BSTJ 20: 2. April 1941: Steady State Solutions of Transmission Line Equations. (Rice, S.O.)

18.7 MB·English

The Bell System Technical Journal

Pol. XX April, 9st Nae

STEADY STATE SOLUTIONS OF TRANSMISSION
LINE EQUATIONS

ornate of atari “he sendy sate volte sd caeents in a
snip vate ine consis of tral palace are debe
in fare This nc tmy or ay nt be sled pon bye cteraly
Sse eldaoted arg telat Asauate ata leis
{enmlinron af tne popagin centr fora sage ecu il
dived ‘Matix expreans ond for ie valage ned cate
‘elves Ike sae nay as cenpeding ingle eal epe™
Slo vate» Le Ja str cha oe dese frog
ihe atgaei s e ogee
Ihre velar aadcore ina compel mmm ean
Make given Thew ila my oe'may aot conn peoenors

AE tsi: Viscose Hose ane of oe yey Ue
Lr taragiion ine, and the tramerelon Une cong of a

umber af identical sates conneetd in tandem. Tae problem disuse
' thi of determining the sted sate eleerca behavior of hese ines
when te termine conditions are given. Ditentaere ane the problem
Df ceterining theremin in wien ssc ite by
arbitrary impress Fle f sore fed feesey oof deterring the
en's prc by generators peed in te otanckes ofthe sections if the
Ti sf the sexual Kihei with which we stall
Te particule concen

Ti deg wil che worm Lassi lige i is Fouad convenat
tpuoduce a matrix F, which sa gesrlaston of the propagation constant
toca shale wie with gro ci, of ora singe cult. Ths engales ws
ternbeain mre expressions forthe curents a wltages whieh ae sar
In fora the singe ret exprsions

Ses ets forthe sae line corre of sane
of syrmetvicalsersions.Tewever, when the sects ar ineytinet real
the curespncting roped soes omnes Lo ye cure te
[loti ane the Fama ace comely one empl de
Emmet ease

This paper lial ft Lo pues eresponding to he two Kil of
lesoemizsion lines, The fist yack dlsesees the wiform line. Aer a
atenent of the bansmissn rll in ate ore, expression for the
Wallages and caveuta age given. Tia methmls of evatating chest ex
pputtone wre ceseined, "The ls f bast spon projeety pessessed by
Fheny unnsmisbn sper, acy thal Li various odes of propegntion
Trove near the soe speed, ‘The 2ocond mctod ix bss unc equations
kick maybe ated by the tema application of a Uncrem due to
Sylvester, Ue feat geze concn with the pros shat thee Lu methods
land eo the cert results

‘Ae tort introc ton de second port iw che difecence et
sione stich govern the amnion ina He coagsed of mtterminst
fevlins. The tections ay cootnin generator. Expressions forthe valle
teeta currass na syaninetcalse‘on lire ine whose setons are
emmerricl are stted aul proved in mies (ie aare ede ns the eae
poningexprecion for Ue uniter line A discussion f he unxymmetr
‘secs line conch the second pert

‘A aketah of she soln of Li unare teaaieion line equtoes by
the elec method i gen in Apoerlis 1 In Appentios Wand IIE
methods tv dsl fa elvig the stmmecial sti Tine Mirence
auctions Those tethoes are sua te the one of Appendix T. The
Method of Appendix IL nes selon constente whi’ ay be obtained from
‘meoduremen seat ene en oa LypicalEeton

Past I
asnonae Tranenusenon Tass

LE Dietict gations

For tie tar of wowenience in writing slowa equations we sball assime
thatthe portion ling under cnaidessice nae of shroe ara CR
tnith glound reson, oF of che patel circus cenotel ly the subrpts
Uh bvend eropestively. The Treat equciens for ths Vine i an athe
ay impress Feld ae

Pala ~ baie + beh

Packets ~ Laie + HO) an

Bee tale = Zab — hal
tan pte a a a
Sits doesn Sod’ ime tet asl seat elec ern
{eourconee tiny “Retina apes fle hen eee fener any a ttle
‘peansuission 1x8 Epvanions 2

and
basa

Vane = Kats = Pion bled i)
Fass — Tare h ats)

shore Zn = Zoe, Var = Toy, tte, The sne deine with the ple

‘hres 1,2 f(s) ae te longitudinal components of the eleti force
Ua the Hnyested hl tthe we soars; ey ACS), fe) ane peel
by the admiteance of the diet Inkoge paths aad the values ofthe im

preted potentials t the wies, JF there ave oer: lskage paths the

Toler lu pat these epstions in mars fo? we nt the clan

Loy
fe), ne
140

Yas Yao
ve ve) an

sel the symmetrical ye rts

[ere Zoe Zoe
2a|20 20 tu
2a Ey De

“The ecm (11) and (12) ay mw be wen a

fons
“9
fo wren
ned sae he sqm:
hae sv mpl el ests 2.3 ie
oan
i ao
aves

Ch La A Pie POE ap Vl 2197p
be es SYST PREMICAL JOURNAL

te analogy withthe oredr ase leads uso put

Pow, rava an

shee Ps square matic petting geneniion of Ye poyagean

Constart. Petting si or Us mame the question of itepeing the

quate toot, we note thet aterbaaging she mows ane! eolamas ia 1® = Z¥
aves

Perea, vavTE ast

wee the primes denote taneniton, 7 ead 2° ae equal an Y aud Z
Tespeatively bess of ter symeaciy, We thus expert P vob azocated
“oth she propagation of # in che sume vy ln Fis aeocated ith the

Dropegation a6
112" Statomont of Renu fora Tniite Zine No Spree Midd

Te isahown thes shen thee f no impressed feld the wltagesand carents
sa any point in ats line extending from + ~ 0 to = = a8
riven by

aa) = Te) — 0)
its = O01 a)
eo) = 2

where #* the agate mitre sel el by the convergent seis mares!

we ee

earn a 10)
pot ar fae)
and eT be the transposed of 7 denoes the ent wails. Zein

Sucre msi andi el the charset Impedance mat
aawizarr? aan,

AiMitonsl expressions of he sane type for 25 ane glen Dy equtions
(14S). The mais eZ, being of He tre of & User imped,
fe symmetric

‘The mation sad Zy maybe compte in aevora ways, th ce
depending pes the cicumgtazces, ‘he et method to be describe is
{euler isn lage aud when she pmpaaion censtanss ofthe
Yevious moves of pray sie Youry aa teach hee. Ta the at
Gf oper ns lines thie stopagicion ronwari« ae gwoype around the
alie fare bers fe the ozs of TALOND res pe over, ‘Tae geooxe
rrethor any be asl forall ee “nla shows for whch ee sr in

as nikal etl Gites s'est minnie ae ee
the fast meth omivege ton soniy to Be of value. However it equies
the solution af an wath ger equstion and the weertietion ol te
modes of propugution where ois the aamcer ot eicutt. Farm — 2
thie ie no handicap and the the! b eqte convenient, Zn shis eae
the emethod i sey selte noes bed by Toba Riordan in ay ur
publishes metro

int Method: Mal:oly che maures % <a vegether to obeaia ZF
CChocee the zumber + a

a

BEL, aa)

Where Fe hemi mates so tat the eet a mln comparison
With.” For mary trpsnssion Lac ite pase wo do this I may Be
tainly oaing the binomial theoret to expan she sane rot in the
foram

va

ws

ey, as)

where eth quae root of 9 whose sealant dnanary oes ace non
agative Tn serving onthe werkt is couveniet sv niece the ones
‘Sothos lees acral n comparison we Unity

ros ay

‘To compute S, fist oupte the mets 2" and then ase the power

QQ 4@4e
+308) BC) + ;

his sores wl esually convegs taply. The nasi eis gan by

here risa nunise, ¢— 4 and © "ie cos ouput Sort

ne GS! Gt
F ae Sh Shs

aan

fs eisined fan "7 Ine ierchonag che sean elanns. “Te
functeistic impedance mais msay be ebtuied from (113)

Barr

ater copting f tem $8 in (18),
a0 ebs SYSTEM PECHENICAD JOURNAL

only &* is minal the folloming erie may be

Or

shore yy andl eave the sun neaiag a above and the ceetciets wre
Computed fons

eet, Gla nets

aac)

Bayt) = 88) =

athe fs corm af the series (#°)" denotes

Second Meta: K€ ard, uy he tgande cs erin of the square
sacri ZV. Tonle to expres thee fancier a. form staf eal
lation we apply Sylvester thea’ The characters marx of 21 js

(ee) = t= 7 39
ehese now! fe mani ab warn ieee «Sed wamber a in the
Fist metaod, We shall supposy thar ZV" a square mats of ondct

fed thatthe ols yf, ofy+7- 9g af eh charactors inst, Le ofthe
‘etermicantal equation

Les) = a
se distnes, Tat the mattis) be the alla of f°) aa devote the
Alexvatve af the catecterists Festi
a 4 ty
Lite = ha oar
Since #98 + havea ifort | G81!" fs ungnl to re for
A, 2uon mm. Sylvmtors thease eu thal IF PIZE is any polyno

in ZY then
pan - Sxghegd am

here Pi i.e sale (ond hs evigts frm oue convention thal pital
Teves dete square watrie). GQ se squate rate

op = PHD
nop = 2, 129
Tee" ‘
When ov — 2, 869) i equal to 1 — Xb
‘nine, BO. Thera ot eterna te 2 a of he ee
Apying (122) wT, ¢ and Z, eventhough they ate not polyrale
in ZV ves relly which may be vera o be Lave

van = Ibi

1.28)
#22 Deter?

ser the eamvmations exten from, rT 40° = wat ras tay 2 9m

sre che square soots ofr}, --- 1% respectively whose el part are nom

Dogative. ye, 7, +++ Yar also the propegation constants af the “owral
rank” af propagation. Some gh: i Levta uu Lao alysis! sgniieance
‘fhe itis yt) by suponsing thet oaly the rh novel made being
yt he raion line, faut ea be empress
Seccalum ntsc ine om rire, he vltage nevus 1,2.
fre pnportions so the first, stein += wey wrenents, rapt ofthe
Column metre, "The earns in inci 12 == ease proportional othe
‘omrenpentng rernete 9 te roe mas
LE" Herts for Any Unifrm Como torent Hilt

‘When the Ieath ofthe line is Site the welaare and catents may be
copied

tet

osha so) ~ sn a Zit)
sink I" 2% + coat 8)

hese Za Pbave the some mennine as bene, he matrices sinh 1%,
Sh eh oP%! ate eymmeri ‘These matrices rah ad sinh x
fre dees ty the seine

coon = EE
j a0

at ,?
soa av = #0

+

et f° fe obtinns Ry interne the ros aa cons af ae a
tnd sb 31” ie obtained similarly from stan 1. Salvin (1.28) fe)
ata gives

fo) ~ cosh 20) inn ob Za)
fo) = sin me Z'(2)-+ cosh aD” He

‘As in ie cae of the fit ze, ae Rave tro wa sompsing the
‘onion fe) ad i) the expressions (1.25) for we) and
ist Methods Choose & muster end compte the rates R515 Zp
as destbed nthe bs sue forthe Indie ze, ‘The marie 2

her: s — yw alee compused from the ies
es
wee

eens

hie to divide #5 by 16, say, compte exp (£5) tous the serie, at
roo 36 ome rn) ci i

thes aban by foor airy wnvgtiatinas. Whe € fe kaomn te

inven can be compted and "blaine fem (116). he byper

Te functions are given by
oak 1
sinkat 412)

heh
aan

nhs loom the ere fiom of ue various maces

only the cofciens a (125) ar voquid we may choose an come
pale Bnd posers! the xsi e/2s. ken the coeicen's fv (1.28)
ae given by

coher = B (BY 8?
EG) "ee (19
sino! = E(B) 22 v

nee isthe transl a Bed the ear coeient eh uetion
= we given by

sais)

29,

so thas (ie 2 =
PeAxsWession sae EpEArIONS 1

Second Methonl: Compare she prngition constants 71. 995 << Fe
and the square mates Me) ven by 1.23) a5 inthe Seoul wht
the fale tin Then

oh ab = 2 Nig oath

Sina 2 = sinh at F*

= ENG aah ane fe)

inh at” 25! ~ sn al" 'Y

Ewiontihaney

bere 399) is ae nsec oF YD, JAE tng deine by (1.28),
Nd the cumonatins extend Seer — 37

‘Wher the uznsmnision lire coi of perfectly conducting wit teu,
on perect insulators over = pevfetly conducting eat the stage! cal
lecrosttis lle are rlta en ne roma equal 2 whe

sling De times the Frequency’ ae the xpos of igh
This atecesting to apoly the Gist mal of serion vo ti te, Tver

hough she pret fist metho ais given in $210, doe noe comer

vh's ese Cee rasta be itl conte that the etre: ans is ina
Weave

wed
Cheering = 4. gives R= a themne $= 0, Ie follows that
Vedat Zante
ns oP = cosh fit
oh at 2, sinh watz
sink 20° 2! — sinh sno

oan

When thse ane put into equations (1.25) the expression forse) al se)
ins perect transi Ise ate a:

9) — cash erast) — A pea,

om as
ie) — ay sinh ey 796) + eo aye
14 Rend fr ny User Fie Impressed Bild
"Tue ciferctal ection o Be sais! im Uns ease are ven by (8)
Aolution whi aces 9) and Hows = = ie

coat a

= [sion ce — or 2ust

[ones

a2
Ho) = sinh a" Za} 4 cab aT)

= [ome orerena + [one oriod

re aties ceeh a7, s1 al" wl Za tips as Ls oes dict
{MDa fh. The elements he foto of augue £0 nok nets
Shelly square mitris) + given byte integeaa af the correspnsing
Janets of C

“ny eases practi inter the rset ld vis exponentially
with wspet co 2 Tena Bgtices I) ands) oy then he 3
pessed as

ax

svhore the 2's aml sae constants and fs te proqagasion constant of
the fpresel eli in tke dimetine of the line. "The iterations i the
xpresane (12) may be esr ith thee

0) = cosh Pa) — sn a 2

Fhe ena bata =z
SN OTT = BO Ba
. ° ast
ta) = sn ot? 77a) cos a1)
f= 20)
- fe ene = ae LA)
snl iat the invest mnie exh. ‘Te nate fo” — DTT
ie teases ot et = COTMN — Gl) sete TE ano of thes
svtrhosy sey FB, hay Saver then Fis mbreary ta evalute te