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

Transcription

1 Physics 6.P. Johnson Spring 6 Using omplx Numbrs in ircuit Analysis viw of th Algbra of omplx Numbrs omplx numbrs ar commonly usd in lctrical nginring, as wll as in physics. In gnral thy ar usd whn som quantity has a phas as wll as a magnitud. Such a situation occurs whn on dals with sinusoidal oscillating voltag and currnt (othr xampls in physics includ optics, whr wav intrfrnc is important, and quantum mchanical wav functions. I want to mphasiz that complx numbrs ar usd to mak calculations asir! Do not b intimidatd by trying to imagin what an imaginary numbr is. Thr is no nd for that. Instad, raliz that thr is nothing imaginary about th phas of a voltag wavform, and thr is nothing particularly complx about working with complx numbrs. Just look at thm for now as a usful tool that you may as wll start gtting usd to. You may also rfr to Appndix B of Horowitz and Hill for a rviw of complx numbrs. prsntations of omplx Numbrs t th symbol z rprsnt a complx numbr, whil x and y ar its ral and imaginary parts: z x y, whr. Th complx conugat of z is * z x y. In gnral, to chang a complx numbr into its complx conugat, simply chang to vrywhr. Thn all of th normal ruls of algbra apply, with th undrstanding that : z ( x y x y xy * z z z ( x y ( x y x y z z ( x y ( x y ( x x ( y y Sinc w want to us complx numbrs to rprsnt phass of wavforms, it is ssntial to undrstand th polar, as wll as artsian, form of a complx numbr. This is no diffrnt from convrsions btwn rctangular and plan polar coordinats, xcpt that instad of labling th axs x and y w labl thm and Im. Think of th complx numbr as a -dimnsional vctor in a plan. Addition of two complx numbrs looks xactly lik vctor addition, ithr graphically or Figur. A complx numbr z shown in th complx plan. x is th ral part of z, y is th imaginary part of z, r is th magnitud of z, and is th phas of z. If you havn t alrady, thn latr you can study th bautiful and smingly magical mathmatics of functions of complx variabls, but thr is no nd for that in this cours. Elctrical nginrs, and our txtbook, us this notation, but physicists and physics txtbooks (and mathmaticians gnrally us th symbol i instad of.

2 Physics 6 Spring 6 algbraically (as in th addition xampl abov. And lngth of th vctor. ooking at Figur, you can s that x r cos y r sin so w can writ our complx numbr as z r x y ( cos r z r cos r sin r sin is ust th whr th last stp maks us of Eulr s formula: cos sin. This ssntial rlation points dirctly to th rason why complx numbrs mak circuit analysis asir. Instad of rprsnting a sinusoidal voltag or currnt as a sin or cosin function, w can rprsnt it as an xponntial. Exponntials ar much asir to work with algbraically! Unlss you lov daling with complicatd trig idntitis, choos th complx xponntial ovr th sin and cosin functions! Hr is a summary of th two rprsntations of a complx numbr: z x y x r cos y r sin z r r x y arctan( y x Kp in mind whn calculating th phas that thr is in gnral an ambiguity of ± π radians, which you hav to rsolv by looking at th signs of both y and x. Th arctan function on your calculator will always rturn an angl in th rang π π. You can avoid this ambiguity if you us th spcial function on your calculator for transforming btwn rctangular and polar coordinats. Also, computr languags usually includ an invrs tangnt function with two sparat argumnts for y and x, which will rturn th corrct valu of in th rang π or π π (.g. ATAN in FOTAN. Basic Algbra with omplx Numbrs Addition and subtraction of complx numbrs ar most asily don using th artsian (rctangular form, for th sam rason that vctors ar most asily addd and subtractd in artsian componnts. z z ( x x ( y y z z ( x x ( y y Howvr, multiplication and division ar most asily don using th polar form, making us of th proprtis of th xponntial function: ( z z r r z r z r ( Nvrthlss, multiplication in th rctangular form is straightforward: z z ( x y ( x y x x y y x y x y. ( (

3 Physics 6 Spring 6 Division can b accomplishd ithr by convrting numrator and dnominator to th polar form and using th quations abov, or by multiplying th numrator and dnominator by th complx conugat of th dnominator. This is an xrcis that is frquntly rquird in circuit analysis: z x y x y ( xx y y ( x y x y z x y x y x y In this way w can sparat th ral and imaginary parts of th ratio, from which w can calculat th magnitud and phas, if ncssary. Do not try to mmoriz such a formula! It is th simpl tchniqu of multiplying th numrator and dnominator by th complx conugat of th dnominator that you should rmmbr. Excuting this tchniqu always guarants that th rsulting dnominator will b ral, with th imaginary numbr apparing only in th numrator. Working with omplx Impdanc oltag and currnt ar always ral, obsrvabl quantitis. In a linar A/ circuit with a sinusoidal stimulus, thy will always hav a form lik ( t cos( t. Th algbraic complxitis com in whn w introduc capacitors and inductors, which produc ± 9 changs in phas. Adding sins and cosins with diffring phass is algbraically painful, rquiring xprtis with trig idntitis. Howvr, if th circuit is dscribd by linar diffrntial quations, thn w can simplify lif by adding an imaginary part to th voltag or currnt: ( t ( t t t cos( sin( with th undrstanding that th obsrvd voltag is ust th ral part of this xprssion. Now, whn you do your circuit analysis you gt to dal with th simpl proprtis of th xponntial function instad of nasty trig idntitis. Whn don, ust tak th ral part of th final rsult, and that is your answr. As you will s, what this procdur will do for you is turn a st of linar diffrntial quations into a st of linar algbraic quations. 3 This works only bcaus th circuit is a linar circuit, dscribd by linar diffrntial quations. Sinc linar quations do not involv any squars, squar roots, and so forth of th voltag or currnt, or multiplication of on voltag or currnt by anothr, th ral and imaginary parts don t gt mixd up. Tak a look at th quations in th prvious sction. Th addition and subtraction quations do not mix up th ral and imaginary parts, but th quations for multiplication and division do. Multiplying a complx numbr by a ral constant also obviously dos not mix up th ral and imaginary parts. Essntially, a linar quation is on that will not mix up th ral and imaginary parts of th voltags and currnts. From a practical standpoint, a linar circuit is on that includs only passiv componnts (rsistors, capacitors, and inductors plus voltag and/or currnt sourcs. No diods, transistors, vacuum tubs, tc. ar allowd. 3 This procdur works for voltag and currnt sourcs that ar sinusoidal (harmonic. Howvr, a nonsinusoidal priodic sourc can b writtn as a Fourir sris of sins and cosins. Each trm in th sris can b tratd by th mthod dscribd hr. Sinc th circuit is linar, th rspons is ust th linar suprposition of th rsponss to th individual harmonic Fourir componnts. 3

4 Physics 6 Spring 6 I Figur. sris circuit. Th rcip for obtaining th stady-stat 4 harmonic rspons of a linar circuit is straightforward. Writ ach non-static voltag or currnt sourc as a complx numbr: or I whr th phas can b takn to b zro if thr is only on sourc. Othrwis th rlativ phass of th sourcs must b takn into account. Thn trat ach passiv componnt as an impdanc: sistor: Z apacitor: Z Inductor: Z Us Kirchhoff s laws to writ a st of linar quations for th currnts and voltag in th circuit, xactly as you would do for a circuit mad up of battris and rsistors. Th only diffrnc is that som of th rsistancs ar imaginary, so what you nd up with is a st of complx linar quations. Solv th quations for th currnts and voltags. This is tdious to do by hand, but kp in mind that a computr can solv an amazingly larg st of complx linar quations in an instant, using standard cannd programs. Many scintific calculators also hav built-in functions for solving sts of linar complx quations. Finally, xprss th rsulting voltags and/or currnts in polar form, from which you can rad off th amplitud and phas of ach currnt or voltag. As an xampl not includd in Horowitz and Hill, lt s analyz th standard sris circuit (Figur which has a voltag oscillator in sris with a rsistor, capacitor, and inductor. Th diffrntial quation for this circuit follows from adding up th voltag changs around th loop: t di Q I, dt t whr is th driving voltag, xprssd as a complx quantity as suggstd abov, with an assumd phas. Using Q Idt, w gt an quation for th currnt: di t Idt I. dt 4 By stady-stat, I man turn all th switchs on and thn wait long nough for th transint bhavior to dampn out and disappar. Usually th wait is vry short, lss than a blink of th y. 4

5 Physics 6 Spring 6 5 This is radily solvd by making th substitution ( t I I, which turns th diffrntial quation into an algbraic quation: I. Th quantity in parnthss is xactly th impdanc that on would gt by using th impdanc ruls listd abov for rsistors, capacitors, and inductors, plus th rul that impdancs in sris simply add up. So, from now on do not bothr to writ down th diffrntial quation! Just assum th ruls for complx impdanc and immdiatly writ down th algbraic quation. To analyz th sris circuit without writing any diffrntial quation, w start with Ohm s aw for a ractiv circuit: Z I with Z. To do th division, I convrt th impdanc to polar form: Z Z with γ arctan arctan Z and and γ. So th currnt is givn by ( γ I Z with γ arctan for th phas of th currnt. This rsult xhibits a rsonanc, with, th natural frquncy of th circuit, bing th frquncy at which th impdanc is minimum (and qual simply to and th currnt is maximum, with a phas shift of zro rlativ to th voltag. Also, γ is a masur of th amount of damping in th circuit and, thus, th width of th rsonanc curv. This rsonanc bhavior is illustratd in Figur 3.

6 Physics 6 Spring 6 Figur 3. sonanc curvs for an sris circuit, with Ohms, µf, and 4mH. A mor complicatd looking xampl is shown in Figur 4, whr th driving it 4 voltag is th ral part of ( t volts, with angular frquncy radians/s. Th impdanc of th inductor is 4 ohms, and th impdanc of th capacitor is. 5 ohms. Th obctiv is to find all th currnts in th circuit and th quivalnt impdanc of th ovrall circuit, as sn by th voltag sourc. In this cas thr ar 4 loops, so w will hav 4 loop quations and 3 nod quations. This gos Figur 4. Exampl of a 4-loop linar circuit. 6

7 Physics 6 Spring 6 4 F i 4 i 3 i i.4mh Figur 5. Th circuit rdrawn with loop currnts. byond th complxity that you will s in homwork, but I throw it in as a dmonstration that th analysis is straightforward and can b formulatd in a mannr that maks a solution by computr fairly asy. I prfr to work with th concpt of loop currnts, in ordr to avoid having to writ down th nod quations. To undrstand this concpt, look at th circuit as rdrawn in Figur 5. Th four loops ar vidnt, and ach is associatd with a loop currnt. Th currnt through th capacitor is clarly i 4, th currnt through th voltag sourc is i, and th currnt through th -ohm rsistor is i 3. Howvr, ach of th othr 4 componnts has two currnts flowing through it. For xampl, th currnt flowing upward through th inductor is i3 i, and th currnt flowing downward through th lftmost rsistor is i i. Now, lt s apply Kirchhoff s loop law to loop #, starting at th lowr lft cornr and procding upwards through th voltag sourc, in th dirction of loop currnt i : ( i i Do th sam for loop #, starting in th lowr lft hand cornr and procding upwards through th -ohm rsistor, in th dirction of th loop currnt i : ( i i ( i i4 ( i i3 4 Th othr two quations, for loops 3 and 4 rspctivly, ar ( i 3 i 4 ( i3 i4 i3 i 4.5 ( i4 i3 ( i4 i Such quations ar asist to dal with if organizd in matrix notation: i 4 4 i i 3.5 i4 Solving ths quations by hand would b tdious and annoying, but doing it by computr with a program lik Mathcad, Mathmatica, or Matlab couldn t b asir. For xampl, in Mathcad lt s call th matrix Z, so th quation looks lik Z I 7

8 Physics 6 Spring 6 Fill th 6 valus into th matrix Z and th 4 valus into, and thn typ I Z and you r don! 5 Th rsult is I Hr is how to intrprt th rsult. For xampl, th currnt i can b writtn in.37π polar form as i 5. 56, so th currnt as a function of tim is i ( t 5.56 cos( t.37π. That is, th currnt passing through th sourc lags bhind th voltag by.37π radians, or about 7 dgrs. Figur 6 shows how th currnt and voltag would look if displayd on an oscilloscop. Th quivalnt impdanc of th circuit, as sn by th sourc, can b calculatd from th ratio of th voltag and currnt of th sourc:.37π Zq. i 5.56 Thus at this frquncy, th circuit looks slightly inductiv to th sourc. Figur 6. Plots of th voltag and currnt of th voltag supply as a function of tim for a supply frquncy of 4 radians/s. Th currnt lags bhind th voltag by svral dgrs. Nonlinar ircuits W hav sn how complx numbrs can mak quick work of linar circuits, by turning a st of coupld linar diffrntial quations into a st of linar algbraic 5 This is not th most fficint way to solv 4 linar quations, but for this purpos, who cars? Th computr will finish th calculation bfor you can say go! 8

9 Physics 6 Spring 6 quations that ar asily solvd with som hlp from a computr. But what about nonlinar circuits? Wll, on gnrally has to analyz a non-linar circuit by making a linar approximation around som initial guss (or around th actual bias point if that is alrady known. That may giv a crummy rsult at first, but usually by itrating this procdur many tims on can arriv at a good approximation to th solution. In any cas, th analysis gnrally boils down to solving (prhaps many tims a st of linar quations, somthing that computrs ar vry good at doing. 6 Th Spic program uss such a procdur. It uss various mathmatical modls of th nonlinar dvics and trats th linar dvics much as w hav hr (xcpt that it can gt mor sophisticatd and includ imprfctions such as lakag currnt in capacitors. First it calculats a bias point, with th tim dpndnc of all of th sourcs turnd off. To do so, it starts with an initial guss for th D currnts vrywhr and maks a linar approximation of all of th modls around that guss. Thn it solvs th complt st of complx linar quations to gt an improvd st of currnts. Thn it maks a nw linar approximation around thos currnts and solvs th st of quations again. Evntually, with som luck, it finds that from on itration to th nxt nothing changs much, at which point it assums that th procdur has convrgd to th physical solution. With th bias point in hand, Spic can thn vry quickly do an A analysis. In such an analysis, th sourcs ar assumd to b sinusoidal and hav vry small amplituds. Spic maks a linar approximation to th circuit around th bias point and calculats th rspons at ach of a larg st of frquncis, without doing any itration. This can vry quickly giv you information on th frquncy rspons of your circuit. A mor involvd analysis is th transint analysis. In this cas, th sourcs ar givn whatvr tim dpndnc and amplitud you ar intrstd in (squar wav, triangl wav, or whatvr. Th amplitud is no longr assumd small, so Spic cannot gt by with a linar approximation with a singl itration. Instad, it must tak small tim stps, such that in ach stp th voltags and currnts don t chang by vry much. Thn a fw itrations can find a nw convrgnc point aftr ach stp. Spic will vary th siz of th stp dpnding on how fast your sourc is changing. As you can imagin, it has to go vry slowly in small stps whn you fd it th dg of a squar wav. For a larg circuit this can tak an normous amount of computr tim, but if you hav th PU cycls, it is worth it, bcaus it will giv you a vry good ida of how your circuit will work bfor you build it. This is spcially important if you ar making intgratd circuits, whr aftr vry nw scrw-up in your dsign you would hav to wait 3 or 4 months, and spnd at last svral tns of thousands of dollars, to find out that it still dosn t work! 6 B awar that in th cas of a nonlinar circuit this procss may not convrg. That is nvr an issu with a linar circuit, for which ust on itration always givs th final solution. 9