1.Estimation of test metrics for the optimisation of analogue circuit testing 1 Ahc`ene Bounceur a Salvador Mir a Emmanuel Simeu a Luis Rolndez a,b aTIMA Laboratory, RMS Group 46, Av. Felix Viallet, 38031 Grenoble Cedex, France. bSTMicroelectronics 850, rue Jean Monnet, 38926 Crolles, France. Abstract The estimation of test metrics such as defect level, test yield or yield loss is important in order to quantify the quality and cost of a test approach. For design-for-test purposes (DFT), this is important in order to select the best test measurements but this must be done at the design stage, before production test data is made available. In the analogue domain, previous works have considered the estimation of these metrics for the case of single faults, either catastrophic or parametric. The consideration of single parametric faults is sensitive for a production test technique if the design is robust. However, in the case that production test limits are tight, test escapes resulting from multiple parametric deviations may become important. In addition, aging mechanisms result in eld failures that are often caused by multiple parametric deviations. In this paper, we will consider the estimation of analogue test metrics under the presence of multiple parametric deviations (or process deviations) and under the presence of faults. A statistical model of a circuit is used for setting test limits under process deviations as a trade-o between test metrics calculated at the design stage. This model is obtained from a Monte Carlo circuit simulation, assuming Gaussian Probability Density Functions (PDFs) for the parameter and performance deviations. After setting the test limits considering process deviations, the test metrics are calculated under the presence of catastrophic and parametric single faults for dierent potential test measurements. We will illustrate the technique for the case of a fully dierential operational amplier, proving the validity in the case of this circuit of the Gaussian PDF. Key words: fault simulation, catastrophic and parametric faults, process deviations, statistical modeling. 1. Introduction The test of integrated analogue and mixed-signal circuits is signicantly dierent from the test of dig- ital circuits. The major dierence stems from the need to consider continuous signals and circuit para- Email addresses: Ahcene.Bounceur@imag.fr (Ahc`ene Bounceur), Salvador.Mir@imag.fr (Salvador Mir), Emmanuel.Simeu@imag.fr (Emmanuel Simeu), Luis.Rolindez@imag.fr (Luis Rolndez). 1 This work has been carried out in the frame of the Euro- pean MEDEA+ project NanoTest. metric deviations, in addition to just catastrophic faults (opens and shorts). For digital circuits, struc- tural testing has provided cost ecient solutions that target the test of catastrophic faults rather than the test of the circuit functionality. Thus, fault cov- erage is the major test metric in this domain, and is somehow independent from the specications. For analogue circuits, the need to consider parametric deviations has lead to the denition of analogue test metrics that take into account also the circuit func- tionality. In other words, even when a parametric fault-based test approach is considered for analogue

2. circuits, test metrics such as fault coverage cannot be calculated without knowing the performance speci- cations [1]. The case of parametric faults has been considered by many authors by simply modifying the nominal values of a design parameter, and considering Monte Carlo simulations. In this way, parametric fault lists have been built in a rather arbitrary way. [1] intro- duced a dierent way of dening parametric faults. A parametric fault is considered as the minimum de- viation of a design parameter that results in a circuit specication being violated. In this approach, para- metric faults are obtained by transient simulations, without recurring to time consuming Monte Carlo simulations. This approach is quite acceptable when faults are considered the result of a single parame- ter deviation, while the other parameters remain at their nominal values. However, it cannot deal prop- erly with the case of device misbehaviour resulting from the combination of multiple small deviations. An early approach to avoid Monte Carlo simula- tion was based on the use of sensitivity analysis to deterministically identify the bounds on circuit pa- rameters [2]. Process information and the sensitiv- ity of the circuit principal components have been re- cently considered in [3] for generating the statistical models of the fault-free and faulty circuits, which is then used for test vector generation. These mod- els are obtained using a statistical approach and a linear estimation, rather than Monte Carlo simula- tions, which can lack accuracy in the case of large parametric deviations. Another statistical approach is considered in [4]. Here, however, parametric faults are injected by swapping transistors, one at a time, by a transistor whose process parameters are shifted by 3 and a sensitivity analysis is performed only in the DC domain. The problem with this approach is again that the misbehaviour resulting from the combination of multiple small deviations cannot be evaluated properly. The consideration of single parametric faults is sensitive for a production test technique if the de- sign is robust. However, in the case that produc- tion test limits are tight, test escapes resulting from multiple parametric deviations become important. In addition, aging mechanisms result in eld failures that are often caused by multiple parametric devia- tions. Thus, for the case of self-test techniques that are supposed to be used during an actual applica- tion (i.e. in the eld), it is mandatory to evaluate the quality of the test approach considering multiple parameter deviations. In this work, we are aiming at dening a proce- dure for optimising test measurements at the de- sign stage, based on the estimation of test metrics. We will introduce a new technique to estimate these metrics for the case of multiple parametric devia- tions, that we will consider the result of process de- viations. The estimation of these metrics will allow us to set test limits for the potential test measure- ments. These test limits will then be used for the evaluation of test metrics under catastrophic and parametric faults. This will be illustrated for the evaluation of a test approach for a fully dierential operational amplier. 2. Test metrics theoretical computation In this Section we will present the dierent test metrics and their theoretical calculation. Next Sec- tion will introduce their practical estimation. 2.1. Denition of test metrics and other terms The test metrics considered for analogue circuits are [1]: Yield Y , Test Yield YT , Yield Loss YL, Yield Coverage YC and Defect Level D where: Y = Proportion of the functional (or good) circuits = P(circuit is functional) YT = Proportion of the circuits that pass the test = P(circuit passes the test) YC = Proportion of the pass circuits that are functional = P(circuit passes the test/is functional) YL = Proportion of the fail circuits that are functional = 1 YC D = Proportion of the faulty circuits that pass the test = 1 P(circuit is functional/passes the test) where a functional circuit is the one for which all its performances are inside their specications and a faulty circuit is the one for which at least one of its specications is violated. The denition of parametric fault coverage will be detailed later. For catastrophic faults, as mentioned earlier, device functionality is not considered and fault coverage is just dened as the ratio of detected faults with respect to the total number of injected faults. We dene next other basic terms: 2 3. Performances: they describe the functioning of a circuit. Knowing the circuit performances allow to decide if the circuit is functional or faulty. Specications: are the set of acceptable values of each performance. Test criteria: they are the measures which are done on the circuit under test. They allow decid- ing if the circuit passes or fails the test. Process parameters: parameters concerning the fabrication process and the physical components. Test parameters: called also test criteria. 2.2. Test metrics theoretical computation Assume that we have n performances and m test criteria. Let A = (A1, A2, , An) be the set of the specications of the performances and B = (B1, B2, , Bm) the test limits (intervals of the accepted values of the test criteria). The test metrics are then calculated theoretically as follows: Y = A fS(s) ds (1) YT = B fT (t) dt (2) YC = A B fST (s, t) ds dt Y (3) D = 1 A B fST (s, t) ds dt YT (4) where, fS(s) = fS(s1, s2, , sn) is the joint probability density of the performances, fT (t) = fT (t1, t2, , tm) is the joint probability density of the test criteria, and, fST (s, t) = fST (s1, s2, , sn, t1, t2, , tm) is the joint probability density of the performances and the test criteria. 2.3. Test metrics computation under Gaussian hypothesis Given a vector X = (X1, X2, ..., Xp)T composed of random variables, where Xj for j = 1, 2, ..., p, is a one-dimensional random variable, the covariance of Xi and Xj is a measure of dependency between these random variables and is dened by: XiXj = Cov(Xi, Xj) = E(XiXj) E(Xi)E(Xj) (5) where E(.) denotes the expected value. If Xi and Xj are independent of each other, the covariance XiXj is necessarily equal to zero. The converse is not true. The covariance of a random variable Xi with itself is the variance: XiXi = Cov(Xi, Xi) = Xi (6) The correlation between two variables Xi and Xj is dened from the covariance as follows: XiXj = XiXj Xi Xj (7) where the standard deviation is dened by Xi = Xi The advantage of the correlation is that it is inde- pendent of the scale, i.e., changing the scale of mea- surement of the variables does not change the value of the correlation. Therefore, the correlation is more useful as a measure of association between two ran- dom variables than the covariance. The correlation is in absolute value always less than 1, close to zero if the random variables Xi and Xj are independent of each other. An empirical estimation of these quantities re- quire a number of observations. Suppose that {xi}n i=1 is a set of n observations of a variable vec- tor X in p . Each observation xi has p dimensions: xi = (xi1 , xi2 , ..., xip ), and it corresponds to an observed value of a variable vector X p . The co- variance of two random variables is then estimated as: VXiXj = 1 n 1 n k=1 xik xjk nxi xj (8) and the variance of a random variable is estimated as: VXi = 1 n 1 n k=1 x2 ik nx2 i (9) The correlation of two random variables is then given by: rXiXj = VXiXj sXi sXj (10) with sXi = VXi . The theoretical covariances among all the random variables can be put into matrix form, i.e. the co- variance matrix: = X1 X1Xp ... ... ... X1Xp Xp (11) 3 4. The estimated (empirical) version of the covari- ance matrix is then given by: S = VX1 VX1Xp ... ... ... VX1Xp VXp (12) Let X be a p-dimension random variable of ex- pected value = (1, 2, ..., p)T and covariance matrix . If X has a multinormal distribution, then X has a probability density function f(x) dened by: f(x) = 1 det(2) exp (x )T 1 (x ) 2 (13) The probability of any subset A p is given by the following multiple integration formula: P(A) = 1 det(2) A1 Ap exp (x )T 1 (x ) 2 dx1 dxp (14) Thus, using the multinormal hypothesis, it is pos- sible to derive the actual probability density func- tions which must be integrated considering the ac- tual boundaries of the random variables in order to compute the test metrics. 2.4. Validation of the Gaussian hypothesis 2.4.1. Classical tests A graphical method can be used to validate that the distribution of the performances and the test criteria is multinormal. This requires to nd graph- ically a tted Gaussian for each performance and each test criterion. It is also possible to use other tests [8] such as the Kolmogorov-Smirnov (KS) test. The KS test compares the empirical distribution function with the cumulative distribution function specied by the null hypothesis. These tests are use- ful for the one-dimensional case. For the multidi- mensional case, we use the following method. 2.4.2. Correlation factors For a multinormal distribution of a set of random variables, it is known that a linear transformation of these variables results in a change of standard devi- ation and a new multinormal distribution while the correlation factors between the random variables re- main constant. It is possible to observe this in our case by increasing the standard deviation of the pro- cess parameters directly in the Design Kit. Thus, for all process parameters, their standard deviation may be increased a factor over their initial deviation. As a result, the standard deviation of performances and test criteria will also change. For example, Fig- ure 1 shows the resulting increase of the standard deviation of a circuit performance. For a xed inter- val of the specications, when the original Gaussian of the performances is changed to an increased one, this gure shows clearly that the probability to gen- erate faulty circuits is greater than the rst one. We can then experimentally see for dierent val- ues of the factor (each value requiring a Monte Carlo circuit simulation) that the correlation fac- tors remain constant up to a certain limit for which the linear transformation is no longer valid. Devia- tions beyond these limits may be considered as sin- gle faults if their probability of occurrence is low enough. In this work, we have applied this technique since it shows that, for our case-study, parametric devia- tions can be rather large while the correlation fac- tors are still constant. This indicates not only that the Gaussian assumption holds, but also that the test limits set under process deviations are valid for multiple and rather large parametric deviations. As a consequence, it justies that we can consider single parametric faults for the evaluation of test metrics under the presence of faults. 2 = ' 1 1 sa1 Original Distribution New Distribution fS (s) Fault-free circuits Faulty circuits b1 gS (s) Fig. 1. Example of an increased performance. 3. Test metrics estimation 3.1. Under process deviations Assuming that the joint probability of the perfor- mances and the test criteria is multinormal. We can use a Monte Carlo circuit simulation to calculate the statistical parameters of a multinormal law (mean and covariance matrix). 4 5. When the number of the specications and test criteria is important (more than 3), the number of integrals for the exact computation of test metrics is too large. The larger the number of integrals, the lower the precision of the computation. To overcome this problem, as the joint pdf of performances and test criteria is assumed multinormal, a simple pro- gram implemented in Matlab (or R) is used to gen- erate about one million 2 instances from the multi- normal distribution using a Monte Carlo technique. Next, the test metrics can be directly estimated us- ing the following estimators: Y D = Number of functional circuits N (15) Y D T = Number of pass circuits N (16) Y D L = Number of fail functional circuits Number of functional circuits (17) DD = Number of pass faulty circuits Number of pass circuits (18) where N is the number of generated circuits. We use the index D to indicate that the metrics are estimated at the design stage using process de- viations. 3.2. Under the presence of faults In this case, we consider only single parametric faults in physical parameters where the test met- rics are calculated using the methodology presented by [1] where partial detectability of the faults is con- sidered. A fault is dened as the minimum value of the process parameter i that causes any perfor- mance specication to fail. This will help to calcu- late the probability of the occurrence pspec i of this fault, which represents the probability that the value of this process parameter is greater than vspec i (Fig- ure 2). We calculate also the probability ptest i to detect a fault in this parameter, which represents the prob- ability of this parameter to be greater than vtest i , where vtest i is the minimum value of the process pa- rameter that causes the test criteria to fail (Fig- ure 2). Considering theses denitions, we can write the analogue test metrics, based on the parameters pspec i and ptest i as follows [1][7]: 2 This takes about 10 seconds using the processor Intel Pentium M, 1.73GHz, memory of 1Go. P(i) pi test vi spec vi test pi spec i Fig. 2. Distribution of a parameter i. Y F = n i=1 (1 pspec i ) (19) Y F T = m j=1 (1 ptest j ) (20) Y F C = GF P Y F = 1 Y F L (21) DF = 1 GF P Y F T (22) where, GF P = n i=1(1max(pspec i , ptest i )), which rep- resents the probability that a circuit is functional and passes the test. The fault coverage F for the parametric faults is calculated using the following equation [1]: FF = n i=1 ln(1 min(pspec i , ptest i )) n i=1 ln(1 pspec i ) (23) We use the index F to indicate that the metrics are calculated under the presence of faults. 4. The CAT platform We have developed a Computer-Aided-Test plat- form for the evaluation of test techniques for ana- logue and mixed-signal circuits [6]. The CAT plat- form, integrated in the Cadence Design Framework Environment, includes tools for fault simulation, test generation and test optimisation for these types of circuits. Fault modeling and fault injection are simulator independent. Figure 3 shows a simplied architecture of this platform. It is composed of three separate tool sets. Fault modelling, fault injection and fault simulation are carried out using the tool set FIDESIM. The results are saved in a database that can be read by the other tool sets, in particular the OPTEVAL tool set for test evaluation and the OPTEGEN tool set for test generation. 5 6. Cadence Database FIDESIM OPTEVAL OPTEGEN Database Optimization Algorithms Results Test Vectors Fault Simulation Test Evaluation Test Generation Fig. 3. Simplied architecture of the CAT platform. Vin + o Vbiasp Vin - o o o o o o o Vbiasp Vcasp Vcasn VCM Vout + Vout - o M2 M1A 200x2.7/2 M1B 200x2.7/2 VDD GND M3A M3B140/2 M4B40/3.2 M4A M5B40x2.7/3.2 M5A M6A 40x2.7/2 M6B 0.2pF 0.3pF 0.3pF0.2pF0.2pF oo VZ Vout + Vout - M7A 10x2.7/1 o Vbiasp M8A 100x2.7/1 M8B M7B R R=3k V1 V2 oVana 0.9V M7C o Vbiasp M8C VREF M10 M9A 60x2.7/1 M9B 50x2.7/1 VZVREF M11BM11A 30/1 M12 M13 o Vbiasp 7/1.6 7/1.6 o VCM M M1BIAS 40/1 RBIAS M5BIAS 10/1 M2BIAS 10/1.6 M6BIAS M3BIAS 20/1.6 M4BIAS 20/1 M8BIAS M7BIAS o Vbiasp M9BIAS 2.5/1.6 M12BIAS M14BIAS o Vbiasn 10/1.6 20/1.6 20/1 M10BIAS 20/1.6 M11BIAS 20/1 10/1 M13BIAS 10/1.6 5/1.6 o Vcasn o Vcasp M1ST o Vcasn M2ST 2/20 10/1 M3ST 10/1 o Vcasp M4ST 10/1 o Vbiasp Start-Up CircuitBias Circuit Amplifier Circuit Common-Mode Control Circuit Fig. 4. Folded cascode fully dierential amplier. The dimensions of each transistor W/L is expressed in multiples of the unity size transistor (WUNIT = 0.28m and LUNIT = 0.18m). 5. Test vehicle The test vehicle is a fully-dierential operational amplier. This amplier has been designed in a 0.18m CMOS technology from STMicroelectron- ics. The amplier is formed of four main blocks: a bias circuit, a start-up circuit, a common-mode control circuit and a dierential amplier circuit. Figure 4 illustrates this circuit. First, we use the statistical analysis to calculate the test limits for the test criteria under process de- viations by calculating the analogue test metrics, in particular the defect level and the yield loss at the design stage. Then, taking into account these test limits, we calculate the fault coverage in order to test the capability of the test technique for fault detec- tion considering both catastrophic and parametric faults. Finally, we will nd the minimal set of spec- ications and test criteria which give the best fault coverage. In order to nd the dierent tted Gaussian dis- tributions of each circuit performance and test cri- teria we performed a Monte Carlo circuit simulation (1000 iterations). The comprehensive set of perfor- mances and test criteria considered is given in Ta- bles 1 and 2, where a1 and a2 represent the speci- cations, that is, the bounds of each performance. The specications of the amplier are not known a priori, since the actual system application of the de- vice is not considered in this work. Thus, we have set ourselves the specication bounds in order to have a high yield at the design stage of Y D = 99.99% when all performances are considered. This requires a tolerance interval of 4.3 for each performance. The test limits of the test criteria will be calculated using the technique which is presented below. Dierent test benches have been used to calcu- late the dierent performances. For example, Fig- ures 5(a,b) show the test benches n 1 and n 7, re- spectively. Each test bench allows the calculation of one or more performances. Table 1 shows the speci- cations with the actual test bench used for the cal- culation. Eight dierent test benches are required for the performances. For the actual test of the fully dierential ampli- er, the measurement of the SNDR 3 of the ampli- er is considered using a sine-wave tting technique described in [5]. The DC Oset of the amplier is also considered as a possible test criterion that can be measured using the sine-wave tting technique. To simulate these test measurements, an additional test bench is required. The total Monte Carlo circuit simulation time of 1000 instances, considering the 8 test benches for the performances and the test bench for the test criteria, 3 Signal-to-Noise and Distortion Ratio. 6 7. (a) (b) Fig. 5. (a) Example of the test bench n1 of the amplier, (b) example of the test bench n7 of the amplier. Specication Condence interval on Performance Test bench a1 a2 (condence level = 99,73%) 1. AD 76.60dB 0.493dB 74.49dB 78.71dB [76.56, 76.65] 2. GBWD 1 330MHz 18.14MHz 252.36MHz 407.64MHz [3.28 108, 3.31 108] 3. Phase Margin D (PM) 63.33 0.45 61.40 65.26 [63.26, 63.35] 4. CMRR 2 42.76dB 1.02dB 47.13dB 38.39dB [42.87, 42.68] 5. PSRR (GND) 3 29.99dB 3.65dB 45.61dB 14.37dB [30.41, 29.73] 6. PSRR (VDD) 4 28.21dB 3.75dB 44.26dB 12.16dB [28.67, 27.97] 7. THD 5 66.19dB 2.38dB 56.00dB 76.38dB [66.25, 66.76] 8. Current Cons. (IDD) 2.48mA 0.21mA 1.58mA 3.38mA [2.45 103, 2.50 103] 9. Intermodulation 6 67.57dB 1.09dB 62.90dB 72.24dB [67.53, 67.76] 10. SR + (CL = 1pF) 7 73.14V/s 5.55V/s 49.38V/s 96.88V/s [7.19 107, 7.31 107] 11. SR (CL = 1pF) 73.14V/s 5.55V/s 49.38V/s 96.88V/s [7.19 107, 7.31 107] 12. In Referred Noise 8 39.22V 0.5V 37.08V 41.36V [3.92 105, 3.93 105] (BW = 20kHz) Table 1 The performances of the amplier with their Gaussian parameters and the specications set at 4.3. Test criterion Test bench SNDR 9 68.85 dB 2.19 dB Oset 0 V 7.69 V Table 2 The test criteria of the amplier with their Gaussian parameters. is 3 hours 4 . The overall process is fully automated using the CAT platform that we have introduced in Section 4. 6. Experimental results 6.1. Validity of Gaussian hypothesis Figure 6(a) shows an example of the distribution of the Gain and Figure 6(b) shows an example of 4 SunOS machine, model SUNW, Sun-Blade-2500-S with a CPU UltraSPARC-IIIi and a memory of 2Go. the distribution of the THD 5 . As we can see, these distributions are very close to the normal one. The same results are obtained for the other performances and test criteria. Monte Carlo circuit simulations have been per- formed for dierent values of between 1 and 3, showing that the correlation factors between perfor- mances and test criteria are maintained practically constant up to a value close to 2. 5 Total Harmonic Distortion. 7 8. 74.5 75 75.5 76 76.5 77 77.5 78 78.5 5 10 15 20 25 30 35 Obtained Gaussian (1000 Monte Carlo simulations) Fitted Gaussian Spec1 :Gain 56 58 60 62 64 66 68 70 72 74 76 5 10 15 20 25 30 35 40 Obtained Gaussian (1000 Monte Carlo simulations) Fitted Gaussian Spec7 :THD (a) (b) Fig. 6. (a) The distribution of the Gain AD of the amplier, (b) the distribution of the THD of the amplier. R = AD GBW D P M CMRR P SRR(GND) P SRR(V DD) T HD IDD Intermod SR + / Noise AD GBW D P M CMRR P SRR(GND) P SRR(V DD) T HD IDD Intermod SR + / Noise 1.000000 0.712648 0.272845 0.357565 0.441979 0.497995 0.864629 0.308574 0.756356 0.783836 0.464466 0.712648 1.000000 0.387835 0.772143 0.780867 0.797906 0.955099 0.828546 0.766403 0.991466 0.062823 0.272845 0.387835 1.000000 0.184315 0.218804 0.236078 0.376087 0.247009 0.333470 0.407271 0.328108 0.357565 0.772143 0.184315 1.000000 0.971420 0.957859 0.725846 0.893844 0.528085 0.756906 0.441985 0.441979 0.780867 0.218804 0.971420 1.000000 0.997131 0.764910 0.804757 0.583147 0.778271 0.294517 0.497995 0.797906 0.236078 0.957859 0.997131 1.000000 0.796851 0.783092 0.615203 0.800639 0.232414 0.864629 0.955099 0.376087 0.725846 0.764910 0.796851 1.000000 0.711331 0.819588 0.980284 0.158534 0.308574 0.828546 0.247009 0.893844 0.804757 0.783092 0.711331 1.000000 0.498811 0.790862 0.476519 0.756356 0.766403 0.333470 0.528085 0.583147 0.615203 0.819588 0.498811 1.000000 0.795849 0.242433 0.783836 0.991466 0.407271 0.756906 0.778271 0.800639 0.980284 0.790862 0.795849 1.000000 0.099168 0.464466 0.062823 0.328108 0.441985 0.294517 0.232414 0.158534 0.476519 0.242433 0.099168 1.000000 Table 3 Correlation matrix R with absolute values. Figures 7(a-j) show the correlation between each performance and THD and Figure 8 shows the cor- relation between the test criterion SNDR and the performance THD, for the values of in the range between 1 and 3. For the case of = 1, as an exam- ple, Table 3 shows the calculated absolute correla- tion matrix for the performances dened in Table 1. As the SR+ has the same values as SR we have eliminated SR from this matrix. Thus, the matrix has 11 lines and 11 columns. Figures 7(a-j) show clearly that for a ratio 2 the multinormal assumption still holds. However, for > 2 this is not true (see for example Figure 7(i)). This corresponds to the case when physical parame- ters experiment large perturbations. In practice, the occurrence probability of such deviations will be ex- tremely low. The large deviations are properly con- sidered using single faults. Notice also in Figure 7(j) that the correlation factor always decreases slightly with the increase of the ratio . This is caused by the fact that performances THD and Noise are prac- tically not correlated. In summary, the results indi- cate that process deviations can be studied properly with the multinormal model. 6.2. Validity of the test metrics estimation under process deviations We need to nd the test limits to separate the faulty circuits from the fault-free ones, as a function of the required test metrics. A trade-o between De- fect level and Yield loss must be considered under process deviations, and this will set the actual test limit. Using the equations presented in Section 2 these test metrics at the design stage can be theoretically calculated. However, in our case, we have 12 speci- cations, and it is not feasible to perform the inte- gration with such a large number of integrals. Thus, we will use the Monte Carlo method of estimation proposed in Section 3.1. 8 9. THD , AD 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1 1,5 2 2,5 3 Abs(Correlation) THD , GBWD 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1 1,5 2 2,5 3 Abs(Correlation) (a) (b) THD , Phase Margin 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1 1,5 2 2,5 3 Abs(Correlation) THD , CMRR 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1 1,5 2 2,5 3 Abs(Correlation) (c) (d) THD , PSRR GND 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1 1,5 2 2,5 3 Abs(Correlation) THD , PSRR VDD 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1 1,5 2 2,5 3 Abs(Correlation) (e) (f) THD , IDD 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1 1,5 2 2,5 3 Abs(Correlation) THD , Intermodulation 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1 1,5 2 2,5 3 Abs(Correlation) (g) (h) THD , SR+ 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1 1,5 2 2,5 3 Abs(Correlation) THD , Noise 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1 1,5 2 2,5 3 Abs(Correlation) (i) (j) Fig. 7. Absolute Correlation between the THD and the other circuit performances as a function of the standard deviation increase ratio . 9 10. THD , SNDR 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1 1,5 2 2,5 3 Abs(Correlation) Fig. 8. Absolute Correlation between the THD and the test criterion SNDR as a function of the standard deviation increase ratio . Defect Level 0 20 40 60 80 100 120 140 160 180 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0 Test Limit ppm Monte Carlo estimation Theoretical Yield Loss 0 100000 200000 300000 400000 500000 600000 700000 800000 900000 1000000 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0 Test Limit ppm Monte Carlo estimation Theoretical (a) (b) Fig. 9. Comparison of the estimated and the theoretical test metrics for the case of two specications and one test criterion: (a) Defect level and (b) Yield loss. In order to see the accuracy of this method, we will rst illustrate a simpler case when only two perfor- mances, Phase Margin and THD, are considered together with the test criterion SNDR. In this case, the metrics presented by (1) to (4) are calculated as follows: Y D = A3 A7 f(s1, s2) ds1 ds2 (24) Y D T = B1 f(t1) dt1 (25) Y D L = 1 GP Y (26) DD = 1 GP YT (27) where GP = A3 A7 B1 f(s1, s2, t1) ds1 ds2 dt1 is the probability that the circuit is functional and passes the test, s1 is the Phase Margin value, s2 is the THD value, t1 is the SNDR value, Ai is the ith specication, B1 is the test limit of the SNDR and f(.) is calculated by (13). The covariance matrix is estimated by S given in Equation (12). For a given test limit, these metrics can be cal- culated exactly in this case, because the number of integrals is small. Figures 9(a,b) show that the estimated (using Monte Carlo algorithm where 1 million circuits are generated for each test limit) and the theoretical values of the metrics are very close for the case of Defect level and Yield loss. For comparison, Figure 10(a) shows the distribu- tions of the Phase Margin and the SNDR for the case of 1000 instances obtained via Monte Carlo cir- cuit simulation and for the case of 1000 instances generated from the multinormal distribution with a Monte Carlo technique. From this Figure, it is clear that both distributions are the same. The same re- sults have been obtained for the other performances and test criteria. In addition, Figure 10(b) shows the generation of 1000 and 1 million circuit instances generated from the multinormal distribution. We can assume that with 1 million instances we will reach the required ppm precision. 6.3. Setting of test limits under process deviations For simplicity, only ki will be considered in the test limits of the criterion i instead of i kii. 10 11. 61.5 62 62.5 63 63.5 64 64.5 65 65.5 56 58 60 62 64 66 68 70 72 74 76 Phase Margin THD Multinormal distribution with Monte Carlo generation Monte Carlo circuit simulation + 1 million instances + 1000 instances (a) (b) Fig. 10. (a) Distribution of 1000 circuits generated by Monte Carlo circuit simulation and from the multinormal law, (b) generation of 1000 and 1 million circuits from the multinormal distribution. 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0 10 20 30 40 50 60 70 80 90 100 kIDD kSNDR DefectLevel ppm 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0 1 2 3 4 5 6 7 8 x 10 ppm 4 kSNDR kIDD YieldLoss (a) (b) Fig. 11. (a) The Defect Level as a function of the SNDR and IDD test limits, (b) the Yield Loss as a function of the SNDR and IDD test limits. 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0 20 40 60 80 100 120 kSNDR kIDD Yield Loss Defect Level ppm 50 55 60 65 70 75 3,9 4,0 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,8 4,9 5,0 Test Limit of SNDR ppm D=YL 4.4 4.1 4.0 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9Test Limit of IDD MIN (a) (b) Fig. 12. (a) The Defect Level and the Yield Loss as a function of the SNDR and IDD test limits, (b) the test limits of the SNDR and the IDD where the Defect Level and the Yield Loss are equal. Test limits can next be set as trade-o between test metrics such as defect level and yield loss. Fig- ure 11(a) shows the Defect Level as function of the test limits kSNDR of the SNDR and kIDD of the IDD. We have considered IDD as a test measure- ment since it results in an improved fault coverage as it will be shown later. Figure 11(b) shows the Yield Loss as function of the test limits of SNDR and IDD. The setting of test limits is always a trade-o be- tween test cost and test quality. In our case-study, we will consider setting the test limits that simulta- neously try to minimise both Defect Level and Yield Loss. Figure 12(a) shows Defect Level and Yield Loss together as a function of the test limits of SNDR and IDD, where the intersection between them is of interest to us. This intersection (points where the Defect Level is equal to the Yield Loss) is redrawn 11 12. 74,38 90,62 51,88 20,62 29,38 53,12 72,5 95,62 71,88 93,75 77,5 84,38 98,12 98,12 32,5 89,38 89,38 0 10 20 30 40 50 60 70 80 90 100 AD GBWD PM CMRR PSRRGND PSRRVDD THD IDD Intermod SR+/- Noise PM&THD&Noise THD&IDD AllPerformances Offset SNDR AllTestCriteria FaultCoverage(%) Performances Test criteria 98,12 SNDR&IDD Fig. 13. Catastrophic fault coverage of the dierent specications and several possible test criteria. in Figure 12(b). We have chosen as trade-o of the test limits of the SNDR and the IDD the minimum of these points which is equal to 55ppm. This results in a test limit of 4.0 for the SNDR and 4.1 for the IDD. 6.4. Condence intervals for test metrics estimation This section deals with the precision on the cal- culation of the test metrics. The precision is given as inferences on the estimated values to ensure that the calculated value do not deviate from the un- known real value more than a certain percentage. We present tolerance intervals that contains the values of the test metrics with a condence level of 99.73% in this interval. However, note that before comput- ing precisions on the test metrics calculations, we rst calculate precisions on the performances mean and the test criteria mean which are estimated do- ing 1000 Monte Carlo simulations of the amplier. The obtained condence intervals of the means of the performances are given on the Table 1. For the case of the precision on calculation of the test metrics, we will calculate the variance of each one of them. For the lower variance, the estimated value is closer to the real one. To calculate the vari- ance for each test metric, we have generated 1000 times samples of one million circuits (amplier) and we have estimated the mean of each test metric. For a condence level of 99.73% the calculated con- dence intervals for these metrics are given by Ta- ble 4. Test metric condence interval (condence level of 99.73%) Yield Y D [99.9877%, 99.9915%] Defect Level DD 55 4ppm Yield Loss Y D L 55 3ppm Table 4 The condence interval on the calculation of the test metrics with condence level of 99.73%. 6.5. Test metrics evaluation under the presence of faults 6.5.1. Case of catastrophic faults We have considered catastrophic faults that result in shorts and opens in all the transistors, resistances and capacitances of the amplier. This results in 160 catastrophic faults. Figure 13 shows the fault cover- age given by the measurement of each performance and several possible test criteria with the test limits xed as explained before. The performances allow to detect 98.12% of faults where the undetected faults occur in the Bias block. On the other hand, the test criterion SNDR allows the detection of 89.38% of the faults. The undetected faults occur also in the Bias block. Maximum fault coverage can be achieved if power consumption (IDD) is considered in addi- tion to the SNDR measurement. 6.5.2. Case of parametric faults We consider parametric faults as a result of a physical parameter deviating beyond an acceptable value. The physical parameters considered include 12 13. the L and W of the PMOS and NMOS transistors and the resistance and capacitance values. We note that L and W of the transistors are not process pa- rameters, and thus their deviations are not included under process deviations. On the other hand, resis- tance and capacitance values deviate under process deviations. In order to calculate the probabilities pspec i and ptest i for each potentially faulty physical parameter, we have to obtain by simulation its limit deviation values. The distribution of each physical parameter is considered as Gaussian with mean equal to the nominal value of this parameter. A standard devi- ation of 10nm is taken for L and W of the transis- tors and 5% for the resistances and the capacitances. Thus, for each varying physical parameter, we have injected in the amplier deviations from -20% to 100%. For each value of a physical parameter, all test benches must be simulated. A dichotomic search is applied to nd the limit deviation for each physical parameter. This process has resulted in the consideration of 180 potentially faulty physical parameters, where only 13 of them result in a specication violation. These faults are listed in Table 5 together with their probabilities. The other faults have a negligi- ble probability of occurrence (pspec i 0). We note here that deviations in transistors that are matched cannot be considered individually. They are consid- ered by injecting the same deviation in all matched transistors. We have seen that faults in all matched transistors have a negligible probability. In order to consider the faulty behaviour resulting from mis- match, it is necessary to use the mismatch option in the Monte Carlo circuit simulation. In this work, we have used the process option for this, but a similar analysis could be performed for mismatch deviations. Using these parameters and Equations (19) to (23), the values of the dierent test metrics are given in Table 6. We consider two cases: in the rst case, all 13 parametric faults are considered. Only 2 of these faults are not detected by the test criteria (SNDR, Oset and Current Consumption). So, the fault cov- erage does not reach 100%. The fault coverage FF is high because the undetected faults have lower prob- ability. Notice that the yield Y F is lower than 60%, lower than the design yield Y D that has a value above 99%. This is because deviations of physical parameters such as L and W are not considered un- der process deviations. The second case of Table 6 does not consider deviations in the physical param- No Component Fault pspec min(pspec, ptest) (Parameter) 1 MP1 (l) +12.21% 0.013996 0.013996 2 MP3 (l) +7.32% 0.093690 0.093690 3 MP5 (l) +22.95% 0.000009 0.000009 4 MP1 (l) 3.18% 0.283305 0.283305 5 MP3 (l) 11.70% 0.017604 0 6 MP18 (l) 15.76% 0.002270 0 7 MN2 (l) 16.23% 0.001736 0.001736 8 MN4 (l) 6.46% 0.122278 0.122278 9 R1 (r) +15.14% 0.001227 0.001227 10 R4 (r) +17.09% 0.000308 0.000308 11 R7 (r) +16.11% 0.000628 0.000628 12 R4 (r) 12.79% 0.005249 0.005249 13 R7 (r) 16.15% 0.000611 0.000611 Table 5 Parameters used to calculate the metrics for the case of parametric faults. eters L and W, but only in the resistance values. In this case, fault coverage FF reaches 100% with a yield Y F above 99%. Since these deviations are also considered as process deviations, we obtain similar results between Y F and Y D . Metric All parametric faults Resistor faults only FF 96.69% 100% Y F 54.56% 99.22% Y F T 55.56% 99.22% Y F C 100% 100% DF 1.98% 0% Table 6 Test metrics values for single parametric faults. 7. Conclusions and Future Work A statistical approach for the evaluation of test metrics at the design stage and the setting of test limits under process deviations is the major contri- bution of this work. The data for this approach is obtained from a Monte Carlo circuit simulation. For a given set of test criteria, test metrics are evalu- ated taking into account the circuit specications, and test limits are set as a trade-o between test cost and test quality. 13 14. The method presented uses a theoretical proba- bility distribution to calculate the dierent test met- rics. Accurate results are obtained when this dis- tribution is known for multivariate random vari- ables, such as the multivariate Gaussian distribu- tion (called a multinormal distribution) used in this work. However, it is not possible to directly compute the test metrics when several performances and test criteria are considered because of the presence of multiple integrals in the calculation. We have shown how to overcome the integration problem by gener- ating a large population of instances from the multi- normal model, and directly estimating the test met- rics from the generated population. We are currently working to extend the technique for arbitrary mul- tivariate probability distributions. Once test limits are set, test metrics under the presence of faults in physical parameters can be cal- culated. Currently, the fault simulation environment does not consider global defects such as threshold voltage deviation and gate oxide thickness. This is currently being included in the environment. A case- study fully dierential amplier has been used to illustrate the use of these techniques for optimis- ing the test approach. We have also shown experi- mentally that when parametric faults are limited to physical parameters already considered during pro- cess deviations, the test metrics calculated at the design stage do not dier signicantly from those obtained under the presence of faults, as required. Acknowledgements The authors would like to acknowledge the fruit- ful discussions on analogue parametric testing with Jean Louis Carbonero, ST Microelectronics, France. References [1] S. Sunter and N. Nagi. Test metrics for analog parametric faults. In Proc. VLSI Test Symposium (VTS), 1999, pp. 226-234. [2] N. Ben-Hamida and B. Kaminska. Analog circuit testing based on sensitivity computation. In IEEE International Test Conferece (ITC), Baltimore, October 1993, pp. 652- 661. [3] K. Saab, N. Ben Hamida and B. Kaminska. Parametric Fault Simulation and Test Vector Generation. In IEEE Design and Test Conference in Europe (DATE), 2000, pp. 650-656. [4] A. Zjajo, J. Pineda de Gyvez and G. Gronthoud. 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