LaPlace Transform in Circuit Analysis

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1 LaPlac Tranform in Circui Analyi Obciv: Calcula h Laplac ranform of common funcion uing h dfiniion and h Laplac ranform abl Laplac-ranform a circui, including componn wih non-zro iniial condiion. Analyz a circui in h -domain Chck your -domain anwr uing h iniial valu horm IVT and final valu horm FVT Invr Laplac-ranform h rul o g h imdomain oluion; b abl o idnify h forcd and naural rpon componn of h im-domain oluion. No hi marial i covrd in Chapr and Scion

2 LaPlac Tranform in Circui Analyi Wha yp of circui can w analyz? Circui wih any numbr and yp of DC ourc and any numbr of rior. Fir-ordr RL and RC circui wih no ourc and wih a DC ourc. Scond-ordr ri and paralll RLC circui wih no ourc and wih a DC ourc. Circui wih inuoidal ourc and any numbr of rior, inducor, capacior and a ranformr or op amp, bu can gnra only h ady-a rpon.

3 LaPlac Tranform in Circui Analyi Wha yp of circui will Laplac mhod allow u o analyz? Circui wih any yp of ourc o long a h funcion dcribing h ourc ha a Laplac ranform, rior, inducor, capacior, ranformr, and/or op amp; h Laplac mhod produc h compl rpon!

4 LaPlac Tranform in Circui Analyi Dfiniion of h Laplac ranform: L { f } F f d No ha hr ar limiaion on h yp of funcion for which a Laplac ranform xi, bu ho funcion ar pahological, and no gnrally of inr o nginr!

5 LaPlac Tranform in Circui Analyi Aid formally dfin h p funcion, which i ofn modld in a circui by a volag ourc in ri wih a wich. f = u f,, Whn =, f = u, which w call h uni p funcion

6 LaPlac Tranform in Circui Analyi Mor p funcion: Th p funcion hifd in im Th window funcion f = u-a f = u-a - u-a a a a

7 Which of h xprion dcrib h funcion plod hr? 5 A. u 5 B. 5u + 5 C. 5u 5 D. 5u 5 5

8 Which of h xprion dcrib h funcion plod hr? A. 8u B. 4u 8 C. 8u 4-4

9 Which of h xprion dcrib h funcion plod hr? A. u u -5 B. u 5 + u + C. u + 5 u

10 LaPlac Tranform in Circui Analyi U window funcion o xpr hi picwi linar funcion a a ingl funcion valid for all im.,, [ u u ] f 4, [ u u 3] 8, [ u 3 u 4], 4 f [ u u ] 4[ u u 3] 8[ u 3 u 4] u 4 u 4 3 u 3 4 u 4

11 LaPlac Tranform in Circui Analyi Th impul funcion, crad o ha h p funcion drivaiv i dfind for all im: Th p funcion f = u Th fir drivaiv of h p funcion df/d Th valu of h drivaiv a h origin i undfind!

12 LaPlac Tranform in Circui Analyi U a limiing funcion o dfin h p funcion and i fir drivaiv! Th p funcion Th fir drivaiv of h p funcion g / dg/d - - gf a [dg/d] a

13 LaPlac Tranform in Circui Analyi Th uni impul funcion i rprnd ymbolically a. Dfiniion: for and d No ha hara undr h g funcion i, which approacha No alo ha any limiing funcion wih h following characriic can b ud o gnra h uni impul funcion: High a Widh a Ara i conan for all valu of

14 LaPlac Tranform in Circui Analyi Anohr dfiniion: du d -a a Th ifing propry i an imporan propry of h impul funcion: f a d f a

15 Evalua h following ingral, uing h ifing propry of h impul funcion. 6 3 d A. 4 B C. 3

16 LaPlac Tranform in Circui Analyi U h dfiniion of Laplac ranform o calcula h Laplac ranform of om funcion of inr: ] [ } {in } { } { } { d d a a a d d d d u u d d a a a a L L L L

17 Look a h Funcional Tranform abl. Bad on h parn ha xi rlaing h p and ramp ranform, and h xponnial and dampd-ramp ranform, wha do you prdic h Laplac ranform of i? A. / + a B. C. / 3

18 LaPlac Tranform in Circui Analyi Uing h dfiniion of h Laplac ranform, drmin h ffc of variou opraion on im-domain funcion whn h rul i Laplac-ranformd. Th ar collcd in h Opraional Tranform abl ingraion by par! ] [ ] [ } { f F d f f d f f d df F F F d f d f d f d f d f d f d f f f f f f L L

19 Now l u h opraional ranform abl o find h corrc valu of h Laplac ranform of, givn ha L{ } A. / 3 B. / 3 C. -/ 3

20 LaPlac Tranform in Circui Analyi Exampl Find h Laplac ranform of a. U h opraional ranform: U h funcional ranform: L L n n n d F f n a a d L a d d 3 d a d a a Alrnaivly, U h opraional ranform: U h funcional ranform: L a 3 a L a f F a 3 L

21 LaPlac Tranform in Circui Analyi How can w u h Laplac ranform o olv circui problm? Opion : Wri h of diffrnial quaion in h im domain ha dcrib h rlaionhip bwn volag and currn for h circui. U VL, CL, and h law govrning volag and currn for rior, inducor and coupld coil and capacior. Laplac ranform h quaion o limina h ingral and drivaiv, and olv h quaion for V and I. Invr-Laplac ranform o g v and i.

22 LaPlac Tranform in Circui Analyi How can w u h Laplac ranform o olv circui problm? Opion : Laplac ranform h circui following h proc w ud in h phaor ranform and u DC circui analyi o find V and I. Invr-Laplac ranform o g v and i.

23 LaPlac Tranform in Circui Analyi Tim-domain Laplac ranform rior: -domain Laplac v Ri L V RI

24 LaPlac Tranform in Circui Analyi Tim-domain Laplac ranform inducor: -domain Laplac v i di L d I L V LI V I L I LI

25 LaPlac Tranform in Circui Analyi Tim-domain Laplac ranform rior: -domain Laplac dv i C d v V L I CV CV

26 Find h valu of h complx impdanc and h ri-conncd volag ourc, rprning h Laplac ranform of a capacior. A. C, V / B. /C, V / C. /C, V / I CV CV

27 LaPlac Tranform in Circui Analyi Rcip for Laplac ranform circui analyi:. Rdraw h circui nohing abou h Laplac ranform chang h yp of lmn or hir inrconncion.. Any volag or currn wih valu givn ar Laplacranformd uing h funcional and opraional abl. 3. Any volag or currn rprnd ymbolically, uing i and v, ar rplacd wih h ymbol I and V. 4. All componn valu ar rplacd wih h corrponding complx impdanc, Z. 5. U DC circui analyi chniqu o wri h -domain quaion and olv hm. 6. Invr-Laplac ranform -domain oluion o g imdomain oluion.

28 LaPlac Tranform in Circui Analyi Exampl: Thr i no iniial nrgy ord in hi circui. Find i and i for > I 4I 9 9 I 4I I I Subiuing, 4 I I 3 [ I

29 LaPlac Tranform in Circui Analyi Rcip for Laplac ranform circui analyi:. Rdraw h circui nohing abou h Laplac ranform chang h yp of lmn or hir inrconncion.. Any volag or currn wih valu givn ar Laplacranformd uing h funcional and opraional abl. 3. Any volag or currn rprnd ymbolically, uing i and v, ar rplacd wih h ymbol I and V. 4. All componn valu ar rplacd wih h corrponding complx impdanc, Z. 5. U DC circui analyi chniqu o wri h -domain quaion and olv hm. 6. Invr-Laplac ranform -domain oluion o g imdomain oluion.

30 LaPlac Tranform in Circui Analyi Finding h invr Laplac ranform: c f F d c Thi i a conour ingral in h complx plan, whr h complx numbr c mu b chon uch ha h pah of ingraion i in h convrgnc ara along a lin paralll o h imaginary axi a dianc c from i, whr c mu b largr han h ral par of all ingular valu of F! Thr mu b a br way

31 LaPlac Tranform in Circui Analyi Invr Laplac ranform uing parial fracion xpanion: Evry -domain quaniy, V and I, will b in h form N D whr N i h numraor polynomial in, and ha ral cofficin, and D i h dnominaor polynomial in, and alo ha ral cofficin, and O{ N } O{ D } Sinc D ha ral cofficin, i can alway b facord, whr h facor can b in h following form: Ral and diinc Ral and rpad Complx conuga and diinc Complx conuga and rpad

32 LaPlac Tranform in Circui Analyi Invr Laplac ranform uing parial fracion xpanion: Th roo of D h valu of ha mak D = ar calld pol. Th roo of N h valu of ha mak N = ar calld zro. Back o h xampl: I I

33 Find h zro of I. I 4 9 A. = 9 rad/ B. = 9 rad/ C. Thr arn any zro

34 Find h pol of I. I 4 9 A. = rad/, = rad/ B. = rad/, = rad/ C. = rad/, = rad/, = rad/

35 LaPlac Tranform in Circui Analyi Exampl: Thr i no iniial nrgy ord in hi circui. Find i and i for >. I I 5; ;

36 LaPlac Tranform in Circui Analyi Exampl: Thr i no iniial nrgy ord in hi circui. Find i and i for >. i L - 5 [5 4 4 ] u Thforcd rpon i 5u Th naural rpon i [ 4 A A; ] u A.

37 LaPlac Tranform in Circui Analyi Exampl: Thr i no iniial nrgy ord in hi circui. Find i and i for >. I I 7; ;

38 LaPlac Tranform in Circui Analyi Exampl: Thr i no iniial nrgy ord in hi circui. Find i and i for >. i L - 7 [ ] u Th forcd rpon i 7u A; Th naural rpon i [ 8.4 A.4 ] u A.

39 LaPlac Tranform in Circui Analyi Exampl: Thr i no iniial nrgy ord in hi circui. Find i and i for >. i i u A u A Chck h anwr a = and = o mak ur h circui and h quaion mach!

40 LaPlac Tranform in Circui Analyi Exampl: Thr i no iniial nrgy ord in hi circui. Find i and i for >. i i u A u A A =, h circui ha no iniial ord nrgy, o i = and i =. Now chck h quaion: i 5 4 i

41 A, h inducor bhav lik A. Inducor B. Opn circui C. Shor circui

42 LaPlac Tranform in Circui Analyi i 5 4 i Exampl: Thr i no iniial nrgy ord in hi circui. Find i and i for >..4 u A u A i 5 5 A i Draw h circui for = and chck h oluion i i 7 7 A Achck! Achck! 48

43 LaPlac Tranform in Circui Analyi W can alo chck h iniial and final valu in h -domain, bfor w bgin h proc of invr-laplac ranforming our -domain oluion. To do hi, u h Iniial Valu Thorm IVT and h Final Valu Thorm FVT. Th iniial valu horm: lim f lim F Thi horm i valid if and only if f ha no impul funcion. Th final valu horm: lim f lim F Thi horm i valid if and only if all bu on of h pol of F ar in h lf-half of h complx plan, and h on ha i no can only b a h origin.

44 LaPlac Tranform in Circui Analyi Exampl: Thr i no iniial nrgy ord in hi circui. Find i and i for >. I I Chck your anwr uing h IVT and h FVT.

45 LaPlac Tranform in Circui Analyi IVT: From h circui, i = and i = I lim i lim I lim lim Achck! 68 I lim i lim I lim lim Achck! 3

46 LaPlac Tranform in Circui Analyi FVT: From h circui, i = 5 A and i = 7 A I lim i lim I 4 36 lim lim Achck! 4 68 I lim i lim I 68 lim lim Achck! 4

47 LaPlac Tranform in Circui Analyi Rcip for Laplac ranform circui analyi:. Rdraw h circui nohing abou h Laplac ranform chang h yp of lmn or hir inrconncion.. Any volag or currn wih valu givn ar Laplacranformd uing h funcional and opraional abl. 3. Any volag or currn rprnd ymbolically, uing i and v, ar rplacd wih h ymbol I and V. 4. All componn valu ar rplacd wih h corrponding complx impdanc, Z. 5. U DC circui analyi chniqu o wri h -domain quaion and olv hm. Chck your oluion wih IVT and FVT. 6. Invr-Laplac ranform -domain oluion o g imdomain oluion. Chck your oluion a = and =.

48 LaPlac Tranform in Circui Analyi Exampl: Find v for >. Bgin by finding h iniial condiion for hi circui. V o V I o A

49 Giv h baic inrconncion of hi circui, hould w u a volag ourc or a currn ourc o rprn h iniial condiion for h inducor? A. Volag ourc B. Currn ourc C. Don mar

50 LaPlac Tranform in Circui Analyi Exampl: Find v for >. Laplac ranform h circui and olv for V. 7.4 I 5n 35. V 35. I n ,5 75 9,765,65.4

51 LaPlac Tranform in Circui Analyi Exampl: Find v for >. V 7 68,5 75 9,765,65 U h IVT and FVT o chck V.

52 LaPlac Tranform in Circui Analyi Exampl: Find v for >. IVT FVT 7 Vchck! 7 9,765, ,5 7 lim 9,765, ,5 7 lim lim lim 9,765, ,5 7 V v V o o Vchck! 9,765,65 lim 9,765, ,5 7 lim lim lim 9,765, ,5 7 o o V v V

53 LaPlac Tranform in Circui Analyi Exampl: Find v for >. V , V Parial fracion xpanion: , , ,5 [ ] ,5 [ ]

54 Whn wo parial fracion dnominaor ar complx conuga, hir numraor ar A. Equal B. Unrlad C. Complx conuga

55 LaPlac Tranform in Circui Analyi Aid look a h invr Laplac ranform of parial fracion ha ar complx conuga co 5.59 ] 6.57 in 6.57 [co 5.59 ] 6.57 in 6.57 [co * f F F

56 LaPlac Tranform in Circui Analyi Th par of h im-domain xprion com from a ingl parial fracion rm: F f 5.59 co 6.57 Imporan you mu u h numraor of h parial fracion who dnominaor ha h ngaiv imaginary par!

57 LaPlac Tranform in Circui Analyi Th gnral Laplac ranform from h abl blow h Funcional Tranform abl F L a b a b a F f co b

58 V Th parial fracion xpanion for V i hown abov. Whn w invr-laplac ranform, which parial fracion rm hould w u? A. Th fir rm B. Th cond rm C. I don mar

59 V Th im-domain funcion for v o will includ a coin a wha frquncy? A. 875 rad/ B. 3. rad/ C. 3 rad/

60 LaPlac Tranform in Circui Analyi Exampl: Find v for >. V Invr Laplac ranform: v 65. co co V Chck a = and : v v 3. co V 3.co... V

61 Thi xampl i a ri RLC circui. I rpon form, rpad blow, i characrizd a: 875 v 3. co V A. Undrdampd B. Ovrdampd C. Criically dampd

62 LaPlac Tranform in Circui Analyi Exampl: Thr i no iniial nrgy ord in hi circui. Find v o if i g = 5u ma. Laplac ranform h circui:

63 LaPlac Tranform in Circui Analyi Exampl: Find V o : ,, , , 4 8 volag diviion CL a op nod V V V V V V V V V o o o o o o o

64 V o.4,, 8 Thi -domain xprion ha zro and pol. A., B., C.,

65 LaPlac Tranform in Circui Analyi Exampl: Chck your - domain anwr: IVT FVT.4 V,,.4 lim,,.4 lim lim lim,, F v V V,,.4 lim lim lim,, F v V

66 Warning hi on ricky! Ju afr =, hr i no iniial ord nrgy in h circui. Thrfor, h capacior bhav lik a and h inducor bhav lik a. A. Opn circui/hor circui B. Opn circui/opn circui C. Shor circui/hor circui D. Shor circui/opn circui

67 LaPlac Tranform in Circui Analyi For = For v.58.4 V chck! v i i V h volag acro a wir!

68 LaPlac Tranform in Circui Analyi Exampl: Parial fracion xpanion: V.4,,, 8.4,,,

69 V,, In h parial fracion xpanion givn hr, and ar A. Boh ral numbr B. Complx conuga C. Nd mor informaion

70 LaPlac Tranform in Circui Analyi Aid find h parial fracion xpanion whn hr ar rpad ral roo. undfind! F

71 LaPlac Tranform in Circui Analyi Aid find h parial fracion xpanion whn hr ar rpad ral roo. How do w find h cofficin of h rm wih u on copy of h rpad roo? F 3 Elimina h wo rm p hi rm! d d F d d d d d d 3 = bcau h drivaiv ill ha + in h numraor = bcau h drivaiv of a conan i = 3 bcau h drivaiv of 3 + i 3

72 LaPlac Tranform in Circui Analyi Aid find h parial fracion xpanion whn hr ar rpad ral roo d d F

73 LaPlac Tranform in Circui Analyi Back o h xampl; find h parial fracion xpanion: V.4,,,,.4,, 6 d d.4,. 4,

74 LaPlac Tranform in Circui Analyi Exampl: Find v for >. Invr Laplac ranform h rul in h -domain o g h im-domain rul: 6 V,.4, v v v 6,.4 V chck! V chck!.4, u V h Laplac abl

75 v o [6,.4, ] u V W hav n hi rpon form in our analyi of cond-ordr RLC circui; i i calld: A. Ovrdampd B. Undrdampd C. Criically dampd

76 LaPlac Tranform in Circui Analyi Exampl: Thr i no iniial nrgy ord in hi circui. Find i if v =.6 in.8 V. Laplac ranform h circui: L.6 in

77 LaPlac Tranform in Circui Analyi Exampl: Find I: I I I

78 LaPlac Tranform in Circui Analyi Exampl: Chck your -domain anwr: IVT FVT. lim. lim lim lim. I i I. lim lim lim. I i I

79 LaPlac Tranform in Circui Analyi Exampl: Parial fracion xpanion: * * I

80 LaPlac Tranform in Circui Analyi Parial fracion xpanion, coninud: d d I

81 LaPlac Tranform in Circui Analyi Exampl: Thr i no iniial nrgy ord in hi circui. Find i if v =.6 in.8 V. i Invr Laplac ranform h rul in h -domain o g h im-domain rul: I co co co co.8 9 u A

82 Which rm of h oluion rprn h forcd rpon? Exampl: Thr i no iniial nrgy ord in hi circui. Find i if v =.6 in.8 V. i [.78.6 co co.8 9] u A A. Fir rm B. Scond rm C. Nihr

83 LaPlac Tranform in Circui Analyi Rcip for Laplac ranform circui analyi:. Rdraw h circui no ha you nd o find h iniial condiion and dcid how o rprn hm in h circui.. Any volag or currn wih valu givn ar Laplac-ranformd uing h funcional and opraional abl. 3. Any volag or currn rprnd ymbolically, uing i and v, ar rplacd wih h ymbol I and V. 4. All componn valu ar rplacd wih h corrponding complx impdanc, Z, and h appropria ourc rprning iniial condiion. 5. U DC circui analyi chniqu o wri h -domain quaion and olv hm. Chck your oluion wih IVT and FVT. 6. Invr-Laplac ranform -domain oluion uing h parial fracion xpanion chniqu and h Laplac abl o g im-domain oluion. Chck your oluion a = and =.

84 LaPlac Tranform in Circui Analyi L Aid How do you invr Laplac ranform F if i i an impropr raional funcion? No hi won happn in linar circui, bu can happn in ohr ym modld wih diffrnial quaion! Exampl: S nx lid! No : O{D } O{N } do no hold!

85 LaPlac Tranform in Circui Analyi 5 3 ; do no hold! } O{N } O{D No : 7 6 u - L L

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