mywbut.com Lesson 13 Representation of Sinusoidal Signal by a Phasor and Solution of Current in R-L-C Series Circuits

Transcription

1 wut.co Lesson 3 Representtion of Sinusoil Signl Phsor n Solution of Current in R-L-C Series Circuits

2 wut.co In the lst lesson, two points were escrie:. How sinusoil voltge wvefor (c) is generte?. How the verge n rs vlues of the perioic voltge or current wvefors, re copute? Soe eples re lso escrie there. In this lesson, the representtion of sinusoil (c) voltge/current signls phsor is first epline. The polr/crtesin (rectngulr) for of phsor, s cople quntit, is escrie. Lstl, the lger, involving the phsors (voltge/current), is presente. ifferent theticl opertions ition/sutrction n ultipliction/ivision, on two or ore phsors, re iscusse. Kewors: Phsor, Sinusoil signls, phsor lger fter going through this lesson, the stuents will e le to nswer the following questions;. Wht is ent the ter, phsor in respect of sinusoil signl?. How to represent the sinusoil voltge or current wvefor phsor? 3. How to write phsor quntit (cople) in polr/crtesin (rectngulr) for? 4. How to perfor the opertions, like ition/sutrction n ultipliction/ivision on two or ore phsors, to otin phsor? This lesson fors the ckgroun of the following lessons in the coplete oule of single c circuits, strting with the net lesson on the solution of the current in the ste stte, in R-L-C series circuits. Sols i or i(t) Instntneous vlue of the current (sinusoil for) I Current (rs vlue) I I φ Miu vlue of the current Phsor representtion of the current Phse ngle, s of the current phsor, with respect to the reference phsor Se sols re use for voltge or n other phsor. Representtion of Sinusoil Signl Phsor sinusoil quntit, i.e. current, i ( t) I sinω t, is tken up s n eple. In Fig. 3., the length, OP, long the -is, represents the iu vlue of the current I, on certin scle. It is eing rotte in the nti-clockwise irection t n ngulr spee, ω, n tkes up position, O fter tie t (or ngle, ω t, with the -is). The verticl proection of O is plotte in the right hn sie of the ove figure with respect to the ngle. It will generte sine wve (Fig. 3.), s O is t n ngle, with the -is, s stte erlier. The verticl proection of O long -is is OC

3 wut.co i ( ) I sin, which is the instntneous vlue of the current t n tie t or ngle. The ngle is in r., i.e. ω t. The ngulr spee, ω is in r/s, i.e. ω π f, where f is the frequenc in Hz or ccles/sec. Thus, i I sin I sinω t I sin πft So, OP represents the phsor with respect to the ove current, i. The line, OP cn e tken s the rs vlue, I I /, inste of iu vlue, I. Then the verticl proection of O, in gnitue equl to OP, oes not represent ectl the instntneous vlue of I, ut represents it with the scle fctor of / The reson for this choice of phsor s given ove, will e given in nother lesson lter in this oule. 3

4 wut.co Generlize cse The current cn e of the for, i ( t) I sin ( ω t α) s shown in Fig. 3.. The phsor representtion of this current is the line, OQ, t n ngle,α ( e tken s negtive), with the line, OP long -is (Fig. 3.c). One hs to ove in clockwise irection to go to OQ fro OP (reference line), though the phsor, OQ is ssue to ove in nti-clockwise irection s given erlier. fter tie t, O will e t n ngle with OQ, which is t n ngle ( α ω t α ), with the line, OP long -is. The verticl proection of O long -is gives the instntneous vlue of the current, i I sin ( ω t α) I sin ( ω t α). Phsor representtion of Voltge n Current The voltge n current wvefors re given s, v V sin, n i I sin ( φ) It cn e seen fro the wvefors (Fig. 3.) of the two sinusoil quntities voltge n current, tht the voltge, V lgs the current I, which ens tht the positive iu vlue of the voltge is reche erlier n ngle, φ, s copre to the positive iu vlue of the current. In phsor nottion s escrie erlier, the voltge n current re represente OP n OQ (Fig. 3.) respectivel, the length of which re proportionl to voltge, V n current, I in ifferent scles s pplicle to ech one. The voltge phsor, OP (V) lgs the current phsor, OQ (I) the ngleφ, s two phsors rotte in the nticlockwise irection s stte erlier, wheres the ngleφ is lso esure in the nticlockwise irection. In other wors, the current phsor (I) les the voltge phsor (V). 4

5 wut.co Mtheticll, the two phsors cn e represente in polr for, with the voltge 0 phsor ( V ) tken s reference, such s V V 0, n I I φ. In Crtesin or rectngulr for, these re, V V 0 0 V 0, n I I φ I cos φ I sin φ, where, the sol, is given. Of the two ters in ech phsor, the first one is tere s rel or its coponent in -is, while the secon one is iginr or its coponent in -is, s shown in Fig The ngle,φ is in egree or r. Phsor lger efore iscussing the theticl opertions, like ition/sutrction n ultipliction/ivision, involving phsors n lso cople quntities, let us tke look t the two fors polr n rectngulr, which phsor or cople quntit is represente. It e oserve here tht phsors re lso tken s cople, s given ove. Representtion of phsor n Trnsfortion phsor or cople quntit in rectngulr for (Fig. 3.3) is, 5

6 wut.co where n re rel n iginr prts, of the phsor respectivel. In polr for, it is epresse s cos sin where n re gnitue n phse ngle of the phsor. Fro the two equtions or epressions, the proceure or rule of trnsfortion fro polr to rectngulr for is cos n sin Fro the ove, the rule for trnsfortion fro rectngulr to polr for is n tn ( / ) The eples using nuericl vlues re given t the en of this lesson. ition/sutrction of Phsors efore escriing the rules of ition/sutrction of phsors or cople quntities, everone shoul recll the rule of ition/sutrction of sclr quntities, which e positive or signe (ecil/frction or frction with integer). It e stte tht, for the two opertions, the quntities ust e either phsors, or cople. The eple of phsor is voltge/current, n tht of cople quntit is ipence/ittnce, which will e epline in the net lesson. ut one phsor n nother cople quntit shoul not e use for ition/sutrction opertion. For the opertions, the two phsors or cople quntities ust e epresse in rectngulr for s ; If the re in polr for s ; In this cse, two phsors re to e trnsfore to rectngulr for the proceure or rule given erlier. The rule of ition/sutrction opertion is tht oth the rel n iginr prts hve to e seprtel trete s C ± ( ± ) ( ± ) c c where c ( ± ) ; c ( ± ) S, for ition, rel prts ust e e, so lso for iginr prts. Se rule follows for sutrction. fter the result is otine in rectngulr for, it cn e trnsfore to polr one. It e oserve tht the si vlues of ' s, ' s n c' s prts of the two phsors n the resultnt one, re ll signe sclr quntities, though in the eple, ' s n ' s re tken s positive, resulting in positive vlues of c' s. lso the phse ngle ' s lie in n of the four qurnts, though here the ngles re in the first qurnt onl. This rule for ition cn e etene to three or ore quntities, s will e illustrte through eple, which is given t the en of this lesson. 6

7 wut.co The ition/sutrction opertions cn lso e perfore using the quntities s phsors in polr for (Fig. 3.4). The two phsors re (O) n (O). The fin the su C (OC), line C is rwn equl n prllel to O. The line C is equl n prllel to O. Thus, C OC O C O O. lso, OC O C O O To otin the ifference (O), line is rwn equl n prllel to O, ut in opposite irection to C or O. line OE is lso rwn equl to O, ut in opposite irection to O. oth n OE represent the phsor ( ). The line, E is equl to O. Thus, O O O O. lso O OE E O O. The eples using nuericl vlues re given t the en of this lesson. Multipliction/ivision of Phsors Firstl, the proceure for ultipliction is tken up. In this cse no reference is eing e to the rule involving sclr quntities, s everone is filir with the. ssuing tht the two phsors re ville in polr fro s n. Otherwise, the re to e trnsfore fro rectngulr to polr for. This is lso vli for the proceure of ivision. Plese note tht phsor is to e ultiplie cople quntit onl, to otin the resultnt phsor. phsor is not norll ultiplie nother phsor, ecept in specil cse. Se is for ivision. phsor is to e ivie cople quntit onl, to otin the resultnt phsor. phsor is not norll ivie nother phsor. To fin the gnitue of the prouct C, the two gnitues of the phsors re to e ultiplie, wheres for phse ngle, the phse ngles re to e. Thus, 7

8 ( ) c C C ) ( where n C c Plese note tht the se sol, is use for the prouct in this cse. C To ivie. to otin the result., the gnitue is otine ivision of the gnitues, n the phse is ifference of the two phse ngles. Thus, ( ) where n / If the phsors re epresse in rectngulr for s n where ( ) ( ) / tn ; The vlues of re not given s the cn e otine sustituting for. s ' s ' To fin the prouct, ( ) ( ) ( ) ( ) c C C Plese note tht.the gnitue n phse ngle of the result (phsor) re, ( ) ( ) [ ] ( ) ( ) C, n c tn The phse ngle, ( ) ( ) ( ) ( ) c tn / / / / tn tn tn The ove results re otine siplifiction. To ivie to otin s To siplif, i.e. to otin rel n iginr prts, oth nuertor n enointor, re to e ultiplie the cople conugte of, so s to convert the enointor into rel vlue onl. The cople conugte of is wut.co 8

9 * In the cople conugte, the sign of the iginr prt is negtive, n lso the phse ngle is negtive. ( ) ( ) ( ) ( ) The gnitue n phse ngle of the result (phsor) re, ( ) ( ) [ ] ( ) ( ) ( ), n tn The phse ngle, tn tn tn The steps re shown here in rief, s etile steps hve een given erlier. Eple The phsor, in the rectngulr for (Fig. 3.5) is, 4 sin cos where the rel n iginr prts re 4 ; To trnsfor the phsor, into the polr for, the gnitue n phse ngle re wut.co 9

10 wut.co ( ) tn tn r Plese note tht is in the secon qurnt, s rel prt is negtive n iginr prt is positive. Trnsforing the phsor, into rectngulr for, the rel n iginr prts re cos 4.47 cos sin 4.47 sin Phsor lger nother phsor, in rectngulr for is introuce in ition to the erlier one, Firstl, let us tke the ition n sutrction of the ove two phsors. The su n ifference re given the phsors, C n respectivel (Fig. 3.6). C ( 4) (6 6) ( 6) (4 6) ( 4) (6 6) ( 6) (4 6) It e note tht for the ition n sutrction opertions involving phsors, the shoul e represente in rectngulr for s given ove. If n one of the phsors 0

11 wut.co is in polr for, it shoul e trnsfore into rectngulr for, for clculting the results s shown. If the two phsors re oth in polr for, the phsor igr (the igr ust e rwn to scle), or the geoetricl etho cn e use s shown in Fig 3.6. The result otine using the igr, s shown re the se s otine erlier. [ C (OC) 0.77, COX 68. ; n ( O) 8.46, OX ] Now, the ultipliction n ivision opertions re perfore, using the ove two phsors represente in polr for. If n one of the phsors is in rectngulr for, it e trnsfore into polr for. lso note tht the se sols for the phsors re use here, s ws use erlier. Lter, the etho of oth ultipliction n ivision using rectngulr for of the phsor representtion will e epline. The resultnt phsor C, i.e. the prouct of the two phsors is C ( ) The prouct of the two phsors in rectngulr for cn e foun s C ( 4) (6 6) ( 4) (4 ) 36 The result ( ) otine the ivision of is ( ) ( ) The ove result cn e clculte the proceure escrie erlier, using the rectngulr for of the two phsors s 4 ( 4) (6 6) ( 4) (4 ) 6 6 (6 6) (6 6) The proceure for the eleentr opertions using two phsors onl, in oth fors of representtion is shown. It cn e esil etene, for s, ition/ultipliction, using three or ore phsors. The siplifiction proceure with the sclr quntities, using the ifferent eleentr opertions, which is well known, cn e etene to the phsor quntities. This will e use in the stu of c circuits to e iscusse in the following lessons. The ckgroun require, i.e. phsor representtion of sinusoil quntities (voltge/current), n lger theticl opertions, such s ition/sutrction n ultipliction/ivision of phsors or cople quntities, incluing trnsfortion of phsor fro rectngulr to polr for, n vice vers, hs een iscusse here. The stu of c circuits, strting fro series ones, will e escrie in the net few lessons.

12 wut.co Proles 3. Use plsor technique to evlute the epression n then fin the nuericl vlue t t 0 s. i t 50 cos 00t sin 00t cos 00t -30 t 0 0 () ( ) ( ) ( ) 3. Fin the result in oth rectngulr n polr fors, for the following, using cople quntities: ) ) ( ) c) )