AC STEADY-STATE ANALYSIS

Transcription

1 EANNG GOAS AC STEAD-STATE ANASS SNUSODS viw basic facs abou sinusoidal signals SNUSODA AND COPEX FOCNG FUNCTONS Bhavior of circuis wih sinusoidal indpndn sourcs and modling of sinusoids in rms of complx xponnials PHASOS prsnaion of complx xponnials as vcors. facilias sady-sa analysis of circuis. PEDANCE AND ADTANCE Gnralizaion of h familiar concps of rsisanc and conducanc o dscrib AC sady sa circui opraion PHASO DAGAS prsnaion of AC volags and currns as complx vcors BASC AC ANASS USNG KCHHOFF AWS ANASS TECHNQUES Exnsion of nod, loop, Thvnin and ohr chniqus SNUSODS x X sin Adimnsional plo As funcion of im X ampliud or maximum valu angular frquncy rads/sc argumn radians " lads by " T Priod x x T, f frquncy in Hrz cycl/sc T f " lags by "

2 BASC TGONOET ESSENTA DENTTES sin sin cos cos sin cos cos cos sin sin sin sin cos cos SOE DEED DENTTES sin sin cos cos sin cos cos cos sin sin sin cos sin sin cos cos cos cos APPCATONS cos sin sin cos cos cos sin sin ADANS AND DEGEES radians π 6 dgrs 8 rads dgrs 8 ACCEPTED EE CONENTON sin sin 9 EANNG EXAPE cos cos 5 ads by 5 dgrs cos 5 6 ags by 5 o cos 5 cos 5 8 ads by 5 o or lags by 5 o

3 EANNG EXAPE v sin 6, v 6cos FND FEQUENC AND PHASE ANGE BETWEEN OTAGES Frquncy in radians pr scond is h facor of h im variabl f Hz 59. Hz sc To find phas angl w mus xprss boh sinusoids using h sam rigonomric funcion; ihr sin or cosin wih posiiv ampliud ak car of minus sign wih cos cos 8 6cos 6cos 8 Chang sin ino cosin wih cos sin 9 6cos 6sin 9 W lik o hav h phas shifs lss han 8 in absolu valu 6sin 6sin 6 v sin 6 v 6sin v lads v by 6 6 v lags v by 5 EANNG EXTENSON i sin77 5 i i i.5cos77 i.5sin77 6 lads lads i i by? by? cos sin 9.5cos77.5sin sin sin 8 i lads i by 55.5sin77 6.5sin lads i by i 65 6

4 SNUSODA AND COPEX FOCNG FUNCTONS arning Exampl f h indpndn sourcs ar sinusoids of h sam frquncy, hn for any variabl in h linar circui h sady sa rspons will b sinusoidal and of h sam frquncy. v Asin i Bsin To drmin h sady sa soluion w only nd o drmin h paramrs B, SS K : di i v d n sady sa i Acos, or i A cos A sin di A sin A cos d A A sin A A cos A A A * / * / A cos algbraic problm A, A Drmining h sady sa soluion can b accomplishd wih only algbraic ools! 7 FUTHE ANASS OF THE SOUTON Th soluion is i A cos A sin Th applid volag is v cos For comparison purposs on can wri i Acos A A Acos, A Asin A A A, an A f A, A A, an i cos an For h currn AWASlags pur inducor, h currn lags h volag. h volag by 9. 8

5 SONG A SPE ONE OOP CCUT CAN BE E ABOOUS F ONE USES SNUSODA EXCTATONS. TO AKE ANASS SPE ONE EATES SNUSODA SGNAS TO COPEX NUBES. THE ANASS OF STEAD STATE W BE CONETED TO SONG SSTES OF AGEBAC EQUATONS... WTH COPEX AABES. y sin sin cos cos A y v A y v Eulr idniy DENTT : ESSENTA cos sin A A and add * / f vrybody knows h frquncy of h sinusoid hn on can skip h rm xpw. A 9 arning Exampl v i Assum v i d di K : d di i d di * / an an an, cos } { } { cos i v sin, cos an, r y r x y x y x r r y x P C - - x y

6 Convring bwn polar and Carsian coordinas Th polar coordinas r and ϕ can b convrd o h Carsian coordinas x and y by using h rigonomric funcions sin and cosin: Th Carsian coordinas x and y can b convrd o polar coordinas r and ϕ wih r and ϕ in h inrval π, π] by: whr aan is a common variaion on h arcangn funcion dfind as PHASOS ESSENTA CONDTON A NDEPENDENT SOUCES AE SNUSODS OF THE SAE FEQUENC BECAUSE OF SOUCE SUPEPOSTON ONE CAN CONSDE A SNGE SOUCE u U cos THE STEAD STATE ESPONSE OF AN CCUT AABE W BE OF THE FO SHOTCUT NEW DEA: SHOTCUT u U { U U NSTEAD OF WTNG u U N NOTATON y cos y } { } U u U y WE WTE... AND WE ACCEPT ANGES N DEGEES U S THE PHASO EPESENTATON FO U u U cos u U cos U U y { cos } SHOTCUT : DEEOP EFFCENT TOOS TO DETENE THE PHASO OF THE ESPONSE GEN THE NPUT PHASOS.

7 arning Exampl arning Exnsions v i di i v d n rms of phasors on has Th phasor can b obaind using only complx algbra. W will dvlop a phasor rprsnaion for h circui ha will limina h nd of wriing h diffrnial quaion. is ssnial o b abl o mov from sinusoids o phasor rprsnaion Acos A Asin A 9 v cos77 5 y 8sin5. Givn f Hz v cos8 6 v cos8 6 Phasors can b calculad using h ruls of complx algbra. A A A A A A A A PHASO EATONSHPS FO CCUT EEENTS ESSTOS v i Phasor rprsnaion for a rsisor Phasors ar complx numbrs. Th rsisor modl has a gomric inrpraion. Th volag and currn phasors ar colinar. n rms of h sinusoidal signals, his gomric rprsnaion implis ha h wo sinusoids ar in phas.

8 NDUCTOS d d laionship bwn sinusoids Th rlaionship bwn phasors is algbraic. For h gomric viw, us h rsul arning Exampl mh, v cos77. Find i 77 A 9 77 i.59 cos A Th volag lads h currn by 9 dg. Th currn lags h volag by 9 dg. 5 d CAPACTOS C d laionship bwn sinusoids C θ C9 C arning Exampl Th rlaionship bwn phasors is algbraic. n a capacior, h currn lads h volag by 9 dg. C F, v cos 5. Find i. 5 C C 9 5 Th volag lags h currn by 9 dg. 6 5 A i. cos 5 A 6

9 EANNG EXTENSONS.5H, A, f 6Hz Find h volag across h inducor f = ω v 75. cos77 6 Now an xampl wih capaciors C 5 F,.6 5, f 6Hz Find h volag across h inducor f C C v 6.66 cos 5 7 PEDANCE AND ADTTANCE For ach of h passiv componns h rlaionship bwn h volag phasor and h currn phasor is algbraic. W now gnraliz for an arbirary -rminal lmn. X sisiv componn X aciv componn NPUT PEDANCE v v i i DNG PONT PEDANCE Th unis of impdanc ar OHS. mpdanc is NOT a phasor bu a complx numbr ha can b wrin in polar or Carsian form. n gnral is valu dpnds on h frquncy. z X X z an Elmn Phasor Eq. C C mpdanc C 8

10 K AND KC HOD FO PHASO EPESENTATONS K : K: v v i i v v v v KC: i i, i,, Phasors! i k i i i i k i i k i, k,,, n a similar way, on shows... Th componns will b rprsnd by hir impdancs and h rlaionships will b nirly algbraic!! 9 SPECA APPCATON: PEDANCES CAN BE COBNED USNG THE SAE UES DEEOPED FO ESSTOS s k k EANNG EXAPE C C s k p k f 6Hz, v 5cos p Compu quivaln impdanc and currn, 5, 5 s, C 5 7.5, 5. 5 C A A s.969. A i.96cos 9. A C 6

11 EANNG ASSESSENT FND i q q C C q A COPEX ADTTANCE G B Simns G conducanc B Sucpanc X X X X X G X X B X Elmn Phasor Eq. mpdanc C C C Admianc G Paralll Combinaion of Admiancs p k k. S C S Sris Combinaion of Admiancs s k C k s s s p.s. S. S s.5.5s

12 EANNG EXAPE p S, 5 6 FND p S p A p A A S p.69.9 S p A p.5.5 p EANNG EXTENSON EANNG EXAPE SEES-PAAE EDUCTONS / S S q.....

13 EANNG EXTENSON FND THE PEDANCE T 6 P P P P.885 P P T PHASO DAGAS Display all rlvan phasors on a common rfrnc fram. ry usful o visualiz phas rlaionships among variabls. Espcially if som variabl, lik h frquncy, can chang. EANNG EXAPE SKETCH THE PHASO DAGA FO THE CCUT Any on variabl can b chosn as rfrnc. For his cas slc h volag. KC : S C C C capaciiv C C ω l NDUCTE CASE CAPACTE CASE induciv 6

14 EANNG EXAPE DAW THE PHASO DAGA FO THE CCUT 77 s. PUT KNOWN NUECA AUES C C S C. DAW A THE PHASOS C is convnin o slc h currn as rfrnc. C DAGA WTH EFEENCE S 9 85 ad valus from diagram! 5 A 5 Pyhagoras C EANNG B DONG C v FND THE FEQUENC AT WHCH v AND i AE N PHASE i.., h phasors for i, v ar co - linalr C PHASO DAGA Noic ha was chosn as rfrnc C and C ar co - linal iff C C 9 6 f.6 rad / s 5. Hz 8

15 EANNG EXTENSON Draw a phasor diagram illusraing all volags and currns A Simplr han Currn dividr DAW PHASOS. A AE KNOWN. NO NEED TO SEECT A EFEENCE. 9 BASC ANASS USNG KCHHOFF S AWS POBE SONG STATEG For rlaivly simpl circuis us Ohm's law for AC analysis; i.., Thruls for combining and KC andk Currn and volag dividr For mor complx circuis us Nod analysis oop analysis Suprposiion Sourc ransformaion Thvnin's andnoron's horms

16 EANNG EXAPE COPUTE A THE OTAGES AND CUENTS q q q S A q A A Compu Ohm's law for, Us currn dividr for, 7.85 EANNG EXTENSON F 85, COPUTE O S O A 5 A A A A THE PAN... COPUTE COPUTE COPUTE COPUTE, S S S S

17 ANASS TECHNQUES PUPOSE: TO EEW A CCUT ANASS TOOS DEEOPED FO ESSTE CCUTS;.E., NODE AND OOP ANASS, SOUCE SUPEPOSTON, SOUCE TANSFOATON, THEENN S AND NOTON S THEOES. COPUTE. NODE ANASS 6 A A NEXT: OOP ANASS OOP ANASS ONE COUD ASO USE THE SUPEESH TECHNQUE SOUCE S NOT SHAED AND o S DEFNED B ONE OOP CUENT OOP : OOP : 6 OOP : CONSTANT : UST FND SUPEESH: 6 6 ESH : /* /* A NEXT: SOUCE SUPEPOSTON

18 SOUCE SUPEPOSTON Circui wih volag sourc s o zro SHOT CCUTED = + Circui wih currn sourc s o zroopen Du o h linariy of h modls w mus hav Principl of Sourc Suprposiion Th approach will b usful if solving h wo circuis is simplr, or mor convnin, han solving a circui wih wo sourcs. W can hav any combinaion of sourcs. And w can pariion any way w find convnin. 5. SOUCE SUPEPOSTON ' " " A COUD USE SOUCE TANSFOATON TO COPUTE " " 6 " 6 A " " ' " " " A 6 " " 6 6 A 5 ' " A 6 NEXT: SOUCE TANSFOATON 6

19 Sourc ransformaion is a good ool o rduc complxiy in a circui... WHEN T CAN BE APPED!! idal sourcs ar no good modls for ral bhavior of sourcs. A ral bary dos no produc infini currn whn shor-circuid + - S a b S a b THE ODES S S AE EQUAENTS S S WHEN mprovd modl for volag sourc mprovd modl for currn sourc Sourc Transformaion can b usd o drmin h Thvnin or Noron Equivaln... BUT THEE A BE OE EFFCENT TECHNQUES 7. SOUCE TANSFOATON S 8 ' 8 Now a volag o currn ransformaion NEXT: THEENN S 5 8

20 THEENN S EQUAENCE THEOE NEA CCUT ay conain indpndn and dpndn sourcs wih hir conrolling variabls PAT A i v O _ a b NEA CCUT ay conain indpndn and dpndn sourcs wih hir conrolling variabls PAT B TH TH v TH i v O _ a b NEA CCUT PAT B PAT A Phasor Thvnin Equivaln Circui for PAT A v TH TH Thvnin Equivaln Sourc Thvnin Equivaln sisanc mpdanc 9 5. THEENN ANASS olag Dividr 8 OC 6 TH 8 5 A NEXT: NOTON

21 NOTON S EQUAENCE THEOE NEA CCUT ay conain indpndn and dpndn sourcs wih hir conrolling variabls PAT A i v O _ a b NEA CCUT ay conain indpndn and dpndn sourcs wih hir conrolling variabls PAT B Phasors i N N N i v O _ a b NEA CCUT PAT B PAT A i N N N Noron Equivaln Circui for PAT A Thvnin Noron Equivaln Sourc Thvnin Noron Equivaln sisanc mpdanc 6. NOTON ANASS TH SC 5 Possibl chniqus: loops, sourc ransformaion, suprposiion B SUPEPOSTON 6 8 SC A

22 EANNG EXAPE FND USNG NODES, OOPS, THEENN, NOTON WH SKP SUPEPOSTON AND TANSFOATON? NODES Noic choic of ground Suprnod consrain : Suprnod KC@ x KC@ Conrolling variabl 6 x 6 6 Adding: 8 OOP ANASS ESH CUENTS AE ACCEPTABE ESH CUENTS DETENED B SOUCES ESH : x ESH : CONTONG AABE : x AABE OF NTEEST : 8 8 8

23 THEENN Alrnaiv procdur o compu Thvnin impdanc:. S o zro all NDEPENDENT sourcs. Apply an xrnal prob TH s " x FO OPEN CCUT OTAGE " x K s " x " x TH ' x TH x 8 OC NOTON Suprnod consrain KC@ Suprnod ''' KC@ : X ''' Conrolling ariabl : x / / SC ''' x A SC ''' x SC SC Now w can draw h Noron Equivaln circui... USE NODES 6

24 NOTON S EQUAENT CCUT TH SC 8 Currn Dividr EQUAENCE OF SOUTONS Using Noron s mhod Using Thvnin s Using Nod and oop mhods EANNG EXTENSON COPUTE USE NODA ANASS / USE THEENN TH 6 6 OC OC 6 OC + - TH TH OC 8

25 EANNG EXTENSON COPUTE USNG ESH ANASS USNG NODES 9 CONSTANT 9 SUPEESH USNG SOUCE SUPEPOSTON 9 9 EANNG EXTENSON COPUTE " ' ". USNG SUPEPOSTON " ' ' 5

26 . USE SOUCE TANSFOATON 6 9 q q q USE NOTON S THEOE SC TH TH 6 9 TH TH SC SC 5

27 EANNG EXAPE Find h currn i in sady sa Th sourcs hav diffrn frquncis! For phasor analysis UST us sourc suprposiion. Frquncy domain SOUCE : FEQUENC r/s Principl of suprposiion 5