AC STEADY-STATE ANALYSIS

Transcription

1 EANNG GOAS AC STEADY-STATE ANAYSS 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 ANAYSS USNG KCHHOFF AWS ANAYSS TECHNQUES Exnsion of nod, loop, Thvnin and ohr chniqus

2 SNUSODS x X sin Adimnsional plo As funcion of im X ampliud or maximum valu angular frquncy rads/sc argumn radians " lads by θ" π T Priod x x T, f frquncy in Hrz cycl/sc T π π f " lags by θ"

3 BASC TGONOETY cos cos cos cos sin sin cos sin sin sin cos cos cos sin cos cos sin sin β α β α β α β α β α β α β α β α β α β α β α β α DENTTES DEED SOE α α α α β α β α β α β α β α β α cos cos sin sin sin sin cos cos cos sin cos cos sin sin DENTTES ESSENTA sin sin cos cos cos sin sin cos π π π π ± ± APPCATONS 9 sin sin π CONENTON EE ACCEPTED ADANS AND DEGEES dgrs 8 rads 36 dgrs radians θ π θ π

4 EANNG EXAPE cos cos 45 cos avanço 45 graus Ou araso d 35 cos 45 cos 45 ± 8 Avanço d 5 ou araso d 35

5 SNUSODA AND COPEX FOCNG FUNCTONS arning Exampl K : di i v d 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 θ i Bsin φ To drmin B,φ SS w only nd o drmin h sady sa soluion h paramrs n sady sa i Acos φ, or i di d A cos A A sin A cos A sin A cos sin A A A A A A algbraic problm, A * / * / A cos Drmining h sady sa soluion can b accomplishd wih only algbraic ools!

6 FUTHE ANAYSS OF THE SOUTON Th soluion is i A cos A sin Th applid volag is v cos For comparison purposs on can wri A A i Acos φ Acosφ, A sinφ A A A A, anφ A A, A For A, φ an i cos an h currn AWAYS lags h volag f pur inducor h currn lags h volag by 9

7 SONG A SPE ONE OOP CCUT CAN BE EY ABOOUS F ONE USES SNUSODA EXCTATONS TO AKE ANAYSS SPE ONE EATES SNUSODA SGNAS TO COPEX NUBES. THE ANAYSS OF STEADY STATE W BE CONETED TO SONG SYSTES OF AGEBAC EQUATONS... WTH COPEX AABES ESSENTA DENTTY : θ cosθ sinθ Eulr idniy y v v cos sin A y Acos φ y Asin φ φ A θ * / and add f vrybody knows h frquncy of h sinusoid hn on can skip h rm xpw A θ

8 arning Exampl v φ i Assum v i d di K : φ d di i d di φ φ φ φ φ φ * / φ an φ an φ an, cos } { } { cos φ φ i v θ θ θ θ sin, cos an, r y r x y x y x r r y x P C

9 PHASOS ESSENTA CONDTON A NDEPENDENT SOUCES AE SNUSODS OF THE SAE FEQUENCY BECAUSE OF SOUCE SUPEPOSTON ONE CAN CONSDE A SNGE SOUCE u U cos θ THE STEADY STATE ESPONSE OF ANY CCUT AABE W BE OF THE FO y Y cos φ SHOTCUT u U θ y Y φ { U θ } { Y φ } θ θ θ φ U u U y Y NEW DEA: U SHOTCUT N NOTATON U NSTEAD OF WTNG u U WE WTE u U θ... AND WE ACCEPT ANGES N DEGEES θ S THE PHASO EPESENTATON FO U cos θ u U cos θ U U θ Y Y φ y { Y cos φ} SHOTCUT : DEEOP EFFCENT TOOS TO DETENE THE PHASO OF THE ESPONSE GEN THE NPUT PHASOS θ

10 arning Exampl v di i v d n rms of phasors on has Th phasor can b obaind using only complx algbra i φ W will dvlop a phasor rprsnaion for h circui ha will limina h nd of wriing h diffrnial quaion arning Exnsions is ssnial o b abl o mov from sinusoids o phasor rprsnaion Acos ± θ A ± θ Asin ± θ A ± θ 9 v cos y 8sin Givn f 4Hz v cos8π 6 v cos8π 6 Phasors can b combind using h ruls of complx algbra θ θ θ θ θ θ θ θ

11 PHASO EATONSHPS FO CCUT EEENTS ESSTOS v i θ θ θ θ Phasor rprsnaion for a rsisor Phasors ar complx numbrs. Th rsisor modl has a gomric inrpraion Th volag and currn phasors ar colinal n rms of h sinusoidal signals his gomric rprsnaion implis ha h wo sinusoids ar in phas

12 θ d φ NDUCTOS d φ laionship bwn sinusoids θ φ Th rlaionship bwn phasors is algbraic For h gomric viw us h rsul Th volag lads h currn by 9 dg Th currn lags h volag by 9 dg arning Exampl mh, v cos377. Find i 377 A A i cos

13 φ d θ CAPACTOS C d laionship bwn sinusoids φ C θ C 9 C Th rlaionship bwn phasors is algbraic n a capacior h currn lads h volag by 9 dg Th volag lags h currn by 9 dg C arning Exampl µ F, v cos34 5. Find i 34 5 C 9 5 C A i 3.4cos34 5 A

14 EANNG EXTENSONS.5H, 4 3 A, f 6Hz Find h volag across h inducor π f π π π 6 v 4π cosπ 6 Now an xampl wih capaciors C 5 µ F, , f 6Hz Find h volag across h inducor π f π C C π π v cosπ 35 π

15 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 z i v i v θ θ θ θ θ NPUT PEDANCE DNG PONT PEDANCE Th unis of impdanc ar OHS C C C mpdanc Phasor Eq. Elmn 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 componn aciv componn sisiv X X X X z an θ

16 K AND KC HOD FO PHASO EPESENTATONS v v 3 v i i i i 3 3 v v v K:,,,3, 3 k i i i i i k k k φ KC:,,3, i v i i i θ 3 3 θ θ θ K : 3 3 θ θ θ Phasors! Th componns will b rprsnd by hir impdancs and h rlaionships will b nirly algbraic!! n a similar way, on shows...

17 SPECA APPCATON: PEDANCES CAN BE COBNED USNG THE SAE UES DEEOPED FO ESSTOS s k k EANNG EXAPE C f s Compu quivaln 6 Hz, v 5cos 3 C impdanc π, and currn 5 3, k p k 5Ω p 3 π Ω, C π Ω, 53. 5Ω s C Ω A A s A i.96cosπ 9. A C 6

18 EANNG EXTENSON FND i 377 Ω C Ω 53.5 q C q q A

19 COPEX ADTTANCE Y G B Simns G conducanc B X Elmn C Sucpanc G X X B X X X Phasor Eq. C X X mpdanc C Admianc Y G Y Y Paralll Combinaion of Y p Y k k Sris Y s Y C k Combinaion of k Admiancs Y. S Y C S Y Y Y s s s.s. S Y p. S.5S Admiancs Y s

20 EANNG EXAPE S 6 45 FND Y p, EANNG EXTENSON Y p Y Y p Y.5.5 S 4 p 8 Yp A Y p S Y p S Y p A A A

21 EANNG EXAPE SEES-PAAE EDUCTONS / Y Y S Y Y S Y Ω Ω q Y Y Y

22 EANNG EXTENSON FND THE PEDANCE T P Y P Y.. Y Y Y Y Y Y Y Y Y Y P Y P Y P P Y.5.5 T

23 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 capaciiv > C < C C C l NDUCTE CASE CAPACTE CASE induciv

24 EANNG EXAPE DO THE PHASO DAGA FO THE CCUT 377 s. PUT KNOWN NUECA AUES C C S C. DAW A THE PHASOS C is convnin o slc h currn as rfrnc DAGA WTH EFEENCE S ad valus from diagram! 3 45 A 45 Pyhagoras > C C 6 45

25 EANNG BY DONG C v FND THE FEQUENCY AT WHCH v AND i AE N PHASE i.., h phasors for i, v ar co - linal C PHASO DAGA C Noic ha was chosn as rfrnc C and ar co - linal iff C C f π Hz rad / s

26 EANNG EXTENSON Draw a phasor diagram illusraing all volags and currns A Currn dividr Simplr han DAW PHASOS. A AE KNOWN. NO NEED TO SEECT A EFEENCE

27 BASC ANAYSS USNG KCHHOFF S AWS POBE SONG STATEGY For rlaivly simpl circuis us Ohm's law for AC analysis; i.., Th ruls for combining and Y KC AND K Currn and volag dividr For mor complx circuis us Nod analysis oop analysis Suprposiion Sourc ransformaion Thvnin' s and Noron's ATAB PSPCE horms

28 EANNG EXAPE COPUTE A THE OTAGES AND CUENTS Compu Us currn dividr for, Ohm's law for, 3 q q 8 8 q Ω S A q A A

29 EANNG EXTENSON F 8 45, COPUTE O S O 3 A 4 45 A A A A S THE PAN... COPUTE COPUTE COPUTE COPUTE 3, S S S

30 ANAYSS TECHNQUES PUPOSE: TO EEW A CCUT ANAYSS TOOS DEEOPED FO ESSTE CCUTS;.E., NODE AND OOP ANAYSS, SOUCE SUPEPOSTON, SOUCE TANSFOATON, THEENN S AND NOTON S THEOES. COPUTE. NODE ANAYSS 6 A NEXT: OOP ANAYSS A

31 . OOP ANAYSS ONE COUD ASO USE THE SUPEESH TECHNQUE SOUCE S NOT SHAED AND o S DEFNED BY ONE OOP CUENT OOP : 3 OOP : OOP 3: UST FND /* A 4 CONSTANT : SUPEESH: ESH 3: /* NEXT: SOUCE SUPEPOSTON 3

32 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

33 3. SOUCE SUPEPOSTON A ' COUD USE SOUCE TANSFOATON TO COPUTE " ' " " " 6 " 6 A " " " " " " 3 6 A " 6 6 A 4 4 ' " 5 3 A NEXT: SOUCE TANSFOATON 6

34 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 WHEN mprovd modl for volag sourc mprovd modl for currn sourc Sourc Transformaioncan b usd o drmin h Thvnin or Noron Equivaln... BUT THEE AY BE OE EFFCENT TECHNQUES

35 4. SOUCE TANSFOATON 8 S Ω ' 8 Now a volag o currn ransformaion NEXT: THEENN S

36 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 v TH TH Thvnin Equivaln Circui for PAT A Thvnin Equivaln Sourc Thvnin Equivaln sisanc mpdanc

37 5. THEENN ANAYSS olag Dividr 8 OC 6 TH Ω A NEXT: NOTON

38 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 Noron Equivaln Circui i N N N for PAT A Thvnin Equivaln Sourc Thvnin Equivaln sisanc mpdanc

39 6. NOTON ANAYSS TH Ω SC Possibl chniqus: loops, sourc ransformaion, suprposiion BY SUPEPOSTON 6 8 SC A

40 EANNG EXAPE FND USNG NODES, OOPS, THEENN, NOTON WHY SKP SUPEPOSTON AND TANSFOATON? Suprnod consrain : 3 Suprnod NODES Noic choic of ground KC@ 3 x KC@ Conrolling variabl x Adding: 4

41 OOP ANAYSS ESH CUENTS AE ACCEPTABE ESH CUENTS DETENED BY SOUCES ESH : 3 x 3 ESH 4 : CONTONG AABE : AABE OF NTEEST : x

42 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 Ω TH x 8 ' x OC TH 4 8

43 NOTON Suprnod consrain 3 KC@ Suprnod ''' 3 KC@ : X ''' 3 Conrolling ariabl : x / / SC ''' x 3 ''' x SC A SC SC Now w can draw h Noron Equivaln circui USE NODES

44 NOTON S EQUAENT CCUT TH SC 8 4 Currn Dividr EQUAENCE OF SOUTONS Using Noron s mhod Using Thvnin s 4 8 Using Nod and oop mhods 8 4

45 EANNG EXTENSON COPUTE USE NODA ANAYSS 3 / USE THEENN TH OC 3 4 OC 6 3 OC - TH TH Ω OC

46 EANNG EXTENSON COPUTE USNG ESH ANAYSS USNG NODES 4 9 CONSTANT 9 SUPEESH USNG SOUCE SUPEPOSTON 4 9 4

47 EANNG EXTENSON COPUTE " ' ". USNG SUPEPOSTON 4 " ' '

48 . USE SOUCE TANSFOATON Ω Ω Ω Ω 6 9 q q q 6 9 6

49 USE NOTON S THEOE SC TH TH TH TH SC 6 9 SC

50 USNG ATAB ATAB rcognizs complx numbrs in rcangular rprsnaion. dos NOT rcogniz Phasors Unlss prviously r-dfind, ATAB rcognizs i or as imaginary unis» z34 z 3. 4.i» z46i z 4. 6.i n is oupu ATAB always uss i for h imaginary uni Phasors cangula r z 45» a45; % angl in dgrs» ara*pi/8, %convr dgrs o radians ar.7854» m; %dfin magniud» xm*cosar; %ral par x 7.7» ym*sinar; %imaginary par y 7.7» zxi*y z i z i;» mpabsz; %compu magniud mp» arranglz; %compu angl in ADANS arr.7854» adgarr*8/pi; %convr o dgrs adg 45 xralz x 7.7 yimagz y 7.77

51 EANNG EXAPE COPUTE A NODE OTAGES Y Y

52 %xampl7p7 %dfin h HS vcor. irzros5,; %iniializ and dfin non zro valus ir*cos3*pi/8**sin3*pi/8; ir5*cospi/4**sinpi/4, %cho h vcor %now dfin h marix y[,,,,; %firs row -,.5,-,,.5; %scond row,-,.5,,-.5; %hird row -.5,,,.5,-; %fourh row,.5i,-.5,-,.5.5i] %las row and do cho vy\ir %solv quaions and cho h answr Echo of Answr v i i i i i ir i.44.44i y Columns hrough 4 Echo of HS i -.i -.i.5.i i.5i Column 5.5i i Echo of arix

53 AC PSPCE ANAYSS Circui rady o b simulad Slc and plac componns Ground s, mrs spcifid AC C Wir and s corrc aribus

54 **** AC ANAYSS TEPEATUE 7. DEG C ************************************************************************ ****** suls in oupu fil FEQ $N_3 P$N_3 6.E.65E E **** 5// 9:3:4 *********** Evaluaion PSpic Nov 999 ************** * C:\ECEWork\rwinPPT\ACSadySaAnalysis\Sc7p9Dmo.sch **** AC ANAYSS TEPEATUE 7. DEG C ************************************************************************ ****** FEQ _PNTP_PNT 6.E.998E E