AC STEADY-STATE ANALYSIS

Transcription

1 AC STEADY-STATE ANAYSS 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 SNUSODA AND COPEX FOCNG FUNCTONS 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 i Bsin φ To drmin h sady sa soluion w only nd o drmin h paramrs B,φ SS n sady sa i Acos φ, or i A cos A sin * / di d A sin A cos * / A A sin A A cos cos A A algbraic problm A A A, A Drmining h sady sa soluion can b accomplishd wih only algbraic ools!

3 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

4 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 : cos sin Eulr idniy v v cos sin y Acos φ y Asin φ * / and add y A φ A f vrybody knows h frquncy of h sinusoid hn on can skip h rm xpw A

5 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 sin, cos an, r y r x x y y x r r y x P C

6 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 THE STEADY STATE ESPONSE OF ANY CCUT AABE W BE OF THE FO y Y cos φ SHOTCUT u U { U y Y } { Y NEW DEA: U U SHOTCUT N NOTATON U φ φ φ u U y Y NSTEAD OF WTNG u U WE WTE u U... AND WE ACCEPT ANGES N DEGEES S THE PHASO EPESENTATON FO U cos u U cos U U Y Y φ y Y cos φ SHOTCUT : DEEOP EFFCENT TOOS TO DETENE THE PHASO OF THE ESPONSE GEN THE NPUT PHASOS }

7 Exampl v di i v d n rms of phasors on has i Th phasor can b obaind using only complx algbra φ 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 cos y 8sin Givn f 4Hz v cos8π 6 v cos8π 6 Phasors can b combind using h ruls of complx algbra

8 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 colinal n rms of h sinusoidal signals his gomric rprsnaion implis ha h wo sinusoids ar in phas

9 d φ NDUCTOS d φ laionship bwn sinusoids 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 Exampl mh, v cos377. Find i 377 A A i cos

10 φ d CAPACTOS C d laionship bwn sinusoids φ C C 9 C C μ F, v cos34 5. Find i 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 34 5 C 9 5 C A i 3.4cos34 5 A

11 .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 5μ F, , f 6Hz Find h volag across h capacior π f π C C π π v cosπ 35 π

12 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 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

13 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 3 3 K : 3 3 Phasors! Th componns will b rprsnd by hir impdancs and h rlaionships will b nirly algbraic!! n a similar way, on shows...

14 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

15 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.5.5S Admiancs Y s

16 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

17 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 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

18 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

19 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 Thvnin's and Noron's ATAB PSPCE horms

20 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 A

21 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

22 3. SOUCE SUPEPOSTON ' ' A " 6 " " " 6 " " " A 6 " A 6 3 " " A 3 5 " ' A " " TO COPUTE TANSFOATON COUD USE SOUCE

23 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

24 5. THEENN ANAYSS olag Dividr 8 OC 6 TH Ω A

25 EXAPE Find h currn i in sady sa Th sourcs hav diffrn frquncis! For phasor analysis UST us sourc suprposiio Frquncy domain SOUCE : FEQUENCY r/s Principl of suprposiion

26 EANNG BY DESGN USNG PASSE COPONENTS TO CEATE GANS AGE THAN ONE PODUCE A GAN AT Kh WHEN C C 5.9μF.59mH