Basic Statistical Analysis and Yield Calculations

Transcription

1 October 17, 007 Basc Statstcal Analyss and Yeld Calculatons Dr. José Ernesto Rayas Sánchez 1 Outlne Sources of desgn-performance uncertanty Desgn and development processes Desgn for manufacturablty A general formulaton to statstcal analyss Tolerance and acceptablty regons Yeld defntons Calculatng the yeld Statstcal dstrbutons and tolerances Nomnal optmzaton vs yeld optmzaton 1

2 October 17, 007 Sources of Desgn-Performance Uncertanty FROM THE DESIGN PROCESS Modelng errors Numercal errors (smulaton errors) FROM THE MANUFACTURING PROCESS Envronmental effects Component agng Fabrcaton mprecson FROM THE TESTING PROCESS Measurement nose Calbraton errors Measurement errors 3 Typcal Desgn and Development Process Start Structure and components Hand calculatons Develop ntal desgn CAD smulatons Manufacture prototype Redesgn Test reject End pass 4

3 October 17, 007 Improvng the Desgn and Development Process Start Structure and components Hand calculatons Parameter statstcs Manufacturng and test models Develop ntal desgn Manufacture prototype Redesgn CAD smulatons, yeld analyss and optmzaton Test reject End pass 5 Desgn for Manufacturablty Start Choose structure and components Assgn ntal parameter values Smulate performance Optmze performance no Performance acceptable? yes Analze yeld Yeld acceptable? no yes End to manufacture... Optmze yeld 6 3

4 October 17, 007 Achevng a Hgh Yeld By numercal methods (yeld optmzaton): for a gven structure, fnd a nomnal soluton that best sut the manufacturng tolerances - $ By developng new structures: fnd a structure (topology, components, materals) less senstve to manufacturng tolerances - $$$$ By controllng (ncreasng precson of) the manufacturng process - $$$$$$$$$$ 7 A General Formulaton to Statstcal Analyss It s assumed that the crcut topology and the component types are already selected by the desgner and are fxed y R t represent the t parameters of the electronc crcut that are subject to statstcal fluctuatons The optmzaton varables are restrcted to a regon X of vald desgn parameters x X R n represent the n optmzaton varables of the electronc crcut to be optmzed y ncludes all the elements n x (n m) The elements n y not ncluded n x are consdered preassgned parameters, contaned n vector p R m, wth m = t n 8 4

5 October 17, 007 A Formulaton to Statstcal Analyss (cont) The parameters of the k-th manufactured devce, outcome y k, are actually spread around the nomnal pont y accordng to ther statstcal dstrbutons and tolerances These parameters can be represented as y = y + j y j j = 1,, N where N s the number of outcomes, and y j represents a random varaton for the j-th outcome 9 A Formulaton to Statstcal Analyss (cont) The crcut responses are denoted by R(y) R r where r s the number of responses to be optmzed The statstcal analyss of a crcut around a nomnal pont y conssts of realzng N smulatons of R(y j )for j = 1 to N (also called Monte Carlo analyss) The number of outcomes N must be suffcently large to have statstcal sgnfcance 10 5

6 October 17, 007 Tolerance and Acceptablty Regons T: Desgn tolerance regon around a pont x (depends on all the sources of uncertanty) A: Desgn acceptablty regon for the responses (depends on the desgn specfcatons) Example: 11 Yeld Defntons The probablty that a manufactured unt (outcome) wll pass ts performance test The probablty that a manufactured unt wll satsfy all ts desgn specfcatons The rato of the number of manufactured unts whch pass performance testng to the total number of unts manufactured (n the lmt, when the number of unts tends to nfnty) The ntersecton between the A and T regons, over T 1 6

7 October 17, 007 Yeld Defntons (cont) The ntersecton between the A and T regons, over T Y = A T T x * : optmal nomnal soluton 13 Yeld Defntons (cont) The ntersecton between the A and T regons, over T A T Y = T x Y* : optmal yeld soluton 14 7

8 October 17, 007 Calculatng the Yeld Usng a mnmax objectve functon U, { Ke ( y) } U ( y) = max K Vector e(y) contans all the error functons wth respect to the desgn specfcatons e(y) s formulated such that a postve element value n e(y j ) mples that the j-th outcome s volatng some desgn specfcaton k 15 Calculatng the Yeld (cont) Each outcome has an acceptance ndex defned by 1, Ia ( y j ) = 0, f U ( y j ) 0 f U ( y ) > 0 If N s suffcently large for statstcal sgnfcance, the yeld Y at the nomnal pont y can be approxmated by usng 1 Y ( y) ( y j ) N N I a j= 1 j 16 8

9 October 17, 007 Probablty Dstrbutons The crcut parameters for the j-th outcome at the nomnal desgn y are gven by y j = y + y j, where y j represents a random varaton for the j-th outcome Each parameter n y follows some probablty dstrbuton functon (PDF), typcally unform, normal (Gaussan) or bnormal If the -th parameter y follows a probablty dstrbuton functon p, then the probablty of y to have a value between y = a and b s b p ( a y b) = p ( y ) dy y = a p( y ) = p ( y ) dy = 1 17 Normal Probablty Dstrbuton Functon 1 p y) = e σ π ( y µ ) ( σ µ average σ standard devaton σ varance Gaussan PDF (µ = 0) σ = 1 σ = σ = Gaussan PDF (σ = 1) µ = 0 µ = µ = -3 p(y) 0. p(y) y y 18 9

10 October 17, 007 Student s t Probablty Dstrbuton Functon ν 1 Γ + y 1 + ν p( y) = νπ Γ ( ν / ) ( ν + 1)/ Student's t PDF ν degrees of freedom Γ gamma functon ν = 1 ν = ν = 3 ν = 30 p(y) y 19 Raylegh Probablty Dstrbuton Functon y = yb e b center (Raylegh) b p( y) p(y) Raylegh PDF b = 1 b = b = y 0 10

11 October 17, 007 Probablty Dstrbutons and Tolerances Unform dstrbuton p( y) p ( y) = C Gaussan dstrbuton 1 = e σ π y σ ( ) p y y(tol) y 0 +y(tol) p( y) σ varance σ standard devaton y(tol) 0 +y(tol) y 1 Calculatng the Number of Outcomes, N We assume that all the statstcal crcut parameters follow a Gaussan probablty dstrbuton functon The number of outcomes N needed to have a certanty c when calculatng the yeld can be obtaned from where Y: estmated (expected) yeld ε: error n the yeld estmaton t: statstcal value wth a probablty c to happen (c s the area under the bell curve between t and +t) [ t( c) ] N = round ( Y )(1 Y ) ε (Meehan and Purvance, 1993) 11

12 October 17, 007 Crcut Nomnal Optmzaton The problem of crcut nomnal optmzaton s formulated as typcally, x x * * = arg mn ( R( x)) x U = arg mn max{ e ( x)} x where all the parameters n the crcut (elements n y) have zero tolerances, x * s the optmal nomnal desgn, R(x * ) = R * s the optmal nomnal response, and U s the objectve functon 3 Crcut Yeld Optmzaton The problem of crcut yeld optmzaton s formulated as Y* x = arg mn U ( x) where some elements of y have non zero tolerances, x Y* s the optmal yeld soluton, R(x Y* ) = R Y* s the optmal yeld response, and U Y s some sutable yeld objectve functon x Y 4 1