Natural and Step Response of Series & Parallel RLC Circuits (Second-order Circuits)

Transcription

1 Naural and Sep Response of Series & Parallel RLC Circuis (Second-order Circuis) Objecives: Deermine he response form of he circui Naural response parallel RLC circuis Naural response series RLC circuis Sep response of parallel and series RLC circuis

2 Naural Response of Parallel RLC Circuis The problem given iniial energy sored in he inducor and/or capacior, find v( for.

3 I is convenien o calculae v( for his circui because A. The volage mus be coninuous for all ime B. The volage is he same for all hree componens C. Once we have he volage, i is prey easy o calculae he branch curren D. All of he above

4 Naural Response of Parallel RLC Circuis The problem given iniial energy sored in he inducor and/or capacior, find v( for. KCL : dv( C d L v( x) dx I v( R Differeniaebohsides o remove he inegral : Divide bohsides by C o placein sandardform: d v( dv( C v( d L R d d v( dv( v( d LC RC d

5 Naural Response of Parallel RLC Circuis The problem given iniial energy sored in he inducor and/or capacior, find v( for. Describing equaion: d v( d LC v( RC dv( d This equaion is Second order Homogeneous Ordinary differenial equaion Wih consan coefficiens

6 Once again we wan o pick a possible soluion o his differenial equaion. This mus be a funcion whose firs AND second derivaives have he same form as he original funcion, so a possible candidae is A. Ksin B. Ke a C. K

7 Naural Response of Parallel RLC Circuis The problem given iniial energy sored in he inducor and/or capacior, find v( for. Describing equaion: d v( d v( LC RC dv( d The circui has wo iniial condiions ha mus be saisfied, so he soluion for v( mus have wo consans. Use s s v ( A e A e V; Subsiue : ( s A e s s s s Ae ) ( s A e s RC s [ s ( RC) s ( LC)] A e A e s [ s ) ( A e LC ( RC) s s ( A e s ) LC)] A e s

8 Naural Response of Parallel RLC Circuis The problem given iniial energy sored in he inducor and/or capacior, find v( for. Describing Soluion: Where s s and s equaion: ( are v( RC) s ( d v( d s Ae A e soluions for LC) s LC he v( RC dv( d CHARACTERISTIC EQUATION:

9 is called he characerisic equaion because i characerizes he circui. A. True s B. False ( RC) s ( LC)

10 Naural Response of Parallel RLC Circuis The problem given iniial energy sored in he inducor and/or capacior, find v( for. The wo soluions o he characerisic equaion can be calculaed using he quadraic formula: s ( RC) s ( LC) ; s, ( RC) ( RC) 4( LC) s, where and ( RC) ( RC LC (he (he RC) ( LC) neper frequency in rad/s) resonan radian frequency in rad/s)

11 So far, we know ha he parallel RLC naural response is given by v( s, A e s A e A. The value of B. The value of C. The value of ( - ) s where and RC LC There are hree differen forms for s and s. For a parallel RLC circui wih specific values of R, L and C, he form for s and s depends on

12 Naural Response Overdamped Example Given V = V and I = 3 ma, find v( for.,5rad/s RC ()(.),rad/s LC (.5)(.) s, o so his is he overdampedcase!,5, 75 (,5) s (,) 5rad/s, s,rad/s

13 Naural Response Overdamped Example Given V = V and I = 3 ma, find v( for. v( Ae 5 Nowwemus use he coefficiens in he equaiono saisfy he iniialcondiionsin he circui : v( dv( d in he equaion A e, in he equaion V, for v( dv( d in he circui in he circui

14 Naural Response Overdamped Example Given V = V and I = 3 ma, find v( for. Equaion: Circui : Equaion: v() v() V A dv() d A A e 5() V 5A A e ( 5) A e,() 5(),A A A (,) A e,()

15 Now we need he iniial value of he firs derivaive of he volage from he circui. The describing equaion of which circui componen involves dv(/d? A. The resisor B. The inducor C. The capacior

16 Naural Response Overdamped Example Given V = V and I = 3 ma, find v( for. Equaion: Circui : i C dv() d ( dv() i d C 5A ( 5) A e 5A dv( C d L v() () R,A,A dv() d 5() (,) A e i C C ().3. 45, C ( i,() L () i R ()) 45, V/s

17 Naural Response Overdamped Example Given V = V and I = 3 ma, find v( for. v( A A e 5 A ; A e Solving simulaneously, Thus, v( 4e, 5 V, for 5A 4, 6e,,A 6 V, for Checks: v() 4 6 V (OK) v( ) (OK) A A 45,

18 Naural Response Overdamped Example Given V = V and I = 3 ma, find v( for. You can solve his problem using he Second-Order Circuis able:. Make sure you are on he Naural Response side.. Find he parallel RLC column. 3. Use he equaions in Row 4 o calculae and. 4. Compare he values of and o deermine he response form (given in one of he las 3 rows). 5. Use he equaions o solve for he unknown coefficiens. 6. Wrie he equaion for v(,. 7. Solve for any oher quaniies requesed in he problem.

19 Naural Response Overdamped Example Given V = V and I = 3 ma, find v( for. You can solve his problem using he Second-Order Circuis able:. Make sure you are on he Naural Response side.. Find he parallel RLC column. 3. Use he equaions in Row 4 o calculae and. 4. Compare he values of and o deermine he response form (given in one of he las 3 rows). 5. Use he equaions o solve for he unknown coefficiens. 6. Wrie he equaion for v(,. 7. Solve for any oher quaniies requesed in he problem.

20 The values of he deermine wheher he response is overdamped, underdamped, or criically damped A. Iniial condiions B. R, L, and C componens C. Independen sources

21 Naural Response of Parallel RLC Circuis The problem given iniial energy sored in he inducor and/or capacior, find v( for. Recap: B e B e v ja ja e A A e j A e j A e v x j x e x j x e e A e e A e A e A e v j s A e A e v LC RC s LC s RC s d d d d d d d d jx jx j j j j d d s s d d d d sin cos ) ( ) ( sin ) ( cos ) sin (cos ) sin (cos ) ( sin cos ; sin cos ) ( : ) ( : ; ; ) ( ) ( ) ( ) (,, ideniy : Noe Euler's where so underdamped, so overdamped, ;

22 When he response is underdamped, he volage is given by he equaion v( Be cosd Be sin In his equaion, he coefficiens B and B are d A. Real numbers B. Imaginary numbers C. Complex conjugae numbers

23 Naural Response Underdamped Example Given V = V and I =.5 ma, find v( for. rad/s RC (,)(.5) rad/s LC (8)(.5) o d v( Be B e so his is he underdampedcase! () cos B e d () cos979.8 B e sin d 979.8rad/s sin V,

24 Now we evaluae v() and dv()/d from he equaion for v(, and se hose values equal o v() and dv()/d from he circui, solving for B and B. The values for v() and dv()/d from he circui do no depend on wheher he response is overdamped, underdamped, or criically damped. A. True B. False

25 Naural Response Underdamped Example Given V = V and I =.5 ma, find v( for. Equaion: Circui : v() B e v() V B V () cos979.8() B e () sin 979.8() B

26 Naural Response Underdamped Example Given V = V and I =.5 ma, find v( for. Equaion: Circui : dv() ( ) Be d ( ) B e dvc () d (.5).5 B 979.8B 98, V/s B i C C () () B d () I C cos979.8() 979.8B e sin 979.8() 979.8B e B V R, 979.8B 98, V/s () () sin 979.8() cos979.8()

27 Naural Response Underdamped Example Given V = V and I =.5 ma, find v( for. v( B e v() B V cos B e d ; sin dv() B d B B 979.8B d I V R 98, C v( e sin V, d B

28 Naural Response of Parallel RLC Circuis The problem given iniial energy sored in he inducor and/or capacior, find v( for. Recap: s ( RC) s ( LC) ; : : :, overdamped, so v( Ae underdamped, so v( B e where s Criically damped, d so s, s ; A e s cos B e d RC sin d ; LC

29 When he response is criically damped, a reasonable expression for he volage is v( Ae A e V, A. True B. False

30 Naural Response of Parallel RLC Circuis The problem given iniial energy sored in he inducor and/or capacior, find v( for. When he circui s response is criically damped, he assumed form of he soluion we have been using up unil now does no provide enough unknown coefficiens o saisfy he wo iniial condiions from he circui. Therefore, we use a differen soluion form: : Criically damped so v( D e s D e s D e D e

31 Naural Response Criically damped Example Given V = 5 V and I = 5 ma, find v( for. RC ()() 5rad/s LC (.4)() 5rad/s o so v( his is he D e criically damped case! D e D e 5 D e 5 V,

32 Naural Response Criically damped Example Given V = 5 V and I = 5 ma, find v( for. Use he iniial condiions from he equaion and from he circui o solve for he unknown coefficiens. Equaion: Circui : v() v() V D () e 5() 5 V D e 5() D D 5

33 Naural Response Criically damped Example Given V = 5 V and I = 5 ma, find v( for. Equaion: Circui : D dv() 5() De d D 5D dvc () V ic () I d C C R , V/s 5D 75, V/s D ( 5)() e 5() D ( 5) e 5()

34 Naural Response Criically damped Example Given V = 5 V and I = 5 ma, find v( for. v( D e v() D 5 V D e 5; 5 dv() D D D 5D d I V R 78, C 5 5 v( 5,e 5e V, D 5,

35 Naural Response of Parallel RLC Circuis Summary The problem given iniial energy sored in he inducor and/or capacior, find v( for. Use he Second-Order Circuis able:. Make sure you are on he Naural Response side.. Find he parallel RLC column. 3. Use he equaions in Row 4 o calculae and. 4. Compare he values of and o deermine he response form (given in one of he las 3 rows). 5. Use he equaions o solve for he unknown coefficiens. 6. Wrie he equaion for v(,. 7. Solve for any oher quaniies requesed in he problem.

36 Sep Response of Second-order RLC Circuis The problem find he response for. Noe ha here may or may no be iniial energy sored in he inducor and capacior!

37 The circui for he parallel RLC sep response is repeaed here. Consider how his circui behaves as. Which componen s final value is nonzero? A. The resisor B. The inducor C. The capacior D. All of he above

38 Sep Response of a Parallel RLC Circuis As : The only componen whose final value is NOT zero is he inducor, whose final curren is he curren supplied by he source. We now have o consruc a response form ha can saisfy wo iniial condiions and one non-zero final value. We can saisfy he final value direcly if we specify he inducor curren as he response we will solve for: i ( L I F (he form of he naural response)

39 Sep Response of a Parallel RLC Circui The problem here is no iniial energy sored in his circui; find i( for. To begin, find he iniial condiions and he final value. The iniial condiions for his problem are boh zero; he final value is found by analyzing he circui as.

40 Sep Response of a Parallel RLC Circui The problem here is no iniial energy sored in his circui; find i( for. : I F 4mA

41 Sep Response of a Parallel RLC Circui The problem here is no iniial energy sored in his circui; find i( for. Nex, calculae he values of and and deermine he form of he response: RC (4)(5n) LC (5m)(5n) 5,rad/s 4,rad/s

42 We jus calculaed = 5, rad/s and = 4, rad/s, so he form of he response is A. Overdamped B. Underdamped C. Criically damped

43 Once we know he response form is overdamped, we know we have o calculae A. d B. s and s C. Nohing addiional

44 Sep Response of a Parallel RLC Circui The problem here is no iniial energy sored in his circui; find i( for. Since he response form is overdamped, calculae he values of s and s : s, 5, 5, 3,rad/s s,rad/s and 5, 4, 8,rad/s, 8, il(.4 Ae Ae A, s

45 Sep Response of a Parallel RLC Circui The problem here is no iniial energy sored in his circui; find i( for., 8, il(.4 Ae A e A, Nex, se he values of i() and di()/d from he equaion equal o he values of i() and di()/d from he circui. From he From he equaion: circui : i L i L () () I.4 A A

46 Sep Response of a Parallel RLC Circui The problem here is no iniial energy sored in his circui; find i( for., 8, il(.4 Ae A e A, From he From he equaion: circui : dil(),a 8,A d dil() vl() V d L L

47 Sep Response of a Parallel RLC Circui The problem here is no iniial energy sored in his circui; find i( for., 8, il(.4 Ae A e A, Solve :.4 and A i L 3mA; ( A,A 4 3e A 8,A A, 8mA 8e 8, ma,

48 i L ( 4 3e, 8e 8, ma, We can check his resul a = and as ; from he equaion we ge A. i L ()=3 ma, i L ( )= B. i L ()=, i L ( )= C. i L ()=, i L ( )=4 ma

49 i L ( 4 3e, 8e 8, ma, If we now wan o find v( for, we need o A. Find he derivaive of he curren and muliply by L B. Find he inegral of he curren and divide by C C. Muliply he curren by R

50 Sep Response of RLC Circuis Summary Use he Second-Order Circuis able:. Make sure you are on he Sep Response side.. Find he appropriae column for he RLC circui opology. 3. Find he values of he wo iniial condiions and he one non-zero final value.make sure he iniial condiions and final value are defined exacly as shown in he figure! 4. Use he equaions in Row 4 o calculae and. 5. Compare he values of and o deermine he response form (given in one of he las 3 rows). 6. Wrie he equaion for v C (, (series) or i L (, (parallel), leaving only he coefficiens unspecified. 7. Use he equaions provided o solve for he unknown coefficiens. 8. Solve for any oher quaniies requesed in he problem.

51 Sep Response of a Series RLC Circui The problem find v C ( for. Find he iniial condiions by analyzing he circui for < : V 5k 9k 5k 5 V (8 V) I A

52 Sep Response of a Series RLC Circui The problem find v C ( for. Find he final value of he capacior volage by analyzing he circui as : V F V

53 Sep Response of a Series RLC Circui The problem find v C ( for. Use he circui for o find he values of and : R L 8 (.5) 8rad/s LC (.5)(),rad/s d 6rad/s underdamped, 8

54 Sep Response of a Series RLC Circui The problem find v C ( for. v v Wrie he equaion for he response and solve for he unknown coefficiens: ( B e () dvc () d v C C C V B B F B 5 V, ( 5e 8 V B d 8 cos6 B e I C B cos V 8 B 66.67e sin 6 5 8B 8 V, sin 6 6B V,

55 Naural Response of Series RLC Circuis The problem given iniial energy sored in he inducor and/or capacior, find i( for.

56 The difference(s) beween he analysis of series RLC circui and he parallel RLC circui is/are: A. The variable we calculae. B. The describing differenial equaion. C. The equaions for saisfying he iniial condiions D. All of he above

57 Naural Response of Series RLC Circuis The problem given iniial energy sored in he inducor and/or capacior, find i( for. KVL : di( L d C i( x) dx V Ri( Differeniaebohsides o remove he inegral : Dividebohsides by L o placein sandardform: d i( di( L i( R d C d d i( R di( i( d L d LC

58 The describing differenial equaion for he series RLC circui is d i( d di( d i( LC Therefore, he characerisic equaion is R L A. s + (/RC)s + /LC = B. s + (R/L)s + /LC = C. s + (/LC)s + /RC =

59 Naural Response of Series RLC Circuis The problem given iniial energy sored in he inducor and/or capacior, find i( for. The wo soluions o he characerisic equaion can be calculaed using he quadraic formula: s ( R where and L) s ( LC) ; s, R (he neper frequency in rad/s) L (he resonan radian frequency in rad/s) LC

60 Naural Response Series RLC Problems The problem given iniial energy sored in he inducor and/or capacior, find i( for. You can solve hese problem using he Second-Order Circuis able:. Make sure you are on he Naural Response side.. Find he series RLC column. 3. Use he equaions in Row 4 o calculae and. 4. Compare he values of and o deermine he response form (given in one of he las 3 rows). 5. Use he equaions o solve for he unknown coefficiens. 6. Wrie he equaion for i(,. 7. Solve for any oher quaniies requesed in he problem.

61 Naural Response Series RLC Example The capacior is charged o V and a =, he swich closes. Find i( for. R L d LC o i( B e so 56 (.) 8 his 8rad/s (.)(.) is he 96rad/s cos96 B e,rad/s underdamped 8 case! sin 96 A,

62 Naural Response Series RLC Example The capacior is charged o V and a =, he swich closes. Find i( for. i( B e 8 Nowwemus use he coefficiens in he equaiono saisfy he iniialcondiionsin he circui : i( di( d in he equaion cos96 B e in he equaion 8 i( di( d sin 96 A, in he circui in he circui

63 The following quaniies used o calculae he unknown coefficiens are defined by differen equaions in boh he series and parallel RLC naural response problems: A. The iniial values of volage or curren from he equaion. B. The iniial values of volage or curren from he circui. C. The iniial values of he derivaive of volage or curren from he equaion. D. The iniial values of he derivaive of volage or curren from he circui.

64 Naural Response Series RLC Example The capacior is charged o V and a =, he swich closes. Find i( for. i( B e 8 Equaion: i() B Circui : i() I cos96 B e B 8 sin 96 A, (same as he parallel case!)

65 Naural Response Series RLC Example The capacior is charged o V and a =, he swich closes. Find i( for. Equaion : Circui : i( di() B d B (same as he parallel case!) d di() vl() vc () vr() V RI d L L L ( ) 56() A/s. 8() 96B B.4.4e 8 sin 96 A,

66 Naural Response of RLC Circuis Summary Use he Second-Order Circuis able:. Make sure you are on he Naural Response side.. Find he appropriae column for he RLC circui opology. 3. Make sure he iniial condiions are defined exacly as shown in he figure! 4. Use he equaions in Row 4 o calculae and. 5. Compare he values of and o deermine he response form (given in one of he las 3 rows). 6. Use he equaions o solve for he unknown coefficiens. 7. Wrie he equaion for v(, (parallel) or i(, (series). 8. Solve for any oher quaniies requesed in he problem.