FIRST AND SECOND-ORDER TRANSIENT CIRCUITS

Transcription

1 FIT AND END-DE TANIENT IUIT IN IUIT WI INDUT AND APAIT VTAGE AND UENT ANNT HANGE INTANTANEUY. EVEN E APPIATIN, EMVA, F NTANT UE EATE A TANIENT BEHAVI EANING GA FIT DE IUIT rcus h conn sngl nrgy sorng lmns. Ehr cpcor or n nducor END DE IUIT rcus wh wo nrgy sorng lmns n ny combnon

2 ANAYI F INEA IUIT WI INDUT AND/ APAIT E NVENTINA ANAYI UING MAEMATIA MDE EQUIE E DETEMINATIN F A ET F EQUATIN AT EPEENT E IUIT. NE E MDE I BTAINED ANAYI EQUIE E UTIN F E EQUATIN F E AE EQUIED. F EXAMPE IN NDE P ANAYI F EITIVE IUIT NE EPEENT E IUIT BY A ET F AGEBAI EQUATIN E MDE WHEN EE AE INDUT APAIT E MDE BEME INEA DINAY DIFFEENTIA EQUATIN DEs. HENE, IN GENEA, NE NEED A E T IN DE T BE ABE T ANAYZE IUIT WI ENEGY TING EEMENT. A MED BAED N EVENIN WI BE DEVEPED T DEIVE MAEMATIA MDE F ANY ABITAY INEA IUIT WI NE ENEGY TING EEMENT. E GENEA APPAH AN BE IMPIFIED IN ME PEIA AE WHEN E FM F E UTIN AN BE NWN BEFEHAND. E ANAYI IN EE AE BEME A IMPE MATTE F DETEMINING ME PAAMETE. TW UH AE WI BE DIUED IN DETAI F E AE F NTANT UE. NE AT AUME E AVAIABIITY F E DIFFEENTIA EQUATIN AND A END AT I ENTIEY BAED N EEMENTAY IUIT ANAYI BUT IT I NMAY NGE WE WI A DIU E PEFMANE F INEA IUIT T E IMPE INPUT

3 AN INTDUTIN INDUT AND APAIT AN TE ENEGY. UNDE UITABE NDITIN I ENEGY AN BE EEAED. E ATE AT WHIH IT I EEAED WI DEPEND N E PAAMETE F E IUIT NNETED T E TEMINA F E ENEGY TING EEMENT Wh h swch on h l h cpcor rcs chrg rom h bry. wch o h rgh nd h cpcor dschrgs hrough h lmp

4 GENEA EPNE: FIT DE IUIT Includng h nl condons h modl or h cpcor olg or h nducor currn wll b shown o b o h orm d d ; olng h drnl quon usng ngrng cors, on rs o conr h H no n c dr d /* d d d d d d ; d d * / d I EXPEIN AW E MPUTATIN F E EPNE F ANY FING FUNTIN. WE WI NENTATE IN E PEIA AE WHEN E IGHT HAND IDE I NTANT s wll b shown normon on h rcon spd o crcu clld h " m consn." Th nl m, o prod o sudy squnl swchngs. sgncn, s rbrry. Th gnrl prsson cn b usd o h

5 FIT DE IUIT WI NTANT UE d ; d d I h H s consn d d Th orm o h soluon s ; Any rbl n h crcu s o h orm ; y nly h lus o h consns _, _ wll chng TANIENT TIME NTANT

6 EVUTIN F E TANIENT AND INTEPETATIN F E TIME NTANT Tngn rchs -s n on m consn Drops.6 o nl lu n on m consn Wh lss hn % rror rnsn s zro byond hs pon A QUAITATIVE VIEW: E MAE E E TIME NTANT E FATE E TANIENT DIAPPEA

7 E TIME NTANT Th ollowng mpl llusrs h physcl mnng o m consn hrgng : + dc d Th modl Assum V b c _, d d Th soluon cn b shown o b d d V V rnsn For prccl purposs h cpcor s chrgd whn h rnsn s nglgbl Wh lss hn % rror h rnsn s nglgbl r m consns

8 NDITIN IUIT WI NE ENEGY TING EEMENT E DIFFEENTIA EQUATIN APPAH. E IUIT HA NY NTANT INDEPENDENT UE. E DIFFEENTIA EQUATIN F E VAIABE F INTEET I IMPE T BTAIN. NMAY UING BAI ANAYI T;.g.,, V... EVENIN. E INITIA NDITIN F E DIFFEENTIA EQUATIN I NWN, AN BE BTAINED UING TEADY TATE ANAYI FAT: WHEN A INDEPENDENT UE AE NTANT F ANY VAIABE, y, IN E IUIT E UTIN I F E FM y, UTIN TATEGY: UE E DIFFEENTIA EQUATIN AND E INITIA NDITIN T FIND E PAAMETE,,

9 I h d q or y s known n h orm Us h d q o nd wo mor quons by rplcng h orm o soluon no h drnl quon y y y d dy W cn us hs no o nd h unknowns d dy, y y Us h nl condon o g on mor quon y HTUT: WITE DIFFEENTIA EQ. IN NMAIZED FM WI EFFIIENT F VAIABE =. y d dy y d dy

10 EANING EXAMPE MDE F FIND. V nl condon d d V V,. AUME / DIFF. EQ. NWN, INITIA NDITIN NWN TEP TIME NTANT dy y d d V s d G m consn s cocn o dr ANWE : * / UTIN or V V /, TEP TEADY TATE ANAYI I nd,, sdy s lu IN TEADY TATE E UTIN I A NTANT. HENE IT DEIVATIVE I ZE. FM DIFF EQ. d V dy s lu d rom d. q. qung sdy s lus V dy IF E MDE I y EN d TEP UE F INITIA NDITIN AT V / V ; /,

11 , FIND V F UE MDE. V d d V nducor INITIA NDITIN TEP V d d TEP TEADY TATE V TEP INITIA NDITIN V AN : EANING EXAMPE ;,,

12 EANING BY DING, MDE. F d d I d d I INITIA NDITIN: TEP TEP I I TEP AN : I

13 , INITIA NDITIN IUIT IN TEADY TATE F MDE F IT I IMPE T DETEMINE MDE F APAIT VTAGE d ; P d d P k 6k k d P k 4V 4V k 6k TEP 6 P F. s TEP TEP 4V 4V 4. [ V ], 4 AN :. [ ma],

14 EANING EXAMPE V> FIND,, TEP : FIND UING TEADY TATE ANAYI d.5 6 V d 6 ; 6V, E NEXT TEP EQUIE E INITIA VAUE F E VAIABE, MDE F. UE V d V F E INITIA NDITIN NE NEED d E INDUT UENT F < AND d 4 [ V ] UE E NTINUITY F E INDUT d UENT DUING E WITHING. d.5 [ A] E TEADY TATE AUMPTIN F < d IMPIFIE E ANAYI d.5 V d 6.5 TEP

15 IUIT IN TEADY TATE < I V V V 4I 4 I 4[ ] V A I 4 4[ V ] MUT F EXAMPE UE EVENIN AUMING INDUT IN TEADY TATE FIND, b 5 5., 4 [ A] 4 8 [ V ] [ V ],

16 EANING EXTENIN FIND, ;, MDE F. UE d d TEP TEP d d c 6 6 F. 6s, [ V ], INITIA NDITIN. IUIT IN TEADY TATE < TEP 8 8[ V ] 6 V 8. 6 [ V ], 9 DETEMINE c

17 EANING EXTENIN FIND, ;, IUIT IN TEADY TATE PI T WITHING MDE F. UE V d d 8 d 9 d TEP s 9 TEP F INITIA NDITIN NE NEED INDUT UENT F < V A TEP [ A] 9 AN : 9 [ A] [ A],

18 UING EVENIN T BTAIN MDE bn h olg cross h cpcor or h currn hrough h nducor rcu wh rssncs nd sourcs b Inducor or pcor prsnon o n rbrry crcu wh on sorg lmn V b s. Volg cross cpcor d d c + nod c c d _ c d d d V V b s. urrn hrough nducor d d b Inducor or pcor Us V d d d d

19 EXAMPE 6 4V Fnd 6 ; 6 6 H Th rbl o nrs s h nducor currn. Th modl s d d And h soluon s o h orm ; N: Inl ondon 6 4V Thnn or > nducor rmnls 6 H 6 6 b s d. ; d..... ;

20 Drmn. ssumpon nd connuy nducor currn 6 4V rcu or < 6 Us sdy s 6 6 o H soluon: oop nlyss 8 Nod nlyss ma 6 nc = h soluon s 6 4V ; Elung + 6. ; 6

21 EXAMPE V For V 6k Fnd, F + - 6k 6k 6k Hnc, h cpcor olg s known h problm s sold Modl or _c d d 6k b + - 6k 6k 6k 6k 6k 6k k * 6V Modl ** or 6 d. d.5.5 F. s Now w nd o drmn h nl lu _c+ usng connuy nd h sdy s ssumpon 6

22 V crcu n sdy s bor h swchng 6k 6k + - 6k 6k 6V onnuy o cpcor olg 6V 6 6V ; ma; 6k D Eq Approch