Chapter Seven ALTERNATING CURRENT - NCERT

1 INTRODUCTIONWe have so far considered direct CURRENT (dc) sources and circuits with dcsources. These currents do not change direction with time. But voltagesand currents that vary with time are very common. The electric mainssupply in our homes and offices is a voltage that varies like a sine functionwith time. Such a voltage is called ALTERNATING voltage (ac voltage) andthe CURRENT driven by it in a circuit is called the ALTERNATING CURRENT (accurrent)*. Today, most of the electrical devices we use require ac is mainly because most of the electrical energy sold by powercompanies is transmitted and distributed as ALTERNATING CURRENT . The mainreason for preferring use of ac voltage over dc voltage is that ac voltagescan be easily and efficiently converted from one voltage to the other bymeans of transformers.

2 Further, electrical energy can also be transmittedeconomically over long distances. AC circuits exhibit characteristics whichare exploited in many devices of daily use. For example, whenever wetune our radio to a favourite station, we are taking advantage of a specialproperty of ac circuits one of many that you will study in this SevenALTERNATINGCURRENT*The phrases ac voltage and ac CURRENT are contradictory and redundant,respectively, since they mean, literally, ALTERNATING CURRENT voltage and alternatingcurrent CURRENT . Still, the abbreviation ac to designate an electrical quantitydisplaying simple harmonic time dependance has become so universally acceptedthat we follow others in its use. Further, voltage another phrase commonlyused means potential difference between two 22 Physics234 NICOLA TESLA (1856 1943)Nicola Tesla (1856 1943) Serbian-Americanscientist, inventor andgenius.

3 He conceived theidea of the rotatingmagnetic field, which is thebasis of practically allalternating currentmachinery, and whichhelped usher in the age ofelectric power. He alsoinvented among otherthings the induction motor,the polyphase system of acpower, and the highfrequency induction coil(the Tesla coil) used in radioand television sets andother electronic SI unit of magnetic fieldis named in his AC VOLTAGE APPLIED TO A RESISTORF igure shows a resistor connected to a source ofac voltage. The symbol for an ac source in a circuitdiagram is . We consider a source which producessinusoidally varying potential difference across itsterminals. Let this potential difference, also called acvoltage, be given bysinmv vt =( )where vm is the amplitude of the oscillating potentialdifference and is its angular find the value of CURRENT through the resistor, weapply Kirchhoff s loop rule ( )t= 0(refer to ), to the circuit shown in Fig.

4 To get=sinmvti R or sinmvitR =Since R is a constant, we can write this equation assinmiit =( )where the CURRENT amplitude im is given bymmviR=( )Equation ( ) is Ohm s law, which for resistors, works equallywell for both ac and dc voltages. The voltage across a pure resistorand the CURRENT through it, given by Eqs. ( ) and ( ) areplotted as a function of time in Fig. Note, in particular thatboth v and i reach zero, minimum and maximum values at thesame time. Clearly, the voltage and CURRENT are in phase witheach see that, like the applied voltage, the CURRENT variessinusoidally and has corresponding positive and negative valuesduring each cycle. Thus, the sum of the instantaneous currentvalues over one complete cycle is zero, and the average currentis zero.

5 The fact that the average CURRENT is zero, however, doesFIGURE AC voltage applied to a In a pureresistor, the voltage andcurrent are in phase. Theminima, zero and maximaoccur at the samerespective 22235 ALTERNATING CurrentGEORGE WESTINGHOUSE (1846 1914)George Westinghouse(1846 1914) A leadingproponent of the use ofalternating CURRENT overdirect CURRENT . Thus,he came into conflictwith Thomas Alva Edison,an advocate of directcurrent. Westinghousewas convinced that thetechnology of alternatingcurrent was the key tothe electrical founded the famousCompany named after himand enlisted the servicesof Nicola Tesla andother inventors in thedevelopment of alternatingcurrent motors andapparatus for thetransmission of hightension CURRENT , pioneeringin large scale mean that the average power consumed is zero andthat there is no dissipation of electrical energy.

6 As youknow, Joule heating is given by i2R and depends on i2(which is always positive whether i is positive or negative)and not on i. Thus, there is Joule heating anddissipation of electrical energy when anac CURRENT passes through a instantaneous power dissipated in the resistor is222sinmp i R i Rt ==( )The average value of p over a cycle is*222sinmpi Ri Rt = <> = <> [ (a)]where the bar over a letter (here, p) denotes its averagevalue and <..> denotes taking average of the quantityinside the bracket. Since, i2m and R are constants,22sinmp i Rt =<>[ (b)]Using the trigonometric identity, sin2 t =1/2 (1 cos 2 t), we have < sin2 t > = (1/2) (1 < cos 2 t >)and since < cos2 t > = 0**, we have,21sin2t <> =Thus,212mpi R=[ (c)]To express ac power in the same form as dc power(P = I2R), a special value of CURRENT is defined and is called, root mean square (rms) or effective CURRENT (Fig.)

7 And is denoted by Irms or I.*The average value of a function F (t) over a period T is given by F tTF t tT( )( )= 10d**<> = = = []=coscossinsin2121221220000 tTt dtTtTTTTFIGURE The rms CURRENT I is related to thepeak CURRENT im by I = / 2mi = 22 Physics236It is defined by22122mmiIii==== im( )In terms of I, the average power, denoted by P is2212mpPi R I R===( )Similarly, we define the rms voltage or effective voltage byV = 2mv = vm( )From Eq. ( ), we havevm = imRor, 22mmviR=or, V = IR( )Equation ( ) gives the relation between ac CURRENT and ac voltageand is similar to that in the dc case. This shows the advantage ofintroducing the concept of rms values. In terms of rms values, the equationfor power [Eq. ( )] and relation between CURRENT and voltage in ac circuitsare essentially the same as those for the dc is customary to measure and specify rms values for ac quantities.

8 Forexample, the household line voltage of 220 V is an rms value with a peakvoltage ofvm = 2 V = ( )(220 V) = 311 VIn fact, the I or rms CURRENT is the equivalent dc CURRENT that wouldproduce the same average power loss as the ALTERNATING CURRENT . Equation( ) can also be written asP = V2 / R = I V (since V = I R)Example A light bulb is rated at 100W for a 220 V supply. Find(a) the resistance of the bulb; (b) the peak voltage of the source; and(c) the rms CURRENT through the (a)We are given P = 100 W and V = 220 V. The resistance of thebulb is()22220 V484100 WVRP=== (b)The peak voltage of the source isV2311mvV==(c)Since, P = I V100 V= ==PIV EXAMPLE 22237 ALTERNATING OF AC CURRENT AND VOLTAGEBY ROTATING VECTORS PHASORSIn the previous section, we learnt that the CURRENT through a resistor isin phase with the ac voltage.

9 But this is not so in the case of an inductor,a capacitor or a combination of these circuit elements. In order to showphase relationship between voltage and currentin an ac circuit, we use the notion of analysis of an ac circuit is facilitated by theuse of a phasor diagram. A phasor* is a vectorwhich rotates about the origin with angularspeed , as shown in Fig. The verticalcomponents of phasors V and I represent thesinusoidally varying quantities v and i. Themagnitudes of phasors V and I represent theamplitudes or the peak values vm and im of theseoscillating quantities. Figure (a) shows thevoltage and CURRENT phasors and theirrelationship at time t1 for the case of an ac sourceconnected to a resistor , corresponding to thecircuit shown in Fig.

10 The projection ofvoltage and CURRENT phasors on vertical axis, , vm sin t and im sin t,respectively represent the value of voltage and CURRENT at that instant. Asthey rotate with frequency , curves in Fig. (b) are Fig. (a) we see that phasors V and I for the case of a resistor arein the same direction. This is so for all times. This means that the phaseangle between the voltage and the CURRENT is AC VOLTAGE APPLIED TO AN INDUCTORF igure shows an ac source connected to an inductor. Usually,inductors have appreciable resistance in their windings, but we shallassume that this inductor has negligible , the circuit is a purely inductive ac circuit. Letthe voltage across the source be v = vm sin t. Usingthe Kirchhoff s loop rule, ( )t= 0, and since thereis no resistor in the circuit,d0div Lt =( )where the second term is the self-induced Faradayemf in the inductor; and L is the self-inductance ofFIGURE (a) A phasor diagram for thecircuit in Fig (b) Graph of v andi versus An ac sourceconnected to an inductor.