FIRST AND SECOND-ORDER TRANSIENT CIRCUITS

Transcription

1 FIST AND SEOND-ODE TANSIENT IUITS IN IUITS WITH INDUTOS AND APAITOS VOTAGES AND UENTS ANNOT HANGE INSTANTANEOUSY. EVEN THE APPIATION, O EMOVA, OF ONSTANT SOUES EATES A TANSIENT BEHAVIO EANING GOAS FIST ODE IUITS ircuis ha conain a singl nrgy soring lmns. Eihr a capacior or an inducor SEOND ODE IUITS ircuis wih wo nrgy soring lmns in any combinaion

2 ANAYSIS OF INEA IUITS WITH INDUTOS AND/O APAITOS THE ONVENTIONA ANAYSIS USING MATHEMATIA MODES EQUIES THE DETEMINATION OF (A SET OF EQUATIONS THAT EPESENT THE IUIT. ONE THE MODE IS OBTAINED ANAYSIS EQUIES THE SOUTION OF THE EQUATIONS FO THE ASES EQUIED. FO EXAMPE IN NODE O OOP ANAYSIS OF ESISTIVE IUITS ONE EPESENTS THE IUIT BY A SET OF AGEBAI EQUATIONS THE MODE WHEN THEE AE INDUTOS O APAITOS THE MODES BEOME INEA ODINAY DIFFEENTIA EQUATIONS (ODEs. HENE, IN GENEA, ONE NEEDS A THOSE TOOS IN ODE TO BE ABE TO ANAYZE IUITS WITH ENEGY STOING EEMENTS. A METHOD BASED ON THEVENIN WI BE DEVEOPED TO DEIVE MATHEMATIA MODES FO ANY ABITAY INEA IUIT WITH ONE ENEGY STOING EEMENT. THE GENEA APPOAH AN BE SIMPIFIED IN SOME SPEIA ASES WHEN THE FOM OF THE SOUTION AN BE NOWN BEFOEHAND. THE ANAYSIS IN THESE ASES BEOMES A SIMPE MATTE OF DETEMINING SOME PAAMETES. TWO SUH ASES WI BE DISUSSED IN DETAI FO THE ASE OF ONSTANT SOUES. ONE THAT ASSUMES THE AVAIABIITY OF THE DIFFEENTIA EQUATION AND A SEOND THAT IS ENTIEY BASED ON EEMENTAY IUIT ANAYSIS BUT IT IS NOMAY ONGE WE WI ASO DISUSS THE PEFOMANE OF INEA IUITS TO OTHE SIMPE INPUTS

3 AN INTODUTION INDUTOS AND APAITOS AN STOE ENEGY. UNDE SUITABE ONDITIONS THIS ENEGY AN BE EEASED. THE ATE AT WHIH IT IS EEASED WI DEPEND ON THE PAAMETES OF THE IUIT ONNETED TO THE TEMINAS OF THE ENEGY STOING EEMENT Wih h swich on h lf h capacior rcis charg from h bary. Swich o h righ and h capacior dischargs hrough h lamp

4 GENEA ESPONSE: FIST ODE IUITS Including h iniial condiions h modl for h capacior olag or h inducor currn will b shown o b of h form d d f ; ( TH Soling h diffrnial quaion using ingraing facors, on ris o conr h HS ino an ac driai d f TH /* d d d d d d ( a( f ( ; ( d f f TH TH ( ( ( f TH ( ( d f TH * / ( d THIS EXPESSION AOWS THE OMPUTATION OF THE ESPONSE FO ANY FOING FUNTION. WE WI ONENTATE IN THE SPEIA ASE WHEN THE IGHT HAND SIDE IS ONSTANT is i will b shown o proid significan informaion on h racion spd of circui calld h "im consan." Th iniial im, o, is arbirary. gnral prssion can b usd o sudy squnial swichings. Th h

5 FIST ODE IUITS WITH ONSTANT SOUES d f TH ( ( ( ( ; f d d TH If h HS is consan d f TH ( ( d f TH ( ( TH f ( ( ( ( TH f ( ( ( TH TH f f Th form of h soluion is ; ( Any ariabl in h circui is of h form ; ( y Only h alus of h consans _, _ will chang TANSIENT TIME ONSTANT

6 EVOUTION OF THE TANSIENT AND INTEPETATION OF THE TIME ONSTANT Tangn rachs -ais in on im consan Drops.63 of iniial alu in on im consan Wih lss han % rror ransin is zro byond his poin A QUAITATIVE VIEW: THE SMAE THE THE TIME ONSTANT THE FASTE THE TANSIENT DISAPPEAS

7 THE TIME ONSTANT Th following ampl illusras h physical maning of im consan harging a capacior S : a S dc S d S Th modl Assum S d d VS b c _, ( d d TH TH Th soluion can b shown o b S TH Wih lss han % rror h ransin is ngligibl afr fi im consans ( V S V S ransin For pracical purposs h capacior is chargd whn h ransin is ngligibl TH

8 ONDITIONS IUITS WITH ONE ENEGY STOING EEMENT THE DIFFEENTIA EQUATION APPOAH. THE IUIT HAS ONY ONSTANT INDEPENDENT SOUES. THE DIFFEENTIA EQUATION FO THE VAIABE OF INTEEST IS SIMPE TO OBTAIN. NOMAY USING BASI ANAYSIS TOOS;.g.,, V... O THEVENIN 3. THE INITIA ONDITION FO THE DIFFEENTIA EQUATION IS NOWN, O AN BE OBTAINED USING STEADY STATE ANAYSIS FAT: WHEN A INDEPENDENT SOUES AE ONSTANT FO ANY VAIABE, y (, IN THE IUIT THE SOUTION IS OF THE FOM ( O y (, > O SOUTION STATEGY: USE THE DIFFEENTIA EQUATION AND THE INITIA ONDITIONS TO FIND THE PAAMETES,,

9 If h diff q for y is known in h form Us h diff q o find wo mor quaions by rplacing h form of soluion ino h diffrnial quaion y( y f y a d dy a W can us his info o find h unknowns f a a a f f a a a a a d dy >, ( y ( y Us h iniial condiion o g on mor quaion y( SHOTUT: WITE DIFFEENTIA EQ. IN NOMAIZED FOM WITH OEFFIIENT OF VAIABE. a f y d dy a a f y a d dy a

10 EANING EXAMPE FIND MODE FO >. ( ( V S iniial condiion d d ( ( VS V, >. ASSUME ( ( S / (DIFF. EQ. NOWN, INITIA ONDITION NOWN STEP TIME ONSTANT dy y f d d ( ( V s d G im consan as cofficin of driai ANSWE : * / SOUTION for > ( V ( V /, > S STEP STEADY STATE ANAYSIS IS S ( and,, > ( ( (sady sa alu IN STEADY STATE THE SOUTION IS A ONSTANT. HENE ITS DEIVATIVE IS ZEO. FOM DIFF EQ. d V Sady sa alu S d from diff. q. (quaing sady sa alus V S dy IF THE MODE IS y f THEN f d STEP 3 USE OF INITIA ONDITION AT ( ( ( f V / V / ( S S ( ;, > (

11 EANING EXAMPE FIND i(, > ( ( ;, > ( V i( i (, > MODE. USE V FO > di VS i( ( d INITIA ONDITION < i( i( inducor i( i( STEP di V S ( i( d STEP STEADY STATE VS i( STEP 3 INITIA ONDITION i ( ANS: VS i(

12 i(, > INITIA ONDITIONS IUIT IN STEADY STATE FO < MODE FO > i ( ( IT IS SIMPE TO DETEMINE MODE FO APAITO VOTAGE ( d ( ( ; P d d ( P 3k 6k kω ( d P 3k ( ( 4V ( 4V 3k 6k STEP 3 6 P ( Ω( F. s STEP ( STEP 3 ( 4V 4V ( 4. [ V ], > 4 ANS: i(. [ ma], > 3

13 EANING EXTENSION FIND O (, > ( ( ;, > i ( MODE FO >. USE d d ( ( ( c d d STEP STEP 3 6 ( (6 Ω( F. 6s (, > ( ( ( 4 3 O ( 6 O 8 3. [ V ], > INITIA ONDITIONS. IUIT IN STEADY STATE < STEP 3 ( 8 8[ V ] ( 6 ( V ( 8. 6 [ V ], > 9 DETEMINE c (

14 ANAYSIS OF IUITS WITH ONE ENEGY STOING EEMENT ONSTANT INDEPENDENT SOUES A STEP-BY-STEP APPOAH THIS APPOAH EIES ON THE NOWN FOM OF THE SOUTION BUT FINDS THE ONSTANTS,, USING BASI IUIT ANAYSIS TOOS AND FOGOES THE DETEMINATION OF THE DIFFEENTIA EQUATION MODE is ( is (, > h sady sa alu h ariabl and can b drmind analyzing h circui in sady sa h using quaion o compu h consans, of is h iniial condiion and proids h scond im consan and can b drmind Thnin across h nrgy soring lmn

15 IUITS WITH ONE ENEGY STOING EEMENT Obaining h im consan: A Gnral Approach ircui wih rsisancs and sourcs a b Inducor or apacior prsnaion of an arbirary circui wih on sorag lmn V TH Thnin TH a b Inducor or apacior apacii Induci ircui ircui TH TH TH a i nod a i i c i V TH c d i _ c d b as. i Volag across capacior TH d d TH d TH TH d TH TH V TH TH b as. urrn hrough inducor TH a di d i i Us V TH THi di di d THi d TH TH i S TH

16 THE STEPS STEP. THE FOM OF THE SOUTION (, > ( ; ( DETEMINE ( STEP : DAW THE IUIT IN STEADY STATE PIO TO THE SWITHING AND DETEMINE APAITO VOTAGE O INDUTO UENT STEP 3: DAW THE IUIT AT THE APAITO ATS AS A VOTAGE SOUE. THE INDUTO ATS AS A UENT SOUE. DETEMINE THE VAIABE AT STEP 5: DETEMINE THE TIME ONSTANT TH TH circui wih on capacior circui wih on inducor STEP 6: DETEMINE THE ONSTANTS, (, ( DETEMINE ( STEP 4: DAW THE IUIT IN STEADY STATE AFTE THE SWITHING AND DETEMINE THE VAIABE IN STEADY STATE.

17 EANING EXAMPE FIND i(, > STEP 3:Drmin i( USE A IUIT VAID FO. THE APAITO ATS AS SOUE STEP : i(, > STEP : Iniial olag across capacior USE IUIT IN STEADY STATE PIO TO THE SWITHING 3V 6 i( 6k 3 ma V 4V i ma i( kω ( c ( c ( 36V (ma(k 3[ V ] NOTES FO INDUTIVE IUIT (DETEMINE INITIA INDUTO UENT IN STEP (FO THE IUIT EPAE INDUTO BY A UENT SOUE

18 STEP 4 : Drmin STEP 5:Drmin i( USE IUIT IN STEADY STATE AFTE SWITHING apacii circui : im consan TH STEP 6:Drmin, (STEP i(, > 6 ( STEP 3 i( ma 3 36 (STEP 4 i( ma FINA ANSWE i( ma i(. 5, > NOTE: FO INDUTIVE IUIT TH OIGINA IUIT TH k 6k. 5kΩ μ F (.5 3 Ω( 6 F. 5s

19 EANING EXAMPE FIND (, > STEP 3: Drmin ( Us circui a. Inducor is rplacd by currn sourc STEP : (, > STEP : Iniial inducor currn Us circui in sady sa prior o swiching 4 I 4mA 6 6 i 6 3 ( I V V 4 V V V [ 3 V ] 6k 3k 8 i ( 3 ma 5 ( 4[ V ] V [ V ] 3

20 STEP 4 : DETEMINE ( USE IUIT IN STEADY STATE AFTE SWITHING STEP 6: DETEMINE, ( 4[ ] (sp 4 V 5 ( (sp [ V ] 3 3 ANS: ( 4, > 3 ( 4[ V ] STEP 5:DETEMINE 4H s Ω TIME ONSTANT Induci TH ircui : 4 6 TH OIGINA IUIT TH Ω

21 EANING EXTENSION FIND O (, > : * / [ V ] [ V ] ( ( [ ] V STEP 3: DETEMINE O ( STEP : ( o, > STEP : INITIA APAITO VOTAGE V O ( V ( < ( 3 ( O ( ( 5[ V ]

22 STEP 4 : DETEMINE O ( STEP 6: DETEMINE, O ( 4 O ( [ V ] 5 O ( 5[ V ] [ 5 V ] ANS: O 4 8 ( [ V ]; > 4 O ( ( [ V ] 5 5 STEP 5: DETEMINE TIME ONSTANT apacii ircui : TH OIGINA IUIT F 8 5 s TH TH Ω

23 STEP 3: DETEMINE O ( FIND O (, > i ( O ( STEP : STEP : ( O, > INITIA INDUTO UENT O i ( O i O O 8 [ V ] 3 ( i ( 4 ( O 4 [ A] 3 8 ( i( [ V ] 3 O

24 STEP 4 : DETEMINE O ( 6: DETEMINE, O ( STEP O ( 6[ V ] (sp 4 8 O ( (sp [ V ] 3 3 ANS: ( 5 O 6 3., > STEP 5: DETEMINE TH O ( ( 6[ V ] TIME ONSTANT Induci TH ircui OIGINA IUIT TH 4Ω 4.5s

25 EANING EXAMPE FIND O (, > STEP : O (, > STEP 3: DETEMINE O ( STEP : DETEMINE i ( i ( 3[ A] i ( 8[ V ] i A 3A 6[ V ] O ( 8[ V ]

26 STEP 4 : DETEMINE O ( O ( 7[ V ] B OPEN IUIT VOTAGE B ' B i A 4 ' 36 B B ( ia * / 4 6 ' ' 4i [ V ], i 4.5[ A] B A STEP 5:DETEMINE induci TH circui V 9[ V ] B A TIME ircui wih TH i ONSTANT dpndn sourcs O S i " A 6A V V O 4 36[ V ] OIGINA IUIT

27 SHOT IUIT UENT i i ''' A S i i NOTE: FO THE INDUTIVE ASE THE IUIT USED TO OMPUTE THE SHOT IUIT UENT IS THE SAME USE TO DETEMINE O ( i 36 ( i 36 ( i i i 4i 6i ''' i A is 36 [ A ] 8 i S 36[ V ] 36/8[ A] O TH 8Ω STEP 6: DETEMINE, O ( 7 (sp 4 O 3 H ( 8 9[ V ] 3 8 (sp 3 s OIGINA IUIT ANS: O 3 ( 7 9 8, >

28 EANING EXTENSION FIND o (, > STEP : O (, > STEP : DETEMINE APAITO VOTAGE AT 4[ V ] V A [ V ] ( ( 4 4 6[ V ] ( ( STEP 3: STEP 4 : DETEMINE DETEMINE O ( O ( ( 6[ V ] O O i O ( O ( 4[ V ] A

29 STEP 5:DETEMINE TIME ONSTANT OPEN IUIT VOTAGE capacii circui O TH O ( TH i O S O O ( 4[ V ] SHOT IUIT UENT Ω V STEP 6:DETEMINE, ( 4 (sp 4 O O ( 6 (sp 3 36[ ] V i S i S 4 A i S 4[ A] A i S TH OIGINA IUIT 4 6Ω 4 6 Ω F s ANS: O ( 4 36, >

30 Inducor ampl FIND O (, > Ω 4Ω STEP : Iniial inducor currn Ω 4Ω H 6V < Ω O i ( 6V < Ω i ( 3A STEP : Form of h soluion STEP 3: Drmin oupu a (inducor currn is consan O ( 3A Ω 4Ω 6V Ω O ( 6V O _ (

31 STEP 4: Find oupu in sady sa afr h swiching Ω 4Ω 6V > Ω O _ ( O ( STEP 5: Find im consan afr swich TH Ω 4Ω 6V > Ω TH TH 8Ω.5s H Ω 4Ω 6V Ω O _ STEP 6: Find h soluion O O ( 6V O ( O. 5 ( 6 ; > ( 6 4 ; > Puls spons ( Sp by Sp

32 PUSE ESPONSE WE STUDY THE ESPONSE OF IUITS TO A SPEIA ASS OF SINGUAITY FUNTIONS < u( VOTAGE STEP > UENT STEP TIME SHIFTED STEP

33 PUSE SIGNA EAN BY DOING i( [ u( u(.]( ma ( [ u( u( ]( V PUSE AS SUM OF STEPS

34 NONZEO INITIA TIME AND EPEATED SWITHING ( ( d d TH ( ( f ; ; f TH ( d ESPONSE FO ONSTANT SOUES This prssion will hold on ANY inral whr h sourcs ar consan. Th alus of h consans may b diffrn and mus b aluad for ach inral Th alus a h nd of on inral will sr as iniial condiions for h n inral

35 EANING EXAMPE FIND THE OUTPUT VOTAGE ( ; > O >.3 ( (.3 " " ' ( O ".3. o.3 (.3 4( 4 o '.4 " V o ( o (.3.( o.3 (.. 4 ; >.3 < ( O ( O ( > ( 9V ' ' o( TH ( 6k k μ F. 4s 8 ' ` ' o ( (9 o ( 8. 4 o ( 4

36 EXAMPE THE SWITH IS INITIAY AT a. AT TIME IT MOVES TO b AND AT.5 IT MOVES BA TO a. FIND (, > ON IS O V - O a b S ( s ( kω μ F.5 O (s Picwis consan sourc EAH INTEVA ONSTANT ( O WHEE THE OUTPUT THE IS SOUE OF THE O ( ' O ( O FOM FO < <.5(swich a b O ' ' ' ( [ V ] ( kω(μ F. s (., < <.5.5 (.5 (.5. O (.5 " " ( ' O O ( " FO >. 5 (swich a a. 5 O " ".985 O ( " (.5., >.5 Th consans ar drmind in h usual mannr o o '

37 USING MATAB TO DISPAY OUTPUT VOTAGE %puls.m % displays h rspons o a puls rspons minlinspac(-.5,,5; %ngai im sgmn linspac(,.5,5; %firs sgmn linspac(.5,.5,; %scond sgmn omin*ons(siz(min; o*p(-/.; %afr firs swiching o-.5*p(-(-.5/.; %afr scond swiching plo(min,omin,'bo',,o,'r',,o,'md',grid il('output VOTAGE', labl('(s',ylabl('vo(v' Firs Ordr

38 THE BASI IUIT EQUATION SEOND-ODE IUITS i i i Singl Nod-pair: Us i S i i i ( i ; i ( d i( ; i d d i i ( ( ( d Diffrniaing d d d d d d S di d S ( Singl oop: Us V S i; i( d ( ; d i d di i i d ( ( ( d di d i d d S S di d (

39 EANING BY DOING WITE THE DIFFEENTIA EQUATION FO (, i(, ESPETIVEY i S S i S ( I S < > MODE FO PAAE d d d d d d d d di S d di d ( ; > S S V ( S < > MODE FO SEIES d i d di d d i d i di d d S d d d i ( ; > S

40 THE ESPONSE EQUATION WE STUDY THE SOUTIONS FO THE EQUATION d d ( a ( a( f ( d d NOWN: ( ( ( d p c THE p paricular soluion complmnary soluion OMPEMENTAY SOUTION SATISIFES c ( d dc a ( a c ( d c IF THE FOING FUNTION IS A ONSTANT f ( A p POOF: FO ( A a A a is a paricular soluion A d p d p p a p a d d ANY FOING c ( FUNTION f ( A A

41 THE HOMOGENEOUS EQUATION d d ( a ( a( d d NOMAIZED FOM ω n ς d d ( ςω ( ( n ω n d d (undampd naural frquncy damping a a ω raio HAATEISTI s n ςω ςω n s ω n n ω n EQUATION a a ς a EANING BY DOING DETEMINE EQUATION, NATUA FEQUENY d d ( 8 ( 6( d d 4 THE DAMPING HAATEISTI ATIO AND OEFFIIENT OF SEOND DEIVATIVE MUST BE ONE s d d ( ( 4( d d HAATEISTI s 4 EQUATION DAMPING ATIO, NATUA FEQUENY d d ( ( 4( d d ςω n ω n ω n ς.5

42 ANAYSIS OF THE HOMOGENEOUS EQUATION NOMAIZED FOM ASE : ς > (ral and disinc roos s s d d ( ( ςω ( ( n ω n d d ASE : ς < (compl conjuga roos s s s ( is a soluion iff ( s ςωn ± jωn ς * s ςωns ωn ( ral s σ ± jωd Iff s is soluion of h characrisic ω d dampd oscillaion frquncy quaion σ damping facor d σ s d s POOF: ( s ; s ( ( A cosωd A sinωd s ( ςωn ± jωd ςωn m jωd d d HINT : m jωd d d s cosω j ( ςω n ( ωn ( ( s ςωns ωn d m sinωd d d ASSUME ( A ja HAATEISTI EQUATION / * s [ ] ςω n s ω n j d s j ( σ ω ( σ ± ωd ( s ςω ASE 3: ς (ral and qual roos n ( ωn ς ωn s ςω n s ςωn ± ς ωn ωn ςωn ( ( B B s ςω ± s n ωn ς HINT : is soluion iff (mods of h sysm ( s ςω s ω AND (s ςω n n n

43 EANING EXTENSIONS DETEMINE THE GENEA FOM OF THE SOUTION s d d ( 4 ( 4( d d HAATEISTI EQUATION 4s 4 ω 4 ω ςω n 4 ς n n s 4s 4 ( s oos ar ral and qual his is a criically dampd (cas 3 sysm ( ( B ( ( B B B s d d ( 8 ( 6( d d 4 Diid by cofficin of scond driai d d ( ( 4( d d ω 4 ω ςω n ς. 5 n n s s 4 ( s 3 s ± oos ar compl conjuga undrdampd (cas sysm σ ςω ; ω ω ς n ( ( d σ n.5 3 ( A cosω A sinω d ( A cos 3 A sin 3 d j 3 ω d

44 Form of h soluion PAAE IUIT Ω, H, F EANING EXTENSIONS WITH SEIES IUIT WITH Ω; H,.5F,F,F d d d d d d d c ( HOMOGENEOUS EQUATION d i di i : / & rplac alus d d d d d i di i s 3 d s ( s d d d ωn ; ςωn ς ωn ; ςωn ς d undrdampd σ 3 ωd ωn ς. criically dampd ordampd A cos A sin discriminan 4 lassify h rsponss for h gin alus of

45 DETEMINING THE ONSTANTS THE NETWO ESPONSE NOMAIZED FOM d d ( ςωn ( ωn ( d d A A s s ( ω ωn n A ( ωn d ( s s d ( A A ( A cosω A sinω ςωn ( d A ( A ωn d d ςωn ( B B ωn A ( B ωn d d ( ςω B B ( ςω A A n n ω d d

46 EANING EXAMPE Ω, 5H, F 5 i( A, ( 4V i i i i i i d ( d i( d d d d d HAATEISTI EQUATION s.5s.5 ± s ( (.5 ω n 4.5 ; ς.5.5 ±.5 STEP 5: FIND ONSTANTS STEP MODE STEP STEP 4 FOM OF SOUTION STEP 3 OOTS To drmin h consans w nd d ( ; ( d IF NOT GIVEN FIND (, i( ( ( ( 4V ANAYZE AT IUIT AT ( d i( ( d d 4 ( ( 5 d (/ 5 (/ 5 4 ; ( ; >

47 USING MATAB TO VISUAIZE THE ESPONSE %scrip6p7.m %plos h rspons in Eampl 6.7 %(p(-p(-.5; > linspac(,,; *p(-**p(-.5*; plo(,,'mo', grid, labl('im(sc', ylabl('v(vols' il('esponse OF OVEDAMPED PAAE IUIT'

48 EANING EXAMPE 6 Ω, H,. 4F i ( 4A; ( V 4 NO SWITHING O DISONTINUITY AT. USE O i( di d ( i( d ( d i di ( i( modl d d d i di 6 ( 5i( d d ω n 5 ωn 5 h. Eq.: s 6s 5 ςωn 6 ς.6 6 ± 36 roos : s 3 ± j4 ωd 3 Form: i( ( A cos4 A i( i( 4A A 4 sin 4 di di TO OMPUTE ( ( ( d d di di ( i( ( ( d d di 3 ( 3i( ( 4A sin 4 4A cos4 d : 3 (4 4A 3 i( (4cos 4 sin 4[ A]; ( i( 3 di d ( ( > ( ( 4cos4 sin 4[ V ]; i( d >

49 %scrip6p8.m USING MATAB TO VISUAIZE THE ESPONSE %displays h funcion i(p(-3(4cos(4-sin(4 % and h funcion c(p(-3(-4cos(4sin(4 % us a siml algorihm o sima display im au/3; nd*au; linspac(,nd,35; ip(-3*.*(4*cos(4*-*sin(4*; cp(-3*.*(-4*cos(4**sin(4*; plo(,i,'ro',,c,'bd',grid,labl('tim(s',ylabl('volag/urrn' il('uent AND APAITO VOTAGE' lgnd('uent(a','apaito VOTAGE(V'

50 EANING EXAMPE V Ω, 8Ω, F, H ( V, i(. 5A h. Eq.: ω 3, ςω 6 ς n s 6 s 9 ( s 3 n 3 ( B B ( ( c ( V NO SWITHING O DISONTINUITY AT. USE O di ( i( ( d ( i ( d d ( d d ( d ( ( ( d d d d d ( ( ( d d d d h. Eq.: s 6s 9 ( 6 ( 9( d d AT ( d i( i ( ( d ( B d d d ( 3( B 3 B d 3 ( 6 ; > ( ( 6 3

51 USING MATAB TO VISUAIZE ESPONSE %scrip6p9.m %displays h funcion (p(-3(6 au/3; ndcil(*au; linspac(,nd,4; p(-3*.*(6*; plo(,,'r',grid, labl('tim(s', ylabl('volag(v' il('apaito VOTAGE'

52 EANING EXTENSION FIND i(, > d i 3 di ( ( i( d d h.eq.: s.5s.5 roos : s,.5 i( ; To find iniial condiions us sady sa analysis for < i ( A Onc h swich opns h circui is sris di 3i( ( ( i( d d Ω di d i( A Ω a > ( V di ( d And analyz circui a i( 4 ; >

53 EANING EXTENSION FIND (, > To find iniial condiions w us sady sa analysis for < V i ( A ( i( ( i( For > h circui is sris di d ( ( ( ( /3 i d i d i di ( 4 ( 3i( d d h. Eq. : s 4s 3 roos : s, 3 i( 3 ; > And analyz circui a i( A di ( ( d i( di ( d ( i( ( 3 ; > ; > 3

54 EANING EXTENSION DETEMINE i, ( ; 8i ( ( ( > ( V Sady sa < V 4V Ω i ( ( Ω di ( i (. 5A 4 ( ( ( 8 ( / 36 i d i d Analysis a d i di ( 9 ( 8i( i ( i (.5( A d d di di h. Eq.: s 9s 8 ( ( ( ( d d roos : s 3, 6 4 ( 8i( ( ( di i ( ; > ( 7 / 3 6 d i( Scond i ( ; > ; Ordr 6 6