FROM SPECIFICATION TO MEASUREMENT: THE BOTTLENECK IN ANALOG INDUSTRIAL TESTING

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1 FROM SPECIFICATION TO MEASUREMENT: THE BOTTLENECK IN ANALOG INDUSTRIAL TESTING R.J. va Rijsige, A.A.R.M. Haggeburg, C. e Vries Philips Compoets Busiess Uit Cosumer IC Gerstweg 2, 6534 AE Nijmege The Netherlas H. Walliga Uiversity of Twete Fac. of Electrical Egieerig Postbox 217, 7500 AE Eschee The Netherlas Abstract The traslatio of the specificatio of a aalog evice ito the ecessary set of measuremets to be carrie out by a iustrial test facility, is iscusse. Algorithms are evelope to compute the umber of testvectors eee to guaratee a certai parameter a to compare several possible testmethos, base o accuracy. A importat iput for these algorithms: the measuremet error is also iscusse. 1 Itrouctio A importat part of the evelopmet of a Itegrate Circuit (IC) for mass-prouctio, is the evelopmet of a Iustrial Test Facility (ITF). The task of the ITF is to guaratee the correct fuctioig of a prouct. Ba proucts must be rejecte by the ITF. For aalog circuits, i orer to fulfil this task, the prouct is compare with the specificatio of the IC. Therefore the evelopmet of the ITF is base o the specificatio of the IC. The traslatio of a specificatio ito the ecessary set of measuremets, to be carrie out by the ITF, is a very eicate task. This is the mai subject of this paper. If we take a look at several specificatios of aalog circuits, we istiguish two kis of specificatios: 7. Fuctioal Specificatio: this part escribes the mai fuctio of the IC. Testig this part is oe very straightforwar. A example is the test if a oscillator oscillates, regarless of the frequecy. 2. Parameter Specificatio: this part escribes the ratigs for several parameters of the IC. Opposite to the fuctioal testig, for parameter testig the accuracy of the test is of high importace. Because the mai testig problems arise for parameter specificatios, the fuctioal specificatio will ot be further iscusse. The accuracy of the parameter specificatio is efie by the measuremet error. Therefore a trae-off betwee test methos ca be mae base o accuracy. The parameter specificatio ca agai be split ito two classes: 7. Sigle Parameter Specificatio: for such a parameter, all the coitios for the measuremets, like iput voltage a -frequecy, temperature, etc. have a preefie value. A example is the specificatio of the DC-gai of a amplifier at 25 C a 10mV iput voltage. 2. Iterval Parameter Specificatio: here, at least oe of the coitios is varyig: for example, the specificatio of the maximum DC-gai eviatio of a amplifier for a rage of iputvoltages at 25OC (liearity). Because the ITF ca oly execute sigle parameter measuremets, the iterval parameter specificatio must be traslate ito several sigle parameter specificatios. This is iscusse i sectio 2. Sectio 3 escribes a metho to compare several testmethos base o accuracy. The basis to etermie the accuracy, the measuremet error will be looke at i sectio 4. The fial coclusios are give i sectio 5. 2 From iterval to Darameter sdecificatio Cosier the measuremet of a liearity factor. I this case, there is a relatioship like: Here, x is the parameter which ca be applie, CH2910-6/0000/0177$01.OO'C 90 IEEE 90 Iteratioal Test Coferece 177

2 a is a arbitrary costat, y is the parameter which ca be measure a p is the parameter which has to be fixe. For several x values it is possible to measure the value of y a compute a value for p. This is illustrate by figure 1. the mea value of y. It is quite logical that a icrease i the umber of testvectors, results i a ecrease of the with of the cofiece iterval. Aother ifluece o the with of the iterval is the measuremet error. If the measuremet error is small, the with of the cofiece iterval will be small. If we take both iflueces ito accout, the cofiece iterval is give by the equatio [l]: b-p = t(-2;a/2) var(b) (3), with t: stuets-t istributio with -2 egrees of freeom a a the loo(l-a)% cofiece iterval for the error i b. The variace of b is calculate from the estimate for p: X- Figure I I this case three poits (1 to 3) are take from a liear curve. Due to the measuremet error, the real value for y will lie somewhere betwee the two ots. The error i the supplie variable x, is eglecte compare to the error i yi. The agle of the curve, which is a measure for p, ca vary betwee $1 a h. If the umber of poits, or testvectors, is icrease, the ucertaity i the preicte curve, which, i this case etermies the liearity error (Ap), is ecrease. Because a o-liear relatio ca easely be trasforme ito a liear oe, the restrictio to a liear relatioship is ot a fuametal oe. For every set of values x, a measure values yi, oe gets a estimate of f3 (writte ow as b). The real value of p will be somewhere ear this estimate value. It is possible to erive a cofiece iterval for the value of p, i.e. this iterval will, with a certai cofiece, for istace 90% certaity, cotai the real value of p. This iterval ca be compute, usig the estimate of p belogig to some set of testvectors (x,,y,). This is escribe by: c (xi-&, /3 + (4). x Wi-X) So, the estimate b is equal to the value of p plus a error compoet. We assume all errors to be equal ormal istribute as N(O,oC2+ou2), with oc2 the variace part which is fully correlate to the other errors, a ou2 the variace part which is totally ucorrelate. Because we oly look at the ifferece betwee two measuremets, the systematic error is elimiate a the variace of b is: x (xi-&, var@) = var ( )= i Wi-X)2 with E, the measuremet error a py 178

3 If we take equal space values of x i the iterval (x, I ", Xm, x) a efie R = IXm -Xm, 1 so ~xl-xl~l 1 = R/(-1) the: umber of measuremets K will become zero. I table 1 some values of 6 with the correspoig values of are give for a 1% relative error i fl, with 99% cofiece. If the relative measuremet error is small, oly three testvectors are eee to compute the value of fl with 1% accuracy. If the measuremet error is relative large, more testvectors are eee. Coclusio: with L = /2 for =eve; = (-1)/2 for =o. The measuremet error U, is i geeral proportioal to the rage which has to be measure, a efie as a percetage (S) of this maximum. Because the rage of y is equal to b*r, this ca be expresse as: ou2 = (SbR)2 (7). If we combie 3,5,6 a 7 we'll get: Ab/b = t(-2;a/2) SK (8)s with K = -l { Z ( +l - i )2 }-l/' 42 2 TABLE 1 Number of measuremets versus the variace of the measuremet error. -- K t(-2;0.005) S 0.01? '/o 0.14% 0.1 8% 0.22% 0.25% 0.28% 0.30% The relative error i the estimate of p epes oly o: the relative error of the measuremet (S) a the umber of measuremet poits (). A icrease i the umber of measuremets will result i a ecrease of K. For a ifiite Regarless of the ature of x a y, it is possible to make a trae-off betwee the accuracy of the measuremet, the umber of test vectors (x,y) a the relative error i the result (Ablb). 3 ComDariso of test methos After the traslatio of the iterval parameters, a testspecificatio of strictly sigle parameter specificatios remais. Now, the specific testmethos for every parameter have to be efie. Because there are may ways to test a specific parameter, a selectio betwee the testmethos must be mae. I this paragraph, a selectio metho base o accuracy is propose. The moel, use to represet the relatio betwee the measuremet a the specifie parameter is base o the moel iscusse i [2]. The moificatio is mae for two reasos: first of all, with the ew moel it is possible to have multiple measuremets for the preictio of oe parameter. This is the case for iterval specificatios, but also for relative parameters like gai. To measure the gai of a amplifier, two measuremets are eee to efie oe parameter (oe for the referece level a oe for the output level). Secoly, correlatio betwee measuremet errors is take ito accout. The propose moel is: S,, = *,/ax, a: Ap, : eviatio i the specifie parameter. AxJ : eviatio i the measure parameter. e, : measuremet error i the parameter 'J ' SI, : relatio betwee the specifie parametervalue pi a the measure parametervalue xj To select the best testmethos, the eviatio of the measure variable Ax is put to zero. I matrix otatio we'll get: 179

4 Ap = S e (1 0) The variace of the iaccuracy Ap is equal to: I[ up2] = S E,, c*st (11) I : I x I ietitymatrix; I is equal to the umber of specifie parameters. S : the matrix of the elemets SI,. ST : the traspose of the matrix S. E,, : the variace-covariace matrix of the measuremet errors e,. The mai target is to miimize the total variace give by: I To compare ifferet testmethos (11) ca be compute for every possible combiatio of the measure parameters. Because this is computatioal iefficiet, a selectiocriterio is erive. Let (13) represet formula (11) for the measure parameters x1... Xk,Xk+l... XJ a (14) for the measure parameters x 1... x ~,xk+l... XJ with LIK: I[ up2] = [ x c ] [ P Q ] [ ( ~)~] QT R 1 [ (WT1 (13) I[ up2] = [ x c ] [ S T ] [ (X )T] TT R 1 (c IT] (14) x : the matrix of the elemets &I, /3x, with 1 IjIK; c : the matrix of the with K+l IjIJ; x : the matrix of the with 1 IjIL; P : the variace - covariace submatrix of e, with 1IjIK; Q : the covariace submatrix of (e,,e,) with 1 IjIK a K+l IiIJ; R : the variace - covariace submatrix of e, with K+l IjIJ; S : the variace - covariace submatrix of e, with 1IjIL; T : the covariace submatrix of (e,,e,) with 1 IjlL a K+l IilJ; The criterio for miimal variace is give by: Usig some staar matrix operatios, the followig equatio is erive from (13), (14) a (1 5): 1 D = 1 [x]p[xit + 2* 1 [x]q[cit - { [x ]S[x IT + 2* 1 [x ]T[cIT } with 1 the summatio of the iagoal elemets of the resultig matrix. (1 6) The first sum cotais the cotributio of the first k parameter measuremet errors to the variace of the errors op2, aalog to (11). The seco sum gives the cotributio of the iteractio betwee these errors a the measuremet errors of the parameters equal for both sets. The term betwee brackets gives this cotributio but ow for the alterative parameter set with the parameters x ~... x ~. The selectiocriterio is: CDc?>O a a coclusio whether it is avatageous to measure x1... xk istea of xfl... X L ca be raw. The efficiecy of this algorithm epes o the ratio betwee the total umber of measure parameters J a the umber of parameters which are ifferet i both sets: K. The reuctio i the total umber of multiplicatios eee to compute D compare to the computatio of (11) irectly, with L=K is equal to: J+l Reuctio == This algorithm is meat to be use i a CAD eviromet. For a reliable output, the variace - covariace matrix of the measuremet error is eee. This aspect will be iscusse i the ext sectio. 4 The measuremet error To fill i the variace - covariace matrix of the measuremet errors, kowlege of the systematic a stochastic error compoets of 180

5 a measuremet is eee. The relatio betwee the variace a covariace elemets a these error compoets are: Because the total testsystem is very complex, the limits for the error compoets have to be set by experimetal ata. If we test several proucts o several ietical testsystems a preictio for the variaces a covariaces of the error compoets ca be mae. The mai questio ivolves the relatio betwee the reliability of the estimate of the variace i relatio to the complexity of the experimet. There are two separate cases which will be iscusse i the ext sectios. 4.1 Eaual error behavior for all testsvstems a Droucts First of all, we assume o sigificat ifferece betwee several testsystems, a o ifluece of the prouct o the error behavior of the measuremet. With this assumptio, it is possible to erive a loo(1 -a)"/o cofiece iterval for the variace of the measuremet error by testig oe prouct o oe testsystem several times. If a relative measuremet is mae, the systematic error compoet is elimiate, a a preictio of Ustoch2 ca be mae. For a absolute measuremet, the variace cosists of both stochastic a systematic errors. If we measure the prouct N times, the sample variace ca be calculate as [l]: 4.2 Noeaual error behavior betwee testsystems a proucts If there is a sigificat ifluece of the prouct a the choice of the testsystem o the error behavior, the followig moel for the measuremet of oe parameter ca be use: xij = p + a, +e,, (22L p : mea value of the parameter x. a, : ifluece of testsystem i; i E [l..i] e, : ifluece of prouct j o testsystem i; j E [l..,] Because a, is iepeet of e, J, the variace of the parameter x is: ux2 = ua2 + 0,2 (23 Here, ua2 represets the variace ue to the testsystem a ue2 the variace ue to the prouctiosprea of the process. I [3] a cofiece iterval for the ratio b=ua2/ue2 for this moel is erive: g, =, / (b, + 1) Sx2 = { C (Xi-k)2 }/(N-1) (20)s -, x, = CX,,/, j=l a the cofiece limits for the variace are [I]: a2 < (-1)SV2 I (-1)SX2 x2(1-2,n) x2(2,n) (21) with ~2(a,N) the ~2-istributio with egrees of freeom. Ufortuately, i most practical cases the assumptio of o ifluece from testsystem a prouct o the measuremet error is violate. I this case the seco metho must be use to erive a cofiece iterval for the variace of the measuremet error. x= c I g, I = C, The solutio of (24) for fp=fl gives the lower bou o b. The upper bou is give by the solutio of (24) for fp=fh. The values of f are fou usig the F-test table with 1-1 a -l egrees of freeom a a cofiece level a: F(fl,I-1,-I)=a/2, F(fh,I-1,-l)=l-a/2. If, =J =a equatio (24) ca be simplifie: 181

6 5 Coclusios To summarize this work, some fial coclusios are give: E -- a *-l * b*a+l I - 1 I a I -x) E j=1 c (XI, -xy -- a * MSa = f, (25) b a+l MSe MSa:. Mea Square of a a MSe: Mea Square of e, F(fl,I-1,(a-l)I)=a/2, F(fh,[-I,(a-l)l)=l-. The with of the cofiece iterval for b ca be calculate: The ratio of mea squares is the estimate for b. I table 2 the value of (l/fl-l/fh) is give versus the umber of testsystems (I) a the umber of proucts (a) for a 90% cofiece level (a=o.lo). As we ca see, for a reliable variace preictio, at least 10 testsystems must be ivolve i the experimet. This may cause practical problems The specificatio of a circuit ca be trasforme ito sigle parameter measuremets, to be carrie out by a iustrial test facility. There is a trae-off betwee the umber of measuremets a the accuracy of the specifie parameter. 2. A computatioal efficiet trae-off betwee testmethos base o maximum accuracy ca be mae. 3. Oe of the aspects eee to make these traeoffs: the measuremet error ca be preicte usig oe of the propose experimets. Here, agai there is a trae-off betwee the complexity of the experimet a the accuracy of the error preictio. 4. To preict the variace of the measuremet error, a rather complex experimet must be carrie out i orer to get a acceptable cofiece iterval for the estimate variace. Refereces [l] C.P. Cox, A habook of itrouctory statistical methos, 87, Joh Wiley & Sos, Ic., ISBN [2] G.J. Hemik, B.W. Meijer, H.G. Kerkhoff, Taste, a tool for aalog system testability evaluatio, Proceeigs of the iteratioal test coferece, 88, pp [3] L.R. Veroore, Statistical iterferece o variace compoets, Agricultural uiversity of Wageige, 88. a=2 a=3 a=4 a=10 a=15 T I=3 1=4 I=5 I=6 1= [=lo [= BLE 2 The with of the cofiece iterva 182