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13. AC Circuits 12th Science 12th - Physics Syllabus 1. Introduction 2. AC Generator 3. Average and RMS values 4. Phasors 5. Different Types of AC Circuits 6. Power in AC circuit 7. LC Oscillations 8. Electric Resonance g. Sharpness of Resonance 10. Choke Coil _Theory Notes Q. State the Faraday’s Laws | Faraday’s First Law - i ic flu: of electromagnetic induction? | Whenever there is a change of magne! | circuit, an emf is induced in the circu! x linked with a closed ” d Law , oo - The way mitude of induced emf produced in the Beek ¢ Cireetly proportional to the rate of change of magnetic circuit. ——— Q. Define the Terms: | a) Induced Current i a) Induced Current The current produced in the coil due toa change LA aararni! b) AC Generator magnetic lines of force passing through the coil 1s c) Alternating Current | current, da) Direct Current | b) AC Generator A generator / A device which converts mechanica, trical energy in the form of alternating current 1s ca l energy into elec- Iled AC generator. c) Alternating Current , she current ao which the polarity of the voltage keeps changing periodically is called the alternating current. d) Direct Current . The current for which has fixed polarit current. y of voltage is called as direct Q. Explain alternating emf | Ans. and alternating current. 1) We know that the source of AC (AC Generator) produces time dependence emf and it varies sinusoidally with time as shown in following graph 3n/2 Qn Fig. Graph of e versus wt 2) The emf generated by AC generator is given by = esinat.........(1) Where @ = peak value of emf w = Angular frequency of rotation of the coil in the AC generator | 3) The current in the circuit connected to this AC generator is similar to that of emf, therefore it is given by, i=fsin(ort a). (2) Where, io = Peak value of current a = Phase difference between the current (i) and the emf (e). 4) The direction of emf is reversed after ever i f r ; r y half revolution of the coil. This type of emf is called the alternating emf and. corre. sponding current is called alternating current. Q. Explain, ; Why moving coil instrument is not used to measure alternating current and/or voltage. Q. What is phasor and phasor diagram. Explain in brief. Q. Explain theory of an AC circuit with resistor and Show that the current and voltage are in same phase. sin’ (wt )dt “HE Rin) = is the same as the heat produced by a DC current of magnitude i for time ; = 42 — Ans. 1) Moving coil instrument (ammeter and voltmeter) measure the average value of current and voltage applied across it. ; 2) The average value of current and / or voltage over a full cycle is zero, 3) Therefore moving coil instruments cannot be used to measure the alternating current and voltage. Ss 4) Hence in order to measure their quantities it is necessary to make use of a property which does not depend upon the changes in the direction of alternating current or voltage. Phasors : : ; Alternating current and alternating emf as a rotating vectors with the angle between them equal to the phase difference between the current and emf. These rotating vectors are called phasors. Phasor diagram The diagram representing a rotating vector quantities varying sinu- soidally with time is called phasor diagram. a) é b) > iinet }reneeeere o> I ie) 10) T,coscot Fig. a) and b): phasor diagrams. 1) The phasor for alternating emf and alternating current are inclined to the horizontal axis at angle wt or wt+a, and rotate in anticlockwise direction. The len; gth of the arrow represents the maximum value of the quantities iy and &. 2) The projection of the vectors on fixed axis gives the instanta- neous value of alternating current and alternating emf. 3) (i= ipsinwt) and (e=esinwt) are the projections taken on Y- axis, which is shown by sine form, Similarly, (i = ipcoswt) and (e = ecoswt) are the projections taken on X- axis, which is shown by cosine form, 4) The representation of the harmonically varying quantities as rotating vectors enable us to use the laws of vector addition for adding these quantities. AC circuit with a pure resistor 1) Suppose a resistor of resistance R is connected to an AC source of emf with instantaneous value of emf (e) is given by e =e sinat... i) Q. Explain, phasor diagram of voltage and current for an AC circuit with resistor. Where é is peak value of voltage and is its angular frequency. R (~) e Fig. (a) An AC Voltage Applied to a Resistor eort Fig. (b) Graph of e andi versus wt 2) Ife is potential drop across the resistance then we write as 3) As instantaneous emf is always equal to instantaneous value of potential drop, the from eqn (i) and eqn (ii) we write as IR =e = sinwt € _ eosinwt RR eo «0 = ipsinwt (since, l= ey. (ili) 4) Comparing iy = pith Ohm’s law, we find the resistors behave similarly for both AC and DC voltage. Hence the behavior of R in DC and AC circuits is the same and R can reduce DC as well as AC equally. 5) From above equations (i) and (iii), it is to be concluded that for resistor there is zero phase difference between instantaneous alternating current and instantaneous alternating emf means they are in phase. And it is shown graphically in above fig(b). Ans. 1) In AC circuit containing R only, current and voltage are in the same phase. 2) Therefore both phasors for i and e are in the same direction making angle wf with OX. 3) The projections of current and emf on vertical axis give their instantaneous values while the phase angle between alternating current and alternating voltage through R is zero as shown in following fig. J sinet Fig. Phasor diagram for a purely resistive circuit Q. Explain, phasor diagram of voltage and current for an AC circuit with pure inductor. Q. Explain the term inductive reactance (X,) j= (coset) L @ «je Og (£_ al =e sin(Z wt) vt = -sin(wi- 4) This is the alternating current developed in purely inductive circuit when connected to a source of alternating emf. . 6) From equation (i) and (iv) we find that the alternating current i lags behind the alternating voltage emf e by a phase of angle 7 /2 radians (90°) or voltage across inductor (L) leads the current by a phase of angle z/2 radians(90°) which is shown in fig(b). Ans. 1) The phasor representing peak emf ¢ makes an angle w/ in anti- clockwise direction from horizontal axis. 2) As current lags behind the voltage by 90°, so the phasor repre- senting i is turned clockwise with the direction of e as shown in fig. e,sinat Fig. Phasor diagram for purely inductive Inductive Reactance (X,) The opposing nature of inductor to the flow of alternating current is called inductive reactance X,. 1) We know that, the current in inductive circuit is given by » ey . bo = OD Gi) 2) According to Ohm’s law, the current in the resistive ac circuit is given by i= Re di) 3) Ifwe compare equation (i) and (ii), we find the conclusion that wl plays a similar role in an inductive ac circuit as resistor in a pure resistor circuit. Therefore X, = wl represents the effective resis- ay offered by inductance L, it is called as inductive reactance X,). “XL = ol = 2afL (since, @ = 2n/T = 2zf) Where, f frequency of AC supply. 4) The inductive reactance (X:) is directly proportional to the induc- tance (L) & the frequency (f) of the alternating current. 5) The dimension of inductive reactance is same as that of resis- tance and its SI unit is Ohm (Q) 6) In DC circuit as frequency is zero (f = 0) therefore inductive reactance is also zero (X, = 0 7) Pure inductor offers zero resistance to DC, i.e. it cannot reduce DC. Thus its passes DC and blocks AC of very high frequency. a Q. An AC source generating @ voltage e = e, sinwt is connected to a capacitor of capacitance C. Find the expression for the current 1 flowing through it. Plot a graph of e andi versus wt uit, the self induced emf opposes the growth, 8) In an inductive circ’ as well as decay of current. $$ A we ae apacitor AC circuit with a pure cap: acitance C connected to an AC nsider a capacitor with capi nnec ’ Source of an emf having instantaneous value is given by C= OSI Whessesee 1) Cc e Fig. (a) An AC source connected a capacitor ct Fig. (b) Graph of e and i versus wt 2) The current flowing in the circuit transfers charge to the plates of the capacitor which produces a potential difference between the plates. As the current reverses its direction in each half cycle, the capacitor is alternately charged and discharged. 3) As qis the charge on capacitor at any given instant t, the potential difference across the plates of the capacitor is given by v= org=Cv......lii) 4) Therefore the instantaneous value of current (i) in the circuit will be i= 4 = Lee (here, V = ¢ at every instant)............(iii) i= fic esinwr) (As, e = esinwt) i= Ce 4 (sinwt) i= WC e.coswt i= Tat) coset i= aezysin(or + z) (since, cos wt = sin( wr + z)) The current will be maximum when sin(wt + ©) = 1, so that i=h, where peak value of current is given by ~ i= eo . = Tac) cn (iv) i= ’sin(or +2) scsconncned (v) 5) Therefore from equation (i) & (ii) it is find that in an AC circuit containing a capacitor only, the alternating current i leads the alternating emf e by phase angle of /2 radians (90°). 8 Q. Compare resistance and reactance. Q. Fora very high frequency AC supply, a capac- itor behaves like a pure conductor, Q. A device Y is connected across an AC source of emf = e sinwt. The current through Y is given as i= ipsin(wt + 2/2) a) Identify the device Y and write the expression for its reactance. b) Draw graphs showing variation of emf & current with time over one cycle of AC for Y. c) How does the reactance of the device Y vary with the frequency of the AC? Show graphically. d) Draw the phasor diagram for the device Y. Ans _— eed _——~Reaetanee istance {| | ___— Resistant 1) Current is affecteq i Cc 1) Equally effective for A‘ ijeduced) but energy ig and DC not consumed (heat is not generated). The energy consumption by a coilis due to its resistive component, 2) Inductive reactance (X,=27fL) is directly proportional and capaci- tive reactance (X- = al is inversely proportional to frequency of the AC. 3) Current opposed by a pure inductor lags in phase while that opposed by a pure capacitor leads in phase by x over the voltage. 2) Its value is independent of frequency of the AC 3) Current opposed by a resistor is in phase with the voltage. Ans. ; . 1) The capacitive reactance for an AC circuit to a pure capacitor is iven by given by , 1 1 Ko = wc 2afC Keo 2) Capacitive reactance is inversely proportional to frequency, means as frequency increases capacitive reactance decreases. 3) Therefore, for a very high frequency of AC supply, the capaci- tive reactance is nearly equal to zero and hence the capacitor behaves like a pure conductor. Ans. a) The emf & current through the device Y is given by,e = esinwt & i= iosin(wt + 7/2). From these equations it is to be concluded that current is leads by the phase of 7/2 with emf. This will be for the device capacitor. Hence the given device Y is a capacitor. 1 1 The reactance for the capacitor is gi =— = ip is given by, X- oC ~ It b) The graph showing variation of emf & current with time over one cycle of AC for Capacitor (Y) Fig. Graph of e andi versus wi 10 Q. Derive an expression for the impedance of an LCR circuit connected to an AC power supply. ce) The reactance of device Y (capacitive reactance) is given by, 1 1 Xe= OG = Infe 1X 0c i readance o frequency. f d) The phasor diagram is as follows, < isin(wt+7/2) e,sinwt Fig. Phasor diagram for purely capacitive circuit Ans. 1) Let a pure resistor of resi L and a pure capacitor of to a source of alternating emf e as shown in following fig. stance R, a pure inductor of inductance capacitance C be connected in series L c oa t ; —" Fig. Phasor diagram for an LCR circuit Q. Explain impedance triangle? Q. An electric lamp is connected in series with a capacitor and an AC source is glowing with certain brightness. How does the brightness of the lamp change on increasing the capacitance? This is the expression for phase difference in LCT series circuit. 3) The alternating current in LCR circuit is represented by / = fo sinwt and emf is e= esin(wit+ ). The three different cases are as follows, a) Case J When X; = X:, then X, — X- = 0 b= tan”! _ [0 )o 2b = tan" X. )= tan (x) tan“'0. -. tang =0 Hence voltage and current are in phase. Thus the AC circuit is non inductive, b) Case II When X, = Xc, then, tang is positive. Hence voltage leads the current by a phase angle #. The AC circuit is inductance dominated circuit. c) Case III When X, = X:, then tan@ negative, therefore ¢ is also negative, Hence voltage lags the current by a phase of angle o. The AC circuit is capacitance dominated circuit. Ans. 1) Consider series LCR circuit to an AC power supply 2) The three phasors can be written as ee = IR, e: = inX: and ec = I0Xc The impedance triangle is as follows, K 10) R A Fig. Impedance triangle 3) The diagonal OK represents the impedance Z of the AC circuit and which is given by Z= /R?+(X,— Xc) The base OA represents the ohmic resistance R and the perpendic- ular AK represents reactance (X; — Xc). 4) The angle ZAOK = ¢, is the phase angle by which the voltage leads the current in the circuit, where, “tang = (4%) Ans. 1) When an electric lamp is connected in series with a capacitor and an AC source, the capacitor offers capacitive reactance which is given by -1__1 Xe= TC 2nfC 2) When capacitance increases, ultimately capacitive reactance decreases. 3) Electric lamp offers resistance while capacitor offers capacitive reactance therefore the combined effect will offers impedance and which is given by Z= /R° + X¢é 4) This impedance decreases with increase in capacitance. Hence the brightness of bulb changes on increasing the capacitance. Q. What is power in AC circuit? Obtain an expression for average power dissipated im purely resistive AC circuit. Q. Obtain an expression for average power dissipated in purely inductive AC circuit. OR Q. When an AC source is connected to an ideal inductor show that the average power supplied by the source over a complete cycle is zero. Power in AC Circuit - . ; As current and voltage changes at every instant in an AC cire Therefore the power in an AC circuit at a given instant is the pro of instantaneous voltage and instantaneous current. — uit duct Average power dissipated in purely resistive AC Circuit ; 1) For pure resistor, alternating and alternating voltage are in same phase, which are given as e = esinwt and i = isinwl Therefore instantaneous power is P = ei ©. P= (eosinwt ipsin wt) wi) 2) Instantaneous power varies with time, hence we consider the average power for a complete cycle by integrating eq (i). work done by the emf on the charges in one cycle Pa = time for one cycle w. P= eoinsin’ wt... r (To, Pat [ eoinsin’ wrar iT Poe Gris % bret eace : Gi) P is also called as apparent power. Average power dissipated in purely inductive AC Circuit 1) In purely inductive circuit the current lags behind the voltage by a phase angle of 7/2. -. As e = eosinat then i = ipsin(wt — z/2) 2) Therefore the instantaneous power is given by, P = ei «P= (esinat)(isin(wt — 2/2)) « P == einsin wt cos wt — work done by the emf on the charges in one cycle Pa = time for one cycle > i [Pat [-evia sin wt cos wt dt ©. Py = T= T 1 enn [2sin @tcos wi dt Pa i a T ;,_[sin2a1 dr T £6 sP =o (‘Sin 2001 dt pp = £0ln(—cos2at ¥ Pa =— $= 985 J . =~ ol (—cos 47x + cos 0 Pa = Sp{ 208 e080) +, p, =—£oin(=1+1 Po =—$r(=54 ~P,=0 Therefore the average power over a complete cycle of AC through an ideal inductor is zero. 14 Q. Explain power factor in LCR series circuit. P= (esineot)(insin(wt + ¢)) +. P= e@insinatsin(wt + d) . P= epin[sinwt cos ¢ + cos wt sin ¢ |sinwt +. P= eio[sin?wt cos b + coswtsing sinwt] work done by the emf on the charges in one cycle vie = time for one cycle [Par [ eoio[sin’ wt cos b + cos wt sing sin wt]dt p=? = — = Pn = OR T * Pay = £11((cos@ sin’ ot at) + (sind [ cosersin ot ar)| J = Here, [ sin‘wt d= £ and [cosarsines dt=0 Py = 210 cos pF oe SS cos oln *. Poy = Va V2 cos d +. Pay = Cmskrms COS P This power is also called as true power. , - The average power dissipated in the AC circuit of inductor, capacitor and resistor connected in series not only depends on rms values of current and emf but also on the phase difference ¢ between them. Ans. 1) We know that the average power dissipated in LCR series circuit is given by Pay = €rmsirms COS D 2) The average power dissipated in the AC circuit of inductor, capacitor and resistor connected in series not only depends on rms values of current and emf but also on the phase difference @ between them. 3) The factor cos ¢ is called the power factor in series LCR circuit “. cos@ = fa rms irms 4) The average power P,, is called true power and the product (Gms X ims)is called apparent power. true power “cosh = apparent power Therefore from impedance triangle we write above equation as true power “cosh =

4) After fully discharged capacitor, the magnetic flux linked with the inductor decreases inducing a current in the direction of earlier current according to Lenz’s Law. This decreasing magnitude of current, charges the capacitor in the opposite direction, Thus the magnetic energy of the inductor begins to change into the electrostatic energy of the capacitor. d Q. Why LC oscillations are damped? Give reason. Q. Obtain an expression oy resonant frequency in series LCR resonant circuit? Draw the graph of series reso- nance curve. | 5) The process continues till the capacitor is fully charged with a polarity which is opposite to that in its initial state. Thus entire 1g ; energy is again stored as 2 % in the electric field of the capacitor, 10 The capacitor begins to discharge again. Thus the process repeats itself. Thus the energy of the system continuously surges back and forth between electric field of the capacitor and the magnetic field of the inductor. This produces electric oscillations of a definite frequency, These are called LC oscillations. Ans. LC oscillations are usually damped due to following reasons. 1) Every inductor has some resistance. This causes energy loss as heat. The amplitude of oscillations goes on decreasing and they finally die out. 2) Ever if the resistance were zero, total energy of the system would not remain constant. It is radiated away in the form of electro- magnetic waves. Working of radio and TV transmitters is based on such radiations. Ans. Nl) Consider an inductance L, Capacitance C and resistance R are connected in series with an AC source. Adjust the frequency of an AC source in such a way that the maximum current will flow through the circuit, such circuit is called as series resonance circuit. L Cc R wana 1+ WW (>) Ce Fig. Series resonance circuit 2) The impedance of the series LCR circuit is given by R’+(Xi— Xe) = fr+(wr-) 3) At very low frequencies, inductive reactance is negligible while capacitive reactance is very high. 4) As we increase the frequency, then inductive reactance increases and capacitive reactance decreases. At some stage the angular frequency (w,),X: — Xe. 20