TRANSIENT ANALYSIS OF ELECTRIC POWER CIRCUITS BY THE CLASSICAL METHOD IN THE EXAMPLES

Transcription

1 MINISTY OF EDUCATION AND SCIENCE OF UKAINE NATIONA AVIATION UNIVESITY EECTICA AND IGHTING ENGINEEING DEPATMENT A.A.ZEENKOV TANSIENT ANAYSIS OF EECTIC POWE CICUITS BY THE CASSICA METHOD IN THE EXAMPES Trag book KYIV 9

2 UDC 6.3(76) evewer G.T.Gorohov PhD, aocate profeor, Seor-cetfc worker of Ukrae Arforce cetfc ceter. Approved by the CSF draftg edtoral board of Electroc ad cotrol ytem ttute, March 9. ZEENKOV A.A. TANSIENT ANAYSIS OF EECTIC POWE CICUITS BY THE CASSICA METHOD IN THE EXAMPES : Trag book K.: NAU, p. The maual TANSIENT ANAYSIS OF EECTIC POWE CICUITS BY THE CASSICA METHOD IN THE EXAMPES teded for the tudet of the eor coure of the electrcal pecalte, ad thoe learg automatc cotrol theory. The am of th book to help tudet to mater the theory ad method of olvg problem appled electrcty. The book cota typcal problem oluto whch gve better ght to the theory ad the phycal ature of varou pheomea, ugget dfferet approache to the problem, ad llutrate the applcato of varou theoretcal prcple. The author ha tred to follow a mddle path betwee rgor ad completee o oe had ad applcato to practcal tuato o the other. The order whch the topc appear that foud motly ucceful log experece of teachg the ubject. Gettg through the coure EECTICA ENGINEEING FUNDATION the tudet may fd th work of atace preparg for the examato. The teacher may alo fd t ueful.

3 CONTENTS PEFACE CHAPTE. CASSICA APPOACH TO TANSIENT ANAYSIS.. Itroducto 6.. Appearace of traet electrcal crcut 8.3. Dfferetal equato decrbg electrcal crcut.3.. Expoetal oluto of a mple dfferetal equato 4.4. Natural ad forced repoe 9.5. Charactertc equato ad t determato.6. oot of the charactertc equato ad dfferet kd of traet repoe Frt order charactertc equato Secod order charactertc equato 3.7. Idepedet ad depedet tal codto Two wtchg law (rule) 34 (a) Frt wtchg law (rule) (b) Secod wtchg law (rule).7.. Method of fdg depedet tal codto Method of fdg depedet tal codto Geeralzed tal codto 43 (a) Crcut cotag capactace (b) Crcut cotag ductace 3

4 .8. Method of fdg tegrato cotat 56 CHAPTE. TANSIENT ESPONSE OF BASIC CICUITS 6.. Itroducto 6.. The fve tep of olvg problem traet aaly 6.3. crcut crcut uder d.c. upply crcut uder a.c. upply C crcut Dchargg ad chargg a capactor C crcut uder d.c. upply C crcut uder a.c. upply C crcut C crcut uder d.c. upply.4 (a) Sere coected C crcut (b) Parallel coected C crcut (c) Natural repoe by two ozero tal codto.5.. C crcut uder a.c. upply Traet C reoat crcut 36 (a) Swtchg o a reoat C crcut to a a.c. ource (b) eoace at the fudametal (frt) harmoc (c) Frequecy devato reoat crcut.5.4. Swtchg off C crcut.. 44 (a) Iterrupto a reoat crcut fed from a a.c. ource 4

5 PEFACE Mot of the textbook o electrcal ad electroc egeerg oly partally cover the topc of traet mple, C ad C crcut ad the tudy of th topc prmarly doe from a electroc egeer vewpot,.e., wth a empha o low-curret ytem, rather tha from a electrcal egeer vewpot, whoe teret le hgh-curret, hghvoltage power ytem. I uch ytem a very clear dfferetato betwee teady-tate ad traet behavour of crcut made. Such a dvo baed o the cocept that teady-tate behavour ormal ad traet are from the fault. The operato of mot electroc crcut (uch a ocllator, wtch capactor, rectfer, reoat crcut etc.) baed o ther traet behavour, ad therefore the traet here ca be referred to a derable. The traet power ytem are characterzed a completely uderable ad hould be avoded; ad ubequetly, whe they do occur, ome very crtcal tuato, they may reult the electrcal falure of large power ytem ad outage of bg area. Hece, the Ittute of Electrcal ad Electroc Egeer (IEEE) ha recetly pad eormou atteto to the mportace of power egeerg educato geeral, ad traet aaly partcular. It wth the belef that traet aaly of power ytem oe of the mot mportat topc power egeerg aaly that the author proudly preet th book, whch wholly dedcated to th topc. Of coure, there are may good book th feld, ome of whch are lted the book; however they are wrtte o a pecfc techcal level or o a hgh theoretcal level ad are teded for top pecalt. O the other had, troductory coure, a wa already metoed, oly gve a uperfcal kowledge of traet aaly. So that there a gap betwee troductory coure ad the above book. The preet book deged to fll th gap. It cover the topc of traet aaly from mple to complcated, ad beg o a termedate level, th book therefore a lk betwee troductory coure ad more pecfc techcal book. The approprate level ad the cocetrato of all the topc uder oe cover make th book very pecal the feld uder coderato. The author beleve that th book wll be very helpful for all thoe pecalzg electrcal egeerg ad power ytem. It recommeded a a textbook for pecalzed uder graduate ad graduate currculum, ad ca alo be ued for mater ad doctoral tude. Egeer 5

6 the feld may alo fd th book ueful a a hadbook ad / or reource book that ca be kept hady to revew pecfc pot. Theoretca / reearcher who are lookg for the mathematcal backgroud of traet electrc crcut may alo fd th book helpful ther work. The preetato of the covered materal geared to reader who are beg expoed to (a) the bac cocept of electrc crcut baed o ther earler tudy of phyc ad / or troductory coure crcut aaly, ad (b) bac mathematc, cludg dfferetato ad tegrato techque Th book compoed of two chapter. The tudy of traet, a metoed, preeted from mple to complcated. Chapter ad are dedcated to the clacal method of traet aaly, whch tradtoal for may troductory coure. However, thee two chapter cover much more materal gvg the mathematcal a well a the phycal vew of traet behavour of electrcal crcut. So-called correct tal codto ad two geeralzed commutato law, whch are mportat for a better udertadg of the traet behavour of traformer ad ychroou mache, are alo dcued Chapter. CHAPTE. CASSICA APPOACH TO TANSIENT ANAYSIS.. INTODUCTION Traet aaly(or jut traet) of electrcal crcut a mportat a teady-tate aaly. Whe traet occur, the curret ad voltage ome part of the crcut may may tme exceed thoe that ext ormal behavour ad may detroy the crcut equpmet t proper operato. We may dtguh the traet behavour of a electrcal crcut from t teady-tate, that durg the traet all the quatte, uch a curret, voltage, power ad eergy, are chaged tme, whle teady-tate they rema varat,.e. cotat ( d.c. operato) or perodcal ( a.c. operato) havg cotat ampltude ad phae agle. The caue of traet ay kd of chagg crcut parameter ad/or crcut cofgurato, whch uually occur a a reult of wtchg (commutato), hort, ad/or ope crcutg, chage the operato of ource etc. The chage of curret, voltage etc. durg the traet are ot tataeou ad take ome tme, eve though they are extremely fat wth a durato of mllecod or eve mcroecod. Thee very fat chage, however, caot be tataeou (or abrupt) ce the traet 6

7 procee are attaed by the terchage of eergy, whch uually tored the magetc feld of ductace or/ad the electrcal feld of capactace. Ay chage eergy caot be abrupt otherwe t wll dw reult fte power (a the power a dervatve of eergy, p ), dt whch cotrat to phycal realty. All traet chage, whch are alo called traet repoe (or jut repoe), vah ad, after ther dappearace, a ew teady-tate operato etablhed. I th repect, we may ay that the traet decrbe the crcut behavour betwee two teady-tate: a old oe, whch wa pror to chage, ad a ew oe, whch are after the chage. A few method of traet aaly are kow: the clacal method, The Cauchy-Heavde (C-H) operatoal method, the Fourer traformato method ad the aplace traformato method. The C-H operatoal or ymbolc (formal) method baed o replacg a dervatve d by ymbol ad a tegral by dt dt Although thee operato are alo ued the aplace traform method, the C-H operatoal method ot a ytematc ad a rgorou a the aplace traform method, ad therefore t ha bee abadoed favour of the aplace method. The two traformato method, aplace ad Fourer, wll be tuded the followg chapter. Comparg the clacal method ad the traformato method t hould be oted that the latter requre more kowledge of mathematc ad le related to the phycal matter of traet behavour of electrc crcut tha the former. Th chapter cocered wth the clacal method of traet aaly. Th method baed o the determato of dfferetal equato ad plttg the oluto to two compoet: atural ad forced repoe. The clacal method farly complcated mathematcally, but mple egeerg practce. Thu, our preet tudy we wll apply ome kow method of teady-tate aaly, whch wll allow u to mplfy the clacal approach of traet aaly. 7

8 .. APPEAANCE OF TANSIENTS IN EECTICA CICUITS I the aaly of a electrcal ytem (a ay phycal ytem), we mut dtguh betwee the tatoary operato or teady-tate ad the dyamcal operato or traet-tate. A electrcal ytem ad to be teady-tate whe the varable decrbg t behavour (voltage, curret, etc.) are ether varat wth tme (d.c. crcut) or are perodc fucto of tme (a.c. crcut). A electrcal ytem ad to be traet-tate whe the varable are chaged o-perodcally,.e., whe the ytem ot teady-tate. The traet-tate vahe wth tme ad a ew teady-tate regme appear. Hece, we ca ay that the traet-tate, or jut traet, uually the tramo tate from oe teady-tate to aother. The parameter ad C are characterzed by ther ablty to tore eergy: magetc eergy w ψ the magetc feld ad electrc eergy w C qv Cv the electrc feld of the crcut. The voltage ad curret ource are the elemet through whch the eergy uppled to the crcut. Thu, t may be ad that a electrcal crcut, a a phycal ytem, characterzed by certa eergy codto t teady-tate behavour. Uder teady-tate codto the eergy tored the varou ductace ad capactace, ad uppled by the ource a d.c. crcut, are cotat; wherea a a.c. crcut the eergy beg chaged (traferred betwee the magetc ad electrc feld ad uppled by ource) perodcally. Whe ay udde chage occur a crcut, there uually a redtrbuto of eergy betwee - ad C-, ad a chage the eergy tatu of the ource, whch requred by the ew codto. Thee eergy dtrbuto caot take place tataeouly, but durg ome 8

9 perod of tme, whch brg about the traet-tate. The ma reao for th tatemet that a tataeou chage of eergy would requre fte power, whch aocated wth ductor/capactor. A prevouly metoed, power a dervatve of eergy ad ay abrupt chage eergy wll reult a fte power. Sce fte power ot realzable phycal ytem, the eergy caot chage abruptly, but oly wth ome perod of tme whch traet occur. Thu, from a phycal pot of vew t may be ad that the traettate ext phycal ytem whle the eergy codto of oe teadytate are beg chaged to thoe of aother. Our ext cocluo about the curret ad voltage. To chage magetc eergy requre a chage of curret through ductace. Therefore, curret ductve crcut, or ductve brache of the crcut, caot chage abruptly. From aother pot of vew, the chage of d curret a ductor brg about the duced voltage of magtude dt.a tataeou chage of curret would therefore requre a fte voltage, whch alo urealzable practce. Sce the duced voltage dψ alo gve a, where ψ a magetc flux, the magetc flux of a dt crcut caot uddely chage. Smlarly, we may coclude that to chage the electrc eergy q requre a chage voltage acro a capactor, whch gve by v, C where q the charge. Therefore, ether the voltage acro a capactor or t charge ca be abruptly chaged. I addto, the rate of voltage chage dv dq, ad the tataeou chage of voltage brg about dt C dt C fte curret, whch alo urealzable practce. Therefore, we may ummarze that ay chage a electrcal crcut, whch brg about a chage eergy dtrbuto, wll reult a traet-tate. I other word, by ay wtchg, terruptg, hort-crcutg a well a ay rapd chage the tructure of a electrc crcut, the traet pheomea wll occur. Geerally peakg, every chage of tate lead to a temporary devato from oe regular, teady-tate performace of the crcut to aother oe. The redtrbuto of eergy, followg the above 9

10 chage,.e., the traet-tate, theoretcally take fte tme. However, realty the traet behavour of a electrcal crcut cotue a relatvely very hort perod of tme, after whch the voltage ad curret almot acheve ther ew teady-tate value. The chage the eergy dtrbuto durg the traet behavour of electrcal crcut govered by the prcple of eergy coervato,.e., the amout of uppled eergy equal to the amout of tored eergy plu the eergy dpato. The rate of eergy dpato affect the tme terval of the traet. The hgher the eergy dpato, the horter the traet-tate. Eergy dpato occur crcut retace ad t torage take place ductace ad capactace. I crcut, whch cot of oly retace, ad ether ductace or capactace, the traet-tate wll ot occur at all ad the chage from oe teady-tate to aother wll take place tataeouly. However, ce eve retve crcut cota ome ductace ad capactace the traet wll practcally appear alo uch crcut; but thee traet are very hort ad ot gfcat, o that they are uually eglected. Traet electrcal crcut ca be recogzed a ether derable or uderable. I power ytem etwork, the traet pheomea are wholly uderable a they may brg about a creae the magtude of the voltage ad curret ad the dety of the eergy ome or mot part of moder power ytem. All of th mght reult equpmet dtorto, thermal ad/or electrodyamc detructo, ytem tablty terferece ad extreme cae a outage of the whole ytem. I cotrat to thee uwated traet, there are derable ad cotrolled traet, whch ext a great varety of electroc equpmet commucato, cotrol ad computato ytem whoe ormal operato baed o wtchg procee. The traet pheomea occur electrc ytem ether by tetoal wtchg procee cotg of the correct mapulato of the cotrollg apparatu, or by utetoal procee, whch may are from groud fault, hort-crcut, a break of coductor ad/or ulator, lghtg troke (partcularly hgh voltage ad log dtace ytem) ad mlar advertet procee. A wa metoed prevouly, there are a few method of olvg traet problem. The mot wdely kow of thee appear all troductory textbook ad ued for olvg mpler problem. It called the clacal method. Other ueful method are aplace ad Fourer

11 traformato method. Thee two method are more geeral ad are ued for olvg problem that are more complcated..3. DIFFEENTIA EQUATIONS DESCIBING EECTICA CICUITS Crcut aaly, a a phycal ytem, completely decrbed by tegrodfferetal equato wrtte for voltage ad/or curret, whch characterze crcut behavour. For lear crcut thee equato are called lear dfferetal equato wth cotat coeffcet,.e. whch every term of the frt degree the depedet varable or oe of t dervatve. Thu, for example, for the crcut of three bac elemet:, ad C coected ere ad drve by a voltage ource v(t), Fg.., we may apply Krchhoff voltage law whch ad the we have v + v + v v(t), C v v v C d dt dt, d + + dt v( t). (.) dt C v( t) (t) C Fg.. After the dfferetato of both de of equato. wth repect to tme, the reult a ecod order dfferetal equato

12 d d dv + + (.) dt dt c dt The ame reult may be obtaed by wrtg two multaeou frt order dfferetal equato for two ukow, ad v C : dv C (.3a) dt C d + + vc v(t). (.3b) dt dv After dfferetato of equato.3b ad ubttutg C dt by equato.3a, we obta the ame (a equato.) ecod order gular equato. The oluto of dfferetal equato ca be completed oly f the tal codto are pecfed. It obvou that the ame crcut uder the ame commutato, but wth dfferet tal codto, t traet repoe wll be dfferet. For more complcated crcut, bult from a umber of loop (ode), we wll have a et of dfferetal equato, whch hould be wrtte accordace wth Krchhoff two law or wth odal ad/or meh aaly. For example, coderg the crcut how Fg.., after wtchg, we wll have a crcut, whch cot of two loop ad two ode. By applyg Krchhoff two law, we may wrte three equato wth three ukow,, ad v C, dvc C + dt (.4a) d + + dt (.4b) d + vc dt (.4c)

13 (t) (t) 3 (t) v( t) C Fg.. Thee three equato ca the be redudatly traformed to a gle ecod order equato. Frt, we dfferetate the thrd equato of dv.4c oce wth repect to tme ad ubttute C by takg t from the dt frt oe. After that, we have two equato wth two ukow, ad. Solvg thee two equato for (.e. elmatg the curret ) reult the ecod order homogeeou dfferetal equato d d C + ( + C ) + ( + ) (.5) dt dt A aother example, let u coder the crcut Fg..3. Applyg meh aaly, we may wrte three tegro-dfferetal equato wth three ukow meh curret: d d + v( t ) dt dt d d + ( + 3) 33 (.6) dt dt dt. C I th cae t preferable to olve the problem by treatg the whole et of equato.6 rather tha reducg them to a gle oe (ee further o). 3

14 v(t) 3 3 C Fg..3 From mathematc, we kow that there are a umber of way of olvg dfferetal equato. Our goal th chapter to aalyze the traet behavour of electrcal crcut from the phycal pot of vew rather tha applyg complcated mathematcal method. (Th wll be dcued the followg chapter.) Such a way of traet aaly the formulato of dfferetal equato accordace wth the properte of the crcut elemet ad the drect oluto of the obtaed equato, ug oly the eceary mathematcal rule. Such a method called the clacal method or clacal approach traet aaly. We beleve that the clacal method of olvg problem eable the tudet to better udertad the traet behavour of electrcal crcut..3. Expoetal oluto of a mple dfferetal equato et u, therefore, beg our tudy of traet aaly by coderg the mple ere C crcut, how Fg..4. After wtchg we wll get a ource free crcut whch the capactor C wll be dcharged va the retace. To fd the capactor voltage we hall wrte a dfferetal equato, whch accordace wth Krchhoff voltage law become dvc C + v c (.7) dt A drect method of olvg th equato to wrte the equato uch a way that the varable are eparated o both de of the equato ad the to tegrate each of the de. Multplyg by dt ad dvdg by v c, we may arrage the varable to be eparated, dv v c c dt (.8) C 4

15 The oluto may be obtaed by tegratg each de of equato.8 ad by addg a cotat of tegrato: dv v dt + c c K, C v(t) C u C Fg..4 ad the tegrato yeld l v c t + K (.9) C Sce the cotat ca be of ay kd, ad we may wrte K l D, we have l v c t + l D, C the t v C c De (.) The cotat D ca ot be evaluated by ubttutg equato. to the orgal dfferetal equato.7, ce the detty wll t t reult for ay value of D (deed: D Ce C + De C ). The C cotat of tegrato mut be elected to atfy the tal codto v c ( ) V, whch the tal voltage acro the capactace. Thu, the oluto of equato. at t become v c ( ) D, ad we may coclude that D V. Therefore, wth th value of D we wll obta the dered repoe 5

16 t v C c( t) V e (.) We hall coder the ature of th repoe. At zero tme, the voltage the aumed value V ad, a tme creae, the voltage decreae ad approache zero, followg the phycal rule that ay codeer hall fally be dcharged ad t fal voltage therefore reduce to zero. et u ow fd the tme that would be requred for the voltage to drop to zero f t cotued to drop learly at t tal rate. Th value of tme, uually degated by t, called the tme cotat. The value of t ca be foud wth the dervatve of v c (t) at zero tme, whch proportoal to the agle c betwee the taget to the voltage curve at t, ad the t-ax,.e., or d dt Ve t C 6 t V C τ C ad equato. mght be wrtte the form v ( t) c t V τ e (.) The ut of the tme cotat are ecod ([τ] [][C] Ω F), o that the expoet t/c dmeole, a t uppoed to be. Aother terpretato of the tme cotat obtaed from the fact that the tme terval of oe tme cotat the voltage drop relatvely to t tal value, to the recprocal of e; deed, at t τ we have v c e,368 (36,8%). At the ed of the 5t terval the voltage le V tha oe percet of t tal value. Thu, t uual to preume that the tme terval of three to fve tme cotat, the traet repoe decle to zero or, other word, we may ay that the durato of the traet

17 repoe about fve tme cotat. Note aga that, precely peakg, t the traet repoe decle to zero fte tme, ce e, whe t. Before we cotue our dcuo of a more geeral aaly of traet crcut, let u check the power ad eergy relatohp durg the perod of traet repoe. The power beg dpated the retor, or t recprocal G, t p C Gvc GV e, (.3) ad the total dpated eergy (tured to heat) foud by tegratg equato.3 from zero tme to fte tme w t t C p dt VG e C V G e C CV. Th actually the eergy beg tored the capactor at the begg of the traet. Th reult mea that all the tal eergy, tored the capactor, dpate the crcut retace durg the traet perod. Example. Coder a umercal example. The crcut Fg..5 fed by a d.c. curret ource, I 5A. At tat t the wtch cloed ad the crcut hort-crcuted. Fd:) the curret after wtchg, by eparatg the varable ad applyg the defte tegral, ) the voltage acro the ductace. 4Ω I mh Fg..5 7

18 Soluto ) Frt, we hall wrte the dfferetal equato: d v + v +, dt or after eparatg the varable d dt. Sce the curret chage from I at the tat of wtchg to (t), at ay tat of t, whch mea that the tme chage from t to th tat, we may perform the tegrato of each de of the above equato betwee the correpodg lmt Therefore, ad or whch reult ( t) I l d t dt. ( t) t I t l ( t) l I ( t) l t, I ( t) e I t. t Thu, ( t) t t Ie 5e, 8

19 or t t 3 τ,5 ( t) Ie 5e, where 4 3 whch reult tme cotat,5 m. Note that by applyg the defte tegral we avod the tep of evaluatg the cotat of the tegrato. ) The voltage acro the ductace, t, ( 5e ) e, V d d v ( t) dt dt 5. Note that the voltage acro the retace v t,5 t,5 4 5e e,.e., t equal magtude to the ductace voltage, but oppote g, o that the total voltage the hort-crcut equal to zero. t.4 NATUA AND FOCED ESPONSES Our ext goal to troduce a geeral approach to olvg dfferetal equato by the clacal method. Followg the prcple of mathematc we wll coder the complete oluto of ay lear dfferetal equato a compoed of two part: the complemetary oluto (or atural repoe our tudy) ad the partcular oluto (or forced repoe our tudy). To udertad thee prcple, let u coder a frt order dfferetal equato, whch ha already bee derved the 9

20 prevou ecto. I a more geeral form t dv + P( t) v Q( t) (.4) dt Here Q(t) detfed a a forcg fucto, whch geerally a fucto of tme (or cotat, f a d.c. ource appled) ad P(t), alo geerally a fucto of tme, repreet the crcut parameter. I our tudy, however, t wll be a cotat quatty, ce the value of crcut elemet doe ot chage durg the traet (deed, the crcut parameter do chage durg the traet, but we may eglect th chage a may cae t ot gfcat). A more geeral method of olvg dfferetal equato, uch a equato.4, to multply both de by a o-called tegratg factor, o that each de become a exact dfferetal, whch afterward ca be tegrated drectly to obta the oluto. For the equato above (equato.4) the tegratg factor e Pdt or e Pt, ce P cotat. We multply each de of the equato by th tegratg factor ad by dt ad obta Pt Pt Pt e dv + vpe dt Qe dt. The left de ow the exact dfferetal of ve Pt (deed, d Pt Pt Pt ( ve ) e dv + vpe dt ), ad thu Pt Pt ( ve ) Qe dt d. Itegratg each de yeld Pt Pt ve Qe dt + A, (.5) where A a cotat of tegrato. Fally, the multplcato of both de of equato.5 by yeld v Pt e Pt Pt Pt ( t) e Qe dt + Ae, (.6) whch the oluto of the above dfferetal equato. A we ca ee, th complete oluto compoed of two part. The frt oe, whch depedet o the forcg fucto Q, the forced repoe (t alo called

21 the teady-tate repoe or the partcular oluto or the partcular tegral). The ecod oe, whch doe ot deped o the forcg fucto, but oly o the crcut parameter P (the type of elemet, ther value, tercoecto, etc) ad o the tal codto A,.e., o the ature of the crcut, the atural repoe. It alo called the oluto of the homogeeou equato, whch doe ot clude the ource fucto ad ha aythg but zero o t rght de. Followg th rule, we wll olve dfferetal equato by fdg atural ad forced repoe eparately ad combg them for a complete oluto. Th prcple of dvdg the oluto of the dfferetal equato to two compoet ca alo be udertood by applyg the uperpoto theorem. Sce the dfferetal equato, uder tudy, are lear a well a the electrcal crcut, we may aert that uperpoto alo applcable for the traet-tate. Followg th prcple, we may ubdvde, for tace, the curret to two compoet ' '' +, ad by ubttutg th to the et of dfferetal equato, ay of the form d dt C we obta the followg two et of equato + dt d dt + v, ' + v ' ' + dt '' d + dt '' C '' + dt C It obvou that by ummato (upermpoto) of thee two equato, the orgal equato wll be acheved. Th mea that '' a atural repoe ce t the oluto of a homogeeou equato wth a zero o the rght de ad develop wthout ay acto of ay ource, ad ' a teady-tate curret a t develop uder the acto of the voltage ource v (whch are preeted o the rght de of the equato).

22 The mot dffcult part the clacal method of olvg dfferetal equato evaluatg the partcular tegral equato.6, epecally whe the forcg fucto ot a mple d.c. or expoetal ource. However, crcut aaly we ca ue all the method: ode/meh aaly, crcut theorem, the phaor method for a.c. crcut (whch are all gve troductory coure o teady-tate aaly) to fd the forced repoe. I relato to the atural repoe, the mot dffcult part to formulate the charactertc equato (ee further o) ad to fd t root. Here crcut aaly we alo have pecal method for evaluatg the charactertc equato mply by pecto of the aalyzed crcut, avodg the formulato of dfferetal equato. Fally, t worthwhle to clarfy the ue of expoetal fucto a a tegratg factor olvg lear dfferetal equato. A we have ee the prevou ecto, uch dfferetal equato geeral cot of the ecod (or hgher) dervatve, the frt dervatve ad the fucto telf, each multpled by a cotat factor. If the um of all thee dervatve (the fucto telf mght be treated a a dervatve of order zero) acheve zero, t become a homogeeou equato. A fucto whoe dervatve have the ame form a the fucto telf a expoetal fucto, o t may atfy thee kd of equato. Subttutg th fucto to the dfferetal equato, whoe rght de zero (a homogeeou dfferetal equato) the expoetal factor each member of the equato mght be mply croed out, o that the remag equato coeffcet wll be oly crcut parameter. Such a equato called a charactertc equato..5 CHAACTEISTIC EQUATION AND ITS DETEMINATION et u tart by coderg the mple crcut whch a ere wtchg o to a d.c. voltage ource. et the dered repoe th crcut be curret (t). We hall frt expre t a the um of the atural ad forced curret + f. The form of the atural repoe, a wa how, mut be a t expoetal fucto, Ae. Subttutg th repoe to the

23 d homogeeou dfferetal equato, whch +, we obta dt t t e + e, or +. (.7a) Th a charactertc (or auxlary) equato, whch the left de expree the put mpedace ee from the ource termal of the aalyzed crcut. Z ( ) +. (.7b) We may treat a the complex frequecy σ + jω. Note that by equalg th expreo of crcut mpedace to zero, we obta the charactertc equato. Solvg th equato we have ad τ. (.8) Hece, the atural repoe t. (.9) Ae Subequetly, the root of the charactertc equato defe the expoet of the atural repoe. The fact that the put mpedace of the crcut hould be equaled to zero ca be explaed from a phycal pot of vew (th fact prove more correctly mathematcally aplace traformato). Sce the atural repoe doe ot deped o the ource, the latter hould be klled. Th acto reult hort-crcutg the etre crcut,.e. t put mpedace. Coder ow a parallel crcut wtchg to a d.c. curret ource whch the dered repoe v (t), a how Fg..6. Here, kllg the curret ource reult ope-crcutg. I Fg..6 3

24 Th mea that the put admttace hould be equaled to zero. Thu, +, or +, whch however gve the ame root ad τ. (.) Next, we wll coder a more complcated crcut, how Fg..7(a). Th crcut, after wtchg ad hort-crcutg the remag voltage ource, wll be a how Fg..7(b). The put mpedace of th crcut meaured at the wtch (whch the ame a ee from the klled ource) or Z ( ) + 3 // 4 //( + ) Z ( ) Evaluatg th expreo ad equalg t to zero yeld ( ad the root + + )( + ) +, eq, eq It worthwhle to meto that the ame reult ca be obtaed f the put mpedace meaured from the ductace brach,.e. the eergy-torg elemet, a how Fg..7(c)

25 z () v 3 4 v 3 4 a b 3 4 z () c. Fg..7 The charactertc equato ca alo be determed by pecto of the dfferetal equato or et of equato. Coder the ecod-order dfferetal equato lke equato. d ( t) d( t) + + ( t) g( t) (.) dt dt C eplacg each dervatve by, where the order of the dervatve (the fucto by telf codered a a zero-order dervatve), we may obta the charactertc equato: + + (.) C Th charactertc equato of the ecod order ( accordace wth the ecod order dfferetal equato) ad t poee two root ad. If ay ytem decrbed by a et of tegro-dfferetal equato, lke equato.6, the we hall frt rewrte t a lghtly dfferet form a homogeeou equato 5

26 d d dt dt d d (.3) dt dt 3 + dt C eplacg the dervatve ow by ad a tegral by a tegral a couter vero of a dervatve) we have ( + ) ( + + ) (ce (.4) C We obtaed a et of algebrac equato wth the rght de equal to zero. I the matrx form (.4a) C Wth Cramer rule the oluto of th equato ca be wrtte a Δ Δ Δ3 3 (.4b) Δ Δ Δ where Δ the determat of the ytem matrx ad determat Δ, Δ, Δ are obtaed from Δ, by replacg the approprate colum ( 3 Δ the frt colum replaced, Δ the ecod colum replaced, ad o forth), by the rght de of the equato,.e. by zeroe. A kow from mathematc uch determat are equal to zero ad for the o-zero oluto equato.4 the determat Δ the deomator mut alo be 6

27 zero. Thu, by equalg th determat to zero, we get the charactertc equato: C or + )( + + 3) + 3 ( + ) 3 + C C Smplfyg th equato yeld a ecod-order equato, eq ξ (.5) C, eqc where , eq, eq ξ ( 3 We could have acheved the ame reult by pectg the crcut Fg..3 ad determg the put mpedace (we leave th oluto a a exerce for the reader). The charactertc equato.5 of ecod order, ce the crcut (Fg..3) cot of two eergy-torg elemet (oe ductace ad oe capactace). By aalyzg the crcut ther traet behavour ad determg ther charactertc equato, we hould alo take to coderato that the atural repoe mght be dfferet depedg o the kd of appled ource: voltage or curret. We have to dtguh betwee two cae: If the voltage ource, t phycal repreetato (.e. wth a er retace coected ere) replaced by a equvalet curret ource (.e. wth the ame retace coected parallel), the traet repoe wll ot chage. Ideed, a ca be ee from Fg..8, the ame crcut A coected (a) to the voltage ource ad (b) to the curret ource. By kllg the ource (.e. hort-crcutg the voltage ource ad opeg the curret ource) we are gettg the ame pave crcut,

28 for whch the mpedace are the ame. Th mea that the charactertc equato of both crcut wll be the ame ad therefore the atural repoe wll have the ame expoetal fucto. V A P a I A P b Fg..8 However, f the deal voltage ource replaced by a deal curret ource, Fg..9, the pave crcut (a) ad (b),.e. after kllg the ource, are dfferet, havg dfferet put mpedace ad therefore dfferet atural repoe. V A P a I A P b Fg..9 8

29 .6. OOTS OF THE CHAACTEISTIC EQUATION AND DIFFEENT KINDS OF TANSIENT ESPONSES.6. Frt-order charactertc equato If a electrcal crcut cot of oly oe eergy-torg elemet ( or C) ad a umber of eergy dpato elemet ( ), the charactertc equato wll be of the frt order: For a crcut + eq (.6a) ad t root eq, τ τ eq, (.6b) where τ a tme cotat. For a C crcut + eq C C τ (.7) where τ eq C a tme cotat. I both cae the atural oluto f t eq t Ae Ae τ, (.8) whch a decreag expoetal, whch approache zero a the tme creae wthout lmt. However, durg the tme terval of fve tme τ the dfferece betwee the expoetal ad zero le tha %, o that practcally we may tate that the durato of the traet repoe about 5τ. 9

30 .6. Secod-order charactertc equato If a electrcal crcut cot of two eergy-torg elemet, the the charactertc equato wll be of the ecod order. For a electrcal crcut, whch cot of a ductace, capactace ad everal retace th equato may look lke equato.,.5 or a geeralzed form d + α + ω (.9) The coeffcet the above equato hall be troduced a follow: α a the expoetal dampg coeffcet ad ωd a a reoat frequecy. For a ere C crcut α / ad ωd ω C. For a parallel C crcut α /C ad ωd ω, whch C the ame a a ere crcut. For more complcated crcut, a Fg..3, the above term may look lke, α eq 3 +, eq C whch actually combed from thoe coeffcet for the ere ad parallel crcut ad ωd ωξ, where ξ a dtorto coeffcet, whch fluece the reoat/ocllatory frequecy. The two root of a ecod order (quadratc) equato.9 are gve a ad the atural repoe th cae α + α ωd (.3a) α α ωd (.3b) t t f( t) A e A e + (.3) Sce each of thee two expoetal a oluto of the gve

31 dfferetal equato, t ca be how that the um of the two oluto alo a oluto (t ca be how, for example, by ubttutg equato.3 to the codered equato. The proof of t left for the reader a a exerce.) A kow from mathematc, the two root of a quadratc equato ca be oe of three kd: egatve real dfferet, uch a >, f α > ωd ; egatve real equal, uch a, f α ωd ; complex cojugate, uch a, α ± jω, f α < ωd, where ω ω d α the frequecy of ocllato or atural frequecy (ee further o). A detaled aaly of the atural repoe of all three cae wll be gve the ext chapter. Here, we wll retrct ourelve to ther hort pecfcato. Overdampg. I th cae, the atural repoe (equato.3) gve a the um of two decreag expoetal form, both of whch approach zero a t. However, ce >, the term of ha a more rapd rate of decreae o that the traet tme terval defed by ttr 5. Crtcal dampg. I th cae, the atural repoe (equato.3) covert to the form f t ( t) ( A t + A ) e. (.3) Uderdampg. I th cae, the atural repoe become ocllatory, whch may be maged a a decayg alteratg curret (voltage) 3

32 αt f ( t) Be (ωt + β) (.33) Here term α the rate of decay ad ω the agular frequecy of the ocllato. Now the crtcal dampg may be terpreted a the boudary cae betwee the overdamped ad uderdamped repoe. It hould be oted however that the crtcal dampg of a more theoretcal tha practcal teret, ce the exact atfacto of the crtcal dampg codto α ωd a crcut, whch ha a varety of parameter, of very low probablty. Therefore, the traet repoe a ecod order crcut wll alway be of a expoetal or ocllatory form. et u ow coder a umercal example. Example. The crcut how Fg. repreet a equvalet crcut of a oe-phae traformer ad ha the followg parameter:.6 H,. H, M.3 H, 6 Ω, Ω. If the traformer loaded by a ductve load, whoe parameter are ld.5 H ad ld 9 Ω, a) determe the charactertc equato of a gve crcut ad b) fd the root ad wrte the expreo of a atural repoe. M ld ld Soluto Fg.. Ug meh aaly, we may wrte a et of two algebrac equato (whch repreet two dfferetal equato operatoal form) 3

33 M ( + ) M + ( + + ld + ld ) The determat of th et of two equato + M det M ( + ) + ( + ) ld ld / / / M ) + ( + ) ( +, / / where, to horte the wrtg, we aged + ld ad / + ld. ettg det, we obta the charactertc equato the form / / / / / M M Subttutg the gve value, we have +,5 +. The root of th equato are:,86,6, whch are two dfferet egatve real umber. Therefore the atural repoe : 86t 6t f( t) A e + Ae, whch cot of two expoetal fucto ad of the overdamped kd. It hould be oted that ecod order crcut, whch cota two eergy-torg elemet of the ame kd (two -, or two C-), the traet repoe caot be ocllatory ad alway expoetal overdamped. It worthwhle to aalyze the root of the above charactertc equato. We may the obta 33 4

34 form: ±, ( M ) / ( / + / ) [ / 4( / ( + M / ) ± ) / (.34) The expreo uder the quare root ca be mplfed to the / / / > ( + ) + 4 M, whch alway potve,.e., both root are egatve real umber ad the traet repoe of the overdamped kd. Thee reult oce aga how that a crcut, whch cota eergy-torg elemet of the ame kd, the traet repoe caot be ocllatory. I cocluo, t mportat to pay atteto to the fact that all the real root of the charactertc equato, uder tudy, were egatve a well a the real part of the complex root. Th very mportat fact follow the phycal realty that the atural repoe ad traet-tate caot ext fte tme. A we already kow, the atural repoe take place the crcut free of ource ad mut vah due to the eergy loe the retace. Thu, atural repoe, a expoetal t fucto e, mut be of a egatve power ( < ) to decay wth tme..7. INDEPENDENT AND DEPENDENT INITIA CONDITIONS From ow o, we wll ue the term wtchg for ay chage or terrupto a electrcal crcut, plaed a well a uplaed,.e. dfferet kd of fault or other udde chage eergy dtrbuto..7. Two wtchg rule (law) The prcple of a gradual chage of eergy ay phycal ytem, ad pecfcally a electrcal crcut, mea that the eergy tored magetc ad electrc feld caot chage tataeouly. Sce the magetc eergy related to the magetc flux ad the curret 34

35 through the ductace, both of them mut ot be allowed to chage tataeouly. I traet aaly t commo to aume that the wtchg acto take place at a tat of tme that defed a t (or t t ) ad occur tataeouly,.e. zero tme, whch mea deal wtchg. Heceforth, we hall dcate two tat: the tat jut pror to the wtchg by the ue of the ymbol,.e. t, ad the tat jut after the wtchg by the ue of the ymbol +,.e. t +, (or jut ). Ug mathematcal laguage, the value of the fucto f ( ), the lmt from the left, a t approache zero from the left ad the value of the fucto f ( + ) the lmt from the rght, a t approache zero from the rght. Keepg the above commet md, we may ow formulate two wtchg rule. Frt wtchg law (or frt wtchg rule) The frt wtchg rule/law determe that the curret (magetcflux) a ductace jut after wtchg ( + ) equal to the curret (flux) the ame ductace jut pror to wtchg ( + ) () (.35a) ψ( + ) ψ() (.35b) Equato.35a determe the tal value of the ductace curret ad eable u to fd the tegrato cotat of the atural repoe crcut cotag ductace. If the tal value of the ductace curret zero (zero tal codto), the ductace at the tat t (ad oly at th tat) equvalet to a ope crcut (ope wtch). If the tal value of the ductace curret ot zero (o-zero tal codto) the ductace equvalet at the tat t (ad oly at th tat) to a curret ource whoe value the tal value of the ductace curret I (). Note that th equvalet, curret ource may repreet the ductace a mot geeral way,.e., alo the cae of the zero tal curret. I th cae, the value of the curret ource zero, ad er retace fte (whch mea jut a ope crcut). 35

36 Secod wtchg law (or ecod wtchg rule) The ecod wtchg rule/law determe that the voltage (electrc charge) a capactace jut after wtchg v c ( + ) equal to the voltage (electrc charge) the ame capactace jut pror to wtchg v c ( + ) v c () (.36a) q ( + ) q () (.36a) Equato.36a determe the tal value of the capactace voltage ad eable u to fd the tegrato cotat of the atural repoe crcut cotag capactace. If the tal value of the voltage acro a capactace zero, zero tal codto, the capactace at the tat t (ad oly at th tat) equvalet to a hort-crcut (cloed wtch). If the tal value of the capactace voltage ot zero (o-zero tal codto), the capactace, at the tat t (ad oly at th tat), equvalet to the voltage ource whoe value the tal capactace voltage V v c (). Note that th equvalet, voltage ource may repreet the capactace a mot geeral way,.e., alo the cae of the zero tal voltage. I th cae, the value of the voltage ource zero, ad er retace zero (whch mea jut a hort-crcut). I a mlar way, a a curret ource may repreet a ductace wth a zero tal curret, we ca alo ue the voltage ource a a equvalet of the capactace wth a zero tal voltage. Such a ource wll upply zero voltage, but t zero er retace wll form a hortcrcut. If the tal codto are zero, t mea that the curret through the ductace ad the voltage acro the capactace wll tart from zero value, where a f the tal codto are o-zero, they wll cotue wth the ame value, whch they poeed pror to wtchg. The tal codto, gve by equato.35 ad.36,.e., the curret through the ductace ad voltage acro the capactace, are called depedet tal codto, ce they do ot deped ether o the crcut ource or o the tatu of the ret of the crcut elemet. It doe ot matter how they had bee et up, or what kd of wtchg 36

37 or terrupto took place the crcut. The ret of the quatte the crcut,.e., the curret ad the voltage the retace, the voltage acro the ductace ad curret through the capactace, ca chage abruptly ad ther value at the tat jut after the wtchg ( t + ) are called depedet tal codto. They deped o the depedet tal codto ad o the tatu of the ret of the crcut elemet. The determato of the depedet tal codto actually the mot arduou part of the clacal method. I the ext ecto, method of determg the tal codto wll be troduced. We hall frt, however, how how the depedet tal codto ca be foud..7. Method of fdg depedet tal codto For the determato of depedet tal codto the gve crcut/etwork hall be pected at t teady-tate operato pror to the wtchg. et u llutrate th procedure the followg example. Example.3 I the crcut Fg.., a traet-tate occur due to the clog of the wtch. Fd the expreo of the depedet tal value, f pror to the wtchg the crcut operated a d.c. teady-tate. V C C Soluto Fg.. By pecto of the gve crcut, we may ealy determe ) the curret through the ductace ad ) the voltage acro two capactace. 37

38 ) Sce the two capactace a d.c. teady-tate are lke a ope wtch the ductace curret ( ) + ) Sce the voltage acro the ductace a d.c. teady-tate zero (the ductace provde a cloed wtch), the voltage acro the capactace v c( ) (). Th voltage dvded betwee two capactor vere proporto to ther value (whch follow from the prcple of ther charge equalty,.e., C vc C vc ), whch yeld: Example.4 v v c c ( ( ) ) ( ( V C ) C + C C ) C + C Fd the depedet tal codto ( ) ad v c ( ) the crcut how Fg.., f pror to opeg the wtch, the crcut wa uder a d.c. teady-tate operato.. 4 I 3 5 C Fg.. 38

39 Soluto ) Frt, we fd the curret 4 wth the curret dvo formula (o curret flowg through the capactace brach) I 4 I // 5 ( + 3 ) Ug oce aga the curret dvo formula, we obta the curret through the ductace I 3 ( + ) ) The capactace voltage ca ow be foud a the voltage drop retace v c( ) (). The example gve above how that order to determe the depedet tal codto,.e., the tal value of ductace curret ad/or capactace voltage, we mut coder the crcut uder tudy pror to the wtchg,.e. at tat t. It uual to uppoe that the prevou wtchg took place alog tme ago o that the traet repoe ha vahed. We may apply all kow method for the aaly of crcut ther teady-tate operato. Our goal to chooe the mot approprate method baed o our experece order to obta the qucket awer for the quatte we are lookg for..7.3 Method of fdg depedet tal codto A already metoed the curret ad voltage retace, the voltage acro ductace ad the curret through capactace ca

40 chage abruptly at the tat of wtchg. Therefore, the tal value of thee quatte hould be foud the crcut jut after wtchg,.e., at tat t +. Ther ew value wll deped o the ew operatoal codto of the crcut, whch have bee geerated after wtchg, a well a o the value of the curret the ductace ad voltage of the capactace. For th reao we wll call them depedet tal codto. A we have already oberved, the atural repoe the crcut of the ecod order, for tace, of form equato.3. Therefore, two arbtrary cotat A ad A, called tegrato cotat, have to be determed to atfy the two tal codto. Oe the tal value of the fucto ad the other oe, a we kow from mathematc, the tal value of t frt dervatve. Thu, for crcut of the ecod order or hgher the tal value of dervatve at t + mut alo be foud. We alo coder the tal value of thee dervatve a depedet tal codto. I order to fd the depedet tal codto we mut coder the aalyzed crcut, whch ha are after wtchg ad whch all the ductace ad capactace are replaced by curret ad voltage ource (or, wth zero tal codto, by a ope ad/or hort-crcut). Note that th crcut ft oly at the tat t +. For fdg the derable quatte, we may ue all the kow method of teady-tate aaly. et u troduce th techque by coderg the followg example. Example.5 Coder oce aga the crcut Fg... We ow however eed to fd the tal value of curret (+ ), whch flow through the capactace ad therefore ca be chaged tataeouly. Soluto We tart the oluto by drawg the equvalet crcut for tat t +,.e. jut after wtchg, Fg..3. The ductace ad capactace th crcut are replaced by the curret ad voltage ource, whoe value have bee foud Example.4 ad are aged 4

41 a I ad V c. 4 () 3 I I () V C () Fg..3 The acheved crcut ha two ode ad the mot approprate method for t oluto ode aaly. Thu, I + G3 Vab + I + (), where G 3. Subttutg Vab Vc + () for Vab we may 3 obta ( )( + G3 ) I I G3V c, or I I G3V c (). + G3 Example.6 A a umercal example, let u coder the crcut Fg..4. Suppoe that we wh to fd the tal value of the output voltage, jut after wtch tataeouly chage t poto from to. The crcut parameter are:. H, C. mf, Ω, Ω, ld Ω, V V ad V 6 V. 4

42 C ld V V a V () V C () () C () ld V () V V b V Fg..4 Soluto I order to awer th queto, we mut frt fd the depedet tal codto,.e., ( + ) ad v c ( + ). By pecto of the crcut for tat t, Fg..4(a),we have ad ( V ) + A, ld + vc () V V. + + Wth two wtchg rule we have v c ( ( + + ld ) ) v () A, ( ) V c ad we ca ow draw the equvalet crcut for tat t +, Fg..4(b).By pecto, ug KC (Krchhoff curret law),we have + + ) V + V + v (). (.37) ld ( c Keepg md that ad ( ), we obta 4

43 V + V + vc () () (),5 A. + Thu the tal value of the output curret,5 A. Note that, pror to wtchg, the value of the output curret wa A, therefore, wth wtchg the curret drop to half of t prevou value. The crcut of th example of the ecod order ad, a earler metoed, t atural repoe cot of two ukow cotat of tegrato. Therefore, we hall alo fd the dervatve of the output curret at tat t +.By dfferetatg equato.37 wth repect to tme, ad takg to coderato that V ad V are cotat, we have d d dvc ( + ld ) +, dt dt dt d dvc ad, ce v ad c, dt dt C d c () v() +. dt t ld C By pecto of the crcut Fg..4(b) oce aga, we may fd v () V + () () 4V Thu, () c ld () d dt t ld (),5 A. 75 A.7.4 Geeralzed tal codto. Our tudy of tal codto would ot be complete wthout meto of the o-called correct tal codto,.e. by whch t look a though the two wtchg law are dproved. 43

44 (a) Crcut cotag capactace A a example of uch a dproval, coder the crcut Fg..5(a). I th crcut, the voltage acro the capactace pror to wtchg v c ( ) ad after wtchg t hould be v c( + ) V, becaue of the voltage ource. Thu, v ( ) v () c + c ad the ecod wtchg law dproved. C V C V V C a b Fg..5 Th paradox ca be explaed by the fact that the crcut Fg.5(a) ot a phycal realty, but oly a mathematcal model, ce t bult of two deal elemet: a deal voltage ource ad a deal capactace. However, every electrcal elemet practce ha ome value of retace, ad geerally peakg ome value of ductace (but th ductace very mall ad our future dcuo t wll be eglected). Becaue, a real wtch, the wtchg proce take ome tme (eve very mall), durg whch the park appear, the latter alo uually approxmated by ome value of retace. By takg to coderato jut the retace of the coectg wre ad/or the er retace of the ource or the retace of the park, coected ere, ad a retace, whch repreet the capactor ulato, coected parallel, we obta the crcut how Fg..5(b). I th crcut, the ecod wtchg law correct ad we may wrte v ) v ( ). c ( + c Now, at the tat of wtchg,.e., at t, the magtude of the voltage drop acro th retace wll be a large a the ource value. A a reult the curret of the frt momet wll be very large, however 44

45 ot ulmted, lke t uppoed to be Fg..4(a). I order to llutrate the traet behavour the crcut dcued, let u tur to a umercal example. Suppoe that a. F codeer coected to a V ource ad let the retace of the coectg wre be about oe hudredth of a ohm. I uch a cae, the pke of the curret wll be I δ /. A, whch a very large curret a V ource crcut (but t ot fte). Th curret able to charge the above codeer durg the tme perod of about, ce the requred 9 7 charge q CV C ad 7 Δq Δ t. Th perod of tme actually equal to the Δ 4 9 tme cotat of the ere C crcut, τ C. From aother pot of vew, the amout of the charge, whch traferred by a expoetally decayed curret, equal to the product of t tal value, I ad the tme cotat. Ideed, we have q dt I t t τ τ e dt I( τ) e Iτ (.38) 7.e., q C, a etmated earler. Th reult (equato.38) jutfe ug a mpule fucto d (ee further o) for repreetg very large (approachg fty) magtude applyg very hort (approachg zero) tme terval, wherea ther product tay fte. Note that the ecod retace very large (hudred of mega ohm), o that the curret through th retace, beg very mall (le tha a teth of a mcroampere), ca be eglected. I cocluo, whe a capactace coected to a voltage ource, a very large curret, te of kloampere, charge the capactace durg a vahg tme terval, o that we may ay that the capactace voltage chage from zero to t fal value, practcally mmedately. However, of coure, oe of the phycal law, ether the wtchg law or the law of eergy coervato, ha bee dproved. 45

46 A a ecod example, let u coder the crcut Fg..6(a). At frt glace, applyg the ecod wtchg law, we have vc(+ ) vc( ) V (.39) v ( ) v ( ) c + c But after wtchg, at t, the capactace are coected parallel, Fg..6(b), ad t obvou that v c ( + ) v c () (.4) whch cotrat to equato.39. V C C V C C a b Fg..6 To olve th problem we hall dvde t to two tage. I the frt oe, the ecod capactace charged practcally mmedately the ame way that wa explaed the prevou example. Durg th proce, part of the frt capactace charge traferred by a curret mpule to the ecod capactace, o that the etre charge dtrbuted betwee the two capactace recprocal proporto to ther value. The commo voltage of thee two capactace, coected parallel, after the wtchg at tat t, reduced to a ew value lower tha the appled voltage V.I the ecod tage of the traet proce th crcut, the two capactace wll be charged up o that the voltage acro the two of them wll creae up to the appled voltage V. To olve th ecod tage problem we have to kow the ew tal voltage equato.4. We hall fd t accordace wth equato.36b whch, a wa metoed earler, expree the phycal prcpal of cotuou 46

47 electrcal charge,.e. the latter caot chage tataeouly. Th requremet geeral but eve more trget tha the requremet of cotuou voltage, ad therefore called the geeralzed ecod wtchg law. Thu, q Σ ( + ) qσ () Cv c( ) (.4) Th law tate that: the total amout of charge the crcut caot chage tataeouly ad t value pror to wtchg equal to t value jut after the wtchg,.e., the charge alway chage gradually. Sce the ew equvalet capactace after wtchg C eq C + C, we may wrte q Σ ( + ) ( C + C) vc(+ ) Cv c( ). Sce, th example, v c ( ) V, we fally have C vc (+ ) vc( ) V (.4) C + C C + C Wth th tal codto, the tegrato cotat ca ealy be foud. It teretg to ote that by takg to coderato the mall retace (wre, park, etc.) the crcut become of ecod order ad t charactertc equato wll have two root (dfferet real egatve umber). Oe of them wll be very mall, determg the frt tage of traet, ad the ecod oe, relatvely large, wll determe the ecod tage. et u ow check the eergy relato th cheme, Fg..6, before ad after wtchg. The eergy tored the electrc feld of the frt capactace (pror to wtchg) w e( ) C Vc () CV ad the eergy tored the electrc feld of both capactace (after wtchg) w e ( ) ( C C) vc ( C ). Thu, the eergy lot 47