10. A.C CIRCUITS. Theoretically current grows to maximum value after infinite time. But practically it grows to maximum after 5τ. Decay of current :
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1 . A. IUITS Synopss : GOWTH OF UNT IN IUIT : d. When swch S s closed a =; = d. A me, curren = e 3. The consan / has dmensons of me and s called he nducve me consan ( τ ) of he crcu. 4. = τ; =.63, n one me consan, he curren reaches 63% of he maxmum value. The me consan ells us how fas wll he curren grow. 5. =, when =, where =. Theorecally curren grows o maxmum value afer nfne me. Bu praccally grows o maxmum afer 5τ. Decay of curren : d 6. When swch S s open a =; = d a =, = a me, = τ e The curren reduces o 37% of he nal value n one me consan.37.e., 63% of he decay s complee. 7. nergy sored n nducor =. hargng of a capacor : 8. When a capacor s conneced o a baery, posve charge appears on one plae and negave charge on he oher. The poenal dfference beween he plaes ulmaely becomes equal o e.m.f of he baery. The whole process akes some me and durng hs me here s an elecrc curren hrough connecng wres and he baery. q 9. Usng Krchoff s loop law + =.. A any me, q = e = Q e.63 V = e ; = e.. The consan has dmensons of me and s called capacve me consan ( τ ).. In one me consan ( τ =), he charge accumulaed on he capacor s q= c b S a decay of curren = growh of curren me me growh of charge me b S a
2 Dschargng of a capacor : 3. When he plaes of a charged capacor are conneced hrough a conducng wre, he capacor ges dscharged, agan here s a flow of charge hrough he wres and hence here s a curren q 4. = 5. q = Qe, where Q = A.. rcus V = e ; ; = e. 6. A =, q=.37q,.e., 63% of he dschargng s complee n one me consan. Alernang curren :. An alernang curren or e.m.f s one whose magnude and drecon vary perodcally wh me.. Alernang curren abbrevaed as ac no A. or a.c 3. The smples ypes of alernang curren and e.m.f have a snusodal varaon, gven respecvely by = sn ω and = sn ω where, are called peak values of curren and volage respecvely and ω s he angular frequency. 4. The me aken by alernang curren o go hrough one cycle of changes s called s perod (T) and π T =. ω decay of charge ω 5. The number of cycles per second of an alernang curren s called s frequency, n = =. T π 6. The phase of an alernang curren a any nsan represens he fracon of he me perod ha has elapsed snce he curren las passed hrough he zero poson of reference. Phase can also be expressed n erms of angle n radans. 7. An alernang curren or e.m.f vares perodcally from a maxmum n one drecon hrough zero o a maxmum n he oppose drecon, and so on. The maxmum value of he curren or e.m.f n eher drecon s called he peak value. 8. The average or mean value of alernang curren or e.m.f for complee cycle s zero. I has no sgnfcance. Hence, he mean value of alernang curren ( ) s defned as s average over half a T / cycle. For posve half cycle = d where = sn ω = =.636 smlarly average value of T π e.m.f =. π 9. The roo mean square (r.m.s) value of an alernang curren s he square roo of he average of durng a complee cycle where s he nsananeous value of he alernang curren. (or) I s he seady curren, whch when passed hrough a ressance for a gven me wll produce he same amoun of hea as he alernang curren does n he same ressance and n he same me. (or) The r.m.s velocy of an alernang volage can be defned as ha drec volage whch produces he same rae of heang n a gven ressance. The r.m.s value of alernang volage s also called as he effecve or he vrual value of he volage. q.37c me
3 T rms = d where = sn ω T rms = = smlarly.77 T rms = d T where = sn ω A.. rcus rms = Volage marked on ac nsrumens s he r.m.s volage,.e. V ac means rms = V.. In any crcu, he rao of he effecve volage o he effecve curren s called he mpedance of he crcu. Is un s ohm.. A dagram represenng alernang volage and curren as vecors wh phase angle beween hem s called phasor dagram.. Purely ressve crcu : A crcu conanng an A. source and a ressor s known as purely ressve crcu. If = sn ω and he curren a a me s, hen sn ω = Here boh volage and curren are n same phase. Insananeous power dsspaon p = = sn ω Average power dsspaon P = rms rms 3. Purely nducve crcu : A crcu conanng an A. source and nducor s known as purely nducve crcu. If = sn ω, he crcu equaon s π = sn where ω = ω d = ; d = sn ωd by negraon we ge d The consan X = ω plays he role of effecve ressance of he crcu. The consan X s called he reacance of he nducor. I s zero for drec curren ( ω=) and ncreases as he frequency ncreases. The curren lags he volage n phase by π / and he quany ω s a measure of he effecve opposon o he flow of A.. The average power consumed n a cycle s zero. 3
4 A.. rcus 9 o 4. Purely capacve crcu : A crcu conanng an A. source and a capacor s known as purely capacve crcu. If = sn ω, he crcu equaon s Q= c = c sn ω by dfferenang; dq π = = sn ω + where = d ω 9 o The curren leads he volage n phase by π /. The quany / ω s a measure of he effecve opposon of alernang curren by a capacor. I s denoed by X and s called capacve reacance X =. ω 5. The peak curren and he peak e.m.f n all he above hree crcus can be wren as = where = for a purely ressve crcu, =/ ω for a purely capacve crcu and = ω for a purely nducve crcu. The general name for s mpedance. 6. If he e.m.f of an A. crcu s represened by = sn ω, he curren can be represened as = sn( ω + φ). For purely ressve crcu φ =; for a purely capacve crcu φ = π / and for a purely nducve crcu φ = π /. The consan φ s called phase facor. 7. seres crcu : The mpedance of he crcu s gven by = + ω. The curren n he seady sae s gven by ω = sn( ω φ) where an φ = + ω ω The appled volage leads he curren by Tan 4
5 A.. rcus ω φ 8. - seres crcu : The mpedance of he crcu s gven by = +. ω The curren n he seady sae s gven by = sn( ω + φ) where The appled volage leads he curren by Tan ω /ω φ 9. seres crcu : rms = + ( ) = + (X X) = + ω ω ω ο ω ω / ω an φ = ; ω = () When ω >, an φ s posve.e., φ s posve n such case e.m.f leads he curren. ω () When ω <, an φ s negave.e., φ s negave n such case e.m.f lags behnd he curren. ω () When ω =, an φ s zero.e., φ s zero n such case curren and e.m.f are n phase ω wh each oher. When X = X or ω =, he mpedance becomes mnmum and hence curren wll be ω maxmum. The crcu s hen sad o be resonance and he correspondng frequency s known as resonan frequency. The resonan frequency=.. The peak curren n hs case s π / ω- ω /ω ω φ 5
6 A.. rcus. Qualy facor of resonance : he selecvy or sharpness of resonan crcu s measured by Q- facor called qualy facor. The Q facor or qualy facor of a resonan crcu s defned as rao of he volage drop across nducor (or capacor) o he appled volage. vol age acros (or ) Q = appled volage Q =. The Q-facor of seres crcu wll be large (or more sharpness) f s low or s large or s low.. Power n A.. crcu : The average power P delvered by A. source n a complee cycle s gven by P= rms. rms cos φ where cosφ s called he power facor of crcu. P also represens he average power delvered n a long me. 6
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