4. Fundamental of A.C. Circuit

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1 4. Fundaenal of A.. icui 4. Equaion fo geneaion of alenaing induce EMF An A geneao uses he pinciple of Faaday s elecoagneic inducion law. saes ha when cuen caying conduco cu he agneic field hen ef induced in he conduco. nside his agneic field a single ecangula loop of wie oes aound a fixed axis allowing i o cu he agneic flux a vaious angles as shown below figue 4.. N Magneic Pole Magneic Flux S Whee, N =No. of uns of coil A = Aea of coil ( ) ω=angula velociy (adians/second) = Maxiu flux (wb) Axis of oaion Axis of oaion Wie oop(onduco) Wie oop(onduco) Figue 4. Geneaion of EMF When coil is along XX (pependicula o he lines of flux), flux linking wih coil=. When coil is along YY (paallel o he lines of flux), flux linking wih he coil is zeo. When coil is aking an angle wih espec o XX flux linking wih coil, = cosω [ = ω]. X ω cosω N Y Y S sinω X Figue 4. Alenaing nduced EMF Accoding o Faaday s law of elecoagneic inducion, d en d ( cos ) e Nd d e N ( sin ) e N sin e E sin Whee, E N N no. of uns of he coil B A B Maxiu flux densiy (wb/ ) A Aea of he coil ( ) f Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5)

2 4. Fundaenal of A.. icui e N B Af sin Siilaly, an alenaing cuen can be expess as i sin Whee, = Maxiu values of cuen Thus, boh he induced ef and he induced cuen vay as he sine funcion of he phase angle. Shown in figue D E F N 9 G 7 S B H 45 A 35 /36 e Figue 4.3 Wfo of Alenaing nduced EMF ω Phase angle nduced ef e 9 e E sin e E 8 e 7 36 e E 4. Definiions Wfo is defined as he gaph beween agniude of alenaing quaniy (on Y axis) agains ie (on X axis). e Apliude + Sine W Tie Apliude + Squae W Tie - - Apliude + Tiangula W Tie Apliude + oplex W Tie - - Figue 4.4 A.. Wfos ycle is defined as one coplee se of posiive, negaive and zeo values of an alenaing quaniy. Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5)

3 4. Fundaenal of A.. icui nsananeous value is defined as he value of an alenaing quaniy a a paicula insan of given ie. Geneally denoed by sall lees. e.g. i= nsananeous value of cuen v= nsananeous value of volage p= nsananeous values of powe Apliude/ Peak value/ es value/ Maxiu value is defined as he axiu value (eihe posiive o negaive) aained by an alenaing quaniy in one cycle. Geneally denoed by capial lees. e.g. = Maxiu alue of cuen = Maxiu value of volage P= Maxiu values of powe Aveage value is defined as he age of all insananeous value of alenaing quaniies ove a half cycle. e.g. = Aveage value of volage = Aveage value of cuen MS value is he equivalen dc cuen which when flowing hough a given cicui fo a given ie poduces sae aoun of hea as poduced by an alenaing cuen when flowing hough he sae cicui fo he sae ie. e.g. s =oo Mean Squae value of volage s = oo Mean Squae value of cuen Fequency is defined as nube of cycles copleed by an alenaing quaniy pe second. Sybol is f. Uni is Hez (Hz). Tie peiod is defined as ie aken o coplee one cycle. Sybol is T. Uni is seconds. Powe faco is defined as he cosine of angle beween volage and cuen. Powe Faco = pf = cos, whee is he angle beween volage and cuen. Acive powe is he acual powe consued in any cicui. is given by poduc of s volage and s cuen and cosine angle beween volage and cuen. ( cos). Acive Powe= P= = cos. Uni is Wa (W) o kw. Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 3

4 4. Fundaenal of A.. icui eacive powe The powe dawn by he cicui due o eacive coponen of cuen is called as eacive powe. is given by poduc of s volage and s cuen and sine angle beween volage and cuen ( sin). eacive Powe = Q= X = sin. Uni is A o ka. Appaen powe is he poduc of s value of volage and s value of cuen. is oal powe supplied o he cicui. Appaen Powe = S =. Uni is A o ka. Peak faco/ es faco is defined as he aio of peak value (ces value o axiu value) o s value of an alenaing quaniy. Peak faco = Kp =.44 fo sine w. Fo faco is defined as he aio of s value o age value of an alenaing quaniy. Denoed by Kf. Fo faco Kf =. fo sine w. Phase diffeence is defined as angula displaceen beween wo zeo values o wo axiu values of he wo-alenaing quaniy having sae fequency. + n Phase ( ) + Posiive Phase () + Negaive Phase (-) - - () = sinω - () = sin(ω+ - () = sin(ω- Figue 4.5 A.. Phase Diffeence eading phase diffeence A quaniy which aains is zeo o posiive axiu value befoe he copaed o he ohe quaniy. agging phase diffeence A quaniy which aains is zeo o posiive axiu value afe he ohe quaniy. Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 4

5 4. Fundaenal of A.. icui 4.3 Deivaion of age value and MS value of sinusoidal A signal Aveage alue Gaphical Mehod Analyical Mehod olage olage Aea Unde he uve /n Tie Tie Figue 4.6 Gaphical Mehod fo Aveage alue Figue 4.7 Analyical Mehod fo Aveage alue Su of All ns anan eous alues Toal No. of alues Aea Unde he uve Base of he uve v v v v v... v cos cos cos. 637 Sin d Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 5

6 4. Fundaenal of A.. icui MS alue Gaphical Mehod Analyical Mehod olage olage s 9 8 /n Tie Half ycle Tie - s - One Full ycle Figue 4.8 Gaphical Mehod fo MS alue Figue 4.9 Analyical Mehod fo MS alue s Su of all sq. of insananeous values Toal No. of alues s Aea unde he sq. cuve Base of he cuve s v v v v v... v s Sin d ( cos ) s d s (sin ) 4 s ( ) 4 s. 77 s Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 6

7 4. Fundaenal of A.. icui 4.4 Phaso epesenaion of Alenaing Quaniies Sinusoidal expession given as: v() = sin (ω ± Φ) epesening he sinusoid in he iedoain fo. Phaso is a quaniy ha has boh Magniude and Diecion. eco aoaion ω ads /s v()= sinω ω oaing Phaso 3 - Sinusoidal Wfo in Tie Doain Figue 4. Phaso epesenaion of Alenaing Quaniies Phase Diffeence of a Sinusoidal Wfo The genealized aheaical expession o define hese wo sinusoidal quaniies will be wien as: v Sin i sin ( ) olage (v) + + uen (i) - - ω AG EAD ω Figue 4. W Fos of olage & uen Figue 4. Phaso Diaga of olage & uen As show in he above volage and cuen equaions, he cuen, i is lagging he volage, v by angle. So, he diffeence beween he wo sinusoidal quaniies epesening in wfo shown in Fig. 4. & phasos epesening he wo sinusoidal quaniies is angle and he esuling phaso diaga shown in Fig. 4.. Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 7

8 4. Fundaenal of A.. icui 4.5 Puely esisive icui The Fig. 4.3 an A cicui consising of a pue esiso o which an alenaing volage v=sinω is applied. icui Diaga v=sinω Whee, = nsananeous olage v = Maxiu olage = olage acoss esisance Equaions fo olage and uen As show in he Fig. 4.3 volage souce v Sin Accoding o oh s law v i sin i i sin Figue 4.3 Pue esiso onneced o A Supply Fo above equaions i is clea ha cuen is in phase wih volage fo puely esisive cicui. Wfos and Phaso Diaga The sinew and veco epesenaion of 4.4 & 4.5. v Sin & i sin ae given in Fig.,i v=sinω i=sinω ω ω Figue 4.4 Wfo of olage & uen fo Pue esiso Figue 4.5 Phaso Diaga of olage & uen fo Pue esiso Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 8

9 4. Fundaenal of A.. icui Powe The insananeous value of powe dawn by his cicui is given by he poduc of he insananeous values of volage and cuen. nsananeous powe p v i ( ) p sin sin ( ) p( ) sin p ( ) Aveage Powe P ( cos ) ( cos ) d 4 P P (sin ) 4 s s The age powe consued by puely esisive cicui is uliplicaion of s & s. 4.6 Puely nducive icui The Fig. 4.6 an A cicui consising of a pue nduco o which an alenaing volage v=sinω is applied. icui Diaga i v=sinω Figue 4.6 Pue nduco onneced o A Supply Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 9

10 4. Fundaenal of A.. icui Equaions fo olage and uen As show in he Fig. 4.6 volage souce v Sin Due o self-inducance of he coil, hee will be ef indued in i. This back ef will oppose he insananeous ise o fall of cuen hough he coil, i is given by di eb - d As, cicui does no conain any esisance, hee is no ohic dop and hence applied volage is equal and opposie o back ef. v -e b di v d di v d Wfo and Phaso Diaga v,i =sinω =sin(ω- 9) di sin d ω sin d di negae on boh he sides, di sin d cos i i cos i sin 9 Fo he above equaions i is clea ha he cuen lags he volage by 9 in a puely inducive cicui. Powe 9 Figue 4.7 Wfo of olage & uen fo Pue nduco 9 Figue 4.8 Phaso Diaga of olage & uen fo Pue nduco The insananeous value of powe dawn by his cicui is given by he poduc of he insananeous values of volage and cuen. nsananeous Powe p vi p sin sin 9 p sin ( cos ) sin cos p ω Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5)

11 4. Fundaenal of A.. icui p sin Aveage Powe sin P d cos 4 cos 4 cos 8 P The age powe consued by puely inducive cicui is zeo. 4.7 Puely apaciive icui The Fig. 4.9 shows a capacio of capaciance faads conneced o an a.c. volage supply v=sinω. icui Diaga i v=sinω q+ q- Equaions fo olage & uen As show in he Fig. 4.9 volage souce v Sin Figue 4.9 Pue apacio onneced A Supply A pue capacio having zeo esisance. Thus, he alenaing supply applied o he plaes of he capacio, he capacio is chaged. f he chage on he capacio plaes a any insan is q and he poenial diffeence beween he plaes a any insan is v hen we know ha, q v q sin The cuen is given by ae of change of chage. dq i d d sin i d Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5)

12 4. Fundaenal of A.. icui i sin i cos / i cos X c o i sin( 9 ) ( ) X c Fo he above equaions i is clea ha he cuen leads he volage by 9 in a puely capaciive cicui. Wfo and Phaso Diaga,i v=sinω i=sin(ω+9) ω +9 ω 9 Powe Figue 4. Wfo of olage & uen fo Pue apacio Figue 4. Phaso Diaga of olage & uen fo Pue apacio The insananeous value of powe dawn by his cicui is given by he poduc of he insananeous values of volage and cuen. nsananeous Powe p vi ( ) p sin sin 9 ( ) p sin cos ( ) p sin cos ( ) sin cos p( ) p( ) sin Aveage Powe sin P d Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5)

13 4. Fundaenal of A.. icui cos 4 cos 4 cos 8 P The age powe consued by puely capaciive cicui is zeo. 4.8 Seies esisance-nducance (-) icui onside a cicui consising of a esiso of esisance ohs and a puely inducive coil of inducance heny in seies as shown in he Fig.4.. i v=sinω Figue 4. icui Diaga of Seies - icui n he seies cicui, he cuen i flowing hough and will be he sae. Bu he volage acoss he will be diffeen. The veco su of volage acoss esiso and volage acoss induco will be equal o supply volage v. Wfos and Phaso Diaga The volage and cuen ws in - seies cicui is shown in Fig. 4.3.,i v=sinω i=sin(ω- ) ω Figue 4.3 Wfo of olage and uen of Seies - icui We know ha in puely esisive he volage and cuen boh ae in phase and heefoe veco is dawn supeiposed o scale ono he cuen veco and in puely inducive cicui he cuen lag he volage by 9 o. So, o daw he veco diaga, fis aken as he efeence. This is shown in he Fig Nex dawn in phase wih. Nex is dawn 9 o leading he. The supply volage is hen phaso Addiion of and. Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 3

14 4. Fundaenal of A.. icui ω = + Figue 4.4 Phaso Diaga of Seies - icui Thus, fo he above, i can be said ha he cuen in seies - cicui lags he applied volage by an angle. f supply volage v Sin i sin Whee olage Tiangle Z pedance Tiangle Powe Tiangle =*Z =* =*X Figue 4.5 olage Tiangle Seies - icui Z X Figue 4.6 pedance Tiangle Seies - icui Appaen Powe,S (A) eal Powe,P (Wa) eacive Powe,Q (A) Figue 4.7 Powe Tiangle Seies - icui ( ) ( X ) X Z whee, Z X Powe Faco Powe faco cos Z P S Z X an X e al Powe P cos e acive Powe Q sin X Appaen Powe S Z Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 4

15 4. Fundaenal of A.. icui Powe The insananeous value of powe dawn by his cicui is given by he poduc of he insananeous values of volage and cuen. nsananeous powe p vi p sin sin p sin sin p sin s in p cos - cos(- ) Thus, he insananeous values of he powe consis of wo coponens. Fis coponen is consan w... ie and second coponen vay wih ie. Aveage Powe cos - cos(- ) d cos - cos( - ) d 4 cos d - cos( - ) d sin(- ) cos cos - sin sin 8 cos - cos P cos P cos - sin4 sin P cos Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 5

16 4. Fundaenal of A.. icui 4.9 Seies esisance-apaciance icui onside a cicui consising of a esiso of esisance ohs and a puely capaciive of capaciance faad in seies as in he Fig i v=sinω Figue 4.8 icui Diaga of Seies - icui n he seies cicui, he cuen i flowing hough and will be he sae. Bu he volage acoss he will be diffeen. The veco su of volage acoss esiso and volage acoss capacio will be equal o supply volage v. Wfos and Phaso Diaga,i v=sinω i=sin(ω+) ω Figue 4.9 Wfo of olage and uen of Seies - icui We know ha in puely esisive he volage and cuen in a esisive cicui boh ae in phase and heefoe veco is dawn supeiposed o scale ono he cuen veco and in puely capaciive cicui he cuen lead he volage by 9 o. So, o daw he veco diaga, fis aken as he efeence. This is shown in he Fig Nex dawn in phase wih. Nex is dawn 9 o lagging he. The supply volage is hen phaso Addiion of and. - = + ω Figue 4.3 Phaso Diaga of Seies - icui Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 6

17 4. Fundaenal of A.. icui Thus, fo he above equaion i is clea ha he cuen in seies - cicui leads he applied volage by an angle. f supply volage O v Sin i sin Whee, Z olage Tiangle pedance Tiangle =Z = A - =(-X) Z - -X Powe Tiangle eal Powe,P (Wa) - Appaen Powe,S (A) eacive Powe,Q (A) Figue 4.3 olage Tiangle of Seies - icui D Figue 4.3 pedance Tiangle Seies - icui Figue 4.33 Powe Tiangle Seies - icui ( ) ( X ) X Z whee, Z X Powe Faco p.f. cos o P Z S Powe Z X an X eal Powe, P cos eacive Powe, Q sin X Appaen Powe,S The insananeous value of powe dawn by his cicui is given by he poduc of he insananeous values of volage and cuen. nsananeous powe p vi p sin sin p sin sin sin sin p p cos - cos( ) Thus, he insananeous values of he powe consis of wo coponens. Fis coponen eains consan w... ie and second coponen vay wih ie. Z Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 7

18 4. Fundaenal of A.. icui Aveage Powe cos - cos(+ ) d cos - cos( + ) d 4 cos d - cos( + ) d sin(+ ) cos cos - sin sin 8 cos - P cos P cos P cos - sin4 sin 4. Seies cicui onside a cicui consising of a esiso of oh, pue induco of inducance heny and a pue capacio of capaciance faads conneced in seies. i v=sinω Phaso Diaga Figue 4.34 icui Diaga of Seies icui uen is aken as efeence. is dawn in phase wih cuen, is dawn leading by 9, is dawn lagging by 9 Figue 4.35 Phaso Diaga of Seies icui Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 8

19 4. Fundaenal of A.. icui Since and ae in opposiion o each ohe, hee can be wo cases: () > () < ase- When, >, he phaso diaga would be as in he figue 4.36 Phaso Diaga - ω ase- When, <, he phaso diaga would be as in he figue 4.37 Phaso Diaga - ω - Figue 4.36 Phaso Diaga of Seies -- icui fo ase > ( ) X X X X Z whee, Z X X The angle by an an an by which leads is given X X X X Thus, when > he seies cuen lags by angle. f v Sin i Sin Powe consued in his case is equal o seies cicui cos. Figue 4.37 Phaso Diaga of Seies -- icui fo ase < ( ) X X X X Z whee, Z X X The angle an an an by which lags is given by X X X X Thus, when < he seies cuen leads by angle. f v Sin i Sin Powe consued in his case is equal o seies cicui cos. Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 9

20 4. Fundaenal of A.. icui 4. Seies esonance cicui Such a cicui shown in he Fig is conneced o an A.. souce of consan supply volage bu having vaiable fequency. i f v=sinω Figue 4.38 icui Diaga of Seies esonance icui The fequency can be vaied fo zeo, inceasing and appoaching infiniy. Since X and X ae funcion of fequency, a a paicula fequency of applied volage, X and X will becoe equal in agniude and powe faco becoe uniy. Since X = X, X X = Z The cicui, when X = X and hence Z =, is said o be in esonance. n a seies cicui since cuen eain he sae houghou we can wie, X = X i.e. = Phaso Diaga Shown in he Fig.4.39 is he phaso diaga of seies esonance cicui. So, a esonance and will cancel ou of each ohe. = The supply volage = ( ) i.e. he supply volage will dop acoss he esiso. Figue 4.39 Phaso Diaga of Seies esonance icui esonance Fequency A esonance fequency X = X f f is he esonance fequency f Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5)

21 4. Fundaenal of A.. icui f f Q- Faco The Q- faco is nohing bu he volage agnificaion duing esonance. indicaes as o how any ies he poenial diffeence acoss o is geae han he applied volage duing esonance. Q- faco = olage agnificaion Q Faco Q Faco S X X f Bu f Gaphical epesenaion of esonance esisance () is independen of fequency. Thus, i is epesened by saigh line. nducive eacance (X) is diecly popoional o fequency. Thus, i is inceases linealy wih he fequency. X X f f apaciive eacance(x) is invesely popoional o fequency. Thus, i is show as hypebolic cuve in fouh quadan. X X f f pedance (Z) is iniu a esonance fequency. Z X X Fo, f f, Z uen () is axiu a esonance fequency. Z Fo f f, is axiu,max Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5)

22 4. Fundaenal of A.. icui Powe faco is uniy a esonance fequency. Powe faco=cos = Z Fo f f, p. f. (uniy) P.F. cos Z X f f -X Figue 4.4 Gaphical epesenaion of Seies esonance icui 4. Paallel esonance icui Fig. 4.4 Shows a paallel cicui consising of an inducive coil wih inenal esisance oh and inducance heny in paallel wih capacio faads. i = cos sin v=sinω Figue 4.4 icui Diaga of Paallel esonance icui Figue 4.4 icui Diaga of Paallel esonance icui The cuen can be esolved ino is acive and eacive coponens. s acive coponen cos and eacive coponen sin. Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5)

23 4. Fundaenal of A.. icui A paallel cicui is said o be in esonance when he powe faco of he cicui becoes uniy. This will happen when he esulan cuen is in phase wih he esulan volage and hence he phase angle beween he is zeo. n he phaso diaga shown, his will happen when = sin and = cos. esonance Fequency To find he esonance fequency, we ake use of he equaion = sin. Z f f sin X X Z Z Z X X f f f he esisance of he coil is negligible, f pedance To find he esonance fequency, we ake use of he equaion = cos esonance, he supply cuen will be in phase wih he supply volage. cos Z Z Z Z Z Bu Z Z because, a The ipedance duing paallel esonance is vey lage because of and has a vey lage value a ha ie. Thus, ipedance a he esonance is axiu. will be iniu. Z Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 3

24 4. Fundaenal of A.. icui Q-Faco Q- faco = uen agnificaion Q Faco sin cos Q Faco sin cos an f Bu f Gaphical epesenaion of Paallel esonance onducance (G) is independen of fequency. Hence i is epesened by saigh line paallel o fequency. nducive Suscepance (B) is invesely popoional o he fequency. Also, i is negaive. B, B jx j f f apaciive Suscepance (B) is diecly popoional o he fequency. j B j f, B f jx X cos,y P.F. B f Z G f -B Figue 4.43 Gaphical epesenaion of Paallel esonance icui Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 4

25 4. Fundaenal of A.. icui Adiance (Y) is iniu a esonance fequency. Y G B B Fo, f f, Y G uen () is iniu a esonance fequency. Y Powe faco is uniy a esonance fequency. Powe faco=cos = G Y 4.3 opaison of Seies and Paallel esonance S.No. Descipion Seies icui Paallel icui pedance a esonance uen 3 esonance Fequency Miniu Z = Maxiu f Maxiu Z Miniu / 4 Powe Faco Uniy Uniy 5 Q- Faco f f f 6 agnifies a esonance olage uen Bhsh M Jesadia -EE Depaen Eleens of Elecical Engineeing (5) 5

r r r r r EE334 Electromagnetic Theory I Todd Kaiser

r r r r r EE334 Electromagnetic Theory I Todd Kaiser 334 lecoagneic Theoy I Todd Kaise Maxwell s quaions: Maxwell s equaions wee developed on expeienal evidence and have been found o goven all classical elecoagneic phenoena. They can be wien in diffeenial