13.1 Circuit Elements in the s Domain Circuit Analysis in the s Domain The Transfer Function and Natural Response 13.

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1 Chaper 3 The Laplace Tranform in Circui Analyi 3. Circui Elemen in he Domain 3.-3 Circui Analyi in he Domain The Tranfer Funcion and Naural Repone 3.6 The Tranfer Funcion and he Convoluion Inegral 3.7 The Tranfer Funcion and he Seady- Sae Sinuoidal Repone 3.8 The Impule Funcion in Circui Analyi

2 Key poin How o repreen he iniial energy of L, C in he -domain? Why he funcional form of naural and eadyae repone are deermined by he pole of ranfer funcion H and exciaion ource X, repecively? Why he oupu of an LTI circui i he convoluion of he inpu and impule repone? How o inerpre he memory of a circui by convoluion?

3 Secion 3. Circui Elemen in he Domain. Equivalen elemen of R, L, C 3

4 4 A reior in he domain iv-relaion in he ime domain:. i R v By operaional Laplace ranform:., I R i L R i R L v L Phyical uni: in vol-econd, I in ampere-econd.

5 5 An inducor in he domain. i d d L v., 0 LI 0 I L I I L i L L i L L v L iniial curren iv-relaion in he ime domain: By operaional Laplace ranform:

6 Equivalen circui of an inducor Serie equivalen: Parallel equivalen: Thévenin Noron 6

7 A capacior in he domain iv-relaion in he ime domain: d i C v. d By operaional Laplace ranform: LC v C Lv, I C C. L i 0 C 0 iniial volage 7

8 Equivalen circui of a capacior Parallel equivalen: Serie equivalen: Noron Thévenin 8

9 Secion 3., 3.3 Circui Analyi in he Domain. Procedure. Naure repone of RC circui 3. Sep repone of RLC circui 4. Sinuoidal ource 5. MCM 6. Superpoiion 9

10 How o analyze a circui in he -domain?. Replacing each circui elemen wih i -domain equivalen. The iniial energy in L or C i aken ino accoun by adding independen ource in erie or parallel wih he elemen impedance.. Wriing & olving algebraic equaion by he ame circui analyi echnique developed for reiive nework. 3. Obaining he -domain oluion by invere Laplace ranform. 0

11 Why o operae in he -domain? I i convenien in olving ranien repone of linear, lumped parameer circui, for he iniial condiion have been incorporaed ino he equivalen circui. I i alo ueful for circui wih muliple eenial node and mehe, for he imulaneou ODE have been reduced o imulaneou algebraic equaion. I can correcly predic he impulive repone, which i more difficul in he -domain Sec. 3.8.

12 Naure repone of an RC circui Q: i, v=? Replacing he charged capacior by a Thévenin equivalen circui in he -domain. KL, algebraic equaion & oluion of I: 0 I C C0 0 R IR, I RC RC.

13 3 Naure repone of an RC circui The -domain oluion i obained by invere Laplace ranform: u e R L e R RC R L i RC RC i0 + = 0 /R, which i rue for v C 0 + = v C 0 - = 0. i = 0,which i rue for capacior become open no loop curren in eady ae.

14 Naure repone of an RC circui 3 To direcly olve v, replacing he charged capacior by a Noron equivalen in he -domain. Solve, perform invere Laplace ranform: C C 0,. R RC 0 v L RC RC e u Ri

15 Sep repone of a parallel RLC Q: i L =? v C 0 - = 0 i L 0 - = 0 5

16 Sep repone of a parallel RLC KCL, algebraic equaion & oluion of : I dc C R L Idc C, RC LC. Solve I L : I L L I dc RC LC LC 6

17 Sep repone of a parallel RLC 3 Perform parial fracion expanion and invere Laplace ranform: I L ma. 3k j4k 3k j4k i L 4u 4 40e 0e 3k j7 e 3k e co 4k j4k 7 u 3k 4 e 4co4k 3in4k u ma. c. c. u ma 7

18 8 Tranien repone due o a inuoidal ource For a parallel RLC circui, replace he curren ource by a inuoidal one: The algebraic equaion change:.,, LC RC LC I L I LC RC C I I I L R C m L m m g. co u I i m g

19 Tranien repone due o a inuoidal ource Perform parial fracion expanion and invere Laplace ranform: I L i L K j Driving frequency * K j K j Neper frequency * K K K e co K. j Damped frequency K co u. Seady-ae repone ource Naural repone RLC parameer 9

20 Sep repone of a -meh circui Q: i, i =? i 0 - = 0 i 0 - = 0 0

21 Sep repone of a -meh circui MCM, algebraic equaion & oluion: I I I I I I I I I I

22 Sep repone of a -meh circui 3 Perform invere Laplace ranform: 336 i 4 // e e u 5 A. 4 i e.4e u 5 7 A.

23 Ue of uperpoiion Given independen ource v g, i g and iniially charged C, L, v =? 3

24 4 Ue of uperpoiion: g ac alone 0., 0. 0, C R C R C C L R R C C L R g g

25 5 Ue of uperpoiion 3. 0 R Y Y Y Y C R C C C L R g. g Y Y Y R Y For convenience, define admiance marix:

26 Ue of uperpoiion: I g ac alone 4 Y Y Y Y Same marix 0 I g, Y Y Y Y Same denominaor I g. 6

27 Ue of uperpoiion: Energized L ac alone 5 Y Y Y Y 0, Y Y Y Y. Same marix Same denominaor 7

28 Ue of uperpoiion: Energized C ac alone 6 Y Y "" Y Y "" C, "" C Y Y C. Y Y Y "" The oal volage i:. 8

29 Secion 3.4, 3.5 The Tranfer Funcion and Naural Repone 9

30 Wha i he ranfer funcion of a circui? The raio of a circui oupu o i inpu in he -domain: H Y X A ingle circui may have many ranfer funcion, each correpond o ome pecific choice of inpu and oupu. 30

31 Pole and zero of ranfer funcion For linear and lumped-parameer circui, H i alway a raional funcion of. Pole and zero alway appear in complex conjugae pair. The pole mu lie in he lef half of he -plane if bounded inpu lead o bounded oupu. Im Re 3

32 Example: Serie RLC circui inpu If he oupu i he loop curren I: I C H R L C LC RC g If he oupu i he capacior volage : C H R L C LC g RC.. 3

33 How do pole, zero influence he oluion? Since Y=H X, he parial fracion expanion of he oupu Y yield a erm K/-a for each pole =a of H or X. The funcional form of he ranien naural and eady-ae repone y r and y are deermined by he pole of H and X, repecively. The parial fracion coefficien of Y r and Y are deermined by boh H and X. 33

34 Example 3.: Linear ramp exciaion Q: v o =? 50u 50/ 34

35 Example 3. Only one eenial node, ue NM: o 000 g o o 6 0, H o g H ha complex conjugae pole: 3000 j4000. g = 50/ ha repeaed real pole: = 0. 35

36 Example 3. 3 The oal repone in he -domain i: o H g The oal repone in he -domain: expanion coefficien depend on H & g j j Y r 4 0 Y pole of H: -3k j4k pole of g : 0 4. v o y r y e 3,000 co4, u u. 36

37 Example 3. 4 Seady ae componen y Toal repone = 0.33 m, impac of y r 37

38 Secion 3.6 The Tranfer Funcion and he Convoluion Inegral. Impule repone. Time invarian 3. Convoluion inegral 4. Memory of circui 38

39 39 Impule repone If he inpu o a linear, lumped-parameer circui i an impule, he oupu funcion h i called impule repone, which happen o be he naural repone of he circui:.,, h H L Y L y H H Y L X The applicaion of an impule ource i equivalen o uddenly oring energy in he circui. The ubequen releae of hi energy give rie o he naural repone.

40 40 Time invarian For a linear, lumped-parameer circui, delaying he inpu x by imply delay he repone y by a well ime invarian:.,,,,,,, u y Y L Y L y Y e X H e X H Y X e u x L X

41 Moivaion of working in he ime domain The properie of impule repone and imeinvariance allow one o calculae he oupu funcion y of a linear and ime invarian LTI circui in he -domain only. Thi i beneficial when x, h are known only hrough experimenal daa. 4

42 4 Decompoe he inpu ource x We can approximae x by a erie of recangular pule rec - i of uniform widh : By having, rec - i /- i, x converge o a rain of impule: q i q. lim rec i i i i i i x x x

43 43 Synheize he oupu y Since he circui i LTI:. 0 0 i i i i i i h x x ;,, y a x a h h i i i i i i

44 44 A, ummaion inegraion:. 0 d h x d h x y if x exend -, By change of variable u=-,. du u h u x y The oupu of an LTI circui i he convoluion of inpu and he impule repone of he circui:. d h x d h x h x y Synheize he oupu y

45 Convoluion of a caual circui For phyically realizable circui, no repone can occur prior o he inpu exciaion caual, {h =0 for <0}. Exciaion i urned on a =0, {x=0 for <0}. y x h x h d. 0 45

46 Effec of x i weighed by h The convoluion inegral y x h d 0 how ha he value of y i he weighed average of x from =0 o = [from = o =0 for x-]. 0 If h i monoonically decreaing, he highe weigh i given o he preen x. 46

47 Memory of he circui If h only la from =0 o =T, he convoluion inegral T y 0 x h d. implie ha he circui ha a memory over a finie inerval =[-T,]. If h=, no memory, oupu a only depend on x, y=x*=x, no diorion. 47

48 48 Example 3.3: RL driven by a rapezoidal ource., H i o i o. u e L h Q: v o =?

49 Example 3.3 v o 0 v i h d. Separae ino 3 inerval: 49

50 Example Since he circui ha cerain memory, v o ha ome diorion wih repec o v i. 50

51 Secion 3.7 The Tranfer Funcion and he Seady-Sae Sinuoidal Repone 5

52 How o ge inuoidal eady-ae repone by H? In Chaper 9-, we ued phaor analyi o ge eady-ae repone y due o a inuoidal inpu x Aco. If we know H, y mu be: y where H j H H j Aco, j H j e j The change of ampliude and phae depend on he ampling of H along he imaginary axi.. 5

53 53 Proof. in in co co co A A A x. in co in co A A A X, in co Y Y A H X H Y r. co.. j A H c c j Ae e j H L y j j. in co in co, where * j j j Ae j H j j A j H j A H j Y K j K j K Y

54 Obain H from H j We can revere he proce: deermine H j experimenally, hen conruc H from he daa no alway poible. Once we know H, we can find he repone o oher exciaion ource. 54

55 Secion 3.8 The Impule Funcion in Circui Analyi 55

56 E.g. Impulive inducor volage Q: v o =? i 0 - =0 A i 0 - =0 The opening of he wich force he wo inducor curren i, i change immediaely by inducing an impulive inducor volage [v=li']. 56

57 E.g. Equivalen circui & oluion in he -domain iniial curren , improper raional

58 E.g. Soluion in he -domain v0 L 60 0e u. 5 To verify wheher hi oluion v o i correc, we need o olve i a well. 00 I , 5 i 4 e 5 u. jump jump 58

59 Impulive inducor volage 4 The jump of i from 0 o 6 A caue i 6, Afer > 0 +, v 5 i o 54 e 5 H i 0e 60 0e conribuing o a volage impule L i. conien wih ha olved by Laplace ranform. 5 5, 59

60 Key poin How o repreen he iniial energy of L, C in he -domain? Why he funcional form of naural and eadyae repone are deermined by he pole of ranfer funcion H and exciaion ource X, repecively? Why he oupu of an LTI circui i he convoluion of he inpu and impule repone? How o inerpre he memory of a circui by convoluion? 60

61 Pracical Perpecive olage Surge 6

62 Why can a volage urge occur? Q: Why a volage urge i creaed when a load i wiched off? Model: A inuoidal volage ource drive hree load, where R b i wiched off a =0. Since i canno change abruply, i will jump by he amoun of i 3 0 -, volage urge occur. 6

63 Example Le o =00 rm, f =60 Hz, R a =, R b =8, X a =4. i.e. L a =X a /=09 mh, X l = i.e. L l =.65 mh. Solve v o for >0 -. To draw he -domain circui, we need o calculae he iniial inducor curren i 0 -, i

64 Seady-ae before he wiching The hree branch curren rm phaor are: I = o /R a =00/ =00 A, I = o /jx a =00/j4. =.9-90 A, I 3 = o /R b =00/8 =50 A, The line curren i: I 0 =I +I +I 3 = A. Source volage: g = o +I 0 jx l =5-.5. The wo iniial inducor curren a =0 - are: i =.9co0-90, i 0 - =0; i 0 =5.co0-6.65, i =35.4 A. 64

65 S-domain analyi The -domain circui i: I 0 = 35.4 A o g = 5-.5 rm By NM: L l =.65 mh o L I l L 0 l g R o a o L L a = 09 a 0, mh a Ll g I0Ra R L L L L 475 j0 j0 R a a l a l 65

66 Invere Laplace ranform Given o j j0, v o e 73co u. 66