ALTERNATING CURRENT (A.C. CIRCUITS)

AENANG UEN (SHO) 3-4 PAGE: AENANG UEN (A.. US) Alternating current An electrical current, agnitude of which changes with tie and polarity reverses periodically is called alternating current (A.) he sinusoidal alternating current (a.c) is expressed as sin t where is the axiu value or peak value or aplitude of a.c. f, where f is called frequency. n ndia f=5hz. Sinusoidal e..f. of an a.c. source is given by ε ε sin ωt where ε is the axiu value or peak value or aplitude of e..f. he pictorial sybol used to represent the a.c. source is shown in figure. Mean or average value of A he average value of ac voltages and current over a coplete cycle of A is zero. Average value of alternating ef and current over a half cycle are and respectively. OO MEAN SQUAE (MS) O UA O EFFEE AUE OF A.. he average value of alternating current or ef over a cycle is zero. herefore we take root ean square value oot ean square value of alternating current is the direct current which produces the sae heating effect in a given resistor in a given tie as is produced by the alternating current. t is denoted by rs, v or by eff. elation between the effective value and peak value of a.c. eans rs apere rs.77 learly r.. s value of an alternating current is 7.7% of peak value. OO MEAN SQUAE AUE OF AN AENANG OAGE t is defined as the steady voltage that produces the sae heating effect in a given resistance in a given tie as is produced by the given alternating ef. t is denoted by give by, rs. Qn. he peak value of an a.c. supply is 3. What is the rs voltage? (b) he rs value of current in an a.c circuit is A. What is the peak current? Ans. (a) Here.. x. rs (b) Here rs A rs.44x =4.4A A. U ONANNG ONY A ESSO onsider a circuit containing a resistance connected to an alternating voltage. et the applied voltage be sin t () f be the current in the circuit at instant t, then the potential drop across will be. According to Kirchhoff s loop rule, sin t f not otherwise entioned, the values of alternating voltages or currents quoted any where are virtual (r..s) values only. For exaple a.c. eans rs volt. An ac of A rs or eff or v. it is

AENANG UEN (SHO) 3-4 PAGE: or or sin t sin t..() where =the axiu or peak value of a.c. Fro eqn() and (), we can understand that the current and voltage are in sae phae A.. U ONANNG ONY AN NDUO onsider a circuit containing an inductor of inductance connected to an alternating voltage. et the applied voltage be sin t () d A back ef is developed across the inductor. According to Kirchhoff s loop rule d sin t d or sin t or d sin t ntegrating, d.sin t. or cos t sin( t /) sint / () [ cost sin( t / )] Where =the peak value of a.c. he ter is called inductive reactance X. On coparing equations () and (), we can understand that, the current lags behind the voltage by an angle / radian. nductive reactance (X ) nductive reactance X f where f is the frequency of a.c. supply. he S unit of inductive reactance is oh( ] For a.c., X f For d.c., f=, so X hus an inductor allows flow of d.c through it easily but opposes the flow of a.c. through it. Qn. A 44H inductor is connected to, 5Hz a.c. supply. Deterine the rs value of current in the circuit.

AENANG UEN (SHO) 3-4 PAGE:3 A.. U ONANNG ONY A APAO onsider a circuit containing a capacitor of capacitance connected to alternating voltage. et the applied voltage be sin t..() q At any instant, voltage across the capacitor is According to Kirchhoff s loop rule q sint or q sin t urrent at any instant is dq d( sint) i cos t or cos t or sin( t /) () where he ter c apacitive reactance =the current aplitude. / is called capacitive reactance X. apacitive reactance X f he S. unit of capacitive reactance is oh( ]. For a.c., X f For d.c., f= X hus a capacitor blocks d.c. ariation of capacitive reactance with frequency apacitive reactance, X f i.e., X f hus the capacitive reactance varies inversely with the frequency. As f increases, X decreases. Figure shows the variation of X with f. Qn. A 6F capacitor is connected to a, 6Hz a.c. supply. Deterine the rs value of current in the circuit. Qn. While perforing an experient in laboratory, a student connected an electric bulb and a capacitor is series to a source of direct current (dc). hen will be the bulb glow steadily? Explain. Qn. What happens when (a) a capacitor is included in a dc circuit. (b) What type of current easily passes through an inductor? (c) Give an explanation for your answer. Qn. An electric bulb B and a parallel plate capacitor are connected in series as shown in figure. he bulb glows with soe brightness. How will the glow of the bulb affected on introducing a dielectric slab between the plates of the capacitor? Give reason to support yours answer. A.. OAGE APPED O A SEES U et an alternating voltage sin t applied to a circuit containing an inductance, a capacitance and resistance connected in series.

AENANG UEN (SHO) 3-4 PAGE:4 According to Kirchhoff s loop rule d q sin t he solution to this equation can be obtained by different ethods. Phasor-diagra solution he phasor diagra for series circuit to which an A voltage sin t is applied is shown below. he current in each eleent is sae. oltage across the resistor is in phase with current. But the voltage across the inductor leads the current by radian and voltage across the capacitor lags behind the current by. et peak value of current in the circuit be. et, and be the peak value of voltage across the resistor, inductor and capacitor respectively. he vector su of, and gives the peak value of the applied voltage. By Pythagorean theore, Substituting the values of or, X X = (X X ) (X X ), and, we have learly, (X X ) is the effective resistance of the series circuit to the flow of ac. t is called ipedence. t is denoted by Z and its S. unit is oh ( ]. hus, ipedence Z X X Phase difference et be the phase difference between the voltage and current, fro figure we can write X X X X tan Special cases i. f X X, then current leads the voltage. ii. f X X, then the current lags the voltage. iii. f X X, then current is in phase with the voltage. ESONANE he current aplitude in an circuit is given by, As angular frequency of alternating ef is increased, goes on increasing. For a particular value of (,say ) goes on decreasing and becoes equal to. hen the ipedence is iniu ( Z ) and the current aplitude becoes axiu ( ]. his phenoenon is known as resonance and the angular frequency is called natural or resonant angular frequency.

AENANG UEN (SHO) 3-4 PAGE:5 Deterination of resonant frequency At resonant angular frequency, X X or he resonant frequency f haracterestics of series resonant circuit. he current is in phase with voltage and power factor is unity (cos = when ). he voltage across is equal to the applied voltage. Qn. A sinusoidal voltage of peak value 83 and frequency 5Hz is applied to a series circuit in which =3, =5.48H, and =796 F. Find the ipedence of the circuit, (b) the phase difference between the voltage across the source and the currents, (c) the power dissipated in the circuit, and (d) the power factor. SHAPNESS OF ESONANE: Q-Factor he sharpness of resonance is easured by a quantity called the quality or Q-factor of the circuit. he Q-factor of a series resonant circuit is defined as the ratio of the resonant frequency to the difference in two frequencies taken on both sides of the resonant frequency at which the current aplitude becoes ties the value at resonant frequency. Matheatically Q-factor can be expressed as e sonant frequency Q = Band wih where and are the frequencies at which the current falls to ties its resonant value, as shown in figure. Q-factor is a diensionless quantity. hus, Q-factor of a series -circuit ay also be defined as the ratio of either the inductive reactance or the capacitive reactance at resonance to the resistance of the circuit. Q / When is low, Q is high and greater will be the sharpness of resonance. uning of a radio receiver he tuning circuit of a radio or is an exaple of resonant circuit. Signals are transitted by different stations at different frequencies. hese frequencies are picked up by the antenna and corresponding to these frequencies, a nuber of voltages appear across the series -circuit. But axiu current flows through the circuit for that a.c. voltage which has frequency equal to f. f the Q-value of the circuit is large, the signals of the other stations will be very weak i.e., circuit will be ore selective. By changing the value of the adjustable capacitor, signal fro the desired station can be tuned in. AEAGE POWE N AN A U Suppose in an a.c circuit, the voltage and current at any instant are given by sin t

AENANG UEN (SHO) 3-4 PAGE:6 and sin( t ) where is the phase angle by which current leads the voltage. he instantaneous power is given by P sin( t ).sin t = sin( t )sin t = cos cos(t ) [ sin A sin B cos(a B) cos(a B) ] he average power in the circuit over a coplete cycle is P av P cos cos(t ) cos cos(t ) = cos = cos ( ) or Pav. cos or P cos av rs rs cos( t ) Special cases. Pure resistive circuit Here voltage and current are in sae phase, i.e., P. cos av rs rs rs rs. Pure inductive circuit Here voltage leads the current in phase by, i.e., Pav rs rs cos hus the average power consued in an inductive circuit over a coplete cycle is zero. 3. Pure capacitive circuit Here voltage lags behind the current in phase by,i.e., Pav rsrs cos hus the average power consued in a capacitive circuit over a coplete cycle is also zero. POWE FAO he average power of an a.c. circuit is given by Pav rsrs cos he product rs rs does not give the actual power and is called apparent power. he factor cos is called power factor of an a.c. circuit. rue power=apparent power x Power factor hus power factor ay be defined as the ratio of true power to the apparent power of an a.c. circuit. Wattless current (dle current) f the average power consued in an a.c. circuit is zero, then the current in a.c. circuit is said to be wattless. his happens in the case of a pure inductor or capacitor. he current is called wattles because the current in the circuit does not do any work. AEAGE POWE ASSOAED WH A ESSO sin ωt and sin ωt nstantaneous power P sin ωt = ( cos ωt)

AENANG UEN (SHO) 3-4 PAGE:7 Hence the average power consued by the resistor over a coplete cycle (i.e., fro t= to t=) is P av P cost ( cost) cost = cos t rs rs =. rs. rs rs AEAGE POWE ASSOAED WH A PUE NDUO sin ωt π and sin ωt cosωt nstantaneous power, sin ωt cosωt P= sin ωt cosωt sin ωt Average power over a coplete cycle, P av P sin ωt sin ωt = sin t hus average power dissipated per cycle in an inductor is zero. AEAGE POWE ASSOAED WH A APAO sin ωt π and sin ωt cosωt nstantaneous power P sin ωt cosωt = sin ωt Average power over a coplete cycle Pav P sin ωt = sin ωt = sin t hoke coil hoke coil is siply an inductor. hoke coil offers a reactance X f to the flow of alternating current. Average power consued by a choke coil is zero. hus, a choke coil reduces current in an a.c. circuit without consuing any power. When an ohic resistance is used, current reduces but energy loss occur due to heating. So choke is preferred. For d.c, f=, so X i.e., choke coil cannot liit direct current. Qn. A choke coil and a bulb are connected in series to an a.c. source. he bulb shines brightly. How does its brightness change when an iron core is inserted in the choke coil? Ans. When the iron core is inserted in the choke coil, the self-inductance increases. onsequently, the inductive reactance, glows dier. X During the first quarter of each current cycle, as the current increases, the agnetic flux through the inductor builds up and energy is stored in the inductor fro the external source. n the next quarter of cycle, as the current decreases, the flux decreases and the stored energy is returned to the source. hus, in half cycle, no net power is consued by the inductor. When the capacitor is connected across an a.c. source, it absorbs energy fro the source for a quarter cycle as it is charged. t returns energy to the source in the next quarter cycle as it is discharged. hus in a half cycle, no net power is consued by the capacitor. increases. his decreases the current in the circuit and bulb

AENANG UEN (SHO) 3-4 PAGE:8 -OSAONS When a charged capacitor is allowed to discharge through a non-resistive inductor, electrical oscillations of constant aplitude and frequency are produced. hese oscillations are called -oscillations. he frequency of oscillation of charge or current f However, the oscillations are usually daped due to the resistive losses in the inductor and dielectric losses in the capacitor. ANSFOME: ransforer is a device used to convert low alternating voltage at higher current into high alternating voltage at low current and vice-versa. in other words, a transforer is an electrical device used to increase or decrease alternating voltage. onstruction t consists of two separate coils of insulated wire wound on sae iron core. One of the coils connected to a.c. input is called priary(p) and the other winding giving output is called secondary (S). heory When an alternating source of ef p is connected to the priary coil, an alternating current flows through it. Due to the flow of alternating current agnetic flux linked with the priary and secondary changes. his produces an ef in the secondary. et NP and N S be the nuber of turns of priary and secondary coils repectively. he iron core is capable of coupling the whole of the agnetic flux produced by the turns of the priary coil with the secondary coil. According to Faraday s law of electroagnetic induction, the induced ef in the priary coil, d p N P.() he induced ef in the secondary coil, d s N S () Dividing () by (), we get s Ns N p p f output voltage is less than the input voltage, the transforer is called step down transforer. n a step down transforer Ns Np and s p. f the output voltage is greater than the input voltage, the transforer is called step up transforer. n a step up transforer Ns NP and s p. For an ideal transforer(in which there are no energy losses), Output power=nput power ss pp Output power ss For a transforer, efficiency, nput power pp For an ideal transforer, efficiency, is %. But in a real transforer, the efficiency varies fro 9-99%. his indicates that there are soe energy losses in the transforer. Energy losses in transforers. opper loss. t is energy loss due to heating of the copper windings due their resistance.. Eddy current loss. t is the energy loss due to heating of the core by the eddy current. his loss can be reduced by using lainated iron core. 3. Hysteresis loss. t is the energy loss due to the heating of the core due to the application of cyclic agnetizing field. t can be iniized by using core aterial having narrow hysteresis loop. 4. Flux leakage. he agnetic flux produced by the priary ay not fully pass through the secondary. his loss can be iniized by winding the priary and secondary coils over one another.