Module 4. Single-phase AC Circuits. Version 2 EE IIT, Kharagpur

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1 Module Single-phase AC Circuis Version EE, Kharagpur

2 Lesson Generaion of Sinusoidal Volage Wavefor (AC) and Soe Fundaenal Conceps Version EE, Kharagpur

3 n his lesson, firsly, how a sinusoidal wavefor (ac) is generaed, is described, and hen he ers, such as average and effecive (rs) values, relaed o periodic volage or curren wavefors, are explained. Lasly, soe exaples o find average and roo ean square (rs) values of soe periodic wavefors are presened. Keywords: Sinusoidal wavefors, Generaion, Average and RMS values of Wavefors. Afer going hrough his lesson, he sudens will be able o answer he following quesions:. Wha is an ac volage wavefor?. How a sinusoidal volage wavefor is generaed, wih soe deail?. For periodic volage or curren wavefors, o copue or obain he average and rs values, and also he ie period.. o copare he differen periodic wavefors, using above values. Generaion of Sinusoidal (AC) Volage Wavefor N B ω A S R Fig.. Scheaic diagra for single phase ac generaion A uli-urn coil is placed inside a agne wih an air gap as shown in Fig... he flux lines are fro Norh Pole o Souh Pole. he coil is roaed a an angular speed, ω n (rad/s). ω n speed of he coil (rev/sec, or rps) N 6 n speed of he coil (rev/in, or rp) l lengh of he coil () b widh (diaeer) of he coil () No. of urns in he coil Version EE, Kharagpur

4 B flux densiy in he air gap ( Wb / ) v b n angenial velociy of he coil (/sec) Magneic Field L L N A θ o θ M A B O θ ω b (a) (b) Fig.. (a) Coil posiion for Fig.., and (b) Deails A a cerain insan, he coil is an angle (rad), θ ω wih he horizonal (Fig..). he ef (V) induced on one side of he coil (conducor) is B l v sinθ, θ can also be ered as angular displaceen. he ef induced in he coil (single urn) is B l v sinθ B l b nsinθ he oal ef induced or generaed in he uli-urn coil is e θ ) Bl b nsinθ Bl b n sinθ E sinθ ( his ef as a funcion of ie, can be expressed as, e( ) E sinω. he graph of e () or e (θ ), which is a sinusoidal wavefor, is shown in Fig..a Area of he coil ( ) a l b Flux cu by he coil (Wb) φ a B l b B Flux linkage (Wb) ψ φ B l b ay be noed hese values of flux φ and flux linkage ψ, are axiu, wih he coil being a horizonal posiion, θ. hese values change, as he coil oves fro he horizonal posiion (Fig..). So, also is he value of induced ef as saed earlier. he axiu value of he induced ef is, dθ E n B l b n φ n ψ ω ψ ψ d Deerinaion of frequency (f) in he ac generaor n he above case, he frequency (Hz) of he ef generaed is Version EE, Kharagpur

5 f ω /( ) n, no. of poles being, i.e. having only one pole pair. n he ac generaor, no. of poles p, and he speed (rps) n, hen he frequency in Hz or cycles/sec, is f no. of cycles/sec no. of cycles per rev no. of rev per sec no. of pairs of poles no. of rev per sec ( p / ) n p N p ω or, f Exaple For a -pole ac generaor o obain a volage having a frequency of 5 Hz, f 5 he speed is, n 5 rps 5 6, 5 rp p For a -pole (p ) achine, he speed should be, rp. Siilarly, he speed of he achine having differen no. of poles, required o generae a frequency of 5 Hz can be copued. Sinusoidal volage wavefor having frequency, Periodic Volage or Curren Wavefor Average value f wih ie period (sec), he curren wavefor shown in Fig..a, is periodic in naure, wih ie period,. is posiive for firs half cycle, while i is negaive for second half cycle. he average value of he wavefor, i() is defined as av Area over half cycle ie period of half cycle i( ) d i( ) d Please noe ha, in his case, only half cycle, or half of he ie period, is o be used for copuing he average value, as he average value of he wavefor over full cycle is zero (). f he half ie period (/) is divided ino 6 equal ie inervals ( Δ ), ( i + i + i + i6 ) Δ ( i + i + i + i6 ) Area over half cycle av 6 Δ 6 ie period of half cycle Please noe ha no. of ie inervals is n 6. Roo Mean Square (RMS) value For his curren in half ie period subdivided ino 6 ie inervals as given above, in he resisance R, he average value of energy dissipaed is given by ( i + i + i + i 6 ) R 6 f Version EE, Kharagpur

6 he graph of he square of he curren wavefor, i ( ) is shown in Fig..b. Le be he value of he direc curren ha produces he sae energy dissipaed in he resisance R, as produced by he periodic wavefor wih half ie period subdivided ino n ie inervals, ( i i i in ) Δ R R n Δ ( i + i + i + i n ) n Δ Δ Area of i curve over half cycle ie period of half cycle Version EE, Kharagpur

7 i d i d his value is ered as Roo Mean Square (RMS) or effecive one. Also o be noed ha he sae rs value of he curren is obained using he full cycle, or he ie period. Average and RMS Values of Sinusoidal Volage Wavefor As shown earlier, norally he volage generaed, which is also ransied and hen disribued o he consuer, is he sinusoidal wavefor wih a frequency of 5 Hz in Version EE, Kharagpur

8 his counry. he wavefor of he volage v (), and he square of wavefor, v ( ), are shown in figures.a and.b respecively. ie period, / f ( ) / ω ; in angle ( ω ) Half ie period, / /( f ) / ω ; in angle ( ω / ) v( θ ) V sinθ for θ ; v( ) V sin ω for ( ω) V av v( θ ) dθ V sin θ dθ cos θ V. 67 V V v dθ V sin θ dθ V ( V cos θ ) dθ V V V ( θ sin θ ).77 V or, V V f ie, is used as a variable, insead of angleθ, V V ω ω ω ω av v( ) d V sin ω d cos ( ω ) V. 67 V ω ω n he sae way, he rs value, V can be deerined. f he average value of he above wavefor is copued over oal ie period, i coes ou as zero, as he area of firs (posiive) half cycle is he sae as ha of second (negaive) half cycle. However, he rs value reains sae, if i is copued over oal ie period. he differen facors are defined as: RMS value.77 V For facor. Average value.67 V Maxiu value V Peak facor. Average value.77 V Noe: he rs value is always greaer han he average value, excep for a recangular wavefor, in which case he heaing effec reains consan, so ha he average and he rs values are sae. Exaple he exaples of he wo wavefors given are periodic in naure.. riangular curren wavefor (Fig..5) ie period ( ) for i Version EE, Kharagpur

9 av d d i 5. ) ( d d i.5775 wo facors of he wavefor are: For facor Average value value RMS Peak facor..5 Average value Maxiu value o noe ha he for facor is slighly higher han ha for he sinusoidal wavefor, while he peak facor is uch higher. i Fig..5 riangular curren wavefor v (sec) Fig..6 rapezoidal volage wavefor rapezoidal volage wavefor (Fig..6) ie period () 8 s Half ie period ( ) s 8 Version EE, Kharagpur

10 v( ) (5 ) 5 v( ) 5( ) for for ; v( ) 5 for ; Please noe ha ie, is in s, and slope, is in V/s. Also o be noed ha, as in he case of sinusoidal wavefor, only half ie period is aken here for he copuaion of he average and rs values. V 5 av v( ) d 5 d 5 d 5 ( ) d ( ).75 V ) V v d (5) d (5) d 5 ( ) ( ) V wo facors of he wavefor are: d ( ) + 5 ( RMS value.85 For facor. 887 Average value.75 Maxiu value 5. Peak facor. Average value.75 o noe ha he boh he above facors are slighly lower han hose for he sinusoidal wavefor. Siilarly, he average and rs or effecive values of periodic volage or curren wavefors can be copued. n his lesson, saring wih he generaion of single phase ac volage, he ers, such as average and rs values, relaed o periodic volage and curren wavefors are explained wih exaples. n he nex lesson, he background aerial required he represenaion of sinusoidal volage/curren as phasors, he recangular and polar fors of he phasors, as coplex quaniy, and he aheaical operaions addiion/subracion and uliplicaion/division, using phasors as coplex quaniy, are discussed in deail wih nuerical exaples. n he following lessons, he sudy of circuis fed fro single phase ac supply, is presened. Version EE, Kharagpur

11 Probles. Wha is he speed in rp of an ac generaor wih poles, o produce a volage wih a frequency of 5 Hz (a) (b) 5 (c) (d) 75. Deerine he No. of poles required in an ac generaor running a, rp, o produce a volage wih a frequency of 5 Hz. (a) (b) (c) 6 (d) 8. Calculae he speed in rp of an ac generaor wih poles, o produce a volage wih a frequency of 5 Hz. (a) (b) 5 (c) (d) 5. Deerine he average and roo ean square (rs) values of he following wavefors. v V / / 5/ (a) +V v / 5/ 8/ / -V (b) Version EE, Kharagpur

12 Version EE, Kharagpur

13 Lis of Figures Fig.. Scheaic diagra for single phase ac generaion Fig.. (a) Coil posiion for Fig.., and (b) Fig.. Periodic curren (i) wavefor (a) Curren (i), (b) Square of curren ( i ) Fig.. Sinusoidal volage wavefor (a) Volage (v), (b) Square of volage ( v ) Fig..5 riangular curren wavefor Fig..6 rapezoidal volage wavefor Deails Version EE, Kharagpur

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

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