Using Complex Numbers in Circuit Analysis and Review of the Algebra of Complex Numbers

Transcription

1 Fom Pofsso obt Johnson, U Santa u, wth pmsson. Phscs 6 Pactcal Elctoncs Spng Usng ompl Numbs n cut Analss and vw of th Algba of ompl Numbs Th pupos of ths not s to vw th algba of concpt numbs and show how th can b usd to smplf analss of lna ccuts. Ths s th basc tho bhnd how PSpc handls lna ccuts and lna small-sgnal appomatons of non-lna ccuts. Th basc tchnqus a also wdl usd n man tps of lna analss found n phscs and ngnng lctcal o mchancal, so f ou a maong n on of thos flds, ou wll nd to bcom famla wth ths tchnqus and concpts soon o lat, vn though w wll not us thm tnsvl n ths cous. I llustat th tchnqu wth a smpl, standad ccut and thn wth a non-tval ampl, and I show how th sam ampl can b asl solvd b PSpc. At th nd I dscuss bfl how PSpc can handl much mo complcatd ccuts, ncludng nonlna ccuts, b tndng th sam mathmatcal tchnqus. ompl numbs a commonl usd n lctcal ngnng, as wll as n phscs. In gnal th a usd whn som quantt has a phas as wll as a magntud. Such a stuaton occus whn on dals wth snusodal oscllatng voltag and cunt oth ampls n phscs nclud optcs, wh wav ntfnc s mpotant, and quantum mchancal wav functons. I want to mphas that compl numbs a usd to mak calculatons as! Do not b ntmdatd b tng to magn what an magna numb s. Th s no nd fo that. Instad, al that th s nothng magna about th phas of a voltag wavfom, and th s nothng patculal compl about wokng wth compl numbs. Just look at thm fo now as a usful tool that ou ma as wll stat gttng usd to. ompl numbs as usd h a quvalnt to th phasos usd fo ths pupos n lmnta phscs ttbooks. Th Fgu. A compl numb shown n th compl plan. s th al pat of, s th magna pat of, s th magntud of, and s th phas of. advantag of callng thm compl numbs nstad of phasos s that ou can mak us of th hopfull famla algba of compl numbs. You ma also f to Appnd B of Hoowt and Hll fo a vw of compl numbs. If ou havn t alad, thn lat ou can stud th bautful and smngl magcal mathmatcs of functons of compl vaabls, but th s no nd fo that n ths cous.

2 Phscs 6 Pactcal Elctoncs Spng psntatons of ompl Numbs t th smbol psnt a compl numb, whl and a ts al and magna pats:, wh. Th compl conugat of * s. In gnal, to chang a compl numb nto ts compl conugat, smpl chang to vwh. Thn all of th nomal uls of algba appl, wth th undstandng that : * Snc w want to us compl numbs to psnt phass of wavfoms, t s ssntal to undstand th pola, as wll as atsan, fom of a compl numb s Fgu. Ths s no dffnt fom convsons btwn ctangula and plan pola coodnats, cpt that nstad of lablng th as and w labl thm and Im. Thnk of th compl numb as a -dmnsonal vcto n a plan. Addton of two compl numbs looks actl lk vcto addton, th gaphcall o algbacall as n th addton ampl abov and n Fgu. And lngth of th vcto. ookng at Fgu, ou can s that cos so w can wt ou compl numb as cos Fgu. Gaphcal psntaton of th addton of two compl numbs. Not how th two numbs add as vctos n -D whn th al and magna pats a smpl addd as = + and = + spctvl. sn sn cos sn s ust th wh th last stp maks us of Eul s fomula: cos sn. Ths ssntal laton ponts dctl to on ason wh compl numbs mak ccut analss as. Instad of psntng a snusodal voltag o cunt as a sn o cosn functon, w can psnt t as an ponntal. Eponntals a much as to wok wth algbacall! Elctcal ngns, and ou ttbook, us ths notaton, but phscsts and phscs ttbooks and mathmatcans gnall us th smbol nstad of.

3 Phscs 6 Pactcal Elctoncs Spng Unlss ou lov dalng wth complcatd tg dntts, choos th compl ponntal ov th sn and cosn functons! H s a summa of th two psntatons of a compl numb: cos actan sn Kp n mnd whn calculatng th phas that th s n gnal an ambgut of adans, whch ou hav to solv b lookng at th sgns of both and. Th actan functon on ou calculato wll alwas tun an angl n th ang. You can avod ths ambgut f ou us th spcal functon on ou calculato fo tansfomng btwn ctangula and pola coodnats. Also, comput languags usuall nclud an nvs tangnt functon wth two spaat agumnts fo and, whch wll tun th coct valu of n th ang o.g. ATAN n FOTAN. Basc Algba wth ompl Numbs Addton and subtacton of compl numbs a most asl don usng th atsan ctangula fom, fo th sam ason that vctos a most asl addd and subtactd n atsan componnts. Howv, multplcaton and dvson a most asl don usng th pola fom, makng us of th popts of th ponntal functon: Nvthlss, multplcaton n th ctangula fom s staghtfowad:. Dvson can b accomplshd th b convtng numato and dnomnato to th pola fom and usng th quatons abov, o b multplng th numato and dnomnato b th compl conugat of th dnomnato. Ths s an cs that s fquntl qud n ccut analss: In ths wa w can spaat th al and magna pats of th ato, fom whch w can calculat th magntud and phas, f ncssa. Do not t to mmo such a fomula! It s th smpl tchnqu of multplng th numato and dnomnato b th compl conugat of th dnomnato that ou should mmb. Ecutng ths tchnqu alwas guaants that th sultng dnomnato wll b al, wth th magna numb appang onl n th numato.

4 Phscs 6 Pactcal Elctoncs Spng Wokng wth ompl Impdanc oltag and cunt a alwas al, obsvabl quantts. In a lna A/ ccut wth a snusodal stmulus, th wll alwas hav a fom lk t cos t. Th algbac complts com n whn w ntoduc capactos and nductos, whch poduc 9 changs n phas. Addng sns and cosns wth dffng phass s algbacall panful, qung pts wth tg dntts. Howv, f th ccut s dscbd b lna dffntal quatons, thn w can smplf lf b addng an magna pat to th voltag o cunt: t t t t cos sn wth th undstandng that th obsvd voltag s ust th al pat of ths psson. Now, whn ou do ou ccut analss ou gt to dal wth th smpl popts of th ponntal functon nstad of nast tg dntts. Whn don, ust tak th al pat of th fnal sult, and that s ou answ. As ou wll s, what ths pocdu wll do fo ou s tun a st of lna dffntal quatons nto a st of lna algbac quatons. Ths woks onl bcaus th ccut s a lna ccut, dscbd b lna dffntal quatons. Snc lna quatons do not nvolv an squas, squa oots, and so foth of th voltag o cunt, o multplcaton of on voltag o cunt b anoth, th al and magna pats don t gt md up. Tak a look at th quatons n th pvous scton. Th addton and subtacton quatons do not m up th al and magna pats, but th quatons fo multplcaton and dvson do. Multplng a compl numb b a al constant also obvousl dos not m up th al and magna pats. Essntall, a lna quaton s on that wll not m up th al and magna pats of th voltags and cunts. Fom a pactcal standpont, a lna ccut s on that ncluds onl passv componnts sstos, capactos, and nductos plus voltag and/o cunt soucs. No dods, tansstos, vacuum tubs, tc. a allowd. It s phaps woth mntonng h that th sam fomalsm, wth th sam advantags of usng compl numbs, woks n mchancs whn dalng wth small, hamonc oscllatons of mchancal sstms. Th cp fo obtanng th stad-stat hamonc spons of a lna ccut s staghtfowad. Wt ach non-statc voltag o cunt souc as a compl numb: o I wh th phas can b takn to b o f th s onl on souc. Othws th latv phass of th soucs must b takn nto account. Thn tat ach passv componnt as an mpdanc ssto: apacto: Inducto: Ths pocdu woks fo voltag and cunt soucs that a snusodal hamonc. Howv, a nonsnusodal podc souc can b wttn as a Fou ss of sns and cosns. Each tm n th ss can b tatd b th mthod dscbd h. Snc th ccut s lna, th spons s ust th lna supposton of th sponss to th ndvdual hamonc Fou componnts. B stad-stat, I man tun all th swtchs on and thn wat long nough fo th tansnt bhavo to dampn out and dsappa. Usuall th wat s v shot, lss than a blnk of th.

5 Phscs 6 Pactcal Elctoncs Spng Fgu. ss ccut. wh n gnal th mpdanc lats th voltag acoss a componnt to th cunt passng though th componnt accodng to a gnalaton of Ohm s law: I Us Kchhoff s laws to wt a st of lna quatons fo th cunts and voltag n th ccut, actl as ou would do fo a ccut mad up of batts and sstos. Th onl dffnc s that som of th sstancs a magna, so what ou nd up wth s a st of compl lna quatons. Solv th quatons fo th cunts and voltags. Ths s tdous to do b hand, but kp n mnd that a comput can solv an amangl lag st of compl lna quatons n an nstant, usng standad cannd pogams. Man scntfc calculatos also hav bult-n functons fo solvng sts of lna compl quatons. Fnall, pss th sultng voltags and/o cunts n pola fom, fom whch ou can ad off th ampltud and phas of ach cunt o voltag. As an ampl not ncludd n Hoowt and Hll, lt s anal th standad ss ccut Fgu whch has a voltag oscllato n ss wth a ssto, capacto, and nducto. Th dffntal quaton fo ths ccut follows fom addng up th voltag changs aound th loop: t di Q I, dt t wh s th dvng voltag, pssd as a compl quantt as suggstd abov, wth an assumd phas. Usng Q Idt, w gt an quaton fo th cunt: di t Idt I. dt t Ths s adl solvd b makng th substtuton I I, whch tuns th dffntal quaton nto an algbac quaton: I. Th quantt n panthss s actl th mpdanc that on would gt b usng th mpdanc uls lstd abov fo sstos, capactos, and nductos, plus th ul that mpdancs n ss smpl add up. So, fom now on do not both to wt down th dffntal quaton! Just assum th uls fo compl mpdanc and mmdatl wt down th algbac quaton. 5

6 Phscs 6 Pactcal Elctoncs Spng 6 To anal th ss ccut wthout wtng an dffntal quaton, w stat wth Ohm s aw fo a actv ccut: I wth. To do th dvson, I convt th mpdanc to pola fom: wth actan actan and and. So th cunt s gvn b I Fgu. sonanc cuvs fo an ss ccut, wth = Ohms, =F, and =mh.

7 Phscs 6 Pactcal Elctoncs Spng wth actan fo th phas of th cunt. Ths sult hbts a sonanc, wth, th natual fqunc of th ccut, bng th fqunc at whch th mpdanc s mnmum and qual smpl to and th cunt s mamum, wth a phas shft of o latv to th voltag. Also, s a masu of th amount of dampng n th ccut and, thus, th wdth of th sonanc cuv. Ths sonanc bhavo s llustatd n Fgu. Analng a Mo ompl na cut A mo complcatd lookng ampl s shown n Fgu 5, wh th dvng t voltag s th al pat of t volts, wth angula fqunc adans/s. Th mpdanc of th nducto s ohms, and th mpdanc of th capacto s. 5 ohms. Th obctv s to fnd all th cunts n th ccut and th quvalnt mpdanc of th ovall ccut, as sn b th voltag souc. In ths cas th a loops, so w wll hav loop quatons and nod quatons. Ths gos bond th complt that ou wll s n homwok o on an am, but I thow t n as a andom dmonstaton that th analss s staghtfowad and can b fomulatd n a mann that maks a soluton b comput fal as. I pf to wok wth th concpt of loop cunts, n od to avod havng to wt down th nod quatons. To undstand ths concpt, look at th ccut as dawn n Fgu 6. Th fou loops a vdnt, and ach s assocatd wth a loop cunt. Th cunt though th capacto s clal, th cunt though th voltag souc s, and th cunt though th -ohm ssto s. Howv, ach of th oth componnts has two cunts flowng though t. Fo ampl, th cunt flowng upwad though th nducto s, and th cunt flowng downwad though th lftmost ssto s. Now, lt s appl Kchhoff s loop law to loop #, statng at th low lft con and pocdng upwads though th voltag souc, n th dcton of loop cunt : Fgu 5. Eampl of a -loop lna ccut. 7

8 Phscs 6 Pactcal Elctoncs Spng 8 Do th sam fo loop #, statng n th low lft hand con and pocdng upwads though th -ohm ssto, n th dcton of th loop cunt : Th oth two quatons, fo loops and spctvl, a.5 Such quatons a asst to dal wth f ogand n mat notaton:.5 Solvng ths quatons b hand would b tdous and annong, but dong t b comput wth a pogam lk Mathcad, Mathmatca, o Matlab couldn t b as. Fo ampl, n Mathcad lt s call th mat, so th quaton looks lk I Fll th 6 compl valus nto th mat and th valus nto, and thn tp I and ou don! 5 Th sult s I H s how to ntpt th sult. Fo ampl, th cunt can b wttn n pola fom as , so th cunt as a functon of tm s.7 cos 5.56 t t. 5 Ths s not th most ffcnt wa to solv lna quatons, but fo ths pupos, who cas? Th comput wll fnsh th calculaton bfo ou can sa go! Fgu 6. Th ccut dawn wth loop cunts.

9 Phscs 6 Pactcal Elctoncs Spng Fgu 7. Plots of th voltag and cunt of th voltag suppl as a functon of tm fo a suppl fqunc of adans/s. Th cunt lags bhnd th voltag b sval dgs. That s, th cunt passng though th souc lags bhnd th voltag b.7 adans, o about 6.7 dgs. Fgu 7 shows how th cunt and voltag would look f dsplad on an osclloscop. Th quvalnt mpdanc of th ccut, as sn b th souc, can b calculatd fom th ato of th voltag and cunt of th souc:.7 q Thus at ths fqunc, th ccut looks slghtl nductv to th souc. na cut