BSTJ 50: 3. March 1971: Spreading Resistance Between Constant Potential Surfaces. (Brooks, R.D.; Mattes, H.G.)

3.2 MB·English

Spreading Resistance Between
Constant Potential Surface

Ty RD. HRODKS and H. G. MATES
Aabryeint evece earn 1, 1m

We divtne Be ayceung sviiance fora shst of hamapnecus ma
tri of ens koe wth ik Samar source ok one ae and th
‘erent clan) ene fhe eatin a pve, and} ale sonrnpalny
isk on the bach plove. A cimabint dry pac exam vase te
ore neti in weed Baseoy condone 3m i plans of the cones
Bresueh eos, 0 ebue oem sell tain te dria al a waneri=
ally atpreted fora vance of seme rotten rooting 0 nirnd
vending resistance our, Uhe tnt ar used ta evawate the sprouting
isitenta onecuntied in a typeat (Lon) soweonsualr mate

‘Thm esi anneiatd wih the sonparaie] eument ow hot
se suavicly repastved spurs ed sake ie teceaned to. a8 conceding
Tsistane, Caloncnn of spreading reise efzen rayirel mn Che
Alglica” ‘crabs somicandietor devices, In pasties, ee:
iran] exzcont Dow inte ul lieun Telwors actaee cantack and
f backeplae cousaet involves he erlelncion of the obzie preahing,
Trsitanne, The heat. fom hetreen a acsive trvzastor ur integrate,
fSeeit ane at exernal bext en iewalves a valeation uf Ua here
Spreating wesistaree sn he dew artis Aba, en @ given ezasture,
The ranean ilnting gigs Hvetly related to sao exec
illo sis

"To ovsen are eid jo thi yapes. The epesding seelcaace 52
acermirel for wt infuils shel ef hmngevens racerel of uniost,
thidkness ho disk eouGnel sour er sert ide al ith enna
callocted (9) tn completes miler ‘suck lune, atl iil nt
fomespanding ick oa the back slaae, D. 1. Kexoody smnlyzed
he rates for cite aptindreal volines but wit: comune bowacury
faucitans? He aasume: x constant fie aver the eoaeas wien ane
efit spnvaingveseranes rca af she maxinuum xsuperaln (or
Duventit on Ue ak, Ta chin paper, we aeeome a onmant eink
“hentia! eves the source, Tia amumption cveute ia mise bey
‘nto in the rouree plane because the potential is eee wer
DL ot cho sutinee, nod the normal deviative ia woeaiid over
the rer of the suriuee Lz the Lining cane of w vlindee wf efiive
tains, these appeats to be some diference between Kenredy's reals
sid tse presented beeen. This appents 20 bo astecate with Ken
Ty's defhiion a senending resistance ia tame of the maximin
Uisk po erin, Hoonsrs, Pe nnsarge, i al on the eeepolation

curves given by ered
[A Gray, ec, cemsdeved certain special eas of the into region
wablom with mixed boundary eoncitions she source plus” Tv
fret, Lay disposed one of two constnirte, Iitaer (i) there ae
ffleion eeperstion aetieen agazte ant sine wo the How dieetulion
fle sik emild be taken equal to the ming Sallspuce esse ar
(2) the dike wae sill enougt co 22 eaneidesed a point cones
Tn csconro, these eansraints again vedaoe the Probl: to one wit &
uae yinbaey ovals he nuts plane, Neither eoneteaiat
is supose in hie pane

Tn This paper, for rach oase, a closed dom iosegre) slut
deve which ia nnmericaly integrates for a targe of gomerie
alice covlling ina univeneal spreading seestance cores, Po» both
fos Hie errs cp-nech the Halfapcoe limit for tek eheete, ané
fe career wayusalieally woprow: tke nrfingeg Tell for very
thin ace, Finally, Uhe resus ane uve te evaluate ‘he spreading
resistances lypie! wf nse eninurtavad i setiomnotar toaknelogg

tae geometry wel even of aortas ate give i Fig 1. Bulag
La Pet's yuan in spline eoordisabes when te potential wi
indoponlent of 9 giver!

elad) — J FRIIC, comb he 1 Csi Ax) Jutko a, a

ze / anast be doleruind by the dounders condisione x: # = 0.
or nocazined boursiny wuncitins ak 2 = 0, #059 ag be fond by
inverting the Tuake traneform? Tx she proweah eas, Moeven, Eee
raized boundary condition mee: be isposed wy the z= Usureoe, Thus,
sentaome ersten m

{00h ah, 02 950
: @
|= [rormcsnnn ts + 6 cot 8).h9 ans =o
‘hiss «dow ogee
ve [imeem 05086:
7 @

J.D, dechson observes that the sehition of thie eo of euntins i

2 Yasin ta
soy = 2 ®
Too
ives = UE [04 (rans 1 PEs eho ah, 6)

st Case 3 Back Pane Groeed

Irae Ges case considered, the goed clive Hew enpletey
rovtalin ar groundst bse’ plane. The bounéary sorvitiony are
fiven in Fig, 2 THis rimctinn le roprezentstive of heat Bow from
fn antezrated cious rough st ins) Feud te a can aang at

CR.

Hig, Plyweal giomesy anf owners een far ean
ie 2a rian Une il rnc a

choo! cantar

Ate = eins.

poe

vee Li austell

Saonte~ coh ie

1 J tedeae, ag
so

si ths) each bw a

vo|

Colin he last eves epsalione ane
intone, yields

secving ow the ities

> eat oes

ba aven Fi oa
hn

ey
sind ily

ayy
“ee Case 2Disk on BosePone 6
‘ihe vase aie the sing worsiie ul id of recs « eoesia wilh
Jue seston Aenea 8 “tnase ery sar Cove 1 asin

Sole sonfgtration sear by. 8. Le ves 0 spreading resi
fewer Ene 2 en be cere tan Case 1 hp sen

od
el - as
16 lee Wa
- an
a Meamesinw Suteki Cans ©
‘Ve snteerls in equations 114) al HTL ean be ealuate nie

ieally fy veshoue aie tatioe AT the unper lit e Gnte ex the
780 aamansvsrin moana soURNA AROK

enc apeore | Sexlde

ie $-Soredig eace Hy etme dae

Sntegoands behave weet the “om tnt, Investigation of the bre
havior ofthe integraude over de entre zeage of ilageation, indicated
the range ool be eit into these rangers ta Oly 0.1 200, end
00 19 «For the 40010 orange, aul uv > 1 the legal for bh
feasts reduce to

[Boe ay
at

i)

1 [sede ro

4s wes evaluated meray on lips] computer sl found to be
(0.96432, Thue = 000661 and in independant a 4/ for af & 1.

or the 9 Ol rango, te sntegrnce tere relsead By eheir eal,
swgement approximations. For = |
sewn westeraves ™

en

ep

ey
0,

oo)

1k follows thatthe sm] argument approsimations forthe integrands
tee a%u, end aj for Caean | and 2, zespectively, and the velhes of
Ihe integrals ove tae range 0 120.1 are a/20e und o/WDe,rospevtvey.
‘This ie good apotonisation Because the integenndn are very Aa
funetione of 3 neat = = 0,

"Thus for muanerie) purposes,

1

= [OP ae ot

5)

ac om]

1

neem oe on
[eo P Siam at oom]

‘These results ld for a/ 1. For aft = 0, the balfspner limiting
se fo Site and infinite w eam be asd to fad wales for Ry and
Res The ress vehi ate} and 4 sor onace Land 2, resgetive'y
This seal fll nee ‘Tom Jackson's wos

“sing the half-space Hix nal ureying out the mimevies istopra-
tiooe for yorioue values of o/s reulle ia Ube waiverel spreading
fesbloner euros ven fs Figs. 2 ake 3

2 Anyi Taite for Case I and?
As the ratio aw becomes lager the components of Ry and Ry due to

ringing broome progreevelysmnler Noplectng ringing Wacrsistanne
fo ether ease i

ke” oy

a
182 “au uta SYSOTACTRORNTCAY.FOCRRAE, SAREE HL

Ron = *. es)

“The svelte given ia this vertion eon be prevented more elowey in
term of eonduseanoss boesuse tke cringing and montane comport
toe in panel ana 2" = O ia linear funaton of afin. Let the oe
‘ductanone corcespuning to Ry il Ry be designated G, and Gy, 10
spectively. For sven a and there abl be more Pingo Cae 1
than in Case 2, and ne finging for. Thu for eny ae,

>> 9)

Also 8 and Ge should anympttieally apenoaat @ as a funetion of e/

ase ulnorcatons are supported by tke curves of 6, G,, and. 6
‘orntn i given i Big. 4. The diffarenoe of fringing eamparaata for
(Cases 1 and'2 are given in Fig- 8. dy and O nee within ope poeent
forai 2 1.

a Typicat Brample
‘The reales given above have been wed to evalunte sprang rae
tancin typeal af thoe> encountered in temicanductorteahnelogs. A

TT Softee [ye

Sp = of RE aeteore

ig, 4 Sure condutuocs wis oud wishes age etneen dak ed
verb onl le an Sevoee di ed aed tito bk a
ig Speeding seine Ry and oe Lee ster
Fed tudeamuy sopra reoynican soem, MARC Jah

10 em sige of eiiom with eimalareontacts wes considered. Kigure 6
ives the calelated R, and, for such a confiurction

An investigation was mage to determine the spring resstanoe in
1 shect af boogencous material of usitera thickaoss with a dick
fontae:couree on ane aide sd with current eolelad () atthe entire
back plane, and (i) aba corespondng disk on the becleplame. These
sages were analyved ail evaluated actly by solution of a dual sot
Dt integra: mations. Tha metkod represented the boundary conditions
fa they physically exis, dn eonizeet to poeviows work, sine, uni
vorzel, spreading resistance curve wne presnted fer enoh onse, These
curves should be useful in designing devices and analysing matrils,
In partionar, the curves sould be wea! to Geteraize eoutuetvity of
‘aboot of remivonduetormorterial ifs thiolrnese i= known Alo, the
curves could be uso divesly in tho ealclation of total expucisance
snd iringing capacisanoe by rlabelicg the ordinates with «/C aactoad,
of Reo end Ler inetons of Or

1. Kewl D, Space, Reeauy Cah Sosa De

Tit agra Pea Se 8 Cate 10)

aon", Sunbird’ Megowst Me Sad Banting, Soot
"Blu baad muclonngd Co Guar A

sadn Be Blatl Bectadynamics Now Yee Wie Wey ad Son,

4 wien "ELS Dizon of Bon