3/16/2012. EE101 Review. Reference Directions

Transcription

1 3/6/ EE eew eerence Drecons uphll: baery downhll: ressor

2 3/6/ eerence Drecons Power and Energy p( ( ( Was w p( d Joules

3 3/6/ eerence Drecons urren s lowng n he passe conguraon I he curren lows oppose o he passe conguraon, he power s gen by p - essors and Ohm s Law a ab ab b The uns o ressance are ols/amp whch are called ohms. The symbol or ohms s omega: Ω 3

4 3/6/ essance elaed o Physcal Parameers ρl A ρ s he ressy o he maeral used o abrcae he ressor. The uns o resy are ohm-meers (Ω-m Power dsspaon n a ressor p 4

5 3/6/ eerence Drecons rcu Laws, olage & urren Dders 5

6 3/6/ Krchho s urren Law The ne curren enerng a node s zero. Alernaely, he sum o he currens enerng a node equals he sum o he currens leang a node. Krchho s olage Law The algebrac sum o he olages equals zero or any closed pah (loop n an elecrcal crcu. 6

7 3/6/ Seres onnecon Seres onnecon 3 3 ( 3 eq 7

8 3/6/ Parallel onnecon eq rcu Analyss usng Seres/Parallel Equalens. egn by locang a combnaon o ressances ha are n seres or parallel. Oen he place o sar s arhes rom he source.. edraw he crcu wh he equalen ressance or he combnaon ound n sep. 8

9 3/6/ 3. epea seps and unl he crcu s reduced as ar as possble. Oen (bu no always we end up wh a sngle source and a sngle ressance. 4. Sole or he currens and olages n he nal equalen crcu. olage Dson oal 3 oal 3 O he oal olage, he racon ha appears across a gen ressance n a seres crcu s he rao o he gen ressance o he oal seres ressance. 9

10 3/6/ urren Dson oal oal For wo ressances n parallel, he racon o he oal curren lowng n a ressance s he rao o he oher ressance o he sum o he wo ressances. Node/Loop Analyss

11 3/6/ Node olage Analyss Node-olage Analyss. Selec a reerence node and assgn arables or he unknown node olages. I he reerence node s chosen a one end o an ndependen olage source, one node olage s known a he sar, and ewer need o be compued.

12 3/6/ Node-olage Analyss. Wre nework equaons. Frs, use KL o wre curren equaons or nodes and supernodes. Wre as many curren equaons as you can whou usng all o he nodes. Then you do no hae enough equaons because o olage sources conneced beween nodes, use KL o wre addonal equaons. Node-olage Analyss 3. I he crcu conans dependen sources, nd expressons or he conrollng arables n erms o he node olages. Subsue no he nework equaons, and oban equaons hang only he node olages as unknowns.

13 3/6/ Node-olage Analyss 4. Pu he equaons no sandard orm and sole or he node olages. 5. Use he alues ound or he node olages o calculae any oher currens or olages o neres. Node/Loop Analyss 3

14 3/6/ Mesh urren Analyss hoosng he Mesh urrens When seeral mesh currens low hrough one elemen, we consder he curren n ha elemen o be he algebrac sum o he mesh currens. Somemes s sad ha he mesh currens are dened by soapng he wndow panes. 4

15 3/6/ hoosng he Mesh urrens Sole or he mesh currens: Mesh urren Analyss 5 ( ( 5 5

16 3/6/ Mesh urren Analyss 5 ( ( 5 Pung he equaons no he sandard orma: Super-Mesh 6

17 3/6/ ombne meshes and no a supermesh. In oher words, we wre a KL equaon around he perphery o meshes and combned. Mesh 3: ( 4( 3 3 ( ( Mesh-urren Analyss. I necessary, redraw he nework whou crossng conducors or elemens. Then dene he mesh currens lowng around each o he open areas dened by he nework. For conssency, we usually selec a clockwse drecon or each o he mesh currens, bu hs s no a requremen. 7

18 3/6/ Mesh-urren Analyss. Wre nework equaons, soppng aer he number o equaons s equal o he number o mesh currens. Frs, use KL o wre olage equaons or meshes ha do no conan curren sources. Nex, any curren sources are presen, wre expressons or her currens n erms o he mesh currens. Fnally, a curren source s common o wo meshes, wre a KL equaon or he supermesh. Mesh-urren Analyss 3. I he crcu conans dependen sources, nd expressons or he conrollng arables n erms o he mesh currens. Subsue no he nework equaons, and oban equaons hang only he mesh currens as unknowns. 8

19 3/6/ Mesh-urren Analyss 4. Pu he equaons no sandard orm. Sole or he mesh currens by use o deermnans or oher means. 5. Use he alues ound or he mesh currens o calculae any oher currens or olages o neres. Théenn Equalen rcus 9

20 3/6/ Théenn Equalen rcus Théenn Equalen rcus oc

21 3/6/ Théenn Equalen rcus sc Théenn Equalen rcus oc sc

22 3/6/ Fndng he Théenn essance Drecly We can nd he Théenn ressance by zerong he sources n he orgnal nework and hen compung he ressance beween he ermnals. When zerong a olage source, becomes a shor crcu. When zerong a curren source, becomes an open crcu. Fndng he Théenn essance Drecly

23 3/6/ Noron Equalen rcus Noron Equalen rcus I n sc 3

24 3/6/ Sep-by-sep Théenn/Noron-Equalen- rcu Analyss. Perorm wo o hese: a. Deermne he open-crcu olage oc. b. Deermne he shor-crcu curren I n sc. c. Zero he sources and nd he Théenn ressance lookng back no he ermnals.. Use he equaon I n o compue he remanng alue. 3. The Théenn equalen consss o a olage source n seres wh. 4. The Noron equalen consss o a curren source I n n parallel wh. 4

25 3/6/ Source Transormaons Maxmum Power Transer The load ressance ha absorbs he maxmum power rom a wo-ermnal crcu s equal o he Théenn ressance. 5

26 3/6/ Superposon Prncple The superposon prncple saes ha he oal response s he sum o he responses o each o he ndependen sources acng nddually. In equaon orm, hs s r r r L T r n Superposon Prncple 6

27 3/6/ Amplers asc Ampler onceps 7

28 3/6/ asc Ampler onceps Ideally, an ampler produces an oupu sgnal wh dencal waeshape as he npu sgnal, bu wh a larger amplude. ( A ( o 8

29 3/6/ Inerng ersus Non-nerng Amplers Inerng amplers hae negae olage gan, and he oupu waeorm s an nered erson o he npu waeorm. Non-nerng amplers hae pose olage gan. olage-ampler Model The npu ressance s he equalen ressance see when lookng no he npu ermnals o he ampler. o s he oupu ressance. I causes he oupu olage o decrease as he load ressance becomes smaller. A oc s he open crcu olage gan. 9

30 3/6/ olage-ampler Model Theenn equalens urren Gan A o o o L A A L 3

31 3/6/ Power Gan G Po P o I I G o Po P A A ( A L Upper case and I ndcae oo Mean Square (MS alues ascaded Amplers 3

32 3/6/ ascaded Amplers o o o o o A o A A The oerall olage gan o cascaded amplers s he produc o he gan o he nddual sages. Ths s also rue or he curren and power gans. Smpled Models or ascaded Ampler Sages Frs, deermne he olage gan o he rs sage accounng or loadng by he second sage. The oerall olage gan s he produc o he gans o he separae sages. The npu mpedance s ha o he rs sage, and he oupu mpedance s ha o he las sage. 3

33 3/6/ Op Amps Ideal Operaonal Amplers 33

34 3/6/ haracerscs o Ideal Op Amps Inne gan or he derenal npu sgnal Zero gan or he common-mode npu sgnal Inne npu mpedances Zero oupu mpedance Inne bandwdh 34

35 3/6/ Summng-Pon onsran Operaonal amplers are almos always used wh negae eedback, n whch par o he oupu sgnal s reurned o he npu n opposon o he source sgnal. Summng-Pon onsran In a negae eedback sysem, he deal opamp oupu olage aans he alue needed o orce he derenal npu olage and npu curren o zero. We call hs ac he summng-pon consran. 35

36 3/6/ Ideal op-amp crcus are analyzed by he ollowng seps:. ery ha negae eedback s presen.. Assume ha he derenal npu olage and he npu curren o he op amp are orced o zero. (Ths s he summng-pon consran. 3. Apply sandard crcu-analyss prncples, such as Krchho s laws and Ohm s law, o sole or he quanes o neres. 36

37 3/6/ The asc Inerer Applyng he Summng Pon onsran 37

38 3/6/ Inerng Ampler A o n Op-Amp rcus 38

39 3/6/ 39 Summng Ampler Summng Ampler A F A A F o A A F o F o F A A A F A A A

40 3/6/ 4 Summng Ampler Inpu ressance seen by A A Inpu ressance seen by Snce he oupu olage does no depend on he load ressance L, he oupu mpedance s zero. Non-Inerng Ampler A n o n n o o n

41 3/6/ Non-Inerng Ampler Under he deal-opamp assumpon, he non- nerng ampler s an deal olage ampler hang nne npu ressance and zero oupu ressance. A o n olage Follower A o n 4

42 3/6/ olage-o-urren onerer n o n F Inegraors and Derenaors Inegraors produce oupu olages ha are proporonal o he runnng me negral o he npu olages. In a runnng me negral, he upper lm o negraon s. 4

43 3/6/ 43 d d d q n c o c o n n c n n ( ( d o n

44 3/6/ Derenaor rcu n dq d d d n o o d d n Derenaor rcu o ( d d n 44

45 3/6/ Inducance and apacance apacor Energy s sored n he elecrc eld ha exss beween he plaes when he capacor s charged. q 45

46 3/6/ apacance o he Parallel- Plae apacor εa A WL d ε 8.85 F m ε ε rε q apacance q ( ( d q( ( ( d ( q 46

47 3/6/ 47 Sored Energy q q d d d d d p w d d p ( ( ( ( ( ( ( ( ( ( q q ( ( ( ( apacances n Parallel ( d d d d d d d d d d d d d d d d eq eq

48 3/6/ 48 apacances n Seres ( ( ( ( ( ( ( ( d d d d d Q d d d eq eq eq Inducance

49 3/6/ Inducance d ( L d L ( ( d ( d ( d ( p d L d Ld L ( w p ( ( ( L( ( d d Seres Inducances ( L eq d d L d d L d L d 3 d d ( L L L 3 d d L eq L L L 3 49

50 3/6/ 5 Parallel Inducances ( ( ( ( ( ( ( L L L L d L L L d L d L d L d L eq eq Muual Inducance Felds are adng Felds are opposng Magnec lux produced by one col lnks he oher col

51 3/6/ Frs Order Transen esponse Transens The me-aryng currens and olages resulng rom he sudden applcaon o sources, usually due o swchng, are called ransens. y wrng crcu equaons, we oban negro-derenal equaons. 5

52 3/6/ Dscharge o a apacance hrough a essance ( d ( ( d d d ( We need a uncon ( ha has he same orm as s derae. s ( Ke Subsung hs n or c ( Kse s Ke s Dscharge o a apacance hrough a essance Solng or s: s Subsung no c (: ( Ke ( ( e 5

53 3/6/ Dscharge o a apacance hrough a essance hargng a apacance rom a D Source hrough a essance d d ( ( S earrangng: d ( ( d Ths s a lnear rs-order derenal equaon wh conan coecens. S 53

54 3/6/ hargng a apacance rom a D Source hrough a essance The boundary condons are gen by he ac ha he olage across he capacance canno change nsananeously: ( ( hargng a apacance rom a D Source hrough a essance Try he soluon: ( K K e s Subsung no he derenal equaon: Ges: d ( ( d ( s K e K s S S 54

55 3/6/ hargng a apacance rom a D Source hrough a essance ( s Ke K s For equaly, he coecen o e s mus be zero: S s s Whch ges K S hargng a apacance rom a D Source hrough a essance Subsung n or K and s: ( K K e s S K e / Ealuang a and rememberng ha ( ( S Ke S K K s Subsung n or K ges: ( K K e s S S e / 55

56 3/6/ hargng a apacance rom a D Source hrough a essance τ ( e s s D Seady Sae ( d d ( In seady sae, he olage s consan, so he curren hrough he capacor s zero, so behaes as an open crcu. 56

57 3/6/ D Seady Sae L ( dl ( L d In seady sae, he curren s consan, so he olage across and nducor s zero, so behaes as a shor crcu. D Seady Sae The seps n deermnng he orced response or L crcus wh dc sources are:. eplace capacances wh open crcus.. eplace nducances wh shor crcus. 3. Sole he remanng crcu. 57

58 3/6/ /L rcus, Tme Dependen Op Amp rcus L rcus The seps noled n solng smple crcus conanng dc sources, ressances, and one energy-sorage elemen (nducance or capacance are: 58

59 3/6/. Apply Krchho s curren and olage laws o wre he crcu equaon.. I he equaon conans negrals, derenae each erm n he equaon o produce a pure derenal equaon. 3. Assume a soluon o he orm K K e s. 4. Subsue he soluon no he derenal equaon o deermne he alues o K and s. 5. Use he nal condons o deermne he alue o K. 6. Wre he nal soluon. 59

60 3/6/ L Transen Analyss Fnd ( and he olage ( ( or < snce he swch s open pror o Apply KL around he loop: S ( ( L Transen Analyss S ( ( d( ( L d S 6

61 3/6/ L Transen Analyss d( ( L d S Try ( K K e s Try ( K Ke s K ( K slk e s S S K K 5Ω S A K L slk s L Transen Analyss ( K e / L ( K e K K ( e / L 6

62 3/6/ L Transen Analyss Dene τ ( e L /τ and L rcus wh General Sources Frs order derenal equaon wh consan coecens d( L ( ( d L d( ( ( d dx( τ x( ( d Forcng uncon 6

63 3/6/ and L rcus wh General Sources The general soluon consss o wo pars. The parcular soluon (also called he orced response s any expresson ha sases he equaon. dx( τ x( d ( In order o hae a soluon ha sases he nal condons, we mus add he complemenary soluon o he parcular soluon. 63

64 3/6/ The homogeneous equaon s obaned by seng he orcng uncon o zero. dx( τ x( d The complemenary soluon (also called he naural response s obaned by solng he homogeneous equaon. Sep-by-Sep Soluon rcus conanng a ressance, a source, and an nducance (or a capacance. Wre he crcu equaon and reduce o a rs-order derenal equaon. 64

65 3/6/. Fnd a parcular soluon. The deals o hs sep depend on he orm o he orcng uncon. 3. Oban he complee soluon by addng he parcular soluon o he complemenary soluon x c Ke -/τ whch conans he arbrary consan K. 4. Use nal condons o nd he alue o K. Second Order Transen esponse 65

66 3/6/ Damped Harmonc Moon Mechancal oscllaor d x dx m b kx d d Elecrcal oscllaor d q L d dq d m q ( s ( Second Order rcus L ( d d ( ( d ( ( α ω L L s Dampenng coecen Undamped resonan requency 66

67 3/6/ Second Order rcus ( ( ( ( ( d α ω d d Derenae wh respec o me: d d ( d( α ω d s ( ( Soluon o he Second-Order Equaon Parcular soluon d d ( d( d α d d omplemenary ( d( α d ω soluon ω ( ( ( 67

68 3/6/ Soluon o he omplemenary Equaon Try s Facorng : ( s αske αs ω Ke haracersc equaon : s Ke x s ( Ke αs ω s s : ω Ke s s Soluon o he omplemenary Equaon α ζ Dampenng rao ω oos o he characersc equaon: s α α ω s α α ω 68

69 3/6/. Oerdamped case (ζ >. I ζ > (or equalenly, α > ω, he roos o he characersc equaon are real and dsnc. Then he complemenary soluon s: c s s ( K e K e x In hs case, we say ha he crcu s oerdamped.. rcally damped case (ζ. I ζ (or equalenly, α ω, he roos are real and equal. Then he complemenary soluon s c s s ( K e K e x In hs case, we say ha he crcu s crcally damped. 69

70 3/6/ 3. Underdamped case (ζ <. Fnally, ζ < (or equalenly, α < ω, he roos are complex. (y he erm complex, we mean ha he roos nole he square roo o. In oher words, he roos are o he orm: s and α jωn s α n whch j s he square roo o - and he naural requency s gen by: ω n ω α jω n For complex roos, he complemenary soluon s o he orm: x c α α ( K e ( ω K e sn( ω cos n n In hs case, we say ha he crcu s underdamped. 7

71 3/6/ rcus wh Parallel L and L d L L ( dq c d d d We can replace he crcu wh s Noron equalen and hen analyze he crcu by wrng KL a he op node: d( ( ( d L ( d L n ( rcus wh Parallel L and d( ( ( d d L derenang : ( d ( d( dn ( ( d d L d d ( d( dn ( ( d d L d L n ( 7

72 3/6/ rcus wh Parallel L and d ( d( dn ( ( d d L d Dampenng coecen α Undamped resonan requency ω L dn ( Forcng uncon ( d d ( d( α ω ( ( d d rcus wh Parallel L and d ( d( α ω ( d d ( Ths s he same equaon as we ound or he seres L crcu wh he ollowng changes or α: Parallel crcu α Seres crcu α L 7

73 3/6/ rcus wh Parallel L and p ( To nd he parcular soluon P ( (seady sae response replace wh an open crcu and L wh a shor crcu. Snusodal Sgnals, omplex Numbers, Phasors 73

74 3/6/ Snusodal urrens and olages m s he peak alue ω s he angular requency n radans per second θ s he phase angle T s he perod ( cos( ω θ m ( cos( ω θ Hambley mxes uns; ω n radans, θ n degrees m Frequency Hz cycles T sec Angular requency ω π radans T sec ω π sn o ( z cos( z 9 74

75 3/6/ oo Mean Square (MS alues rms T T ( d I rms T T ( d P ag rms P ag I rms MS alue o a Snusod rms m The rms alue or a snusod s he peak alue dded by he square roo o wo. Ths s no rue or oher perodc waeorms such as square waes or rangular waes. 75

76 3/6/ 76 Phasor Denon ( ( cos uncon : Tme θ ω : Phasor θ Phaser Phasor Arhmec sn(45 5cos( sn(6 cos(6 6 j j j Z Z j j j Z Z Z Z Z Z j j j Z j j j Z o o o o o o o o o o o o

77 3/6/ Addng Snusods Usng Phasors Sep : Deermne he phasor or each erm. Sep : Add he phasors usng complex arhmec. Sep 3: oner he sum o polar orm. Sep 4: Wre he resul as a me uncon. Phase elaonshps To deermne phase relaonshps rom a phasor dagram, consder he phasors o roae counerclockwse. Then when sandng a a xed pon, arres rs ollowed by aer a roaon o θ, we say ha leads by θ. Alernaely, we could say ha lags by θ. (Usually, we ake θ as he smaller angle beween he wo phasors. 77

78 3/6/ Phase elaonshps To deermne phase relaonshps beween snusods rom her plos ersus me, nd he shores me neral p beween pose peaks o he wo waeorms. Then, he phase angle s θ ( p /T 36. I he peak o ( occurs rs, we say ha ( leads ( or ha ( lags (. omplex Impedances-Inducor L ( L m sn( ω θ dl ( ( L d ωl m cos( ω θ I Z L L L m ωl θ 9 m θ ( ωl 9 I jωl ωl 9 L jωli L L Z L I L 78

79 3/6/ omplex Impedances-Inducor jωl I L L Z L jωl ωl 9 o L Z I L L 79

80 3/6/ omplex Impedances-apacor Z I Z j ω j ω ω o 9 I 8

81 3/6/ Impedances-essor I 8

82 3/6/ Fourer Analyss, Low Pass Flers, Decbels Fourer Analyss 8

83 3/6/ Fourer Analyss All real-world sgnals are sums o snusodal componens hang arous requences, ampludes, and phases. Fourer Analyss 4A 4A 4A sn( ω sn(3ω sn(5ω... π 3π 5π sq ( 83

84 3/6/ Flers Flers process he snusod componens o an npu sgnal derenly dependng o he requency o each componen. Oen, he goal o he ler s o rean he componens n ceran requency ranges and o rejec componens n oher ranges. Flers 84

85 3/6/ Transer Funcons The ranser uncon H( o he wo-por ler s dened o be he rao o he phasor oupu olage o he phasor npu olage as a uncon o requency: H ( ou n Transer Funcons The magnude o he ranser uncon shows how he amplude o each requency componen s aeced by he ler. Smlarly, he phase o he ranser uncon shows how he phase o each requency componen s aeced by he ler. 85

86 3/6/ Transer Funcons Magnude Phase Deermnng he oupu o a ler or an npu wh mulple componens:. Deermne he requency and phasor represenaon or each npu componen.. Deermne he (complex alue o he ranser uncon or each componen. 86

87 3/6/ 3. Oban he phasor or each oupu componen by mulplyng he phasor or each npu componen by he correspondng ranser-uncon alue. 4. oner he phasors or he oupu componens no me uncons o arous requences. Add hese me uncons o produce he oupu. 87

88 3/6/ Lnear crcus behae as hey:. Separae he npu sgnal no componens hang arous requences.. Aler he amplude and phase o each componen dependng on s requency. 3. Add he alered componens o produce he oupu sgnal. Frs-Order Low Pass Fler n I jπ n ou I jπ jπ jπ ou H ( jπ j( / n π Hal power requency 88

89 3/6/ Frs-Order Low Pass Fler H H ( ( j ( o ( arcan ( ( H arcan Frs-Order Low Pass Fler For low requency sgnals he magnude o he ranser uncon s uny and he phase s. Low requency sgnals are passed whle hgh requency sgnals are aenuaed and phase shed. 89

90 3/6/ 9 Frs-Order Low Pass Fler L j L j L j H j L j L n ou n ou n π π π π π / ( ( ( ( I I ( ( ( H H H n ou n ou ou ou n ou n ou

91 3/6/ ascaded Two-Por Neworks ( H ( H ( H ( H( H ( d d d H ode Plo, Hgh Pass Fler 9

92 3/6/ Frs-Order Low Pass Fler H H ( ( j ( o ( arcan ( ( H arcan ode Plo or a Frs-Order Low- Pass Fler A ode plo shows he magnude o a nework uncon n decbels ersus requency usng a logarhmc scale or requency. H ( H ( log d ( 9

93 3/6/ ode Plo or a Frs-Order Low- Pass Fler H H ( ( ( log H ( d log( log log log log log ( ( ( ( ( / ( ( Asympoc ehaor o Magnude or Low and Hgh Frequences H For For ( d << >> log H H ( d log( ( log log d 93

94 3/6/ Magnude ode Plo or Frs- Order Low Pass Fler Asympoc ehaor o Phase or Low and Hgh Frequences H H ( ( Tan 9 45 j ( ( ( << >> o Tan ( 94

95 3/6/ Phase ode Plo or Frs-Order Low Pass Fler. A horzonal lne a zero or < /.. A slopng lne rom zero phase a / o 9 a. 3. A horzonal lne a 9 or >. 95

96 3/6/ Frs-Order Hgh-Pass Fler ou ou n j π j( / j( / n j π where n π jπ jπ n H H Frs-Order Hgh-Pass Fler ( H ( ( ou n j ( ( ( j( 9 Tan o ( 9 o o ( 9 ( Tan ( Tan ( 96

97 3/6/ Frs-Order Hgh-Pass Fler H ( ( o H ( 9 Tan ( ( Frs-Order Hgh-Pass Fler H H ( ( ( log H ( d log( log( log( log log log log ( ( ( ( ( ( / 97

98 3/6/ Asympoc ehaor o Frs-Order Hgh-Pass Fler H For For H For For ( ( log( log ( << H ( log( >> H ( log( log( ( 9 << >> o arcan H H ( 9 o ( o ode Plos or he Frs-Order Hgh- Pass Fler For << For >> H( log( H ( log( log( ( H ( log( log ( For << H( log( For >> H ( log( log( o H ( 9 arcan o For << H ( 9 o For >> H ( H ( ( log( log ( For << H ( log( For >> H ( log( log( H ( For << For >> o 9 arcan H H ( 9 o ( o 98

99 3/6/ 99 Frs-Order Hgh-Pass Fler / ( / ( ( / / ( / ( / / n ou n n n ou j j H L where j j L j L j L j L j π π π π π Hgh Pass Flers, nd Order Flers, Ace Flers, esonances

100 3/6/ Seres esonance Z s ( jπl j π For resonance For he reacance resonanceo : he nducor and he capacor cancel: π L π π L (π L Seres esonance Qualy acor Q S Q Subsue L Q s s eacance o π (π L ( π nducance a resonance essance rom π L

101 3/6/ Seres esonance jq Z L Q and L Subsue L L j L j j L j Z s s s s ( ( ( π π π π π π π π Seres esonance Q Tan Z Q Z s s s

102 3/6/ jq jq jq Z s s s s s s s s / ( I I Seres esonan and-pass Fler ( jq s s Seres esonan and-pass Fler

103 3/6/ Seres esonan and-pass Fler Seres esonan and-pass Fler H L Q s L H 3

104 3/6/ Parallel esonance Z p ( j π j( π L A resonance Z P s purely resse: ( π L jπ j π L Parallel esonance Qualy acor Q P Q Subsue L Q P P eacance π L (π ( π o nducance a resonance essance rom π L 4

105 3/6/ 5 Parallel esonance ( ( jq L jq L j L j j L j j Z P P P ( ( π π π π π π π Parallel esonance jq Z P P ou I I ou or consan curren, aryng he requency

106 3/6/ 6 Ideal Flers Second Order Low-Pass Fler jq jq L L j j H L L j j j L j j Z Z Z Z S S n ou n n n L ou / ( ( π π π π π π π π π

107 3/6/ 7 Second-Order Low-Pass Fler ( ( ( ( ( ( ( ( ( 9 Q Q H Q Tan Q Q jq jq H S s s S s s s o n o u Second Order Low-Pass Fler

108 3/6/ Second Order Hgh-Pass Fler A low requency he capacor s an open crcu A hgh requency he capacor s a shor and he nducor s open Second Order and-pass Fler A low requency he capacor s an open crcu A hgh requency he nducor s an open crcu 8

109 3/6/ 9 Second Order and-ejec Fler A low requency he capacor s an open crcu A hgh requency he nducor s an open crcu Frs-Order Low-Pass Fler o j j Z Z H j Z j j Z Z Z H π π π π π / ( ( ( A low-pass ler wh a dc gan o - /

110 3/6/ Frs-Order Hgh-Pass Fler o j j j j j j j Z Z H Z j Z Z Z H π π π π π π π / ( / ( ( ( A hgh-pass ler wh a hgh requency gan o - / Hgher Order Flers n n n n j H H H H / ( ( ( ( ( (

111 3/6/ uerworh Transer Funcon uerworh lers are characerzed by hang a parcularly la pass-band. H ( H ( n Magnec rcus, Maerals

112 3/6/ Magnec Feld Lnes Magnec elds can be sualzed as lnes o lux ha orm closed pahs The lux densy ecor s angen o he lnes o lux Magnec lux densy gh-hand ule

113 3/6/ Flux Lnkages and Faraday s Law Magnec lux passng hrough a surace area A: φ da A For a consan magnec lux densy perpendcular o he surace: A The lux lnkng a col wh N urns: λ Nφ Faraday s Law Faraday s law o magnec nducon: e dλ d The olage nduced n a col wheneer s lux lnkages are changng. hanges occur rom: Magnec eld changng n me ol mong relae o magnec eld 3

114 3/6/ Lenz s Law Lenz s law saes ha he polary o he nduced olage s such ha he olage would produce a curren (hrough an exernal ressance ha opposes he orgnal change n lux lnkages. Lenz s Law 4

115 3/6/ Magnec Feld Inensy and Ampère s Law µ H H 7 Magnec W A b m µ 4π µ µ r elae µ Ampère s Law: H dl eld permeably nensy Ampère s Law The lne negral o he magnec eld nensy around a closed pah s equal o he sum o he currens lowng hrough he surace bounded by he pah. 5

116 3/6/ 6 Magnec Feld Around a Long Sragh Wre r I H r I H I r H Hl π µ µ π π Flux Densy n a Torodal ore NI NI H NI H Hl π µ π π

117 3/6/ Flux Densy n a Torodal ore φ λ A µ NIr Nφ µ NI πr π µ N Ir Magnec rcus In many engneerng applcaons, we need o compue he magnec elds or srucures ha lack sucen symmery or sragh-orward applcaon o Ampère s law. Then, we use an approxmae mehod known as magnec-crcu analyss. 7

118 3/6/ 8 magneomoe orce (mm o an N-urn curren-carryng col I N F relucance o a pah or magnec lux µa l F φ Analog: olage (em Analog: essance Analog: Ohm s Law Magnec rcu or Torodal ol I Nr NI r r A l r A l µ φ µ π π µ µ π π F F

119 3/6/ Adanage o he Magnec-rcu Approach The adanage o he magnec-crcu approach s ha can be appled o unsymmercal magnec cores wh mulple cols. A Magnec rcu wh elucances n Seres and Parallel Fnd he lux densy n each gap 9

120 3/6/ a a a a a a c b a a b c b a b a oal c b a c oal A A dder curren N φ φ φ φ φ φ φ ( A Magnec rcu wh elucances n Seres and Parallel Muual Inducance & Transormers

121 3/6/ Inducance and Muual Inducance Denon o nducance L: L Flux lnkages curren λ Subsue or he lux lnkages usng λ Nφ L Nφ Inducance and Muual Inducance Subsung L φ N N

122 3/6/ Faraday s Law olage s nduced n a col when s lux lnkages change: dλ d( L e d d L d d Muual Inducance Sel nducance or col L λ Sel nducance or col L λ Muual nducance beween cols and : M λ λ

123 3/6/ Muual Inducance Toal luxes lnkng he cols: λ λ λ λ ± λ ± λ Muual Inducance urrens enerng he doed ermnals produce adng luxes 3

124 3/6/ rcu Equaons or Muual Inducance λ λ ± M e e L ± M L dλ d d L ± M d d d dλ d d ± M L d d d Transormers an be used o sep up or sep down ac olages 4

125 3/6/ 5 ( ( N N ( ( N N ( ( p p Ideal Transormers Mechancal Analog ( ( ( ( F l l F N N l l N N

126 3/6/ Impedance Transormaons Z N L Z L I N 6