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# Microwave Engineering EECS 723

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This 55 page Class Notes was uploaded by Melissa Metz on Monday September 7, 2015. The Class Notes belongs to EECS 723 at Kansas taught by James Stiles in Fall. Since its upload, it has received 30 views. For similar materials see /class/186778/eecs-723-kansas in Elect Engr & Computer Science at Kansas.

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Date Created: 09/07/15

EECS 723Microwave Engineering Teacher quotBoquot7 do you even know your mufpI39co on fobes BarT quot We I know of Them Like BarT and his mulTiplicaTion Tables many elecTrical engineers know of The concest of microwave engineering Concest Such as characTerisTic impedance ScaTTering parameTers SmiTh CharTs and The like are familiar buT ofTen we find ThaT a compleTe Thorough and unambiguous undersTanding of TheSe concest can be somewhaT lacking Thus The goals of This class are for you To 1 ObTain a compleTe Thorough and unambiguous undersTanding of The fundamenTaI concest on microwave engineering 2 Apply These concest To The design and analysis of useful microwave devices AlmosT all The devices we sTudy will be boTh linear and Time invarianT Thus almosT all our analysis will have aT iT rooT The maThemaTics of linear TimeinvarianT sysTems CerTainly all elecTrical engineers know of linear sysTems Theory BuT iT is helpful To firsT review TheSe concest To make Sure ThaT we all undersTand whaT This Theory is why iT works and how iT is useful FirsT we musT carefully define a linearTime invarianT sysTem HO THE LINEAR TIMEINVARIANT SYSTEM Linear sysTems Theory is USeful for microwave engineers becaUSe mosT microwave devices and sysTems are linear aT leasT approximaTely HO LINEAR CIRCUIT ELEMENTS The mosT powerful Tool for analyzing linear sysTems is iTs eigen funcTion HO THE EIGEN FUNCTION OF LINEAR SYSTEMS Complex voTages and currenTs aT Times COUSe much head scraTching leT39s make Sure we know whaT These complex values and funcTions physically mean HO A COMPLEX REPRESENTATION OF SINUSOIDAL FUNCTIONS Signals may noT have The expliciT form of an eigen funcTion buT our linear sysTems Theory allows us To relaTively easily analyze This C036 as well HO ANALYSIS OF CIRCUITS DRIVEN BY ARBITRARSI FUNCTIONS If our39 man sysfem is a linear39 circui39l39 we can apply basic cir39cuif analysis To defermine all ifs eigen values HO THE EIGEN SPECTRUM OF LINEAR CIRCUITS The Linear Time Invarianf System MOST of The microwave devices and networks lha r we will study in This course are bo rh linear and Time invarian l39 or approxima rely so Le r s make sure lha r we unders rand wha r These Terms mean as linear Timeinvarian r systems allow us To apply a large and helpful mathematical roolbox 39 A 0 a LINEARITY V gt Ma lhema ricians of len speak of opera l or s which is ma thpeak for any ma lhema rical opera rion lha r can be applied lo a single elemen l39 eg value variable vec ror matrix or function operafors operafors operafors For example a func l ion fx describes an opera rion on variable X ie fx is opera ror on X EG fyy23 97 27 yXX Moreover we find that functions can likewiSe be operated on For example integration and differentiation are likewise mathematical operations operators that operate on functions EG my dy W lexdx foo A special and very important class of operators are linear operators Linear operators are denoted as y where l symbolically denotes the mathematical operation And y denotes the element eg function variable vector being operated on A linear operator is any operator that satisfies the following two statements for any and all y 1 Y1Y2 y1 y2 2 a y a y where a is any constant Fr39om TheSe Two sTaTemenTs we can likewise conclude ThaT a linear39 oper39aTor39 has The pr39oper39Ty ay1 13 0 1b 2 wher39e boTh aand b are consTanTs ESSenTially a linear39 oper39aTor39 has The pr39oper39Ty ThaT any weighTed Sum of soluTions is also a soluTion For39 example consider The funcTion 7 97 2 27 AT7 1 g7 1212 andaTT2z gf224 NowaTT123 wefind 91223 6 24 9192 More generally we find Tha r 97 1 7227 1 72 227 1 27 2 97 197 2 and ga7 207 2027 ag7 Thus we conclude Tha r The func rion g7 27 is indeed a linear func rion Now consider this func rion yx mx b Q Buf fvaf s ve equafon of a line 7770f masf be a near fund0n Ply77 A I39m not sure Ie r39s find ou r We find Tha r yaxmaxb 0 mx b bu r ayx amxb amxab therefore yaxayX 11 Likewise yXt We mm Xob mx Hm 1 but yXtyXo quot Xl b 39Xe 1 mxmx22b therefore yXt We yXtyXe II The equation of a line is not a linear function Moreover you can show that the functions are likewise nonlinur kernernber linear operators need not be ctions Consider the derivative u operator which operates on functions dfx k Note that 0 a fX a gX fx9XJTT and also 0 0 ED fX 0 7 We thus can conclude that the derivative operation is a linear operator on function fx 0397 X T fx You can likewise show that the integration operation is likewise a linear operator If y dy 11V M But you will find that operations Such as ide are not linear operators ie they are nonlinear operators We find ThaT mosT maThemaTical operaTions are in facT non linear Linear operaTors are Thus form a small subseT of all possible maThemaTical operaTions Q Vkes I f 397ear operaTors are so rare we are we was ng our me ear77q abow Them A Two reasons Reason 1 In elecTrical engineering The behavior of mosT of our fundamenTal circuiT elemenTs are described by linear operaTors linear operaTions are prevalenT in circuiT analysis Reason 2 To our greaT relief The Two characTerisTics of linear operaTors allow us To perform TheSe maThemaTical operaTions wiTh relaTive ease Q How Is performI79 a linear opequot0770 enser Tho7 perform 3979 a nonlinear one A The SecreTquot lies is The reSUlT ay1 by2 a y1b y2 NoTe here ThaT The linear operaTion performed on a relaTiver complex elemenT ay1 byz can be deTermined immediaTely from The resulT of operaTing on The quotsimplequot elemenTs yl and y239 To see how This mighT work leT39s consider some arbiTrary funcTion of Time V7 a funcTion ThaT exisTs over some finiTe amounT of Time 739 Le v7 0 for 7 lt 0 and 7 gt 739 Say we wish To perform some linear operaTion on This funcTion ppm 9 Depending on The difficulTy of The operaTion L andor The complexiTy of The funcTion v7 direchy performing This xal operaTion could be very painful ie A approaching impossible InsTead we find ThaT we can ofTen expand a very complex and sTressful funcTion in The following way vltfgta0woltfgtw1ltrgtazwzltrgtm anew where The values a are consTanTs ie coefficienTs and The funcTions 11 7 are known as basis funcTions For example we could choose The basis funcTions 117 7 for n20 ResulTing in a polynomial of variable 7quot v7 aoqfazf2asf3mZan 7 quot0 This signal expansion is of course know as The Taylor Series expansion However There are many oTher useful expansions ie X many oTher USeful basis 91 7 The key Thing is ThaT The basis funcTions w 7 are independenT of The funcTion v7 ThaT is To say The basis funcTions are selecTed by The engineer ie you doing The analysis The SeT of SelecTed basis funcTions form whaT39s known as a basis WiTh This basis we can analyze The funcTion v7 The resulT of This analysis provides The coefficienTs a of The signal expansion Thus The coefficienTs are direchy dependenT on The form of funcTion v7 as well as The basis USed for The analysis As a reSUlT The SeT of coefficienTs 01azas compleTely describe The funcTion v7 Q I don f see why fIS quotexpoman of fund0n of v7 395 he ofu 397 jusf 00k5 76 0 07 more work 7 0 me A Consider whaT happens when we wish To perform a linear operaTion on This funcTion v7 a w 7 a Liv7m Look whaT happened InsTead of performing The linear operaTion on The arbiTrary and difficulT funcTion V7 we can apply The operaTion To each of The individual basis funcTions Q And 7 1a7 1s supposed 7 o be easien A IT depends on The linear operaTion and on The basis funcTions y 7 Hopefully The operaTion 11 7 is simple and sTraighTforward Ideally The soluTion To 11 7 is already known Q 01 yea1 ike I in going 7 o ge7 so lucky I 971 sure in a my cincuif anaysis probems evaua 7 ing L 11 7 wi be ong frusfnafing and painful A Remember you geT To choose The basis over which The funcTion V7 is analyzed A smarT engineer will choose a basis for which The operaTions 11 7 are simple and sTraighTforward Q Bu7 I m s l confused How do I choose wna7 basis y 7 7 0 use and how do I anayze 7 1e funcfion V 7 7 0 de fennine 7 1e coefficienfs a 9 A Perhaps an example would help Among The most popular basis is This one e42 O s 7 s 739 W and 27m an 7 e a so 6 039 e 27rnr V7 Zaejrj forOsrsT 7700 quotV The asfu39l39e among you will recognize This signal 5 2 A expansion as The Fourier Series V is Q Yes jusf why is Fourer anayss so 39 pre vaem A The answer reveals iTself when we apply a linear operator To The signal expansion ciao 0 i a le241 NoTe Then ThaT we musT simply evaluaTe 27 L elrlf for all n We will find ThaT performing almosT any linear oper aTion L on basis funcTions of This Type To be exceeding simple more on This laTer TIME INVARIANCE Q Thaf s rig17V You said f1a7 masf of The mcmwave devices fhaf we w sfudy are approximate0 39near39 f IrIeInvarl39anf de Vices Whaf does me invariance mean A From The sTandpoinT of a linear oper aTor iT means ThaT ThaT The operaTion is independenT of Time The resulT does noT depend on when The oper aTion is applied IE if xr y7 Then lxvir wo ir where r is a delay of any value The devices and neTworks ThaT you are abouT To sTudy in EECS 723 are in facT fixed and unchanging wiTh respecT To Time or aT leasT approximaTely so As a resulT The maThemaTical operaTions ThaT describe mosT buT noT all of our circuiT devices are boTh linear and Time invarianT operaTors We Therefore refer To These devices and neTworks as linear TimeinvarianT sysTems Linear CircuiT ElemenTs MosT microwave devices can be deSCribed or modeled in Terms of The Three sTandard circuiT elemenTs 1 RESISTANCE R W 2 INDUCTANCE L QQQ 3 CAPACITANCE C I For The purpOSes of circuiT analysis each of TheSe Three elemenTs are defined in Terms of The maThemaTical relaTionship beTween The difference in eecTric poTenTial v7 beTween The Two Terminals of The device ie The volTage across The device and The currenT 397 fowing Through The device We find ThaT for TheSe Three circuiT elemenTs The reIaTionship beTween v7 and 397 can be expresSed as a 627 gt linear operaTor 11va W M R VRU R cer Ve7 9 027 0 L iv6rerc quot V50 6 aim W g I W 0 7 w W o 1 Lima W Z I n0quot 0 7 3 L 700 v 7 L o cgixxrm W L djf 6 Since The circuiT behavior of TheSe devices can be expreSSed wiTh linear operaTors TheSe devices are referred To as linear circuit elements Q We fhaf s 5711pe enough bm whaf abom an eemem formed from a composife of 1656 fundamenfa eemenfs For example for exampe how are v7 and 39 7 reafea in Me CFCUf beOMP 397 CL II6 II 13 7 v7 29 W L R C A If Turns ouT ThaiL any circuif consfrucfed entirely with linear circuiT elemen rs is likewise a linear system Le a linear circui r As a reSUIT we know ThaiL ThaiL There must be some linear operator ThaiL rela res v7 and 397 in your example 2 1 2 V0 The circuiT above provides a good example of a singlepor l39 aka onepor39l39 nefwork We can of courSe consfrucf nefworks wifh Two or more porTs an example of a 39l39wopor39l39 network is shown below 40 I 639 fa oil I 0 147 L R Vz7 000 C O Since This circuiT is linear The relaTionship beTween all volTages and currenTs can likewiSe be expreSSed as linear operaTors eg 2191 2 V2 7 52mm 2 V2 7 222 2 V2 7 Q Vkest Whaf woua fhes e ned4 opequotafors39 for fhs CFCUI39 be How can we defernine Them A IT Turns ouT ThaT linear operaTors for all linear circuiTs can all be expreSSed in precisely The same form For example The linear operaTors of a singleporT neTwork are v7 13 7 l 937 7quot r39dr39 r yv7 l M 7 v0 0397 In oTher words The linear operaTor of linear circuiTs can always be expreSSed as a convoluTion inTegral a convoluTion wiTh a circuiT impulse funcTion 97 Q BU J39us39f whaf is fhs quotcfquotcuff I39mpus39e response quot9 A An impulse responSe is simply The response of one circuiT funcTion ie 397 or v7 due To a specific sTimulus by anoTher39 ThaT Specific sTimulus is The impulse funcTion 67 The impuISe funcTion can be defined as 77 1 5 7 50 25173 a 139 Such ThaT is has The following Two properTies 1 67 O for f 0 2 5rdr1o The impuISe responses of The oneporT example are Therefore defined as 927 i V7 7 57 and i 7 Vf f Meaning simply ThaT 920 is equal To The volfage funcTion vf when The circuiT is Thumped wiTh a impulse currem ie 1f 60 and 0 is equal To The currem ir when The circuiT is Thumped wiTh a impulse volfage ie vf 5a Similarly The relaTionship beTween The inpuT and The oquuT of a Two pom neTwork can be expressed as an aim MM WW where NEW Vi757 NoTe ThaT The circuiT impulse response musT be causal noThing can occur aT The oquuT unTil someThing occurs aT The inpuT so ThaT gf0 for fltO Q ykes I reca eVaLafing CD VDLfD infegras fa be messy difficuf and sfressful Surey fhere is an easier way fa describe iI7ear CIfCLifs A Nope The convoluTion inTegral is all There is However we can use our linear sysTems Theory Toolbox To greale simplify The evaluafion of a convoluTion inTegral The Eigen Function of Linear Time Invariant Systems Recall that that we can express expand a timelimited signal with a weighted summation of basis functions V7 Zanvn0 wher39e v7 0 for39 7 lt0 and 7 gt7quot Say now that we convolve this signal with some system impulse function 97 vr i 90 7 vow 2 j 9r 7quot go My 0 7quot 0 j 97 7quot My 0 7quot Look what happened Instead of convolving the general function v7 we now find that we must simply convolve with the set of basis functions Q Huh You say we mu5397 quotsimpy convove 7 he 53967 07 basis fund077539 V17 Why woua 7 h5 be any whiper A Remember you geT To choose The basis 117 If you39re smarT you39ll choose a seT ThaT makes The convoluTion inTegral quotsimplequot To perform Q 3717 don 7 I 7 57 need 7 0 know 7 he expI39c7 form 07 g 7 before I 3977 egem y 70056 117 A NoT necessarily The key here is ThaT The convoluTion inTegral r LivquotHF 97 7quot M7quot 0 7quot is a linear TimeinvarianT operaTor Because of This There exisTs one basis wiTh an asTonishing properTy These special basis funcTions are 6 for O s 7 s 739 917 where a O for 7 lt O 7 gt T Now inserTing This funcTion geT ready here comes The asTonishing parT inTo The convoluTion inTegral 2M I904 w dr and using the substitution u r 7 r39 we get H Haw 2W or I 9a WM 71 a 4 aw J39 u W 71 w Tow W 0 i 522 Doesn39t that ustonish Q I m mfonishedomjbyhowldu you o 2 How is this result any m quotMonshinquot than any of the onion supposedly quotusefuf things you ve been filing Us A Note that the integration in this ranquot is O convolution the 39ntegrul is simply uvulu that depends on II but not time 5aquot t W it i As a result convolution with this sp eciulquot set of basis functions can always be expre 39 ens f9 7 7U ejwnfdf ejwnr 6 a ernf gt lt gt The remarkable Thing abouT This resulT is ThaT The linear operaTion on funcTion Vn7 expjwn7 resulTs in precisely The same funcTion of Time 7 save The complex mulTiplier 6m IE libquot731 5a V17 ConvoluTion wiTh Vn7 expjwn7 is accomplished by simply multiplying The funcTion by The complex number6wn NoTe This is True regardless of The impulse response 97 The funcTion 97 affecTs The value of 6a only Q Bg dea Areff There 0 7 5 of ofher fundans fhaf woua safsfy The equa on above equa on A Nope The only funcTion where This is True is MU 6 This funcTion is Thus very special We call This funcTion The eigen funcTion of linear TimeinvarianT sysTems Q Are you sure fhaf There are no ofher egen fundans A Well sorT of Recall from Euler39s equaTion ThaT ejwnf 05 07 j 5 07 IT can be shown ThaT The sinusoidal funcTions cos 07 and 5m 07 are likewise eigen funcTions of linear TimeinvarianT sysTems The real and imaginary componenTs of eigen funcTion expjw7 are also eigen funcTions Q Whaf abet7 The 53967quot of values 6 0 9 Do They have any sgm39 cance or imporfancePP A AbsoluTely Recall The values 6a one for each n depend on The impulse response of The sysTem eg circuiT only 6a i Mowquot dr Thus The seT of values 6a compleTely characTerizes a linear TimeinvarianT circuiT over Time 0 s 7 s T We call The values 6a The eigen values of The linear TimeinvarianT circuiT g i Q 0 Pana exfer a eyeI sfuff 7715 393 3 39 Imyn be in feres ng I39fyau re a 39 39 mafhema a39an buf is if 07 a usefu fa us eecfrca engineers a A IT is unfaThomably useful To us elecTrical engineers Say a linear TimeinvarianT circuiT is exciTed only by a sinusoidal source eg vs7 cos maf Since The source funcTion is The eigen funcTion of The circuiT we will find ThaT aT every poinT in The circuiT boTh The currenT and volTage will have The same funcTional form ThaT is every currenT and volTage in The circuiT will likewise be a perfecT sinusoid wiTh frequency wall Of course The magniTude of The sinusoidal oscillaTion will be differenT aT differenT poinTs wiThin The circuiT as will The relaTive I phase BUT we know ThaT every currenT and 7 volTage in The circuiT can be precisely expressed as a funcTion of This form Acosaa7 a Q Isn f f1s preffy obvious A Why should iT be Say our source funcTion was insTead a square wave or Triangle wave or a sawTooTh wave We would find ThaT generally speaking nowhere in The circuiT would we find anoTher currenT or volTage ThaT was a perfecT square wave eTc In facT we would find ThaT noT only are The currenT and volTage funcTions wiThin The circuiT differenT Than The source funcTion eg a sawTooTh They are generally speaking all differenT from each oTher We find Then ThaT a linear circuiT will generally speaking disTorT any source funcTion unless ThaT funcTion is The eigen funcTion ie an sinusoidal funcTion Thus using an eigen funcTion as circuiT source greale simplifies our linear circuiT analysis problem All we need To accomplish This is To deTermine The magniTude A and relaTive phase 0 of The resulTing and oTherwise idenTical sinusoidal funcTion A Complex Representation of Sinusoidal Functions Q 50 you Say for exampe if a 39near twoport crcut is driven by a 539nu5390390 a Source wth arbitrary frequency a then the output w be I39dentl39cay 5nu5390390 a ony with a d fferent magntua e and rea tve phaSe o Il o 0 v1t l1cosaat 01 8 L R v2t lm2 cosaat 02 C O How do we de termne the unknown magnitude lm2 and phase 02 of this output A Say the input and output are related by the impuISe r39esponSe 9t v2t zv1r j gt r v1t390 t39 We now know that if the input were instead l17 6 Then v27 LieW 6w0 emquot where we i Dimerquot dr Thus we simply muITiply The inpuT v10 2 e b by The complex eigen value 6w0 To deTer39mine The complex oquuT v2 7 v27 6 00 emquot Q You professors a rve me crazy wfn a fns mafn nvovnq complex 1 e rea ana lmagnary vofage fundans In The ab I can only generafe and measure real valued vofages and real valued vofage fundans lofage Is a real valued physical paramefer A You are quiTe correcT VolTage is a realvalued parameTer39 expressing elecTr39ic poTenTial in Joules per39 uniT charge in Coulombs Q So a your complex formuafons and complex egen values and complex egen fundans may a be sound mafhema fl39cal absfrac ons bu7 aren 7 They worfhless To us elecfrl39cal engineers who work In The quotrealquot wora pun Infena ea A Absolu rely no r Complex analysis ac rually simplifies our analysis of realvalued vol rages and cur r en rs in linear cir cui rs bu r only for39 linear cir cui rsl The key r ela rionship comes from Euler39s Identity 5 cosan J39sinan Meaning Re 5 cos an Now consider a complex value C We of course can wr i re This complex number in Terms of if real and imaginary par rs Cajb aReC and bImC BUT we can also wr i re if in Terms of HS magnitude C and phase q wher e CCC aZbZ q ran391 Thus The complex function 65 is Ce b Cejw em lclejfw cosao7 0 j sinao7 0 Therefore we find cos 0107 0 Re CeW Now consider again The realvalued volTage funcTion l 10quot le C05 739 01 This funcTion is of courSe sinusoidal wiTh a magniTude lm1 and phOSe 1 Using whaT we have learned above we can likewise express This real function as l 10quot Z le C05 739 01 Re l1 em where V1 is The complex number V1 le 6 Q I see A real valued sinusoid has a magnfua e andphase jusf 39ke complex number A single compex number 1 can be used 7 0 specfy bafh of Me fundamenfa rea vauea parame fers of our 5397u500 lm 0 Whm I 0 07 7 see 395 how fIS heps us 397 our CFCUf anayss A ffer a39 v27 6 00 11 6 whcI men75 v27 6 00 Re l1 e quot quot Whm Men 395 16 real valued 0117 pr v27 of our fwo porf I76 fwork when 16 07pr v17 395 16 real valued 577u50b v 7 21 cos 7 1 m1 C01 7 Relle f A Lef39s go back To our39 original convolufion infegr al v20 390 r39v1r39dr39 700 If l 10quot le C 3aa7 01 Re l1 61W Then v27 I9 7 7quot Re l1 e u0 7quot Now since The impuISe funcTion 97 is realvalued This is really impor39fam if can be shown That v27 I9 7 7quot Re l1 e fl0 7quot w f I I Rej 97 7quotl1eJ quot quot0 7quot Now applying whaT we have previously learned 7 I v27 Rej 97 7quotl1eJ quot quot0 7quot Re 97 7quot eja fa f 2 Re 11 5 00 em Thus we finally can conclude The r39ealvalued oquuT v27 clue To The realvalued inpu l39 V1 Z 7111 C 3aa7 01 Re l1 emf is v27 Re 12 emquot lm2 05 0707 02 where V2 50 V1 The really important r39eSUIT here is The lasT one 0 OI V1 lml 3030107 1 8 L R V20 2 Re 600 V1 ejwaf C O The magniTude and phase of The oquuT sinusoid expressed as complex value V2 is r39eIaTed To The magniTude and phase of The inpuT sinusoid expressed as complex value 11 by The sysTem eigen value 600 6wa mix Therefore we find ThaT really ofTen in elecTr39ical engineering we 1 USe sinusoidal ie eigen funcTion sources 2 Express The voTages and cur39r39enTs cr39eaTed by TheSe sources as complex values ie noT as real funcTions of Time For39 example we mighT say 3 20quot meaning 13 20 20 21 2 v3 7 Re2o eJ erv 20 cos 007 Or IL 2 30 meaning IL 2 20 30 e gt 39L 7 Re 30 e ew quot 30 cos 007 7r Or V j meaning 14 j 10 6 2 vs 7 Re 1Oe 2ewv 10 cosaa7 7 Remember if a linear cir cuiT is exciTed by a sinusoid eg X eigen funcTion expJ39aof Then The only unknowns are The magniTude and phase of The sinusoidal currenTs and volTages associaTed wiTh each elemenT of The cir cuiT These unknowns ar e compleTely described by complex values as complex values likewise have a magniTude and phase We can always quotrecoverquot The realvalued voTage or cur r enT funcTion by muTipying The complex value by expLwof and Then Taking The real par T buT Typically we don39T afTer39 all no new or unknown infor maTion is revealed by This oper aTion c n a 639 0 v1 3L R Vz6wav1 6 5 Anal sis o Circuits riven by Arbitram Functions Q What happens if a 39near circuit is excited by some fundan that is not an quoteigen fundan quot9 Isn t binting our anayss39 fa sMusab s foo restrictive A Not as restrictive as you might think Because sinusoidal functions are the eigenfunctions of linear timeinvariant systems they have become fundamental to much of our electrical engineering infrastructure particuary with regard to communications For example every radio and TV station is assigned its very own eigen function ie its own frequency m It is very important that we use eigen functions for electromagnetic communication otherwise the received signal might look very different from the one that was transmitted WiTh sinusoidal funcTions being eigen funcTions and all we know ThaT receive funcTion will have precisely The same form as The one TransmiTTed albeiT quiTe a biT smaller Thus our aSSUmpTion ThaT a linear circuiT is exciTed by a sinusoidal funcTion is ofTen a very accuraTe and pracTical one Q 57 39 we offen find a CIfcuf ThaT 39539 nof39 driven by a 53939nu5390390 a source How woua we anayze fhl39s crcuf A Recall The properTy of linear operaTors ay1 by2 a y1b y2 We now know ThaT we can expand The funcTion v0 a0 wora1w1ra2 w27 W and we found ThaT zvr cl w 7 0 lm 7 Finally we found ThaT any linear operaTion y 7 is greale simplified if we chOOSe as our basis funcTion The eigen funcTion of linear sysTems emquot for O s 7 s 739 2 917 where a nE j O for 7 lt O 7 gt T so ThaT 11D 7 50 6M Thus for39 The example 0 var g L R W 6 5 We follow TheSe analysis sTeps 1 Expand The inpuT funcTion v17 using The basis funcTions vquot 6XPJ39wn7 00 f f f 117 lcleJ quot141er1 141er2 E l1 6 quot quot700 where 1T V771f V17 e J quotfa7 2 EvaluaTe The eigen values of The linear39 sysTem 60 2 I90 67 0quotquot 0 7 0 3 Perform The linear operaTon The convoluTion inTegr al ThaT r39elaTes v2 7 To v17 V2 7 lib10 an ejwf quot00 Z 2 K11 6aquot ejw f 1700 8 Summarizing V2 7 Z Z Vnz em wher39e Vnz 6a an and T 00 412Iv17 e quotquot0 7 6a zjg7 67 0quotquot 0 7 0 0 0 6 9 00 0 w V17 Z Vni 6W 8 L R V20 2 E 510 an emf C O As sTaTed earlier The signal expansion USed here is The Fourier Series Say ThaT The TimewidTh 701 The signal v10 becomes infiniTe In This caSe we find our analysis becomes 1 w v27 EIl2aeJ da 700 where l2w5wl1w and v1wTv1rewdr 6wT9re dr The signal expansion in This caSe is The Fourier Transform We find ThaT as 739 gt oo The number of discreTe sysTem eigen values 6a become so numerous ThaT They form a conTinuum 6w is a conTinuous funcTion of frequencya We Thus call The funcTion 60 The eigen specTrum or frequency response of The circuiT Q You claim 7 ha7 a 773 fancy ma7 hema7 cs ea egen fund0775 ana egen vaues make anayss 07 397ear sysfems ana away7 s much easier ye7 7 0 appy 7 hese 7 ecmaues we musf defermfne 7 he egen vaues 0r egen spec7 rum 6w Tg7 e quotquot0 7 6wfg7 e m 0 7 Nel39fher 0f 7 hese opequot0770775 00k of all easy And 777 add77077 7 0 performI79 7 he m 7 eqra7 077 we musf somehow a e 7 ermI7e 7 he impulse func on g 7 07 7 he 397ear sy57 em as we Jusf how are we supposed 7 0 0 0 107 A An insighTful quesTionl DeTermining The impul3e responSe g7 and Then The frequency responSe 60 does appear To be exceedingly difficulT and for many linear sysTems iT indeed is However much To our greaT relief we can deTermine The eigen specTrum 60 of linear circuiTs wiThouT having To perform a difficulT inTegraTion In facT we don39T even need To know The impu3e responSe 97 The Eigen Spectrum of Linear Circuits Recall the linear operators that define a capacitor 0 v 7 4mm W 6 05 12903 W g I W 0 7 700 We now know that the eigen function of theSe linear time invariant operators like all linear timeinvariant operartors is expja 7 The question now is what is the eigen spectrum of each of theSe operators It is this spectrum that defines the physical behavior of a given capacitor For 1167 2 expja 7 we find 20 LAC VCUH aleof 0 7 2 J39aJC39 6M JusT as we expecTed The eigen funcTion expjaf survives The linear39 oper39aTion unscaThed The cur39r39enT funcTion f has precisely The same form as The volTage funcTion 17 expLwf The only difference beTween The currenT and volTage is The mulTiplicaTion of The eigen specTrum denoTed as 65W 39 7 g 17 em 2 65m 61 Since we jusT deTer39mined ThaT for39 This caSe 397 06 61quot iT is evidenT ThaT The eigen specTr39um of The linear39 oper39aTion dV 7 I0 Lglvltrgt1 cd E 550 2 ja c a c e172 1 So for39 example if Vf 1 cos 0 p 2 Re elwewaf we will find ThaT 3 14 e e quot 550 ym enem a1639 8 VM eJ ewgf 01639 m ew em Therefore I39c 7 Re 01639 mejlw em 06 VmCOSa07 aC39 Vmsinaa7quotgp Hopefully This example again emphasizes ThaT These real valued sinusoidal funcTions can be compleTely expr39esSed in Terms of complex values For39 example The complex value 16214quoter means ThaT The magniTude of The sinusoidal volTage is lC l and iTs r39eIaTive phaSe is z 6 0 The complex value I6 V6 2 066 V6 likewise means ThaT The magniTude of The sinusoidal currenT is lIcl 6wlvcl 2061 And The r39eloTive phase of The sinusoidal currenT is 475 45 azlc 2A We can Thus summarize The behavior39 of a capaciTor wiTh The simple complex equaTion I6 J39wc c o Ic jaClc V C 066 V6 6 0 Now eT39s r39eTur39n To The second of The Two new oper39aTor39s ThaT de3cr39ibe a capaciTor W 12960 g I W 0 7 700 Now if The capaciTor currenT is The eigen funcTion I 7 expjaf we find ejwf ejwf 07 1 M 1M e where we 033ume 39 7 2 oo O Thus we can conclude That 1 jwf J06 e Hopefully if is evidem ThClT The eigen spectrum of This linear oper aTor is Leg61m 2625061507 J39aJC39 01639 01639 1 l I c J39QCC Q Waf a second I57 7 fIS essen a y 16 same F65U7 05 16 one o ervea for operaforll 7 5360 1 J39 Lem And so A IT39S precisely The same For boTh oper aTor s we find This should noT be surprising as boTh operaTors L and 1 relaTe The currenT Through and volTage across The same device a capaciTor The raTio of complex volTage To complex currenT is of course referred To as The complex device impedance Z 1 Z I An impedance can be deTermined for any linear TimeinvarianT oneporT neTwork buT only for linear TimeinvarianT oneporT neTworks Generally speaking impedance is a funcTion of frequency In facT The impedance of a oneporT neTwork is simply The eigen specTrum 630 of The linear operaTor 3 2 1 2 V0quot Z 2 620 NoTe ThaT impedance is a complex value ThaT provides us wiTh Two Things 1 The ratio of The magnitudes of The sinusoidal volTage and cur r enT Zl III 2 The difference in phase beTween The sinusoidal volTage and cur r enT AZ 2 A l AI Q Whm abow le 760quot opequotafar yv7 r 9 A Hopefully if is now evidenT To you Thai Now r eTur ning To The other 39l39wo linear cir cuiT elemenTs we find and you can verify 39l39l lCl l39 for r esisTor39s l3 1e7 e7 gt 5 w1R lRO H 1020 gt 5 w R and for inducTor s L L 1 Lyman w 2 6y w M 1 i7 W 2 62 w Jul meaning ZR RReJ and 4 L 72 Now noTe ThClT The r39eloTionship Z NIK forms a complex Ohm39s Lawquot wiTh regard To complex cur r enTs and voh oges AddiTionally ICBST IT Can Be Shown Thcn Kirchoff39s Laws are likewise valid for complex cur r enTs and volfoges 2520 340 II where of course The summoTion r epr esenTs complex addition As a resulT The impedance ie The eigen specTrum of any oneporT device can be deTermined by simply applying a basic knowledge of linear circuiT analysis ReTurning To The example I C gt I I 0 V 1 L R 2 a And Thus using ouT basic circuiTs knowledge we find Z 26 ZRZL ZM R ij Thus The eigen specTrum of The linear operaTor LED 7 V7 For This oneporT neTwork is 630 2 6 R ij Look whaT we did We were able To deTermine 62w wiThouT explicile deTermining impulse response 937 or having To perform any inTegraTions Now if we acTually need To defermine The volTage funcTion v7 creaTed by some arbitrary curremL funcTion 397 we inTegraTe v7 illon 0 Ia ejma a i I t Rquot Jam Ia elmda where I 0 T re a r OTherwise if our curremL funcTion is Timeharmonic ie sinusoidal wiTh frequency a we can simply relaTe complex curremL I and complex volTage l wiTh The equaTion VZI QCRJ39aLI Similarly for our 39l39wopor39l39 example o Il a v1 L 9 we can likewise defermine from basic circuiT Theory The eigen spectrum of linear operaTor Lab1 V2 7 ZLZR J39aJLHR Z Z Z 1 C Lquot I c JaJLHR J39w 6210 2 so l39l lCl l39l V2 521011 or more generally 1 00 39wf W I 6210 vim e1 dw 27x 700 where 140 I V17 e d7 Finally a few imporTanT definitions involving impedance and admiTTance Re Z i ResisTance R ImZ i ReacTance X Re V i Admi ance 6 Im V i Su3cepTance B Therefore ZRjX V6jB BUT be careful Alfhough 1 1 y 6 393 J R JX Z keep in mind Thcn Gabi and B R X

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