BSTJ 10: 3. July 1931: Mutual Impedance of Grounded Wires Lying on the Surface of the Earth. (Foster, Ronald M.)

5.1 MB·English

‘Mutual Impedance of Grounded Wires Lying on the
Surface of the Earth *

By RONALD M. FOSTER.

HE cautual impslance of trond cieeuits may De
tertain reals obtaines by A. Smaumeteld? who hes deve
formule fr He slestrie and mete fk a ce earth ard in the air
tlue to Aodizmtal nd weve! electeie and magnetic anconns siete
Stehe eueface of She earth, Pore resent problem we use his formu
Te for the lvl field in te earth ue toa horizontal ectic conte,
lace this doublet aay be reared 2a share clement #8 of wie of
eglgile siameter carrying a rite current, Ae the el of this
presene tes we shall shi Fen the sane formula Zor tee sutua
Finoelance may be abtaines etl rom hrs prices
‘Somunefeet ones rectangular ermmiates (595) and the co
sponding eslncieal coordinates 7,2) the surfnce of the eal
coud fat, being they pla. sad the x ave evening upward i
fhe air. The doublet je at she erign, ana its as along the ss
‘Then renianponents of the Herzrin yeeer Sin tbe carh fs 0) om:
‘which the electric el is etermined are"

o

+ Prone by la 8 Eagan ting of the Amen 2g. emt
ep lees by Sema rd ws No
SSPE ant” eh
Beh
NW

a MpGp,

ee oe
ose the sine acor His omitted thoughout Jos the Bonet
Facto tender tr, cog he eonstans a se He pase
‘Sistans th teen a ine sf te woes soy he
fe ryt atari Hessaieusiene ven y Suan

a (e+ ies),

where cand 4 ate the citectcie am stants of che earth and of Tees
reepectively, i the comluctviy uf He earth, enue union, ad

Ciethe yrladly of fight. Toth amelia tke perc licy 8 taken
tics, Seo
o No FRY ANE OE

snd Cis constant meacuring the clectic moment ef te doublet

‘We ans replace theslowble hy «shor element of wire dS earying a
ceurens Un tthe same time we sestme that « ane oy are both
eslizible, 2 that all apiaremene curren are neglected. This is a
Saypiicetion which is ordinal mace a= 4 feet approximation at
power frequencies fr the shorter twanemisson Hines. Thea, falccue
Iie eke. electromagnerie units in mich che sneuhutively af the
fcvth ©, sad wotigg thac we have changed the sign of 4, formal
1) (6) beceme

= tie 95,
fc) 6,
16) a

sad the stant Cis ued that

ay

1 eur X elletive leg of dau

=
Bn
‘Suattusing from {7-(41) in (L]A(8} we Inve, thereon

pe
peer
its (EO OY
densi Loe tae tae |

0 a!

as mae.

res

a

bE

with ab
“The itepral P is well known while Qi evaluated by a suitable
transformation of 8 Resrer inegeal® dle) = Jai) an
SUPENH es) ave the Bese! fanetions of the fest and sem]
for iuazinasy argorments eee by GN, Wateor Te reducing
Te to thie foro we ue the fens ceyation* Za to obtain the

sehr
4 BY sens + tere) — 0.
(iS gh) sos + antes

“Te eomponwas of the eetse free in dhe earth are obtane rm

by te formula

on i= grad abv a — 0,

sb pts,“ Sas Gutaey Ce 08 Fae

syangsHtscul fovree nee os he Mic afTabiel wake — hrmbect

ie lethal ef a gs
SEAR i, pees
CRSA: 6 (SRO tata of 218
MUTUAL IMPEDANCE OF GROUNDED WIRES tit

and we dius obtain By, Ma B, in te compact foun

18) Fa By Ba

where P and Q ate given by (1S) snd (16), In deriving this form we
se the fact Pst satan the wave equation

79,0

00.

At the surlace of che earth (2 ~ 0) the electric force tase sim le
form
wis) ny

© amy (t)]>
ew have he pesos fo he detest Bed
re AG, ane te Sn!

The mucual ines 4 letween tio inalesinal lest dS
and ds is nove writen dws asthe rato ofthe resulting eee force in
fone eles ka ie curren inthe rer, with ign reversed:

(20) du

eos

SHE) ana wea}

‘hese ® an! 6 ac the angles whic Une elements eS ond d5 make with,
fend e= 0 ie the sugle oe) rove with cath other

Tatsgration over the vo wkee Sand gives a gereral formula fos
‘hw matualfapedaace of grounded wites vig om the murfeoe of the
on

29 Bons PP [cu C2) SE + memes

~ LS lavas

Api =e ane fast
5 BT wn

SEA Man, ofa

EAE cl
an nex SvgrEAE FHCHNIeAE JOURNAL

“the dace

on ptt drone

approaches unity a4 approaches etn, ai! Zz thes agrens with the
7 : — oe

a | °

ae ZY ne

tty hme a
yl eevien

pad nto = yl eda
Alirectewerent mmol impelince es ebven by G. A. Compete
Iatrodcing this factor, which is a Function of yr oaly, into the re
factance tet for the drat curent smal impedance becwees tors
Clements dS and ds given the genera wxpmessien forthe muti
slice ecreeeponding tote propagetion coesant y. isinteret'ng
flo ta determine for any gives value of, the vakation of the factor
(2) for ineccasing values o'r. ‘This shun very clay in Fig. ty
‘where the veal and imaginary parss of (21 are alerted for increming

values cf #— Jay = Iekai!"7. The rl prt we note, decreases
pidly from the tial vale uty 987 ienetse, while the imaginary
artic always negative, deeeasng from 2ec0 to & rina valor

‘approsimataly = 0.3 Tor #8) ne Lae ineresing towstde aez
alhougn i eoes aQe approach sero ply os the neal part does

“The Rest three terms ia the expansion of Ze for lot tvequencite
are given by

tmfanee as given by G. A. Canpbel
"The Stet term isthe expassin o! Zs far longstaght wie S ad

== Teing the pose distance fromm de ws S, an Ye angle Between és
land S. Rul) =~ dell) iv the Bevel fnneton of the second
Lind for inaginery argument es defined hy G. N. Watsoa, In ob=
ais ing G4) From 221) we use the derivative with rej va © of 186

aga [eto nin

which is a special cas of she image weal ahove fn evaluating Q, with
{anual poise

renga, Be ttn fn Ct” Sytem
CON, Wataon op. eter 8.
“The expreaion in snnce rach in (24) isthe maul mpedance
dient porelle! co on infinite wire se» pastve distace 3 from the
(rive, Ie agrees with the results pled independently by F.
Pollaceek*. Re Caren atl CG Habeslind, a has been emplnyed
fy a in obtain turmeric results sor 17. Pollacaele has aleo i=
‘esipated the ease of fn gourd circuits of Gite length

“The anutual impels Wye Between a short grounded cirenit aS
sul a umcercleckiee stall ap sre do, 08 the sure ut the
fart ie gh Lye formula

2) aay SHEET Bb Sy FHT

where 6 is the angle which dS’ makes with ree line fom dato a8,
This may be ebtaited fom Sertinefeld's formal for he. oxizoacal
Clestie one due to avert agate astenaa, 1 it may he obined
bean apnleation of Stokes’ Chevean €0 Formula (20) abo,
‘Be a furcher applvatinn of Stoker's theme e may ebesin the
rutual impedance hereren two eountercirkwisesiall loops dl and
anes

date
Tak

(28) a At + 990 4 Se + elem — 9]

‘hig eeeult wight alan be derived from Sommerfeld forma fr che
Senieal magretic forse due toa veriel magnetic antenna,

‘We shall now iligate breSy how the etme valve of #28 given in
(18) above maybe abrained dirty, thosgh mune Ishorinusly,érom
fast prac, Iz his mehial xe stort (som the fundamental
selon ™

en oe)

of the wave equation

a yg

Punta eet ng cen
te eee paneer

PE maa th pe 851 2s 26
shin i scsind by the eleceric ovo in the sary 9 = (ited ie
the yenpazation constane fr plane waves which vary with the time as
0 The pararetes fm, m seta the relation

9) Phoebe ae

In the aie, theme eqwatons hold, ue wich the prosgation constant
“yea zero, and we note chat the solution in the we most be chose
to vanish at an inbrice Reighs, wile che eveth the tution must
‘ons tan inknite depth,

Fur eanveniece in this method we star wih ashore staighs wire of
leageh 2a Tving along the + axis, Lacs allowing to anproich mer
Thus we suppose that te current Je entens the eacth at the point
{20,03 mal Teves i at the point (a, 0. 0), The acon wl he
mitted, however, thoughout the following work, ‘The surrest ow
in hie sjstem is americnl with espe tothe vere plane thigh
the wire, thes pve, and <9 symmetial, bus Wt sign revere,
‘with respect to the vertical plane aot, co the: wire at ils dint,
the y= plane. Ther if we rep the ehree parametecs 1, of
hy So iulepealest grarmerers such chat

em re

formula (29) is demsicllysatshedy ane we can thes replace the for
solutions «°° by cher sorresparting esprvesona i teiie of ines
and cosines, nay,

sin sasin ym sinsweos.%, — eoe-tusin 9, en snes ye

‘The above considecations of syauncty will iba for eich com
ponent of the eleetrie fore, all But one of these form. With there
raining set aise we build up, by means of Ue Focrerf
Negra a ever expression for any. possible steady harmaic use
tian. "Hence we wy write down the general sulisina far the total
clectuic Loves in the ean (2 © Mh ae follows,
wtese the positive eign fs conn inthe expamentia tem containing 2
Sree tietcluion must vanish at an infinite death, «being negative
the earls ane that value of the radial i Len whieh fs positive
Fea part fs Py ane atkitzicy funecions ef their argument, to be
{levermined by the physica conditions of che problem

Thr the a {@ <=) we aay Formate the corresponding solutions for

the total elec once a6

35) ain awain dade,

a0) aia ey con pwd

-wese the propagation consti ao in the air he negative sl fs
Chen ithe eaposersial term containing + sir the schation must
avi tan inliate bluht = being positive inthe as and 2, Pax F.
re ashtccay Smet of eit aeSUEORs

"To determine ses. x echitary fetin we 2eed sx independ:
lations. amir then, wo of theee relations are nisined. by
Uieng Le feet chat the diverge of Tae eect force wither in she
fuarth or in thea sequal to vero, tel is

ay means of this we e'etin Fro (81) 3,

on Wh tO ANTE | FA = 0,
and from (34) (6),

os = aoe

Since the Ins ienstel eompanents of the eletre fore ane continuous
tthe ucla of the ear (2 = 0) we ace chat we musta have, From
G1) ant (4,

38) Pea Pn

and from (82) and (35),

0 ny
MUTEAL sumunaweR OF GROENDED WiRRS 4

We may obtain a filly relation from che fat chat the eurent 7
flows thiough the earth from une gionadisg gent tw te others TO
1 fw chs ace fee Us compute the eral curse flowing wat theoug’
Five faces of a rectangular pian iy the earthy the sat face wing a
restange in the suiace of the warih sirunlag the grnunng int

9, 0), the pom extending from #= ef toa ets from
BD qtoy= 9 and fom? — $107 = 9, ‘The eompoments
‘heetectric fone being given hy (41)-(4, and being the eruductvity
he earth, we obtai fer his current he expreesien

au Oe

her sinplfying by meane of the divergence condition (27). “Thie
urea iain nt thrash the prea i Ff the face ia te mace of
the earth ineles only he ome gvouning pint ey 0.6), I men
(it iclides beth steading peinte: tet ie, Ce above integral 41
equals Tit «a, burt equals aera sf 22 <4, for any postive value
vis rely weed that che Bnurer integral

» 7

asthe dese penton rcs me mt ve

‘To ulin the wre ndtinal retion which e noes, we nike we
fof the faerthat the eurrene Flows thegh the seg wie from ne
counding point to tae othes. Let am fategrate the magacte fore
‘round a rectangle in @ plane pecpeadicucas to the ‘wie, sats
perpendicular tn the x ae, the receangle extenling farm =
to y= y and fcom =~ w=, the pach of integration being
Lukes ie eositse diectcn owing slang the posisive diction of
tes avis ard then equate this hvegral re tines te ootal cre
thiceding, the rectangie, The compouenia of the magnetic, lore
which we none, Hy onl Maced feos the ast Nyt eae = — ta,
that iy

a iat, .
2B, ty
45) itt, = Me SE,