BSTJ 50: 4. April 1971: Statistical Circuit Design: The Optimum Assignment of Component Tolerances for Electrical Networks. (Karafin, B.J.)

8.4 MB·English

Statistical Circuit Design:

The Optimum Assignment of Component
Tolerances for Electrical Networks

By BJ. KARAVIN
lasaauspt revived Dror 15, 200)

‘The prediction of wonusetving yield af am alec cient, given
the tolerances and statictin of the components, ie a atrightforward
procure using Monte Caro tolerance anclyys, Tho iverse problem,
onely, specifying thr component tolerances that pres the ches
ot anton’, 8 more dffnd! act hea received Witte attention is the
erates, Fa Bive poper nw aigorthin ts presmeted that acto this
‘robin jor ianiBeant clos of metverka

"The elgorithn i linited to civ for which one hundred percent
ial 1s sourht sand those eomprnante nea satstclly vutependt,
es ikuete ects. The one handed percent yield eeeumplin ised
fo raduee the wumter of poss Llorance choices variation of the
branch end bows atcateny Phat St own to be perkseiary ficient
thew used to make an uplinain election. Tio-dimensonnl per
formance conto, worat-eae, col Mente Cerlo computations are
also wre av not of the procedor

“hn example ecu is demonstrated foe which Qe alrithn pro
duced 9 tosroner wanigment eubelanGally checper than that pro-
joted by the designer

‘Monte Carlo tolerunce analysis has proven to bes used tool in
cevalvating the ofeess of cumponeat teleranees and. exvizonmentl
SPerietione on elvis ciesui: pertonwarer. ‘The method involses
“Temettuctine” sacnples of the eteus inside the computer using ele-
rent va'ves that obey Uhe auatatecturing statisti, analyzing Uae
eoples, ane. foraimg, erie) dietetbabion 02 pesiorznnee. One
coun oneoate af the pices is the predietion of yiell

‘Mone Cav tolerance wvalyeie, aenectoth referred ta a2 TAP
1295 ym na eva mEOTOAE ZONMEAT, APL Rt

(Tolerence Analysis Progesure), is an open-loop structure, If we ex
Anse she sayin whibh i is woe, we find that for discrete eeeite
the drainer supplies the TAP pupa with » tof exponent ler
noose has charon Based on hrealnoant mesnurements, en? anf
ivity oF eors-onse analysis, ar tome wther approximate techie
‘AS the eonelason of the ‘PAD rin, he observer yield oa is Inet with
tne of 80 sitnations:

(@) Yield is (oo Jom, Wish thi rel the disgner noms be mst
huge his tolermeee. Usfortutely, TAP gives him lite
iacormution ss to which tulersneey vo change and ly how me,

(i) Vild ie ndoqusle- Hee the designer my be sali ey the
esign but he obiaiza Title help in determining whether a
cheaper (laser) set of rloranens right not give equally antic.
faolony i

‘TAP, ther, is open-loop in thet ie too for pedieting Field given
sel of tolerorn hh Te Il seb of tolerates or ang changes to
hak aot ree provided by the denignee wll arsnioner from
TAD, [Tt e insomating 20 note thc for Fatogrte neta the designer
hs litle ‘raedons 3 specifying tolerances or sacking. In this ease
TAP is used to check tht desigos themselves are udequute, given the
‘exponent tracking thst es bo ebieved i integra’ prodttion.

‘This paper diteusses en algorithm for eloing the ‘TAP loop. "The
program embodying this algorchm vil secepta cuit lapolgy, wom
‘nal cemeat velue, « Agere of mest for eieult perfornance asd data
relating werent cost to ema tolerance, and specify the act of
tolerances that will proce the chess. menaced ncwork tha,
has 100 yesoanb a pric) viel. (By © prue! yield we noun Ube per
‘omtese of those manufactured cirenits all of whose components
‘within tale-anee ite acl which wre wit oorvely thu: amet epee
cautions. 10) pervent priori yield iuplee thet any memnfcetued
rout thal file met speeiations hos ther eorsponent outed
ff tolerence 0° n wicing error} The tocbriques deseribed! re eppl
‘ble ab prsent only to dire sireuite, ic, cirelte whose the
Parameter satiate lnlepevint. Work 5 im progress ta erent
evaile wth comme puracioters,

‘The svete seeribod below ia a canhination of (atimensions
performarce coniours @ wethed rhe author hae sino learned is a
‘aration ow Sie are xa bowed tee igi? ans-wort-euse come
putations und repetitive TAP computations
‘rman SOLER ier

Ta thi paper the cheapest netwusk is define! a the network with
100 persent © pricw yk svoh that the sea of The ese of be wom
ponents cetaprisng that network is noe greeter than tle sum of este
For evesy ot actork wih 100 pereone a print yield, We are ee
feroed ith a nebwark o? fixe! tepoiogy anal xed nominal cleme
YValucs ior which ve went (a vose a fverance vot, each of hose
Clements, yy the tolerance cwen fr iret porate, ny. Assoc
ste with each parameter foleranre, = @ east, Cr where Cx-) i
the ene uretio of the th eat parareter

"Por each pararcter thr vesigoer tas @ Tiited susnber of ieee
tolereners ftom whieh to ehoose. Theron, will conider five
tuniwene of posible realizations, each realaation being characterized
by 9 aniqie tweranes yoolar| We will nucber Ue reliralone in the
Universe 12, oy and refer to eaea roizbin 7 its sociated
{oleranes vedtor =, We next Sofi the oe,

(= 1% anaton & has 100 porvento price yield

an he serie se,

a=W léeal

“Then o cheapest network has tlerange vector + eatisiyng te cond
ion:

«8 Seow: terallkee,

sete the sircuit unr emy se» eugene

(One could inmgine eer esiteria hy whieh the ehcepest network,
night he define! Tr partvnle, ane ight fd a netwark eos function
That lsadod off cl ngsinet cos cl result, eamonen. sclvaying,
for distanting networks, Kowover, copsieration of such trndeot
Trsisee wrione qertions abun! manuartuning practice, not to mention
the cueation ef slowing svh-rarginalcireuite to fv installed nthe
Brome ofthese queens, she feowing dicusson is hmited to
tireuite for whieh 100 person price yield is require. A® we shall
feo, this uasumpticn ie eeral to the propose Lolerancespesifiestion
reethod
128 ram may evens mmeanien, SoURAL, amt NL

Tt is nsaned that the cree with whith we ere dling have a
sevlar perommanee, J, that is # corpuable Jeneion af the eeu
peromcer values; © ominel design wish periormuace measure, J

‘and an allowable dearedation from xomintl, « sorh that Uy +
marke a bourdsry sepersting secepttsle circuits srom unaeozptubie

We now exaaine thy r-inensinal paraneler deviation
‘apace hoor origin is defi ax the nominal parameter velar and
fae of whore kee je che preack dev'nion of ove gmeaneler ftom
sis nominal valve, We postulate the exclence of « connected region,
inti the ogi, seh that all eneuits bl with parameter values
eoreeenbel ly poitls inside she engion are aeecpiable ane Une ote
fige the roxion are eauooepal, We eal Us Ue Texiva of Anoop
bility, Rut Wer mow wative thee in this same span ca tolerance
vector, je wsocinet with an aatiensianal pellelepiped dat we
fall the Regien of Peeibilty.” To ocbor ordeal crete Bult of
fompanente wit tolerances sperified he uve yusaneter intion
‘ators inside tho puralelopiped we are calling the Resion of Pos
baty,

“To clay these ders we extend nn oxtmple proven in Ral, 1
Pique ta shows e simp voltage divider Tle tranive Fimotion ie geen
bp T = 1/(L + teu), ais inp ratange fs = By + Ry, With
nominal on-ob resistors, the zominal bricafer funelion and inp
Fxisanos ane 0.6 and 20 repeosvely. Suppose the dorgn sseifestons
tall for 046-5 7" 0.53 and 185 5 5 215, Then the Region of
Aeoeptabiity Uk) shuva in Fig. Th i bounded by fur Tine ench of
tric ie the Ino ofall pninte prodting networks ceuetly cating
fe of the for spcifeation Tonite. Figure th also shows the Regian
fof Bossi (Rp) whew Ue voltage divider ir bolt with 6 pereant
fomponenta. Notes that Ry intersects the parameter devistion axes
tusks pareent. Novi further that all posible xatvoria are arooptable,

2, Rr is lollly waned within Ry In Big. Le a naH Be that i the
renult fusing 10 perbant resittor for 2, end s 9 percent reeetor

ene} for emnpnof nuhine Cano sha frgun ce
Tousis tues, The ngenmetie age con te heled oak ronahes che mee
RISE! Lally, neat all camponan Se daly adept
ston asenna i)

TEs fe,
‘ NER Pop,

mw A

ES

RES drat vais vale de 43

for fis superinpesod ovo B,. Notice that some aetiurs in
econ guava fy my they ave inside ftp bat outside Ze, Hoos,
the tolerance nels (10,3) ss acts asember of acd enol bo wed
for dh exami neon

"Ths comclaion egy tow ene i that im omer for» eer Lo
ve 10D pervs print wed me rst heee My © By Further
ssa wt ace testreting our alten to nek tit se 100 pe
cent prin yield, our ta sto choo the eeapo uleranoe Yeetor
Unt wl aati Ube soncition ep Sy

we cnuld ezaraeters tw Rgguet of Acreptcilte, the tase of
clheeing a tolesenee vector thst profes the cheapest aetwoek ‘ould
toe reduced tothe problem of Sting the “largest” parallelepiped le
thet is contained thing - ONolive tbat our arsumetions of 100 per
font a petor ld and 10D ponent eampaiont arting ebvite the nexd
toeonaer composcnt distibitions, sone esneeted with are the
Fit of eomportont devia.) Taieneunately, characterization of Ry ix
fn impractice! tuk; we an trsye ya eaten of dimensionality.”
[Byer ts wore as simyll mop a8 4 agp, te woul, for wn
1250 cy en wena TRONNTEAT SOORNAE, APR Ut)

work of way 15 pansneters, have over $2.00 vertnes, The number of
fhsetion evalutions rove ts Lonte Uhm serine wou be exter
rly lange. When rv errs that the (we — 1) dinesinnal acs
bovaiding Ay are Tkely to be quite onmples, dhe magnitude of the
problem becomes stoesotng

‘The algorithm thu watd to ole de toloranen vostor that wil
produce the chaapert network, 7, bagi snl x oonaderabon of
redimensionnl subspaces of Ry We wl require that two-mensional
rkepasot of Zee inside the enrresoning aepaare wf RW wil
Te ne eres of techniques to inane shat ep R,

"The slgoritha for nding ¢ cheapest network is brcadly outlined
in Fig 2. Dror ne take-up ete ofthe major books in detail

41 Posie Conaeaine

‘The frst ctap in cbtaining peirvico oumstetin's on component tol-
seanoes is tp enmpute tae minimson aaterept for each paramter.
(Call this, forthe sth perametor. Drie, thi ia the minum devin-
tion seh pamimetor exp tnorgo, keryng all her paramore fixed
it nominal sue, before the nolwork epevfionn wre viele

‘Next we eamsider the ()petinrmanee oortian! fre The m pacnractoe
rotwork. These connate the Dundares af the txostimeraignat
subupuees of My, huldiag (9 — 2) parasoiere fixed nt nominal. Hoek
tonto determines what tolerance paite mr allowable forthe watable
rumors Ta hor, tach tolenvien par defies « teokangle The
Piri llowsble if that rettange ie within the prefranoe eotour

‘The designer i premumad to have specified s discrete sat of obtainable
toverareos for eats component. Cal fis set S, forthe sth companent.
We wil mnatar the isambans oF 8, jn devsonding order and ter to
thonn e@ ray tuys ters WhATe my, 2 the mmber of obtainable
foleronece foe the ith component. We will frm table for cach pair
fF paramaies, say (9), of te fe.

For eae of ehsaquent eompo‘atin, che neeemetar onthe left wil be
The ene with the saler sina: inereepl Nota meine wee
wecrong MORAN MR ssenomtEs wat

ie 2olraon oer wai

axily exist in Jus stl, Piet we emt show ‘olerances “hat wm
Tages chan che sino interop, nee wey will auey produce res
langle nat eousained within tie (ne) conteus (Fpl. Then for tae
feat ny 2 [do il the largest esa ofS, chat proves 9 ner
tangle imice Ty, std thin (oleranre ie estore ano Ube table uppoibe
teas ait ofthe tbl ae

find if poremotor roy olerane +, parameters miy B49 ry OF
Say emaller toleranes, THe amised tarotigh the esrents of 8, Aoding
fhe comrapondisg tee alowabse eleranee within S, until we bave
feted re am ite companion inthe table
1282 ram amy svemmae maceccen 9oumtsd, APR I

‘An example should clarify Wee ideas, By for the voltage divider of
Fig Ia ie reproduc in Fig 8, Sie (ho valage divider has only bo
Inananeers only one performace euatou exists atl this i ‘lente
to Re Suppose thar 2, and I, have che rar se cf ofzsiaable talr-

8 = 115, 10,5,5, 11

igure 8 indicate Ly > “1 percent Ly = 18 pazoont. We WE thore-
foe carsiderp, tobe the left side of the table, To begin to fll the tela
‘wonatice sy = Ii pareent > [ly snd is theretore elminsted fom oon
fidoraton. rg = 10 percents entered ae thea. enty iz te left-hand
chun. Stcring at the pointe iv Fig. (10 pevent,O} ar (—10 per-
text, 3, rhe yrlial venions are nay deerrsioe bow “te Ra
ray be varied before apeieations are exceed. To the tons quad=
ant, speifieations ave muted lafare AR, = 2 peveent. Thevefore
an =f perch is mau the fl yht-haral ray in the kale, Cam=
PUtadisnclly we alternate betwecs. AR. = 10 portent and AR, = —10
Pereer: end take stnall steps in AP lth pstvsly aed negatively ne
fil the apostistons ate violaled (shawn as slo L on Pig, 3). The
horizontal line ofthe (10,1) restangle eve then explored until as has
Teen stn 10 rg = & purer, See no seleion fue are
Vielete, Uhecutey [10's eotid (ste> 2a Tg. 8}. Note ve are
fceuming that performance costaurs are siply connected.) We shea,
texplarsvertiet ly Irom the four points an wich step 2 ended aa fine
Ua he nest entry ine table (8, 8}. We continue in cis way until
pron qouteNon ASSERT 1283

‘the table shows in Vig. 3 is complete, wd we have explared the com
ines gorneter of all slomable teleance reeanales, Note that
Fhteriog a. tom fpeveenh 0 R povtenn resulted fo ineteased
allowatie toler for.

‘Phi peogess is repose for egch of the @) parameter pats. Th
abl foreach parameter por (y») ie denoted 1. Tae table yivex
turn pmireise gonstrane on tlermere. The next sep ia our effort 0
ad ft fo find the ohoapest netwenk slisfyny tho pairwise cose
ruins, Le, to Rnd oleramos voetor, rsh lements atisfy
ta lane contre parse wal ir the Fheapest ef ll el olerauce
vetore, Having four! sf

ore wil shen ary to diseneor rt ec.
Teifs? = Hy € re il Gnd the next chenpest tolerance
tector sutafing the ynirwite eonstavile, ele, until we Sind oe Buk
We mamber of. TT we define a ee

ah = fr | tsi rie opnstrais

then ib elesr thal @, 2 @ andy Acorn elorsens of in der
tf inrensing ent, te fiatehoeen cleat of @, thot is so an element
fof isin Facto

42 Roe an Bout Tel

1 thie seecion ws desesbe 9 variation on the Urvnel and bound
‘echniqu Hak 3 und bo fr the easiest element of «7%. Coxe
Mer the tina tree struc shower in ge. Te nanos the
Tit column represent the slemants of 8, thr obtainable tolerances
for pe, From etch soe in the fret eokirn we have branches lesing
fivallthe elements of & , the obtesnable tolerances fr gst, Finally
Ihe tat elaeon hos al he elements oS, reveesed for every econ
the (r= 1) precadig rement tenner, Those are [TS my nee
fn the but eam ca an istiealrmber of pat Form the Tet
fae of the tres to the vight, Each poh asocsted with « choles of
elonos vector ed bener bas aa weeded cost, © — Dit C4.
"Ph tosk of Sading or then the tore of nding » yl hrongh the
tre Bern left to eight auch thn

£9) The soleenee pair weeciated with every node pi of the path it

conssien. with the tahlod eoetesin‘s

(Gi) Cis minim.
With cach novls ws cam setociate partial cost, Por a nods selected in
Ahlunin fy ave have a parla! ret OF Dery CG) where vy is the
folertnoe sete in column Fates ¢-< Fran Ue Lolewarons oleate
for Tho iniliat yar: of the path,

ive for Poading the Choagest Networe
129k ue mau averese tecmrstcan roman, aram vent

A flowchart fo seleting the eheegest uth whose asoeited ©
fn show ie Fig. 5, To words, se bogn ay che let ard eseese fore ths
Jnrgset (heaps!) tolorarc not renter than Ly . We tess aye wero
the eve picking the th element tolcnes ua

n= Min alr,

‘where the faretion Py (ni pies ube sight enizy oppeite vin
th table, Pie. After eae’ ach wrrotion wo soups partial com,
(C5 fr the portion of fae pach elena ao as, Twa eecurenets will low
us wo elininate int pat ofthe tree with initial pathy ae vin

© Acoinmn 5, nme oF more 2:4(¢.) does not exist, This tenes
that any r with the frst G2 ~ 1) components, my

4 tang oh, The ngs ooet, itil
fs: at ©, ie repaoad by the eat of the Set complete path
Found, ani is upeated as sheaper toleraneovesters emerge fron
the process.

ro)