LaPlace Transforms in Design and Analysis of Circuits Part 3: Basic Parallel Circuit Analysis

Transcription

1 LaPlace Tranfrm n Degn and Analy f Crcut Part : Bac Parallel Crcut Analy Cure N: E-7 Credt: PDH Thma G. Bertenhaw, Ed.D., P.E. Cntnung Educatn and Develpment, Inc. 9 Greyrdge Farm Curt Stny Pnt, NY 98 P: (877-8 F: ( nf@cedengneerng.cm

2 LaPlace Tranfrm n Degn and Analy f Crcut Part by Tm Bertenhaw Bac Crcut Analy - Parallel Crcut Smple Tw Lp In Part, LaPlace technque were ued t lve fr the put n mple ere reactve crcut. Th part wll examne the technque ued n apprachng the lutn t tw and three lp parallel crcut wth reactve cmpnent. Parallel crcut that cntan a number f lp beynd three wll nt be exampled, a any number f lp can be reduced t tw r three by the ue f Thevnan' r Nrtn' therem, r by merely extendng the prcedure develped here (althugh the math huekeepng get ntene. Cnder the fllwng tw lp ytem: Z Z n Z Z Wrtng the crcut equatn fr each lp; reveal that there are tw equatn n tw unknwn that charactere the crcut. n ( ( There are everal technque avalable fr lvng ytem f lnear equatn that have the ame number f equatn a unknwn, and f the, we wll ue Cramer' Rule thrugh th ectn (If yu want a quck refreher n the ue f Cramer' Rule fr lvng ytem f lnear equatn, ee Appendx A. Suppe we have the fllwng value fr thrugh

3 f.f Crcut Wrtng the equatn fr the tw lp; ( n dt dt dt Quetn: Hw wa the factr dt develped? Snce the defnng equatn fr capactr behavr v c cdt. C c dv C dt c, t fllw that We can prbably reduce the labr (and anguh f the abve tw crcut equatn are cnverted t ther LaPlace tranfrm. Lkng at the tranfrmed crcut; / / the crcut equatn are re-wrtten a; ( ( ( ( n (

4 Re-wrtng thee equatn a matrce; ( n ( ( Frt, fnd the determnant. Det. (.7. The tak t fnd (put taken acr rghtmt capactr. a a functn f the drver, and n th cae let ( ( Hwever, frt we'll fnd the mpule repne t gan an dea f the frm f the tranent repne f the crcut. Becaue ( the nly current nvlved n fndng ( we need nt lve fr (, except t atfy curty. Slvng fr ( ; Det. (.7. and the put vltage repne t a current mpule ( n turn then (.7..t.7t ( e e ( t.7 Ex Indcatng that the tranent repne, fr all practcal purpe, er at (recall that n prevu mdule e wa defned t be ab er. t 9 ecnd

5 Eq. Ampltude Tme n Sec. Suppe that a drver f n(.t appled a n (t. (Often parameter are chen, n th cae the frequency, t exaggerate me apect f the repne fr llutratn purpe. Then ( equal Det... Snce ( (, then (.... (.7. (.7.(. (.7.( (f yu need a refreher n Partal Fractn Expann, partcularly wth repect t fndng factr n cmplex denmnatr, refer t LaPlace Tranfrm n Degn and Analy f Crcut Part and fnally ( t.e.7e.n(.t. t.7t Ex.

6 Eq..8.. Ampltude Tme n ec. Agan, pleae nte that the put cntan tw dtnct cmpnent: the tranent repne and the teady tate repne. The tranent repne the tme lmted cntrbutn f the "natve" r mpule repne (ampltude mdfed by the nature f the drver. The teady tate repne ha the charactertc f the drver and cntnue untl the drve remved. All repne wll cntan thee tw cmpnent. The RLC Anther, and very gnfcant, crcut the analg f the ere RLC netwrk; a expected t a parallel RLC netwrk; ften knwn a a "parallel tank" crcut. It prperte are uch that t preent a very hgh mpedance at the renant frequency renderng the crcut very ueful n flterng and frequency determnatn applcatn. Lke ther clac crcut th ne can al be mplemented ung actve cmpnent. Hwever t an undertandng f the repne that ught at th tme and nt the technque nvlved n mmckng reactve cmpnent wth actve cmpnent. The degn technque wll be develped n ubequent mdule wthn th ere. I t I I /C R L by npectn, Crcut t

7 C c L R Snce c, we can take advantage f that and wrte the crcut equatn a C L R r L R LC CR t t LC L R R L LC Suppe the fllwng crcut ext, and that we wh t knw t mpule repne. (See Appendx C fr a mple but relatvely effectve current drver I t I I /.. Puttng me fleh t the tranfer functn ( (. ( ( ( ( ( 99.87(.7 ( t t t ( t e c(99.87t.7e n(99.87t e n(99.87t 87 Ex

8 Ex. 8 Ampltude (lt Tme n multple f. The tank exhbt a charactertc "rngng" that deterrate ver tme n accrdance wth the ytem tme cntant. Tak: dentfy the tw cmpnent that determne the tme cntant; what the theretcal reult f a hm retr? Quetn: hm' pble under nrmal ambent cndtn? It wrth ntng the magntude f the put n repne t δ (t. Parallel tank generate cnderable vltage at r very near ther renant frequency and, a a degner, ne mut alway bear n mnd the cnequence t the remanng crcutry. It a quetn f the phyc f an nductr: Aume a ttal crcut current A n( ωt where the magntude f renance, jx L jx C AZ n a functn f ttal crcut mpedance Zttal d L L ; why L large? dt, and we al knw that Suppe th crcut drven by t n(t, then t Z t. At ( ( ((. ( ( 99.87( (( 8 ( ( t e n(99.87t n(t 8 t Eq. It wuld prbably be prudent t reduce the ttal current nput t. ntead f a drvng ampltude f. 7

9 Eq. Ampltude Tme n mec. d Bear n mnd, frm abve, that L L L ( jωl, meanng that the magntude f L dt drectly prprtnal t the magntude f bth L & ω. The gr effect that cnderable vltage can be bult acr an nductr, whch may preent a haard t the remanng crcutry. Of cure there are technque t cntrl and/r lmt the magntude, but that dcun fr a later tme. A an ade, and a a general cmment: whle cnderable effrt mantaned t mntr the crrectne f all the calculatn, ft tme what can g wrng wll g wrng. Therefre, f yu dcver an errr, pleae d nt hetate t cntact the cmpany and/r the authr. Cnder the fllwng crcut Three Lp Crcut Z Z Z n Z Z Z Crcut 8

10 Wrtng the lp equatn ( n ( ( Wrtng the ytem equatn n matrx ntatn ( ( ( n Wrtng the determnant ung the ether the example r the defntn cntaned n Appendx A ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. Det Det Slvng fr thrugh ( ( ( ( (.. Det Det n n n. Det n ( ( (.. Det Det n n. Det n 9

11 ( (.. Det Det n n. Det n Three by three matrce get mey when there are reactve cmpnent n the crcut, and even meer at greater dmenn. But a lng a the fundamental f the matrx lutn prce are undertd, t recmmended that yu rert t the ue f a cmputer r calculatr fr lutn t ytem greater than X. Naturally, the ade are nt eental, merely cnvenent. Fr practce, cnder the fllwng crcut n / / The determnant matrx cllectng term and factrng (curtey f truty HP9 - I cannt be ure, but I wuld gue the nternal rne ue the Newtn-Raphn (r varant theref methd f fndng the rt.

12 Det. 7 (.98 (.9 (. Aume that taken acr the nductr, n that cae L ( ( Det. n ( ( and n (.98 (.9(. n n (.98 (.9( (..98 (.9(..8 n (.9*.98. ( The mpule repne f ( t.t.9t.9e n(.98t..e Eq.

13 Eq. Ampltude Tme n ncrement f. Suppe the crcut drven at a frequency near renance n - then.8 (.98 ( (.9(..9. * ( ( t.t.9t.8n( t.e n(.98t 9..9e Eq.

14 Eq. Ampltude Tme n.ec A ery Gd Tank Crcut r The Strange Cae f a Puled Drver Suppe we wh t buld a hgh qualty tank crcut fr frequency determnng purpe, and further uppe that frequency megahert (.8 megarad/. We knw that ± j the renant frequency (n rad/ when R (an mpble cndtn n LC practcal realty, but frequently ued fr pedaggcal r rugh etmatn purpe and that the frequency f cllatn (n rad/ ( R appendce D & E. ± j when R (ee LC L Further uppe we buld the crcut a a parallel tank wth n phycal retr n the crcut. We wll mdel t a f the DC retvty f the nductr and cmpnent lead mall; n the rder f hm. Sme degn cnderatn wll urface. In addtn the drve wll be a repettu pule (the analy f the repne wll requre me thught t recncle the math t realty. A drve f the nature decrbed abve pee a pectrum f frequence, hwever fr llutratve purpe we wll gnre the harmnc cntent at preent. The develpment f the pectrum a ubject fr Furer analy; we wll hld that effrt n abeyance fr the purpe f th mdule. Jut bear n mnd that the actual put wll cntan a range f frequence the majrty f whch wll be attenuated by the flterng prperte f the tank. Chng the lumped value f. hm f lead retance and. hm f DC nductr retance, and further lettng C pf; then L. mh, verfy that th cmbnatn

15 yeld an ω 8 rad/ and that X L X C. Quetn: what happen t the renant frequency f the capactr and the nductr value vary by ± % n prductn? De th cndtn preent an envelpe f frequence? Hw culd th be rectfed t meet a pec f ω 8 ± %? The abve crcut ha tw lp current whch we wll call t C L C & L ; and nce t.. t.x X.( 78.9 ( 79.9X ( 78.9 t ( ( X ( 9. ( 9. (.8X (.8X (.8X ( 9. (.8X Fndng the mpule repne 9.t ( t X e n(.8x t Obvuly a magntude f vlt unupprtable n a mall gnal crcut. Lke a 9 tranmtter there wuld be "arc and park". There nthng wrng wth the math; the mprbable reult a functn f a practcal mpblty. δ (t an ntellectual cntruct utable fr ntructnal purpe. A practcal apprxmatn f δ (t a fnte ampltude pule f a few nan ecnd duratn, that al pee me fnte re and fall tme. Whle the mpule repne hwn abve yeld a "peek under the tent" t de nt yeld an undtrted mrrr f ampltude realty.

16 The tranfer functn, a already develped t LC( L R R L LC C( R L R L LC re-arrangng C R L LC t ( If we gnre the re and fall tme, and mdel k ( u( t u( t, where a f a k few nan ecnd duratn and the ampltude, we get much cler t a practcal reult. We get even cler f the re and fall tme are ncluded. The entre expren fr uch a drver mght be R L t a t {( tu( t tu( t β tu( t χ tu( } k t ε Where β < χ < ε and χ β a, all n the rder f a few nan ecnd. The cntant mdfe the lpe f t. k k t -k t a There are trade-ff n ether cenar n the frm f an unwanted frequency pectrum. and al perhap n the frm f mre flterng by addng an addtnal tank r tank. Addng addtnal tank nt necearly underable, a that narrw the bandwdth and prvde ncreaed electvty (anther tpc fr a later dcun, but nt n th

17 mdule. A alway n realty we deal n trade-ff and lk fr the bet cmprme. One f the trade-ff that even thugh the abve mdel mght mre accurately reflect the actual drvng pule, t' mathematcal mplementatn fal t reflect the realty f the put. Fr the ake f mdelng and llutratn and mplcty the fllwng an example wrked fr a drver cntng f a tep f nec duratn... R. R e C L C L R R ( ( L LC L LC 8 When tranent repne the ue we huld wrk nly wth the frt expren t the rght f the equal gn a the ecnd wll be a mrrr mage, albet wth a gn change and a tme ffet. That' the rub, t erve t cancel the put qute pbly befre the crcut ha had tme t ettle. The rean fr that that the math mdel fal t fllw realty n th cae. Eentally we are treatng a δ (t a a fnte ampltude pule wth fnte duratn t lmt the predcted put ampltude t the neghbrhd f realty. The abve mdel predct an put tp cncdent wth a drver tp, a the drver mdeled a a tep. S, knwng that the tank wll cntnue t cllate untl exhauted by the tme cntant rule f (tranent repne, treat the drver a a mathematcal mpule and gnre the drver duratn WHEN t much hrter than tme cntant. Of cure f the duratn f the drver greater than tme cntant then mdel t a we dd abve. When the drver a tme f extence much maller than tme cntant the LaPlace tranfrm methd de nt handle that a well a ther methd. Why bther wth mdelng the drver a a hrt duratn tep at all? The phyc f a tank are uch that when current flw ntated, t create a back and frth tranfer f charge between the nductr and the capactr, much lke the wng f a pendulum. Intally bth the nductr and capactr are uncharged. When current flw ntated, the nductr ret the change, but the capactr charge. Eventually the rate f change f charge n the capactr dmnhe a the nductr begn t charge. When the drver ceae, the nductr ret the change and cntnue t charge at the expene f the capactr. Eventually the capactr dcharged and current flw mmentarly ceae. The nductr' magnetc feld cllape at that pnt, rechargng the capactr n the ppte plarty. The prce cntnue, back and frth tranfer f charge (cllatn untl r le exhaut the crcut (expreed a a tme cntant. The drver duratn determne the amunt f charge avalable t create current flw, a dt ; f C cure the lmt n the ntegral are the duratn f the drvng tep. In th cae C c C. The um ttal f all f th dcun that f yu che t apprxmate an actual mall duratn drver a an ampltude lmted δ (t, yur put predctn wll be much cler t the actual cae.

18 Anther, and n many way mre atfyng, way f lkng at tank current t vuale the varu flw cndtn ung Krchhff' current law. The nly aumptn we need make that when current nt flwng frm the drver (bear n mnd that we are mdelng wth a current drver, lttle r n current can flw nt the drver;.e., t behave a an pen. Cae : Intal current flw; reactve cmpnent nt yet charged. Cae : Current flwng frm bth the drver and the nductr. Cae : Current flwng frm bth the drver and the capactr. Cae : Drver ff; capactr dchargng nt the nductr. 7

19 Cae : Drver ff; nductr dchargng n t the capactr. A the abve are the nly pble cae f current flw, t farly eay t vuale what gng n wth the drver ether n r ff. The majrty f the tme Cae & predmnate a the drver generally n nly fr a mall duty cycle. can al be re-wrtten a C( R L. R L LC 8 ( e wth value... A ( 9. (.8X ( 9. (.8x.... * (.8X ( 9. (.8X ( 9. (.8x A Bell wuld ay "FAPP" (Fr All Practcal Purpe ( t e 9.t n(.8x t Our math predct a magntude f ; that becaue the nput wa mdeled by manpulatng unt tep. But the nput a n wde pule that de nt have the tme avalable t cmpletely charge the crcut (f the crcut cmpletely dcharge n tme cntant. t wll charge n tme cntant; recall that C dt C. A Furer tranfrm apprach wuld be a better apprach n generatng a mdel f the repne; but Furer tranfrm yeld a frequency pectrum and heren we cncerned wth the tme repne. B Jhn S. Bell, 98-99, Phyct extrardnare 8

20 Al an ntantaneu plarty reveral at 8 ecnd expected frm the mdel becaue we have aumed an deal drver wth ntantaneu re and fall tme. d d Intantaneu "anythng" are generally nt pble. Snce L L and fr an dt dt ntantaneu change (a dcntnuty,.e., the drver ha tw dfferent multaneu value the mdel fal. That mut be kept n mnd t make the fnal reult meanngful. That anther rean why we fall back n FAPP. FAPP ften an engneer' bet frend. A lttle bt f drvng current yeld an ftme magnfcent put, metme mre than the crcut ntended fr and addtnal crcutry requred t keep thng ane. Neverthele, even a drver that a tmed and equenced unt tep de nt yeld an undtrted pcture f realty, but t much cler than the mpule repne. A lk at the tme cntant ndcate that the put ha decayed t.7% f max at ab m. Suppe t dered that the crcut never be allwed t decay belw 9% f max. What the nterval needed between drvng mpule? S the crcut puled every.7m. e 9.τ.9 9.τ ln e ln.9 τ.7m In the cae abve, the drver n fr n and cycle tme.7m; the duty cycle ab.%. That a pretty decent tank. 9. In the abve example, remember the tme cntant actually, a the unt n 9. R MUST be t. 9. repreent L, therefre the tme cntant L - de th R expren have unt f t? Ye, t de. Nt develped n th dcun the bandwdth f the tank and pectrum f the put. Bth tpc are f crtcal nteret t the analyt and t the degner. Nether can be gnred n practce. They have been tactly gnred n th dcun a the ntentn here t fcu n the tme repne. Dcun and develpment f tank bandwdth and ablty t dcrmnate agant unwanted frequence wll be delayed untl a later mdule. Neverthele, t truly the frequency cntent f the drvng functn that caue ympathetc cllatn n the crcut. That apect ha been ttally gnred n th 9

21 dcun a t nt needed n the develpment f a mple tme repne. But a we hall ee n ubequent mdule frequency a majr player, ften the 8 pund Grlla, n crcut tablty/ntablty cnderatn.

22 Tranfrm f (t F ( K t Ke σ K n( ω t K c( ω t σ Ke t n( ωt Ke t K K σ Kω ω K ω Kω ( σ ω σ c( ωt K ( σ ( σ ω ( 7 δ (t 7a* Kδ (t K 8 Ku( t a Ke a 9 f '( t F ( f ( f ( t dt F ( af ( t bg( t af ( bg( t at te ( Table a * K preerved fr practcal crcut rean, nt fr theretcal rean a K apprxmately equal t It very mprtant t undertand that t be able tranfrm any F ( t an f (t, F ( mut be reduced t ne f the frm far develped. If t nt n ne f thee frm t cannt be perated n untl t. Study the rght hand de frm, they dentfy the left hand de.

23 Appendx A Cramer' Rule Refreher Th appendx nt ntended a a tutral, but rather a a refreher fr the that want a remnder f hw the prce prceed. Suppe there a ytem f equatn repreentng a mple tw lp crcut cntanng tw unknwn and tw equatn, uch a a b a b n In the abve cae & are the unknwn and everythng ele knwn. Then ung the ntatn f Algebra we can re-wrte thee equatn a a a n b b In rder t lve fr the tw unknwn, frt the determnant (Det. fund by multplyng the element f left t rght dagnal, and then ubtractng the multplcatn f element n the rght t left dagnal. Det. ab ba Next, t fnd, the rghtmt clumn cntanng & cntanng a & a. n ubttuted fr the clumn n b b Then fnd the determnant f that new matrx. n b b Next dvde by the Det., the reult the value f. S b b n a b b a

24 The term b er, and ncluded nly fr cmpletene a the drver n the ecnd lp nt alway er. Fnally lve fr by ubttutng & fr the clumn cntanng b & b. n a a n and dvde by Det. a a a b n b a Pleae bear n mnd that the ecnd lp may have a drver and therefre the left hand de f the decrbng equatn wll nt be er. Al, there n cntrant n the matrx element t be real, n fact n actual crcutry they are mre than ften cmplex. Al becaue the rule f LaPlace Tranfrm par allw addtn, the whle et f equatn may be wrtten n the '' dman. Optnal: Fr a hrt dcun f why th technque wrk, ee appendx B. An example; Z Z n Z Z Frm the abve, n ( ( The determnant ( ( r Det.

25 Then ( n Det. and Det. ( n Aume: n,, Then Det. 7 and and Suppe, then. 8 Fr a three lp crcut, there wll be three equatn and three unknwn. Z Z Z n Z Z Z The prcedure fr any matrx greater than a x a n the abve example, extended and mdfed lghtly. In the cae f the three lp crcut there a three clumn and three rw; ( ( ( ( ( ( n The determnant matrx wll be

26 ( ( ( At th pnt a mdfcatn ccur. There are three clumn, and therefre there mut be three term fr bth the left and rght hand dagnal. A frequent crutch that wrk t merely cpy clumn & t the rght f the matrx; that yeld three cmplete dagnal n each drectn. T extend the rule, an nxn matrx requre n dagnal n each drectn. ( ( ( ( ( the determnant then ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( A an example, uppe that three equatn n three unknwn are. The matrce are The determnant 7 ( ( ( ( (( ((( ( (( ( ( ( (( (

27 The mnu gn n tw. merely mean that t flwng n a drectn ppte t the ther Fr matrce greater than three rw by three clumn (x, the labr ge up gnfcantly, and the ue f a calculatr uch a an HP9 r a cmputer prgram mlar t MATLAB very helpful. Hwever fr the that enjy the labr the fllwng rule are ffered: The determnant the algebrac um f all the pble prduct where: a. each prduct ha factr f ne element, and nly ne, frm each rw and clumn b. a plu gn agned t each prduct f the number f clumn nvern even, nvern beng defned a even. A mnu gn agned t a prduct that ha an dd number f clumn nvern. An nvern, by llutratn, that f the natural rder f cuntng and a prduct frmed frm clumn then t cntan tw nvern; t change t, the mut mve tw place t the rght. ha x nvern a the mve three place, the mve tw place and the mve ne t create. Thee rule are mply the prcedure ued fr a x extended t an nxn. There are ther, equally vald technque frm ur lde-rule day, uch a Gauan Elmnatn, Matrx Invern and the ue f Mnr, but all n all nce yu undertand the fundatn f the prce the ue f a gd calculatr ncredbly labr and errr avng. Of cure t the undertandng f thee technque that frm the fundatn fr the prgrammng n the calculatr' ROM lbrary.

28 Suppe Ax By P Cx Dy Q Appendx B Fundatn then By P Ax and y P Ax B then P Ax Cx D Q and becme CBx DP ADx BQ B whch, n turn becme x( CB AD BQ DP r x PD AD BQ BC Ung Cramer' rule the matrce fr the tw ntal equatn are x P Q A C B D B D PD BQ AD BC Fr the wth nfnte tamna th prcedure can be extended t any number f equatn that pe the ame number n unknwn. But the authr, beng a member f Layhd Incrprated, ue mechancal mean nce the thery etablhed. 7

29 Appendx C Cmmn Bae Amplfer Cnder the fllwng crcut R e c C E e be cb R lad B n cc Th crcut cnt f an NPN trantr, frward baed emtter t bae ( and revere baed cllectr t bae (. Typcally n the rder f apprxmately.7 vlt, the current thrugh R e, be be be I e n.7 R e The phyc f the trantr are uch that the current thrugh the cllectr and hence thrugh R alway I (true wthn the manufacturer peratng lad charactertc range fr the partcular trantr type. Therefre adjutng I e, whch n turn cntrl the lad current. In hrt e n adjut I. 99 c I e Obvuly th make the current thrugh the lad utterly dependent n dependent upn the value chen fr R e & n. I e, whch n turn The abve a very prmtve vern f a cmmn bae amplfer, and degn cnderatn f cuplng, mpedance, bandwdth, emtter retance, etc., have been utterly gnred a t fcu n the current generatr effect at the cllectr. The mt cmmn pnt f cnfun that f the cllectr current 99% f the emtter current, what ha happened t hm' law n the cllectr t bae lp. Nthng actually. Krchhff' vltage equatn fr that lp 8

30 I R cc e lad cb A the trantr actve, and cntraned by t' phyc, the vltage law. cb adjut t accmmdate 9

31 Appendx D Renance defned t ccur when X X. Fr a ere crcut the ttal mpedance L C R t jx L jx C It clear at renance the mpedance at mnmum and depend nly upn R t. In a parallel crcut the abve um the denmnatr f the expren f ttal mpedance. Agan t clear that t a mnmum at renance. A mnmum denmnatr create a maxmum mpedance. A a rule f thumb at renance: a a ere crcut apprache a hrt, and b a parallel crcut apprache an pen. Snce X L jπfl, and X C j π fc, at renance πf L πf C nce π f ω we can re-wrte a fllw ω LC an fnally ω LC

32 Appendx E A quadratc ha nly tw pble et f rt; a. bth real (may be equal r unequal r b. a cmplex par. Cnder the quadratc th rt are R L LC R ± L R L LC whch lead t R R ± L L LC when the dcrmnant, the rt are real, and when the dcrmnant < the rt are cmplex and may be re-wrtten cnvenently a R ± L j LC R L