ECE Connections: What do Roots of Unity have to do with OP-AMPs? Louis Scharf, Colorado State University PART 1: Why Complex?
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1 ECE Conncion: Wha do Roo of Uni hav o do wih OP-AMP? Loui Scharf, Colorado Sa Univri PART : Wh Compl?. Curioi, M favori curioi i : π π ( ) π π ECE Conncion: Colorado Sa Univri Ocobr 007
2 . Quion, Bu wha abou h quion bfor u, If vrhing w build and maur i ral, wh do w u compl anali? And, wha do h compl roo of uni hav o do wih ral OP-AMP? 3. Anwr. Y, vrhing w build and maur i ral. Bu vrhing w build ha wo ral channl (ihr acual or virual) and h wo channl ar convninl rprnd wih on compl channl. Thi i h boomlin anwr. For man of h ampl o follow, compl anali implifi our lif. For on of h ampl, hr would b no lif wihou compl anali. ECE Conncion: Colorado Sa Univri Ocobr 007
3 4. Th Gnral Ida, Rplac wo ral channl wih on compl channl. : ral z : compl : ral z + θ r z z Th ral par of cod for (or and for) h -channl, and h imaginar par of cod for h -channl. Wha do h doubl arrow man? (, ) z : z + r θ : r +, θ arcan / (, ) z : R z z + z *, z z * ; r coθ, r inθ z + i h unral opraor. Or, i i h unimaginabl opraor? ECE Conncion: Colorado Sa Univri Ocobr 007
4 z : 4. Th Gnral Ida (coninud): : ral : ral compl z + Ral mporal mod (or ignvcor) Ral mporal mod (or ignvcor) coω inω Compl mporal mod (or ignvcor) coω + inω ω : ral : ral z : compl Ral paial mod (or -coordina of a fild Ral paial mod (or -coordina of a fild Compl rprnaion of a dimnional fild : ral : ral z : compl Ral caual par of a ignal Ral ani-caual par of a ignal Compl rprnaion of a wo-idd ignal ECE Conncion: Colorado Sa Univri Ocobr 007
5 5. Eampl : Phaor (ECE 0, 3, 33, 34, 444, 457). Phaor rprn ral ignal. ( On phaor rp. Aco( ω R A θ + θ ) ω ω θ A Acoθ coω A θ ω + A Ainθ inω θ +ω ω ω A θ Two phaor rp. coω A θ u coθ v inθ inω ( Ral modulaor rp. w u + ECE Conncion: Colorado Sa Univri Ocobr 007 ω v co ω + z( inω Compl modulaor rp.
6 5. Eampl : Modulaor and Dmodulaor (ECE 3, 3, 33, 33, 4, 43, 444, 457). Phaor gnraliz o im varing phaor. coω coω u( LPF u( v( ( LPF v( inω TX: Baband informaion-baring ignal, modulad o RF inω RX: RF ignal dmodulad o baband o rcovr info. w( R ( LPF w( ω ECE Conncion: Colorado Sa Univri Ocobr 007 ω ω w ( u( + v( co ω + inω Compl mod/dmod
7 ECE Conncion: Colorado Sa Univri Ocobr Eampl 3: -Dimnional Fild and Polarizaion (ECE 34, 34, 444, 457). Phaor alo gnraliz o mimachd phaor. Th produc Liaou figur, or llip. Sum of wo linarl polarizd fild Sum of lf and righ circularl polarizd fild B B A A A A E E z ω ϕ ω ϕ ω θ ω θ θ ω θ ω Im R ) in ( ) co( ) ( ) ( ) ( z( Imag ( -coordina R ( -coordina
8 7. Eampl 4: Filr Dign (ECE 0, 3, 3, 33, 33). Th n pol will hav o b organizd ino caual abl pol and ani-caual abl pol. Thi ugg ha righ-half plan pol ar no all bad. Mor o com. End of Par, dmonraing h powr of compl rprnaion for rprning ral ignal and ral fild. ECE Conncion: Colorado Sa Univri Ocobr 007
9 PART : Polnomial, Roo of Uni, ODE, and OP-AMP +. : Richard Fman (Nobl QED) likd h quaion π + 0 0,,,, π bcau i id oghr. Bu w know zro of h fir ordr pol +. i u h ingl ECE urn zro ino pol b conrucing a raional ranfr funcion: H ( ) h( u( + In our circui and m cour (ECE 0, 3, 3, 33, 33), w dcrib filr b h ranfr quaion. To a H ( ) /( + ) d ( / d + ( ( i o a. π Y ( ) H ( ) X ( ) ( + ) Y ( ) X ( ), which i o a Tha i, b urning zro of a polnomial ino pol of a ranfr funcion, w conruc h cofficin of an ordinar diffrnial quaion from h cofficin of a pol. Thi ugg ha good diffrnial quaion, or good circui, hould b conrucd from good polnomial. Good pol. ECE Conncion: Colorado Sa Univri Ocobr 007
10 + +. : L inch along o h pol, which i h pol ha fir forcd compl numbr ino our licon. Th corrponding ranfr funcion i H ( ) + ( + )( ) / + / + h( in u( Thi i ECE 0, 3, 3, and M 60, 6, 340. W hav variou dcripion of hi ranfr funcion. h( π H ( ω) ω ECE Conncion: Colorado Sa Univri Ocobr 007
11 +. (coninud): From h ranfr quaion, w can driv h ordinar diffrnial quaion for hi m, al an analog compur wiring diagram, and dign an OP-AMP circui: Y ( ) H ( ) X ( ) d ( + ) Y ( ) X ( ) + ( d ( ( d ( / d ( ( 0 Thi i h conncion bwn M 60, 6, 340, ECE 0, 3, 3, 33, 33. ECE Conncion: Colorado Sa Univri Ocobr 007
12 . + (coninud): To h circui dignr, h OP-AMP i h hing. Bu o a m dignr, h ranfr funcion i h hing. X H () Y H X H () H () A a pol, cod for ODE; a a compl impdanc, map ino. ( ( H ( ) X H ( ω) Y H X H ( ω) ( ω ( H ( ω), cald b h ignvalu H ( ω). A a compl frqunc rpon, map h ω ignvcor ino h ignvcor h( h h ( ( A an impul rpon, map ino ( ( h )(. Mak no miak: Th OP-AMP do i. Howvr, h ranfr funcion dcrib h ODE obd b h OP-AMP. Th compl frqunc rpon rval frqunc lcivi b lling u ha i i a if w ar mulipling frqunc componn in h frqunc domain, and h impul rpon plain im domain moohn b lling u ha i i a if w ar convolving ignal in h im domain. ECE Conncion: Colorado Sa Univri Ocobr 007
13 +. (coninud): In circui and m, filring i don in h im domain, and alkd abou in h frqunc domain. W nvr acuall build H ( ω). Rahr, w build a circui o m pcificaion on H ( ω). Bu in cohrn opic, w u ln o Fourir ranform inpu, mulipl b h pcral mak H ( ω), o g Y ( ω) H ( ω) X ( ω), and invr Fourir ranform wih anohr ln o g (. Tha i, in cohrn opic, filring i acuall don in h frqunc domain, and alkd abou in im domain. Th impul rpon i calld h poin prad funcion. h( Thing can g vn fancir in cohrn opic, whr a prim i ud o para frqunc componn ino pac, whr frqunc domain filring i applid o conruc vr hor pul. Thi ECE 457. ECE Conncion: Colorado Sa Univri Ocobr 007
14 . + (coninud): Whr do im domain moohn com from, and frqunc domain lcivi?, ( Anwr: A vr im mu b o mooh ha i, plu i cond drivaiv i zro. Coin do i. For mor gnral diffrnial quaion, linar combinaion of dampd compl ponnial do i. ( ( A om frqunci, h rpon of o an inpu, ω ω ω naml d / d + ω A + A, ha o much druciv inrfrnc bwn h wo componn ha h circui can uppor larg ampliud A whil ming h uni inpu conrain. A h frqunci, h circui or m ha larg gain. Convrl,. So, vn filring i, a boom, an inrfrnc ffc. Inrfrnc i bhind vrhing: focuing in camra, opical imag formaion, polarizd gla, radiaion parn in annna, circui, conrol, and communicaion. 0 A ω ω ω ω A ω ω 0 A 0 ω A ω ECE Conncion: Colorado Sa Univri Ocobr 007
15 . + (coninud): Ar an of h inigh uful for m or channl which do no ob ODE? H () Anwr: Y. W r o modl non-raional ranfr funcion, wih raional approimaion, a illurad blow: G() H G H G H G H G Sri: qualizr for communicaion. (ECE 3, 3, 4, 43) Paralll: modl followr for conrol. (ECE 3, 3, 4, 4) ECE Conncion: Colorado Sa Univri Ocobr 007
16 + n 3. : L g rckl. Thi polnomial ha h nh roo of - for i zro. Th zro could no b idnifid wihou compl anali. n k, k,,..., n H () W lik h LHP pol a pol of a caual and abl filr. Acuall, hr i a lo o lik abou h RHP pol: h ar pol of an ani-caual and abl filr. In fac, bcau of h mmri, if w u h LHP pol o conruc a caual ranfr funcion H (), h RHP pol will * * auomaicall build h ani-caual ranfr funcion H ( ). Thi grouping of pol could no b don wihou compl anali. H () i h ranfr funcion for a caual D/A or ranmi filr in a cll * * phon and a caual D/A or nhi filr in a digial conrollr. H ( ) i h ranfr funcion for an ani-caual A/D or rciv filr in a cll phon and an ani-caual A/D or anali filr in a digial conrollr. Evr cll phon and vr digial conrollr wih i could hav a non-caual A/D. ECE Conncion: Colorado Sa Univri Ocobr
17 + n 3. (coninud) : L conruc h ranfr funcion H () from h LHP pol, and u h compl mmr of h pol o facor H () ino wo cond ordr cion: H ( ) H ( ) H ( ) ζ + ζ Thi facoring could no b don wihou compl anali. Each of h ranfr funcion cod for an ODE: + ( ζ + ) Y ( ) X ( ) d ( / d ζ d( / d + ( ( Each ODE cod for an analog compur wiring diagram: ζ ζ Thi analog compur wiring diagram hould cod for an OP-AMP circui. Thi i ECE 3, 3, 33, 33. ECE Conncion: Colorado Sa Univri Ocobr 007
18 + n 3. (coninud) : L rdraw h analog compur wiring diagram: k k k k Thi i fdback. ECE 3, 3, 33, 33, 4, 4. L build i up, from crach. ECE Conncion: Colorado Sa Univri Ocobr 007
19 + n 3. (coninud) : L idnif an OP-AMP circui wih ach of h uccivl mor complicad fdback diagram. W ar don. ECE Conncion: Colorado Sa Univri Ocobr 007
20 4. Summar : Singl compl channl cod for dual ral channl, hrb clarifing our undranding of modulaion and polarizaion. Roo of uni cod for a ranfr funcion. Thi ranfr funcion ma b facord ino i caual and ani-caual par. Th caual par cod for an ODE. Thi ODE cod for an analog compur wiring diagram. Thi analog compur wiring diagram cod for an OP-AMP circui. Good OP-AMP circui com from good polnomial. You nam h ffc ou ar afr and I will argu for inrfrnc ffc a h undrling bai for wha ou can achiv. Inrfrnc ffc ar rvald b phaor. In raional m of RLC and OP-AMP, h inrfrnc ffc ar drmind b polnomial. ECE Conncion: Colorado Sa Univri Ocobr 007
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