we note that the gain parameter is 2 and is unitless (or we could say that the units are V/V). Later in the book, we will see that controlled sources are very useful in modeling transistors, amplifiers, and electrical generators, among other things. Independent Current Sources An ideal independent current An ideal independent current source forces a specified current to flow through itself. source forces a specified The symbol for an independent current source is a circle enclosing an arrow that gives current to flow through itself. the reference direction for the current. The current through an independent current source is independent of the elements connected to it and of the voltage across it. Figure 1.33 shows the symbols for a dc current source and for an ac current source. If an open circuit exists across the terminals of a current source, we have a contradictory circuit. For example, consider the 2-A dc current source shown in Figure 1.33(a). This current source is shown with an open circuit across its terminals. By definition, the current flowing into the top node of the source is 2 A. Also by definition, no current can flow through the open circuit. Thus, KCL is not satisfied at this node. In good models for actual circuits, this situation does not occur. Thus, we will avoid current sources with open-circuited terminals in our discussion of ideal networks. A battery is a good example of a voltage source, but an equally familiar example does not exist for a current source. However, current sources are useful in construct- ing theoretical models. Later, we will see that a good approximation to an ideal current source can be achieved with electronic amplifiers. 2A 3 sin (100pt) A Figure 1.33 Independent current (a) Dc current (b) Ac current sources. source source  Section 1.6 Introduction to Circuit Elements 25 + 3vx vx 2iy iy − Voltage-controlled Current-controlled current source current source (a) (b) Figure 1.34 Dependent current sources. The current through a dependent current source depends on a current or voltage that appears elsewhere in the circuit. Dependent Current Sources The current flowing through a dependent current source is determined by a current The current flowing through a or voltage elsewhere in the circuit. The symbol is a diamond enclosing an arrow that dependent current source is determined by a current or indicates the reference direction. Two types of controlled current sources are shown voltage elsewhere in the in Figure 1.34. circuit. In Figure 1.34(a), we have a voltage-controlled current source. The current through the source is three times the voltage vx. The gain parameter of the source (3 in this case) has units of A/V (which we will soon see are equivalent to siemens or inverse ohms). If it turns out that vx has a value of 5 V, the current through the controlled current source is 3vx = 15 A. Figure 1.34(b) illustrates a current-controlled current source. In this case, the current through the source is twice the value of iy. The gain parameter, which has a value of 2 in this case, has units of A/A (i.e., it is unitless). Like controlled voltage sources, controlled current sources are useful in con- structing circuit models for many types of real-world devices, such as electronic amplifiers, transistors, transformers, and electrical machines. If a controlled source is needed for some application, it can be implemented by using electronic amplifiers. In sum, these are the four kinds of controlled sources: 1. Voltage-controlled voltage sources 2. Current-controlled voltage sources 3. Voltage-controlled current sources 4. Current-controlled current sources Resistors and Ohm’s Law The voltage v across an ideal resistor is proportional to the current i through the resistor. The constant of proportionality is the resistance R. The symbol used for a resistor is shown in Figure 1.35(a). Notice that the current reference and voltage polarity reference conform to the passive reference configuration. In other words, the reference direction for the current is into the positive polarity mark and out of the negative polarity mark. In equation form, the voltage and current are related by Ohm’s law: v = iR 26 Chapter 1 Introduction v v = iR + i v R i − (a) Resistance symbol (b) Ohm’s law Figure 1.35 Voltage is proportional to current in an ideal resistor. Notice that the references for v and i conform to the passive reference configuration. The units of resistance are V/A, which are called ohms. The uppercase Greek letter omega () represents ohms. In practical circuits, we encounter resistances ranging from milliohms (m) to megohms (M). Except for rather unusual situations, the resistance R assumes positive values. (In certain types of electronic circuits, we can encounter negative resistance, but for now we assume that R is positive.) In situations for which the current reference direction enters the negative reference of the voltage, Ohm’s law becomes v = −iR This is illustrated in Figure 1.36. The relationship between current direction and voltage polarity can be neatly R i included in the equation for Ohm’s law if double-subscript notation is used. (Recall that to use double subscripts, we label the ends of the element under consideration, + v − which is a resistance in this case.) If the order of the subscripts is the same for the Figure 1.36 If the references current as for the voltage (iab and vab , for example), the current reference direction for v and i are opposite to enters the first terminal and the positive voltage reference is at the first terminal. the passive configuration, Thus, we can write we have v = −Ri. vab = iab R On the other hand, if the order of the subscripts is not the same, we have vab = −iba R Conductance Solving Ohm’s law for current, we have 1