Sandu, D. D. - Probleme moderne de tehnica impulsurilor

Probleme
moderne

de tehnlca

!mpulsurl lor

-Lr':
CCC2 ^i9

Lucroreo ,,Probleme moderne de tehnico irnpul'
surilor" troteozli ospecte legote de generoreo 9i- for-
moreo innpulsurilor de duroto foorte scurt6 9i cu fron-

tuii ropiie. Principolele probleme obordote sint :

comutoreo dispozitivelor semiconductoore, inclusiv o
celulelor cu TECMOS complementori 9i o diodelor tu-

nel (DT) ; circuite de impulsuri cu DT si linii de intir-

ziere, circuite integrote logice ;i memorii cu semicon-

cluctori integrote, iircuite iu diode Boff (cu-ocumulore
de soSrcinint din)'ciilucsirecurieteolliozgdicrieieccuednistepoozbititvineuGteunpne' plon

mondiol (peste 300 trimiteri bibliogrofice) precum 5i

r"irttot" originole ole outorului ; lucroreo este bogot

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V ivttua

friile spa,tiale, calaulutoare electroniae ultrara'ptd,e, Jizica particulelor elemen-
tare, reocli'i.Ttermonucteare d,irijate, rad,i,oloca,tia etc.

Lucrarea este 4,mpd,r!i,td, in pase cwTti,tole dupd, cum u,rmeazd,:
Capitrilul tr tratenzd, ctteaa probleme generale referitoare la procesele

tranzi,torii, din li,ni,i,le de transm'isi,e (4,nt4rzi,ere), utilizirtdtt-se transformd,ri,le
Laplaae. Ii.ind, si,steme cu pararnetri, d,istribui.bui,li, Iittiile de |nt4rzi,ere

consti,tuie elemente id,ea'l,e pentru, Jormarea impu,Isurilor de clurattv Joarte
sourtd, gi cu Jronturi abru,pta. 84nt prezentate succesir: forma generald, o, ecu,-
a!,i,ilor, metoda u,ndelor mobile, ,intercanectarea ;i, cu,plurea lini,ilor.

in cagti,tolut 2 sLnt Ttrezentate teoria generald, ;i particti,l,aritd,lite de

construc,tie a, transformatorilor da imptulsuri cu linii d,e [tfi,rziere I de asemenea
sl,nt desari,se succint li,ni,ile neu,n'iforme.

in capi,tolul 3 se stud,i,a,sd, qte scurt Ttrocesele d,e comtttqre a d,,isgtozi,-
tiaelor serniaond,uctoare bi,gtolare (d,iod,e, tranzistori, tiristori) ;i a celor cu

efeat de ci,mp ; se insi.std, asuprd tranz'istor'ilor cu eJect de ctmp ar:|nd, sarc'ind,
integratd, ;r, asupra celulelor f ormate din tranzistor'i TnC MO{! comTtlementar'i.

Capitolul 4 prezi,ntd, o tratrare detaliatd, a procesel,or de comutare a

d,i,od,elor tunel pi, a cirau'itelor d,e impulsuri, echipate cu astrJel de d,iod,e. Datoritii

aostului, scd,zut, tsitezei, m,uri d,e comuture pi simplitiif,i.i, circttitelor, d,iodele

twnel consti,tu,te prin encelenld, elentente da generare a imgtltlsurilor d,e d'wratd,
Joarte scurtd. Aaantaje deosebite mani,festd, aircu,itele rezultale 'prin asocie-

rea diod,elor tunel pi, a l,intitor d,e tnltrzi,ere; rLsupra lor autorul s-a, oprit

pe larg, prezenttnd, teor'ia i{agumo-Blr,'imura pi o serie d,e rezultate origi,nalo

propri,i,.

in caTtitotul, 5 si,nt enpuse pri,nci,pii ;i scheme p'rin care se reali.zeazd,

4mbund,td!,irea lronturilor i,mpulsurilor. Si,nt d,escri,se circu,ite care conlin
combi,nali,i. d,e d,iod,e tunel, combi.na!,ii de d,iode tunel pi tranzistori, biTtolari,

ciroui,te d,e comutare cu ferite atstnd, ciclul Ttysterezi,s dreptunghiular (CHD),
getteratori de im,pulsuri, au reaclie |ntf,rziatd, .,si cu tranzistori, da aualanpd,.

in capitotut 6 si,nt incluse unele probleme speciale d,e tehnica'imgtulsu-
ri'l,or, gi, anulne : cirauite de tmgtulsuri, cu d,iod,e Bo.ff (cu acumu,lare de salr-

oi,nd,), oi,roui,te 'i,ntegrate logi,ce, mernori'i integrate aw semicond,uator'i pi' cirau-

i,te d,e i,mpwlsuri ou, iliod,e Gu,nn. Me'mariile integrate cw semico'ruduatori aon-

sti,tuie o realizare d,e airf 6n d,omeniul calcu,lntoarelor electronice portabile

fi(i,nclusi,a eele 'instalato pe si,stentele mobile cuTn ar narele cosm,iee) d,e capa-

ci,tote med,ie sau, ntare pi, d,e aitezd' rid,i.catd,, ele remarctndn-se pr'intr-wn aon-

sLt",nn efrtrenx de redus. Diod,ele Gunn sdnt disptozitiue carye 4n momerfiuL de

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3.3.2. Procesele de comutare directd 74
3.3.3. Procesele de comutare inversd 78
3.4. Comutarea tranzistorilor cu efect de clmp (TEC) .82
3.4.1. Constmciia qi parametrii principali ai TEC 82
3.4.2. Cornutarea tranzistorilor TECMOS cu sarclntr rezistivi 87

3.4.3. Circuite de comutare cu tranzistori TEC\IOS avlnd sarcini 90
,,integrati"
ot
3.4.4. Procese de comutare a tranzistorilor TECMOS cu sarcine
,,integrati" 98
100
3.4.5. Procese de comutare a inversorului logic cu tranzistori TECMOS
103
complementari
103
Bibliograf ie 103
108
Capitolul 4' Comutarea D ?, Circuite ile impulsuri cu D? 9i linll ile lntirzlere 712
4.t.
Caracteristici generale Ei parametrii DI 7t2
4.2. 4.1,1. Construclia gi parametrii D?
4.3. 4.1.2. Nletode de aproximare a caracteristicilorstatice ale D? 714
Probleme de stabilitale a circuitelor cu D? 115
4.4. 4.2.1. Stiri de echilibru 115
4.2.2. Stabilitatea circuitelor cu DT
t23
Procese de comutare a DI t25
4.3.1. Comutarea prin semnale de curent
4.3.2. Comutarea prin semnale de tensiune 128
4.3,3. Comutarea prin semnale implrlsulare 128
Circuite c,t DT $i bobine de inductan{i 131

4.4.1. Circuite monostabile 742
L42
4,4.2. Circuite astabile 144
151
4.4.3. Circuite bistabile 160
166
4.5. Circuite cu DT qi linie de intlrziere
4.5,1, Monostabil cu D? si linie de lntirziere 169
4.5.2. Muttivibrator cu DT Ei linie de intlrziere. Procese fizice 169
169
4.5.3. Teoria Nagumo-Shimura 177
181
4'6' Circuite at DT gi douii linii de intirziere 186
193
Bibliografie
193
Capitolul 5. Formarea ql generarea impulsurilor ile durati foarte mio[' Scherne 199
205
eu ilioile, ?-B, Iinii tls lntlrziere 9i ferite CIID
zto
5.1. Variante de circuite cu DT
5.1.1. Comportarea D? avlnd sarcini neliniari 2to
5.1.2. Procesele de comutare a dispozitivului cornpus din D? Ei 7'B
5.1.3. Particularititi ale circuitului compus din doui DT 216
217
5,1.,tr. Exemple de schene
K' Circuite cu ?'B

5.2.1. Consideralii asupra metodelor de imbundtifire a fronturilor

imPulsurilor

5.2.2. Circuite cu ferite CHD Si TB
5.2.3. Circuite de irnpulsuri cu tranzistori de avaianqi

5.3. Circirite de impulsuri cu reaclie lntlrziatd
5.3.1. Introducere teoretici asupra generatorilor de impulsuri cu

reactie lntlrziati
5,3.2. Exemple de generatori cu reactie intlrziati

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i<-
-_-_aFx-ig-_.f 1.1.

se procedeaz5, astfel: ecualia (1.1) se deriveazd in raport ca u, iar (1.2j
insumare gi rearanjarea termenilor se obline
in raport cu t; prin

o'* : LC 9A:Yt,z+ (Rc+LGt9! +ne u. (1.3)

dnz dI

Dacd se urmd,reqte eliminarea lui u, prin calcule similare so obline

!:L: LCd:Lf +(Rc + Le'\ldJt * net. (1.4)

dnz

Ultimele doud, ecualii se numesc ecua!'iile telegrafiptilor (uneori qi ecu-
aliile (1.1) qi (1.2) se numesc ecualiile telegrafiqtilor).- Pentru studiul pro-
ceselor din iiniile de transmisie este necesar sd, se rezolve numai o ecua.tie,
reexzpurlteds,iad,iunnperiimsinagiunrebamzda,rpimroi,prrlieeteSx,elilmorpleuvi?d!eun,tet) ;_a doua md,ri-
gd,sindu-se (1.1) qi (1.2) sau din similtitudinea ecualiilor de dualitate a,

me i,(n, t) (1-.3) qi (1.4).
ecualiilor

Existd, o serie de cazuri cind. ecualiile telegrafiqtilor capd,td, o formd,
mai simpld.
pierd,eri, : R: 0, sGpe-ci0fi.caInlii,
vine a) Linie fd,rd, acest caavzeareilnalviaed,(e1r.e3)edcue-- i
(in continuare, in absenla altor se Ya
iJ
alia tensiunilor)
t
o0'ruz_L""Cg0:fy_Lw' ozu. (1.5)
0t2' i_'

untle r.o:1/l[t,C arcdimensiunile unei viteze qi reprezintl" aiteza de propa- L
in linie ecualia (1.5) este o ecualie de undd,, unidimensio-
gare undelor ; E
cl
nald,. ^
{
LG s:emb-)nEaCLlin(acip,oelnicfddai,trtdilaadIilinesttaroavrisrseiuidnseei.)pDlrionapciaadg,Ssp,eapnreaummlineeiqetrtieiqisf5ep,sredtec, iddfiicesitteoscarsttaiisutfnalaic contlilia
un +l
d-eoarece
,l
distanla
I"r
fi) ctL o anumit5, atenuare insd, pistrind. forma semnalului incident. Notind.
Ui-
:,to I IV-I-C qi a : l/nG se poate scrie m

RC + LG :2RC :Zalw (1.6)

il

9l

ut?r I+dsForqs r .os u€r+ o p raopg.rrmurmr 8reo.rrrrin€"rn+cuoed"'s"a"lJt #re,1u:r],lgoJf-,tailfufiaTu.riaguilfiaTditre,li.JrnJi:u&rF"afri (g'r)

'rrJo?rzu€4. pull

n.r1ued 14u 'aceldeT JoIrJKrrrJoJSu€J+ €po+aru a?sa p+rpulds-el elluog 'e{uu1
acaJu
"]jnya-FIzcsrvu{geaazn4leclezIa+"vrnoetsdr+ooEcsoencs(pEporouirruIpeolse'orru?o{1ulueuoo);u{u.€rpoemldud.xr.oer*ldc.Ke.;rdrfrosrrpJ"o.or;Jne"r"pquron8Jls'ooJ",4KrT+fI.runo[rrszp€onJo+dreJs."or"nq#.eqr#+stu"ornarlinld"urceelut.msr:ectCp:Jporcr.s€iats;iorlod:pc*uonUnuc
ospre{.rrpuu.roudceas+o€op+.g?+cc€oJJsrrp+tsps.rnagpseooor€;rcd €IiIP
u1aur1-srs'po83u(..proKle+plqrue{r"ere1l olm"reuaF- n{n1os raun
pmc -OISU
o yzvazlyqn os
'eldurrs_1eur oloc olrJnz€c uI JErqc I pucglp op ruoJ+xo pualqo.rd o a,nltlsuoc -ndot
KBelpceor+{+.tarserruudqroqelnp€s.argraJ.cronapdqsonrc€fpi ao'el+eriorr{zruuoad(l"aeryalurdprnf.s.rrroadppro{e+ron€lca,rolrtrso€.rp5oenJqcEur^aIEolezproo{uJ+aoo.trLue) yrKiipcprJspoEupco,riE{uepn.orco+odool€u
o1JoEd^.IeOrZsOruJrsOu€pJ.qg.CroSptsICJo€lprrou+{oIeIIp:r{oe+nncceso,luenlacporslaupooc+alunlurmrFpa"€r {ornp uud elualoza;r {s'r)
13
+ua.rorrpu1 -nca

a:e1de1 Jol!JgruJoJsueJl -ap (
EeJEzlllln u1.rd .ro;1;u;1 .ro11ien>e €aJpllozo1 .Z.L.t
9rnJo
(orr) 'atzn{tpt apr T,yinnca rcun er{uor;rumos pur^ts
'(l'r)
*?eg0U:Wzr0
€ a1
ouraop (9.1) rr{enca i elulquc gunq
op JoT1€I:opzlInnucu,opFdxp€csocapolno'f(luqgn€fTc)cnlecpr'uB{eox1cae.tror,cpupr.+irupdaag.rlcXrrmcJnn;posupla,arauJrceasnatuupra,+z7€€@co+psoroo\?J -lJBrt
€q.JBoJ'0 {ariut
-o.rd
'0: : _p -nca
-ctsJ un qaezetd a?+srgo JFoorrnul{nqod-ery+uelr{"droepc{x.uae.nrncc'rr.Srap.r;rooprdunprisgujoelluelrl{nuqgr.erolsrrspizercoacep
(r'r )
o+so runtsro?s1p
eulrq
(a'r) -61'vO 6-01_, 6-r t.)
(s'r) tn"g (e'r)
rnr7
Ta k'r)

eur{qo os (r'T) erfenca ur ?In
9o?ysBi
(l'n)rn lao-e : (l'n)n

er{e1e.r ur"rd plrm;ep (gtn)rn er{cuny purcnpoJ+ul

(l'r) 'nzn * ?e 0x _14 _0?, _6 frow
nzg
ne _nr_6-'-n, z6'gw I 6

?ullop (g'1) er{unce rS

eii"ciu"a'oliri1tO".-AiA"'apit"rli"ecuinni1ndiatot"eptroacnu*sdcfoeurrmivvaaarrtieea6Ip-i,aiaaiplilanacl,eec^csrelieretrscaetndsrefionzromrhaS'5p, oianrptoecicupuvraianliridamifbeeitrloaednfe-'
1ic;lla"ss1eicoae2p.fi.i;ic;E;eaco1unaiioliuaiullo(gi nretrb,dalr)niifcsegefroieirinnm(rlira,aarle?peo)Ior-s.t,readipuialnapacnleircee5cin,urentrroazasupncolsturfatotetre-lmeculiaU,rlailespraiuau1nb,1acil.ptaulPaulcepen; rtseirnecuveoedatrerfsllnaient-.
"f"i.""-,*'."5","idri1nriaI6creste c;,i'aie avantaje qi inconveniente, asupra cd,rora ne oprim

pe scurt.

td,n"c"",euviif,c'fogieevirrl.saei"aiaenr"Brixt)reaiu"iiscb:"tpd"toia,lailna"aotupiirttuveca"lar.titnsricrituraeioaerendbacdc,doei.luenafiPliacsniiluerieiaa.lanqanltt$itgreeelutsiiiminpnddetir,e'rtiiiifdlneeni,remcned:rntd,pee!d_teiglr"iia.raforaiiluacvn.ildaeraas,etfrcreoeepurfmednedenera,itettorroarueriaadfIeard-ie,ntreaeiumtplepallararmclrvare,iatniieadrnaicliarerueerbaaac6i!tpl'nadicaot,rdr-tittroneogcnerriaucsaduipuaors!onui'in.dirqe-tt,i

elu(n, r)f : I; tt@, tle-" d, : U(r, s), (1.11)

pentru ecualia (1.5) se poate scrie

:d'? I/(r, s) s2 a@, s) \lrut*, o*) u"(n, o*)1, (1.12 )
--9 10'
cln2

unde ru'(r, 0*) :l*,u@' t)],:0..

f"er*lttti""v(;i^,"i(;ir"sdr-l,,i-gerefili[nlst)s,p,.tt-e:"Ufaleift6muis"c9ointniee-sbtXroupl1nialntadueCdruicatari(,eetatmres,c(oa.eictn-sxrdtbicre),,u]eiutc.nud)ctuaseircatitdaveael,itptreraempdopderrpietniitofnleaeecarrurmtaepeepavrinnilfamliiacicraoarirealecinri,zbeasocaiotl5lcaavr,odtnairitnttsanSe;snai,lnto(rsaS2utrfno:)"q'dtro(sei-m1neerlo)Sprirners,rarztiedein-pioegglo6-mSr,'air5,,n,setrateet-epcqoecltudageduceriaavten'rralcaieianlrcan)ivaoss;ceinbfcl.rido(if,slee3aire--;-)
ii(r"tae*b".s*rfiicoobir!t)m*CXCio,o"ninifinouLrtemrar-tpoirlaeaceaceceseutcceauilliiare6el5ic,a,eit,liJgedienc,biiJrrreaiacrle5pian,oIlirc,tsalieftceeurreeonvail-ailtinizraoeiiad,arb,ezidola('r1da_ricn.,1ea2dar)|es"6'ftianinnittpSrr'rao,inppeorucirnutti&lcrielu?ielrareolisaIae-

9"[ U(r, s)] _-\[- U(n, s) e-P* dn : U(P, s). (1.13)

Aplicintl aceastd, transformare ecualiei (1'12) se obline

- : --p'pzu ( p,sy U(0, s) U'(0, t) - rf U @, s) #l,U (p, 0) +- D'(p, 0).J, (1' t4)

t6

fttr) LL

( ra' r) :.(+o:rLtg I Q.n)ne1 ( g'n),n)

[#]'((+0 (n),n-(+0 'tr)n,s - (s rr)prs :

l;+1'-((.+0 .fr)?L (s'r )11 s :

(oe'r)

as1s1r1,eso ar*r-s 1od os ( rr r ) e1{ lulyag,ep .;';+,;t',3n";.trff :#fi,*""*"

ra,r*zrae:rd eyru;tod o?IurJur loro'l It? lertieop, rnlnz*c t?o'r'Bp'roqY (r'1) tS (e'r)
[-lkorprr1JSoJrySruar.BrfJa?1at?11Klrl?oJgoruuo!1F'.ruoriu1;ir4ou;nIIc?euIr:rE'1uer ytlqur'uz-er:.rrd:\ia1s lJa'u:JgrE:jJ_ts*dI":+cseacld€euTI 'ro['r

gr1u11u1 er"u13unl op !!u!l't'L't

'r{u;rluecuoc

rrne-uKuiI.brrr1n(ri?rprurrlollssrr;reilr;ucr,ruaru"zrerJ"rl,slT1ojrrqI''ru(g"rJrnLaers",ls,3rroo+urFul.e.l1o,rohzqrOap+o?dlp(ulipzJs)aea+,,laJtarsnsnt.OoaroosBcnp;-eKl)froopls,udJilncJruau*noEpcto?lcKul?icuZoelp'(argAqlzrlIcdlruuIzaornrIcpotrIi:rJcu?nrrr1osp;Jd)norqtnd'rsp+z1ocGKleauurpuE+1J;rraqarotIsi+osenpltssluunsur.nzrra1.|r,taerrau;tcsoaagralltraJugsosJucsuooIuelu!o,€;+uuK1rr50IieSaourplnqd_pIlqr:ruuSddor(IuKisSnpuInerlusr9rsrr'istcoier?co)pp9rliJIeaIInae1u+loqeccelJurr€Jrruuaulrf.Ioso1Julp'nloJr1y3trr,:is+dttoIi.al9uouK(Ie'lgtoIeInuslso+m1csctq;.cIprttel€llriooId.i1luosotep4u.atpuolrrtlprJou9aFerclodpofs.orcelI,r€equte_ecl;sJ;oe€I1drcroeu+d,IqrFutl3aouoEod^guorsnIutlr+I+reroeqpdSrnutcpocusroaroopup"Ilte+?oreqalcrdsdglJ-;rltdaraa'asple(1ropagtuu(usclIrqtnn€p'rp'.ucodretgeo1I-lulutB)uppesl2sr4c.capcrl

(e rr) '[(s'r)n]r-5: Q'r)n'

(err) 'lG'il)nl,js : 6'r)n

I nc 1.rodel u1 rode r nc lrodeJ uI II+u1 G 'd)n rlrurJprrr osJolur

ocu1du.1 JoIIJFruJoJSue,rl ee.recrlcle uurd eur{c1o as (1 'r)ii, K+IJop el{n1og

(rrr) :-"or,fr(s"nlt)d (s.d)n
(s '0),r? *(s'O)nd
-

euriclo es (s(d)n nc ltodu;r q (tI'I) "er{unce puplozog
(r)nf'S :
(eT'r) l( 1'r),nl%: ( 0' d), n'l( 6' d) A
11
(grr)
:f:.1{*o,nf] (s'0),n 'l(t 'O)nlg : (s'O)n

l are,rlrqre oluts?suoc gtps ($td)?nc lroder u1 ar{enca P+suac€ tq
pci;rqeFle er{unca o o+sa ol€c

IJtilizind aceste egalitS,!i' prin aplicarea transformd,rii I-,aplac

ecualiei (1.3) rezultd,

W -6r,1-R)1Gf sc) u: -lRC +LG+slClu(n, 0a)*LCu'(r,0+).

dr:

(1.23)

in ultima ecualie se noteazii (7.24)

yz(s) : (-E f sI) (C * sC)

iqf'aei caseelrnenililm:r0ti'n(+n[;,m0ad)rriemszeeualptdo,ua'te(ne, foe*c)t,uac,alrieninnucl este imediat evident5. Elirnina-

seama de ecualia (1.2) in care

Gu,(n,0a) i Cu'(r,O+) : - di( r.0-)' (1.25)

1l iI)

Introducind u.ltimele douf, relalii in (1.23) se obline ecuatia

dzu 2u : L cli(r' 0*) c(n * sr) u(r,la), (1'26)
d.* -
;7-r

care servegte drept punct cle plecare pentru toate pr^oblemele referitoa,le

Ieastleininiidleepdeendtreanntsmileiiire.dParredceizphinrndec[d,.e1s(s, )dseecinud-me etiqmtep.fu,nclia d,e propctl1are,

Pentru curentul i(n, t) se procedeazd, in mod similar ; prin aplicarea
transformS,rii Laplace directe in raport cu variabila f, din ecuafia (1.4) se
obline

dd2m1 z- y' 2r: c du'(f' o+) - Lglsc) i.(r,oa), (r.27)

tln

urrde I(n, s) : 9li(r, t)1. de lungirne infiniti, (iig. 1'2), la
0 se aplicd, o treapt5, unitari de
intlareCaoncSsi,dreeiar[r(nr:o0)li,nilea mcleomtraennstuml isdie:

r--I o(tl ,\I^":--_------_--.--_
-)!- -nt

-0Lt --r x=0 y:!**
Fig. 1.2

tensiune tEt(inc,utr)e:ntuolu(in) ,lat)c. liSsteanclearer rS,spunsul (valorile instantanee ale
tensiunii de cap5,tul emili,tor.

Presupunem condilii la limit[, nule in raport cu variabila I

u(n, O) : 0, di(n, 3)/da; : 0 ; (1.28)

t8

6t

(le'r) I ntrs-o (s/f) : G 'n)n r

zur lsaae uI '0 : O '0 : auraop (Zg'f) urftncg
y I unptat'd p"t'p{ arur1 Pa
'nls : J' 'e.Ie1nc11;red rt:.:rzvc IoJ+
RJopIsuoc os oJ€nul+uoo uI
'IoIqI IB rcrzlJ r* rcr.rleruoe8 I-r;rleureled ep F+Eu*u
-ro+op o+so ro €ar€oI,BA
i pl"li e pbl1.siia1cu:rec g,jLuepedrur er{cun;glurze"rde"r

(ge'r) (cs * p)l@s + u)/l: (s)"2

opun

(se'r) (s)"2 rP p Is + 2I:(S',l)f . ^\_
3-G:- ',r)n
(rf 's), I-

uro.\€

:(s 'n)n 1n1 ersa;rdxo ap ri 0 (0 'r)p ?1lu4l el er{rpuoc op +uoD pulqf,

(ru'r) (-sjq'r)-np - :(6'r)27 - (s'r)rrs*(s'n)ru

-r1du i (1'1) er{lanca u1 ileeptiueerurpys{"qpio"ofulors?q+?c+ogc4zouprrepIptsoeraucouoldll€doceslo+€KuooSltJ?unrIucKror+uJssr?ulxu€goJd1'.(lp'Zu'ltc)

l:t4enca *f "p'puru.rod
'1,7-e(s/r)li-s : [(s 'n)afvg : Q b)n
(ee'r)

-qo as gsJo.\ul eae1du.1 €or€urJoJsue1;1r1unprusulcor?idtsv o'fatuT€'+iutes+rssou;rctlxsaoJa.€ruol,uLleepuurni

(ze'r) ''^-e(sh) : @tn)2

u01:"r4g's+/1€Ip: o}urrr rcop 's/t : (s else (66'1)

rar{ence er{n1os or€rrrr11 E+Inzar €nop €'OE)l4?Iptturooact?U"IC4i6;";
eruud op €ur?as pu1ul+ !

'+rurr ouyuyt (7 'a)n 'Qo <- 7-: fr

(re'r) ,(l)o : (I'1)n 'O: r

,t €Iq€rrts.{. BI aJtsolIJoJoJ K?rruII q oIITi

-rpuoc yzaozl;r+n es e.rul5elul op Jololune?suoc t?oJ€ulrrrJo+op nJ+uod

(og'r) ',xo1i't"-oY:@tn)p

(oa'r) rsruJoJ op 'erfnlos Pur-l'€

'o:nrL- -frt)

*,

ouraop (gZ'f) erfunce

prin aa-ptleicnasreiuantiriainnsfPorumndc,truiilLraplace invelse se gzise;te valoarea instanta-

neLr

u(r, t) - E-1[(1/s) e-t"i'] : o(t- rlut). (1.38)

tlltinra relalie ar,ati c.i in linie este_prezentil o.t,ndd' progresird' de
de la stinga spre clieapta cu vit'eza ar' Intr'-adevi'r
tensiune cat'e sde *proIp'a:gid. in linie ra"e-riita o tel,siune de amplitudine egaiii
la un lroment

ttl1.1--u(xil r__-| ulx't) t

I I

0twLA> lr Li -r
/W
(o)
{bl

Fig. 1.3

cn 1 pentru n lwtr,rsi egalii cu zero pentru n >ulr(fig.l'3 a)' Pe de altS' parts
<-r^1numit5 aistanlX"n:1r:tten-siu.ncz1
l:i este zeio pentru t{rr/al qi va fi

egirlil cu 1. pent'u t> rrf w (fig. 1.3 b). djn c(r1rz.3ul5l)insieeiobid,seearlve5Z, c"[: r(r's)
tliferri Referindu-ne la variatia curentului in
R":
d.e U(n, s) nurnai pr,in'parametruL 2,.
: VTP este o r.ezisten!5, puri qi se numegte rezistenla caracteristicfl. Prin
ulnral'e expresia curentului este

i(r, t) : s*1[1(r, s)] : (1/l?") o(t-nlto), (1.3e)

avind aceea;i lege d.e varialie ca qi tensiunea. caz expresia funcliei
b) Linie
Jd,rd d,istorsi*ni : RC : LG. in acest
de plopagare i,(s) este

:y(s) [@+;f)G+;O : (n*sr) lfC]T (1.10)

ial rela-tia (1.32) capritS, forma

(J(u, s): (Us) s-rn1'/ c1r"-ts1w (1.41)

Ili,spu1s1l in tensiule a1 tiniei se obfine din ultirna relalie plin apli-
c:ll'oll tlzr,nsformiilii Laplace invelse

u,(r,t): S-1[(1/s)e- xn{c1r e-n'tuf : s-rn!qLt(t-nlw); (1.'+2)

acest rispnns clifelzi de (1.38) numili prin factoml tle atenu^lu o-rRYcrt.
Dacil in loa de treaptS, unitar[ c1e tensiune o(l), la intrarea liniei ln

momentul I : 0, se aplicd, o frrnclie oarecare /(l) astfel incit

:Lr(0, s) sl./(r)l : ?(s), (1.43)

li.rlp "n-?J +:1e)y.ra-1:(r)elra 'to*-" 2t1 :1c);'ra

-;'
z

e111ie1e.r ugd a11u13aP 1u1s
or€ora ap ^EJeluouelduoc elicun; 1s (o1e1111qtqo.ld ap ulurFelur) ar"oie'ep elfcu"g G'

??ILL I (t'a)?

; llr: :tvtrl,"*-e
(rg'r) l, 11cs/1, -a slrlt g I c Lllr-5

?-'(rn1n1T:ur8r.ro 'R nc

polr{:1nuru1 o" Uio"rnotqco a+so rrur5eurr tso,rttalJop) trul5'Brur rr.rp^Irop €uroJ
-oa+ ap BruBos ourf os osJaaur eculdul roluurroJsutsJ+ InInJIuc n,l+uod

(og'r) : (s)7

"", 11" -e uslC 11 @

also rnp+uoJnc € pleuor{erado urse.rdxg

,.(ffi) :(ol'r) """ r-s:Lc-v:11'-a(s/r)l Qh)n'

ifl ortsoro op g,reluaurold
o-uarorrc{cuurfncyunroJuFnzuEazollsrlJno,otsulseyaretuldoedxl oIo1+dEeru,r,pTopJSuu;t,stJt+pIlnuriniuceltusocdxruotruooeduriuoc

(el'r) 'cxs1r-o (s/r) : G 'r)n

ouraap (Zg'f ) tt{enco rer
J. orts e"re8udo.rcl ap er{cuny (olurxu alqec) zl?c +saD€ IU
'00g: :AO,:0 trso.rdxo
: rtpd,pcs ap piunnnpuoc tw ti po'L\cnprt1,au aru'r7 (c
T : olrLu
(rt.r)'@lr-lo ltcjax-o Ilell : L,t,,-s 7!?lte,-o (rlt) sla llh-a : Q'r)!

tfJl"tiit"" *

oour?+u€?sur €or€oltsa aurlqo os Ksro.nrrr ooeld'e1 €ar*ul,roJsun"r

(srr) 'mtsr-o-rrolLer-o(slt)rlE/: @b)1

rf1:o-u?rsp"og(rK.raletpsporuopnrclul(gKlug.IrT'flJo)IJaeeIrru{ueIr1nIenruzt€o.rcpd'ur+ouIurm'€OonscuTJpaou++:lEsullpCo+ppIaIJuIRnerJr${oq1r+opocurEoou_sJrcl (eIstUrst+(drnJ.)stEg7rIJnpsDIYn-T(r+?:g+ertJrstpot'3frodn')c+Cfzsuovr{_cup1uual.ctrt

rrceurn{Jc,rur3nr;uerusgJleou1rsFrpuoRJ€EuJrJoorJurrlysoap1mdfnr.rtIunrInEuAe.rpa+gucI rnldcx+ets1rzd,4u+;ullsIo3atuteIl'elx1,e-ar1sIngrdo?ec'lls?oJ
uud lenuoqts o?-qo t(t)/ p;ru.r1rq.re er'icun; o garldu os o.l€Jlur €I pulc rcu'n1e 'n

e{uelsrp tsI rurusJo+slp ?JgJ rorurl Insundsg;r pc g1u;rt et'i'tl1eJ larm?Ul

(srr) t;;:r.- -r' (m I 7) o tm t r tl

;--:, €o,tsru'orsutsI? rtsI

(w'r) 'ntsr-otrclx'-o(s),.{: ,L-o (s),tr: G 'n)n

our^ap (OZ'f) totfunce er{n1os Icunle

Reprezentarea, graficd, a ms,rimilor u(n,l) ,5i i(r, t) in funclie de argu-
rrrentul [tlnzRC pentiu o tlistanl;[ dat5, r, este arS,tatii in f igura 1.4 cle unde
ruenzutlitmd, pc['ctelensoiule^;nt,eealeqiimcuproerntatunlt.siIn)tinpuatecernaisct:idicsatuozrs5i,ocnaabteielienttirmanps,oecexaisntiicnetl

0.8

0,6

i(x".t)

0,/. -J--"--___

56

Fig. 1.4

mseulnltctalercc[odnidni.t1iaocRinClo:c cu sarcini inductive pentru a ne aplopia cit mai
-tG a liniei fri'ri' distorsiuni'

1.2. Linii de lungime finitd. Metoda

undelor mobile

1'2''r' Forma

i,"lli,ili,,3'"'""t"'"'

in aplicaliile referitoare la tehnica impulsurilor ln general se utili-
uear^ finir- de lungime finitd, (numite adesea linii de intirziere)' Ca urmare
ecualiile liniilor
d'?9(.{, s) _ \?(J(n, s) : o,
(1.52)

u,r"

d.21(r, fs) _ yrl(r, s) : o (1.53)

iLn2

(scrise pentru cond.i!,ii la limitd, nule in raport cu valiabila f),. vor- ar-ea
ii""ir"id?,ii"ti*"""f*o"um(an,mt)aiqcioim(ap;licl)atcdo,;nsgSlit,suirieeaparocebsletomr esoinluelriieqnit apoi a valorilor
mai laborioase

d.ecit in cazal liniei de lungime irfinitd '

zs
ZL

r

1;--*-r---;'

Fig. 1.8

Presupunem o lianviientdl eimlupnegdiamnelaz(in{iige.r1n[.5Z)joI,acainretragerenesfeeazaiiplfjucdn,culi.an
g"o"ru-trl" at ietrsiuo"

.__rrffi

?,

(zs'r ) 'zl"z*t _- "_?zz*-1'z2- - ru,

(rg'r) 'zl"z-T
(og'r)
(oq'r) €oruurlsu{ oJtsc uI

'O:txo7 - u,*oYrJ urour{qo opun op

'Gto{ -,,-oy) U l, Z) : txa{ }- r,_oV

(s'ilr'z : $'l)n

: or.Ics
r-oluod es rS 2 :
tr r;tto-Le, ,96 an|e;. op olau.roq nrluecl J€Furrs po111 uI
'r>uJ >
€unrcp1o+ ! rerurl E a.rrJqur op olauJoq [I orxo1Jor op 1nlrrorcrJnoi p1oT""*an*

-o2zlr"zz*-TT "zioz
-(sg'r) tr,
z-bz-
'r

I?AUIIJEIII OJ€C UT

(zs'r) 'q"zrlwoz:B"J-v

IuolBocs opun op

(gs'r) '@f v)*@-v)(zloz): (s'o)n +$ 'o)fz: (s)fl

ar"rcs aleod os r"l g : tr urortt? ,/f oJtsJ+trr ap olouJoq 12I ou-npuuoJog

'(a.ru;r8elur

op o1uu+suoa glurzarde:r g ti l. alrrurr?ru) (gg'f ) nrisa"rdxa o.re (s)'7 epun

z(s)"
-(gg'r) '
(s)B (s)trr

".,"o "r_s

EruroJ ore (gg'f ) ;og{,uo;nc v
eleriua,royrp rar{enco erfnlos (Se'f) ti (f'f ) olrr{unco op +uoc pulq$

gurrurolap es ri s €Ilqur.rts^'tropoelpeqrutsnrruelonr€cr+lrroqrdeegl lucluKn?;Iulur1g1su(1sa)gpr'i(isp)u1or cepuulpn

(ru'f ) '"1e (s)g * ,1-o (s)f : G 'n)n

r?rrrroJ ep erfnlos e;ru (ag't) rolrunlsua+ u plur{uoro;rp ur{encq

psunfl: I'c?o orxouoJ Knou o
rJorns op alauJoq u1 p5edord alulcaga;r alrsa.r6o:Id
apun tsA K+E?toIroJ r?pun o.r€J+ur alauJoq e,rds
rios E+ts?caIJoJ o+so
roun BruroJ qns oJ€J+ur ap
ounrsuo+ ap tspun a,nSar ep olouJoq u1 pug5unfe i (elueprcur) elconp elrsal8

7-o.rd opun roun tsruJoJ qns ,AZ e;rrier eeeqp"ndaiualeor;uur;J1,or'tqsarea(;pgrd' rsor,ugfcr5puenndylol+;drdvm$ues7aZ,7p-61p1o6uJ71€dJae+p.urir1t

op olouroq e1 plearlde ounlsuol ep

-uepadurr I a reutc:rvs K+ts+couoc else

r-e1pl<'ezDlin,i<tnd,r1ec.olaelfiiicleie(n1tu.5l 7d)e.si reflexie la bornele de ie;ire a lirriei I totdeauna
(1.61) oblinem valorile lui -4 rezaltil
;i B ;

,4 : J9' (sZ)oZ*'2" e"'- eYI Z"I 1r---f'noIl,e-zvr , (1.63)
=- L (s) Ls+t,2z".
I'o f.re -rl

- k #*r,IB rAe'2r I :-E( s ) (1.6'1)
-Z

Introclucind aceste I'alori in lelal;iile (1.54), (1.55) se obtin explesiile :

ur r. s) : -Ets l - Jt c*t'' Fl't-"(t'-l, (l.uJ)
Z-" 1 -
Zo = lof,e-2Yl (1.66)

/(,r. s): -Ets)'zo*2" 1-fol,e-zrr

in cazul genera,l (Ilo + O, lt + 0) este e-\tlem de dificii[ gi,sirea

transformateloil",aplace. invelse a m[,rimilor fI(n. s) qi I(r' s)1 aclesea acest

inconvenient se in1i,tur5, prin efectuarea unei dez\-olt[ri in serie cle telmeni
exponenliali a numitorului [4]

1- ln l,g-zri :1-r fofre-zrt' aTfil?e-4Y,+ . . . -rl6lfe-2nvt - n!:ol6l1's-t""'.
( 1 .GZ)
finind cont c1e ultima relatie, expresiile (1.65) qi (1.66) devin
U(r,s): a(s) -;e[e-v' ' rre-r(2r-o)1,i f3alu-'""', (1.68]

I(r, s): uU, *Zle-"'- f; 'f,e-Yrzr-'r1 lffl7 g-zort (1.69)

1.2.2. Metoda undelor mobile

se poate arSta ci, in orice punct r al liniei de lungime finitS, cu coefi-
cienlii dd relfexie 1,, lo diferilf de zero, tensiunea insta.ntanee este
teztl-

tatu.i suprapunerii unei unde irrcidente (clirecte) qi-a unei unde reflectate ;
la fel, o'itoarea instantanee a curentului in punctul r este rezultatul supra-
punerii untlei de curent dilecte qi a celei reflectate.
R : A, G : 0, de
Oonsiclerii'm in acest scop o linie f5,r5, pierderi,
intrare este conectat un generator dj tensiune avincl
lungime I (fig. 1.6) l Ia iar la iegire este conectat5 sarcina I?r. Presupunind
rezisten!,a interrrd, Ro

97 pr
pr
\gt'I) '@lr * t6-t)'JtL * @ln-t)"ry : [(s 'n)n7rs : (7tn)n e

rr{e1e.r rorur+In E ?sJoaur aceldel tsltsruJoJsutsJ+ op {-
E+"sp o+So iolull Its r qcuna un-J?ul IlufiSuo+ E aauts?urE+Sul €oJ€oI€A
_I
&n) (mtr-*)s-a'a tS'S* *t,,-o T : $ tn)0
-I
u;x61*Tic"d'*da";rr";uE,&irjroler.u;pLll r1€,iTifiirr".n'fr"uo"c1nur*o"*"1lipoui;tl1tuf"'ilegsdlocp"i"!ip;p,orreiie(la-g"u'u:if,;"s6ro'of1)rq+e)+1e1e*)11ir."cf0p"uu1urnbrd"sc'c'rer1nu1ierrdxEltuuroIsaepeI4u.'"nI:'7n1rs3l6tuos+eaaftrrlo',oB.o'U?e1prnu'O7toon:up'€aIn+eu7crleuuoun7zl"vrn(tocvgrqedlseeo?'rrectdsrrcscl
(
s's ---------' (s';t),
(err) . -l- 61tt-a''l - {
(iltx-Lzrs-a6JL
ij
eur{qo eS '((l)o : (ila ountsuo+ ap o+t3'}run E+.$.er+ o
1
gcrlde os oJeJ+uI €I) sh- : :(s)g- | r'1rr1 "'I'o'L:l: t'l It rsaa-o'itrf'1 K : ory
II
,(rU*oU)rT:ry1 triepru'url no1'ut:l{g*rnyu:ree.rglxroierrlRonuopluoalJap.roo'r'Ilululi spun roTrrt e a;rt?3rdo'rcl
1
"pi"arir{ Jlse " ariqer tlrul+InuI
(
ftr'i 'n!,ruz-Q,1.f9.r o:u **r,r-al"fvr.a* (ostl)E: G 'r)n
K lmt@-tz)s-a

pldurrs IBru gruroJ o p1gdec (gg'f) erinlos

(u'r) ig: 7'0'l' ZfP

(s 'r)O - (s ,rlnrp
",

s'r'Ftg

Z (l'x)l + l.

our^ap (Zg'i e1{,ence 'a1ur1 uI Jolopun s e"re5edord ep ezolrl o+so oz opun

'clryi: "'[ : "z

(ol'r ) tmis: CT11s : ,L

r.urro; Pqgdvc "Z 1S I P1
eprsardxe gc Pull.rosqo r* o1nu ? €IIqr:uBA nc lrode,r uI K+Iur11 tl1 p{rpuoc

Primul termen reprezintS, o undX incidentfl (directi) care se pro-
pag5, in sens pozitiv, de la bornele de intrare spre cele de ieqire, avind
viteza w: d.nldt; al doilea termen reprezintd, o und5, reflectatS, c&ro so,
propag5, in s:ensdnnledgt.atPivr,indeurlma abroernreelelaldiae ieqire spre cele de intrare, avind
(1.?5) se poate serie sub
viteza -w forma,

u(n, t) : ut(ff, t){n',(n, t) (1.76)

u.nde uu@, t) : LR"l(R,+ R)l o(t-rlw).

u,(r,t): ln"l(n, + n)W,o(l - 2r { rlw). (L.77l"

In mpuondctanraalolgl,inliineiinstel seama de relalia (1.69), pentru curentuL
poate scrie
dintr-un

i,(n,t) : iu@, t) | i,(u,t), (1.78}

unde

:i a(n, t) lr | (Rs + R")) o(t - n I w), i,,: - lI | ( R o + R")l l, o(t -2r * n I w).

(1.7e)

Se poate trage conclazia c:a valoarea instantanee a tensiunii in orice
punct r al liniei poate fi descompusd in doud, und,e m,obile; unda mobilS,
(progresivd,) incitlentS, u{n, t) ;i unda mobild, (progresivi,) reflectatd,

u,(n,, t). L,a fel, valoarea instantanee a curentului in oricepunct r al liniei
poate fi descompusi in doui, unde mobile, una incidentd ir(q, l) ;i una
reflectatii, i,(r, t).

b) Interaalu||+Ac.in acest c'azn,:1 $ik0:1f fof,e*z"t; in acest

interval treapta unitate de tensiune aplicatd, la bornele de intrare 71' se
propagd, pin5, la bornele de ie;ire 22', este reflectat[ qi se propag5, spre
bornele de intrare unde ajunge dupri timpul 2t ; aici este reflectatd, din
nou qi se propag[, spre bornele de ie;ire etc. linind seama de valoarea
lui ko, relalia (1.73) devine

a@, s) : as1..1s1a lol,e-2'") s-"1* + I' t1*fuI-l e-2"";lre-s(2r-rtw). (1.80)

-

Aplicind transformarea Laplace inversd, g5,sim valoarea instantanee

a tensiunii

u(n, t) :kl -6 (t |n I w) lo l, o( I - -2 a n I w)l lk f''lo(t -2'c {

* u lw) * fo l, o(l-4r I n lw)1. (1.81)

Primul termen reprezint5, o und5, incident5 (directd,) deoarece se

propagi, in sens pozitiv avind. viteza w: d.nldt; aceastdundd, are dou5,
inilial5, caracterizatd,prin argumentul
componente : unda incidentri (d-tu-pndnl,wlwc)e.)
,si unda incident5, sesundard, caracterizatd prin argumentul (t-2r
Ifnda incidentd, secundard, a rezultat din unda incident5, iniliatd,
aceasta a parcllrs linia intii in sens pozitiv, a fost reflectatd, la borneio de
ieqire apoi a parcurs linia in sens negativ qi a fost reflectatdla bornele de

intrare.

26

-:ffif:@:ffi,?ssffi**

LZ

'i"u: 'g?lts1couoc o+so oJrsor op alorrJoq neci Isouzn"ar.sEucor+,}sorJpo?Icn€Jruoc+€ui;uo€upeo5d1lutsl+ocJotsuo6ctso\+rJsaII
gu"requr t{uoqsrza.i

I'r 'Flg

j

oJ€J+uI t?I JoeI r"g pcrlsr"re\cvJtsD'efuupedurr oJz l-EruII: (l'f 'Fl+)ft.1n1cund
u+i"IoI+qBo*?"n€"+coc"eo5ulsorSecroorlus+lruiuprslduluveJpis'+oalerov;lprcsur€€caorzurrr;rlrlteoso+poJulg+eorrrr'dolrpr+oKIalraIuoErJuJuIrpxI{ouI+uuouo[cIScl'nZ"eZop$pIln*qororcZrZug+nerIodpcp{tnr.1sucuerSoKpe3IeIe?d"(rI.rgJ1udIp"le?IclEo€urprI
eoEuIxeovraIuJcoauJoopJo-p.oru+odiuJeoorJu,IapolsuopeurysSul e€elJue?nfierox'eapourao1lcasJsro"ae+c1[oIcrud-.orpptl5^cua1ala1iu+etD3cupInotgdctlnuuzr1rv.oceueoe+llus;oac;1rtBoodup-Iao'r+reIcJIaorrpIuuoru[l
(f ea;11iel rEI tsoJll+cauoc nBS o+IJeJIp ocl+slJolctsJuc
frufJo?In111 I€111 13 ) 11gn leun lrrp+cauocro+rrl Eo+ts?rseceu a,redu tsasopY
gnop e
e{uepedu4 nc

e;eldepe aP apolel^l

'ersrursuPJ] aP Jolllull eaJellauo)Jalul 'e 'L

(rs'r) '(s)'? + (s)'?: (s)tf: (s'?)1
'(s)'"n +(s)z?n : (s)t,?:(s't)'2
(se'r)
: uroar? (l: r) a.uiai ep olauJoq nlluod
(gs'r) ! (s)t'.f * :(s)ro1 (s)tl : (s'O)1
(ru'r) '(s)t'n + (s)r? : (s)tn : (s'O),?

eeru.pn1iure*6rd(odduplerr.or1"rgoeacpe€p-p'ii,cu1rup,!serl)-3rr1s{eff1e"o1p;8r (,o?in;{nroz?poJdoi)i:oirr&uel"?1pIO,r"r1{' a',Ju-Tn'0;?+;#ueiisrtx"e?Hgo'ol."'"rI'o""fpnrgf'"r{Ju4fef"1"c"ell;epi?ouogn

(es'r) '(s'n)'I*$'n)ul:$b)7

(as'r) (s'n)'n * G'r)'n: $'r)2

: €urroJ qns osrrcs IJ +od (OS'f ) l* (SS'f ) opr{e1e.r ortsrllrtt
u1.rrl 'ru oclJo ru+uod apqelel uFr?J sns rtsur esndxa olo+ualoprsuoC

'(nlr * LV-il lueun-8ru op Krospuncos E+E+
-colJar upun ri (mlr * tp6gs-u?e)cvluIeXurpnl5n.rpu_o:p -tutllexreuzra,rldli-Kp?utsp+tcsolJralrrlutsfipauun : o+uou
suos uI
-odiuoc gnop-ele ppun
p8udold as boe.iuoep F+€+coIJor Kpun o plurze;rla;r uorrJo+ tsogop IV

rezistenta tRenrs.:iuRne"ze. (Pt)r:esuop(ul)n.eI1nn c5, generatolul firnizeazd o treaptd,
unita,te rle
punctul de interconexiune se pot scrie

relaliile :'ltrt ut { ttr, i, : ia * ir,

(1.88}

uuirnnf di,lei:n?ira,-8, "2ir,.;Predrpeinr-eadzsieenftmdin,eilnuieenadcaoaeudfieclioetencntrsreilugdnaeel,irteir,efllieslepxieeucutailviuud:nedRec"iu.,.dr,eeun,ttfei,ntrs:ainrrsn-mReicsifntlr

punettrl de colres.iune rt'e e.rpresiir

-rr R"z- R"t (1.8e)

Lzr- -R- "'*R" '

iar coeficientul de reflexie pentru uncla de curent este lz e: ste- 1". Limi-
nnde timpul de
tindu-ne pentru moment la intervalul de 0 + 2c' r,
propagale in linia 7, se pot elx,p;rimqtaimrneidl,"riurn, i:le a,, qi f, ii,n: funclie
f, qi coeficierrtul de rellexie |,tl,, ;i de mr qi
- l,fr deci

tt,:(1+ f,) 'u,i:Ttu'u"i, i,:(1-1,)iu:y,.1 . (1.e0)

lf:irinile l,u qli l,u reprezintri coeficienlii de transmisie pentru ten-
sirrne, r'espectiv curent: daci R"zlR," avern lu)0 qi llu)1, l-,r(0; drlcti
}?,rz<R"r a\rem lu{0 Fi l,r<1, 1,, }0.
2o) fnterconectalea cu rii,mificalii. Considerii,rn cii la bornele de ieqire

:lle unei linii rle transmisie sint conectate douii sall rnai multe linii identice
I aceste linii sint adaptate la iesire ;i dorim ca in punctul de cone-
cu- prima realiz[,rn o adapta,re perfect. O variantil posi]tilri tle rcalizare :u
xiune s[,
lelelei de adaptale in punctul de coenxiune este ar:itatii' in figura 1.8 au

Icionniielectactonusntingdendeinrattororndseoaimnepudlesucrai baiuvincdoaRxiaol.: La intrarea liniei (1) este
sX afl5m va-
R"; clorim
lorile rezistorilor din leleaua de adaptare qi amplitudinea impulsurilor

6Z

'(enrrFunl r{leooce n€ (8) rS (3) eputl)
o
€lEAcApE o.r€oIEA op oqar+rc€d€c gurcJ€s g r3+ts+couoc olso €IoJpc ee.rrSer
€I '(f) €rurl od F1€+callar EPun aP E?€Inu€ ed oe+€+calJoJ t?pun
qa^ Z erurl

(ql

i rgnlcnpur Tulcr€s o eadrv,rgz'rleflrorrturo.rpaxFap+'ogsorfug(n?l:+dtsuJgr1a3.xoa+sualpqs 6(aa'1.)re'el{d.urenI5pI reJulauppIts(euqnnupau{olcl
pzea uf rr.rpldup'tl
re

urp ,rolrro+srza.r olrrolts11 'a.niar a;rds F5eclo;rd os nu crluru '(r) ls (a) alltult
guoE6KTo1rprLc{;urpr;dr€orvlroaJ,apzuplaaruvoIu.p,rugoruau"zrargusr'ailzouuur,lesnraaIer.lrinppa-aniuurges.,1luruea,'l1rtuarten?uo1sBlndo'Jn€eSrn?opacq+rllucur?rK1oSet,iIrtgcurua(o.iaIoIulaJ)s('aatd3e;eFl )nguaeu€rlalaplt:IurrUpdeuleu.nIlIrIddnsoltuntlpsCuic(q(ualaarocpuu)ru\iurI.arn:qso.a8r5corzxoseyraoIoIreuu.r"ro.uuOadrJclrsrpor8sapooau?p+8+nnjuuEocIlodn+_oKroc+euap+.fcorIcudJ1laooa^nIirJJad6unos,.tcrrulnoa1oolnulsopa1€lool€cscu;uu1guontassunlreclusaduruo+'s(orogu1+€+u1)i
$eui-epc,1ad nc auoruaru ep ayJncoltl ui csaull+q os IJnztsc ep 1a;1se) e.4ie1
BI lrnorrc+.rncs uI a?sa ortsc '(a) e1uII nc E+€?cauoc a+-sa (1) 'urw'61'I(uu.rnFrJ

r4 alup 1u1s oldurero uB1no'(pal:l1rclutltal raopl1oriolltluoosltalour+lppHuI+Sron+IoIlJ.suuoacruFlgdIDupJtessr)eez{lyueea1r

eluod es rrlzeCa?soDt?
ae-c;1atst.sorcJrlJroolerlJiprue'1rolBFIznrezIurEyoD.'Iyanalcsle;otorcid{rr{anescrlsunaueDreyllpoeu,JrrnSrrorrrilcaanuprl?ea;srlauuudolocnciatl laeu1+ounlocrT.tr;uonalselsFgprllmqsotSxogpIooJ(r.ugJouIInunr€I
glnIKinhn'S;coteruutll+lue-rlIuzeIer7re'udll-to"r.rfLzilolooqae;-cirgneenu7stgnrs21auc7olq;,ressr18rnsa:lear'oqfaprdo,urucre.edirlrr:.u;strl,,ntras8rud'nagr;lplln'nuugoGsllcrnluo*nl?y+Ar-dozlt'?quon:o1eJrprai1tJzsrt[errJner'rlroK+.z.Ilrss?oo:olaBcnautTotsr,nJSJraloIpuupnJlIcreao]opadpuIy;zato;TetKuralpala1IccBr.B_lrrleZloarsepnIir)I;ute€tprrt:oasusrlpuJgep'olalald,,sd_r:l11ustt€epnor{sreler;:eru€€c.ateroonq4.1uur8ul8lrtrpslpdt'rnlnriageggsuptolt+ltdr"dIsruIsurJqSSl?xPr€aleo+Jcr
ap olalcund u1 ,rer tlg:"n o?-\o Inlnslndrur uourpnylduru;l lnlcund ut
|ap--(pl1aruo'upFnnnodo1lpsurao"lUllgnf r:o)IiuraZe1rrsul(eraltl-lapu*lecgqrune)nz-ldrlr-ule2lpun1'fyc irutrlln.alrnnlnl-esEsllrrlnsnltucsdFlru?uerrnKtres+IneouOuuaBrrpdpJrSnr'Iyp'?lllBdldlSuSurart,sY{IJeo'Klte+lS"l!doUrn€Easp:a)ecZoUaluEgV.or:+r+*iIsennorz:eaA1r

1.4. Cuplarea liniilor de transmisie

1.4.1. Cuplai slab, Ecualii generale; modul
comun gi modul diferenlial

In numeroase aplicalii cum ar fi interconectarea circuitelor integrate
interconexiunile d.intre diferitele unitd'li ale calcul.a-
rapide, cablajete qi telecomunicalii etc., apar cuplaje nedorite intre liniile
toarelor electionice, ad'esea pot
de transmisie; d.eqi, in general, cuplajeie parazite sint slabe,
utile, sau Semnale patazit'e
ap5,rea d.eformd,ri'impoi'tante ale semnalelor
g"o""*azd, inforrialii false. Se pot realiza qi cuplaje dorite intre linii;

"aic,re"stea sint de obicei cuplaje strinse.

a) Procese Jizice. Mod'rtl colnun pi' modul diJeren!'ial

in stcraonpsuml pisrieecsizelrfaiicnoolsiuenriielodreqiipportoeczeesesliomrpclieficaautoloacre1a: cuplajul intre
(1) se presu-
liniile de
pun linii trifilare, f5,r5, pierd.eri, omogene gi uniforme; (2).lungimea liniilor
rapirt cu diatttetrol concluotorilor qi distanla dintre ei;
este foarte mare in uiconductor tl]r. qi unul retur (conductoruf retw poate
(3) fiecare linie are de potenlial
este un mod zero) ;
i(i4)"om*oodoulladeceplerodpoaugda,rleiniai sub forrna unei benzi metalice trans-
undelor d,e tensiune qi curent
versal electromagoJti"; (5) liniile sint nedispersive (toate frecvenlele din
spectrul semnalelor se propagd, cu aceeaqi vit'ez6')'
consid.erS,m liniile / gi 3 clin figura 1.10 identice trebuie precizat c5'
(pe unitate de lungime) -t
fiecare linie are parametrii 6istribufui specifici caracteristicd,
rJzistenla R" :{W.
si C, viteza a* p"op"gr""- * :ltQn qi

Fig. 1.10

Dfriecafieto;rricmito5,epfecicuniptirniuatujmulloudmieterinnedttr,uIuaiceluieinndtmreouledtu,uainqldci,i_dineJcndl{,teqd;iocpialpepasaurcapitmuatneeitnardi_dmcidsu,ttulraiablmdu,iolCimrsaep.neNtugel-

(so'r) tg
(ro'r)
,o? _"? - "? tvp _ try:4n
.Pg{"t:h, t?n),cn-rn

?I'oIt?n*Wpnpzo'o"noapnw.u'trtnoseeerlz1oaulaSlci{reouanlulrnl{oncrernasnuruuuurooqellsrousuVsolrpcmcc'"Ennumr+?raraVour'zrnccvnazn+oe1€1tl1nelu1.ulotrn.rrJdr{pouo"orelo;pou.szrqlrpeuJnyzaEnsru.IdpUrnesrTd{meoeIsrl1rpleounuz.oarpeuo+zonacsedrr".suresrsdo.uFreeJunt..ndorr8?'eocorIod"r'Jr"onlodg'll,!rzn+r'+oduuo,spntdru'a1tdgo1lpszcclnolerypr1ed'oo'gnlrscqurpne,rdot.fnornfluui{ltc.eoaunon.ewun.rm'r$o"n€rpoc1Lr

(go'r) W_tltrBrgfc)l: n* ,\WJ4)8/L _,*

(elueprcur epun

€I rurJaJal ou ropc 0< :ilL zsnloz'o(W+ra+1€Iu)rlnQu n*erne)s:) rf1u1s e.ruFtdo.rd ep olazo+Il

\cqc '(W -r)l?-n) 1u1s rr{n1os Ioryc oI€

(so'r) o- :_+zT-6 n--zW-6L0 -zT 2.tr4[

- *"frtL (wq ,tn)z

re-crrsa{oepdn'c1leuuiese;runarcl{tqlrdSoneacusnfflisuu:rlol?anodo}pppouElal+plorutNznvcn'(a,ar,lruIeeFydcedeVo.zeruvdo5Jeo+dpsou"orodzruooo+ppI^roJpsnnp[o6op]utusK1?)nsorexp.ortoJFpotdqa

-o.rd ep Jolozo?ra .roprse.rdxa tsoJtsulrrrJa+ap alrur.rad e1e1r1eFe tsurI+IO

(rc'f) '"(n - zmT) : "(Q -'"m76),

qo'F c(, + c)cl"rcelr:lrlquFt(azroc'czn*pocp )eclel(o'cd *o,s?()ZO:1'f)?tS+(€f+Oo'uf) olouralsls

€-s apun

'*ljr,*t!q):"n

(eo'r)

'mlFlq*t!n):rn

: e+tsocs as rr{enca op Tno+srs 1nlrnl1n urq

Q,a'r) I8x: (zn-tn)zrC0x

I :ex+tx znc(b
[ ,1- ,!: rncm

(ro'r) Jrn* * r.trym: zry

IrlW* { r176y1- rp

(gplnp : n) epr{u1a;r eurcs lod es 'rp Tz{uelsrp 8"rnc.red olopun lp InItsA

-ro?r4 u1 pc r* # Ksrcsq€ ep pueld t1 sunfe n€ +uornc r* aunrsuel op olopun I

1") Mod,ul de propagare conrun. Dacd, se consider5, modul de propagare
c:oretts2p:unzMd",t,oirr:viteizz:ei ar", sint evid.ente egalitXlile (uo:O, iu:91 i ,tL1:
i"; rezistenla caracteristicd, pentru modul comun

este

uc :R"":lt@+-MW> R":llffi. (1.ee)

i"

Deci pentru mod"ul comun und.ele d.e tensiune si curent au aceeaqi

ao:(mLlinp+ileMitue)dc1ihn2iev,aqlCeins,t5:e,m2(nCfigi.n.T'aIi.nm1ia1baeel)echalivivnianildie;nprt5ea,zruaalmtrdee, tccr5oii,trdcdeiuslectrtdioboruuudlil,itulsinrpiefioccirfomicniastdtitudi:ien
cond.uctori tur ai liniilor componente iar conductorul retur este format

din concluctorii retur ai iiniiior componente care se realizeazd, de obicei

sub forma unei benzi metalice de poten,tial zero.

2') Modut de propagare diferen,tiat. In cazul modului de propagare
diferenlial corespunzd,tor vitezei zod sint evidente realitd,lile (u":0, i":0);
Mtertis:tic-,51,rpbelnratr:u (mWo-duulzd)1i2fe, rire:nlia-ilze,steid: Ur- ir)12; rezistenla c tac-

?:n"o :ll@=WI@ +z4A<R":VW. (1.100)

1,d

Deci pentru modul tliferenlial und.ele de tensiune in cele dou5, linii

tea:auCcumrhpapivac*laielCtluiepadnqr1iiitmnd2ae,.emL(iqfpiilnigliinsit.aueiildm.ei1innca1eehribvsoca)ipolseuanesndvmetuSinnc;,etdarroeoreprpzuuauclslrotreadnem,;dtcuIuea5rct,rftfceiooielrrslumsepli eadutouctnuirdfddiecfi,noilelrLinmcdaoieain:t2ccduodu(nrLicenstontci-rttouuanliMeudr,ea)uo,ctcuetCloeirnraduiaqe:lil

celei de a doua linii.

Fig. 1.11

3") Decuplaj optim,. Interacliunea parazltra dintre cele doud, linii este

deosebit de supS,rd,toare atunci cind vitezele celor doud, moduri de propa-
gare) rilc si r,uu, d.iferd, mult. Se realizeazd' ttn decuplaj optimal atunci cind"
't0".: 111o1 aceast[ egalitate impune anumite condilii asupra parametrilor

specifici. finind cont d"e relalia (1.96), se obline

lrlL: c,,l(c * cn). (1.101)

32

ss 97 A99 io-8

(oorr) 'onrs('cz*il@-'il: #

(sorr) '"n"sc@*il: #

l11l epr{unco olrcs lod es lTse;rdxe ?nop olorur+In qCI

ilfl :-(z,o1r)'z a lztg y7 Fc *
#"s
+'n "s l"' cr - Ft 0 + ilwJ

(go;r)l "n"sL"'cr- (z'c*c)wl +rn rsl"cw - 6c+ ilTl : 0

rr"nn"o

K+Inzor mlnuro+srs e1la rr{ence Knop olorur?In urc fielzl7 'ngltlg ogrse.rdxe
r "I "n'r2op +uoc puiuli 1$ 'rn nc a.rodur m ruo+srs urp rr{ence e1eu4ld rugarraq
's rA ep rr{cun; luls a-f epun

(sorr) *Q11g(zrg* C) rnzr1s- : #-

(rorr) :t.z2 zres _ln(zrC + C)s A-

(eorr) '"rrs *'rws : frg

'no

(aorr) w-"rws * :r7"7s n0

€IrqtsrJtsl nc grode"r rrl gllu4l q rr{rpuoc) epr{unce er.rcs qod es (z(ror1'5nulr)'3

zt'r'31.t

*lP.'l' E

a+uol€alqco rorurl I€ xp +uomolo un n.rluad r(1et{ueqodtqce) pseru €I +ts+couoc
rn+or op ln.ro?cnpuoc nc r* t.rep.rerd pr-e; 'oarluapl oilu-rl pulroplsuo3

apydnc rolxlu.rl o1n a1nr,0ua6 ay,r,inncg {q

unde s-a notat: (J": (UI+ Uz)iz, au : (Ut - Ur)12, I": (11* Iz)12,

Id: g1-I2)12.

Solu.tiile ecua.tiilor (1.108), (1.109) sint

U": At (s) g-""1'o * -42(s) s""/o", (1.110)
Ua : B{s) e-srtwa+ Bz(s) s'"r*a, (1.11U

unde ?tc qi zou au expresiile (1.96). Referindu-ne la curenli se obline

I" : (If R,") l,ao(s) "-srtwo lA2(s) e6'r'o1, (1.112)

Io : (IlRra) tBt(s) I 82e"t*of, (1.113)

"-srtwa

IiCnro,ncIsarta,rleinnste-ioRnlel"ud, ld$iiselieeRian("mt1ue.ag1ar1dau0er)eer-xeAplar('e1lisA.i1liei"1l,e3eB)(xr1ip,s.o9tBe9t,n)tfseireeddsinepetteteracerltmiiaYiUnte5("1,p,.1deU0inn0ot),,rc.uoI"n(,JdrIi,luiiAleq,ilaqUiliy1mr,i41t522.,,
ale sistemului studiat. Yalorile instantanee ale tensiunilor
concrete ur(r, t), ur(n, t) qi r,r(r, t), uu(r, t),, i"(n, qi
curenlilor t), i2(n, ,) sau u"(n,
qi fu(r, i) se oblin prin aplicarea transform[rilor Laplace inverse. t)

c) Condilii de cuplui slab

fn cazul general al cuplajului dintre doui linii, unda care se propagi,
linia induce in linia 2 unde care la rindul lor induc unde care se propagS,
in linia -Z se admite c5, unda din linia
in etc.In cazul cuplajului slab I induce
1
in linia 2 unde de amplitudine foarte mic5, care au o influenld neglijabil5,
EdicnCultairnllpiiaileI/.(1CC.o1dn0(d2il)1i-il(;e1i'd1ne0a5cc)uepdslaet jvcisnalazbasvinetmdel?scccri=se de inega-
'latiDtscdu=,plirt/aetDunnd\d:L'et0<lo.\r
R"a: R",

}at _ sLI1, (1.114)
0n -

L#: -sMIr-sLI2, (1.115)

#:-s(c+ C,z)rJr, (1.116)

_dl, : s0pUl - (ClCd sUz. (1.117)

Ar

Procedind similar ca in caztl general din relaliile (1.114) qi (1.116)

scoatem

-t:t' U 1 L(C+Clr)s'(/r:0 (1.118)

A"'

34

(ralr) (

+,*,,,-.['rr'] :1ra)o-'v nfaw-'n ] I+:['r,r, . :*]T - :dr (

eur{qo es z7' n;rtruod 'a;reolpzundsa.roc o1alncpc pulnlcaJo ';tspulrs pom q {

(gzrr) llrrltrdaV-r*t"r-?tv"l 1'*t -: 'n:'re|7' - - "I (
(
(srrr) ,
a
((tff'f) er{qa;r op +uoc putuli) elsa tT In+uarnC _r

',,r,,4 |f'zmvsdTI tr-O-tt + za I + 9

r,6,,-o l,r' # f*-'" :] zd,a. +zon: zn (
I
q (rzrr) er{m1e.r our{qo as IR€uzIvJolqrLlI!vfctZosyizl foI plSrAtyluorcriupaurctruJotio3'(fZf'f)
purn+r?sqns

ftzrr) 't61rsQfr zfl * t*1or-qgtY : za 2

g;re1ncr1;red tr{n1os a+rupts (ZZyil ursaldxe ut.rd
alSarncopl os 14 elec ug 1de;rp IuqIIIau rS pu.rur{uoc rar{,ence er{n1os .re1

(szrr) '"*rrnoz{* t*rrs-arg : zoC

€ruroJ qns orrcs os 1de;rp Inrqruau TrFJ (IUT'I) rarienca mrinlog

kzrr) '16160z[ l t*pr-QrY : r[)

plu"raue8 €rrrroJ e.ru (961'1) rar{enca et{n1og

'zrg7lffg* c)rn:, d '("t,J-. c)izrc:n '(arc * c)T /,lr : roa geqou "e-s

(rzrr) "l (r*d )" Im, ln zno

(oarr) ;: -s zn-'t

6 146/

'o:'n tzs- :Yrczg

€-LrrroJ qns esnd IJ +od rr{unco Enop olaurr+In

(arrr) 'nrs l"rcr-Frc+ c)r{l : "nrs("c+ c)T - 2fi0
zAzA

tuour{qo (Z,rf'f )- (grf 'f) alrr{u1e"l urp J€I

vrrd.e wLL:1[f'11C+ CCrornl s-tannt"eleeds.eteinrteezgirsatreenAlat, cAarr,aBct,erBis,tisced,de^termlininied,i
7 in prezenla liniei 2.
din condiliile la limit5, concrete ale sistemuiui studiat. Yalorile instantanee
ale tensiunilor qi curenlilor se oblin prin aplicarea transformd,rilor Laplace
inverse relaliilor (I.I22), (1.125), (1.126) qi (1.127).

Un studiu detaliat al cuplajului slab, precum qi numeroase exemple

de aplicare pot fi gd,site ln lucrarea [9], fascicula 1.

1.4.2. Cuplai strins, Ecualii 9i scheme echivalente

Cuplajul siab este stud.iat in principal pentru a gdsi metode adecvate
de ini5turare a influenlelor ned.orite intre diferite linii de transmisie.
Cuplajul strins, d.in ccounmtraa, rsefirecaulizpeloazriSi ,diniremcotidviv, otriat nqisaforermaaptloicraiilici udirleinctiei,

in diferite domenii
stud.iul diafoniei intre linii strins cuplate etc. Trebuie subliniat cd, studiul
lcionmiipfldert5a, lpcieurpdlaejrui,luiidestnritnicseeqstieuenxiftorrmmed.e complicat chiar in cazul a doud,

a) Ecuali,ile Eenerale ale cuTtlo,julu'i strtns

7") Yiteze de propagare. Conform lucrdrilor 1d9o, u7d2f,iinceiai mai potrivitd,
metodd, pentru studiul cuplajuiui strins dintre de transmisie
este cea lsazatd, pe utilizarea calculului matricial care permite determinarea
schemelor echivalente qi relelelor cle adaptare pentru diferite moduri de

propagare.

Se considerd, doud linii fd,r5, pierderi qi identice, cuplate strins intre
ele; dispunerea liniilor este ca cea din figura (1.10). Funclionarea sistemu-
lui considerat este descrisd, (1.102) (1.105), pe care le punem
sub forma matriceald, de ecua.tiile -

l-Xl: srtrr r1r, (1.128)

in care s-a notat: l-#]: s tortur, (1.12e)
(1.130)
: :t ur
ll ;'ll, ,', ,

l[; ll

rlr :V* *"1, Lct :l'::;' ;ft,,il

Din relaliile (1.128) gi (1.129) se scoate imediat

l#l: s2 rtrl tcl tul' (1.131)

36

Lt (r

eng{cq.uv€equqar.esotzqzaect"sdr ororSuori+eqsmoa6qausu1nu1o1+"grpo-uN€iaur€nlclro€€ursruuonsrcuooalr++odapeup.lcr+rquldo€l('eI()lZrq):c:(aa(q;,oqg1fg'tflIa..FuI $a.)88,IgJu1)nI?-I"uurgr{uorIl"ourd+r (o

et'l'31.r (

{ql (

t U
-
i2
O
-,-e:es(Jn;goa(lc.b!r€:?re6oJog9pI"€'a-)o:,olt'one!luu"/,[rosJnelop,lnXnqneuaJwrolq1:pu,oi|"lopp@noqo)?.lunSualod?pIqe'!Icp'laspe'p.FEg:'rIp;Jdnr)enopzcp"grraeorlpdeq'ftrt9uuslJnavoEyl{Jds0?loz!y?nar1uu;arqlys:ewgqro'l(rnyl(*n'8aaalur.,sql€lsc*raIuon-azErnuJ+u€,pIan)lrq?.rrl+ccr(ougg1luuo(-rt"cua'6tr^2puEr{ur)flrtl:rA€,r otr
ri'?+E+caIJoJ epun p+uoprcur €pun pmziJo+ctsJtsc .1er,1ua.leyrp
mlnpolll l.rolpzundsaJoc (.r7rr+o) ruaru.ral rop !glelcalJoJ rBoc rS gguaprcur €

€: 1p{ucunlqpsruprzIurJoarr+rroa€+Jn€lt1e'udnueuror{c.uomc {'nupao(r6ugIr'.Tro)1rpazr{uenndasoa:trogclo(l,d*7rur"o+oc)err{una1uougrol rop O

(serr) 'Fcz * c)Or -r)llt:on : zn 'c(w *r)/.lr : :"01, r(L opun 9

(serr) fi(zaTxlg : z1'ImlsT - t.t 'p

* OIIJ tl

-o1v^ .r, nrluad ur{qo es er{e1a;r €rur+In arp '"(rmlT) Ec "a(rronploila rudo;rd oIrJoI€. . o
rrl puia11 a
nc II urpro op Ecrl"erlpd eog"rleur o olsa lCl lfl
'[o] : lcl WJ - Lrl"sl"r
(rerr)

aur{qo as (ee I'I) er{unca urp '0 : [n]

li: li:r',?I€utsq "erfnlos puru,urrs jlJ 0ll LII +€+ou €-s opun

tii:
'iol : lnl {lcllz) - Lrl"sl""t
(eerr)

v,+IrLzor (fef'f ) q (ZSI'I) erfela.r purcnpo4ul 'e1t1dnc "roFIuII

o1u g+Iu4I e1 epr{rpuoc mp purruro?ap os [op] 1ec1.4€rrr alo+u?Iuola ar€c ur

(llccJ'r) : :ll1.', il vnr 'ron t."s tal

€ruroJ ap orinlos o ollurpts er{unce €url+111

generator echivalent de curent (fig. 1.13 c), unde il: hlR". Este evident

c5, linia este adaptatd, la ieqire cinil Rn: R".
Considerim dou5, Iinii (f5,rd, pieriteri, identice) cuplate (fig. 1.10).
Adaptarea la iesire tlebuie realizatd" alit pentrLi modul comun icniiflEigpi einntflu.
modul diferenlii,l. Existd" doui tipuride refele cie adaptare:
Ne referim lntii la reteaua in II. Yaiorile elementelor trebuie astfel deter-
minate incit adaptarla s5, fie asiguratd, concomitent, pentru cele dou5,
mmooddnulric.oCrnounnrl(ilfiiag.d1e.1aadaa)pitmarpeu(naebsRortb:tieRt,ot:aldR,,,"li,pasastfreelflceaxipilorirn) pentru
-R, nu

(b)

Fig. 1.14

trece crrrent ri,; condilia de adaptare pentru modul diferen.tial impune ca
releauaformatd, din R^,R1:R2:R", s[ allsoarbi total acest moC (fig. 1.fJa.)'

uadrmiclat, rZewu-lreRlnetauuaal ' Rdain:i"f"igdueraun7d.e74sccoaastiegmur5R, na:d2aRp"t"aRre"daf(pRe"n,-tRru"ua).mPberilen

moduri de propagare. Fecind un ralionament similar pentru releaua

D

I

u_
+

,TttT-{it--

(ql bl (cl
Fig. 1.15

de adaptare in f (fig. 1 :.75Raz, b:) se obfine schema din figura 1.15c untle,
I1"4, R1:(R"r- R"d) f2.
din calcule rezulti, Rl

BIBLIOGRAFIE

1. D.K. Cnusc, ,42a/ysis oi linear sisfe,?s, Addison-1Y.-slc), Fubl. Co-,rp., I'lcatling, llassachu-
setts-London, 1959.

2. L,A. \{otnovrcr, I-.G. i{sr-rclxxo, Impul'snctitt tehtil:a, Sor.. Radio, l\Ios}iva, 1954.
3. A. BanN,l, Itigh-speed pulse circuits, \\riley-Interscience John Wiley and Sons, New York-

London- Sydnslr-'I'oronto, I97O.
4. G. Mrrzcr:n, J.P, V,c.enn, trIectronique tles impulsions, Tome II, l\Iasson eL Cie, Paris,

1966.

5. A.l{. KuculEv, N. S. Gor,uenv6 Osttoog radioelektronil:i, IzC. Iinerghia, N{oskl'n, 1969.

6E

'(816 r)

gg '8 "Jl{ateolpeu 'qnJEz 'l\iraplrz ttull afrultu6Dtlro.rJ.Yalg''Irxrlcst{ 'I 'A 3Nrr^uog 'I's 'BI
'(oz,Or) Opor '0t 'm "uot+\alg .I 'lolpEt{ 'ndt\ oao4saprJputlt' tpr.4tarad
putlqD[sa)trlqaptpollDtau qfiytidt1yo Dalsloas a.1?snpaaouoloA 'Ao-r]'.rors-r$ 'd 'A 'prso51 'I'V 'lI

'(OfOr) eer 't '?a "uorl{alg I
'loIpEU '11,toa7 frcotdon afiuuarqzl 'lpopand ttnt1 afiao:1soyod ' ;rxs,\ox'rvrj 'I 'V '.to('d,{uN 'I'CI 'gI

'(osor) 't('N
'[[IH-ld,Er0ctr{ 's:Fomlau uollDctunurutoc puo \nyfip .toJ sautp uors.sru/:rupJJ 'xorrvJ{ ':I'H 'gt
'&Sgil Zt 'gHI 'rord 'shldno) ctpufotuo4cala lntrolparrQ 'u.r-\I'ro 'J1'g 'tI
'696I'slred'aI) t? uossEtrt
.I ap sau6t1 sap anbtsfrLlrl 'J-ir^ru{) 'd 'tI
aruoJ 'sacuanbVlJ salnnttr n4ln.p 7a sacuanbV.tJ salnrt,l .(t96I) eunf

.lA,a.r^au vcu 'saurl uots'slursurJl Taptod aldg\nut Jo stsfiloun utDurop-aurlJ 'v,\rwar4iv 'H 'zI
'(Z,SO1)'Sf 'J''d uo 'su€rJ gggl 'suta7sfis lnltttp ut (asrou) rylnlsso.rl 'rrvC 'I 'II

'696I 'ultJcg'Fe1.ran'pu>1y'sanbrrrtlca| aslnd puocesou?N s^uvr.S 'jI'cNrar:rl{'IA. '0I

'246I 's.iJEd'ar3 la uossEr\I 'U 'I'JSEC'Ii\ auroJ'suorslndurr sap anbtttot|cala'auflvr\'d'I' '6
'q96I'81trs

-rr4u8oy-'1'1,.1 'ttrH-A\EJl etr11 'sutrol aaam frttrtl4tms Ttun 7n1t6tp'a.s/nd 'aoI,,J 'H 'NvJ^r1-rrtr\i'f '8
'Za6I 'AelX 'u{luqal 'pzl'altuqat tous;ndut od qrucoantdg '('par) ,ra-r.Loxvl 'N',\ .L
'6q6I !BA{soI{ rolpeg '^oS 'DaFtor}sn afius1ndu1 'rxorlll 'S 'vI '9

CA.PITOLUL 2

c

Transformatorul de impulsuri
cu linii de intirziere.
Linii neuniforme

a

2.1. Teoria transformatoru lu i
lin ii fdrd pierderi

Transformatorii de impulsuri au multiple utilizS,ri in producerea, tlansmi-
terea qi rnodificarea formei impulsurilor. ExistS, cloud, variante construc-
(1) transformatori cu dou5, infXsur5,ri si miez de feriti, (clasici) si
tive :
(2) transformatori cu linii de lransmiiie.
teazi,fnbacnadzauldteranfrsefcovremnaleto:ruilnudi ucclatasnic!5e, xdisetisoc5s,pedrieri,dceafpaacctoitr*ilicapraeralizmitie-
intre spire sau intre inf5,gur5,ri si rliferitele obiecte metalice apropiate,
varialia permeabilitdlii magnetice a feritei cu frecvenla etc. DacS un astfel

de transformator este conectat in sistemul de transrnisie d,e la generator

spre sarcinS, a unor impuisuri irleal clreptunghiulare a c5,ror spectru conline

o infinitate de frecvenle, impulsurile vor fi distorsionate. in primul rincl
permeabilitatea miezalui de feritX, scade rapid pentru frecvenle mai mari

de sute d.e megaherli, in a1 doilea rtnd capacitS,!,iie parazrte inductivitS,-
lile de scd,p5,ri pot prod.uce rezonanle locale care limiteazd, sever caracteris-
ticile de foecven!5, ale transformatorului.

Transformatorul cu linii de transmisie, care reprezint5, o combinalie
reuqit5, intre propriet5,lile liniiior de transrnisie qi cele ale transformatoru-
lui, tnlS,turd, in mare parte inconvenientele specificate rna,i sus, oblinindu-se
caracteristici de frecevenld net superioare. $i trebuie subliniat cd, aaeste
1mbun5,t5,!iri esenliaie se ob!'in cu ajutorui unor construclii relativ simple.
InductivibXlile qi capacitS,liie parazite sint absorbite in impedanla caracte-
risticd a liniei de transmisie; evident lungimea concluctoriior trebuie sfl,
fie relativ mare ln laport cu cliametrul qi rlistanlele dintre ei, a'Jicd, sistemul
sil constituie o linie 1ung5,. Dar in cazul iiniilor lungi apar tirnpi de intlr-
ziere foarte mari, iar in unele cazuti dirnen,qiunile rnecanice clevin inaccep-
tabile. Scurtarea liniilor nu. poate depi,qi iimita cind timprLl de propagare
d.evineOepgraolpcruiedtautreatafoaimrtpeutrimsupluoirt1ainnt5c,azacotrnatnrasrfoarrpnaar tdoirsiitoorrsciuuni
nedorite.

linii este
aoeea c5, d"imensiunile 1or mecanice se micsoreazd, ottatd" cu micsorarea
duratei impulsurilor; acest lucru le confer6, avantaje esen!,iale in tehnica
irnpuls',rrilor de durat[, foarte scurt5,.

Transforrnatorii cu linii pot fi : (a) f[rd, miez de ferit5, si cu miez de
feritS, gi (b) cu linii bifiiare (inciusir- iinii formate din cloi sonductori izolali
qi rS,sucili) sau cu aable coaxiale. Principatele tipuri de transformatori
intiinili in praci;ic5, sint :latrpaonrstfo1r)m, tartaonruslfoirnmveartsoorrul(ddeeraimppoerttla-Ln,)te,
tra,nsfor-
matorul neinversor (d,e (adaptor

t?

g--(rpeorrqdJ4Serxuuld€EtaJInI+rI1a€ealaeRr.irl+cdaculronpnJdoyectnsoaz€lrrJusuuJfrsnorJrniaaserulnnnqonoS."Jur+ollu)p-fejdrlear{u1.mruuprlnlu'leJoEsoc?aa€"pgJIutnsr1oo4npJ{"rsrrbuu1tonsrpc"Ji?a{oiuul€"oel usooJJS1oe1Jq.)+qur,,r[li:"6e1+p.."?6"?eoce{ou1;ruen*op8*c

z'z '3lg

?'1trIn-grnuInBeJnJaI1Js€nou+dIpcunlure€IJudoEyJ'I?rcnroocosussll+rnae€{rIeoa.u"ct+r,ruEle1oFsdr._"+u?trrosatrsoc1?ppJund-ln_enopl.fT+ec-creuoea"loqlzr+c"Jdegocusnuea1or:odeopl.toF^r(sp{qar"u^renlpIr€arqs1Tuospcai..+oalo66airc+rlaEpan.ecaFlau[pnpll_rnoruoypr'1Fl)qoceuu.orcprool;aasurcqpo!xucnac.oturrds:lpaprocrs1p'ucaoI1ds.rnZor'nlBu;Jsn.rTlt5foolquspreul.Jee]datl+rmJcIorrruqurl"JnIrnraordiIislceJotsns;sapgrrulSpocunro3"rool;epeIoJrnlpji+zoJreJ;,rruordlElou{ne+JrzqpnaBnqo€o;rralqruIruurIlrJrirspurJEyo.u,saIrrJIoeinIoJSyeafudcJuerlx1lo€uopeon+lJidru+uuu{?eoinl."osraor€reooscus+iIu6lrtmrsaoo?oii+u$pJccI

+-Juo"JrsSuJtosJI+Iru+{Sorcts"eJ-n€+co+JuaopprraroJoo+lcsngpu'RoscEu!I (ea1p-15u..rSeuqe)8 glryu"ore?lconppuzaorcururpp;rpr{ymr1arle1sldunoac
surJ+s I1uII ?nop urp ?EruJoJ ..ros,ro,rur Jo+t3tuJoJsui3J+ un rupJoprsuo3

I-Jrnaol?soEd.lnerrs"lrr,us1un€;er+"r'eo.prediluunr relnaldno ,",r**'i,tJ":l*,$i:3htltJi#?"Xfi?H

no InJo+ErrrJoJsu,rsJ+ rJKJcnI ra+SacB ru.roJuoC

l'z '51.q

{!t!

['Uff€In'Oclcts]tsoJir[J5,F]rcan"riqqe1 'fd+€'f]i.oln:!zl a€par(?1rDI4lqi uI ?rcr.roJ pollr ur K+,r?zrlBreuaF o,trsa
nc
mlnJoq.Buuroysue;r1 Erroe,Jr
'(g+l.ral n? rJo?EurJoJSu€4 pzaazrlr+rL as eCllce.rd
JoilJlrzEc uelelr.rofur uI) K+IJOJ ap zorur EJp]_nES nmop3t-rots;rrexcEeorcg'ej1leqeiuovnpoaoaurJlraJetspi
o+EzrIoJ lJ +od ola
-TJIq IFII nc orJ 'nrdrcur.ld in

r1i (2) impedir,nfa tle sarcin5, lrebuic si fie adapbatS, cu iurPectranla tle
a transforrnaborului.
ies' ire Din figura 2.2. rezultra c5, aceste condilii sint satisfflcute cind este

satisfd,cutl egalitatea

Rot: R"a * RrR"a : Rr,z. (2.1)
Rr * R,a

erii se1"Y)aPnri,omtauttlteral s(te)u: 0<e(rt<)1120: (txs{>"(l)a'",O)abd)s.eirnvinvd-edcedr'etat'u'$s)irn:pli0ficSte,nrisi.iusncirlei-

de intrarc colespunzS,toare celor doufl ]noduri, comun qi difelen!,ial, sint

: + -u,?(t) (rt\l ullu)) 12 uf,rt(t) 12, (2.2)

: - :ttfft{fi (ulh) ul:r\)12 \LLt)ft)12 (2.3)

(exnonentii din parantez[ se referl la numiirul traseului). Cornponentele

Lleitle'tt)c"qirerrs9p'1et;c*t"iv^pirdoiparlgecl ispre bornele tle ieqire, unde ajung clupd, interva-

u,[!t - u,,(t - r") : uo(t - r")12, (2.4)

ug) : u6ft - r) : -nt,"{t, r)12. (2.5)

Undele inciclente pe fiecare linie, la bornele a' si b' sint

:u,1f;, ut!) + uh' : (Il2)lu"(t - r,) J- tt,(t' - r)1, (2.6)
:u'l[) -u,tlt -x,rlt) Ql2)lu"(t - r") - u,(t - r)]' (2'7)

La bornele a' ,si b' au loc reflerii; notintl prin a$) ,si 'm[]t tensiunile
chpl" primul traseu, se poate scrie
existente Ia bornele ti,'rb'

tr,[!"l 4u,f),:ultt, (2.8)
(2.s)
:u,l.!ul -1 u,fll u.f;),

F ig. 2.:l

unde u,!1j,, r.lfl) 1epr.6sintd, tensiurrile reflectate spre_intrare la sfirqitul Pligtl-
lui traseu. Tensiirnite uf,l't, uul!f>;'tse:0caicruelzeualztS[ 'diimnesdcihaetma echivalentri ard'tatd'

in figura 2.3a. intrucit

:xr::), - ut'), : (-1 2)lt+,(t - t") { u"(t - taf. (2.10)

42

st

{oa'z) '[(', - ct 1)onty6 1- (116 - I.)"n(t:f - f) -
-
(o|a)
- (.6- t)"n ('x + r)l(r/r-):6n - <"1n: rtin

'l('" - "t - t7)"'n6 ('tz - ?)"n (tx - r) *
* ("6 - ?)"n ('x I r)l(t/r) - : *,8tu dtn : ("\"

G+uaprcrn olapun tsaaa uroa orurl a.I€corJ ad

(Srr) 'l('Z - il"n +(o'-'t- ?)"nf(X - fXlir-): (r" - ,t)€fr:
(3m

Q,t'Z) 'L(rt-"- ?)"n I ("6 - t)"n)(.Jf + f)(l/f-) : (," - t)6n: ufrb

Rctpv (pL lrgcedsa.r "r, aplorz.rllq nr a.r€J?ur e1 Funlr olapu11

@t'z) 'l("- *ilo,n (,'- il"nl('X - 1J}lr-) : -Lftjn ,(rjttl(zlt):
<f,n

Gt'z) 'le'- ?)"n + (':"- tr)"wl('n + il{rlt-): *Lfrin ,(ri"ltr,lil: €p

-.roq a.rds paedo;rd os ortsc .1er{ue.ro;rp r* uru'oc rnlnpo"' ;:#ji*"t"tt-3}X:
a1runlsuofr 'Joololosapcuon.rdIBnncosF€J?qt.sp{?oaF.ootpU a?d)ec0url ,,'.ena, s,ontay"rriar ap olau.roq
1)u €[
orxalJal ap naryop) IV e,

ftr'z) 'jr : (in'li\jn

R?Inzar (Of'Z) n? (ZI'6) pulr€duroo

Gtz) '(o'E I "E)l(g - "t) : {o'u * "u)loa :'}r

krz) apun

'f("" - g)"n J- ('t - l)"nf (ZlrX-) :

-- irr"nrT - - -"'1){o'- tr -'in',\.". : ili't

eur{Lqo

os (6'a) urfela;r eli li (f'A) o.ru1d'epr ap er{rpuoc op .Brusas Putulfr

(rrz) (n"u * tu)l'"Eou * e"g *'"u zaY: (?-"?)r"g: d?L

(n"E + ou)lou'(r\nz- illnz

lnxvr,dt -r.ciulir*.r5d 1n"ro1u,raues ap a+so ,, ffll;'"*u VP,f J"I?lii:"1.:;r;3r;i

pugcqde a '0xza[ 'l€nspnpord na "g1eFe elso .,iJn, €aunrsuoiL

L,a borneLe d.e intrare au loc d.in rrou ref lexii ; tensiunile existente
Ia bornele a, b d,upi al doilea traseu' sitrt :

+ :ul?\ tr!2) xr{z), (2'2L)

: uL\,+11t7') u'fu) (2'22)

und,e tr,f2j qi rrl? sint tensiunile reflecbate spre bornele d.e iesire a'. b'.Ien-
siunile uf) ti rrt) :s; aohtatS, i1figura
Lrorcoi"*r; din scnefa, Lchivalent5,
2'3b' intrrtcit 'tl6 or t'ezuiti

ttf] : - ulil : (1/4)[(1 + Kr) tto(t - 2'") - (1 - Kr) t "('t - 2') +

-l 2Krtr"(t - a" - aa)f. (2'23)

Tensiunea uff' are expresia-(', i" - inunde ri' este d'at d'e genera-

toru,l 2rLll iar in d,e generatonil =2tt'|f/)

uf) : Rorio: Urt RrR"df(Rr * ' Q'24)

r1,s1 -T tr"s * R"o)

lininc]contd.erelaliile(2'1)qi(2.13)ultimarelalied.evine

: -ul,"t xt'lt) KPtff;\ ; (2.25)

cornparind cu ccua!'ia (2.21) se oblrne (2.26\

t[2] : - Kru{:|) ;i ulZ) : K{rfr\.

unde 3in)cAidtterenit,eleaptreanster'uLl,ce?l,e0 dSoui.dS' m3otod: uUrni:d'ele reflectate'rrf'zj qi. ,rr,['z/, d,evin

ufi:(tlz)lufl *uQl, (2.27)

: -u(f;\ (1 l2)lttf) u','] l' (2.28)

Ele ajnng ia bornele de ieqire a";b' ctJ' intirzierile respective r" qi tu

adic5, uLl):u'lBt(t-r"\ (2.2e)

u,l}):uf,t1t-r), (2.30)

*uainlpec"lteuen*Cniso[i3i,no)s(n2giidl.ioe2rru3i$nt3iad) isab"ero"eu;";x,"tP-;*p;rri,em;iz;*[u'u-littridtne,it"d{t9it"nlengcrli,e!riaile9asq'eldiiear'ee, us(a2ef}'l2p6s6o,lt)ir'uasl'"nc}r;sieflaoerrmxipnardetuoslriiulsel5u'ugi'eanAf'e/srtaafreleel

pentru bornele de int'rare se obline :

:'tr" uLD + uL4 *'rr;tr + ... (2.31)

ua: ut) * u? + +u(ut) ... (2.32)

U,e'z) st

.U6gT: ?cg l* O Of : *[ ou1{qo os opun op
,(N*T)n"T:z,r'[-r6'V

u€erlKscueJ.oor+rr.rur+n-drstelsnipudtesugrarsprez+op.rrnlrpculcoro+otrusaTnpuo+oaJrnsu4uIsBpuJou+cpn€ec-:srOn(16krug5o31te1.1rreo:u:yeJ8gr[azt?iryLrutrr:rc"'.errelsls"pfis
orri€rojr

: :-ereuo8
rJ€Soc_ou
o.reldepe ep er{rpuoc r..zrlvat e nrqued fc +'eg€lsuoc e-s leluernr.redxg

p'z'Eta

lD)

- 6lz I|5
,(xoda
plz!1s

uolJDa

ltour

S''---:'u'(9g8uncrlu^)o'rn1l,,o6(yn0:1€ppg)rQotau€Teufdc:o,"taa:xSrconoo:o!(tpfepdcgrtyIye)€xtL"+€unroo€uo+rCercrtr"g+FeslpJreeJlsutoc+pwnunrcluegp+or.gopnny{s1ncqlulevs.eescoeJlunlepoldo5rpe1olT.+ro1orsorIcnu€1{rppJ+saI(1Ioa1llu_d(dnc11lgIrosd€ge4frcouo.s8qeram+nneupuus)dr.+n-am.rarsroFq{ol?rIeqqrfnpd1eileoJadoclpoorernpp1encl+s"spss-I€roeqrrorerxer(ruuJeopqoqopJrrldpod+"{ree.ofdJoecc.oc^SnZsncqqlpurru.qeilrdu€8dsugoJusgrosr+cpyoi*r)petucrs€opIe;lcr,a1ae-eolrJclnyZ8€+n1€rrlercJrllJ+{suelon+eeusa.opmrmurprluiSuoco5oeiceerico;na1"eeerrnulpnueonme"rr?r.pusgoqdrc,n+u-olunsngs.u,glcseou.pgqrruolu!deecd.cr@zlsCupocA's1rgoqT6eav.,rVdr.py€or:r.yrXrdZ1cgSj,
rurrrpolor{nrpnuqorlsCuo'[cZ_]+uulsi Ion+Jo€+p€?ruuolsJSou+€uJ€+so-€Jzats+oqruJeoleJqouecurrrru.rqadrxneorp.rgoxro;i.lroec11rts Jn+

(ge'z)

"'* * (x*rL9T('z-r) _ (rt _ 9I

Gtz) r) ('x
Gtz) -(eq
r)on ?)nTn + .*

T)

vv
* (ot - !)"n *,X-fI_- ('" - ?)"n='XL -T :Prn - : 13,17
:h 0)n) our{qo os sns reur eqrspF eprse.rdxe pulcnpor+ul

"'+ $n+ ({P+ ($n-ton

"' + <3n + tLin -r (Ifu :'"tt,

: rrroa€ an*er ep olouJoq n.rluad Jqrrms poru uf

(ee'z) . . . + (pr - "t - ?)otL;- - (t 6-t),rO=-

(orz - ?)'n=-9- :Q,ytr ctL

Lx
- T- -* 0)"n' -

qos sns lem elrsp8 ogrse"rdxe rr{e1o.r gnop olorm+In eulfqo
pulcnpo.r1ul

icao"ovdrnea"eatJcflll,To*ascr*tued.o,b]tuouqdniieregusiinsmctudreebup,glcritineet"iiltairi"rni'tt,*aic!iii,5rle,i"piai""nrg"oteripieixu-treiip!e-rutn*ianrd#itcp,itiuaud-llspeIuribnodl'uaiieDfiue.iicrTezInd-n1qdif1siluea0torc,ie;trii.s1vdiiEo.tPiantioatd[errrriaiatedlaep.lllebinaciicumueeripmsJ2ut?ual'2csirruefeealgsuvpttieae't

(vl

1,0

(v)

1,

(v)
0.5

0.

0

-0.5

-i'ob

Fig. 2.5 Fig' 2'6

;f"inT;f;i";is";if;p;;.ea;;c"5iiapnlipJiarie"infdr(osspnkpitiinuir"a-lueiaiioir"ali";intti;irt.ierl,mirtii"lo"uar","i(m";feioigieiio.zmud2bro.i6ulCo)n,rSt,.iod,d!ai[etltTn.ofeelriiriti5.tedi,ns, ne(cjaneaclzi1aeualsalgtl{iacntoafioirzloramercsetstieicvuiemtrrtappetru'isalne-t
tii-tmiono-n-ai"nil-tp-eioi*la-trDs"riciaaia"icgit*frutilptaup-uclfruauaingirbl.c?ittuemIrame"tei"t*dri;reti';iioo"a;r,I"e)-'q.dp"ei"L.t;";cii6^totn""o"suiittgiri"uiaiciottl"iulvanrcgeg6isAmmteerph-no,senipnceiennssttealcirroespciunaalt-fudifr;eieax,anlaentemtcaeeptnrftolirinrnameniezsauofzonlaSri---'

2.2' Transformatori cu miez de feritd

;;;,t; ;";-illnciatIKufturaJ;,:i-,,lbil)gfii-o,;n.l5ica"rtDicn",io..1fa#ciatP"nl"xtrerrueprtiir"n"on,ad"-;ctrre;rrsioiau,eil;i!gn6airt^iitr,iier"laab,baOf,o"tru"z;reJ-nl".ffpiJeii;;gi-re"il^pee;)ittaciU;cde,i,eofle*(dniaz';"aidetn:tue.a'in+citnl.rtiei1tiaaegpu'l,le$airr-eaeTrmKcdarezrieeenza-usidaitvi1iteeptearlq'alaanlicpdtrirpcaegirrlnoeiird.netpurzm'rtieneainrp,aenrncrlgeiiltolnaaeimndnglsziguiuea.duc"ucmi9al.ri(t.mp2draD^,al'ii:ii1pasmcini3tcludm,ps5)olcsrf,purlirooussn.eunel'riszlums6fiuuiinmnucltaitslit'd^peiientlrf'onuezu(vctlprdirasuadunro,l'itil.fnr,zutaaiKrtcaccitee1iiteu-t'-,'

L3

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