FUNDAMENTALS OF
`
`ELECTRIC CIRCUITS
`
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`VOLTSERVER EXHIBIT 1029
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`VOLTSERVER EXHIBIT 1029
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`
`Chapter‘l
`
`Basic Concepts
`
`
`
`1.1
`
`
`_
`
`Introduction
`
`Electric circuit theory and electromagnetic theory are the two funda—
`mental theories upon which all branches of electrical engineering are
`built. Many branches of electrical engineering, such as power, electric
`machines, control, electronics, communications, and instrumentation,
`are based on electric circuit theory. Therefore, the basic electric circuit
`theory course is the most important course for an electrical engineer-
`ing student, and always an excellent starting point for a beginning stu—
`dent in electrical engineering education. Circuit theory is also valuable
`to students specializing in other branches of the physical sciences
`because circuits are a good model for the study of energy systems in
`general, and because of the applied mathematics, physics, and topol-
`ogy involved.
`In electrical engineering, we are often interested in communicating
`or transferring energy from one point to another. To do this requires an
`interconnection of electrical devices. Such interconnection is referred
`
`to as an electric circuit, and each component of the circuit is known as
`an element.
`
`An electric circuit is an interconnection of electrical elements.
`
`A simple electric circuit is shown in Fig. 1.1. It consists of three
`basic elements: a battery, a lamp, and connecting wires. Such a simple
`circuit can exist by itself; it has several applications, such as a flash—
`light, a search light, and so forth.
`A complicated real circuit is displayed in Fig. 1.2, representing the
`schematic diagram for a radio receiver. Although it seems complicated,
`this circuit can be analyzed using the techniques we cover in this book.
`Our \goal
`in this text
`is to learn various analytical
`techniques and
`computer software applications for describing the behavior of a circuit
`like this.
`
`+ 9 V' (DC)
`
`
`Antenna
`
`
`
`
`
`Figure 1.1
`A simple electric circuit.
`
`Electret
`microphone
`
`Page 8 of 37
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`Ins—“u... 4 n
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`Page 8 of 37
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`

`

`1.3
`
`Charge and Current
`
`5
`
`Electric circuits are used in numerous electrical systems to accom-
`plish different tasks. Our objective in this book is not the study of
`various uses and applications of circuits. Rather, our major concern is
`the analysis of the circuits. By the analysis of a circuit. we mean a
`study of the behavior of the circuit: How does it respond to a given
`input? How do the interconnected elements and devices in the circuit
`interact?
`We commence our study by defining some basic concepts. These
`concepts include charge, current, voltage, circuit elements, power, and
`energy. Before defining these concepts. we must first establish a sys-
`tem of units that we will use throughout the text.
`
`' ;
`
`Systems of Units
`
`As electrical engineers, we deal with measurable quantities. Our mea-
`surement, however, must be communicated in a standard language that
`virtually all professionals can understand. irrespective of the country
`where the measurement is conducted. Such an international measurement
`language is the International System of Units (SI), adopted by the
`General Conference on Weights and Measures in 1960. In this system,
`there are seven principal units from which the traits of all other phys-
`ical quantities can be derived. Table 1.1 shows the six units and one
`derived unit that are relevant to this text. The 81 units are used through-
`out this text.
`One great advantage of the SI unit is that it uses prefixes based on
`the power of 10 to relate larger and smaller units to the basic unit.
`Table 1.2 shovvs the SI prefixes and their symbols. For example, the
`following are expressions of the same distance in meters (m):
`
`600,000,000 mm
`
`600,000 or
`
`600 km
`
`9'
`
`
`
`The SI prefim_
`PrefixMultiplier Symbol
`
`
`
`
`
`10'“
`10is
`1012
`10°
`
`:03
`
`102
`#3-,
`10" 2
`10‘3
`10—6
`10—9
`10— ‘2
`10— '5
`10' ‘3
`
`exa
`peta
`tera
`giga
`
`kilo
`
`mm
`2:1:
`centi
`milii
`micro
`nano
`pico
`femto
`atto
`
`E
`P
`T
`G
`
`k
`
`h
`2a
`c
`m
`p.
`n
`p
`f
`a
`
`“L3
`
`
`
`.-
`
`Charge and Current
`The concept of electric charge is the underlying principle for explain-
`ing all electrical phenomena. Also, the most basic quantity in an elec—
`tric circuit is the electric charge. We all experience the effect of electric
`
`Six basic SI units and one derived unit relevant to this text.
`Quantity
`Basic unit
`' Symbol
`
`Length
`meter
`or
`Mass
`kilogram
`kg
`Time
`second
`s
`Electric current
`ampere
`A
`Thermodynamic temperature
`kelvin
`K
`Luminous intensity
`candela
`cd
`Charge
`coulomb
`C
`
`Page 9 of 37
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`

`
`
`Chapter 1
`
`Basic Concepts
`
`charge when we try to remove our wool sweater and have it stick to
`our body or walk across a carpet and receive a shock.
`
`Charge is an electrical property of the atomic particles of which mat—
`ter consists, measured in coulombs (C).
`
`We know from elementary physics that all matter is made of funda-
`mental building blocks known as atoms and that each atom consists of
`electrons, protons, and neutrons. We also know that the charge 9 on an
`electron is negative and equal in magnitude to 1.602 X 10—19 C, while
`a proton carries a positive charge of the same magnitude as the elec—
`tron. The presence of equal numbers of protons and electrons leaves an
`atom neutrally charged.
`The following points should be noted about electric charge:
`
`1. The coulomb is a large unit for charges. In 1 C of charge, there
`are l/(l.602 X 1049) 2 6.24 X [0'8 electrons. Thus realistic or
`laboratory values of charges are on the order of pC, nC, or ,uC.‘
`2. According to experimental observations,
`the only charges that
`occur in nature are integral multiples of the electronic charge
`.9 = —i.602 X10“"‘C.
`3. The law of conservation of charge states that charge can neither
`be created nor destroyed, only transfetTed. Thus the algebraic sum
`of the electric charges in a system does not change.
`
`We now consider the flow of electric charges. A unique feature of
`electric charge or electricity is the fact that it is mobile; that is, it can
`be transferred from one place to another, where it can be converted to
`another form of energy.
`When a conducting wire (consisting of several atoms) is con-
`nected to a battery (a source of electromotive force). the charges are
`compelled to move; positive charges move in one direction while neg-
`ative charges move in the opposite direction. This motion of charges
`creates electric current. It is conventional to take the current flow as
`
`the movement of positive charges. That is, opposite to the flow of neg-
`atjwe charges. as Fig. 1.3 illustrates. This convention was introduced
`by Benjamin Franklin (1706—1790), the American scientist and inven—
`tor. Although we now know that current in metallic conductors is due
`to negatively charged electrons, we will
`follow the universally
`accepted convention that current is the net flow of positive charges.
`Thus,
`
`Electric current is the time rate of change of charge, measured in
`
`amperes__(_A).
`
`Mathematically, the relationship between current 1', charge q, and time i is
`
`
`
`(1-1)
`
`' However, a large pawer supply capacitor can store up to 0.5 C of charge.
`
`
`
`
`
`Battery
`
`Figure 1 .3
`Electric current due to flow of electronic
`charge in a conductor.
`
`A convention is a standard way of
`describing something so that others in
`the profession can understand what
`.
`‘ we mean. We will be using IEEE con—
`ventions throughout this book.
`
`Page 10 of 37
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`Page 10 of 37
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`

`

`
`
`1.3
`
`Charge and Current
`
`1"
`
`
`
`meanest-v,
`
`. _ ..
`
`.-__..
`
`_
`
`.._'_j'_".
`
`Andre-Marie Ampere (17754836), a French mathematician and
`physicist, laid the foundation of electrodynamics. He defined the elec-
`tric current and developed a way to measure it in the 18205.
`Born in Lyons, France, Ampere at age 12 mastered Latin in a few
`weeks, as he was intensely interested in mathematics and many of the
`best mathematical works were in Latin. He was a brilliant scientist and
`a prolific writer. He formulated the laws of electromagnetics. He in-
`vented the electromagnet and the ammeter. The unit of electric current,
`
`
`the ampere, was named after him.
`
`The Burndy Library Collection
`at: The Huntington Library,
`San Marino. California.
`
`———~—-—--—-———————._.__________________________________
`
`where current is measured in amperes (A), and
`
`l ampere = l coulomb/second
`
`The charge transferred between time to and r is obtained by integrat-
`ing both sides of Eq. (1.1). We obtain
`
`(1.2}
`
`The way we define current as i in Eq. (1.1) suggests that current need
`not be a constant-valued function. As in
`of the examples and prob-
`lems in this chapter and subsequent chap 'rs suggest, there can be sev-
`eral types of current; that is, charge can vary with time in several ways.
`If the current does not change with time, but remains constant. we
`call it a direct current (dc).
`
`
`
`Adlrectcurrent (dc) isacurrentthatremainsconstantwithtime.
`
`By convention the symbol I is used to represent such a constant-current.
`A time-varying current is represented by the symbol :2 A common
`form of time—varying current is the sinusoidal current or simmering
`current (ac).
`
`3p
`
`r
`
`(b)
`An _alternatlns current (arc) is a current thatvaries sinusoidallywith time-.3
`.
`.
`_
`.
`Figure 1.4
`Such current is used in your household to run the air conditioner, m0 common types of current: (a) direct
`refrigerator, washing machine. and other electricappliances. Figure 1.4
`current (dc). (b) alternating current (ac).
`
`Page 11 0f37
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`Page 11 of 37
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`

`

`
`
`tn
`
`Chapter 1
`
`Basic Concepts
`
`5 a f/ _5A
`"
`/
`
`i
`
`it!)
`Figure 1 5
`C‘mvemioéa] current flow. (a) positive
`CHI—mm flaw, (in negative current flow.
`
`(b)
`
`shows direct current and alternating current; these are the two most
`
`common types of current. We will consider other types later in the
`book.
`Once we define current as the movement of charge. we expect can
`rent to have an associated direction of flow. As mentioned earlier, the
`direction of current flow is conventionally taken as the direction of pos—
`.
`_
`.
`.
`~
`row: charge movement. Based on tlns convention. a current of 5 A may
`be represented positively or negatively as shown in Fig. 1.5. In other
`words, a negative current of —5 A flowing in one direction as shown
`in Fig. 1.5(b) is the some as a current of +5 A flowing in the opposite
`direction.
`
`
`Example 1 .1
`How much charge is represented by 4.600 electrons?
`
`Solution:
`Each electron lnts
`
`lot]? X 10"“) C. Hence 4.600 electrons will have
`
`—l.602 >< 10"” C/electron >< 4,600 electrons : —7.369 >< 10‘”3 c
`
`Practite Problem 17?
`
`Calculate the amount of charge represented by six million protons.
`
`Answer: +9612 x 10‘” C.
`
`
`Example 1.2
`The total charge entering a terminal is given by q = 5r sin 411-: mC.
`Calculate the current at r = 0.5 3.
`
`Solution:
`
`.
`dq
`d -
`.
`1
`.
`t= tr = 7(3t51n41fl‘) mUs = {5 Sin 4111’: + 2091‘: cos 4111) nrA
`(
`At r = 0.5.
`
`1': Ssin 211' +107rcos 2n = O +1017 = 31.42mA
`
`. .'Pt:actice Problem 1 .2
`
`If in Example 1.2. q = (10 — [De—2’) mC. find the current at r = 1.0 5;.
`
`Answer: 2.707 tnA.
`
`
`Page 12 of 37
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`Page 12 of 37
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`

`

`
`
`1.4
`
`Voltage
`
`
`
`Determine the total charge entering a terminal between r= l s and
`r = 2 s if the current passing the terminal is r = (3:2 _ r) A.
`
`Solution:
`
`2
`
`Q=J id“! (hi—ad:
`
`2
`
`l
`
`2
`
`l
`
`=(8—2)(l—E)=5.5C
`
`1
`
`
`
`'3‘”
`
`Practice???
`
`
`The current flowing through an element is
`
`0<r<l
`I__{4A,
`Calculate the charge entering the element from r = 0 to r = 2 s.
`
`4:223,
`
`t>l
`
`Answer: 13333 C.
`
`
`
` _. Voltage
`As explained briefly in the previous section, to move the electron in a
`conductor in a particular direction requires some work or energy trans-
`fer. This work is performed by an external electromotive force (emf),
`typically represented by the battery in Fig. 1.3. This emf is also knowu
`as voltage or potential deference. The voltage nab between two points
`a and b in an electric circuit is the energy (or work) needed to move
`a unit charge from a to b; mathematically,
`
`val: g £5)—
`dq
`
`{.3
`
`(1.3)
`
`~ where w is energy in joules (J) and q is charge in coulombs (C). The
`voltage nab or simply v is measured in volts (V), named in honor of
`the Italian physicist Alessandro Antonio Volta (1745—1827), who
`invented the first voltaic battery. From Eq. (1.3). it is evident that
`
`1 volt = l joulefcoulomb = l newton-metedconlomh
`
`
`
`Figure 1.6 shows the voltage across an element (represented by a
`Irectangular block) connected to points a and b. The plus (+) and minus
`‘(—) signs are used to define reference direction or voltage polarity. The
`“as can be interpreted in two ways: (1) Point a is at a potential of U95
`
`Figure 1 .6
`Polarity of voltage nab.
`
`Page 13 of37
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`J.
`
`I
`J
`
`I |
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`

`

`‘l a
`
`Chapter 1
`
`Basic Concepts
`
`
`
`Historical
`
`
`
`Alessandro Antonio Volta 0745—1827), an ltalian physicist,
`invented the electric battery-—which provided the first continuous flow
`of electricity—and the capacitor.
`Born into a noble family in Como, Italy, Volta was performing
`electrical experiments at age 18. His invention of the battery in 1796
`revolutionized the use of electricity. The publication of his work in
`1800 marked the beginning of electric circuit theory. Volta received
`many honors during his lifetime. The unit of voltage or potential dif—
`
`ference, the volt. was named in his honor.
`
`The Burndy Library Collection
`at The Huntington Library.
`San Merino. California
`M—
`
`
`
`volts higher than point b, or (2) the potential at point a with respect to
`point b is ugh. It follows logically that in general
`val? : _Ulm
`(1'4)
`For example, in Fig. 1.7, we have two representations of the same volt-
`age. In Fig. 1.7(a). point a is +9 V above point 12‘. in Fig. 1.7(b), point b
`is ~9 V above point rt. We may say that in Fig. 1.7(a). there is a 9—V
`voltage drop from o to b or equivalently a 9-V voltage rise from b to
`a. In other words, a voltage drop from a to b is equivalent to a volt— '
`age rise from b to a.
`J
`Current and voltage are the two basic variables in electric circuits.
`The common term signal is used for an electric quantity such as a cur-
`rent or a voltage (or even electromagnetic wave) when it is used for
`conveying information. Engineers prefer to call such variables signals
`rather than mathematical functions of time because of their importance
`in communications and other disciplines. Like electric current, a con-
`stant voltage is called a dc voltage and is represented by V, whereas a
`sinusoidally time-varying voltage is called an ac voltage and is repre-
`sented by u. A dc voltage is commonly produced by a battery; ac volt-
`age is produced by an electric generator.
`
`____________________._..._-_-—-—-——--————-~--
`1.5
`i Power and Energy
`Although current and voltage are the two basic variables in an electric
`circuit, they are not sufficient by themselves. For practical purposes,
`we need to know how much power an electric device can handle. We
`all know from experience that a lOO-watt bulb gives more light than a
`fill-watt bulb. We also know that when we pay our bills to the electric
`utility companies. we are paying for the electric energy consumed over
`a certain period of time. Thus, power and energy calculations are
`important in circuit analysis.
`
`+
`
`9v
`
`r—wa
`[l
`
`L—Eb
`
`(a)
`(bi
`Figure 1 1
`_
`Twp equivalent representations ofthe
`sarne voltage nab: (a) Pointa is 9V above
`901mb; (bl 90m 3315 43" abOVe Pm!“ 0-
`Keep in mind that electric current is
`! alwaysmngsanelementandthat
`.. electric voltage is always across the
`element or between two points.
`
`'
`
`Page 14 of 37
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`Page 14 of 37
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`

`

`
`
`11
`
`1.5
`
`Power and Energy
`
`To relate power and energy to voltage and current. we recall from
`physics that:
`
`Power is the time rate of expending or absorbing energy, measured-in
`watts (W).
`
`We write this relationship as
`
`(1.5)
`
`where p is power in watts (W). w is energy in joules (J), and r is time
`in seconds (5). From Eqs. (1.1), (1.3), and (1.5), it follows that
`
`01'
`
`(Ir 7 dc} . dt
`
`or
`
`p = vi
`
`(L6)
`
`(1.7)
`
`The power p in Eq. (1.7) is a time—varying quantity and is called the
`instantaneous power. Thus, the power absorbed or supplied by an ele-
`ment is the product of the voltage across the element and the current
`through it. If the power has a + sign, power is being delivered to or
`absorbed by the element. If, on the other hand, the power has a — sign,
`power is being supplied by the element. But how do we know when
`the power has a negative or a positive sign?
`Current direction and voltage polarity play a major role in deter-
`mining the sign of power. It is therefore important that we pay atten-
`tion to the relationship between current i and voltage U in Fig. 1.8(a).
`The voltage polarity and current direction must conform with those
`shown in Fig. 1.8[a) in order for the power to have a positive sign.
`This is known as the passive Sign convention. By the passive sign con-
`vention, current enters through the positive polé‘iity of the voltage. In
`this case, p = +vi or of > 0 implies that the element is absorbing
`power. However, if p = —vt' or vi < 0, as in Fig. 1.80)), the element
`is releasing or supplying power.
`
`Passive sign convention is satisfied When the current enters th'roiigh '
`the positive terminal of an element and p = +vr'.
`If the current enters
`through the negative terminal, p = —w‘.
`
`'
`
`Unless otherwise stated, we will follow the passive sign conven~
`tion throughout this text. For example, the element in both circuits of
`Fig. 1.9 has an absorbing power of +12 W because a positive current
`enters the positive terminal in both cases. In Fig. 1.10, however,
`the
`element is supplying power of +12 W because a positive current enters
`the negative terminal. Of course, an absorbing power of —12 W is
`equivalent to a supplying power of + 12 W. In general,
`
`+ Power absorbed = —Power supplied
`
`Page 15 of37
`
`----—-O
`+
`
`m”.
`
`I!
`
`“5
`p =+vi
`
`(a)
`
`+
`
`ti
`
`—
`p = *u
`
`(b)
`
`Figure 1.8
`Reference polarities for power using the
`passive sign convention: (a) absorbing
`power, (13} supplying power.
`
`When the voltage and current directions
`, .
`,. conform to Fig. 1 .8 (b), we have the 6C"-
`avenge contention and p = +vr'.
`
`3 A
`_..
`
`3 A
`.—
`
`+
`
`4v
`
`_
`
`"
`
`4v
`
`+
`
`(a)
`
`(b)
`
`Figure 1.9
`Two cases of an element with an absorbing
`powerofl2W:(a}p = 4 X 3 =12W,
`(b}p=4x3= 12w.
`
`3A
`«om—-
`
`3A
`—-
`
`+
`
`4v
`
`_.
`
`—-
`
`4v
`
`+
`
`(a)
`
`(b)
`
`Figure 1.10
`Two cases of an element with a supplying
`powerof 12W: (3) p = —4 x 3 =
`—12w,(b)p = —4 x 3 = —12w.
`
`Page 15 of 37
`
`

`

` 1‘2
`
`Chapter ?
`
`Bastc Concepts
`
`to fact? the [air of conservation afenergr must be obeyed in any
`electric circuit. For this reason.
`the algebraic sum of power in a cir-
`cuit. at any instant of time. must be zero:
`
`:5 2;;20 l
`?______
`i
`
`(1.8)
`
`This again confirms the fact that the total power supplied to the circuit
`must balance the total prwtrer absorbed
`From Eq. (M1).
`the energy absorbed or supplied by an element
`from time rt. to time 2‘ is
`
`w
`
`ll pd! = i ain’t
`in
`‘ J u
`
`(1.9)
`
`Energy is the capacity to do work, measured in joules (J).
`
`The electric power utility companies measure energy in watt-hours
`(Wh). where
`
`l Wh .: 3.600J
`
`Example 1.4
`
`to flow
`An energy source forces a constant current of 2 A for It) s:
`through a light bulb. If 2.3 k] is given off in the form of light and heat
`energy. calculate the voltage drop across the bulb.
`
`Solution:
`
`The total charge is
`
`The ‘voltage drop is
`
`Aq=iAr=2>< l0=20C
`
`23 ><103
`Am
`U = __
`AT;
`20
`
`llSV
`
`.-.'_Pr_actice Problem 1.4“
`
`To move charge :5; from point a to point 1') requires —30 J. Find the
`voltage drop um, if: (a) q = 6 C. (b) q = —3 C.
`
`Answer: (a) —5 V, (b) 10 V.
`
`
`.:_,.-_-_;Exampte 1.5
`
`Find the power delivered to an element at r = 3 ms if the current enter-
`ing its positive terminal is
`
`.r' : 5 cos oflnrA
`
`and the voltage is: (a) o = 31’. {b} v = 3 di/dr.
`
`Page 16 of 37
`
`
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`Page 16 of 37
`
`

`

`m
`
`1.6
`
`Circuit Elements
`
`15
`
`
`1.6
`Circuit Elements
`
`As we discussed in Section 1.1, an element is the basic building block
`of a circuit. An electric circuit is simply an interconnection of the ele-
`ments. Circuit analysis is the process of determining voltages across
`(or the currents through) the elements of the circuit.
`There are two types of elements found in electric circuits: pas-
`sive elements and active elements. An active element is capable of
`generating energy while a passive element is not. Examples of pas-
`sive elements are resistors. capacitors. and inductors. Typical active
`elements include generators. batteries, and operational amplifiers. Our
`aim in this section is to gain familiarity with some important active
`elements.
`
`important active elements are voltage or current
`The most
`sources that generally deliver power to the circuit connected to
`them. There are two kinds of sources;
`independent and dependent
`sources.
`
`An ideal independent source is an active element that provides a
`specified voltage or current that is completely independent of other
`circuit elements.
`
`In other words. an ideal independent voltage source delivers to the
`circuit whatever current is necessary to maintain its terminal volt-
`age. Physical sources such as batteries and generators may be
`regarded as approximations to ideal voltage sources. Figure 1.1]
`shows the symbols for independent voltage sources. Notice that both
`symbols in Fig.
`[.1 1(a) and (b) can be used to represent a dc volt-
`age source, but only the symbol
`in Fig. l.]l(a) can be used for a
`time-varying voltage source. Similarly, an ideal independent current
`source is an active element that provides a specified current com—
`pletely independent of the voltage across the source. That is, the cur-
`rent source delivers to the circuit
`hatever voltage is necessary to
`maintain the designated current. Th symbol for an independent cur—
`rent source is displayed in Fig. 1.12. where the arrow indicates the
`direction of current r'.
`
`
`
`O
`
`u (21;
`
`
`
`———o
`
`r—wo
`
`+l
`V
`
`(a)
`
`(b)
`
`Figure 1.1 1
`Symbols for independent voltage sources:
`(a) used for constant or time-varying volt-
`age. (b) used for constant voltage (dc).
`
`An ideal dependent (or controlled) source is an active element in
`which the source quantity is controlled by another voltage or current.
`
`Figure 1 .1 2
`Symbol for independent current source.
`
`Dependent sources are usually designated by diamond-shaped symbols,
`as shown in Fig. 1.13. Since the control of the dependent source is
`achieved by a voltage or current of some other element in the circuit,
`and the source can be voltage or current. it follows that there are four
`possiblehtypes of dependent sources, namely:
`
`i. A voltage-controlled voltage source (VCVS).
`2. A current-controlled voltage source (CCVS).
`3. A voltage-controlled current source [VCCS].
`4. A current-controlled current source (CCCS).
`
`(a)
`
`(b)
`
`Figure 1.13
`Symbols for: (a) dependent voltage
`source. (b) dependent current source.
`
`Page 17 of 37
`
`Page 17 of 37
`
`

`

`——__
`
`30
`
`Chapter 9
`
`Bast: Laws
`
`' introduction
`2.1
`introduced basic concepts such as current. voltage, and
`Chapter I
`power in an electric circttit. To actually determine the values of these
`variables in a given circuit requires that we understand some funda-
`mental laws that govern electric circuits. These laws. known as Ohm’s
`law and Kirchhoff‘s laws, form the foundation upon which electric cir—
`cuit analysis is built.
`in addition to tltese laws. we shall discuss some
`in this chapter.
`techniques commonly applied in circuit design and analysis. These tech-
`niques include combining resistors in series or parallel. voltage division.
`current division and delta-to—wye and wyeito—delta transformations. The
`application of these laws and techniques will be restricted to resistive
`circuits in this chapter. We will finally apply the laws and techniques to
`real—life problems of electrical lighting and the design of do nteters.
`
`M 2
`
`Ohm’s Law
`.2
`Materials in general have a characteristic behavior of resisting the [low
`of electric charge. This physical property, or ability to resist current. is
`known as resistance and is represented by the symbol R. The resist—
`ance of any material with a uniform cross-sectional area A depends on
`A and its length E, as shown in Fig. 2.1(a). We can represent resistance
`(as measured in the laboratory). in mathematical form.
`R :- p-l-
`A
`
`(2.1)
`
`
`
`where p is known as the resistivity ol'the material in ohm—meters. Good
`conductors, such as copper and aluminum. have low resistivities, while
`insulators. such as mica and paper, have high resistivities. Table 2.1
`presents the values of p for some common materials and shows which
`mate'rials are used for conductors. insulators. and semiconductors.
`The circuit element used to model the current—resisting behavior of a
`material is the resistor. For Lhe purpose of constructing circuits, resistors
`are usually made from metallic alloys and carbon compounds. The circuit
`
`_ R
`
`eststivities of common materials.
`
` Material Resistivity (El-m) Usage
`
`Silver
`[.64 X 10‘8
`Conductor
`Copper
`L72 X [0—:1
`Conductor
`Aluminum
`2.8 X ltl‘R
`Conductor
`Gold
`2.45 x to “
`Conductor
`Carbon
`4 x 10'5
`Semiconductor
`Germanium
`47 X 10—3
`Semiconductor
`Silicon
`6.4 X It)2
`Semiconductor
`Paper
`10'”
`Insulator
`Mica
`S X illi‘1
`Insulator
`Glass
`It)I 3
`Insulator
`Teflon
`3 X 10[2
`Insulator
`
`
`(:t
`|
`+ J.
`
`_.__._
`
`3:
`
`1
`l
`1
`U
`(bi
`
`Material With
`
`resistivity p
`
`Cross—sectional
`area A
`
`tut
`
`Figure 2.1
`(at Resistor. [b] Circuit symbol for
`resistance.
`
`Page 18 of 37
`
`Page 18 of 37
`
`

`

`
`
`31
`
`53.?
`
`(3th Law
`
`symbol for the resistor is shown in Fig. 2.1(b), where R stands for the
`resistance of theresistor. The resistor is the simplest passive element.
`Georg Siifion Ohm (1787—1854), a German physicist, is credited
`with finding the relationship between current and voltage for a resis-
`tor. This relationship is known as Ohm’s law.
`
`Ohm’s law states that the voltage vacross a resistor is directly proper
`tional to the current r’ flowing through the resistor.
`
`That is,
`
`t) 0:
`
`f
`
`(2-2)
`
`Ohm defined the constant of proportionality for a resistor to be the
`resistance, R. (The resistance is a material property which can change
`if the internal or external conditions of the element are altered, e.g., if
`there are changes in the temperature.) Thus. Eq. (2.2) becomes
`
`1.: = ER
`
`(2.3)
`
`which is the mathematical form of Ohm’s law. R in Eq. (2.3) is mea-
`sured in the unit of ohms. designated (1. Thus,
`
`The ren'sr‘ance 1? of an element denotes its ability to resist the fiowof
`electric current; it is measured in ohms (it).
`
`We may deduce from Eq. (2.3) that
`
`l
`R Z ‘T
`
`(2.4)
`
`so that
`
`l (I 2 1 WA
`
`To apply Ohm‘s law as stated in Eq. (2.3), we must pay careful
`attention to the current direction and voltage polarity. The direction of
`current i and the polarity of voltage 0 must conform with the passive
`
`
`'- firetcnsatf‘fi
`
`in 1826
`Georg Simon Ohm (1787—1854), a German physicist,
`experimentally determined the most basic law relating voltage and cur—
`rent for a resistor. Ohm‘s work was initially denied by critics.
`Born of humble beginnings in Erlangen, Bavaria, Ohm threw himw
`self into electrical research. His efforts resulted in his famous law.
`
`He was awarded the Copley Medal in 1841 by the Royal Society of
`London. In 1849, he was given the Professor of Physics chair by the
`University of Munich. To honor him, the unit of resistance was named
`the ohm.
`
`
`
`
`
`eSSPLviaGettyImages
`
`we
`
`
`
`Page 19 of 37
`
`Page 19 of 37
`
`

`

`
`
`32
`
`Chapter 9
`
`Basic Laws
`
`sign convention. as shown in Fig. 2.1{b). This implies that current flows
`from a higher potential to a lower potential in order for o = r'R. If cur-
`rent flows from a lower potential to a higher potential. I) = —i R.
`Since the value of R can range from zero to infinity.
`it is impor—
`tant that We consider the two extreme possible values of R. An element
`with R = t) is coiled a short Circuit. as shown in Fig. 2.2(a). For a short
`circun,
`
`o = ER = t)
`
`(2.5)
`
`showing that the voltage is zero but the current could be anything. in
`practice, a short circuit is usually a connecting wire assumed to be a
`perfect conductor. Thus,
`
`A short circuit is a circuit element with resistance approaching zero.
`
`Similarly. an element with R = x is known as an open cirr'tu‘r, as
`shown in Fig. 2.2(b). For an open circuit,
`
`r =
`'
`
`It] —- =
`1'
`r
`’->- it
`
`(i
`
`.
`(26
`
`)
`
`indicating that the current is zero though the voltage could be anything.
`Thus,
`
`An open circuit is a circuit element with resistance approaching infinity.
`
`A resistor is either titted or variable. Most resistors are of the fixed
`type. meaning their resistance remains constant. The two common types
`of fixed resistors (wirewound and composition) are shown in Fig. 2.3.
`The composition resistors are used when large resistance is needed.
`The circuit symbol in Fig. 2.l(b) is for a fixed resistor. Variable resis—
`tors have adjustable resistance. The symbol for a variable resistor is
`shown in Fig. 2.4(a), A common variable resistor is known as a th’JfiL’R'
`riometer or pot for short, with the symbol shown in Fig. 2.4(b). The
`pot is a three-terminal element with a sliding contact or wiper. By siid-
`ing the wiper. the resistances between the wiper terminal and the fixed
`terminals vary. Like fixed resistors. variable resistors can be of either
`wirewonnd or composition type. as shown in Fig. 2.5. Although resistors
`like those in Figs. 2.3 and 2.5 are used in circuit designs, today most
`
`
`
`
`
`
`
` “—13,-.71: 4’
`
`r' = (IJ' R = [l
`f
`_Q-;I
`
`ta}
`
`he's;— t.= t.
`i
`r
`I- R :m
`
`1.
`
`I
`
`1;.)
`___3._g
`
`(bi
`
`Figure 2.2
`(a) Short circuit (R = O). (b) Open circuit
`(R = w).
`
`
`
`
`
`
`(h)
`
`Figure 2.3
`Fixed resistors: (a) wirewound type,
`(b) carbon fihn type.
`Courtesy ochch America.
`
`l.
`{a}
`
`(b)
`
`(b)
`(a)
`
`
`Figure 2.4
`Circuit symbol for: (a) a variable resistor
`in general, (b) a potentiometer.
`
`Figure 2.5
`Variable resistors: in} composition type. (b) slider pol.
`Courtesy of Tech America.
`
`Page 20 of 37
`
`Page 20 of 37
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`

`

`2.2
`
`Ohm’s Law
`
`33
`
`
`
`©EricTomevJAmy
`
`Figure 2.6
`Resistors in an integrated circuit board.
`
`circuit components including resistors are either surface mounted or
`integrated. as typically shown in Fig. 2.6.
`It should be pointed out that not all resistors obey Ohm’s law. A
`resistor that obeys Ohm‘s law is known as a linear resistor. It has a
`constant resistance and thus its current-voltage characteristic is as illusi
`Hated in Fig. 2.?(a): Its i-t) graph is a straight line passing through the
`origin. A nonlinear resistor does not obey Ohm’s law. Its resistance
`varies with current and its i«v characteristic is typically shown in
`Fig. 2.7(b). Examples of devices with nonlinear resistance are the light
`bulb and the diode. Although all practical resistors may exhibit nonlin—
`ear behavior under certain conditions, we will assume in this book that
`all elements actually designated as resistors are linear.
`A useful quantity in circuit analysis is the reciprocal of resistance
`R. known as conductance and denoted by G:
`
`(2.7}
`
`
`
`‘
`1
`.
`
`The conductance is a measure of how well an element will con»
`duct eiectric current. The unit of conductance is the mito (ohm spelled
`backward) or reciprocal ohm, with symbol ‘0, the inverted omega.
`Although engineers often use the mho, in this book we prefer to use
`the Siemens (S), the SI unit of conductance:
`
`,
`
`Thus.
`
`[S =io=1AN
`
`(2.8)
`
`Conductance is the ability of an element to conduct electric current;'
`it is measured in mhos (U) or Siemens (5).
`
`The same resistance can be expressed in ohms or siemcns. For
`example. 10.0 is the same as 0.1 S. From Eq. (2.7), we may write
`
`The power dissipated by a resistor can be expressed in terms of R.
`Using Eqs. (1.7) and (2.3),
`
`i = Go
`
`(2.9)
`
`
`‘
`
`(h)
`
`Figure 2.1
`The H: characteristic of: (a) a linear
`“3315‘“? (b) ‘1 “mime” res-“‘05
`
`U:
`2
`p = m‘ = i R z 35
`
`(2-10)
`
`.
`
`The pOWer dissipated by a resistor may also be expressed in terms of
`G as
`
`'2
`
`=vszuio=i
`G
`
`p
`
`(211)
`
`We should note two things from Eqs. (2.10) and (2.11):
`
`I- The power dissipated in a resistor is a nonlinear function of either
`current or voltage.
`2. Since R and G are positive quantities, the power dissipated in a
`resistor is always positive. Thus, a resistor always absorbs power
`from the circuit. This confirms the idea that a resistor is a passive
`element, incapable of generating energy.
`
`
`
`Page 21 of 37
`
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`Page 23 of 37
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`Page 24 of 37
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`Page 25 of 37
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`
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`Page 26 of 37
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`Page 26 of 37
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`Page 27 of 37
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`Page 28 of 37
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`Page 29 of 37
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`Page 29 of 37
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`

`

`We note 513.1 ”I?"
`
`f .-= —m. “U3,
`
`Page 30 of 37
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`Page 30 of 37
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`

`

`156
`
`Chapter ?
`
`First»0rder Circuits
`
`the voltage decreases is expressed in terms of the time constant,
`denoted by r. th