1/22/2007 EECS 723 intro 2/3

Transcription

1 1/22/2007 EES 723 iro 2/3 eraily, all elecrical egieers kow of liear sysems heory. Bu, i is helpful o firs review hese coceps o make sure ha we all udersad wha his heory is, why i works, ad how i is useful. Firs, we mus carefully defie a liear-ime ivaria sysem. HO: THE INEAR, TIME-INVARIANT SYSTEM iear sysems heory is useful for microwave egieers because mos microwave devices ad sysems are liear (a leas approximaely). HO: INEAR IRUIT EEMENTS The mos powerful ool for aalyzig liear sysems is is eige fucio. HO: THE EIGEN FUNTION OF INEAR SYSTEMS omplex voages ad curres a imes cause much head scrachig; le s make sure we kow wha hese complex values ad fucios physically mea. HO: A OMPEX REPRESENTATION OF SINUSOIDA FUNTIONS Sigals may o have he explici form of a eige fucio, bu our liear sysems heory allows us o (relaively) easily aalyze his case as well. Jim Siles The Uiv. of Kasas Dep. of EES

2 1/22/2007 EES 723 iro 3/3 HO: ANAYSIS OF IRUITS DRIVEN BY ARBITRARY FUNTIONS If our liear sysem is a liear circui, we ca apply basic circui aalysis o deermie all is eige values! HO: THE EIGEN SPETRUM OF INEAR IRUITS Jim Siles The Uiv. of Kasas Dep. of EES

3 1/21/2007 The iear Time Ivaria Sysem.doc 1/13 The iear, Time- Ivaria Sysem Mos of he microwave devices ad eworks ha we will sudy i his course are boh liear ad ime ivaria (or approximaely so). e s make sure ha we udersad wha hese erms mea, as liear, ime-ivaria sysems allow us o apply a large ad helpful mahemaical oolbox! INEARITY Mahemaicias ofe speak of operaors, which is mahspeak for ay mahemaical operaio ha ca be applied o a sigle eleme (e.g., value, variable, vecor, marix, or fucio)....operaors, operaors, operaors!! For example, a fucio f ( x ) describes a operaio o variable x (i.e., f ( x ) is operaor o x ). E.G.: 2 ( ) 3 ( ) 2 ( ) f y = y g = y x = x Jim Siles The Uiv. of Kasas Dep. of EES

4 1/21/2007 The iear Time Ivaria Sysem.doc 2/13 Moreover, we fid ha fucios ca likewise be operaed o! For example, iegraio ad differeiaio are likewise mahemaical operaios operaors ha operae o fucios. E.G.,: dg ( ) f ( y ) dy y ( x ) dx d A special ad very impora class of operaors are liear operaors. iear operaors are deoed as [ y ], where: * symbolically deoes he mahemaical operaio; * Ad y deoes he eleme (e.g., fucio, variable, vecor) beig operaed o. A liear operaor is ay operaor ha saisfies he followig wo saemes for ay ad all y : 1. [ y1 + y2] = [ y1] + [ y2] 2. ay = a [ y], where a is ay cosa. Jim Siles The Uiv. of Kasas Dep. of EES

5 1/21/2007 The iear Time Ivaria Sysem.doc 3/13 From hese wo saemes we ca likewise coclude ha a liear operaor has he propery: [ ] [ ] ay1 + by2 = a y1 + b y2 where boh a ad b are cosas. Esseially, a liear operaor has he propery ha ay weighed sum of soluios is also a soluio! For example, cosider he fucio: [ ] = g ( ) = 2 A = 1: g ( = 1) = 2( 1) = 2 ad a = 2: g ( = 2) = 2( 2) = 4 Now a = 1+ 2 = 3 we fid: g ( 1+ 2) = 2( 3) = 6 = 2+ 4 = g + ( 1) g ( 2) Jim Siles The Uiv. of Kasas Dep. of EES

6 1/21/2007 The iear Time Ivaria Sysem.doc 4/13 More geerally, we fid ha: ad ( + ) = 2( + ) g ( ) g a = ( ) g ( ) = g + = 2a = = 1 2 a 2 ag ( ) Thus, we coclude ha he fucio g ( ) 2 liear fucio! Now cosider his fucio: y ( x) = mx + b = is ideed a Q: Bu ha s he equaio of a lie! Tha mus be a liear fucio, righ? A: I m o sure le s fid ou! We fid ha: ( ) ( ) y ax = m ax + b = a mx + b bu: ( ) = ( + ) ay x a mx b = amx + ab Jim Siles The Uiv. of Kasas Dep. of EES

7 1/21/2007 The iear Time Ivaria Sysem.doc 5/13 herefore: ikewise: bu: ( ) a y( x) y ax!!! ( ) ( ) y x + x = m x + x + b = mx + m x + b 1 2 ( ) + ( ) = ( + ) + ( + b ) y x y x mx b mx = mx m x2 2 b herefore: ( ) ( ) ( ) y x + x y x + y x!!! The equaio of a lie is o a liear fucio! Moreover, you ca show ha he fucios: are likewise o-liear. 2 ( ) 3 ( ) f y = y y x = x Remember, liear operaors eed o be fucios. osider he derivaive operaor, which operaes o fucios. df( x) dx f ( x ) d dx Jim Siles The Uiv. of Kasas Dep. of EES

8 1/21/2007 The iear Time Ivaria Sysem.doc 6/13 Noe ha: ad also: d d f ( x ) d g ( x ) f ( x ) g ( x) dx + = + dx dx d d f ( x) af( x) a dx = dx We hus ca coclude ha he derivaive operaio is a liear operaor o fucio f ( x ) : df( x) dx f ( x) = You ca likewise show ha he iegraio operaio is likewise a liear operaor: ( ) ( ) f y dy = f y Bu, you will fid ha operaios such as: 2 dg ( ) d y ( x) dx are o liear operaors (i.e., hey are o-liear operaors). Jim Siles The Uiv. of Kasas Dep. of EES

9 1/21/2007 The iear Time Ivaria Sysem.doc 7/13 We fid ha mos mahemaical operaios are i fac oliear! iear operaors are hus form a small subse of all possible mahemaical operaios. Q: Yikes! If liear operaors are so rare, we are we wasig our ime learig abou hem?? A: Two reasos! Reaso 1: I elecrical egieerig, he behavior of mos of our fudameal circui elemes are described by liear operaors liear operaios are prevale i circui aalysis! Reaso 2: To our grea relief, he wo characerisics of liear operaors allow us o perform hese mahemaical operaios wih relaive ease! Q: How is performig a liear operaio easier ha performig a o-liear oe?? A: The secre lies is he resul: [ ] [ ] ay1 + by2 = a y1 + b y2 Noe here ha he liear operaio performed o a relaively complex eleme ay1 + by2 ca be deermied immediaely from he resul of operaig o he simple elemes y 1 ad y 2. Jim Siles The Uiv. of Kasas Dep. of EES

10 1/21/2007 The iear Time Ivaria Sysem.doc 8/13 To see how his migh work, le s cosider some arbirary fucio of ime v ( ), a fucio ha exiss over some fiie amou of ime T (i.e., v ( ) = 0 for < 0 ad > T ). Say we wish o perform some liear operaio o his fucio: v ( ) =?? Depedig o he difficuly of he operaio, ad/or he complexiy of he fucio v ( ), direcly performig his operaio could be very paiful (i.e., approachig impossible). Isead, we fid ha we ca ofe expad a very complex ad sressful fucio i he followig way: v ( ) a ψ ( ) a ψ ( ) a ψ ( ) a ψ ( ) = = = where he values a are cosas (i.e., coefficies), ad he fucios ψ ( ) are kow as basis fucios. For example, we could choose he basis fucios: ( ) for 0 ψ = Resulig i a polyomial of variable : Jim Siles The Uiv. of Kasas Dep. of EES

11 1/21/2007 The iear Time Ivaria Sysem.doc 9/13 = = = v ( ) a a a a a This sigal expasio is of course kow as he Taylor Series expasio. However, here are may oher useful expasios ψ ). (i.e., may oher useful basis ( ) * The key hig is ha he basis fucios ( ) idepede of he fucio ( ) ψ are v. Tha is o say, he basis fucios are seleced by he egieer (i.e., you) doig he aalysis. * The se of seleced basis fucios form wha s kow as a basis. Wih his basis we ca aalyze he fucio v ( ). * The resul of his aalysis provides he coefficies a of he sigal expasio. Thus, he coefficies are direcly depede o he form of fucio v ( ) (as well as he basis used for he aalysis). As a resul, he se a, a, a, compleely describe he of coefficies { } fucio v ( )! Q: I do see why his expasio of fucio of v ( ) is helpful, i jus looks like a lo more work o me. A: osider wha happes whe we wish o perform a liear operaio o his fucio: Jim Siles The Uiv. of Kasas Dep. of EES

12 1/21/2007 The iear Time Ivaria Sysem.doc 10/13 = = = = v ( ) a ψ ( ) a ψ ( ) ook wha happeed! Isead of performig he liear operaio o he arbirary ad difficul fucio v ( ), we ca apply he operaio o each of he idividual basis fucios ψ. ( ) Q: Ad ha s supposed o be easier?? A: I depeds o he liear operaio ad o he basis fucios ψ ( ). Hopefully, he operaio [ ψ ( )] is simple ad sraighforward. Ideally, he soluio o [ ψ ( )] is already kow! Q: Oh yeah, like I m goig o ge so lucky. I m sure i all my circui aalysis problems evaluaig [ ψ ( )] will be log, frusraig, ad paiful. A: Remember, you ge o choose he basis over which he fucio v ( ) is aalyzed. A smar egieer will choose a basis for which he operaios [ ψ ( )] are simple ad sraighforward! Q: Bu I m sill cofused. How do I choose wha basis ψ ( ) o use, ad how do I aalyze he fucio v ( ) o deermie he coefficies a?? Jim Siles The Uiv. of Kasas Dep. of EES

13 1/21/2007 The iear Time Ivaria Sysem.doc 11/13 A: Perhaps a example would help. Amog he mos popular basis is his oe: ad: ψ T 2π j T e 0 T = 0 0, T 1 1 2π j T a = v ( ) ψ ( ) d = v ( ) e d T T T 0 0 So herefore: 2π j T v ( ) = a e for 0 T = The asue amog you will recogize his sigal expasio as he Fourier Series! Q: Yes, jus why is Fourier aalysis so prevale? A: The aswer reveals iself whe we apply a liear operaor o he sigal expasio: 2π 2 j π j T T v ( ) = a e = a e = = Jim Siles The Uiv. of Kasas Dep. of EES

14 1/21/2007 The iear Time Ivaria Sysem.doc 12/13 Noe he ha we mus simply evaluae: for all. e 2π j T We will fid ha performig almos ay liear operaio o basis fucios of his ype o be exceedig simple (more o his laer)! TIME INVARIANE Q: Tha s righ! You said ha mos of he microwave devices ha we will sudy are (approximaely) liear, ime-ivaria devices. Wha does ime ivariace mea? A: From he sadpoi of a liear operaor, i meas ha ha he operaio is idepede of ime he resul does o deped o whe he operaio is applied. I.E., if: he: x ( ) y ( ) = where τ is a delay of ay value. x ( τ ) = y ( τ ) Jim Siles The Uiv. of Kasas Dep. of EES

15 1/21/2007 The iear Time Ivaria Sysem.doc 13/13 The devices ad eworks ha you are abou o sudy i EES 723 are i fac fixed ad uchagig wih respec o ime (or a leas approximaely so). As a resul, he mahemaical operaios ha describe mos (bu o all!) of our circui devices are boh liear ad imeivaria operaors. We herefore refer o hese devices ad eworks as liear, ime-ivaria sysems. Jim Siles The Uiv. of Kasas Dep. of EES

16 1/21/2007 iear ircui Elemes.doc 1/6 iear ircui Elemes Mos microwave devices ca be described or modeled i erms of he hree sadard circui elemes: 1. RESISTANE (R) 2. INDUTANE () 3. APAITANE () For he purposes of circui aalysis, each of hese hree elemes are defied i erms of he mahemaical relaioship bewee he differece i elecric poeial v ( ) bewee he wo ermials of he device (i.e., he volage across he device), ad he curre i ( ) flowig hrough he device. We fid ha for hese hree circui elemes, he relaioship bewee v ( ) ad i ( ) ca be expressed as a liear operaor! R( i ) v ( ) i ( ) R Y R = R = vr( ) R i ( ) v ( ) R i ( ) R Z = = R R R + vr( ) R Jim Siles The Uiv. of Kasas Dep. of EES

17 1/21/2007 iear ircui Elemes.doc 2/6 i ( ) dv ( ) v( ) i( ) d Y = = 1 i( ) = v( ) = i( ) d Z v ( ) + + v( ) ( i ) 1 ( ) ( ) ( ) v = i = v d Y di( ) i( ) v( ) d Z = = Sice he circui behavior of hese devices ca be expressed wih liear operaors, hese devices are referred o as liear circui elemes. Q: Well, ha s simple eough, bu wha abou a eleme formed from a composie of hese fudameal elemes? For example, for example, how are v ( ) ad i ( ) relaed i he circui below?? Jim Siles The Uiv. of Kasas Dep. of EES

18 1/21/2007 iear ircui Elemes.doc 3/6 + i ( ) Z i ( ) v ( )??? = = v( ) R A: I urs ou ha ay circui cosruced eirely wih liear circui elemes is likewise a liear sysem (i.e., a liear circui). As a resul, we kow ha ha here mus be some liear i i your example! operaor ha relaes v ( ) ad ( ) Z i ( ) v ( ) = The circui above provides a good example of a sigle-por (a.k.a. oe-por) ework. We ca of course cosruc eworks wih wo or more pors; a example of a wo-por ework is show below: 1( i ) 2( i ) + + v1( ) R v2( ) Jim Siles The Uiv. of Kasas Dep. of EES

19 1/21/2007 iear ircui Elemes.doc 4/6 Sice his circui is liear, he relaioship bewee all volages ad curres ca likewise be expressed as liear operaors, e.g.: 21 v1 ( ) = v2 ( ) i ( ) v ( ) = Z i ( ) v ( ) = Z Q: Yikes! Wha would hese liear operaors for his circui be? How ca we deermie hem? A: I urs ou ha liear operaors for all liear circuis ca all be expressed i precisely he same form! For example, he liear operaors of a sigle-por ework are: ( ) = ( ) = ( ) ( ) v i g i d Z Z ( ) = ( ) = ( ) ( ) i v g v d Y Y I oher words, he liear operaor of liear circuis ca always be expressed as a covoluio iegral a covoluio wih a circui impulse fucio g ( ). Q: Bu jus wha is his circui impulse respose?? Jim Siles The Uiv. of Kasas Dep. of EES

20 1/21/2007 iear ircui Elemes.doc 5/6 A: A impulse respose is simply he respose of oe circui v ) due o a specific simulus by fucio (i.e., i ( ) or ( ) aoher. Tha specific simulus is he impulse fucio δ ( ). The impulse fucio ca be defied as: δ ( ) π si 1 τ = lim τ 0 τ π τ Such ha is has he followig wo properies: 1. δ ( ) = 0 for 0 2. δ ( ) d = 10. The impulse resposes of he oe-por example are herefore defied as: ad: gz( ) v ( ) i ( ) = δ ( ) gy( ) i ( ) v ( ) = δ ( ) Jim Siles The Uiv. of Kasas Dep. of EES

21 1/21/2007 iear ircui Elemes.doc 6/6 Meaig simply ha g ( ) is equal o he Z volage fucio v ( ) whe he circui is humped wih a impulse curre (i.e., i ( ) = δ ( ) ), ad g ( ) is equal o he Y curre i ( ) whe he circui is humped wih a impulse volage (i.e., v ( ) = δ ( ) ). Similarly, he relaioship bewee he ipu ad he oupu of a wo-por ework ca be expressed as: where: v ( ) = v ( ) = g ( ) v ( ) d g ( ) v ( ) 2 v ( ) = δ ( ) 1 Noe ha he circui impulse respose mus be causal (ohig ca occur a he oupu uil somehig occurs a he ipu), so ha: g ( ) = 0 for < 0 Q: Yikes! I recall evaluaig covoluio iegrals o be messy, difficul ad sressful. Surely here is a easier way o describe liear circuis!?! A: Nope! The covoluio iegral is all here is. However, we ca use our liear sysems heory oolbox o grealy simplify he evaluaio of a covoluio iegral! Jim Siles The Uiv. of Kasas Dep. of EES

22 1/22/2007 The Eige Fucio of iear Sysems 1/7 The Eige Fucio of iear,time-ivaria Sysems Recall ha ha we ca express (expad) a ime-limied sigal wih a weighed summaio of basis fucios: v ( ) a ψ ( ) = where v ( ) = 0 for < 0 ad > T. Say ow ha we covolve his sigal wih some sysem impulse fucio g ( ): ( ) ( ) ( ) v = g v d ( ) ψ ( ) = g a d ( ) ψ ( ) = a g d ook wha happeed! Isead of covolvig he geeral fucio v ( ), we ow fid ha we mus simply covolve wih he se of basis fucios ψ. ( ) Jim Siles The Uiv. of Kasas Dep. of EES

23 1/22/2007 The Eige Fucio of iear Sysems 2/7 Q: Huh? You say we mus simply covolve he se of basis fucios ψ ( ). Why would his be ay simpler? A: Remember, you ge o choose he basis ψ ( ). If you re smar, you ll choose a se ha makes he covoluio iegral simple o perform! Q: Bu do I firs eed o kow he explici form of g ( ) before I ielligely choose ψ ( )?? A: No ecessarily! The key here is ha he covoluio iegral: ψ ( ) ( ) ψ ( ) = g d is a liear, ime-ivaria operaor. Because of his, here exiss oe basis wih a asoishig propery! These special basis fucios are: jω e for 0 T 2π ψ ( ) = where ω = T 0 for < 0, > T Now, iserig his fucio (ge ready, here comes he asoishig par!) io he covoluio iegral: Jim Siles The Uiv. of Kasas Dep. of EES

24 1/22/2007 The Eige Fucio of iear Sysems 3/7 ω ω ( ) = j j e g e d ad usig he subsiuio u =, we ge: jω jω( u) g ( ) e d = g ( u) e du ( ) 0 jω j u ( ) ω ( ) ( ) = = e g u e du + jω jω u e g ( u) e du 0 See! Does ha asoish! Q: I m asoished oly by how lame you are. How is his resul ay more asoishig ha ay of he oher supposedly useful higs you ve bee ellig us? A: Noe ha he iegraio i his resul is o a covoluio he iegral is simply a value ha depeds o (bu o ime ): ( ω ) ( ) jω G g e d 0 As a resul, covoluio wih his special se of basis fucios ca always be expressed as: Jim Siles The Uiv. of Kasas Dep. of EES

25 1/22/2007 The Eige Fucio of iear Sysems 4/7 g e d e G e ( ) = = ( ω ) jω jω jω The remarkable hig abou his resul is ha he liear ψ = exp jω resuls i precisely he operaio o fucio ( ) [ ] same fucio of ime (save he complex muliplier ( ) I.E.: ψ ( ) G ( ω ) ψ ( ) = ovoluio wih ψ ( ) exp[ jω ] G ω )! = is accomplished by simply muliplyig he fucio by he complex G ω! umber ( ) Noe his is rue regardless of he impulse respose g ( ) (he fucio g ( ) affecs he value of G ( ω ) oly)! Q: Big deal! Are here los of oher fucios ha would saisfy he equaio above equaio? A: Nope. The oly fucio where his is rue is: j ( ) e ω ψ = This fucio is hus very special. We call his fucio he eige fucio of liear, ime-ivaria sysems. Q: Are you sure ha here are o oher eige fucios?? Jim Siles The Uiv. of Kasas Dep. of EES

26 1/22/2007 The Eige Fucio of iear Sysems 5/7 A: Well, sor of. Recall from Euler s equaio ha: j e ω = ω + ω cos j si I ca be show ha he siusoidal fucios cos ω ad si ω are likewise eige fucios of liear, ime-ivaria sysems. The real ad imagiary compoes of eige fucio exp jω are also eige fucios. [ ] Q: Wha abou he se of values G ( ω )?? Do hey have ay sigificace or imporace?? A: Absoluely! Recall he values G ( ω ) (oe for each ) deped o he impulse respose of he sysem (e.g., circui) oly: ( ω ) ( ) jω G g e d 0 Thus, he se of values G ( ω ) compleely characerizes a liear ime-ivaria circui over ime 0 T. We call he values ( ) G ω he eige values of he liear, ime-ivaria circui. Jim Siles The Uiv. of Kasas Dep. of EES

27 1/22/2007 The Eige Fucio of iear Sysems 6/7 Q: OK Poidexer, all eige suff his migh be ieresig if you re a mahemaicia, bu is i a all useful o us elecrical egieers? A: I is ufahomably useful o us elecrical egieers! Say a liear, ime-ivaria circui is excied (oly) by a siusoidal source (e.g., vs( ) = cosωo ). Sice he source fucio is he eige fucio of he circui, we will fid ha a every poi i he circui, boh he curre ad volage will have he same fucioal form. Tha is, every curre ad volage i he circui will likewise be a perfec siusoid wih frequecy ω o!! Of course, he magiude of he siusoidal oscillaio will be differe a differe pois wihi he circui, as will he relaive phase. Bu we kow ha every curre ad volage i he circui ca be precisely expressed as a fucio of his form: Acos ( ω + ϕ ) Q: Is his prey obvious? o Jim Siles The Uiv. of Kasas Dep. of EES

28 1/22/2007 The Eige Fucio of iear Sysems 7/7 A: Why should i be? Say our source fucio was isead a square wave, or riagle wave, or a sawooh wave. We would fid ha (geerally speakig) owhere i he circui would we fid aoher curre or volage ha was a perfec square wave (ec.)! I fac, we would fid ha o oly are he curre ad volage fucios wihi he circui differe ha he source fucio (e.g. a sawooh) hey are (geerally speakig) all differe from each oher. We fid he ha a liear circui will (geerally speakig) disor ay source fucio uless ha fucio is he eige fucio (i.e., a siusoidal fucio). Thus, usig a eige fucio as circui source grealy simplifies our liear circui aalysis problem. All we eed o accomplish his is o deermie he magiude A ad relaive phase ϕ of he resulig (ad oherwise ideical) siusoidal fucio! Jim Siles The Uiv. of Kasas Dep. of EES

29 1/21/2007 A omplex Represeaio of Siusoidal Fucios.doc 1/8 A omplex Represeaio of Siusoidal Fucios Q: So, you say (for example) if a liear wo-por circui is drive by a siusoidal source wih arbirary frequecy ω o, he he oupu will be ideically siusoidal, oly wih a differe magiude ad relaive phase. + + ( ) = cos( ω + ϕ ) v V 1 m1 o 1 R ( ) = V ( ω + ϕ ) v2 cos o m2 2 How do we deermie he ukow magiude V m2 ad phase ϕ 2 of his oupu? A: Say he ipu ad oupu are relaed by he impulse respose g ( ): v ( ) = v ( ) = g ( ) v ( ) d We ow kow ha if he ipu were isead: j 0 v ( ) = e ω 1 Jim Siles The Uiv. of Kasas Dep. of EES

30 1/21/2007 A omplex Represeaio of Siusoidal Fucios.doc 2/8 he: where: ω ( ) = = ( ω ) v e G e j 0 jω0 2 0 jω0 G ( ω0 ) g ( ) e d 0 j 0 Thus, we simply muliply he ipu v1 ( ) = e ω by he complex eige value G ( ω 0 ) o deermie he complex oupu v ( ): j 0 ( ) = ( ω ) v G e ω 2 0 Q: You professors drive me crazy wih all his mah ivolvig complex (i.e., real ad imagiary) volage fucios. I he lab I ca oly geerae ad measure real-valued volages ad real-valued volage fucios. Volage is a real-valued, physical parameer! A: You are quie correc. 2 Volage is a real-valued parameer, expressig elecric poeial (i Joules) per ui charge (i oulombs). Q: So, all your complex formulaios ad complex eige values ad complex eige fucios may all be soud mahemaical absracios, bu are hey worhless o us elecrical egieers who work i he real world (pu ieded)? Jim Siles The Uiv. of Kasas Dep. of EES

31 1/21/2007 A omplex Represeaio of Siusoidal Fucios.doc 3/8 A: Absoluely o! omplex aalysis acually simplifies our aalysis of real-valued volages ad curres i liear circuis (bu oly for liear circuis!). The key relaioship comes from Euler s Ideiy: jω e = cosω + j siω Meaig: Re jω { e } = cosω Now, cosider a complex value. We of course ca wrie his complex umber i erms of i real ad imagiary pars: { } { } = a + j b a = Re ad b = Im Bu, we ca also wrie i i erms of is magiude ad phase ϕ! = j e ϕ where: 2 2 = = a + b ϕ = a 1 b a 0 Thus, he complex fucio j e ω is: Jim Siles The Uiv. of Kasas Dep. of EES

32 1/21/2007 A omplex Represeaio of Siusoidal Fucios.doc 4/8 e = e jω jϕ jω 0 0 = e jω + ϕ 0 e ( ω ϕ) si ( ω ϕ) = cos + + j Therefore we fid: cos jω0 ( ω0 + ϕ) = Re{ e } Now, cosider agai he real-valued volage fucio: ( ) = cos ( ω + ϕ ) v V 1 m1 1 This fucio is of course siusoidal wih a magiude V m1 ad phase ϕ 1. Usig wha we have leared above, we ca likewise express his real fucio as: ( ) = cos ( + ϕ ) v V ω = Re Ve ω 1 m1 1 where V 1 is he complex umber: j { 1 } = j 1 V1 Vm 1e ϕ Q: I see! A real-valued siusoid has a magiude ad phase, jus like complex umber. A sigle complex umber (V ) ca be used o specify boh of he fudameal (real-valued) parameers of our siusoid ( Vm, ϕ ). Jim Siles The Uiv. of Kasas Dep. of EES

33 1/21/2007 A omplex Represeaio of Siusoidal Fucios.doc 5/8 Wha I do see is how his helps us i our circui aalysis. Afer all: j o v = G ω V e ω ( ) ( o ) ( ) 2 1 which meas: j o ( ) ( ωo ) { } v G Re V e ω 2 1 Wha he is he real-valued oupu v2( ) of our wo-por ework whe he ipu v1( ) is he real-valued siusoid: ( ) = cos( + ϕ ) v V ω = Re Ve ω 1 m1 o 1 j o { 1 }??? A: e s go back o our origial covoluio iegral: If: he: ( ) = ( ) ( ) v g v d 2 1 ( ) = cos( + ϕ ) v V ω = Re Ve ω 1 m1 o 1 j o { 1 } jω o ( ) = ( ) { } v g Re V e d 2 1 Now, sice he impulse fucio g ( ) is real-valued (his is really impora!) i ca be show ha: Jim Siles The Uiv. of Kasas Dep. of EES

34 1/21/2007 A omplex Represeaio of Siusoidal Fucios.doc 6/8 jωo ( ) = ( ) { } v g Re V e d 2 1 = ( ) 1 jωo Re g V e d Now, applyig wha we have previously leared; 2( ) = ( ) 1 jωo v Re g V e d = 1 ( ) jωo Re V g e d jωo { 1 ( ω0) e } = Re V G Thus, we fially ca coclude he real-valued oupu v2( ) due o he real-valued ipu: is: where: ( ) = cos( + ϕ ) v V ω = Re Ve ω 1 m1 o 1 j o { 1 } jωo ( ) = Re{ V e } v 2 2 ( ω ϕ ) = V cos + m2 o 2 ( ω ) V = G V 2 o 1 The really impora resul here is he las oe! Jim Siles The Uiv. of Kasas Dep. of EES

35 1/21/2007 A omplex Represeaio of Siusoidal Fucios.doc 7/8 + + ( ) = cos( ω + ϕ ) v V 1 m1 o 1 R ( ) = Re ( ω ) j o { o } v G V e ω 2 1 The magiude ad phase of he oupu siusoid (expressed as complex value V 2 ) is relaed o he magiude ad phase of he ipu siusoid (expressed as complex value V 1 ) by he sysem eige value G( ω o ): V2 V 1 = G ( ω ) Therefore we fid ha really ofe i elecrical egieerig, we: o 1. Use siusoidal (i.e., eige fucio) sources. 2. Express he volages ad curres creaed by hese sources as complex values (i.e., o as real fucios of ime)! For example, we migh say V 3 = 2.0, meaig: ( ) { ω } o V = 2.0 = 2.0e v = Re 2.0e e = 2.0 cosω j 0 j 0 j 3 3 o Jim Siles The Uiv. of Kasas Dep. of EES

36 1/21/2007 A omplex Represeaio of Siusoidal Fucios.doc 8/8 Or I = 3.0, meaig: ( ) { } ( ) o ω π j π Re 3.0 j π j ω I = = e i = e e = 3.0 cos + o Or V s = j, meaig: { } ( ) j( 2) j( 2) j o Vs j 1.0e π π ω = = vs ( ) = Re 1.0e e = 1.0 cos ωo + π 2 * Remember, if a liear circui is excied by a siusoid (e.g., eige fucio exp jω0 ), he he oly ukows are he magiude ad phase of he siusoidal curres ad volages associaed wih each eleme of he circui. * These ukows are compleely described by complex values, as complex values likewise have a magiude ad phase. * We ca always recover he real-valued volage or curre fucio by muliplyig he complex value by exp j ω0 ad he akig he real par, bu ypically we do afer all, o ew or ukow iformaio is revealed by his operaio! + + V 1 R ( ω ) V = G V 2 o 1 Jim Siles The Uiv. of Kasas Dep. of EES

37 1/21/2007 Aalysis of ircuis Drive by Arbirary Fucios.doc 1/6 Aalysis of ircuis Drive by Arbirary Fucios Q: Wha happes if a liear circui is excied by some fucio ha is o a eige fucio? Is limiig our aalysis o siusoids oo resricive? A: No as resricive as you migh hik. Because siusoidal fucios are he eige-fucios of liear, ime-ivaria sysems, hey have become fudameal o much of our elecrical egieerig ifrasrucure paricularly wih regard o commuicaios. For example, every radio ad TV saio is assiged is very ow eige fucio (i.e., is ow frequecy ω )! I is very impora ha we use eige fucios for elecromageic commuicaio, oherwise he received sigal migh look very differe from he oe ha was rasmied! j ψ ( ) e ω Jim Siles The Uiv. of Kasas Dep. of EES

38 1/21/2007 Aalysis of ircuis Drive by Arbirary Fucios.doc 2/6 Wih siusoidal fucios (beig eige fucios ad all), we kow ha receive fucio will have precisely he same form as he oe rasmied (albei quie a bi smaller). Thus, our assumpio ha a liear circui is excied by a siusoidal fucio is ofe a very accurae ad pracical oe! Q: Sill, we ofe fid a circui ha is o drive by a siusoidal source. How would we aalyze his circui? A: Recall he propery of liear operaors: [ ] [ ] ay1 + by2 = a y1 + b y2 We ow kow ha we ca expad he fucio: v ( ) a ψ ( ) a ψ ( ) a ψ ( ) a ψ ( ) = = = ad we foud ha: = = = = v ( ) a ψ ( ) a ψ ( ) Fially, we foud ha ay liear operaio ψ ( ) is grealy [ ] simplified if we choose as our basis fucio he eige fucio of liear sysems: Jim Siles The Uiv. of Kasas Dep. of EES

39 1/21/2007 Aalysis of ircuis Drive by Arbirary Fucios.doc 3/6 0 < 0 > jω e for 0 T 2π ψ ( ) = where ω = T for, T so ha: j ( ) G ( ω ) e ω ψ = Thus, for he example: + + v1( ) R v2( ) We follow hese aalysis seps: 1. Expad he ipu fucio v1 ( ) usig he basis fucios ψ ( ) = exp[ jω ]: jω jω jω jω 1 = = 1 = v ( ) V e V e V e V e where: T 1 jω V 1 v1( ) e d T = 0 Jim Siles The Uiv. of Kasas Dep. of EES

40 1/21/2007 Aalysis of ircuis Drive by Arbirary Fucios.doc 4/6 2. Evaluae he eige values of he liear sysem: ( ω ) ( ) jω G g e d = 0 3. Perform he liear operao (he covoluio iegral) v : ha relaes ( ) 2 v o ( ) 1 v ( ) v ( ) = 2 1 jω = V 1 e = jω = V 1 e = = = 1 ( ω ) V G e jω Summarizig: j v ( ) V e ω = 2 2 = where: ( ω ) V = G V 2 1 ad: T 1 jω jω V 1 = v1( ) e d T G ( ω ) g ( ) e d 0 = 0 Jim Siles The Uiv. of Kasas Dep. of EES

41 1/21/2007 Aalysis of ircuis Drive by Arbirary Fucios.doc 5/6 + + j ω v1( ) = V 1 e R = j v2( ) = G1( ω) V1 e ω = As saed earlier, he sigal expasio used here is he Fourier Series. Say ha he imewidh T of he sigal v1 ( ) becomes ifiie. I his case we fid our aalysis becomes: + 1 = jω v2( ) V2( ω ) e dω 2π where: V ( ω ) = G ( ω) V ( ω) 2 1 ad: + jω jω V ( ω ) = v ( ) e d G ( ω ) g ( ) e d = The sigal expasio i his case is he Fourier Trasform. We fid ha as T he umber of discree sysem eige values G ( ω ) become so umerous ha hey form a coiuum G ( ω ) is a coiuous fucio of frequecyω. Jim Siles The Uiv. of Kasas Dep. of EES

42 1/21/2007 Aalysis of ircuis Drive by Arbirary Fucios.doc 6/6 We hus call he fucio G ( ω ) he eige specrum or frequecy respose of he circui. Q: You claim ha all his facy mahemaics (e.g., eige fucios ad eige values) make aalysis of liear sysems ad circuis much easier, ye o apply hese echiques, we mus deermie he eige values or eige specrum: jω ( ω ) = ( ) ( ω ) ( ) G g e d 0 + jω G = g e d Neiher of hese operaios look a all easy. Ad i addiio o performig he iegraio, we mus somehow deermie he impulse fucio g ( ) of he liear sysem as well! Jus how are we supposed o do ha? A: A isighful quesio! Deermiig he impulse respose g ( ) ad he he frequecy respose G ( ω ) does appear o be exceedigly difficul ad for may liear sysems i ideed is! However, much o our grea relief, we ca deermie he eige specrum G ( ω ) of liear circuis wihou havig o perform a difficul iegraio. I fac, we do eve eed o kow he impulse respose g ( )! Jim Siles The Uiv. of Kasas Dep. of EES

43 1/21/2007 The Eige Specrum.doc 1/12 The Eige Specrum of iear ircuis Recall he liear operaors ha defie a capacior: Y dv ( ) v( ) i( ) d = = Z 1 i( ) = v( ) = i( ) d We ow kow ha he eige fucio of hese liear, imeivaria operaors like all liear, ime-ivaria exp jω. operarors is [ ] The quesio ow is, wha is he eige specrum of each of hese operaors? I is his specrum ha defies he physical behavior of a give capacior! For v ( ) exp[ jω] =, we fid: i( ) = Y v( ) jω de = d = j e ( ω ) jω Jim Siles The Uiv. of Kasas Dep. of EES

44 1/21/2007 The Eige Specrum.doc 2/12 Jus as we expeced, he eige fucio exp[ jω ] survives he liear operaio uscahed he curre fucio i ( ) has precisely he same form as he volage fucio v = exp jω. ( ) [ ] The oly differece bewee he curre ad volage is he G ω muliplicaio of he eige specrum, deoed as ( ) jω i ( ) = v ( ) = e = G ( ω ) e Y Sice we jus deermied ha for his case: ( ) = ( ω ) i j e ω j Y jω Y. i is evide ha he eige specrum of he liear operaio: dv( ) i ( ) = Y v ( ) = d is: j 2 G ( ω) = jω = ω e π Y!!! So for example, if: ( ) = cos ( ω + ϕ) v V m = Re o jϕ jωo {( Vm e ) e } we will fid ha: Jim Siles The Uiv. of Kasas Dep. of EES

45 1/21/2007 The Eige Specrum.doc 3/12 Therefore: o ( m ) ( ωo) ( m ) = jϕ jω jϕ jωo Y V e e G Y V e e = π 2 ϕ ( ω ) ( m ) j j jω e V e e ( ( )) 2 + = ωvm e e ( ϕ + π 2) i ( ) = Re ωv e e j π ϕ jω { } j jω m ( ωo ϕ π ) = ω V cos + + m = ω V si m o ( ω + ϕ) Hopefully, his example agai emphasizes ha hese realvalued siusoidal fucios ca be compleely expressed i erms of complex values. For example, he complex value: o 2 o o V = j Ve ϕ m meas ha he magiude of he siusoidal volage is ad is relaive phase is V = ϕ. V = V, m The complex value: I = G ( ω ) V = Y j π 2 ( ωe ) likewise meas ha he magiude of he siusoidal curre is: V Jim Siles The Uiv. of Kasas Dep. of EES

46 1/21/2007 The Eige Specrum.doc 4/12 I = G ( ω ) V Y m = G ( ω ) V Y = ωv Ad he relaive phase of he siusoidal curre is: I G ( ω ) V = + = π 2 + ϕ We ca hus summarize he behavior of a capacior wih he simple complex equaio: I = jω V Y ( ) ( ω ) I = j V = j π 2 ( ωe ) V + V Now le s reur o he secod of he wo liear operaors ha describe a capacior: 1 Z v ( ) = i ( ) = i ( ) d Now, if he capacior curre is he eige fucio i ( ) = exp[ jω ], we fid: Jim Siles The Uiv. of Kasas Dep. of EES

47 1/21/2007 The Eige Specrum.doc 5/12 Z where we assume i ( = ) = 0. jω 1 jω e = e d 1 = e jω jω Thus, we ca coclude ha: Z 1 = = e GZ( ω ) e e jω jω jω jω Hopefully, i is evide ha he eige specrum of his liear operaor is: G ( ω ) Z 1 j 1 j ( 3π 2 ) = = = e jω ω ω Ad so: V 1 = I jω Q: Wai a secod! Is his esseially he same resul as he oe derived for operaor Y?? A: I s precisely he same! For boh operaors we fid: V 1 = I jω Jim Siles The Uiv. of Kasas Dep. of EES

48 1/21/2007 The Eige Specrum.doc 6/12 This should o be surprisig, as boh operaors Y ad relae he curre hrough ad volage across he same device (a capacior). The raio of complex volage o complex curre is of course referred o as he complex device impedace Z. V Z I A impedace ca be deermied for ay liear, ime-ivaria oe-por ework bu oly for liear, ime-ivaria oe-por eworks! Z Geerally speakig, impedace is a fucio of frequecy. I fac, he impedace of a oe-por ework is simply he eige specrum GZ ( ω ) of he liear operaor Z : V = ZI + V I Z Z i ( ) v ( ) = Z = G ( ω ) Z Noe ha impedace is a complex value ha provides us wih wo higs: Jim Siles The Uiv. of Kasas Dep. of EES

49 1/21/2007 The Eige Specrum.doc 7/12 1. The raio of he magiudes of he siusoidal volage ad curre: V Z = I 2. The differece i phase bewee he siusoidal volage ad curre: Z = V I Q: Wha abou he liear operaor: v ( ) i ( ) Y =?? A: Hopefully i is ow evide o you ha: G Y ( ω ) = 1 1 G = Z Z ( ω ) The iverse of impedace is admiace Y: Y 1 I = Z V Now, reurig o he oher wo liear circui elemes, we fid (ad you ca verify) ha for resisors: R Y R Z R v ( ) i ( ) G ( ω ) 1 R R = R Y = R i ( ) v ( ) G ( ω ) R R = R Z = Jim Siles The Uiv. of Kasas Dep. of EES

50 1/21/2007 The Eige Specrum.doc 8/12 ad for iducors: Y v ( ) i ( ) G ( ω ) = Y = 1 jω Z i ( ) v ( ) G ( ω ) jω = Z = meaig: 1 j 0 ZR = = R = R e ad Y R 1 j ( π 2) Z = = jω = ωe Y Now, oe ha he relaioship V Z = I forms a complex Ohm s aw wih regard o complex curres ad volages. Addiioally, IBST (I a Be Show Tha) Kirchoff s aws are likewise valid for complex curres ad volages: I = 0 V = 0 where of course he summaio represes complex addiio. Jim Siles The Uiv. of Kasas Dep. of EES

51 1/21/2007 The Eige Specrum.doc 9/12 As a resul, he impedace (i.e., he eige specrum) of ay oe-por device ca be deermied by simply applyig a basic kowledge of liear circui aalysis! Reurig o he example: + I V Z = I V R Ad hus usig ou basic circuis kowledge, we fid: Z = Z + Z Z = + R jω R 1 j ω Thus, he eige specrum of he liear operaor: Z i ( ) v ( ) = For his oe-por ework is: G ( ω ) = 1 j ω + R jω Z ook wha we did! We were able o deermie G ( ω ) wihou Z explicily deermiig impulse respose g ( ) perform ay iegraios! Z, or havig o Jim Siles The Uiv. of Kasas Dep. of EES

52 1/21/2007 The Eige Specrum.doc 10/12 Now, if we acually eed o deermie he volage fucio i, we v ( ) creaed by some arbirary curre fucio ( ) iegrae: jω v ( ) = GZ ( ω) I( ω) e dω π jω = ( jω + R jω) I( ω) e dω 2π where: + jω I ( ω ) i ( ) e d = Oherwise, if our curre fucio is ime-harmoic (i.e., siusoidal wih frequecy ω ), we ca simply relae complex curre I ad complex volage V wih he equaio: V = ZI 1 ( j ω R jω ) = + Similarly, for our wo-por example: I + + V 1 R V 2 Jim Siles The Uiv. of Kasas Dep. of EES

53 1/21/2007 The Eige Specrum.doc 11/12 we ca likewise deermie from basic circui heory he eige specrum of liear operaor: is: v ( ) v ( ) = G ( ω) Z Z = = R 21 Z 1 + Z ZR jωr + jωr jω so ha: V = G ( ω) V or more geerally: where: + 1 = jω v2( ) G21( ω) V1( ω) e dω 2π + jω V ( ω) v ( ) e d = 1 1 Fially, a few impora defiiios ivolvig impedace ad admiace: Re Im { Z} { Z} Resisace Reacace R X Jim Siles The Uiv. of Kasas Dep. of EES

54 1/21/2007 The Eige Specrum.doc 12/12 Re { Y } Admiace G Im { Y } Suscepace B Therefore: Z = R + jx Y = G + jb Bu be careful! Alhough: 1 1 Y = G + jb = = R + jx Z keep i mid ha: 1 1 G ad B R X Jim Siles The Uiv. of Kasas Dep. of EES