BSTJ 50: 4. April 1971: Statistical Circuit Design: Large Change Sensitivities for Statistical Designs. (Butler, E.M.)

7.6 MB·English

Sta

ical Ci

cuit Design:

Large Change Sensitivities for
Statistical Design

By EMOBUTEER §
{Massuips moon Novo 96, A

A Monte Cavto ald $e wn avaiyns 1 Me, sence that for azeciiod
tolerances, eorerlavos, ate, ongivie etrivatios nf mecswes of
performance ave oblacned. x sqproark i prevented ahi aderaes
esl tothe sisoes prob, ha of determsnang the Polemnces, ca
lations, ete, weassory f0 alle arveptasie performance te=
wn Phe approach te based on the enatept af fange change seri
‘ew orbick wee poopored ea 3 wesseve of veveitvity for sfaUetas
Geaign, The aporoach speitenlsy wiilrsue denign probleme such as
peeing laoruners, detonstizing @ nomiunt devin, secopnsing the
Dossbiiie for end specifying tuming amazon matching proce,
ed verifying that a dvign v0 nitent With expected stabsia! cx

elation between paramere, We preset nn exnogue iustrating cee

rat of these eppicatine

Realetieaysters rd eet sown ame: apo for the feet ch
sant tralisavions of paper sssigne ame sedan achieved, Tie Bel
Syecemt ie pastcuarly eeaitve ty tie problem gor anly teense af

ohysiral and soonorsie elmils sn reuntTaetre, Eat als Deewuse
OT Ue vised Se envizonnisate fa whieh Uncrate sak aperete, The
Ulfcle of varie desig paramctors, whieh ate nsually modeled

fs random varies, san be investigated via a Monte Catlo study.
However, « Monin Carlo study ie en anclvals i the renee hak for
pected probability density functions of design parameters fopesied
bp snomiral” vale, teranee.corelation, ete.) an ements
‘ion for various ons or Performance mcesures is found. The inser
problem, that of fv sewinal value, tleranges, end correlation
inorder to obtain an scope yerferzanoecatribuion, has roveived
1210 ym va sre THLMSEAL JOLAAL, APRTE 10M

ns. This paper ceeribes ex wpprowsh lds

relatively Hite nll
to this problem of “esing Ue Toop sfound telerenco wnalysis

There are threo signifoan points about thie approach. Firs the
approach does at rly un fine or seeondeorder approximations? Like
Mone Cera, no attempt is rude vo upprosimsle mexsures of per-
fortanee, Second, the spprund cae wremumedate muliple-epeciten-
tions, Thind, the implemensstion of the ayoronch ix feasible with a
sompuier ard se the lchniquee ray be thousht of us vomputer ade
to meatstcal dese.

“The wpproaeh i based ox four ascumptions

(There iste a dasignerspecifed sealer performance criterion,
“J shish edequatety relict the goodness of x design and which
is a oontimanus fonetion of the design parameter. A method
To forming «sgl ertesion froma maany einai utrated in
the sams

(i) There is a designerspecifed value of this eritrion beyond
hid dasigns are not ansepeable

(Gi) There iw own nna ign which is woe in terns

ff the prfornnve criterion

Degition: ‘oe cogion of wreuptabilte, Ra, 6 defined to be w aon
Insed raion in persmeter epee wth that the nominal design ie in
‘and sith that fra ealirtin in the womespunding performance
ie ene

Fiolly, we make 8 fourth nasurption

(9) Au relations indo Ry wre valle poo: ny 0 pay ail
decision enn he msde for aah aleation

‘Thee dona ae trated in theme parameter, to cxterion example
fn Hig. L where J; is the sear value ofthe th performance cxterion,
i the aomine! desiga parameter yale, J? i ue sh pexformane
‘St mousinal and oi the allowable oration in J, from 72." Sine
there is more than oo spuifntion which ant ie rel, the region of
sonepeabity, Hi, i the Interetion of the individual neon of n=
fcaptabley for cach To thie oaample, inp [as p= ML

eng le lm ny in ar bande! Tied
"Firtore have tx aayesoas is wutrol ory to eschew Qra-ander sensitive

sepgircin ope mesh oe ek lr =

diva au iui esate i Te onal semen
ase ean suerte wu

{

ler she wove wescworions, che region at saeoptability is cea
operas ube contex: of statietval desga, the 38, im spel az
feminal volute, tolevaneesy te. Ts Tat in Th shape of thie region
Sand the placerient othr wane ini walker thas the sale of the
Pesforinanen at nora that 3 impart, Pe exe, ite vendo
tleviatwews fly ralfacion st i the above exesople weze uiterly
‘nate snl ayneirie alwa’ “ho sorinal, shen a nemaeal deen
loeatod ial my berwcon osines < sne b sould be beteer tam pa
coum of y(t anor awa ol-nnae, Paheninae, in thie eontent

of staisenl design the vonery af ma-iisily Ger ana sew mB NS
Seiseh i introsused in Seaton TT, Avpliarions of zip eneiice ine
oration to eos he inp jn tolesenee analyse an dieuaaed

ection ILL arden exampie ie given in Sensi TV.

2 Interop

‘Suppor Lint wll ll parasatens Hast at seminal exsopt the Kt,
‘We deine the upper ower) insercapt of parameter & to be the value
Jn ponent sevinlen fen nnoninal af portmetar for whish the per-
ecmeave is unaereplble far thelist ine aa peter i inert
(doercaed) fo nominal. Parameter values are exprizeed in percent
deviation frum nosiml for nasoneble geal. Pointe ed an the
upper ead Toner intereple fr the one parang example Fig.
‘We denote the interrnts by
1212 ra aun svsomag TRoMGHL JOUR, AML LOL

Ti, 91, = 02 = upper intorecpt of puremeter & with respect
{ penZormnes criterion, nominal design p,
fend ellomable performioer degrufation #

Sinilur notation hols Zor

"Te interop ane simply mre of how ar a ingle parameter
can devia withis the togion of weeping Tf the ft intarepts are
Sl then being in il, i voy "emilive” to the kth parameter, and
‘ioe vera. Using this observation us molivabin, ti Following meas
cf welt ie proposed.

AY,

6) — Li = upper lange shangs sensitivity of parameter &
swith sexpert la Jy pad «

Similan,
Bey,

A ingle Inge change seuitivity for perametar & ean be defined us the
‘weximum of 1, Fa
an Peforens Canin

‘The ioteroryis provid iammatinn owt how far a snale parameter
cnn vary mil all ters axe fad nomial Deft he spsiatinns
Sie ot inet, This iden ean be entended to two srarlere For pair
fof dose marunsters, © tine (or lins) of constant, Just ceaeprable
Devformunoe penvidee an indeaton about usr da (wp punters oth
are sie allanmoualy around nomine! before spevfeutions are not mek
Tr fart, pofasrase comiows tre a pair of purumctors is defined to
‘be thin info linge wich daeebes the edgy uA restrict lo nb.
Miwpssional aubapioe defined by thor pucumctem while all ulher
Iurarnlet any ld fees at zemiaal. Aan, ah poeamoter valu i
ented interme of pereunt eviati Swom ie nomad value

‘Tee content of « performance eunlaur enn be lasrated wth a smple
seam, Consider tho to perumcter voltage cider shown in ig. 2
‘whem A — 2 — 1 The tue fanobun, 7 given

Tm RR. + 1),

he inp resets, ia given by BB, Ry Em

"= 0

Tce ses denote vein quate
ages eats #29: Tors 1213

hk

Tig. 2-8 vole dee enuple

Suppave the design speciistions esl for 049 23 1 = O51 and
18 2 R= 22. The region uroupl seminal where the first speifien-
Winn is mgt i the shaded are i Tig, and he region where the secon
tpeifiecton ia met ie the eroet-hateld un. The tgion where both
Spreifirstions are met is the intoreetion of Use rims. The edge of
ie seg He the perfomance vonbous. The pointe where
the contour eroses the axee ate the ilorer pts
"Notion Pit she performace corzeus Ins sharp vores heesuse of
‘the mullite desigr aproteatiene. "The ousiealurepesifsw ion hich
feotemsincs gh inteept 7 3 estion of & perZoraee wots sa
to be domi: nt ib coint oF pons, Foe example, che wpe le
fart of the contour iz Tp. ie eerie hy she T= O speticn
ti
N poxtarmanes contour ens intron on providing “Seoond-
11 ren ama, svscuse uavusuen, Jommvat, ARN

condes" lags clsnge vemiivily information since it icieates how 2
fed Z- forone parazneter wl ang fue a change in the nominal vale
DT the aecond paranceter

28 Comparison with Classical Senate:

‘One ean engao that lange clive aeasilvitios ae similar to ft-
onder sensitivities in te sete thal wny one parameter is varied while
tier art held fixe, Ther is, however, x Fondanental diference ix
fanpreach, The lnler suites are proportional to the change in
performance duc to vmilar changes in the individual parameters. The
Tange ehange sensitivities, on the othor have, are ase a the change
ttvesnry In each parameter to bring about « perio (but similar)
Change in perfrmence, Hina, it shunld be ole! Kha J is Linear
fuveton af the pararacers, the two sensiliritee ate similar. This
‘ends oreence to she defntien of Inege chenge senstivity se propar
tional to the inyorse of tae interes

Tei intoroting Uo vole to ange ching and lassi! aensitivities
each provide a chacactesizien, ‘The intensple weal performance
bontomte (fist- ard eovoad-order large change senssivities) provide
I ctaracesaation of 2, in ere ati Hnenshi paramos space
hile the Aes. seo tomas tae Taylor series of peeformice een,
hominal (feat- end. sccend-ordcr clusion) seuitivitiee) provide a
sharantaiastion af ts porierunee ner wowing,
tention 's fer thonid bel Zooued on Ry rusher than on the per
Tormanee sl aomical. 7

we large esa eonsitvisie provide o «ba760-
terimmtion of Bq. and betes, auexeute 6) parwrieer vensivty for
ttallalea! design

ea Preliainary Remaots

Infornution provided hy lange eoange eenstiviton oan enbenes a
designer's Insight int problem. This can be especially important
shen eomplicniel specifeations exist and intuition becomes, har
Dnerted, One aright question the ammount of useful information derivable
froma pesformanoe eunlaars nee they tepeeent fq for only puis of
[ranians, Ts shoul be painted wit, Inver, that cleeteial properties
{eno depen on passsncters 3» pairs slew RC produts end resi
tortie.

In this ection, eoural peso aplicacions of lenge change sensi-
sivigy information $0 resign probleme are diseased The problems
sagow on exer enero: 1216

‘which ave wissen inalie dessin + nominal design, speityine
tolerances, rexenwixing the need for end srei'ying tuning and/or
bhatehing dering hata ceri ievonsatent ith known alas =
‘al comeltion ur Irur¥ing

2 Desansiaing a Desi

sa The Problem

‘We have seen in the exumnple nf Fig 1 tht the design el tolerate
Incyee arcs vatacors ov ona’ if tho nomial were coated
it fe That iy we enald destitive Gs lange chase sense) the
tlsign by pling the naval hal way betwesn the intaruzpte eater
thas al pt Werestene thie notion to A’ simensions and base out
Ineamre of being rere in Zt on the Ilerernt,o ecuvatealy, on
the lage champ seni.

Wo have ieestyz tec em algoithny co wilavalicay cesnsiiee em
intial deega ‘sil tics the erferunee speiflestone bn: [8 not
eceeewily eanoered. Tn pertinent aoszvalios whieh iuPuonwed the
feral of ox ian re

(6 Th wn ohange tke aomita! suey of more than anv parameter
rimaltsngodaly iv om alle co cantar bus only on intewsk
Jiformacir, ik ie goeeble to move outside of Ry haat veel

faneider the hypotheties txcmnureter example deser'bed by
ite pertwemnoe soatour ia Fg Tee crvcer both py and ps
J216 im nnn, svete tecmoss sOURSAL, APACE SA

Smultanoausly based onthe intereepts at point (ot, we move
to point. vich is outide a

(@) The inteoopts for w purtieulr poremeter ame smple fanetions
of thet paramcter’s nominal value, but they can be eneplionted
fren noucontnus funelign of Uhe nominal values Af other
panamactere. For exumple, in Tig. 4 the upuerinlerat of pi
Yaroster I ior p) ~ pf) ia ot continuous function of ps

tAlewrtivo jor Desens

Tn we of the ahve @ummente, i wes decided to ierace towards
sacs desig by changing oly ane nominel €& a dize using ©
Suaple algaitum, We bave token a2 our mensare of bing entered,

Fig) — max TIT, 9,8 + TAT, B,| = axe,

ae Ai

seks
SSuppcer wo etere at p anal tat the aren, Rip) — Ry ie wtburable
ta paramoier Fie, Af = &, Let ws ell py ahe "x0et weal”
swe conser pasanota: fe and! corapute the sew stror Bip’) = F. Nate
thet eneaing py msuree ey 0, bus the intareepes fr the other neem
feces een, end robstly wil, etange, LE ie os har B, pr mores
four new starting pon: with cavor Band we thes eter ins neue
‘which i the corent ween fend, {It etme bo 7 ner pe oon
feo.) THB" not lea han B, fe meane ths: venterng pe ve alloted
toler eres enough Ie cio © eget ernn, We thor mp Fe
bk hntfruy Telus oy pron satae aed ile vigil value, and
gin porate hs sr. This pros sontine unl lower error bas
Hen foul rs gc ay Ten sng seven ten ab wie,
int sho cor is wceeied al te algoslhusvtaais rer, Note shat i
Ui pes the worst eer wil ot be paainearb

"The wit Tox ee interned rrp pen ell
onsvnue (Byperimertal Cernig Progearo an sone eeslte 2. bo
‘given in Seotion TY. Nu claim are mame wba eonvengeney of the
lorichm:; rater, ite erent live i te simplicity.

a Speiiying Tolorncee
‘Typical approaches t specifying toleaness in the past have been
cither to st toleanecs rouealy snvereels proportional to Hrs-ovdcr

<Wr bre cemed yuma ia de yay di: tet fa Tie
{hiaal-ar low ave ould era) proporsanal en fae
SARE Cuan SSRIS a

sensitivities or to seh then to the tightest, qvalable, The former ap~
prose’ cecuires line appreaisiatioas while the later can Be Unnors-
fatty exgensive The iseente an performace ents provide 0
Thsigner ah idorracion bout Live far parameters oa: deviete and
Stay wii Ry. This i wht m cesigver nally noeds to better specify
falerances, Furthermore, 100 cereal vied e desied, the beeps
fue eowlrs provide upper beats sl palviee constants cee

tively on feasible pareanetor tolerances Tais Lat has been nie 20
hap solve version ofthe ein ms nezsace speciation probe
fem; the metho and reslee are preted 9 anocher paper in this

4 Pavaetor Correlation with Resp tothe Peformance Specifeations

Consider again the perlurmanee cortonr ehewn in Fig. 8 for the
sultye divider example, The shape is an indiewton {hat the two
store are “eargeates” wilh rspee! co the doen specifications
re juratans ace qlitatively deo! be cooled with respect
to the performance speitatien the resion af seecpsbiliy for one
peranrter (specie yy ite imezecp) yond sxeney on che vale
Biche second paravecer T the contour wre retanguler eng parallel
tivthe as, {ae ory parwnete'e ond he uncorelAted, This sereletion
Safortalinn ca be met a a

Fist resign tea be evehuntes, oy detevnng whee poranetes
ceoreelata wit tepnct to inn perfersanen speeitcationsw eonsisent
Soh staishoa peraratereonslabiurs. A sigh oud be Saeosiive
{ota in ays should take edvuctage 0° knows atatieal eorrelae
tion for example, it nul be serirable if Ry gad He in tia vollagy
sider exemple tracked ach other (ar emul be ebosts to track)
Seeause of mianaciee and/or uavinimment, Tn sdaiten, at 38 sue
ested thas if one i iwedtigating design with ettiten! correlation
Tretween mony paraetes # migit be wlvantegcous to find intercepts
thd contours ‘or fhe independent aad eoralnton determining random
variables sino hey are really Lhe cesign parameters

‘Second, probleins of specifying tuning wr matching are inberertly
linked to {he tolerance spesitestion prablen; parameter corelation
sith reapest eo perm sce spetifientions ian indieasion that xing
Gr nstthing igh: be sesiree. Tn alten, the rontoat:aformetion
ten von howto rach paraiesr¥s ang or ae pees lo which
to tune, The ease of turing, fo enorme wie e oaninan slong
the “long” ide of he convo sche vieron to whic Uo vine. Tn the
Crvvaqe divide exem.2 sho wink? he 236 transfer foneion seca
1218 nam aguy avereNe mrevoeuT comma, APRIL tT

tion, Por the eso of mtd, eorrinvon oun ino th she pats
foresee speeifentions dezend strongly an 0” are moet sant t, &
pastiolareombinscion oF the co parzmerats such ce cei ratio. Prop-
fri of parmanos entity slic “vlate eeteis comes cents
shane te pareulue centiatione of Ue sp parieten are given
Useatens However, Ge of the trons icportest rapes aw sated
Jn he Anpendix. Design sssmders pmanmahly would be matehea
ecoring to the partitler eortinusion. Tor easmple, Lie eoalour for
the voltage divider Hes alang era eontains the 43° line nd s0 ane
should consider matehing te rio of Ue (we resistor to heir naninal
tio (see Section A.2. Finally is should be pointed out that ihe eon
tou information ean he is Lo suse sequel tuning oF matehing
procedures since contours for tn! or mee devin might sage
forther tuning or matching

6 Computation of Prfrmanes Contours

‘A programm, comsou, has been written to oompuie interests and
zerformance wort lon ane suppligt subout'hes wih empate
design performence. The proarara hos boez wean for aa inteactive
(CDC 800 feilty. Uline teape aapleya of “he contour: as well as
Galeoms plete ate options sich eampcement priser otipat, A
Monte Casto prograr::o estate pel ie aleo part of cho entire pack
fun. Tus, «sor can very imszadiabaly whether any eaangee ade
based or contours or interecpre de, ‘a fac, inerouse the yiold, The
choice cf parameter pate for which performance con:oure exe co be
campated sup othe nse.

‘The interoente are found via a seares and the contours are found vie
1 performanee eontour following algorithm. Sinee many performance
valuation ate neerseary, the epee of computation depend eteongly
fn the time required for @ eevull analysis, For the example to be
preven, « general purpase analyse routine fo lade natok wae
tee to anelye the rot nd te nonganatina of performance con=
tour typeally tole ox the osder of afew serones, No advansage mas
tween of the fact thee coring che soaveh for itzzeapts or sexzputetion
of m eoxlour, ony one or a siveit parameter values ane changed
Setiveen eat enalsce

41 Petimivany Remeres
a the Jet sation we indies ced that large ebange eonitvity
ation cold heed deanetin an inal sign, to help eect