Chapter 11: Electrical Engineering - University of Waterloojzelek/teaching/syde361/0066ch11.pdf · 11 Electrical Engineering ... This section presents the fundamental laws of circuit

11Electrical Engineering

11.1 Introduction 11.2 Fundamentals of Electric Circuits

Electric Power and Sign Convention • Circuit Elements and Their i-v Characteristics • Resistance and Ohm’s Law • Practical Voltage and Current Sources • Measuring Devices

11.3 Resistive Network AnalysisThe Node Voltage Method • The Mesh Current Method • One-Port Networks and Equivalent Circuits • Nonlinear Circuit Elements

11.4 AC Network AnalysisEnergy-Storage (Dynamic) Circuit Elements • Time-Dependent Signal Sources • Solution of Circuits Containing Dynamic Elements • Phasors and Impedance

11.1 Introduction

The role played by electrical and electronic engineering in mechanical systems has dramatically increasedin importance in the past two decades, thanks to advances in integrated circuit electronics and in materialsthat have permitted the integration of sensing, computing, and actuation technology into industrialsystems and consumer products. Examples of this integration revolution, which has been referred to asa new field called Mechatronics, can be found in consumer electronics (auto-focus cameras, printers,microprocessor-controlled appliances), in industrial automation, and in transportation systems, mostnotably in passenger vehicles. The aim of this chapter is to review and summarize the foundations ofelectrical engineering for the purpose of providing the practicing mechanical engineer a quick and usefulreference to the different fields of electrical engineering. Special emphasis has been placed on those topicsthat are likely to be relevant to product design.

11.2 Fundamentals of Electric Circuits

This section presents the fundamental laws of circuit analysis and serves as the foundation for the studyof electrical circuits. The fundamental concepts developed in these first pages will be called on throughthe chapter.

The fundamental electric quantity is charge, and the smallest amount of charge that exists is the chargecarried by an electron, equal to

(11.1)

As you can see, the amount of charge associated with an electron is rather small. This, of course, hasto do with the size of the unit we use to measure charge, the coulomb (C), named after Charles Coulomb.However, the definition of the coulomb leads to an appropriate unit when we define electric current,

qe 1.602 10 19– coulomb×–=

Giorgio RizzoniOhio State University

©2002 CRC Press LLC

since current consists of the flow of very large numbers of charge particles. The other charge-carryingparticle in an atom, the proton, is assigned a positive sign and the same magnitude. The charge of aproton is

(11.2)

Electrons and protons are often referred to as elementary charges.Electric current is defined as the time rate of change of charge passing through a predetermined area.

If we consider the effect of the enormous number of elementary charges actually flowing, we can writethis relationship in differential form:

(11.3)

The units of current are called amperes (A), where 1 A = 1 C/sec. The electrical engineering conventionstates that the positive direction of current flow is that of positive charges. In metallic conductors, however,current is carried by negative charges; these charges are the free electrons in the conduction band, whichare only weakly attracted to the atomic structure in metallic elements and are therefore easily displacedin the presence of electric fields.

In order for current to flow there must exist a closed circuit. Figure 11.1 depicts a simple circuit,composed of a battery (e.g., a dry-cell or alkaline 1.5-V battery) and a light bulb.

Note that in the circuit of Fig. 11.1, the current, i, flowing from the battery to the resistor is equal tothe current flowing from the light bulb to the battery. In other words, no current (and therefore nocharge) is “lost” around the closed circuit. This principle was observed by the German scientist G.R.Kirchhoff and is now known as Kirchhoff ’s current law (KCL). KCL states that because charge cannotbe created but must be conserved, the sum of the currents at a node must equal zero (in an electrical circuit,a node is the junction of two or more conductors). Formally:

(11.4)

The significance of KCL is illustrated in Fig. 11.2, where the simple circuit of Fig. 11.2 has been augmentedby the addition of two light bulbs (note how the two nodes that exist in this circuit have been emphasizedby the shaded areas). In applying KCL, one usually defines currents entering a node as being negativeand currents exiting the node as being positive. Thus, the resulting expression for the circuit of Fig. 11.2 is

Charge moving in an electric circuit gives rise to a current, as stated in the preceding section. Naturally,it must take some work, or energy, for the charge to move between two points in a circuit, say, frompoint a to point b. The total work per unit charge associated with the motion of charge between two

FIGURE 11.1 A simple electrical circuit.

qp +1.602 10 19– coulomb×=

idqdt------ C/sec( )=

inn=1

N

∑ 0 Kirchhoff’s current law=

i i1 i2 i3+ + + 0=


points is called voltage. Thus, the units of voltage are those of energy per unit charge:

(11.5)

The voltage, or potential difference, between two points in a circuit indicates the energy required tomove charge from one point to the other. As will be presently shown, the direction, or polarity, of thevoltage is closely tied to whether energy is being dissipated or generated in the process. The seeminglyabstract concept of work being done in moving charges can be directly applied to the analysis of electricalcircuits; consider again the simple circuit consisting of a battery and a light bulb. The circuit is drawnagain for convenience in Fig. 11.3, and nodes are defined by the letters a and b. A series of carefullyconducted experimental observations regarding the nature of voltages in an electric circuit led Kirchhoffto the formulation of the second of his laws, Kirchhoff ’s voltage law, or KVL. The principle underlyingKVL is that no energy is lost or created in an electric circuit; in circuit terms, the sum of all voltagesassociated with sources must equal the sum of the load voltages, so that the net voltage around a closedcircuit is zero. If this were not the case, we would need to find a physical explanation for the excess (ormissing) energy not accounted for in the voltages around a circuit. KVL may be stated in a form similarto that used for KCL:

(11.6)

where the vn are the individual voltages around the closed circuit. Making reference to Fig. 11.3, we cansee that it must follow from KVL that the work generated by the battery is equal to the energy dissipatedin the light bulb to sustain the current flow and to convert the electric energy to heat and light:

or

FIGURE 11.2 Illustration of Kirchhoff ’s current law.

FIGURE 11.3 Voltages around a circuit.

1 volt1 joule

coulomb---------------------=

vnn=1

N

∑ 0 Kirchhoff’s voltage law=

vab vba–=

v1 v2=


One may think of the work done in moving a charge from point a to point b and the work donemoving it back from b to a as corresponding directly to the voltages across individual circuit elements. LetQ be the total charge that moves around the circuit per unit time, giving rise to the current i. Then thework done in moving Q from b to a (i.e., across the battery) is

(11.7)

Similarly, work is done in moving Q from a to b, that is, across the light bulb. Note that the word potentialis quite appropriate as a synonym of voltage, in that voltage represents the potential energy between twopoints in a circuit: if we remove the light bulb from its connections to the battery, there still exists avoltage across the (now disconnected) terminals b and a.

A moment’s reflection upon the significance of voltage should suggest that it must be necessary tospecify a sign for this quantity. Consider, again, the same dry-cell or alkaline battery, where, by virtue ofan electrochemically induced separation of charge, a 1.5-V potential difference is generated. The potentialgenerated by the battery may be used to move charge in a circuit. The rate at which charge is movedonce a closed circuit is established (i.e., the current drawn by the circuit connected to the battery) dependsnow on the circuit element we choose to connect to the battery. Thus, while the voltage across the batteryrepresents the potential for providing energy to a circuit, the voltage across the light bulb indicates theamount of work done in dissipating energy. In the first case, energy is generated; in the second, it isconsumed (note that energy may also be stored, by suitable circuit elements yet to be introduced). Thisfundamental distinction required attention in defining the sign (or polarity) of voltages.

We shall, in general, refer to elements that provide energy as sources, and to elements that dissipateenergy as loads. Standard symbols for a generalized source-and-load circuit are shown in Fig. 11.4.Formal definitions will be given in a later section.

Electric Power and Sign Convention

The definition of voltage as work per unit charge lends itself very conveniently to the introduction ofpower. Recall that power is defined as the work done per unit time. Thus, the power, P, either generatedor dissipated by a circuit element can be represented by the following relationship:

(11.8)

Thus, the electrical power generated by an active element, or that dissipated or stored by a passiveelement, is equal to the product of the voltage across the element and the current flowing through it.

(11.9)

It is easy to verify that the units of voltage (joules/coulomb) times current (coulombs/second) are indeedthose of power (joules/second, or watts).

FIGURE 11.4 Sources and loads in an electrical circuit.

Wba Q 1.5 V×=

Powerworktime------------ work

unit charge---------------------------charge

time--------------- voltage current×= = =

P VI=


It is important to realize that, just like voltage, power is a signed quantity, and that it is necessary tomake a distinction between positive and negative power. This distinction can be understood with referenceto Fig. 11.5, in which a source and a load are shown side by side. The polarity of the voltage across thesource and the direction of the current through it indicate that the voltage source is doing work in movingcharge from a lower potential to a higher potential. On the other hand, the load is dissipating energy,because the direction of the current indicates that charge is being displaced from a higher potential to alower potential. To avoid confusion with regard to the sign of power, the electrical engineering communityuniformly adopts the passive sign convention, which simply states that the power dissipated by a load isa positive quantity (or, conversely, that the power generated by a source is a positive quantity). Anotherway of phrasing the same concept is to state that if current flows from a higher to a lower voltage (+ to –),the power dissipated will be a positive quantity.

Circuit Elements and Their i-v Characteristics

The relationship between current and voltage at the terminals of acircuit element defines the behavior of that element within the circuit.In this section, we shall introduce a graphical means of representingthe terminal characteristics of circuit elements. Figure 11.6 depicts therepresentation that will be employed throughout the chapter to denotea generalized circuit element: the variable i represents the current flow-ing through the element, while v is the potential difference, or voltage,across the element.

Suppose now that a known voltage were imposed across a circuitelement. The current that would flow as a consequence of this voltage,and the voltage itself, form a unique pair of values. If the voltageapplied to the element were varied and the resulting current measured, it would be possible to constructa functional relationship between voltage and current known as the i-v characteristic (or volt-amperecharacteristic). Such a relationship defines the circuit element, in the sense that if we impose any prescribedvoltage (or current), the resulting current (or voltage) is directly obtainable from the i-v characteristic.A direct consequence is that the power dissipated (or generated) by the element may also be determinedfrom the i-v curve.

The i-v characteristics of ideal current and voltage sources can also be useful in visually representingtheir behavior. An ideal voltage source generates a prescribed voltage independent of the current drawnfrom the load; thus, its i-v characteristic is a straight vertical line with a voltage axis intercept corre-sponding to the source voltage. Similarly, the i-v characteristic of an ideal current source is a horizontalline with a current axis intercept corresponding to the source current. Figure 11.7 depicts this behavior.

Resistance and Ohm’s Law

When electric current flows through a metal wire or through other circuit elements, it encounters acertain amount of resistance, the magnitude of which depends on the electrical properties of the material.Resistance to the flow of current may be undesired—for example, in the case of lead wires and connection

FIGURE 11.5 The passive sign convention.

FIGURE 11.6 Generalized repre-sentation of circuit elements.


cable—or itelements exhto be dissipaaccording to

that is, that value of the

The resistinverse of reelement (shoproportiona

It is oftenThe symbol

FIGURE 11.7

FIGURE 11.8

0066_Frame_C11 Page 6 Wednesday, January 9, 2002 4:14 PM

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may be exploited in an electrical circuit in a useful way. Nevertheless, practically all circuitibit some resistance; as a consequence, current flowing through an element will cause energyted in the form of heat. An ideal resistor is a device that exhibits linear resistance properties Ohm’s law, which states that

(11.10)

the voltage across an element is directly proportional to the current flow through it. R is the resistance in units of ohms (Ω), where

(11.11)

ance of a material depends on a property called resistivity, denoted by the symbol ρ; thesistivity is called conductivity and is denoted by the symbol σ. For a cylindrical resistancewn in Fig. 11.8), the resistance is proportional to the length of the sample, l, and inversely

l to its cross-sectional area, A, and conductivity, σ.

(11.12)

convenient to define the conductance of a circuit element as the inverse of its resistance. used to denote the conductance of an element is G, where

(11.13)

i-v characteristics of ideal sources.

The resistance element.

V IR=

1 Ω 1 V/A=

v1

sA-------i=

G1R--- siemens (S), where 1 S 1 A/V= =

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Ohm’s lawbecause of imaterials. Tyapply at verbehavior eveand current

The typicof cylindricamercially avAnother comrating for reand televisiocode associaFor exampleyellow (b3 =

In Table 1show the thbrown-black

TABLE 11.1 Common Resistor Values (1/8-, 1/4-, 1/2-, 1-, 2-W Rating)

Ω Code Ω Multiplier kΩ Multiplier kΩ Multiplier kΩ Multiplier

10 B12 B15 B18 B22 R27 R33 O39 O47 Y56 G68 B82 G

FIGURE 11.9


©2002 CRC P

s law can be rested in terms of conductance, as

(11.14)

is an empirical relationship that finds widespread application in electrical engineeringts simplicity. It is, however, only an approximation of the physics of electrically conductingpically, the linear relationship between voltage and current in electrical conductors does not

y high voltages and currents. Further, not all electrically conducting materials exhibit linearn for small voltages and currents. It is usually true, however, that for some range of voltages

s, most elements display a linear i-v characteristic.al construction and the circuit symbol of the resistor are shown in Fig. 11.8. Resistors madel sections of carbon (with resistivity ρ = 3.5 × 10–5 Ω m) are very common and are com-ailable in a wide range of values for several power ratings (as will be explained shortly).monly employed construction technique for resistors employs metal film. A common power

sistors used in electronic circuits (e.g., in most consumer electronic appliances such as radiosn sets) is W. Table 11.1 lists the standard values for commonly used resistors and the color

ted with these values (i.e., the common combinations of the digits b1b2b3 as defined in Fig. 11.9., if the first three color bands on a resistor show the colors red (b1 = 2), violet (b2 = 7), and 4), the resistance value can be interpreted as follows:

1.1, the leftmost column represents the complete color code; columns to the right of it onlyird color, since this is the only one that changes. For example, a 10-Ω resistor has the code-black, while a 100-Ω resistor has brown-black-brown.

rn-blk-blk 100 Brown 1.0 Red 10 Orange 100 Yellowrn-red-blk 120 Brown 1.2 Red 12 Orange 120 Yellowrn-grn-blk 150 Brown 1.5 Red 15 Orange 150 Yellowrn-gry-blk 180 Brown 1.8 Red 18 Orange 180 Yellowed-red-blk 220 Brown 2.2 Red 22 Orange 220 Yellowed-vlt-blk 270 Brown 2.7 Red 27 Orange 270 Yellowrg-org-blk 330 Brown 3.3 Red 33 Orange 330 Yellowrg-wht-blk 390 Brown 3.9 Red 39 Orange 390 Yellowlw-vlt-blk 470 Brown 4.7 Red 47 Orange 470 Yellowrn-blu-blk 560 Brown 5.6 Red 56 Orange 560 Yellowlu-gry-blk 680 Brown 6.8 Red 68 Orange 680 Yellowry-red-blk 820 Brown 8.2 Red 82 Orange 820 Yellow

Resistor color code.

I GV=

14--

R 27 104× 270,000 Ω 270 kΩ= = =

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In addition to the resistance in ohms, the maximum allowable power dissipation (or power rating)is typically specified for commercial resistors. Exceeding this power rating leads to overheating and cancause the re

That is, the it, as well as application

Example 1

A common gauge. Straias a functiomeasuremenconductor o

If the conduchange, andwill decreassince the lengiven by the

and since th

the change i

where R0 is tvalue of G f

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©2002 CRC P

sistor to literally start on fire. For a resistor R, the power dissipated is given by

(11.15)

power dissipated by a resistor is proportional to the square of the current flowing throughthe square of the voltage across it. The following example illustrates a common engineeringof resistive elements: the resistance strain gauge.

1.1 Resistance Strain Gauges

application of the resistance concept to engineering measurements is the resistance strainn gauges are devices that are bonded to the surface of an object, and whose resistance variesn of the surface strain experienced by the object. Strain gauges may be used to performts of strain, stress, force, torque, and pressure. Recall that the resistance of a cylindricalf cross-sectional area A, length L, and conductivity σ is given by the expression

ctor is compressed or elongated as a consequence of an external force, its dimensions will with them its resistance. In particular, if the conductor is stretched, its cross-sectional areae and the resistance will increase. If the conductor is compressed, its resistance decreases,gth, L, will decrease. The relationship between change in resistance and change in length is gauge factor, G, defined by

e strain ε is defined as the fractional change in length of an object by the formula

n resistance due to an applied strain ε is given by the expression

he resistance of the strain gauge under no strain and is called the zero strain resistance. Theor resistance strain gauges made of metal foil is usually about 2..10 depicts a typical foil strain gauge. The maximum strain that can be measured by a foilut 0.4–0.5%; that is, ∆L/L = 0.004 to 0.005. For a 120-Ω gauge, this corresponds to a change of the order of 0.96–1.2 Ω. Although this change in resistance is very small, it can be detectedf suitable circuitry. Resistance strain gauges are usually connected in a circuit called the bridge, which we analyze later in this section.

Short Circuits

ient idealizations of the resistance element are provided by the limiting cases of Ohm’s lawance of a circuit element approaches zero or infinity. A circuit element with resistance zero is called a short circuit. Intuitively, one would expect a short circuit to allow forflow of current. In fact, metallic conductors (e.g., short wires of large diameter) approximate of a short circuit. Formally, a short circuit is defined as a circuit element across which thero, regardless of the current flowing through it. Figure 11.11 depicts the circuit symbol forrt circuit.

P VI I2RV2

R-----= = =

RL

sA-------=

G∆R/R∆L/L-------------=

e ∆LL

-------=

∆R R0Ge=

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Physicallypurposes, hoFor exampleelectrical po24 gauge wir

A circuit expect no cuopen circuitillustrates th

TABLE 11.2 Resistance of Copper Wire

AWG Size Number of Strands Diameter per Strand Resistance per 1000 ft (Ω)

FIGURE 11.1

FIGURE 11.1

FIGURE 11.1


©2002 CRC P

, any wire or other metallic conductor will exhibit some resistance, though small. For practicalwever, many elements approximate a short circuit quite accurately under certain conditions., a large-diameter copper pipe is effectively a short circuit in the context of a residential

wer supply, while in a low-power microelectronic circuit (e.g., an FM radio) a short length ofe (refer to Table 11.2 for the resistance of 24 gauge wire) is a more than adequate short circuit.element whose resistance approaches infinity is called an open circuit. Intuitively, one wouldrrent to flow through an open circuit, since it offers infinite resistance to any current. In an

, we would expect to see zero current regardless of the externally applied voltage. Figure 11.12is idea.

24 Solid 0.0201 28.424 7 0.0080 28.422 Solid 0.0254 18.022 7 0.0100 19.020 Solid 0.0320 11.320 7 0.0126 11.918 Solid 0.0403 7.218 7 0.0159 7.516 Solid 0.0508 4.516 19 0.0113 4.7

0 The resistance strain gauge.

1 The short circuit.

2 The open circuit.

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In practicpath amounhold, howevdown at a stwo conductbe generatedignite the airideal open a

Series Resi

Although eleto combinatiwith parallelanalysis. Parsection and resistors: theimportant to

For an exto resistors R

DefinitionTwo or more

The threeof current redefining the

which is alsodivider.

FIGURE 11.1


©2002 CRC P

e, it is not too difficult to approximate an open circuit; any break in continuity in a conductingts to an open circuit. The idealization of the open circuit, as defined in Fig. 11.12, does noter, for very high voltages. The insulating material between two insulated terminals will breakufficiently high voltage. If the insulator is air, ionized particles in the neighborhood of theing elements may lead to the phenomenon of arcing; in other words, a pulse of current may that momentarily jumps a gap between conductors (thanks to this principle, we are able to-fuel mixture in a spark-ignition internal combustion engine by means of spark plugs). Thend short circuits are useful concepts and find extensive use in circuit analysis.

stors and the Voltage Divider Rule

ctrical circuits can take rather complicated forms, even the most involved circuits can be reducedons of circuit elements in parallel and in series. Thus, it is important that you become acquainted and series circuits as early as possible, even before formally approaching the topic of networkallel and series circuits have a direct relationship with Kirchhoff ’s laws. The objective of thisthe next is to illustrate two common circuits based on series and parallel combinations of voltage and current dividers. These circuits form the basis of all network analysis; it is therefore master these topics as early as possible.

ample of a series circuit, refer to the circuit of Fig. 11.13, where a battery has been connected

1, R2, and R3. The following definition applies.

circuit elements are said to be in series if the same current flows through each of the elements. resistors could thus be replaced by a single resistor of value REQ without changing the amountquired of the battery. From this result we may extrapolate to the more general relationship

equivalent resistance of N series resistors:

(11.16)

illustrated in Fig. 11.13. A concept very closely tied to series resistors is that of the voltage

3 Voltage divider rule.

REQ Rn

n=1

N

∑=

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Parallel Re

A concept acircuit conta

DefinitionTwo or morelements. (S

N resistor

or

Very often iresistors wit

where the syThe gener

Example 1

The Wheatscircuits. Theis an unknowlatter circuit

FIGURE 11.1


©2002 CRC P

al form of the voltage divider rule for a circuit with N series resistors and a voltage source is

(11.17)

sistors and the Current Divider Rule

nalogous to that of the voltage may be developed by applying Kirchhoff ’s current law to aining only parallel resistances.

e circuit elements are said to be in parallel if the same voltage appears across each of theee Fig. 11.14.)s in parallel act as a single equivalent resistance, REQ, given by the expression

(11.18)

(11.19)

n the remainder of this book we shall refer to the parallel combination of two or moreh the following notation:

mbol signifies “in parallel with.”al expression for the current divider for a circuit with N parallel resistors is the following:

(11.20)

1.2 The Wheatstone Bridge

tone bridge is a resistive circuit that is frequently encountered in a variety of measurement general form of the bridge is shown in Fig. 11.15(a), where R1, R2, and R3 are known, while Rx

n resistance, to be determined. The circuit may also be redrawn as shown in Fig. 11.15(b). Thewill be used to demonstrate the use of the voltage divider rule in a mixed series-parallel circuit.

4 Parallel circuits.

vn

Rn

R1 R2… Rn

… RN+ + + + +-------------------------------------------------------------------vS=

1REQ

-------- 1R1

----- 1R2

----- … 1RN

------+ + +=

REQ1

1/R1 1/R2… 1/RN+ + +

-----------------------------------------------------------=

R1 R2…|| ||

||

in

1/Rn

1/R1 1/R2… 1/Rn

… 1/RN+ + + + +----------------------------------------------------------------------------------------- iS Current divider=

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The objectiv

1. Find vS. No

2. If R1

Solution1. First,

voltagthesethe soThusvoltagvoltag

Final

This r2. In or

equat

which

Practical

Idealized moof practical models thatpractice. Cothe source i

FIGURE 11.1


©2002 CRC P

e is to determine the unknown resistance Rx.

the value of the voltage vad = vad – vbd in terms of the four resistances and the source voltage,te that since the reference point d is the same for both voltages, we can also write vab = va – vb.

= R2 = R3 = 1 kΩ, vS = 12 V, and vab = 12 mV, what is the value of Rx?

we observe that the circuit consists of the parallel combination of three subcircuits: thee source, the series combination of R1 and R2, and the series combination of R3 and Rx. Since

three subcircuits are in parallel, the same voltage will appear across each of them, namely,urce voltage, vS.

, the source voltage divides between each resistor pair, R1-R2 and R3-Rx, according to thee divider rule: va is the fraction of the source voltage appearing across R2, while vb is thee appearing across Rx:

ly, the voltage difference between points a and b is given by

esult is very useful and quite general, and it finds application in numerous practical circuits.der to solve for the unknown resistance, we substitute the numerical values in the precedingion to obtain

may be solved for Rx to yield

Voltage and Current Sources

dels of voltage and current sources fail to take into consideration the finite-energy naturevoltage and current sources. The objective of this section is to extend the ideal models to are capable of describing the physical limitations of the voltage and current sources used innsider, for example, the model of an ideal voltage source. As the load resistance (R) decreases,s required to provide increasing amounts of current to maintain the voltage vS (t) across

5 Wheatstone bridge circuits.

va vS

R2

R1 R2+----------------- and vb vS

Rx

R3 Rx+-----------------= =

vab va vb– vS

R2

R1 R2+-----------------

Rx

R3 Rx+------------------–

= =

0.012 1210002000-----------

Rx

1000 Rx+------------------------–

=

Rx 996 Ω=

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its terminal:

This circuit the load, in

Figure 11vS, in series voltage sour

It should so that the c

A similar current soursource, consinfinity (i.e.

A good curra desirable c

FIGURE 11.1

FIGURE 11.1


©2002 CRC P

(11.21)

suggests that the ideal voltage source is required to provide an infinite amount of current tothe limit as the load resistance approaches zero..16 depicts a model for a practical voltage source; this is composed of an ideal voltage source,with a resistance, rS. The resistance rS in effect poses a limit to the maximum current thece can provide:

(11.22)

be apparent that a desirable feature of an ideal voltage source is a very small internal resistance,urrent requirements of an arbitrary load may be satisfied.modification of the ideal current source model is useful to describe the behavior of a practicalce. The circuit illustrated in Fig. 11.17 depicts a simple representation of a practical currentisting of an ideal source in parallel with a resistor. Note that as the load resistance approaches, an open circuit), the output voltage of the current source approaches its limit,

(11.23)

ent source should be able to approximate the behavior of an ideal current source. Therefore,haracteristic for the internal resistance of a current source is that it be as large as possible.

6 Practical voltage source.

7 Practical current source.

i t( )vS t( )

R-----------=

iS max

vS

rS

----=

vS max iSrS=

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Measuring Devices

The Ammeter

The ammeteflowing throevident for o

1. The aresist

2. The abe m

The Voltm

The voltmedifference inelement who

1. The v2. The v

else iinfini

Figure 11.19Once aga

considering resistance toopen circuitmodels for t

FIGURE 11.1

FIGURE 11.1


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r is a device that, when connected in series with a circuit element, can measure the currentugh the element. Figure 11.18 illustrates this idea. From Fig. 11.18, two requirements arebtaining a correct measurement of current:

mmeter must be placed in series with the element whose current is to be measured (e.g.,or R2).mmeter should not resist the flow of current (i.e., cause a voltage drop), or else it will not

easuring the true current flowing the circuit. An ideal ammeter has zero internal resistance.

eter

ter is a device that can measure the voltage across a circuit element. Since voltage is the potential between two points in a circuit, the voltmeter needs to be connected across these voltage we wish to measure. A voltmeter must also fulfill two requirements:

oltmeter must be placed in parallel with the element whose voltage it is measuring.oltmeter should draw no current away from the element whose voltage it is measuring, ort will not be measuring the true voltage across that element. Thus, an ideal voltmeter haste internal resistance.

illustrates these two points.in, the definitions just stated for the ideal voltmeter and ammeter need to be augmented bythe practical limitations of the devices. A practical ammeter will contribute some series the circuit in which it is measuring current; a practical voltmeter will not act as an ideal but will always draw some current from the measured circuit. Figure 11.20 depicts the circuithe practical ammeter and voltmeter.

8 Measurement of current.

9 Measurement of voltage.

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All of theoperation ofby a circuit ammeter.

Figure 11preceding psimultaneouthe load.

11.3 R

This sectionintroduced aillustrate var

The Nod

Node voltagapplication the voltage (usually—buOnce each nto determinexpressed in Figure 11.22

FIGURE 11.2voltmeter.

FIGURE 11.2


©2002 CRC P

considerations that pertain to practical ammeters and voltmeters can be applied to the a wattmeter, a measuring instrument that provides a measurement of the power dissipatedelement, since the wattmeter is in effect made up of a combination of a voltmeter and an

.21 depicts the typical connection of a wattmeter in the same series circuit used in thearagraphs. In effect, the wattmeter measures the current flowing through the load and,sly, the voltage across it multiplies the two to provide a reading of the power dissipated by

esistive Network Analysis

will illustrate the fundamental techniques for the analysis of resistive circuits. The methodsre based on Kirchhoff ’s and Ohm’s laws. The main thrust of the section is to introduce andious methods of circuit analysis that will be applied throughout the book.

e Voltage Method

e analysis is the most general method for the analysis of electrical circuits. In this section, itsto linear resistive circuits will be illustrated. The node voltage method is based on definingat each node as an independent variable. One of the nodes is selected as a reference node

t not necessarily—ground), and each of the other node voltages is referenced to this node.ode voltage is defined, Ohm’s law may be applied between any two adjacent nodes in ordere the current flowing in each branch. In the node voltage method, each branch current isterms of one or more node voltages; thus, currents do not explicitly enter into the equations. illustrates how one defines branch currents in this method.

0 Models for practical ammeter and

1 Measurement of power.

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Once eachat each nodeinto the nod

Figure 11.23The syste

equations. Hsince it is usindependenrequired to nodal voltag

The noda

Node Volt

1. Select2. Defin3. Apply

voltag4. Solve

In a circuit

The Mes

In the mesh defines the the voltages the directionFigure 11.25

FIGURE 11.2analysis.

FIGURE 11.2


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branch current is defined in terms of the node voltages, Kirchhoff ’s current law is applied. The particular form of KCL employed in the nodal analysis equates the sum of the currentse to the sum of the currents leaving the node:

(11.24)

illustrates this procedure.matic application of this method to a circuit with n nodes would lead to writing n linearowever, one of the node voltages is the reference voltage and is therefore already known,

ually assumed to be zero. Thus, we can write n – 1 independent linear equations in the n – 1t variables (the node voltages). Nodal analysis provides the minimum number of equationssolve the circuit, since any branch voltage or current may be determined from knowledge ofes.l analysis method may also be defined as a sequence of steps, as outlined below.

age Analysis Method

a reference node (usually ground). All other node voltages will be referenced to this node.e the remaining n – 1 node voltages as the independent variables. KCL at each of the n – 1 nodes, expressing each current in terms of the adjacent nodees.

the linear system of n – 1 equations in n – 1 unknowns.

containing n nodes we can write at most n – 1 independent equations.

h Current Method

current method, we observe that a current flowing through a resistor in a specified directionpolarity of the voltage across the resistor, as illustrated in Fig. 11.24, and that the sum ofaround a closed circuit must equal zero, by KVL. Once a convention is established regarding of current flow around a mesh, simple application of KVL provides the desired equation. illustrates this point.

2 Branch current formulation in nodal

3 Use of KCL in nodal analysis.

iin∑ iout∑=

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The numbAll branch cbe shown. Stematic procin applying

Mesh Curr

1. Definconve

2. Apply3. Solve

In mesh aconfusion inare using th

One-Port

This generaland is partiFig. 11.26 is

Thévenin

This sectionof an equivain terms of equivalent canalysis, yousources and appears agaicircuits fall

FIGURE 11.2

FIGURE 11.2

FIGURE 11.2


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er of equations one obtains by this technique is equal to the number of meshes in the circuit.urrents and voltages may subsequently be obtained from the mesh currents, as will presentlyince meshes are easily identified in a circuit, this method provides a very efficient and sys-edure for the analysis of electrical circuits. The following section outlines the procedure usedthe mesh current method to a linear circuit.

ent Analysis Method

e each mesh current consistently. We shall always define mesh currents clockwise, fornience. KVL around each mesh, expressing each voltage in terms of one or more mesh currents.

the resulting linear system of equations with mesh currents as the independent variables.

nalysis, it is important to be consistent in choosing the direction of current flow. To avoid writing the circuit equations, mesh currents will be defined exclusively clockwise when we

is method.

Networks and Equivalent Circuits

circuit representation is shown in Fig. 11.26. This configuration is called a one-port networkcularly useful for introducing the notion of equivalent circuits. Note that the network of completely described by its i-v characteristic.

and Norton Equivalent Circuits

discusses one of the most important topics in the analysis of electrical circuits: the conceptlent circuit. It will be shown that it is always possible to view even a very complicated circuitmuch simpler equivalent source and load circuits, and that the transformations leading toircuits are easily managed, with a little practice. In studying node voltage and mesh current may have observed that there is a certain correspondence (called duality) between currentvoltage sources, on the one hand, and parallel and series circuits, on the other. This dualityn very clearly in the analysis of equivalent circuits: it will shortly be shown that equivalentinto one of two classes, involving either voltage or current sources and (respectively) either

4 Basic principle of mesh analysis.

5 Use of KVL in mesh analysis.

6 One-port network.

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series or parbegins with

The ThévenAs far as a loresistors, mawith an equ

The NortonAs far as a loresistors, mawith an equ

Determina

The first steresistance prto zero and cin the circuiopen circuitNorton) equ

Computat

1. Remo2. Zero 3. Comp

equivplace

For example

FIGURE 11.2

FIGURE 11.2

FIGURE 11.2


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allel resistors, reflecting this same principle of duality. The discussion of equivalent circuitsthe statement of two very important theorems, summarized in Figs. 11.27 and 11.28.

in Theoremad is concerned, any network composed of ideal voltage and current sources, and of lineary be represented by an equivalent circuit consisting of an ideal voltage source, vT , in seriesivalent resistance, RT.

Theoremad is concerned, any network composed of ideal voltage and current sources, and of lineary be represented by an equivalent circuit consisting of an ideal current source, iN, in parallelivalent resistance, RN.

tion of Norton or Thévenin Equivalent Resistance

p in computing a Thévenin or Norton equivalent circuit consists of finding the equivalentesented by the circuit at its terminals. This is done by setting all sources in the circuit equalomputing the effective resistance between terminals. The voltage and current sources presentt are set to zero as follows: voltage sources are replaced by short circuits, current sources bys. We can produce a set of simple rules as an aid in the computation of the Thévenin (orivalent resistance for a linear resistive circuit.ion of Equivalent Resistance of a One-Port Network:

ve the load.all voltage and current sourcesute the total resistance between load terminals, with the load removed. This resistance is

alent to that which would be encountered by a current source connected to the circuit in of the load.

, the equivalent resistance of the circuit of Fig. 11.29 as seen by the load is:

Req = ((2 || 2) + 1) || 2 = 1 Ω.

7 Illustration of Thévenin theorem.

8 Illustration of Norton theorem.

9 Computation of Thévenin resistance.

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Computin

The Thévento the open-

This statecircuit voltagThévenin voconsisting ofvoltage acros

Computin

The computvoltage. The

DefinitionThe Nortonreplaced by

An explanarbitrary onits Norton e

It should the Norton short circuit

Experimen

Figure 11.32arbitrary neattention, billustrates ththe computamarks in thcurrent” arequantities.

FIGURE 11.3

FIGURE 11.3


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g the Thévenin Voltage

in equivalent voltage is defined as follows: the equivalent (Thévenin) source voltage is equalcircuit voltage present at the load terminals with the load removed.s that in order to compute vT, it is sufficient to remove the load and to compute the open-e at the one-port terminals. Figure 11.30 illustrates that the open-circuit voltage, vOC, and the

ltage, vT, must be the same if the Thévenin theorem is to hold. This is true because in the circuit vT and RT, the voltage vOC must equal vT, since no current flows through RT and therefore thes RT is zero. Kirchhoff ’s voltage law confirms that

(11.25)

g the Norton Current

ation of the Norton equivalent current is very similar in concept to that of the Thévenin following definition will serve as a starting point.

equivalent current is equal to the short-circuit current that would flow were the loada short circuit.ation for the definition of the Norton current is easily found by considering, again, an

e-port network, as shown in Fig. 11.31, where the one-port network is shown together withquivalent circuit.be clear that the current, iSC, flowing through the short circuit replacing the load is exactlycurrent, iN, since all of the source current in the circuit of Fig. 11.31 must flow through the.

tal Determination of Thévenin and Norton Equivalents

illustrates the measurement of the open-circuit voltage and short-circuit current for antwork connected to any load and also illustrates that the procedure requires some specialecause of the nonideal nature of any practical measuring instrument. The figure clearlyat in the presence of finite meter resistance, rm, one must take this quantity into account intion of the short-circuit current and open-circuit voltage; vOC and iSC appear between quotatione figure specifically to illustrate that the measured “open-circuit voltage” and “short-circuit, in fact, affected by the internal resistance of the measuring instrument and are not the true

0 Equivalence of open-circuit and Thévenin voltage.

1 Illustration of Norton equivalent circuit.

vT RT 0( ) vOC+ vOC= =

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The follow

where iN is t

Nonlinea

Descriptio

There are a current in ai-v character

FIGURE 11.3

FIGURE 11.3


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ing are expressions for the true short-circuit current and open-circuit voltage.

(11.26)

he ideal Norton current, vT the Thévenin voltage, and RT the true Thévenin resistance.

r Circuit Elements

n of Nonlinear Elements

number of useful cases in which a simple functional relationship exists between voltage and nonlinear circuit element. For example, Fig. 11.33 depicts an element with an exponentialistic, described by the following equations:

(11.27)

2 Measurement of open-circuit voltage and short-circuit current.

3 i-v characteristic of exponential resistor.

iN iSC 1rm

RT

------+ =

vT vOC 1RT

rm

------+ =

i I0ea v, v 0>=i I0, v ≤ 0–=

ress LLC

There existsrelationshipobtain a clo

One apprelement as aApplying KV

To obtain thcurrent, ix, iIf, for the mfollowing sy

The two pamethod of c

11.4 A

In this sectioby sinusoidasignals in thused in hou

Energy-St

The ideal repractical eleca dissipative much in theenergy storaenergy in an

The Ideal

A physical capolarized by

FIGURE 11.3a linear circu


©2002 CRC P

, in fact, a circuit element (the semiconductor diode) that very nearly satisfies this simple. The difficulty in the i-v relationship of Eq. (11.27) is that it is not possible, in general, tosed-form analytical solution, even for a very simple circuit.oach to analyzing a circuit containing a nonlinear element might be to treat the nonlinear load, and to compute the Thévenin equivalent of the remaining circuit, as shown in Fig. 11.34.L, the following equation may then be obtained:

(11.28)

e second equation needed to solve for both the unknown voltage, vx, and the unknownt is necessary to resort to the i-v description of the nonlinear element, namely, Eq. (11.27).

oment, only positive voltages are considered, the circuit is completely described by thestem:

(11.29)

rts of Eq. (11.29) represent a system of two equations in two unknowns. Any numericalhoice may now be applied to solve the system of Eqs. (11.29).

C Network Analysis

n we introduce energy-storage elements, dynamic circuits, and the analysis of circuits excitedl voltages and currents. Sinusoidal (or AC) signals constitute the most important class ofe analysis of electrical circuits. The simplest reason is that virtually all of the electric powerseholds and industries comes in the form of sinusoidal voltages and currents.

orage (Dynamic) Circuit Elements

sistor was introduced through Ohm’s law in Section 11.2 as a useful idealization of manytrical devices. However, in addition to resistance to the flow of electric current, which is purely(i.e., an energy-loss) phenomenon, electric devices may also exhibit energy-storage properties, same way a spring or a flywheel can store mechanical energy. Two distinct mechanisms forge exist in electric circuits: capacitance and inductance, both of which lead to the storage of electromagnetic field.

Capacitor

pacitor is a device that can store energy in the form of a charge separation when appropriately an electric field (i.e., a voltage). The simplest capacitor configuration consists of two parallel

4 Representation of nonlinear element init.

vT RTix vx+=

ix I0ea nx, v 0>=

vT RTix vx+=

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conducting or Teflon). F

The preseDC current;present at that the two cwhich is timis proportio

where the pdevice to acc(F). The faraor picofaradto the capac

Thus, althouvoltage will current. The

If the aboacross a cap

Equation (1up until the

FIGURE 11.3

∗ A dielectelectric field.


©2002 CRC P

plates of cross-sectional area A, separated by air (or another dielectric∗ material, such as micaigure 11.35 depicts a typical configuration and the circuit symbol for a capacitor.nce of an insulating material between the conducting plates does not allow for the flow of thus, a capacitor acts as an open circuit in the presence of DC currents. However, if the voltagee capacitor terminals changes as a function of time, so will the charge that has accumulatedapacitor plates, since the degree of polarization is a function of the applied electric field,e-varying. In a capacitor, the charge separation caused by the polarization of the dielectricnal to the external voltage, that is, to the applied electric field:

(11.30)

arameter C is called the capacitance of the element and is a measure of the ability of theumulate, or store, charge. The unit of capacitance is the coulomb/volt and is called the faradd is an unpractically large unit; therefore, it is common to use microfarads (1 µF = 10−6 F)s (1 pF = 10–12 F). From Eq. (11.30) it becomes apparent that if the external voltage applieditor plates changes in time, so will the charge that is internally stored by the capacitor:

(11.31)

gh no current can flow through a capacitor if the voltage across it is constant, a time-varyingcause charge to vary in time. The change with time in the stored charge is analogous to a relationship between the current and voltage in a capacitor is as follows:

(11.32)

ve differential equation is integrated, one can obtain the following relationship for the voltageacitor:

(11.33)

1.33) indicates that the capacitor voltage depends on the past current through the capacitor, present time, t. Of course, one does not usually have precise information regarding the flow

5 Structure of parallel-plate capacitor.

ric material contains a large number of electric dipoles, which become polarized in the presence of an

Q CV=

q t( ) Cv t( )=

i t( ) Cdv t( )

dt------------=

vC t( ) 1C--- iC dt

∞–

t0

∫=

ress LLC

of capacitorfor the capa

The capacito

The significagiving rise tois sufficient

From theand parallelillustrated into the same

Physical curation yieldorder to inctightly rolledillustrates ty

FIGURE 11.3

FIGURE 11.3


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current for all past time, and so it is useful to define the initial voltage (or initial condition)citor according to the following, where t0 is an arbitrary initial time:

(11.34)

r voltage is now given by the expression

(11.35)

nce of the initial voltage, V0, is simply that at time t0 some charge is stored in the capacitor, a voltage, vC (t0), according to the relationship Q = CV. Knowledge of this initial condition

to account for the entire past history of the capacitor current. (See Fig. 11.36.) standpoint of circuit analysis, it is important to point out that capacitors connected in series can be combined to yield a single equivalent capacitance. The rule of thumb, which is Fig. 11.37, is the following: capacitors in parallel add; capacitors in series combine accordingrules used for resistors connected in parallel.apacitors are rarely constructed of two parallel plates separated by air, because this config-s very low values of capacitance, unless one is willing to tolerate very large plate areas. In

rease the capacitance (i.e., the ability to store energy), physical capacitors are often made of sheets of metal film, with a dielectric (paper or Mylar) sandwiched in-between. Table 11.3pical values, materials, maximum voltage ratings, and useful frequency ranges for various

6 Defining equation for the ideal capacitor, and analogy with force-mass system.

7 Combining capacitors in a circuit.

V0 vC= t t0=( ) 1C--- iC dt

∞–

t

∫=

vC t( ) 1C--- iC td V0 t ≥ t0+

t0

t

∫=

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types of capif a sufficien

Example 1

As shown in

where ε is thtion. The peby a distanclarge plate acircuits. Ondevices thatthe air gap bother to an can obtain t

where C is (variable) didisplacemenment. For sm

The sensichange in di

Thus, the secapacitance becomes stearea equal to

TABLE 11.3 Capacitors

Material Capacitance Range Maximum Voltage (V) Frequency Range (Hz)


©2002 CRC P

acitors. The voltage rating is particularly important, because any insulator will break downtly high voltage is applied across it. The energy stored in a capacitor is given by

1.3 Capacitive Displacement Transducer and Microphone

Fig. 11.26, the capacitance of a parallel-plate capacitor is given by the expression

e permittivity of the dielectric material, A the area of each of the plates, and d their separa-rmittivity of air is ε0 = 8.854 × 10–12 F/m, so that two parallel plates of area 1 m2, separatede of 1 mm, would give rise to a capacitance of 8.854 × 10–3 µF, a very small value for a veryrea. This relative inefficiency makes parallel-plate capacitors impractical for use in electronic the other hand, parallel-plate capacitors find application as motion transducers, that is, as can measure the motion or displacement of an object. In a capacitive motion transducer,etween the plates is designed to be variable, typically by fixing one plate and connecting the

object in motion. Using the capacitance value just derived for a parallel-plate capacitor, onehe expression

the capacitance in picofarad, A is the area of the plates in square millimeter, and x is thestance in milimeter. It is important to observe that the change in capacitance caused by thet of one of the plates is nonlinear, since the capacitance varies as the inverse of the displace-

all displacements, however, the capacitance varies approximately in a linear fashion.tivity, S, of this motion transducer is defined as the slope of the change in capacitance persplacement, x, according to the relation

nsitivity increases for small displacements. This behavior can be verified by plotting theas a function of x and noting that as x approaches zero, the slope of the nonlinear C(x) curveeper (thus the greater sensitivity). Figure 11.38 depicts this behavior for a transducer with 10 mm2.

Mica 1 pF to 0.1 µF 100–600 103–1010

Ceramic 10 pF to 1 µF 50–1000 103–1010

Mylar 0.001 to 10 µF 50–500 102–108

Paper 1000 pF to 50 µF 100–105 102–108

Electrolytic 0.1 µF to 0.2 F 3–600 10–104

WC t( ) 12--CvC

2 t( ) J( )=

CeAd

------=

C8.854 10 3– A×

x----------------------------------=

SdCdx------- 8.854 10 3– A×

2x2----------------------------------– pF/mm( )= =

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This simpdenser) micrchange in cacircuit. An esimplified foof two fixed material) aninlet orificesit is measurinterminals b exists, the twcloser to the

This behacircuit illustvout, is precisfrom zero wtransducer. W

The Ideal

The ideal intypically mamaterial, sho

FIGURE 11.3

FIGURE 11.3


©2002 CRC P

le capacitive displacement transducer actually finds use in the popular capacitive (or con-ophone, in which the sound pressure waves act to displace one of the capacitor plates. Thepacitance can then be converted into a change in voltage or current by means of a suitablextension of this concept that permits measurement of differential pressures is shown inrm in Fig. 11.39. In the figure, a three-terminal variable capacitor is shown to be made up

surfaces (typically, spherical depressions ground into glass disks and coated with a conductingd of a deflecting plate (typically made of steel) sandwiched between the glass disks. Pressure are provided, so that the deflecting plate can come into contact with the fluid whose pressureg. When the pressure on both sides of the deflecting plate is the same, the capacitance between

and d, Cbd, will be equal to that between terminals b and c, Cbc. If any pressure differentialo capacitances will change, with an increase on the side where the deflecting plate has come fixed surface and a corresponding decrease on the other side. vior is ideally suited for the application of a bridge circuit, similar to the Wheatstone bridgerated in Example 11.2, and also shown in Fig. 11.39. In the bridge circuit, the output voltage,ely balanced when the differential pressure across the transducer is zero, but it will deviatehenever the two capacitances are not identical because of a pressure differential across the

e shall analyze the bridge circuit later in Example 11.4.

Inductor

ductor is an element that has the ability to store energy in a magnetic field. Inductors arede by winding a coil of wire around a core, which can be an insulator or a ferromagneticwn in Fig. 11.40. When a current flows through the coil, a magnetic field is established, as

8 Response of a capacitive displacement transducer.

9 Capacitive pressure transducer and related bridge circuit.

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you may recof the wire voltage droptime-varyinthe followin

where L is c

Henrys are used.

The indu

If the curren

then the ind

Inductors inconnected in

Table 11.4mechanical

FIGURE 11.4


©2002 CRC P

all from early physics experiments with electromagnets. In an ideal inductor, the resistanceis zero, so that a constant current through the inductor will flow freely without causing a. In other words, the ideal inductor acts as a short circuit in the presence of DC currents. If a

g voltage is established across the inductor, a corresponding current will result, according tog relationship:

(11.36)

alled the inductance of the coil and is measured in henry (H), where

(11.37)

reasonable units for practical inductors; millihenrys (mH) and microhenrys (µH) are also

ctor current is found by integrating the voltage across the inductor:

(11.38)

t flowing through the inductor at time t = t0 is known to be I0, with

(11.39)

uctor current can be found according to the equation

(11.40)

series add. Inductors in parallel combine according to the same rules used for resistors parallel. See Figs. 11.41–11.43. and Figs. 11.36, 11.41, and 11.43 illustrate a useful analogy between ideal electrical and

elements.

0 Iron-core inductor.

vL t( ) LdiL

dt-------=

1 H 1 V sec/A=

iL t( ) 1L--- vL dt

∞–

t

∫=

I0 iL t = t0( ) 1L--- vL dt

∞–

t0

∫= =

iL t( ) 1L--- vL td I0 t ≥ t0+

t0

t

∫=

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TABLE 11.4 Analogy Between Electrical and Mechanical Variables

FIGURE 11.4tor and analo

FIGURE 11.4

FIGURE 11.4


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Mechanical System Electrical System

Force, f (N) Current, i (A)Velocity, µ (m/sec) Voltage, v (V)Damping, B (N sec/m) Conductance, 1/R (S)Compliance, 1/k (m/N) Inductance, L (H)Mass, M (kg) Capacitance, C (F)

1 Defining equation for the ideal induc-gy with force-spring system.

2 Combining inductors in a circuit.

3 Analogy between electrical and mechanical elements.

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Time-Dependent Signal Sources

Figure 11.44 illustrates the convention that will be employed to denote time-dependent signal sources.One of th

appear frequA periodic s

where T is thencounteredsawtooth waavailable sigamplitude, a

As stated dependent ssinusoid is d

FIGURE 11.4

FIGURE 11.4


©2002 CRC P

e most important classes of time-dependent signals is that of periodic signals. These signalsently in practical applications and are a useful approximation of many physical phenomena.ignal x(t) is a signal that satisfies the following equation:

(11.41)

e period of x(t). Figure 11.45 illustrates a number of the periodic waveforms that are typically in the study of electrical circuits. Waveforms such as the sine, triangle, square, pulse, andves are provided in the form of voltages (or, less frequently, currents) by commercially

nal (or waveform) generators. Such instruments allow for selection of the waveform peaknd of its period.in the introduction, sinusoidal waveforms constitute by far the most important class of time-ignals. Figure 11.46 depicts the relevant parameters of a sinusoidal waveform. A generalizedefined as follows:

(11.42)

4 Time-dependent signal sources.

5 Periodic signal waveforms.

x t( ) x t nT+( ) n 1, 2, 3, …= =

x t( ) A cos wt f+( )=

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where A is definitions o

where

The phase sreference corestrict the acosine wave

It is imposecond) to dper second,

Average an

Now that a measuremenof measuremsuring the mtakes into accthe average vchosen) peri

where T is tto computin

FIGURE 11.4


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the amplitude, ω the radian frequency, and φ the phase. Figure 11.46 summarizes thef A, ω, and φ for the waveforms

(11.43)

hift, φ, permits the representation of an arbitrary sinusoidal signal. Thus, the choice of thesine function to represent sinusoidal signals—arbitrary as it may appear at first—does notbility to represent all sinusoids. For example, one can represent a sine wave in terms of a

simply by introducing a phase shift of π/2 radians:

(11.44)

rtant to note that, although one usually employs the variable ω (in units of radians perenote sinusoidal frequency, it is common to refer to natural frequency, f, in units of cyclesor hertz (Hz). The relationship between the two is the following:

(11.45)

d RMS Values

number of different signal waveforms have been defined, it is appropriate to define suitablets for quantifying the strength of a time-varying electrical signal. The most common typesents are the average (or DC) value of a signal waveform, which corresponds to just mea-ean voltage or current over a period of time, and the root-mean-square (rms) value, whichount the fluctuations of the signal about its average value. Formally, the operation of computingalue of a signal corresponds to integrating the signal waveform over some (presumably, suitablyod of time. We define the time-averaged value of a signal x(t) as

(11.46)

he period of integration. Figure 11.47 illustrates how this process does, in fact, correspondg the average amplitude of x(t) over a period of T seconds.

6 Sinusoidal waveforms.

x1 t( ) A cos wt( ) and x2 t( ) A cos wt f+( )= =

f natural frequency1T--- cycles/sec, or Hz( )= =

w radian frequency 2pf radians/sec( )= =

f 2p ∆TT

------- radians( ) 360∆TT

------- degrees( )= =

A sin wt( ) A cos wtp2---–

=

w 2pf=

x t( )⟨ ⟩ 1T--- x t( ) td

0

T

∫=

A wt f+( )cos⟨ ⟩ 0=

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A useful me

Note immedEq. (11.47), of the squarorder to obt

Solution

The major dinductors is as opposed circuit of FigApplying KV

Observing thto obtain

Equation (1differential e

FIGURE 11.4

FIGURE 11.4element.


©2002 CRC P

asure of the voltage of an AC waveform is the rms value of the signal, x(t), defined as follows:

(11.47)

iately that if x(t) is a voltage, the resulting xrms will also have units of volts. If you analyzeyou can see that, in effect, the rms value consists of the square root of the average (or mean)e of the signal. Thus, the notation rms indicates exactly the operations performed on x(t) inain its rms value.

of Circuits Containing Dynamic Elements

ifference between the analysis of the resistive circuits and circuits containing capacitors andnow that the equations that result from applying Kirchhoff ’s laws are differential equations,to the algebraic equations obtained in solving resistive circuits. Consider, for example, the. 11.48 which consists of the series connection of a voltage source, a resistor, and a capacitor.L around the loop, we may obtain the following equation:

(11.48)

at iR = iC, Eq. (11.48) may be combined with the defining equation for the capacitor (Eq. 4.6.6)

(11.49)

1.49) is an integral equation, which may be converted to the more familiar form of aquation by differentiating both sides of the equation, and recalling that

(11.50)

7 Averaging a signal waveform.

8 Circuit containing energy-storage

xrms1T--- x2 t( ) td

0

T

∫

vS t( ) vR t( ) vC t( )+=

vS t( ) RiC t( ) 1C--- iC dt

∞–

t

∫+=

ddt---- iC dt

∞–

t

∫ iC t( )=

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to obtain the following differential equation:

where the arObserve t

that this is nexample, wethe followin

or

Note the simexcept for thof either equ

We can gelement can

where y(t) rconsist of coential equat

Consider Application

Equation (1derivative. Tobtain:

FIGURE 11.4


©2002 CRC P

(11.51)

gument (t) has been dropped for ease of notation.hat in Eq. (11.51), the independent variable is the series current flowing in the circuit, andot the only equation that describes the series RC circuit. If, instead of applying KVL, for

had applied KCL at the node connecting the resistor to the capacitor, we would have obtainedg relationship:

(11.52)

(11.53)

ilarity between Eqs. (11.51) and (11.53). The left-hand side of both equations is identical,e dependent variable, while the right-hand side takes a slightly different form. The solutionation is sufficient, however, to determine all voltages and currents in the circuit.

eneralize the results above by observing that any circuit containing a single energy-storage be described by a differential equation of the form

(11.54)

epresents the capacitor voltage in the circuit of Fig. 11.48 and where the constants a0 and a1

mbinations of circuit element parameters. Equation (11.54) is a first-order ordinary differ-ion with constant coefficients.now a circuit that contains two energy-storage elements, such as that shown in Fig. 11.49.of KVL results in the following equation:

(11.55)

1.55) is called an integro-differential equation because it contains both an integral and ahis equation can be converted into a differential equation by differentiating both sides, to

(11.56)

9 Second-order circuit.

diC

dt------- 1

RC-------iC+ 1

R---

dvS

dt-------=

iR

vS vC–R

--------------- iC CdvC

dt--------= = =

dvC

dt-------- 1

RC-------vC+ 1

RC-------vS=

a1dy t( )

dt------------ a0 t( )+ F t( )=

Ri t( ) Ldi t( )

dt----------- 1

C--- i t( ) dt

∞–

t

∫+ + vS t( )=

Rdi t( )

dt----------- L

d2i t( )dt2

------------- 1C---i t( )+ +

dvS t( )dt

--------------=

ress LLC

or, equivalently, by observing that the current flowing in the series circuit is related to the capacitorvoltage by i(t) = CdvC /dt, and that Eq. (11.55) can be rewritten as

Note that aland (11.57)

where the gcapacitor voequation wican therefor

Phasors a

In this sectiocomplex num

Phasors

Let us recalwhose arguamplitude ois therefore

1. Any s

and a

2. A phpeak signa

3. Whensinus

Impedance

We now anaThe result wthe same nowill be exten


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(11.57)

though different variables appear in the preceding differential equations, both Eqs. (11.55)can be rearranged to appear in the same general form as follows:

(11.58)

eneral variable y(t) represents either the series current of the circuit of Fig. 11.49 or theltage. By analogy with Eq. (11.54), we call Eq. (11.58) a second-order ordinary differentialth constant coefficients. As the number of energy-storage elements in a circuit increases, onee expect that higher-order differential equations will result.

nd Impedance

n, we introduce an efficient notation to make it possible to represent sinusoidal signals asbers, and to eliminate the need for solving differential equations.

l that it is possible to express a generalized sinusoid as the real part of a complex vectorment, or angle, is given by (ω t + φ) and whose length, or magnitude, is equal to the peakf the sinusoid. The complex phasor corresponding to the sinusoidal signal Acos(ω t + φ)defined to be the complex number Ae jφ:

(11.59)

inusoidal signal may be mathematically represented in one of two ways: a time-domain form

frequency-domain (or phasor) form

asor is a complex number, expressed in polar form, consisting of a magnitude equal to theamplitude of the sinusoidal signal and a phase angle equal to the phase shift of the sinusoidall referenced to a cosine signal. using phasor notation, it is important to make a note of the specific frequency, ω, of the

oidal signal, since this is not explicitly apparent in the phasor expression.

lyze the i-v relationship of the three ideal circuit elements in light of the new phasor notation.ill be a new formulation in which resistors, capacitors, and inductors will be described intation. A direct consequence of this result will be that the circuit theorems of section 11.3ded to AC circuits. In the context of AC circuits, any one of the three ideal circuit elements

RCdvC

dt-------- LC

d2vC t( )dt2

------------------ vC t( )+ + vS t( )=

a2d2y t( )

dt2-------------- a1

dy t( )dt

------------ a0y t( )+ + F t( )=

Ae jf complex phasor notation for A wt f+( )cos=

v t( ) A wt f+( )cos=

V jw( ) Ae jf=

ress LLC

defined so fresistance. Tdependent reexcitation. Fimpedance ftreats the cirideal circuit

Let the so

without lossLet us exam

The impephasor curre

The impe

FIGURE 11.5


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ar will be described by a parameter called impedance, which may be viewed as a complexhe impedance concept is equivalent to stating that capacitors and inductors act as frequency-sistors, that is, as resistors whose resistance is a function of the frequency of the sinusoidaligure 11.50 depicts the same circuit represented in conventional form (top) and in phasor-orm (bottom); the latter representation explicitly shows phasor voltages and currents andcuit element as a generalized “impedance.” It will presently be shown that each of the three

elements may be represented by one such impedance element.urce voltage in the circuit of Fig. 11.50 be defined by

(11.60)

of generality. Then the current i(t) is defined by the i-v relationship for each circuit element.ine the frequency-dependent properties of the resistor, inductor, and capacitor, one at a time.dance of the resistor is defined as the ratio of the phasor voltage across the resistor to thent flowing through it, and the symbol ZR is used to denote it:

(11.61)

dance of the inductor is defined as follows:

(11.62)

0 The impedance element.

vS t( ) A wt or VS jw( )cos Ae j0°= =

ZR jw( )VS jw( )I jw( )

------------------ R= =

ZL jw( )VS jw( )I jw( )

------------------ wLe j90° jwL= = =

ress LLC

Note that thmagnitude o“impede” cuat low signalto behave man inductor of this magn

The impe

where we hadependent cfrequency, aan open circnegative, sincapacitor ar

The impeproblems, bfor DC circuimpedance o

where R is cX has been iThe examplcircuit.

Example 1

In Example a parallel-plcapacitor wa

FIGURE 11.5plex plane.


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e inductor now appears to behave like a complex frequency-dependent resistor, and that thef this complex resistor, ωL, is proportional to the signal frequency, ω. Thus, an inductor willrrent flow in proportion to the sinusoidal frequency of the source signal. This means that frequencies, an inductor acts somewhat like a short circuit, while at high frequencies it tendsore as an open circuit. Another important point is that the magnitude of the impedance ofis always positive, since both L and ω are positive numbers. You should verify that the unitsitude are also ohms.dance of the ideal capacitor, ZC( jω), is therefore defined as follows:

(11.63)

ve used the fact that 1/j = e–j90° = –j. Thus, the impedance of a capacitor is also a frequency-omplex quantity, with the impedance of the capacitor varying as an inverse function ofnd so a capacitor acts like a short circuit at high frequencies, whereas it behaves more likeuit at low frequencies. Another important point is that the impedance of a capacitor is alwaysce both C and ω are positive numbers. You should verify that the units of impedance for ae ohms. Figure 11.51 depicts ZC( jω) in the complex plane, alongside ZR( jω) and ZL( jω).dance parameter defined in this section is extremely useful in solving AC circuit analysis

ecause it will make it possible to take advantage of most of the network theorems developedits by replacing resistances with complex-valued impedances. In its most general form, thef a circuit element is defined as the sum of a real part and an imaginary part:

(11.64)

alled the AC resistance and X is called the reactance. The frequency dependence of R andndicated explicitly, since it is possible for a circuit to have a frequency-dependent resistance.es illustrate how a complex impedance containing both real and imaginary parts arises in a

1.4 Capacitive Displacement Transducer

11.3, the idea of a capacitive displacement transducer was introduced when we consideredate capacitor composed of a fixed plate and a movable plate. The capacitance of this variables shown to be a nonlinear function of the position of the movable plate, x (see Fig. 11.39).

1 Impedances of R, L, and C in the com-

ZC jw( )VS jw( )I jw( )

------------------ 1wC-------- e j– 90° j–

wC-------- 1

jwC----------= = = =

Z jw( ) R jw( ) jX jw( )+=

ress LLC

In this example, we show that under certain conditions the impedance of the capacitor varies as a linearfunction of displacement—that is, the movable-plate capacitor can serve as a linear transducer.

Recall the

where C is (variable) ddetermined

so that

Thus, at a fiproperty mawas shown aas a consequby a correspotwo resistorThe bridge i

Using pha

If the nomingiven by

where d is tcapacitors (i

FIGURE 11.5ment transdu


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expression derived in Example 11.3:

the capacitance in picofarad, A is the area of the plates in square millimeter, and x is theistance in millimeter. If the capacitor is placed in an AC circuit, its impedance will beby the expression

xed frequency ω, the impedance of the capacitor will vary linearly with displacement. Thisy be exploited in the bridge circuit of Example 11.3, where a differential pressure transducers being made of two movable-plate capacitors, such that if the capacitance of one increased

ence of a pressure differential across the transducer, the capacitance of the other had to decreasending amount (at least for small displacements). The circuit is shown again in Fig. 11.52 where

s have been connected in the bridge along with the variable capacitors (denoted by C(x)).s excited by a sinusoidal source.sor notation, we can express the output voltage as follows:

al capacitance of each movable-plate capacitor with the diaphragm in the center position is

he nominal (undisplaced) separation between the diaphragm and the fixed surfaces of then mm), the capacitors will see a change in capacitance given by

2 Bridge circuit for capacitive displace-cer.

C8.854 10 3–× A

x---------------------------------=

ZC1

jwC----------=

ZCx

8.854 jw A--------------------------=

Vout jw( ) VS jw( )ZCbc x( )

ZCdb x( ) ZCbc x( )+----------------------------------

R2

R1 R2+-----------------–

=

CeAd

------=

CdbeA

d x–----------- and Cbc

eAd x+------------= =

ress LLC

when a pressure differential exists across the transducer, so that the impedances of the variable capacitorschange according to the displacement

and we obta

Thus, the ou

Reference

Irwin, J.D., 1Nilsson, J.WRizzoni, G.,

RidgeSmith, R.J. a1993. The EBudak, A., PVan Valkenb


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in the following expression for the phasor output voltage, if we choose R1 = R2.

tput voltage will vary as a scaled version of the input voltage in proportion to the displacement.

s

989. Basic Engineering Circuit Analysis, 3rd ed., Macmillan, New York.., 1989. Electric Circuits, 3rd ed., Addison-Wesley, Reading, MA. 2000. Principles and Applications of Electrical Engineering, 3rd ed., McGraw-Hill, Burr, IL.nd Dorf, R.C., 1992. Circuits, Devices and Systems, 5th ed., John Wiley & Sons, New York.

lectrical Engineering Handbook, CRC Press, Boca Raton, FL.assive and Active Network Analysis and Synthesis, Houghton Mifflin, Boston.urg, M.E., 1982, Analog Filter Design, Holt, Rinehart & Winston, New York.

ZCdb

d x–8.854 jwA------------------------ and ZCbc

d x+8.854 jwA------------------------= =

Vout jw( ) VS jw( )d x+

8.854 jwA------------------------

d x–8.854 jwA------------------------ d x+

8.854 jwA------------------------+

--------------------------------------------------------R2

R1 R2+-----------------–

=

VS jw( ) 12-- x

2d------

R2

R1 R2+-----------------–+

=

VS jw( ) x2d------=

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Chapter 11: Electrical Engineering - University of Waterloojzelek/teaching/syde361/0066ch11.pdf · 11 Electrical Engineering ... This section presents the fundamental laws of circuit