UNIT 1 CIRCUIT ELEMENTS AND KIRCHHOFF'S LAWS

Structure 1.1 Introduction

0 bjectives

1.2 Voltage, Current and Power 1.2.1 Rep~.esentation 1.2.2 References and Conventions 1.2.3 Standard Prefixes

1.3 Sources 1.3.1 Ideal Voltage Sources 1.3.2 Ideal Current Sources 1.3.3 Practical Sources 1.3.4 Dependent Sources

1.4 Passive Circuit Ele~nents and their Properties 1.4.1 Resistance 1.4.2 inductance 1.4.3 Capacitance 1.4.4 (:onstructional Aspects

1.5 Intercoonectio~i Constraints 1.5.1 Kirchhoff's Current Law (KCL) 1.5.2 Kirchhoff's Voltage Law (KVL)

1.6 Signal Wavefon~~s 1.6.1 Waveform Representation 1.6.2 Step Function 1.6.3 Periodic Signals 1.6.4 Effective Value

1.8 Answers to SAQs

1 1 INTRODUCTION

An electricd circuit, also called an electrical ~~etwork is formed through the interconnection of various circuit elements. The behaviour of the circuit dcpcnds as much on the inallner of interconnection of the various e le~~ler~ts as on the characteristic properties of the ele~nents themselves. In this Unit, we begin our study of circuits by leanling about these two separate aspccts. After a brief review of the concepts of electric current and voltage, we define the nature of each circuit elen~ent in ter~tE of the voltage and/or current at its terminals. We the11 proceed to discuss the two Kirchhoff's laws, which specify the ways in which the various element voltages and currents are constrained by the pattern of il~tercolinection of the circuit elements i.e., by the topology of the circuit configuration. Finally we will get acquainted with the representation of a time varying sigl~al (voltage or current) and the characteristics of some il~lporta~lt types of such signals.

0 bjectives

After a study of this unit, you should be able to

exp la i~~ the concepts of voltage and current and the notation used for their representation,

distinguish between active and passive circuit elements,

identify the tenuinal relations of current slid voltage sources and of resistors, inductors and capacitors and make simple calculatior~ therefrom,

establish the inlplication of Kirchhoff's voltage and current laws and form the pertinent equations,

describe the ~ ~ ~ e a n i n g of thc wavefonn of a signal and identify the waveforms of a step function and of a general periodic function, and

deduce the average, peak and effective values of a givcn periodic signal.

lotrodudio~~ to Circuits

VOLTAGE, CURRENT AND POWER

1.2.1 Representation The two fu~ldalnelltal electrical quantities used to describe the couditiols in an electrical circuit are thepotenrinl differerzce and thc current. As its very ualne implies, the luniler indicates the difference of electric potential across a pair of points. This quantity is nlore 1

conl~nonly termed voltage as its unit of measurcnlent is the volt. We use the sy~nbol V to 1 4

indicate potential difference.

Note : The colnnlon practice in E~iglisli litcraturc is to use the sylnbol V to represent potential difference or voltage. However, the adoption of the same symbol V to indicate a certain quantity (potential difference) as well as its unit of measurement ?

(volt) is liot a healthy practice. 11 can lead to awkward expressiolls like V = 4 V, j where the left-hand side stands for the potential difference across a specitied pair ol' terminals, the right-hand side indicating its value as 4 volts. The continental practice is to use the synlbol U for potential difference and in fact IS 3722 (Part 2) recolnnlends the use of U. We however succulnb to the expediency of u s i ~ ~ g the salne synibol (viz. V) for voltage, as in lilost other text books in English so as to avoid confusion. To remove the awkwi~rdness in expressiolls of the type cited, we shall uqe appropriate subscripts on the left-hand side, wherever warranted.

Electricil current (I) on the other hand is a nleasure of the rate of flow of electrical charge (Q) through a section of the conductor at any point in Ihe circuit, and is measured in amperes (A), one ampere being equivalent to a flow rate of one coulomblsec. You call visualise the two quantities V and I as annlogous to pressure difference and llow ralc in a water distribution system. It is interesti~lg to observe that the product of the two spccified variables in either of these systelns connotes the rate of transfer of energy and therefore has the dimelaiola of power (P) which is lneasured in ternis of a conillion unit watt (W), one watt representi~~g an energy transfer rate of one joule per second. Incidentally, a joule is also referred to as a watt-sec.

In electrical com~nunication, informatiou is trallsmitted in the for111 of a voltage or a current and the ten11 signal is used to indicate either of these quantities. Practice has sanctified the use of this tern1 as a co~nnion descriptio~i of voltage or current irrespective of the context in which it appears.

To distinguish between general signals whose values can change with time (tiine-varying signals) and those which are constant, we adopt the following coliventiolls :

(a) Lower case letters v and i are used lo denote instantaneous values of time-varying signals. They are also solile times indicated as v (t) or i (t) to e~nphasise their depelide~ice on tinie t.

(b) Capital letters V, I arc used to indicate values of steady unvaryilig signals. The 1;lttcr are rrferred lo as d.c. sign;lls. Capik~l letters V, I are also used to indicate soma agrecd constant i~ttributc of a lime varying signal (e.g., rnls value of a periodic signal ;IS wc shall see later).

The delineation of the inanner in which a signal varies with time is called waveform repre- sentation. The wavefor~ns of the voltages available fro111 a household electric power socket and from the ter~iii~lals o l thc dry cell in your pocket calculator are shown in Figure 1.1.

(a1 ( b ) Fig. 1.1 : Wa~eforms of voltage available

(a) from Ihe hou.;ehold power socket. and (h) from a 1.5 V dry cell

1.2.2 References and Conventions Circuit Elements and Kirchhoff s Laws

When current i passes through a circuit ele~ile~lt having the teriiiinals n and h, we need to indicate the direction of current flow. This is straightforward if the direction of flow is always from, say, a to 1). In a general situation however the direction of current through a given element may change from time to time. Secondly, even if the current direction does not change, the actual direction may not be known at the beginning of the solution of a problem. What we can do therefore is to inark an arrow beside the circuit elelllent in its diagrammatic representation to indicate what is tenlied as the reference direction for the current. If the actual current direction coincides with the reference direction, i will be taken to have a positive numerical value. Conversely, a 11eg;itive value of i is indicative of the fact that the current flow is opposite to the marked reference direction. Two ways in which the current reference direction may be indicated in a circuit diagram are shown in Figure 1.2(a).

(a) Current reference

v t v -

b a 0 0

(h) Voltage reference

Fig. 1.2 : Alrernative ways of indicating Current and Voltage references

111 a sinlilar way, we need to indicate the reference direction for the potential difference between two terminals in general and between the two tcnni~lals of a circuit element in particular. Figure 1.2(b) illustrates two common practices in this regard. The negative sign '-' is often omitted in the first convention, if thc ter~lli~lal to which it refers is otherwise clear. Note that the arrow in the second co~lve~~tioil of Figure 1.2(b) i~ldicates the direction of voltage drop i.e., the tenilinal correspo~~di~lg to the tail of the reference arrow is take11 to be at a higher potential than that associated with the tip of the arrow. Another alternative which we shall find convenient to use is to indicate the potential difference betweell the ter~ninals a and 1) as v,, where v,, is taken as positive if there is a drop fro111 a to b (i.e., a is positive with respect to 0 ) and negative otherwise. With this double subscript notation, there is no need to exhibit the voltage refere~icc 011 the circuit diagfi~m. However the two te r~ni~~als of the circuit elelllent have to be distinctly labelled.

You should cle&ly understa~id the difference betweell the reference directions and actual voltage/current directions. The actual voltage polarity or current direction at ail i~istii~lt may or nlay not agree with the marked reference. If it does, the signal is taken to have il positive numerical value. If not, i t has a negative value.

Figure 1.3 shows a circuit e le~ne~it carrying a current i(t) and having a voltage drop v (t) across it in the direction of current. The product v (t) i (t) iudicatesp ( 1 ) the instanta~leous value of power nhsorhed (rate of absorption of energy) by the circuit

t V

element. If the reference sign of either v or i is reversed, the product v i would indicate the instanta~leous value of power supplied by the elenie~it to lhe resl of the electrical Fig. 1.3 : Combined Voltage and network in which it is connected. ('urrent References

Example 1.1

Let the wavefornls of voltage v ( t ) and current i (t) associated with the circuit elenlent of Figure 1.3 be as shown in Figure 1.4.

The following inferences call be drawn :

When t = 1 s, the tenninal a is at 1 V higher potential than b i.e., v,,, = 1 V, a current of 1 A flows fro111 n to h, and a power of 1 W is absorbed by the element.

When r = 7 s the terminal b is at I V higher potelltin1 thau a i.e. v,,, = - 1 V, or vbn = 1 V, a current of 1 A flows fro111 n to b and the powerp absorbed by the elenlent is (- I) x (1) = -1 watt, i.e., a power of 1 watt is delivered by the elenle~rl.

introduction to Circuits

Fig. 1.4 : Waveforms of v and i in Example 1.1

1.2.3 Standard Prefixes

Coi~sidering the whole range of applications in electrical engineering, the basic units used for various electrical quantities become either too snlall or too large in many coi~texts. For instance, the basic unit of power which is one watt is too large to indicate the power dissipatcd in an electronic device which is best given in tenns of ~nilliwatts W). On the other hand, it is too snlall to specify the rating of a large electrical power generator which is best expressed in niegawatts ( l o 6 W). Similar considerations apply to other electrical units as well. Thus, to facilitate the use of co~~venient nunlbers in expressi~~g quantities which have a large range, we use the standard prefixes indicated in Tabie 1.1 to signify the pertinent ~nultiples/sub~nultiples of the basic units.

Table 1.1 : Standard Prefixes for Units

Prefix

giga I inega

kilo

hecto

pico

SAQ 1 Referring to Figures 1.3 and 1.4, draw appropriate inferences about the voltage, current and power at t = 2, 3 and 5 s in the liianner of Exainple 1.1.

SAQ 2 Referring to Figures 1.3 and 1.4, s'ketch the wavefonn of the power absorbed by the element for 0 < t c 8 and identify the intervals of time in which the element is delivering power. What is the total energy absorbed by the element in the interval O < t < 8 ?

1.3 SOURCES

We shall first introduce ourselves to two types of circuit elenlents which provide life to an electrical circuit. These are the voltage and current sources. They are capable of supplying net electrical energy to the network to which they are co~ulected ant1 h e ~ ~ c e are called active elements. 111 the absence of active elements, all currents and voltages in a network tetd to be zero and the network may be regarded as dormant.

1.3.1 Ideal Voltage Sources

Circuit Elements and KirchhoWs Laws

An ideal voltage source which is represented as in Figure 1.5 is a 2-tenninal element which maintains the potential difference between its terminals a and b as a specified function of time vs (t).

For example if

i I for 11 5 - < n + l , n = 0 ,1 ,2 ,... T

= 0 for t c 0, Fig. 1.5 : An ideal vol~age source

then the tenninal voltage vab (t) of the source will have the wavefonn shown in Figure 1.6. The symbol e, standing for electromotive force (emf), is also often used to indicate the voltage of a source instead of the symbol v.

Fig. 1.6 : Waveform for v, = 10 ( -- ) for n s I< n + I ; . - 0 , l . z ,... T

I A voltage source is said to be a d.c. source (direct current source) when 1% (t) is a constant i.e., if the tenninal voltage of the source has an unvarying ~rlagnitude and polarity. A

1 d.c. source is coillnlonly represented by the fanliliar sy~llbol of a d.c. battery shown in

/ Figure 1.7 and has a tenninal voltage wavefonll of the type shown in Figure 1 I@). The battery voltage is usually represented by the IetterE; the positive polarity

I corresponds to the longer of the two lines in the symbol. i A source elenlent is nonllally employed in a circuit to deliver power to the rest of the network and therefore it is conventional with these elements to take the reference direction for current in the direction of voltage rise through the element, as shown in Figures 1.5 and 1.7. With this choice of reference directions the product vi represents Fig. 1.7 : A d . ~ . voltage source the power supplied by the source. Note that this amnge~rle~l t is in contrast to the pair of references shown in Figure 1.3, which r e m a i ~ ~ ~ as the preferred cho~ce for circuit ele~nents which are not sources of power.

11

Introduction to Circuits The term ideal voltage source implies that the temlinal voltage will be ~naintained as the specified function of tiine (e.g, as in Figures 1.1 aud 1.6) irrespective of the value of currei~t supplied by it. The latter value gets fixed by the rest of the circuit to which the source is connected. h~ other words, an ideal voltage source inaintains its terminal voltage as a specified function of time, no inatter to what load it is coiu~ected.

1.3.2 Ideal Current Sources The counterpart of the voltage source is the current source, which has the characteristic of supplying through its tennii~als a current which is a specified function of time. An ideal current source is denoted as in Figure l.X(a). A less coil~~nonly used representation is that show11 in Figure 1.8(b). Again, the ideality of the source arises from the property that the strength of current through its terminals is maintained no matter to what load it is connected and thus irrespective of what voltage exists at its terminals. There is no specific distinguishing symbol for a d.c. current source similar to that for a d.c. voltage source.

Fig. 1.8 : Alternative representations of a Current source

The exp;~nsion of d.c. is direct current. It lnay thus appear to be farcical to use a term like d.c. current, which on expansion should strictly read direct current current. However such usage is sanctified by practice and the understanding is that the pair of letters d.c. is an adjective used to ii~dicate unidirectional uiivarying quantities (current, voltage or power).

1.3.3 Practical Sources Ideal voltage and current sources do not exist. In practice the tennilla1 voltage of a voltage source depei~ds on the current it is called upon to supply. Similarly the terminal current of a current source varies with the voltage it is called upon to establish by virtue of its connection in a circuit. However over liniited ranges of output current and output voltage respectively, there exist practical devices which approxiinate the ideal voltage source or the ideal current source. Furtherinore, you will learn later on how to model a practical source as a co~nbiiiation of an ideal source and other circuit elen~ents. Thus the co~~cepts of an ideal voltage source and an ideal current source are useful ones.

Exainples of practical voltage sources range froin the dry cells used for portable torch lights and transistor radios and the lead-acid storage batteries used in autoinobiles (vide Figure 1.9(a)) to the giant-sized ge~ierators in large electric power generating stations. Electronically regulated d.c. power supply units conlnlo~lly used in electrical laboratories function essentially as ideal voltage sources upto a specified nlaximu~n value of the output current. Current sources, on the other hand are less common. One example is the photoelectric cell, which can be nod el led as a current source whose strength is a function of the light falling on it. Another is a transistor which under certain operating conditior~. provides a nearly constant current output between its emitter and collector terminals, as shown in Figure 1.9(b). There also exist electronically operated laboratory units which provide a constant d.c. current with varying output voltage below a certain ~ n a x i ~ n u ~ n value.

Terminal voltage

Circuit Elements and KirchhoE's Laws

0 5 v collector - emitter voltage --c

Fig. 1.9 : Terminal characteristics of ( a ) a practical d.c. Voltage source (12-V Automohile hattery) (h) a practical d.c. Current source (Transistor)

1.3.4 Dependent Sources

The voltage and current sources described in the preceding paragraphs are termed independent sources because their strengths are i~idepe~idently prescribed and do not depend on the value of ally other electrical quantity in the circuit in which they are connected. In nod el ling electronic devices, it is f o u ~ ~ d useful to illcorporate voltage and current sources whose strengths depend on either the voltage or the currelit that would prevail at a different locatio~i in the overall circuit configuration. These are called dependent or controlled sources. They nlay he coilsidered to have 2 tenninal pairs, the input terminal pair which senses the controlling quantity (voltage or current) and the output ter~iii~ial pair constituti~ig the teniiinals of the de.pendent source. Dependent sources colne in four fonns as set forth in Table 1.2. For each one of these elements, we shall assume that the stre~igth of the co~itrolled or dependent source is proportional to the controlling quantity, as i~idicated in the table. The audio amplifier in a radio receiver call be niodelled in terms of a voltage controlled voltage source. A siiiall voltage developed across its input ter~iii~ials through the preceding stage serves as the controlli~ig signal to drive the depe~ide~it voltage source to furnish to the loudspeaker iin enlarged or amplified versioi~ of the input signal.

Table 1.2 : Dependent Sources

Type of De endent 4' ource

Voltage Controlled Voltage Source (VCVS)

Voltage Co~itrolled Current Source (VCCS)

Current Coutrolled Voltage Source (CCVS)

Current Controlled Current Source

1 (CCCS)

Representation Input (Controlling

Quantity)

Output (Controlled Quantity)

l n t r o d ~ ~ d i o ~ ~ to Circuits Exiimple 1.2

I A practical d.c. voltage source has the v - i characteristic shown in Figure 1.10. Can it be co~~sidered an ideal voltagr source? What is the maximum power that the source can deliver to a load co~l~~ec ted to it ?

I I\ Solution " The ter~ni~lal voltage stays co~ l s t a~~ t at 10 V for a load current of 0 to 100 mA. For

this range of load current, the source can be considered to be an ideal one. The nlaximunl power that the source call deliver is the ~ n a x i ~ n u ~ ~ ~ value of the vi-product

0 0 100 110 on the characteristic. This occurs at a current of 100 mA and is equal to

I / m A - 10 x 100 x = 1 watt.

Fig. 1.10 : Source for Example 1.3- Example 1.3

A comn~ercially available power source has the v - i characteristic show11 in Figure 1.1 1. C ~ I I ~ I I I ~ I I ~ 011 its characteristic as a voltagelcurrent source.

Fig. 1.11 : Source for Example 1.3.

Solution

This is an interesting unit which fu~~ct io~ls as a voltage source or a currellt source. It behaves as an ideal 60 V d.c. voltage source for a current output from 0 to 3.33 A and as an ideal d.c. current source of 10 A for an output voltage from 0 to 20 V. The n~aximum power that call be delivered in either node is 200 W. For the intermediate zoue of (20 V . . . 60 V) and (3.33 A . . . 10 A) it can not be co~~sidered either an ideal voltage source or an ideal current source. It however delivers 200 W of power when operating iu this zone.

SAQ 3 Fill up the blanks in the following statements :

(a) An ideal ' source ~naintai~ls its terminal ' at the specified value irrespective of the current it is called upon to deliver.

(b) An ideal d.c. source has a variable tennilla1 voltage.

SAQ 4 State if the followi~~g assertio~~s are true or false :

(a) An ideal current source with no restrictio~ls on its voltage range should be cauable of deliveri~~g illfinite power.

(b) The dry cell used in your pocket calculator call be referred to as a current source.

SAQ 5 Calculate the power supplied or absorbed by the independentldepende~lt sources in the circuits of Figure 1.12 (a) and (b).

Circuit Elemeuls and Kirchhotl's Laws

Fig. 1.12 : Circuits for S A O 5

1.4 PASSIVE CIRCUIT ELEMENTS AND THEIR PROPERTIES

From experimental observatio~i of the behaviour of aclual electric devices, several characteristic relationships between terminal voltages and currents are identified. Various circuit elements are then defined based on their voltagecurrent (v-i) relationships. Chief among them are.the three linear passive two-terminal circuit elements viz., the resistance element, the inductance. element and the capacitance element. They are called linear elements because their tenninal v-i relationships are either li~iear algebraic or differential equations. They arepassive since they either dissipate or store electrical energy and are not capable of delivering net electrical energy to the rest of the circuit in which they are incorporated. In the sequel we shall have a closer look at the characteristics of these elements.

1.4.1 Resistors A linear resistance element or a resistor is one which maintains a proportio~iality relationship between the voltage across its termi~~als and the current passi~ig through it in the direction of the voltage drop. A resistor is i~idicated by either of the represealations shown in Figure 1.13, of which we shall use the one in Figure 1.13(a) in the rest of our work.

i i t ) - + v (1)

Fig. 1.13 : Alternative representations of a Resistor

The terminal v-i relationship of a linear resistor is sketched in Figure 1.14 and is mathematically represented by the equation, '

where R is a constant and is referred to as the value of resistance dthe device. R is measured in ohms, one ohm being equal to 1 voltlampere and is represented by the sy~iibol P i n abbreviated' form. The unit is in honour of the Genna~i scie~itist George Ohm who deduced through experiments the relationship known as Ohm's law which is embodied in

' / /

Eq(l.1). Resistance is a measure of the oppositio~i to the flow of curre~~t in the sense that 0 L L a

more the resistance, the more.woulrl be the potential difference (voltage) required to drive I -- the same amount of current througn the device. Consider the analogy of pumping water

Fig 1 14 Tcrn i~na l rel.tilui~ through a pipe line network. The ~liore the frictio~ial resista~ice offered by the pipe l i~~es , the 5 ~ ~ ~ ~ ' o i A l l n e ~ r more would be the pressure difference ~ieeded to sustai~ra give11 flow rate. Electrical resisior

resista~lce offered to the flow of current is a~ialogous to the frictio~ial resista~ice offered by the pipe lines in the above exaniple. As an alter~iative to the expressio~i in Eq. (1.1), we call formulate the v-i relation of the device as

Iutrodudiou to Circuils where G is tenned the conductance, nleasured in sie~ile~ls (abbreviated as S). A few decades earlier, i t was the practice to refer to this unit as the nlho, i.e., oh111 spelt backwards. Obviously G is the reciprocal ofR and >Illy resistor having a resistance ofR ohms can also be regarded as a devicc having a co~~ducta~ice of R-' or 1IR siemens. One implicatio~~ of Eq. (1.1) i~nd (1.2) is that i and v cilll be time varying quantities and the equations are valid if i and v are ~lleasured at the same instant. As a consequeqce, if the resistor has a voltage havi~lg ill1 arbitrary wavefor111 applied ilcross it, the currelit through the resistor should have an exactly silililar waveform scaled by a factor R.

Resislors which are not linear have characteristics different from that shown in Figure 1.14. So111e nlay have approxi~iiately linear characteristic over a specified range of currents, others niay he radically different. But all of the111 satisfy a ge~ieral relation of the fon~ i v = f (i) i.e, v (t) is some fu~lction of the currelit i (t) at that instant. We shall not pursue the characteristics of no~llinear resistors further at this point.

Fro111 Eq. (1.1) and (1.2), it call be seen that the power absorbed by the resistor is given by

This power is dissipated in the resistor irreversibly and is converted into heat. You should particularly note that the power dissipated is proportio~lal to the square of the current or voltage and that the power dissipated is 11on-negative for either direction of current flow (for both positive and negative valies of current).

If v is a furlctio~~ of time, thelip is likewise il tunction of time and the expression in Eq. (1.3) gives the instanti~neous value of power i.e., the value of power at the instant the voltage has the value v or the current has the value i. On the other hand, when the voltage applied across the resistor is constant i.c. v (t) is a d.c. quantity V, then the power dissipated in the resistor is also a colstant. In this case,

Finally, let us familiarise ourselves with the nlealli~ig of the two terms short-circuit and open-circrrit betweell two terminals. They correspo~ld to v = 0 and i = 0 respectively. I11 the case of a short circuit, the voltage betweell the te~~iiinals is zero irrespective of the current passing hetween them. On the other h a d , an open circuit constraills the current to be zero, irrespective of the voltage across the tenniuals. A switch for example represents an open circuit when i t is open and a short circuit when it is closed as shown in Figure 1.15.

(a) Switch (I?) Switch closed (c) Switch open (Short circuit) (Open circuit)

Fig. 1.15 Depiction of S h o r ~ circuil and Open circuit conditions of a swilch

Example 1.4

An electrical heating ele~llent is designed to colsulne 250 W with 230 V d.c. applied to it. The heating ele~lle~it call be assu~iled to be a resistor. What is the resistance of the heating element at the operating condition'? If the heating ele~nent ca 11 be safely operated upto a power of 300 W, what is the maximum permissible voltage that can be applied to it ?

Solution

Let V' be the maxinium pennissible voltage correspo~ldillg to a dissipation of 300 W. Since power dissipation for ;I given resistance vanes as the square of the applied voltage we have

Circuit Elernents and KirchhoWs Laws

leading to

This example shows that just 10% overvoltage can lead to 20% increase in power in a resistor with consequent increase in the heat produced. As many electrical devices are vulnerable to high temperatures, one should take adequate precautions to avoid overvoltages or overcurrents beyond a inarginal degree.

1.4.2 Inductors Iilductatlce is the property of a device, usually a coil of wire, to set up magnetic flux (4) when carrying a current. An element which has the property of inductance is called an inductor.

In literature a distinction is inade between a pure irtductor which has only the inductance property and an irrducror which has substantially the inductance property but Itlay also have other conlponent properties like resistance. For the sake of siinplicity we shall ordinarily use the tenn inductor to iinply apure irzductor in the above sense. Where necessary, we shall nlake a particular mention if the inductor is to be regarded as not pure.

In a linear inductor, the flux linkages h (product of the magnetic flux 4 measured in webers and the number of turns N with which it links) is proportional to the current in it. That is,

(111 Eq. (IS) , we have made the sitnplifyiilg assuniptioi~ that every tun1 encloses or ''links" the total flux.)

I The proportionality constant L is called the inductance parameter of the device. It inay __C

a- b be defined as the flux linkages per unit current and is measured in henrys (abbreviated as H). The circuit symbol for an inductor is as show11 in Figure 1.16. + v

Fig. 1.16 : R~~,ILYCIII.I~IOII ot an inductor If the current i is time varying, there is an induced voltage in the inductor which

;~ccordi~ig to Faraday's law of inductiou is given by d(N4)ldr and reduces to L dildr by virtue of Eq. (1.5). This induced voltage acts in a direction to oppose the change of i. If in the inductor of Figure 1.16, dildt were positive, the induced voltage in it should tend to drive a current the other way round in an atteinpt to arrest the growth of current. That is, the inductor should act like a source with tennilla1 n having a positive polarity with respect to terinir~al b. For the voltage reference chose11 in Figure 1.16, the above ailalysis leads to the following tenninal v-i relation of an inductor.

You are urged to consider the case of decreasing curre~~t and convince yourself that the above equation with the same sign is valid for this co~itingeilcy as well. Froin the circuit viewpoint, hereafter we will not be concen~ed with Faraday's law of induction and induced voltages. Rather it is the terminal voltage and the current through the inductor thqt are of interest to us. Eq. (1.6) states that the voltage drop i11 an inductor is proportional to the rate of change of current (in the direction of the voltage drop, as seen froin the reference signs of Figure 1.16) and it is this equation which we shall adopt as the cardinal property of a linear inductor.

When v is given and one wishes to find the associated i, Eq. (1.6) can be recast as

where the last expression renloves the arbitrariness associated with the indefinite integral in the middle expression.

Let us now consider the power input into the inductor, which is obviously

Introduction to Circuits

It is clear from Eq (1.8) that the iiistantaiieous value o f p can be positive or negative, dependii~g on whether i2 is increasing or decreasing. Negative power intake inlplies that the inductor is supplying power to the rest of the circuit. This may appear puzzling at first sight. Let us postpone an examination of this issue for the moment but proceed to calculate the energy input into the inductor. Recall that power is rate of transfer of energy and therefore energy is integral of power. IC w, is the energy input illto the inductor during an interval 0 to t when the current is changed fro111 0 to a value i, we have

This is the energy stored in the inductor when it carries a current i and does not depend 011

the nianner in which the current has been built up to the value i. If i2 decreases, it is clear that the stored energy also decreases. Wheu i = 0, the entire stored energy gets depleted. In ally time interval in which i2 value is decreasing, the inductor returns some energy to the rest of the circuit or in other words its power input is negative. This explains our earlier observation following Eq. (1.8) concerning negative power. (Incidentally, the energy stored in the inductor can be regarded from the electrolnagnetic field theory point of view as the energy stored in the m::gnetic field created by the current in the inductor.)

The foregoing observations demonstrate the fact that an illductor is an energy storage element and not an energy dissipating element like a resistor. It is a passive element since it cannot give out Inore energy than what is stored in it initially.

The behaviour of an inductor when carrying a steady unidirectional current (d.c. current) merits special attention. In this case dildt = O which from Eq. (1.6) implies that the voltage drop across the inductor is zero. Consequehtly in d.c. circuits under steady state (circuits in which all elements carry d.c. currents), all inductors have zero voltage across them and can be therefore regarded as short circuits.

Example 1.5

Find the voltage variation across a 2-H inductor carrying (a) a current i l (b) a current iz as shown in Figure 1.17. Find the energy stored in the inductor at t = O and at t = 6 in both cases.

Fig. 1.17 for Example 1.5

Solution

The wavefortns of vl = 2 d i l /d t a~ld v2 = 2 d i2/d t are sketched in Figure 1.18. Note that (i) v l and v2 are equal for 0 c r c 4 even though il z i2, as the rates of variatio~~ of i l and i2 are equal (ii) vl = v2 = 0 tor 4 c t < 6 as the currents are steady and (iii) vl is negative for 6 < t < 8 even though i l is positive since i l is decreasing.

1 w ~ 2 (0) = x 2 x ( -4 )2 = 16 joules : -

(a)

Circuit Elentents and KirchhofY's Laws

Fig. 1.18 for Example 1.5

1.4.3 Capacitors Capacita~ice is the property of a device (usually a pair of co~lduct i~~g surfaces called plates separated by a dielectric) to store electric charge w h e ~ ~ a potential difference is established across its terminals. An ele~ile~it which has this property is called a capacitor. 111 a linear capacitor, the charge q stored on the capacitor plates is proportioiial to the voltage v across the capacitor. Thus,

where the co~ i s t a~~ t of proportio~~ality C is called the capacitance parameter of the device and inay be def i~~ed as the charge stored per unit applied volli~gc.,Thc unit of capacitance is the farad (abbreviated as F). The circuit sy~nbol for il capi~cilor is shown in Figure 1.19.

This represe~~tatioii closely follows the physical co~ifiguratio~~ of a parallel platc capacitor show11 in Figure 1.20. When v,, is positive, the inside surklccs of plalcs o a~id 11 acquire

I - a

equal and opposite surface charges of q and -q creating all electric field in the space l b between the plates. --- -

+ v

? Electric Field Fig. 1.19 : Representation of a

Fig. 1.20 : ('harges on a capacitor

When v is a varying quantity, q would also vary in sympathy. To establish this varying charge, a currellt has to flow on to the capacitor plates through the co~ l~~ec t i i~g leads and this co~lstitutes the tennilla1 curreut i. As i is the rate of transfer of charge, we have

This is the cardii~al circuit equation for a capacitor. A positive i tends to illcrease the positive charge on plate a. At the same tiine the charge on plate b should beconle more negative requiring an equal current i to flow out from plate h. Thus even though there is no coliductioi~ cum,nt inside the capacitor, one iilay assume from the point of view of ter~lliilal behaviour, that a cur re~~t i flows right through the capacitor from one tenilinal to the other

capacitor

Introduction to Circuits as in a resistor or an inductor.

Note the duality of the tenninal relationships of a capacitor in Eq. (1.10) and Eq:(l . l l ) and the tenninal relationships of an inductor in Eq. (1.5) and Eq. (1.6). If the symbols A, v, i, L are replaced by their dual quantities q, i, v, C the corresponding equations relevant to a capacitor follow fro111 those for an inductor. Watch out for more such dual relationships in the sequel. The knowledge of duality helps you to anticipate and foretell the relations in one donlaill or at least check their validity when you have the knowledge of similar relations in the other domain.

We note that the current in a given capacitor is determined only by the rate of change of voltage across it and not on the magnitude of the voltage. If the capacitor has a d.c. voltage across it, it is obvious that dvldt = 0 and consequently the terminal current is 7xro. Thus a capacitor can be regarded as an open circuit in a d.c. circuit under steady conditions.

Reverting to the case of varying currents and voltages, the voltage v across a capacitor can be expressed in tenns of the current i through it in the following manner :

I

where v(0) = q(0)lC is the capacitor voltage at t = 0.

The power and eliergy inputs into a capacitor can be found on similar lilies as for an inductor. The power input p is given by

It can be argued that an uncharged capacitor, collsequently.one with zero tenninal voltage, has zero energy stored in it. Starting from this state,at t = 0, let the voltage be raised to v in time r,. The energy input illto the capacitor in the process is given by

'\ t 1 t 1 1 w, = J O P d t = S O ~ m d t = d ( v 2 ) = - C I L joules.

2 dr 2 (1.14)

This is then the energy stored in the capacitor when it is charged to a potential of v volts. The energy reniains at this value as long as v is kept unchanged. The entire stored energy is released hack to the circuit when the capacitor voltage is reduced to zero. Like an illductor the capacitor is an energy storage element and not an energy dissipating element. The energy stored depends on the square of the applied voltage unlike in the case of an illductor where it depends on the square of the current. The eliergy expression in Eq. (1.14) can be shown to agree with the energy stored in the electrostatic field set up by the charges on the capacitor plates.

Capacitors and iiiductors can act as energy sources for short periods of time but they can not give out Inore energy than what is originally stored in them. As they can not supply net energy, they come under the class of passive elements.

Example 1.6

If a voltage v = (4 t 3 sin t) volts is applied across a 2-F capacitor, find ;III expression for the current through the capacitor as a function of time and the nlaxin~unl and ~ n i ~ i i ~ n u ~ n values of energy stored in the capacitor.

Solution

d v d i = C - = 2 - (4 + 3 sin t) = 6 cos t aniperes d t d t

1 wc = - C v2 = ( 4 + 3 sin t )2 joules 2

The niaximum and nlinintum values of wc occur at the maximum and minimum 2 values of v i.e., when sin t = +1 a11d - 1 respectively. Thus (wc),,,, = 49 joules and

(MJ~),;, = 1 joule.

1.4.4 Constructional Aspects Circuit Elenieuts aud KirchhofPs Laws

L Resistor The electrical resistance of a device depends on the area available for current flow, the length of the current path and a characteristic property of the conducting nlaterial known as specific resistance p. For a resistor with a regular geonlelry like a wire where the cross-sectional area for current flow is constant along the entire length, the resistance is

t I given by

t where 1 is the length of the path in nletres, A is the cross-sectional area in 111' and p is the specific resistance (also known as resistivity) in ohm-~netres. The resista~ice also varies with tenlperature and for the nlajority of the useful materials, can be put in the foml

R2 = Rl [ I + u ( t 2 - t l ) ] ,

I where R2 and R1 are resistance values at te~ilperatures t2 and t , expressed in "C and u is what is tenned the temperature coeficietit of resistatlce of the co~iducting niaterial.

I Standard resistors and those intended for large power dissipation are wire wound. They enlploy a wire of a suitable metal or alloy wound around an insulating piece of small cross-section. They are expensive and are not suitable for large resistance values and for high freqeuncy applications as needed in electronic circuits. Resisters for co~nnlon

i electronic circuit applications are usually of the carbon-film type or the carbon-composition

i type, both of which employ carbon as the conducti~~g medium. The carbon film resistor conlprises a crra~nic core on which a thin film of carbon is deposited. The carbon

i conlposition resistor consists of a solid conducting rod fornied from a nlixture of carbon particles and a binding i~~sulating material.

Resistors are specified in tenns of their nonlinal resistance value, tolerance in percent and I penrlissible power dissipation in watts. A resistor of 100 Q nonlinal value and 5% tolerance 1 can be expected to have an actual resisl:~nce value in the range 100 Q + 5% or 95 Q to I 105 Q. The ~io~ninal resistance values o l resistors used for electronic circuits are

standardised so that the different units to be stocked may be linlited to a reasonable nun~ber. 1 In the case of 5% tolerance resistors, the 100 S2-resistor is expected to cater to the resistance ! range 95 Q - 105 Q. The next higher value of nominal resistance is fixed as 110 Q , which

provides for the 105 - 115 Q range approximately. A decade nnge is covered by 24 distinct values, each standard value being 1 0 ' ' ~ ~ = 1.1 times its next lower value. Nornlally available values of resistance range fro111 a few ohnls to a few nlegohtns. The physical size of a resistor depends on its power rating and not on its resistance value, higher valued resistors of the sanle physical size having a lower current carrying capacity.

Inductor

An illductor is essentially built as a coil of good conducting nlaterial wound around a11 air *-m-

core or an iron core in which the flux IP is established. Other factors being the same, the inductance value of a linear inductor varies as the square of N, the ~lunlber of lur~ls in the coil, because @ a N and h = N @ a N' for a given current. The iron-core serves to provide a Fig : Representat i0o

larger value of inductance for the same inductor size but has the disadvantage of providing a an iron-cored inductor

~iolilinear @ vs i relationship, which means that the i~iductance value varies with the value of current. An iron cored inductor is so~nelimcs indicated as in Figure 1.21 to highlight the presence of iron. Inductance values nonnally available range fro111 microhenrys to henrys.

Figure 1.22 shows the construction of an inductor with a toroidal iron core. The inductallce value for this configuration is

N2A henrys, L - K R T

where p, is the relative penneability of iron for the particular operating condition,A is the I I radial cross sectional area of the core, 1 is the nlean circunlferential length of the core, N the I I

number of turns and p,, = 4 IT x 10-' H/m is the penneability of air. Fig. 1.22 : Induclor with a toroidal core configuration

Introduction to Circuits Capacitor

Ally two co~~ducting objects separated by a dielectric or i~~sulating medium constitutes a capacitor. A parallel plate capacitor like the one shown in Figure 1.23(a) with the space between the plates filled with a dielectric of perinittivity E has a capacitai~ce value given by

A C = E - farads, d

whereA is the area of plates in in2 and d the distance of separatioi~ of plates in 111. To enhance the effective area of the capacitor plates within a compact overall size, the multiple plate arrangement show11 iu Figure 1.23(b) call be used. Specificatio~ls of a capacitor should indicate its capacitance value, tolerance and voltage rating (maxiinum voltage that can be applied). Capacitors are categorised in terms of the dielectric employed. Air capacitors are i~or~nally used for variahle capacita~~ces in which the overlap area between the plates is varied. Dielectrics co~n~nonly e~nployed are mica, paper, ceramics and pliistics like polystrene. These materials enhance the value of the capacitance and the voltage rating. The typical co~lstructio~~ of a paper capacitor is illustrated in Figure 1.23(c). This co~nprises a sandwich of two metal foils with an i~~sulating layer of paper in between, which is then rolled up illto a cyli~~drical fonu. Nonual capacitor sizes vary froin a few pF to a few pF with voltage ratings of up to a few kV. There also exist electrolytic capacitors which provide large values of capacitances up to thousands of pF within a reasonable size. They have certain l i~~~ i t a t i o i~s and are used for specific applications requiriug large values of capaci t i~ce at low openting voltage levels. - Plates of Area A -7

~ielect6c of permittivity E

(a) Parallel Plate Capacitor

Paper

Terminals

I (b) Multi Plate Capacitor

(c) Tubular Paper (?apacitor

Fig. 1.23 : Examples o f capacitor construction

Example 1.6 2 Find the resistance at 20°C of 1 k~i l of copper wire of cross sectional area 0.1 cili , if

the specific resistance of copper at this t e ~ n erature is 17.3 x Q-111. What would 0" be its resistance at 3S°C if a = 0.0043 per C ?

Solution

Example 1.7

Deduce the units for p, y and E.

Solution

Q-111, Hlm, Flm. They ca 11 be deduced fro111 Eq. (1.15), (1.16) and (1.17).

Example 1.8

If glass is i~itroduced as the dielectric in a parallel plate air capacitor and the plate separation doubled, how is the capacita~ice value modified'? Glass has a relative pennittivity of 8 with respect to air.

Solution

SAQ 6 , A voltage v (t) having the wavefonn shown in Figure 1.24 is app.lied across (a) a 100-& resistor and (b) a 1- yF capacitor. Sketch the waveform of the resulting current in the two cases.

Circuit Elen~ents and KimhbolT's Laws

Fig. 1.24 for SAO 6

SAQ 7

The currelit in a 100 mH inductor is u~~ifonllly reduced from 2 A to zero in 2 1 s .

What would be the voltage across the inductor in this interval of time ?

SAQ 8 For the 10% tolerance class of resistors, identify the next two higher standard resistance values above 10 &. Round off the values to the nearest integer.

Introduction to Circuits SAQ 9 Fill up the bla~iks in the followi~ig s tate~i le~~ts :

A linear relationship is maintained between and current iu a resistor, betweell voltage and in an inductor and between current and in a capacitor.

An inductor is equivalent to when a direct currellt is passed through it.

A capacitor is equivale~it to when a d.c. voltage applied across it.

The size of a carboll co~i~positio~i resistor depe~ids on its 6

SAQ 10 What is the advantage of an iron-cored inductor ovcr all air-cored one ?

We have so b r acquainted ourselves with the various circuit e l e ~ n e ~ ~ t s and their ter~ilinal relations. These relatio~is are expressed in terllls of the voltage across the ter~niiials of the e l e ~ ~ ~ e ~ ~ t andlor the currellt passing though the element. For example v = 2 V is the ter~ili~lal relatiori of a d.c. lead acid cell; i = C dvldt is the ter~ili~ial relation of a capacitor.

We now tun1 our attention to the co~istrai~its i~i~posed on the ele~ile~it voltages and currelits by virtue of their interconnection. These constraints are e l e~ l l e~~ t indepelident and depend purely on the way the various elelne~its are co~u~ected together, the scheme or pattenl of i~~terco~l~~ect ion of the ele~lle~its being referred to as the topology of the network. To eniphasize this fact, liialiy ~ietwork e le~i ie~~ts in what follows, will not be ide~itified by their specified syuibols but will be represented by the C ~ I I I I I I ~ I I sy~ilbol of a rectangle.

To facilil;~le the trratnient in the followi~ig section, let us defii~e ;I tiode as a point where two or niore 2-terminal elen~e~its (branches) are co~i~~ected together and a loop as ;I closed for~iied by the elemenls. Referring to the network of Figure 1.25, n, b, c, are ex;~niples of its nodes. The closed path foniled by ele~lie~lls XI, X , , X,, and X, is an exalllple of a loop.

1.5.1 KirchhofYs Current Law (KCL) Kirchhoff's current law states that the algebraic su~ii of the curre~its lenvit~g a node at ally instant through all the branches co~i~iected to it is zero. Co~isider the node c in Figure 1.35.

Fig. 1.25 : Network used to illustrate KCL and KVL

KCL states that

- i, + jZ - i3 + i4 = 0.

i, and i, are currents leaving the node and are co~lsidered positive, whereas i, and i3 which enter the node are taken as negative. This is the sigiiificance of the term algebraic iii the state~nent of KCL. KCL is a conseque~lce of the fact that electric charge call not continually accu~nulate at a node and is valid irrespective of the nature of elements incident at the node iii question.

1.5.2 Kirchhotrs Voltage Law (KVL) Kirchhoff's voltage law states that the algebraic sum of voltage drops reckoned in a specific direction arouiid a loop is zero at every instant of time.

To illustmte the law for the loop b c f e in the network of Figure 1.25, let us coilsider the voltage drops in the clockwise direction of traversal around the loop. We then have

VI - v2 - v3 + v', = 0

For the reference signs adopted, v, aid v, are voltage rises in the cyclic directio~l chosen and hence are take11 as uegative. This is the sigllificalice of the tern1 algebraic in the statc~ncnt of KVL. Alter~iatively, one could state that the su~rl of voltage drops around a loop equals at every instant the suirl of voltage rises. Again, as with KCL, no restrictions apply as regards the nature of elelnents or the nature of the voltage siguals. KVL is a state~ne~it of the fact that the electric field is conservative, the effect of varying ~naguetic field in the loop having been separately considered in the fonn of voltage drops across inductors.

Example 1.9

Given that El = 12 V, E2 = 6 V , vs = 10 sin 400 t, find i in the circuit of Figure 1.26.

Circuit Elements nod KirchhofPs Laws

1

Fig. 1.26 : Circuit for Example 1.9

Solution

Applying KVL to the loop abcd,

Example 1.10 Show that il + i2 = 0 in the circuit of Figure 1.27.

Fig. 1.27 : Circuit for Example 1.10

introduction to Circuits Solution

Let the curreilt throughX3 frolny to r be i3 and let the current frolnp to q through X, be 4. Froill KCL at node r, we see that the current ill X2 from r to q is also i3.

Now on applicatioi~ of KCL, we have

i, + i, + i, = 0 at node p aiid

iz - i, - i, = 0 at node q

Addition of the above two equations yields the required result.

SAQ 11 Could the statement of KCL as given at the begiililing of 1 S . 1 be altered by replaciilg the word leaving by the word enteritig ?

1.6 SIGNAL WAVEFORMS

A wide variety of signals is met with in electrical engineering. As the final topic in the Unit, which is devoted to the preliininaq~ aspects of network analysis, we shall now look at the ways in which a signal is represented and acquaint ourselves with sollle inlportant categories of signals aiid their characteristics.

1.6.1 Waveform Representation A sigilal (voltage or current) is defined by its value at every instant of time. One way of doing this is through an analytic expression for the same. An alternative method is a graphical repr~selitation. This is called wavefor111 representation and the displayed variation of the signal with respect to time is called the waveform of the signal. Figure 1.28 gives exalnples of wavefonns of a signal. Waveform representation in this manner gives a better physical feel of the signal characteristics than an analytical expression. Secondly the time variation of the signal niay ]lot he always amenable to represelitatioii by an analytic expression. Figure 1 .%(a) is an example of this situation. Such a wavefonn could be the recording of a signal obtained 011 an iilstruiilei~t knowil as cathode ray oscilloscope.

01 I I

lo 2 4 t l s --.

(b) Fig. 1.28 :Waveform representation of signals

Furthennore, even i f il lnathenlatical expression is possible, it may be curnbesome as a single expression may not fit the entire sigilal variation. For example a mathematical

representation of the current signal in Figurc 1.28(b) would be. C h i t Elements and Kirchboffs Laws

The waveforrn represe~itation of a signal 3s a graph is therefore useful. Wc suggest that you acquire the habit of plottilig the wavefonl~ even when you havc on hand convenierit analytical ways of treating the signal. This gives you a better ~nental image of its characteristics. III what follows, you will be introduced to some specific and important waveforms.

i t =

I t 1.6.2 Step Function

0 for t < 0

4e- ' f o r O s t C 2 0 for 2 5 t i 3

4 '2- ('- for 3 s t < 5 0 for 5 5 t

( Consider a simple d.c. circuit (Figure 1.29) comprisilig a d.c. voltage source, a switch S and , a resistorR. Kirchhoff's voltage law forthe loop gives us

When the switch is open, the current in the circuit is zero and 11e11ce v, = 0. Therefore for ) this condition, we havc

vR=O; v s = E Whcn the switch is closed, it is equivalent to a short circuit. For this condition,

Now if thc switch is closed at t = 0, we have,

0 for t < O v~ = { E for r > O

The wavefonn of v, therefore exhibits a junip (or step) at t = 0, and has different constant values fort < 0 and fort > 0.

A step fu~iclion, first introduced by Oliver Heaviside to describe sucll wavefonils, can be S

conveniently used to describe the foregoing behaviour of v,. Formally, a unit step function designated as u ( t ) is defined as

u(r) = 0 for t < O 1 for t 2 0

A sketch of ii (t) is given in Figure 1.30. Strictly speaking, the value of u ( t ) at t = 0 may be - defined in several ways. Wc shall take it as 1 for the sake of definiteness. Fig. 1.29 : A simple d.c. circuit

Fig 1.30 : The Unit Step function u (t)

With the above definition, the voltage vR ill the circuit of Figure 1.29 can be written as v, = E 11 (t). The unit step fuirclion u (t) and its associated fulictions can be used with advantage to describe I l lRl lY d i s c o ~ i t i ~ ~ u o ~ s wavefortiis. Even though u (t) has a coilstant value fort > O it should not be confused with a d.c. function of unil magtiitude. The latter, represented in Figure 1.31, has a constant value for the entire time duration unlike a step function.

Fig 1.31 : Waveform of a d.c. signal of uni t magnirude

In@odudiaa to Circuits Example 1.11

Express the voltage waveforms of Figure 1.32 (a), (b) and (c) by appropriate step functio~ls.

Solution

vl is a unit step delayed by 4 seconds. It can be written as u (t - 4) since it is zero for t - 4 < 0 and is equal to 1 for t - 4 2 0. v2 is 5 times vl. Hence vz = 5 u (t - 4). vj call be viewed as the su~ll of 5 u ( t - 4) and - 5 u ( t - 5). Thus v3 = 5 [u (t - 4) - 11 (t - 5)].

Incidentally, the wavefomls of Figure 1.32(a) and (b) lnay be referred to as delayed step functions while wavefonns such as those in Figure 1.32(c) where the signal is llollzero o~lly for a s11lall i~iterval of time are called pulse wavefonns.

01 4 tls -

( c )

Fig. 1.32 : Waveforms for Example 1.11

SAQ 12 Sketch the wave for111 of a voltage sig~lal give11 by

1.6.3 Periodic Signals A periodic signal f (t) is one which repeats itself identically at regular il~tervals of time, say

, , L'! T, as in Figure 1.33. Formally, a periodic signal f ( t) satisfies the relation,

f ( t ) = f ( t + T ) for all r, (1.19)

where T is a co~lsta~lt and is said to he the period of the signal. 111 other words, the sig~ial has@entical values at two points sepanted by a time interval T. It is obvious that the same valites are obtai~led at i~~tervals of ZT, 3T. . . also but we take the s~ilallest such iliterval as the period. If the cycle of signal variation over one period is known, the11 we can extrapolate the signal for all tilne by repeating the cycles endlessly.

Fig. 1.33 : A periodic signal

The number of periods in one seco~ld is called the frequency f of the sigual and is measured in a unit called hertz (abbreviated Hz) in tlo~iour of the German scientist Hertz. Thus

(You may be interested to note that a few decades earlier, it was the practice to express frequency in cyclesper secottd, abbreviated as CIS, a unit which is now replaced by Hz.)

Some exarnples of cornll~o~ily used periodic wavefor~lls and their names are give11 in Figure 1.34. Of these the si~lusoidal wavefonll is the most iillporta~~t as this is the wavefonn of the voltages and currents in all electrical power generation and distribution systems.

(a) Synimelrical Square wave

(h) A Sawtooth wave

i l i

TI1 ( c ) A Pulse train

(d) A Sinusoid

Fig. 1.34 : Examples of periodic waveforms

Circuit Elements and Kirchhotrs Laws

Introduction to Circuits Peak and Average Values

Each periodic signal varies in a characteristic fashio~l over olle period. It would be convenient if we can associate with each signal one single value which is a measure of the strength of the signal duri~ig the period. One quantity which suggests itself for this purpose is the peak value or the amplitude of the signal. One should then indicate both positive and negative peakvalues in case of waveforms which are not symmetrical about the time axis. Generally, the peak value is the strength of the signal at one or possibly niore isolated values of time and does not represent its intensity over sustained time intervals. It appears therefore Inore reasonable to adopt the average value of the signal as the criterion. The average of v (t) over one period is given by

Adopting this criterion, the average values of the sawtooth and the pulse train of Figure 1.34 are readily seen to beAl2 and AdIT respectively. However the average values of the sy~nnietric square wave and the si~iusoidal wave become zero b e ~ i u s e the areas under the positive and negative loops of the wavefonn ance l each other out. To associate a zero average value with all such syrnnietric wavefor~rls carries no i~lfonnatio~i at all of the strength of the signal. For such wavefonils, where V,,, = 0, we make a departure from the conve~~ t io~~a l definition and take the average of the absolute value of the signal. That is,

For the square wave of Figure 1.34, we have

TI? If we had taken the avenge value over the positive half cycle i.e. 2lT I A d t , the same

0

result would have been obtained. This is one alteniative approach to the calculatio~l of absolute average value, which is therefore so~iietinles called half cycle average. Convi~lce yourself that this is a valid procedure. Applying this technique to the sinusoid, we have

That the absolute average value (or half cycle average value) of a s i~~usoid of peak valueA is 2AIn is an illlportant result which we shall make use of later.

Each of the three parameters viz., the peak value, the average value, the absolute average value of a periodic waveform that we have discussed so far is useful in some application or other. For example, the satisfactory performance of insulation in an electrical machine depends on its ability to withstand the peak value of voltage to which it is subjected. When a d.c. meter is used to measure a periodic voltage or current, the meter reading would be the average value which ilicidentally is also called the d.c. co~nponent of the signal. On the other hand, meters k~iown as rectifier type nleters respond to the absolute average values. Useful as they are, liolle of the three parameters is indicative of the power available from the signal when coli~iected to a resistor. Since by far the most significant use of electricity is in the context of power and energy, we should look for another parameter which is nleani~igful in the related applications. It is this, the effective value of a periodic voltage or current, that we shall study in the next section.

Example 1.12 Find the period, frequency, peak value and absolute average value of the periodic voltage wave in Figure 1.35.

Fig. 1.35 : Waveform lor Example 1.12

Solution

T = X m s ; f = = 1 2 5 H z x X l o - - '

Vprnk = 5 V. Both positive and negative peaks have the salile magnitude.

W e shall calculate the average over the positive half cycle to find Vabs To avoid integration, we call find the area uilder the curve for the interval 0 . . . 4 111s and divide it by the base viz 4 Ins to get the average value.

1 1 ~ ~ 5 x 1 + 5 ( 3 - 1 ) + z ~ 5 ( 4 - 3 ) Vah, aw = v = 3.75 V

4

SAQ 13 The general expression for a sinusoidal voltage is Vm sill (2nfr + €I), where Vm is the maximum o r peak value. A particular siilusoidal voltage v ( t ) has a period of 50 ms, ail absolute average value of 31.8 V aiid ail illstai~ta~leous value of zero at t = 0. Find the expressioil for v ( t ) explicitly.

1.6.4 Effective Value Consider a source of periodic voltage v ( t ) of period T to be coil~iected to a resistor R as shown in Figure 1.36.

The iilstai~taileous power dissipated in R is given by

which varies froin instalit to instant but repeats itself identically at iiltervals of T. Let us calculate the average value o f p ( t ) over one period.

Now the above value of average power is sigiiificai~t because the energy dissipated in R over any interval tx equals Pa jx strictly if t, is a multiple of T and approxiiiiately so even if tx is not a inultiple of T. The error in the approxiinatioil becoines progressively smaller as the value of tJT increases and is less thaii 1 % for tx > 100 T. Hence the average power over a

Circuit Elen~enls and KircbhoWs Laws

Fig. 1.36 : A periodic voltage source connected lo a resistor

Introduction to Circuits period is taken as the rate of energy dissipation in R and one simply uses the syinbol P to indicate it. A second point to note is that the above integration need only be carried out over a co~ltinuous time duration of T seconds, and not necessarily commencing from t = 0, as p (t) is periodic. We then have

where t, is arbitrary.

Let us now ask the questioil What value of d.c. voltage when applied across R causes dissipatiora of the same amouru ofpower as the givenperiodic voltage ? Let us call this voltage Vef Obviously the power dissipated with this d.c. voltage is V&I R. Equating this to the expression in Eq. (I.%), we have

The above is known as the Effective Value of the given periodic voltage. Formally it is defined as that value of d.c. voltage which dissipates the same power in a fixed resistance as the given periodic voltage. Now the expression forP can be written as

Note the process of arriving at Veff for a given v (t). We first square v (t), then take the ineail

of the square of v (t) which is - + 2 d t and finally find the square root of the T fl

mean of the square of v (t). For this reason V4is also called root-mean-square value or r m s value and indicated as V,,.

In a similar fashion the effective value o~nns-value of a periodic current i (t) can be defined as that value of d.c. current which when passed through a given resistance produces the saine power dissipation as the periodic current in question. Hence we have the following expressioils for Ieff and P.

I

Note that expressions for power in Eq. (1.26) and (1.28) are identical with those in a d.c. situation. It is exactly to arrive at such an arrangement that we have defined the effective values of v ( t) and i ( t) in the manner we did.

Let us now calculate the effective value of the siilusoidal voltage V,,, sin (2 nfr + 8)

Thus

This is a very important relation which we shall make use of extensively in our later work.

Example 1.13

Find the nns value of the periodic voltage of Figure 1.34 (c).

Solution

Instead of carrying out the i~~tegratiou mathematically let us arrive at the answer through the nns concept. The square of v (t) is as sketched in Figure 1.37.

Fig. 1.37 for Exampla 1.13

x d Mean value of (t) over the interval 0 - T = - T

Root mean square value of v (r) = = A volts

SAQ 14 A sinusoidal current i (t) = 10 sill 100 n t is passed through a 100 S2 resistance. Find the energy dissipated in'the resistance in 1 second.

SAQ 15 Find the effective value of the sawtooth voltage wave of Figure 1.34(b).

1.7 SUMMARY

Voltages and currents are the variables of primary interest ill an electrical circuit. Collectively they may be called signals. I11 this Unit which deals with the preliminaries needed for circuit analysis, you have learnt the uature and characteristics of these signals and how they together influe~rce a third variable of interest viz, power. In particular, you have noted the following :

The d is t i~ lc t io~~ betweell d.c. s i g~~a l s which are denoted by capital letters V a~id I and general time varying signals indicated as v and i. - The ilnportance of indicating reference signs for voltages and curre~its on the circuit diagnms, without which the expressions for the signals become ambiguous.

The two alternative methods of specifying signals viz nlathe~natical expressions and waveform representation.

Circuit Elements and Kirchhofl's Laws

Introduction to Circuits Some irnportaiit categories of signal waveforms like the step and pulse functions, the square and sawtooth waves, and the siiiusoid.

The nature of periodic sig~:?.ls and their important attributes like period, frequency, peak value., average value, absolute average value and effective value, the last one being of great significance ill power related issues.

In a given circuit, the currei~t aiid voltage values are fixed by the terminal relations of the circuit elenie~its on one hand and by the topology (mnauner of interconnection of elenlents) 011 the other. As regards circuit elelrients, you were iritroduced to the i~iiportant classes :

a) the active elelne~~ts colnprising the independelit voltage1 curreilt sources whose tennilla1 voltageslcurrenls are held at the spccified values irrespective of the load 011 theill aiid the four types of dependent sources, whosc values are controlled by voltages/currents at their input ter~ninal pairs and

b) the passive circuit elements viz the energy dissipating resistor clelnent with the tennilla1 relation v = Ri, and the nou-dissipative elements, inductor anrl capacitor with the terminal relations v = L dildf and i = C dv / dt, which are capable of storing and rclcasillg energy.

To gain an u~lderstanding of the coilstrailits iniposed by the interconnection, you were iiitroduced to the two following laws which hold in a circuit at every instant of time alitl are i~~dependent of the nature and size of thc elelnents constitutiilg the circuit.

The algebraic sun1 of currents leaving a node through all branches incident at the node is zero (Kirchhoff's current law : KCL)

The algebraic sum of voltage drops counted in a cyclic sense around a closed path is zero (Kirchhoff's voltage law: KVL)

We are now in a position to take up methods of circuit analysis and this we shall do in the succeeding units.

1.8 ANSWERS TO SAQs

SAQ 1 :

At t = 2 s, v,b = 2 V i.e., a is at 2 V higher potential than h, there is no current in the elelnent and power is neither absorbed nor delivered.

At t = 3 s, v0b = 1 V, acurrent of 1 A flows from h to o and a power of 1 W is delivered by the elenlent.

At t = 5 s , vflb = - 1 V or vba = 1 V i.e., b is at 1 V higher potential tllali o, a ctlrre~it o l 1 A flows fro111 1, to n and a power of (- 1) (- 1) = 1 W is ;ibsorbed by the elenlent.

SAQ 2 :

For the reference signs indicated, the product vi indicates ilstantaaeous power absorbed by the element. Its value as a function of time is sketched in the figure shown.

For 2 < t < 4 and 6 < t < 8, the product v i has a negative value, which implics that the elenlent is actually supplying power during these two intervals of time. Power bcing rate of transfer of energy, energy is the integral of power. The energy absorbed by the elenlent during O < t < 8 is therefore the net area under the p vs t curve for 0 < t < 8, which fro111 synilnetry can be seen to be zero.

Figure for Answor to SAO 2 : Variation of power

SAQ 3 : 1. voltage. 2. voltage, 3. curre111

SAQ 4 :

(a) True (b) False

SAQ 5 :

a) The voltage sourcc ~ilai~itains V,, z 2 V. The curre~~t sourcc I, drives 2 A out of t e n ~ l i ~ ~ a l n through Vs and receives 2 A through ter~iiilial b.

Power delivered by currcllt source = 2 x 2 = 4 watts. This is positivc since current flow through the elemelit is in the direction of voltage rise.

Power absorbed by voltage source = 2 x 2 = 4 watts. Curre~lt flow through voltage source is in the direction of voltage drop. So power is absorbed.

b) Ix=I51

Since n and b are co~u~ected together they are at the same potential and Vab = 0

Power delivered by current source 1 = IsI x O = 0

Power delivered by de,pendent source = (4 = 4 I,1 IS2 = 8 W

Power delivered by current source 2 = (- 4 I,) Is2 = - 8 W

Alten~atively we can say that the power absorbed by currellt source 2 is S watts.

It can be secii fro111 thc above exaiiiple that the depende~~t source supplies 8 W to the extenral circuit through its output terminals, while receiving zero power through its input (controllil~g branch at>). Such an action is representative of an ideal amplifier.

SAQ 6 :

Circuit Elements and KircbhoB's L a w s

I Figure for Answer lo SAQ 6

SAQ 7 :

The ~iegetive sign implies that the terminal through which the current leaves the illductor will be positivc with respect to the other terminal.

SAQ 8 :

If R, and R2 are the next higher values,

RI (1 - 0.1) = 10 (1 t 0.1) and R2 (1 - 0.1) = RI (1 t 0.1)

R 1 = 1252;R2= 1 5 Q

SAQ 9 :

(1) voltage (2) rate of change of current (3) n t e of change of voltage (4) a short-circuit (5) all open circuit (6) power rating.

SAQ 10 :

Larger i~lducta~lce for the same size.

SAQ 11 :

Yes. It aniounts to reversing the signs of all the currer~ts in the current surn~uation. Since the latter is equal to zero, the statement is valid.

SAQ 12 :

Figure for Answer to SAO 12

SAQ 13 :

1 f = - = = 20 Hz 50x10- '

2 31.8 x n Vabs = - Vm = 31.8 a Vm = -------- =

7r 50 v

2

v (0) = V,, sin 8 = 0 * 8 = 0 or x

Hence, v (i) = 50 sin 40 ;c i or 50 sill (40 ;c / t n)

SAQ 14 : 10 2 100

A ; P = I R = --- x 100 = 5000 W 4f 7 - Energy dissipiatcd in 1 s = 5000 joules

SAQ 15 : A t v = - for O < t < T

T