Worst Case Circuit Analysis An Overview (E E C T R N I P S/C I I T S To .

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Worst Case Circuit Analysis - An Overview(E1ectroni c P arts/Circuit s To1erance An a1y si s)Walter M. Smith0Design and Evaluation, Inc. 0 Laurel SpringsKey Words: Worst Case Analysis, Electronic Parts/Circuit Tolerance Analysis,Worst Case Scenario, Worst Case PartVariations, Sensitivity Analysis, Extreme Value Analysis, Root-Sum-Square, Monte Carlo Analysis.2 - WCCA is normally a contractrequirement on most Hi-Re1 Militaryand Space Programs.3 - WCCA is "THE WAY" to designelectronic circuits given a WorstCase Part Variations data base,WCCA is relatively easy for circuitdesignerdanalysts as well as aneconomical approach and if thecircuit "passes" we're finished ourjob and should get through testeasier than normal.SUMMARY & CONCLUSIONSThis paper has been prepared utilizing material fromDesign and Evaluation, Inc.'s (D&E) Training Course andEngineering Handbooks on Worst Case Circuit Analysis(WCCA). The WCCA Training Course was developed byD&E and upgraded under contract to NASA's Jet PropulsionLaboratory to standardize an approach to performing WorstCase Circuit Analysis and to exemplify the methodologiesand tools for completing a customer acceptable Worst CaseCircuit Analysis final report. The WCCA metholdologiesdiscussed herein for the development of a Worst Case PartsVariation Data Base, Sensitivity Analysis and themathematical approaches of the Extreme Value Analysis(EVA), Root-Sum-Square (RSS) and Monte Carlo Analysisfor solving circuit equations and combining variables havebecome an accepted industry standard over the last eightyears.1. INTRODUCTIONWORST CASE CIRCUIT ANALYSIS(WCWWHAT IT IS0EVALUATIONOF CIRCUITS FOR TOLERANCE TO DRIFT0The objective of this paper is to introduce the basicphilosophy and concepts of Worst Case Circuit Analysis(WCCA) and to exemplify these principles.The crux of this paper are the concepts of electronicpiece-part parameters variations beyond their initial tolerancedue to aging and environmental influences, the mathematicalsensitivity of circuit performance to these variations, and thenumerical methods, both non-statistical and statistical, ofhandling the numerous variables affecting circuitperformance. The subjects of Sensitivity Analysis, and threemathematical techniques, namely Extreme Value Analysis,Root-Sum-Square and Monte Carlo Analysis are discussedherein.The first questions to be addressed are "What isWorst Case Analysis ?" and "Why do it?" Figure 1.0 sums upthe "What". The "Why" is three fold:1 - To design reliability INTOhardware for long term trouble-freefield operation - reliability againstdrift.326-@0A RIGOROUS MATHEMATICALEVALUATION OF A CIRCUIT'SPERFORMANCEATTRIBUTES AGAINST PERFORMANCETOLERANCE LIMITS, UNDER SIMULTANEOUSEXISTENCE OF ALL THE MOST UNFAVOWLECONDITIONSBEING AT RELIABLE LIMITSPART PARAMETER VARIATIONS - WORST CASEENVIRONMENTALEXTREMES, TEMP.ETCINPUTPOWER0INPUT STIMULI, UPPERLOWER LIMITS0LOAD VARIATIONS AT EXTREMES0I/F INTERFERENCES AT MAXIMUMSEVALUATION OF PARTSOVER STRESSES IN WORST CASE CONDITIONSIMPROPER APPLICATIONSFIGURE #10-7803-3112-5/96/ 5.00 0 1996 IEEE1996 PROCEEDINGS Annual RELIABILITY and MAINTAINABILITYSymposium

The next questions to be answered are "How doesWCCA fit into a Reliability Program?", "What is the Value ofWCCA?" and "What is the Return on Investment?" Figures 2and 3 answer the first two questionsThe "Return on Investment" is difficult to assess foranalysis tasks especially if a mission is 100% successful. It isrecognized that the success of the mission of a complexsystem such as a satellite or the Space Shuttle Program istruly due to the collective efforts of various teams of peopleand technical disciplines all working and coordinating effortsand resources over a long time, toward the common goal of"success". However, ROI for a Worst Case Circuit AnalysisProgram has been addressed within a major governmentagency for a space program involving the launching of asatellite to orbit a neighboring planet. This satellite has beenoperating successfully for several years. For this program, aWCCA program was conducted over a four year period,involving fifty five boxes, and at a cost of two and a halfmillion dollars. This WCCA effort uncovered and correctedtwenty two major discrepancies, and sixty overstressconditions of parts of which eight were corrected andthe remaining were accepted as minor. Two digital unitswere proven inadequate and redesigned. The WCCA Programcosts less than 1% of total program costs and avertednumerous potential failures of three hundred million dollarsof hardware. The results economicallyjustified the cost.2 - OVERVIEW OF WORST CASE CIRCUIT ANALYSISThe basic approach to performing a WCCA on aschematic for an electronic board is to breakdown the circuitinto simple functional block and perform a WCCA on eachblock. The analyst should first document a detaileddescription of each block. He must then develop the WorstCase variations of all critical parameters for all parts utilizedin the circuit, to arrive at a Worst Case MAX and a WorstCase MTN for each part parameter. This subject will beaddressed in a following section. The analyst must thenestablish the performance requirements for the critical circuitattributes for each block and show analytically whether thecircuit attributes actual performance values either meet or"fails to meet" the requirements, utilizing the WC MAX andWC MIN Values discussed above. Finally, the analyst shouldVALUE OF WCCAHOW WCCA FITS INTO A RELIABILITY PROGRAMDESIGN OBJECTIVES OF ELECTRONICS FORAERO/SPACEFLIGHT PROJECTSDESIGNS HIGH RELIABILITY INTO CIRCUITSPROVIDES MARGIN AGAINST WC DRIFT OFPARTS OVER LIFE AND MAS ENVIRONMENTS.1 - DESIGN PERFORMANCE MUST REMAIN WITHINSPECIFIED TOLERANCE LIMITS OVER ENTIRE MISSIONLIFE(NO DRIFT FAILURES)m:WORST CASE CIRCUIT ANALYSIS (WCCA)2 - ALL PIECE PARTS MUST SURVIVE THE MAX LEVELS OFSTRESS IMPOSED ON THEM(NO CATASTROPHIC FAILURES)m:ELECTRICAL STRESSDERATINGANALYSIS3 - ASSESS MISSION EFFECTS IF A FAILURE ISSHORT TERM "RETURN ON INVESTMENT"REDUCES NUMBER OF HARDWAREDESIGN ITERATIONS DURINGDEVELOPMENT PHASEREDUCES TIME IN TESTREDUCES NUMBER OF "DESIGNCHANGE NOTICES" AFTER DRAWINGRELEASEENCOUNTERED(ALLEVIATE ADVERSE EFFECTS)m:FAEIJRE MODES & EFFECTSANALYSIS-4 DESIGN MUST RECOVER FROM "LOGIC UPSETS" WHICHWILL OCCUR DUE TO TRANSIENT RADIATION INSPACE(FAULT TOLERANT CIRCUITS)m:SINGLE EVENT UPSET ANALYSISNOTE:TO DEVOTE MOST OF ONE'S RESOURCESTO SEVERAL OF THESE AREAS, TO THENEGLECT OF THE OTHERS, YIELDS APRODUCT ASSURANCE VOID -HIGH PROJECT RISK!LONG TERM "RETURN ONINVESTMENT"INCREASES PRODUCTION EFFJCIENCYTROUBLE FREE FIELD OPERATIONLONGLIFECUSTOMER SATISFACTION ANDCONFIDENCEWCCA IS COST EFFECTIVE IN BOTHTHE SHORT AND LONG RUNFIGURE 2996 PROCEEDINGS Annual RELIABILITY and MAINTAINABILITY SymposiumFIGURE 3327

show how all the functionally blocks "play together" to meetthe overall unit (board) requirements in the Worst Case.-BUT will the circuit survive the real worldenvironments to which it will be exposed during its mission,in the Worst Case? What is that were looking for.2.1 CLASSICAL EVALUATION OF A BAND-PASSFILTER (BPF)2.2 CREATING A WORST CASE SCENARIOTo illustrate the performance and results of aWCCA, we will utilize an example of a real WCCA whichwas performed on the Band-Pass Filter shown in Figure 4.The amplifier gain at the center frequency (Afo) will be thecircuit attribute selected for analysis.Were looking to determine if there is any margin leftin the circuit after we calculate the Worst Case MAX andWorst Case MIN performance and compare these values tothe upper and lower tolerance limits which cannot beexceeded per the circuit design specifications, see Figure 5. G i N%5c3Vi 12--.\hl\r-10 i ri"r' le-3'Given that U6 is an ideal OP-AMP(Rm MARGIN03,ROUT 0, AVOL O0), it does not enter into the equationfor gain, which is1-R1 c 2R5 R(')-2%1 1 1Rq Rg 'R6 IRRt Rg ('21')%'21-&The specified requirement for the minimum Afo isVOLT. The nominal and initial tolerancevalues for the resistors (R) and capacitors (C) are as follows:C1 , C 2 CYR20 (1500pF, l?'o)7.0RI , R 2 , R3 RNR5O(15KR,5Io/o),R4 RNRSOHf40.2KR, Cl%)R5 RNRSOH(1OKR, lo/o),R6 RNRSOH (1.21KR, ?1%)Substituting the nominal part values into equation 1yields Afo 11.08 V/V which shows the result to be "inspec". For a more pessimistic answer we could use the piecepart initial tolerance values ( 1%) which would yield Afo 7.84 V/V, again "in-spec". It should be noted that the initialtolerance values for the R s and C's are the most pessimisticvalues which the analyst has available to him and this is howcircuits are typically designed and analyzed.328FIGURE 5Normally, the Worst Case calculations must beperformed twice i.e. to assess WC performance against theupper and lower spec limits. For the Band Pass Filter wehave a one sided requirement i.e. the minimum gain is all weare interested in. We cannot go below the required minimumgain of7.0 V/V in the Worst Case.2.2.1 THE WORST CASE, PART VARIATIONS, DATABASE (WCDB)Two facts of life: first, when a part vendor specifiesan initial tolerance (procurement tolerance), he is merelyguaranteeing that when you buy the part and when youreceive it in-house, that all the parts in each lot will fallwithin the initial tolerance specified ( 1% for the R's and C'sof the Band Pass Filter). He is NOT guaranteeing that thepart will remain within this tolerance band forever. Fact #2,the part will drift beyond the value it has, after you select itfrom stock and power it up and expose it to the limits of allthe mission environments. In many, many, cases, especiallythose involving long missions, the part will drift beyond itsinitial tolerance, which we used in designing the circuit.Worst Case Analysis philosophy stacks up each maximumpossible drift value contributed by each mission environment,i.e. they are added algebraically to the initial tolerance. SeeFigure 6.1996 PROCEEDINGS Annual RELIABILITY and MAINTAINABILITY Symposium

1VAC'JUM7,,/-ORIF7DUE TO LOW TEMP,-?RIFrWORST CASELIMITS (Yo)(MIN) (MAX)WEORlFT DUE/mL:fF\*a%-20%I-PART PARAMETER VARIATIONS FOR WORST CASE DRIFT(SUPERIMPOSED)ORIF: SUETOHARDEXAMPLE. CAPACITOR, CLRNOMINAL VALUE- PART PARAMETER VARIATIONSINITIAL TOLERANCE VS WORST CASE LIMITS(ACTUAL DATA FROM DATA BASE BK SATELLITE PROG.BEYOND INITIAL TOLERANCE1200 pFFIGURE 6For Worst Case Circuit Analysis, it is assumed thatwhen you select a part from stock, it is at its initial tolerancevalue already, which it may or may not be but it is certainlypossible and we are creating a Worst Case Scenario. Nextwe assume for WCCA, that all parts in a circuit are at theirmax drift values (on top of initial tolerance).SIMULTANEOUSLY. Now, admittedly, this is stretchingthings but it is a Possible scenario in the worst case. A morelikely scenario, is that some combinations of parts will be atdrift values beyond their initial tolerance and probably theywill not all be at the maximums possible drift. Survival in aworst case scenario of all parts being at their MAX driftvalues simultaneously, assures survival in any scenario ofpossible combinations of part variations to any degree. If youanalytically calculate circuit performance under the WorstCase Scenario described above and still have margin in yourcircuit relative to not exceeding the specified upper and lowerperformance tolerance limits, you will have a rock soliddesign against part variations. This is the philosophy ofWorst Case Circuit Analysis and this is how electroniccircuits and systems are designed for most military andaerospace applications requiring high reliability.WORST CASE CIRCUIT ANALYSIS DESIGNSRELIABILITY INTO ELECTRONIC CIRCUITS.CAPACITOR CKRO6, 1.0 pfCAPACITOR 10,000 pfCAPACITOR CSRCAPACITOR FIXED ELEC. 1200 ,ufCAPACITOR TANT. NON-SOLID. 7200RESISTOR. ANN 118 'NRESISTOR, RWR 2 WRESISTOR. N W K PREC WWRESISTOR. CC 114W RCRRESISTOR. CC 112 W RCR 20XFMR. ? U S E (INDUCTANCE)XFMR. CURR. SHARING (INDUCTANCE)NOUCTOR. LINEAR (INOUCTANCE)JIOOE. FET CURR. REG (IO)IIOOE, z. vzoIIOOE. FAST REC .5c15.444.542.9c1.48 1.55c5.50-17.6c12.4-16.4c10.3c38.2-48.3-46.4 c36.3-38.2 16.2-27.9 c32.6-9.6 c7.7-44.7 41.9FIGURE 7The actual task of developing the Worst Case PartsData Base is a significant part of the work and cost involvedin performing a Worst Case Circuit Analysis. The goal ofthis task is to develop a document consisting of Worst CaseData Base (WCDB) Worksheets, as shown in Figure 8, for allelectronic parts utilized on a program. There worksheetsmust be filled out by an engineer experienced with electronicparts. These worksheets, serve as a quantitative assessmentof the dominant sources of variability for each part type forthe environments and life of the program mission.GENERATE A DOCUMENT CONTAININGWORKSHEETS OF WC "PARTVARIATIONSSAMPLE WCDB WORKSHEZ FORMAT[COMPANY NAME I ?ROGWM NAMEI[PARTS DATA*SE FOR WORST USEPARAMETER'8lAS?'a1-INITIAL TOLERANCE AT 25'CnoterIANALYSIS1RANDOM DATA?'aSOURCBCCMMUJTSFigure 7 shows a comparison of some typical piecepart initial tolerance values vs. the Worst Case limits. Thistable was compiled from an actual Worst Case Data Base fora major commercial satellite program. The Worst Case limitsshown were used in the actual designs of the satellite whichhas been operating successfully for approximately five yearswith five to go to complete its mission.1996 PROCEEDINGSAnnual RELIABILITY and MAINTAINABILITY SymposiumFIGURE 8329

The Worst Case Data Base will serve as a program uniformreference source to assure that all Worst Case CircuitAnalyses performed on the program (and there are many onlarge programs) utilize identical source data. You do notwant all design engineers on a program developing their ownW.C. Data Base. Once developed within a company, theWorst Case Data Base can be maintained, expanded andtailored for usage on other programs. The "how to develop"aspects of a Worst Case Data Base takes approximately fourhours in the WCCA Training Course mentioned in theSummary at the beginning of this paper and hence is farbeyond the scope of this paper which is only a brief overviewof a complex task. At this point the reader should simplyunderstand what the Worst Case Date Base is and theparamount need for its development and availability to alldesign engineers and analysts on a program who are involvedwith Worst Case Circuit Analysis.versa. It's predictable. A random variation is totallyunpredictable in which way the part value changes regardlessof which direction the environment changes. These bias andrandom variations can be different for different part types,e.g. a particular environment change may cause a bias changeof one part parameter type but a random change for another.Some drift effects may even be a combination of bias andrandom variations (e.g. Temp. Coef. 100 10 PPMIOC).The question being posed is "How do we combineBias and Random Variations to arrive at the WC MAX andthe WC MIN part parameter values. One acceptabIe method,and the recommended method unless dictated otherwise, onnumerous military programs is as shown in equations 2 and 3i.e. add biases algebraically and Root-Sum-Square (RSS) therandom variations. RSS is a statistically correct manner to STATISTICS OF PART VARIATIONSDRIFT DUE TO AGEING (RANDOM)DRIFT DUE TQ LO TEMP. (BIAS)\NOMINALDRIFT DUE TO AGEING (RANDOM)1DRIFT DUE TO HI TEMP. (BIAS)DRIFT DUE TO HARDVACUUM (BIAS)DRIFT DUE TORADIATION (BIAS)W.C. MIN.(-70.4%)W.C. MAX.INITIALTOL.-20% 20%( 51.4%)EXAMPLE: CAPACITOR, CLRNOMINAL VALUE 1200pfFIGURE 9One final important consideration in the development of aWorst Case Data Base which the reader should be aware of, isthe manner in which we can statistically combine theindividualized part variations (temp., life, radiation etc.),once the drift limits have been determined, to arrive at theWC MAX and WC MIN for the parameters. Looking ofFigure 8, notice that there are columns for entering thevariations due to the environments as either a "BIAS"variation or a "RANDOM" variation. The manner in whichthese variations are combined becomes a program choice ifnot dictated by the customer, and it does indeed make adifference. Figure 9 above illustrates examples of bias andrandom variations. A bias variation simply means that theparameter changes (increases or decreases) in the samedirection as the environment changes (increases ordecreases), e.g. if temperature increases, the part parametervalue increases (positive temperature coefficient) and vice330combine random variables.W.C. MIN NOMINAL VALUE - x NEG. BIASES- JE(RANDOMS)Z ( 2 )W.C.MAX NOMINAL VALUE z POS. BIASES { Z(RANDOMS):()Some agencies and prime contractors require that the randomterms be treated as biases and hence added algebraically asshown in equation 4 and 5.W.C. MIN NOMINAL - L NEG BIASES -I:RANOCMS( 4 1.W.C. MAX NOMINAL Z POS BlASES E RANDOMS( 5 ).This method creates a worst, worst case scenario and isknown as the Extreme Value Method (EVA) for combiningpart variations. Later, we will also discuss EVA and RSSanalysis of circuits for WCCA.1996 PROCEEDINGS Annual RELIABILITY and MAINTAINABILITY Symposium

2.2.2 OTHER CONTRIBUTING FACTORS TO A WORSTCASE SCENARIOAdditional factors contributing to a Worst CaseScenario which must be taken into account are the interfaceconnections, i.e. the bodcircuit input power, input signalsand loads, all of which have specified tolerance limits aroundthe nominal values. In performing a WCCA, all theseinterface values must be set to their limit and in the direction(k)which causes the biggest problem for the circuit attributeunder analysis.2.3 SENSITIVITY ANALYSISReferring back to equation #1 for the gain of theBand-Pass Filter, we stated that substituting the nominal partvalues for all the R's and C's into equation #1would yield again of 11.08 V/V and that substituting the initial tolerancevalues for all the R's and C's into equation #1 would yield again of 7.84 V/V. Using nominal values, it was a straightforward substitution of the part values. However, using initialtolerance values, which have an algebraic sign (k) with eachpart value, we had to make a choice of either the " " value orthe "-'I value for each part to drive the gain to a minimum.The problem arises as to what combination of the "parts"MAX and MIN values will yield the circuit parameter MAXand MIN values? We must determine the circuit sensitivitydirection response (i.e. circuit value increases or decreases)for the directional change ( or -) for each part. Worst CaseCircuit Analysis requires performing this same circuit"Sensitivity Analysis" since we are dealing with part MAXand MIN values. This analysis is absolutely mandatory sinceone wrong sign for any part will totally void the Worst Casesolution. The classical solution for determining the sign ofthe sensitivity for each part when solving for either the circuitparameter Worst Case MAX or Worst Case MIN is to takethe partial derivative of the circuit equation with respect toeach part individually. That will yield the "sign" of the partwhich must be used. For the Band-Pass Filter, the equation is2N aA 5Afo AfO(NOMIi liPj- 'i-WM)(6)Fortunently, many circuit simulators allow one to performthis sensitivity analysis. If such a program is not immediatelyavailable, there are other means to determine sensitivity e.g.substitute small incremental changes in each part individually(holding all other part values constant) and solve the circuitequation to see in which direction it changes (increases ordecreases). You could also sweep the gain over a fairly widerange of each part value and display these graphically asshown in figure 10.ISunIIFIGURE 10Notice in figure 10 that as C1 increases the circuit gain (Afo)increases (positive sensitivity for Cl) and as C2 increases thegain Afo decreases (negative sensitivity for C2).2.4 WORST CASE EVALUATION OF THE BAND-PASSFILTERTo evaluate the Worst Case minimum gain at thecenter frequency (Afo) for the Band-Pass Filter (Figure 4 andequation l), we need to determine the WC MAX and WCMIN for the R's and C's. This was done and is illustrated infigure 11.1996 PROCEEDlNGS Annual FELlABILITY and MAINTAINABILll'Y Symposium33 1

-Notice the significant difference between thenominal solution (11.08 V N ) and the initial tolerance (7.84VN) and the Worst Case (5.76 V/V) solutions.W.C. ,w -2:7W.C. MIN. I -29%C.f.20.JNR50. W.C.w- 4 3 %W.C. MIN. -2F.II. .-.II.Note that we do not have to drive all the R's and C's to theirWorst Case limits to cause Afo to fall below 7.0 V N , AfoWC 5.76 V N . There are numerous combination of onlyseveral parts exceeding their initial tolerance which wouldcause the gain to fall below 7.0 V N .The circuit approach taken, i.e. substitution of piecepart WC MAX and WC h4IN values into the circuit equationis called the Extreme Value Analysis (EVA). there are twoother circuit techniques which can be utilized for a WorstFIGURE 11Case Circuit Analysis. These are discussed briefly in the nextAll variations were treated as biases. Note that Vi and Vo of section.Figure 4 do not enter into equation 1. If they did we wouldhave to set them at their max or min tolerance also. We nowmust determine the directional ( or -) sensitivity of each 2.5 Alternate Techniques for performing Worstpart. This was doing using a simulator which performedCase Circuit Analysis (WCCA)sensitivity analysis. See figure 12.Two alternate approaches for performing a WCCAwillnowbe discussed. They are the ROOT-SUM-SQUAREPART VALUESENSITIVITYPART VALUEFOR A MAX.(RSS)Analysisand the Monte Carlo Analysis (MCA). Bothf0A 0PlPCItechniques are valid approaches. These two subjects consumePART(pf/C)(Pf/s4about an hour and a half in the WCCA Training Coursec:. i s m gf14X1532mentioned in the Summary and hence can be given only ac2. 1500 pf? 47015-32brieftreatment herein. Both techniques yield results whichR 1 . 1 5 i(n14.35515.3A5are more optimistic then the Extreme Value Analysis (EVA)R2.1SKa14.55515.345solution. Discussion of both techniques herein use an83.15 illl14,355i5.345R4.40.2 ,U39.27541.125example of a simple voltage divider circuit with four resistors,as. 70 icn97'7010.230(RI R4) and two internal voltages sources (VI and V2).R6. 1210 n11827 238The equation for the output voltage is given in-(2)fequation 7.FIGURE 12Substituting the WC MAX and MIN values into equation 1for Afo and in the proper direction dictated by the sensitivityanalysis yields Afo 5.76 V/V which fails the minimum gainrequirement of 7.0 V N by quite a bit. The Afousing allnominal or all initial tolerance values passed the requirementof 7.0 V/V. This was a real example from an actual WCCA.See figure 13.3321996 PROCEEDINGS Annual RELIABILITY and MAINTAINABILITY Symposium

ROOT-SUM-SQUARE (RSS)MONTE CARLO ANALYSIS PROCESSRSS is the statistical technique for combiningstandard deviations (a).The RSS approach is based on the “Law of largenumbers” (Central Limit Theorem) which states that if alarge number of variables are statistically combined theresulting distribution is a normal distribution independent ofthe form of the distributions of the variables which arecombined.e 3E?EATESLY SELECT %NOCM S C S OF VAL’JES.THIS ?ROOUCESA RANDOMLY cncsm E ? L X A OFTHE CIRCUIT FUNCTIONAL PEgFCAMANCE DISXBUTTCNex. -FlRRS;SELZCN (n I 1),A-;I1IIiA A m A AAThe determination of the standard deviation(0) of a normal distribution for any circuit attribute bymathematically combining the standard deviations of eachpiece-part based on the magnitude of the sensitivity of thecircuit attribute to the value of the piece-part is therefore avalid statistical approach.IICALCULATE Vo .FOR EACH SET VIA THE CIRCUIT WUATlONThe normal distribution of the output curve for Vohas a standard distribution related to the standard deviationsof the part parameters as shown in equation 8.The standard deviation of the output variable Vo willbe identified as OT . Multiplying the solution of equation 8by three will yield us the 3 0 (99.7%) value for Vo which isherein defined as the Worst Case value.I3E?%T ‘n’ TlMES (n4 PARAMETc3S)SORT AND GROUP SESULTS0GENERATE Vo HISTOGRAM OF FftEQUENCY DISTFIIBLJTION(8)FIGURE 14MONTE CARLO ANALYSIS (MCA)The Monte Carlo Analysis is hereby defined as:The empirical determination of the statistical distribution ofany circuit attribute by the repeated evaluation of thatattribute under various circuit conditions in which the valuesof each piece-part are randomly selected.The MCA process is illustrated in Figure 14.From the area of a histogram, one can now calculatethe circuit Mean and Standard Deviation (0). The 30(99.7%) value is again defined as the Worst Case Value.Fortunently, there are numerous simulators available whichperform a Monte Carlo AnalysisCONCLUSIONSElectronic production hardware requiring reliableoperation over a period of time should never be built based oncircuits which were designed utilizing only the nominal orinitial tolerance values of the piece parts. Part values willdrift over the life and environments of a mission after beingassembled onto circuit boards. Worst Case Circuit Analysisis not a major deviation from the classical circuitdesigdanalysis which electronic engineers normally performin their daily work, given that the Worst Case Part Variationsare developed or made available to the designer. In general,the required information to develop a Worst Case Parts DataBase (analytically) is available or can be extrapolated orestimated with rationale.Worst Case Circuit Analysis on electronic circuitsand systems has been performed for many years and the1996 PROCEEDINGS Annual RELIABILITY and MAINTAINABILITY Symposium333

approach and analysis methods described herein have beenacceptable to government agencies and major primecontractors.ACKNOWLEDGMENTWe wish to acknowledge the efforts of Mr. HarryPeacock, who, while serving as Technical leader forReliability Analysis for NASA's Jet Propulsion Laboratories(JPL), personally managed and contributed significantly tothe modification of Design and Evaluation, Inc.'s "WorstCase Circuit Analysis" Training Course while it was beingupgraded under contract to JPL.REFERENCES1. Design and Evaluation, Inc., Worst Case Circuit AnalysisTraining Course, Design and Evaluation, Inc., LaurelSprings, New Jersey, 1989. Taught at NASA's Jet PropulsionLaboratories, NASA Johnson Space Center, NASA MarshallSpace Flight Center, NASA Lewis Research Center, HarryDiamond Labs, Rocketdyne, Canoga Park, Ca, FordAerospace (Loral) Palo Alto, Ca., Martin Marietta Corp.,Orlando, Fl., GE Aerospace (Lockheed Martin) Camden, NJ,and Syracuse, NY,Honeywell, Phoenix, Az and Minn, Minn,Ford Motor CO, Dearborn, Mi and numerous seminarsconducted by D&E open to the public since 1989.'I2. Design and Evaluation, Inc, "Worst Case Circuit Analysis"Handbooks (5 Volumes) 1989, Design and Evaluation, Inc.,Laurel Springs, NJBIOGRAPHYWalter M. SmithDesign and Evaluation, Inc.1451 B Chews Landing RoadLaurel Springs, NJ 08021 USATele: 609-228-3800 (work), 609-227-8020 (Hm)Fax:609-228-9687Walter M. Smith is President and General Managerof Design and Evaluation, Inc @&E) which is a Reliability,Maintainability, Worst Case Analysis consulting company.He has held this position since 1976 to present. His personalexperience includes over thirty (30) years in both ReliabilityEngineering and the analysis of electrical and electronicsystems. Mr. Smith previously spent ten years with theGeneral Electric Co., in Phila., Pa where his primeresponsibility was analyzing and improving the Design andReliability of missile arming and fuzing systems, missileelectrical power systems. Telemetry systems and navigationand guidance systems.Mr. Smith holds a BSEE from Drexel University andhas done graduate work at the University of California, SanDiego and Penn State University.3341996 PROCEEDINGS Annual RELIABILKY and MAINTAINABILITY Symposium

Key Words: Worst Case Analysis, Electronic Parts/Circuit Tolerance Analysis,Worst Case Scenario, Worst Case Part Variations, Sensitivity Analysis, Extreme Value Analysis, Root-Sum-Square, Monte Carlo Analysis. SUMMARY & CONCLUSIONS This paper has been prepared utilizing material from Design and Evaluation, Inc.'s (D&E) Training Course and

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