Fast search agorithm for tolerance design

Y. Lin S.W. Foo

Indexing terms: Search-space smoothing, Tolerance design

Abstract: A new fast search technique based on search-space smoothing is proposed for the tolerance design of electronic circuits. By smoothing and enlarging the sampling region of Monte Carlo analysis so that focus is placed on the global features of the acceptance region, the authors looked at the solution space from a larger perspective and reduced the number of local optimal points to be investigated. After the target area is identified the algorithm zooms in to focus on the detailed features of the target area. The yields on spaces with different degree of smoothing are computed and finally the most probable global solution is obtained. The algorithm is simple and efficient. The proposed method is applied to different electronic circuits. Results show that the computational efficiency and the resultant yield achieved are significantly better than the popular centres-of-gravity method.

1 Introduction and terminology

The component values of mass-production circuits vary according to the degree of control of tolerance values. The combined effect of the variation of the constituent components may be such that the circuit performance becomes out of specification and has to be rejected. The manufacturing yield is then reduced to that frac- tion of the mass-produced circuits that are within spec- ification. The aim of design centring is to maximise the manufacturing yield for a given set of tolerance values of the components.

Of the optimisation techniques for design centring, iterative local search methods are the most commonly used. Of the iterative local search methods used in tol- erance design, the centres-of-gravity (COG) method [l] is the most popular and the most robust. For this method, Monte Carlo sampling is first made within the tolerance region, the centre of gravity of the parameter vectors of the pass circuits is then computed. The nom- inal design centre, the parameter vector pT, is then moved towards the COG of the pass circuits. This proc- ess is repeated until the desired yield is achieved or 0 IEE, 1998 IEE Proceedings online no. 1998 1592 Paper first received 15th November 1996 and in revised form 27th May 1997 The authors are with the Department of Electrical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260

IEE Proc.-Circuits Devices Syst., Vol. 145, No. 1, February 1998

until the incremental improvement in yield becomes very small.

The number of circuit simulation required for COG method is large and hence computational load is heavy. In the past 20 years many techniques, have been pro- posed to improve the COG method. These include par- ametric sampling [2, 31, common poinls scheme [4] and simplicial approximation [5 ] . Most of the schemes aim at reducing the number of samples required for each iteration by reusing some of the points in the previous iteration.

In this paper a new approach to improve the COG method is proposed. The method, which we call auto- focus method, aims to reduce com]x~tational effort through reduction of the number of iterations. The proposed method is also compatible with the improve- ment schemes based on reuse of sampled points and hence can be combined with these methods to further reduce computational load.

1. I Monte Carlo yield estimation For notational convenience it is useful to define the various regions of interest in the component space. Consider a circuit in which there are m variable compo- nents pl, p2, ..., p , with tolerances tl, t2, ..., t,. The tol- erance region denoted by RT is the set given by RT = (a1 p , - t , 2 p p I p , + t,; i = 1,2, ..., m}. And the accept- ance region denoted by R A is the set given by RA = (a\ LJ 2 Jl(p) 5 U , j = 1, 2, ..., n} , where LJ. and V, are, respectively, d e lower and the upper specification lim- its of the circuit’s responses at the jth point of interest. The feasible region RF, where the circuits meet the specifications, is then the intersection of the tolerance and acceptance regions. Mathematicall~y, RF= RA n Rp

An estimation of the manufacturing yield Y is then given by the following m-fold integral

Y = s,, @(PI,. . . ,Pm)dPl . . . dPm

where ..., p,) is the joint PDF of the values of the m variable components. The Monte Carlo method is commonly used to estimate this yield as direct integral evaluation of Y is not practical. For the Monte Carlo method, component values are randomly generated based on their probability distributions. The circuit performance is then assessed either through circuit analysis or computer simulation. To obtain statistically meaningful results a sufficiently large number of sam- ples must be assessed. The manufacturing yield is then estimated as the percentage of circuits satisfying the performance requirements. Mathematically, the yield Y is given by

number of pass circuits total number of circuits tested

Y =

19

2 Autofocus method

0 Initial

region of

2.1 Concept For typical search space, different neighbourhood structures result in different terrain surface structures and different numbers of local minima. The e ness of a heuristic local search algorithm is a function of the number of local minima in the search space. The fewer the number of local minima, the more effective the local search algorithm 161, For the method pro- posed in this paper the number of local optima is delib- erately reduced at the beginning of the search by smoothing the structure of the search space. The basic concept is explained as follows. Assume that there is a search space with many local minima as shown in Fig. 1, where a solution point could easily be trapped.

se search space is developed to approximate the original fine search space and hence some local minima are eliminated and the number of local minima is reduced.

tolerance region RTo

acceptability RA

search space

global min. Graphical illustration of search space smoothing Fig. 1

p2 t I 1 enlarged tolerance region R,,

I 1 I

F Pi

Fi .2 Graphical illustration of autofocus searching process of circuit w i g two designable parameters

The method can be compared with photographing a specific part of an object (see Fig. 2). It first starts at a nominal design associated with a nonzero yield simi- larly to the camera focusing on a certain part of the object. The next step is to enlarge the view area so that a larger view including the object is captured. The cross-hair in the viewfinder is then moved to the centre of the object. In autofocus tolerance design, we enlarge the sampling area by increasing the variances of all the designable parameters to cover a large search space. The design centre is then relocated to the centre of the pass circuits. A new search space is created with minima close to the most probable global minima.

The next step is to focus the view area on the specific part of the object so that it occupies a larger percentage of the picture frame. In circuit optimisation, adjust- ments of the magnitude of the variances are then made to include a finer structure of the search space and to include more localised features.

A photographer usually repeats the readjustment process until the object occupies a reasonable percent- age view area. Similarly, in the autofocus algorithm, gradual adjustment is made until the search is made on the original search space to obtain the global solution.

2.2 Autofocus algorithm for design centring The objective of design centring is to move the set of nominal values of the components, the design centre, towards a point that maximises the total yield while keeping the tolerance of the components unchanged. The autofocus method aims to reach these optimum nominal values of components in a small number of iterations. Details of the method are described as fol- lows.

First the following parameters are set: the number of Monte Carlo samples N to be used for each iteration, the initial focus factor p which is a real number, the initial design centre pTo = p j ... p,O) and the asso- ciated tolerance values tr = ( t l , t l , ... t,) where IYZ is the number of designable parameters, Based on the toler- ance values and the types of probability distribution functions of the parameters, the variances of the distri- butions are determined, At the first iteration, the vari- ances are multiplied by the initial vafue of the focus factor p, a number larger than 1. N samples of parame- ter values are then generated using Monte Carlo sam- pling of the stretched probability distributions of the parameters. The performance of each of the corre- sponding circuits is determined and checked against the given specifications. The yield is then determined as the percentage of circuits that satisfy the specifications. Any known design centring method can then be applied to maximise the yield in the coarse search space. For example, if the COG method is used the cen- tre of parameters corresponding to the pass circuits is chosen as the new design centre.

At the end of the first iteration p is reduced and the process is repeated using the new design centre and the new set of variances. Simulations show that the desired effect can be achieved with p decreasing linearly to 1 in 3 to 4 steps. Note that p = 1 corresponds to the origi- nal search space. After /3 is decreased to 1 the search process i s continued in the original search space until a

value of yield is obtained or until the number ions has exceeded a predetermined limit.

3 Example of application

3,l Example I To illustrate the method we choose the re tial divider as the first example, shown in circuit is chosen as it is simple to understand and the region of acceptance and region of tolerance can be portrayed on a graph. Let the upper and lower limits on output voltage given by

5.5 v > Vent > 4.5 v These are transformed, respectively, into boundaries A and B in the parameter space as shown in Fig. 3b. Sim- ilarly, the constraints on input resistance given by

1 2 0 0 > R,, > 8 0 0

20 IEE Isroc.-Ctrcuzts Devices Sysl , Vol I4S, No. I , February 1898

are transformed, respectively, into boundaries C and D. Together, the four boundaries define in parameter space the acceptance region RA corresponding to the specifications of output voltage and input resistance.

a b Fig.3 Two-resistor circuit and acceptance region RA in parameter space corresponding to upper and lower specifications on output voltage and input resistance a Circuit b Acceptance region

In our simulations, 500 Monte Carlo samples are used at each iteration. The variances are calculated by multiplying the original variances by the focus factor. The value of the focus factor p is set respectively to 2.5, 1.75 and 1 for the first three iterations. Thus, from the third step onward the estimated yield is the yield corre- sponding to the original variances of parameters. Simu- lation using p = 1 is continued till the tenth iteration.

For comparison the design centring process of the circuit is also carried out using the popular COG method with the same number of sampling points and tolerances.

To ensure that the results are not biased by the choice of initial points, four experiments were carried out using the following four corner points of the RA, (66, 54), (54, 66), (36, 44) and (44, 36). The components are assumed to have uniform distribution with toler- ance = 12. The starting nominal design values of RI , R2, its corresponding yield, the yield at the third itera- tion (step), its corresponding RI, R2 values, the maxi-

mum yield achieved in ten iterations and its corresponding RI, H 2 values are tabulated in Table 1.

6 2 R. C. 10.47nF C,

a

frequency(H2)

b Filter circuit and its specifications m pequmcy domain Fig.4

a Filter circuit b Specifications

3.2 Example 2 To further illustrate the performance of the method we present the results of the application of the method to the filter circuit commonly cited in literature on toler- ance design [5] . The circuit and its desirable response limits are shown in Fig. 4. For this circuit it is required that the attenuation values at a set of chosen frequen- cies fall within specified limits. The ,attenuation as a function of frequency f is given by

A ( f ) = 20 log( I V , (2x990) /K (2.irf) 1)

Table 1: Comparison of yields obtained from autofocus and COG methods with different starting points

Yield improvement YO autofocus over COG COG

Initial

yield 3rd step Max 3rd step Max 3rd step Max Experiment

yield yield yield yield yield yield

1 0.288 0.722 0.98 0.98 0.992 26.3 1.2 RI 66 59.6 55.8 51.6 51.9 R2 54 54.6 54.3 51.8 51.9

2 0.238 0.758 0.95 0.982 0.99 22.8 4 RI 54 53.9 54.5 50.8 51.1 R2 66 59.8 56 51.4 51

3 0.28 0.798 0.935 0.97 0.976 17.7 4.2 RI 36 42.2 45 49.7 49.8 R2 44 44.9 45.1 50.5 50.4

A 0.27 0.764 0.945 0.97 0.976 21.2 3.1

c1 44 44.3 45,3 50.2 50.2 c3 36 42.4 45.4 50.4 50.4

Average 0.269 0.7605 0.9525 0.9755 0.9835 22 3.1

IEE Proc -Czrcuzts Devises Syft., Vol 145, No 1, Februavy 1998 21

Table 2: Comparison of yields obtained using autofocus and COG methods

Yield improvement % Autofocus CoG autofocus over COG

COG Initial yield Experiment

Max 3rd step Max 3rd step Max yield yield yield yield yield

0.866 0.91 0.91 12.9 4.8 9.51 9.3 9.3 16.99 16.16 16.16 34.7 35.84 35.59 96.72 94.94 95.39

3rd step yield

0.792 9.7 17.24 32.77 96.67

1 0.42 c1 10 c 3 18 c 4 30 c5 95

2 0.638 c 1 15 c 3 10 c 4 30 c5 95

0.826 15.06 10.25 32.13 97.71

0.88 0.865 0.96 4.5 15.17 13.24 13.04 11.11 12.06 12.51 33.66 34.28 34.24 96.77 102.1 97.49

8.3

7.1 5 0.206 c 1 13 c 3 18 c 4 35 c 5 90

0.66 11.89 16.43 35.51 95.66

0.854 0.875 0.92 24.5 11.17 10.873 10.61 15.24 14.6 14.79 34.72 35.44 34.91 96.45 91.64 91.44

3.4 6 0.82 c 1 7 c 3 18 c 4 35 c5 90

0.874 0.82 0.905 -5.2 7.31 7.94 7.89 18.05 17.39 17.49 37.17 37.01 36.52 87.59 89.43 90.38

0.862 7.13 18 36.38 89.73

7 c 1 c 3 c 4 c5

0.296 17 13 40 100

0.66 0.856 16.35 15.29 12.62 11.88 35.67 34.62 99.52 97.32

0.875 13.94 12.03 35.2 95.86

0.89 24.5 3.8 13.72 12.47 35.31 95.74

8 c 1 c 3 c 4 c5

0.228 17 13 40 80

0.596 0.856 15.33 14.16 12.33 12.14 37.89 35.18 83.93 90.79

0.835 14.14 12.53 34.83 94.25

0.905 28.6 5.4 13.81 12.47 34.57 95.73

9 c 1 c 3 c 4 c 5

0.008 10 7 30 90

0.59 0.856 10.54 10.93 10.03 11.59 32.97 33.39 94.64 96.15

0.598 0.84 11.95 11.21 16.71 15.59 36.35 35.06 94.1 96.36

0.81 12.28 10.73 33.02 105.7

0.86 27.1 0.4 12.24 1 1.48 32.87 105.41

10 c1 c 3 c 4 c5

0.074 13 18 42 90

0.83 10.58 14.54 36.15 90.3

0.91 27.9 7.6 10.45 14.62 35.23 90.51

Averaae 0.35 0.70 0.86 0.85 0.91 17.19 4.90

and the constraints are as fo1lows: $1 = 45 - A(170) 5 0 4 2 = 49 - A(350) 5 0 4 3 42 - A(440) 5 0 $4 = A(630) - 4 5 0 4.5 = -0.05 - A(630) 5 0 4 6 = A(680) - 1.75 5 0

4 7 = -0.05 - A(680) 5 0 4s = A(1800) - 1.75 5 0 4 9 = -0.05 - A(1800) 5 0

The design is characterised by four designable parame- ters denoted by pT = (Cl, C,, C,, C,). For this example ten experiments were carried out using different start- ing nominal values. Truncated gaussian distributions with standard deviation o = 9, tolerance = 27, and ini-

IEE Proc.-Circuits Devices Syst., Vol. 145, No. 1, February 1998 22

tial value of p = 2.5 are used. The initial yield, yield obtained at the third iteration, maximum yield obtained in ten steps and their corresponding design centres p T = (Cl, C3, C4, C,) are tabulated in Table 2.

From Tables 1 and 2 it can be seen that for most of the experiments the maximum yield obtained in ten iterations using the autofocus method are higher than that obtained using the COG method. It can also be observed that the average yield achieved at the third iteration using the autofocus method is much higher than that obtained using the COG method. In other words the rate of convergence of the autofocus method is higher than that of the COG method. This is more easily observed from the graphical presentation in Fig. 5.

60

cc" 50

40 1 1

-

-

-

0.8

t i i 01 0 2 4 6 8 10 Com arison of yields achieved at duerent iterations of autofocus Fi .5

an! COG met/&: starting nominal design is (13, 18, 42, 901 +-+ antofocus method 0-0 COG method

90

80

70

BO

50 oc"

40

30

20

I I

IT1 RTO

30 40 50 60 70 80 90 100

RI

Autofocus: sizes and locations of R, and trajectory of design cen- Fig. 6 tres for ten iterations Starting nominal design is [66, 541; Monte Carlo sampling size = 500 points

The sizes and locations of R, and the design centres of the ten iterations relative to RA using the autofocus method and the COG method are plotted in Figs. 6 and 7. The autofocus method captures the major part of RA in three steps while the corresponding movement using the COG method is much slower. From Table 2, Figs. 6 and 7 it can also be observed that for the COG method the nominal design values of (Cl, C,, C,, C,) at the third iteration is strongly related to the initial nominal

design values, whereas for the autofocus method the nominal design values at the third step are very much different from the initial nominal design values, espe- cially if the initial yield is low. This means the autofo- cus method is more adaptable in the selection of nominal design centres.

70 t

30 t 30 40 50 80 70 80 90 100

RI

Fig.7 design centres for ten iterations Starting nominal design is [66, 541; Monte Carlo sampling size = 500 points

Conventional COG: sizes and locations of RT and trajectory of

4 Conclusion

The method proposed, the autofocus method for toler- ance design, has aimed to maximise the yield in a small number of iterations. Results have shown that the method significantly outperforms the .most commonly used local search algorithm, the COG algorithm, both in the number of iterations needed to achieve a given yield improvement and in the value of the maximum yield obtained.

The method is simple, efficient and effective. In addi- tion, the method is compatible with other known local search methods, hence further improvement can be achieved by combining the autofocus technique with other heuristic local search algorithms.

5 Acknowledgment

The authors would like to thank the National Univer- sity of Singapore for providing the University Research Grant to support the research presented in this paper.

6 References

1 SPENCE, R., and SOIN, R.S.: 'Tolerance design of electronic cir- cuits' (Addison-Wesley, 1988)

2 BATALOV, B.V., BELYAKOW, Y.N., and KURMAEV, F.A.: 'Some methods for statistical optimization of integrated microcir- cuits with statistical relations among the par,zmeters of the com- ponents', Sov. Microelectron. (USA) , 1978, 7, (14), pp. 228-238 SINGHAL, K., and PINEL, J.F.: 'Statistical design centring and tolerancing using parametric sampling', ZEEE Trans., 1981, 01s- 28, (7), pp. 692-702

4 STEIN, M.L.: 'An efficient method of sampling for statistical cir- cuit design', IEEE Trans., 1986, CAD-5, (l), pp. 23-29

5 DIRECTOR, S.W., HACHTEL, G.D., and VIDIGAL, L.M.: 'Computationally efficient yield estimation procedures based on simplicial approximation', ZEEE Trans., 1978, CAS4.5, (3), pp. 121-130

6 GU, J., and HUANG, X.: 'Efficient local search with search space smoothing: A case study of the traveling salesman problem (TSP)', IEEE Trans. Syst. Man Cybern., 19514, 24, (5), pp. 728- 735

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