34 IEEE TRANSACTIONS ON RELIABILITY March

EMPIRICAL PARAMETER VARIATION ANALYSIS FOR ELECTRONIC CIRCUITS

STUART KLAPP, MEMBER, IEEE

Summary-Procedures for empirical param- The objective of parameter-variation analysiseter variation analysis are presented which should is to improve circuit designs from the point ofbe useful in practical programs of design analysis view of drift sensitivity so that the occurrences ofand reliability improvement for electronic circuits. drift failure can be reduced. Specifically, the tech-The objective of the technique is to estimate the nique enables the design engineer to evaluate hiseffect that drifts in component-part parameters design and his choice of component parts in termswill have upon circuit performance. This esti- of possible parameter drift, so that he will be ablemate is obtained by application of formal statistical to identify areas where improvement is needed.and/or worst-case analysis methods in conjunction Use of such analysis techniques leads to higherwith systematically made circuit 'breadboard' actual reliability of electronic circuits, althoughmeasurements and data from component-part test the improvement will not always be evident inprograms. The mathematical basis for the tech- simple reliability prediction calculations per-nique is presented, along with a description of formed by addition of failure rates. This methodmethods to simulate part-parameter drifts and of estimating reliability often results in overlyother suggestions for conducting the required optimistic predictions since catastrophic failuresbreadboard measurements. are the only mode considered. These estimates

are valid only if the design has been carefully ex-amined from the point of view of drift sensitivity,

INTRODUCTION preferably by use of a formal technique of varia-bility analysis such as one of the methods to be

Electronic circuits can fail to perform in a discussed in this paper.satisfactory manner for either of two reasons: Variability-analysis techniques combine data

on the variation in component-part parameters1) Catastrophic failure in which one or more with information on the sensitivity of the circuit

component parts (e. g., resistors or transis- design to such variations in order to determinetors) may fail completely by exhibiting open- the statistically anticipated variation in circuitcircuit or short-circuit characteristics. performance. Component-part-variation data are

2) Drift failure in which parameters (e. g., re - becoming available from various test programs,sistance, or transistor "h" parameters) of and are not generated as part of the analysis tech-one or more component parts may drift in nique. The procedures to be discussed are used tovalue to such an extent that the circuit no estimate the sensitivity of the circuit design tolonger operates satisfactorily, even though part-parameter changes and to combine this infor-no component part has completely failed. mation with component-part data to estimate circuit

variability.Both of these kinds of failures must be prevented ifreliability is to be assured, although the need to re-duce drift failures is sometimes overlooked. Theerror of this kind of oversight was indicated early ARIATY ANALYSISin the history of formal reliability programs by the VARIABILITY ANALYSISAdvisory Group on Reliability of Electronic Equip- Both mathematical and empirical methods ofment (AGREE) when it reported that drift failures Bothavematinlund emp methodsofappear to be the dominant cause of field malfunc- analysis have been under development for sometions in electronic equipment [1]. time to be used in estimating and reducing the

probability of drift failure of electronic circuits [2] .Manuscritreceied Apri,1963.In the mathematical methods a mathematical model,

The author is with the SCM Corporation, Data Proc- or system of equations, is used to represent the be-essing Systems, Oakland, Calif. He was formerly at the havior of the circuit. Component-part parametersBattelle Memorial Institute, Columbus, Ohio. in the model are varied systematically, and the

1964 KLAPP: PARAMETER VARIATION ANALYSIS 35

equations are solved to determine the response of cludes estimates of the variance of the importantthe circuit to these variations. A digital computer circuit performance variables such as circuitmay be used to mechanize both the parameter- voltages, currents, gains, etc. The method isvariation procedure and the solution of the based on the statistical propagation of variancecircuit-model equations. Special routines for equation2 which, for the present purposes, canpart-parameter variation have been developed be written as follows:to obtain maximum information from the circuitmodel [3] -[5]. 2 (NaV2 2

The use of these mathematical methods depends V . aPIupon the availability of a suitable mathematical i= i imodel or system of equations to represent the cir-cuit. These models involve the use of linear ap- N-1 Nproximations, and are, therefore, not suitable for + 2 aL / a ause with nonlinear Class "B" or Class "C" ampli- i=1 aji\ Pi2M Pj) ij PiPjfiers, or with digital or switching circuits whenthe nonlinear transition between linear states of whereoperation is a critical factor. Furthermore, con-siderable engineering time and cost are involved 2in writing the equations, even for linear circuits. a = the variance of one output variable, V, ofFor these reasons, there has been considerable the circuit. This variance is the quantityemphasis on empirical methods which use a we wish to obtain from the analysis. Thephysical breadboard of the circuit rather than a output variables of interest might includemathematical model as the basis for the circuit voltages and currents, or transferanalysis [6] - [ 8] . functions such as gain.

Most of the empirical techniques now in use 2are informal in their parameter-variation pro- aP = the variance of the ith component-partcedures and, therefore, do not provide a compre- i parameter, Pi (resistance, capacitance,hensive estimate of circuit drift sensitivity. This transistor "h" parameter, etc.). Thispaper describes how certain formal procedures of quantity comes from component-part testparameter variation, which were developed for use data.with the mathematical methods of analysis, can beadapted for use with the empirical or testing ap- pi. = the correlation coefficient (-1 . p . + 1)proach. The particular parameter-variation between parameters Pi and Pj which canmethods to be used are the Moment Method [4] often be assumed to equal zero. Thisand the Worst-Case Method [5]. These formal quantity also comes from component-parttechniques offer many advantages since they com- test data.bine the procedural formality of the mathematical a vmethods with the wider applicability and lower a p - the partial derivative relating changes incost of empirical methods. 1 the circuit output variable, V, to changes

in the ith component-part parameter, Pi.

THE EMPIRICAL MOMENT METHOD In performing the Empirical Moment method thepartial derivatives are estimated numerically from

The Empirical Moment method for circuit- the expressiondesign variability-analysis is based upon a similartechnique previously developed for use with mathe- av AVLmatical models [4]. *This technique requires a P. A P.component-part parameter-variation data from"life tests" which are expressed in terms of where AV is the measured change in a circuit per-parameter variance1 at a particular point in time. formance variable V due to a known change APi,The information obtained from the technique in-

1 2 This relationship is discussed in most standard textsVariance is a statistical quantity which measures the on applied statistics. For example, see A. Hald, "Sta-dispersion or "spread" of a distribution of values. tistical Theory With Engineering Applications, " JohnThe square root of variance is termed "standard Wiley and Sons, Inc., New York, N. Y., pp. 117-118;deviation" or sigma. 1952.

36 IEEE TRANSACTIONS ON RELIABILITY March

which has been made in component-part parameter extent to which circuit behavior is sensitive toPi. This measurement is performed on a circuit changes in each component-part parameter, canbreadboard with all component-part parameters help the design engineer determine where preci-other than Pi held at their nominal values. Par- sion components are needed, and where lowertial derivatives estimated in this manner are priced wide-tolerance parts will be satisfactory.substituted into the propagation of variance The propagation of variance equation can be anequation, together with variances from component- aid to the designer in determining the causes ofpart test data, and the formula is evaluated to de- excessive circuit performance variance since thetermine the variance of one circuit output variable. designer can study this equation and determineThis process is then repeated for each output a\2 2variable of interest for the circuit. which uP.)uJ terms are contributing most

Component-part test data are frequently given i iin terms of limits for parameter values. Variances heavily to circuit variance. He can then decidecan be approximated from such data by use of the whether to reduce the large terms by using morefollowing formula: stable component parts (reduction of (X2 ) or by

P.1

2 FUpper Limit-Lower Limit] 2 modifying the circuit configuration in such a waya = L ~- ~ 6 j that circuit sensitivity to parameter changes is

This relationship assumes that the upper and lower reduced (reduction of (a PY))limits represent approximately the + 3a points ofthe distribution of values.

It is important to note that the propagation ofvariance formula gives an estimate of the statisti-cal variance of the circuit when all part param- THE EMPIRICAL WORST-CASE METHODeters are varying concurrently in a mannerrepresentative of actual drifts. This is true even The Empirical Worst-Case method is also re-though the partial derivatives themselves are ob- lated to a corresponding technique for use withtained by varying only one part parameter at a mathematical models [5]. It differs from thetime. The partial derivatives provide information Moment method in that it is not statistical inas to the sensitivity of the circuit to part-parameter nature. That is, it does not consider the proba-changes, while part data provide information as to bilities associated with various possible com-the amount of change to be expected in the part binations of component-part parameter drifts.parameters. When these two kinds of information In worst-case analysis, all component-partare combined in the propagation of variance form- parameters are simultaneously set to their end-ula, the result is useful information on circuit of-service tolerance limits in the combinationperformance variability. which is most cumulative in effect to obtain

The values of circuit performance variances worst-case maximum and minimum values forobtained by this method represent estimates of the each circuit performance variable. The resultstatistical variability of the circuit performance obtained from this worst-case analysis is a yesvariables. The square root of the variance, the or no answer to the question, "Will the circuitstandard deviation, becomes an indication of "rms perform satisfactorily under any possible com-error." An excessive variance indicates that the bination of component -part parameter drifts? "circuit will not behave properly under conditions No statistical information is obtained, but whenof component-part-parameter variation. In some the answer is yes, one has complete assurancecases it may be useful to assume that the distribu- of freedom from drift failures. When thetion of the performance variables is approximately answer is no, however, one must not neces-normal in order to estimate the probability of a sarily conclude that the circuit is unsatisfactory,circuit performance variable falling outside any since the low probability of occurrence associatedpredetermined circuit tolerance limits. with certain combinations of part-parameter drifts

In addition to this performance variance infor- has not been considered.mation, the Moment method of analysis provides While worst-case analysis is often suitable forsupplemental information for the circuit designer digital-type circuits, it is usually difficult or evento aid him in modifying his design if this is neces- impossible to design analog-type circuits so theysary. The partial derivatives, which show the will operate under all worst-case conditions. Im-

1964 KLAPP: PARAMETER VARIATION ANALYSIS 37

posing worst-case requirements on analog circuits MEASUREMENT PROCEDUREScan often lead to overdesign and unneeded circuitcomplexity with the result that the reliability may In order to conduct an empirical circuit analysis,actually be reduced due to the increased probability whether moment or worst-case, it is necessary toof catastrophic failure of a complex circuit. There- construct a special-purpose breadboard circuitfore, worst-case analysis should be applied with model which permits component-part parametersdiscretion primarily to digital circuits and to the to be varied, and to provide instrumentation withdc bias criteria of some analog circuits. The sta- suitable sensitivity and accuracy as required intistical Moment method is more applicable to the the analysis. The procedures outlined in this sec-over-all behavior analysis of most analog circuits. tion have proven to be workable and convenient.

In worst-case analysis the part-parameter in-formation from life tests is presented in terms of General Concepts Employedmean value, upper-limit value, and lower-limitvalue. This type of part data is often more readily Certain basic concepts and ideas as listed belowavailable than the variances (and correlation co- are employed in the construction of the breadboardefficients) required for the Moment method. In circuit model and in providing the related instru-some cases the form of the available component- mentation:part data may dictate the method of analysis to beused. 1) Modular breadboard construction is used,

To perform an Empirical Worst-Case analysis Circuit elements (e. g., resistors, capaci-the engineer sets up a "breadboard" circuit with tors, and transistors) are constructed asthe part parameters at the worst possible com- separate modular units complete with pro-bination of tolerance-limit values, and then vision for parameter variation. Thesemeasures the performance of the circuit under modules can be held in a "library" andthese conditions. If the performance is satisfac- used in the assembly of more than onetory under these conditions, then it can be assumed circuit model.to be satisfactory under all conditions of parameter 2) High-low-nominal switching is used fordrifts. In order to avoid setting up all possible allpatarmetrs Provisin is madecombinations of limit parameter drifts, the engineer all part parameters. Provision iS maden . . ...................to instantaneously switch (rather than con-can use the signs of the partial derivatives which tinuously vary) all variable part parametersrelate circuit performance changes to changes tween nominal arigh ort valueterin part parameters as an aid in determining which between nominal and high or low values. Forcombination of parameter drift limit values will be example aw100-hm resior odul mightmost cumulative and, therefore worst in the effect ovid swit ale ale o5 ohms (high),on circuit performance. For example, suppose he 100 ohms (nominal), and 95 ohms (low). Thison circuit pefrmne.Frxenables changes in output to be convenientlywishes to set up the part-parameter drift combina- g ytion which will give the lowest circuit gain. If in- determined by merely switching the partcreasing resistor R1 reduces gain (sign of partial parameter while viewing a meter connected

across the output voltage. Drift effects inderivative is negative) then the high-limit value the circuit or instrumentation do not maskshould be used for Rl. If increasing resistor R2 the partial derivative being measured whenincreases gain (sign of partial derivative is posi- this procedure is used.tive), then the low limit value should be used forR2. This procedure is repeated for each 3) Simulators are provided for semiconductorcomponent-part parameter in the circuit. Then devices. These simulators permit inde-the worst-case combination of drifts is set into pendent variation of parameters (e. g., hfethe breadboard and the measured gain is the and hie) by using auxiliary circuits in con-worst-case minimum. The entire procedure is junction with a "nominal" device. Therepeated to obtain worst-case maximum and nominal semiconductor device providesminimum limits for each circuit performance the basic characteristic as exhibited byvariable of interest. For example, one might the simulator, including nonlinear effects,wish to obtain maximum and minimum limits while the auxiliary circuits are used tofor gain, input impedance, transistor bias and modify these characteristics so that pa-power dissipation, etc. rameters can be adjusted as desired.

38 IEEE TRANSACTIONS ON RELIABILITY March

4) Differential instrumentation is employed sistor base, thereby lowering the effective hfe asfor use with the Moment method. Since observed from the external terminals. The ele-this method requires the experimenter to ments which make up the bypass path were de-determine changes in circuit performance termined experimentally as a combination whichdue to changes in input parameters (partial gives a nearly constant current division ratioderivatives), the instrumentation require- when placed in parallel with the transistor base.ments emphasize sensitivity to small This ratio, and therefore hfe, is varied by ad-excursions in output level, rather than justing the potentiometer labeled "hfe." Theaccurate absolute measurements. For input resistance hie as seen externally may beexample, it may be possible to measure varied by adjusting the potentiometer labeledchanges in output voltage of a fraction of "hie" which is in series with the transistora per cent, although the absolute meter input resistance.accuracy may be only 1 or 2 per cent. To use the simulator, the operator first

selects either a nominal or a high gain speci-The Transistor Simulator men of the same type of transistor he wishes

to simulate and plugs this transistor into theA transistor simulator has been developed simulator. He then adjusts the simulator for

which enables the user to vary transistor param- the desired set of parameters while viewingeters hfe and hie as seen from the external ter- the simulator characteristics on a transistorminals of the simulator. This simulator is curve tracer or while measuring the parametersbased on a refined version of the original idea on some other suitable instrument. The simu-proposed by Morgan [8]. The circuit used is lator can then be plugged into the circuit bread-shown in Fig. 1, where the "B," "E," and "C" board, and the switches on the simulator willterminals represent the external base, emitter, enable the operator to select between the originaland collector terminals, and the internal tran- and the preadjusted value for each transistor pa-sistor is a nominal sample of the same type as rameter (hfe and hie).is being simulated. The simulator is constructedas a plug-in module, and may be removed from Instrumentationthe circuit and plugged into a "transistor curvetracer" for adjustment. The circuit shown is for The basic circuit used for measurement of dcan n-p-n transistor; diode and battery polarities voltages for determination of the partial deriva-would be reversed for a p-n-p version. tives is a differential voltmeter as shown in Fig. 2.

The adjustment of hfe is effected by providinga bypass path in which some of the base currentas seen externally is allowed to bypass the tran- Senstte dc Meter

to be Measured Reguloted Voltage

T Supply

Nominal Specimenw \/arisble hie of Transistor

Norm o c Fig. 2-DC instrumentation

h 500SLw <b oEle The regulated voltage supply is set at the nominalvalue of the voltage being measured, so that changein output may be read directly from the voltmeter.

Variable h The same arrangement is used for making acNormal ' hfe measurements, except that the ac signal is first

s;Germonium converted to a proportional dc potential by usinga peak-reading rectifying circuit.

hfe 0

RESULTS

+ The Empirical Moment and Worst-Case methods15v as presented in this paper have been applied suc-

cessfully to a nonlinear push-pull power amplifierFig.1-The transistor simulator. circuit. In this particular circuit the worst-case

1964 KLAPP: PARAMETER VARIATION ANALYSIS 39

limits on gain and input impedance as well as dc complete circuit design analysis information isbias points were within circuit specification; and, essential for high-reliability programs and isas would be expected for a circuit with satisfactory certainly desirable in any design project.worst-case limits, the variances of the circuit out-put parameters were also satisfactory.

These techniques in a less fully developed formhad previously been applied to three circuits at an ACKNOWLEDGMENTearly stage in their design. One of these circuits,a detector, was found to be completely satisfactory; The author would like to express his apprecia-another circuit, an amplifier, was found to have ex- tion for the support of the Aeronautical Division ofcessive sensitivity to parameter changes (high Minneapolis-Honeywell Regulator Company whichvariance) so that a design revision was needed; a sponsored work related to this paper at Battellethird circuit, a high-gain amplifier, was found to Memorial Institute, Columbus, Ohio. Specialoscillate under certain conditions of parameter recognitition is due to D. Mark of Battelle whodrift and was also revised. This program of de- made important suggestions related to this effort.sign analysis was conducted with the close cooper-ation of the circuit design engineers and wasconsidered to be a design tool to aid them inevaluating and improving their circuits and in REFERENCESdemonstrating the suitability of their finaldesigns. [1] Office of the Assistant Secretary of Defense,

Advisory Group on Reliability of ElectronicEquipment (AGREE), "Reliability of Military

CONCLUSIONS Electronic Equipment," Task Group 2 Rept.,App. A; 1957.

The feasibility of a formal empirical approachto circuit design variability analysis has been [2] D. M. Larson, and B. J. Grinnell, "A Comi-demonstrated. This technique consists of system- parison of Methods of Drift Reliabilityatic collection and analysis of data from a special Determination," Proc. Seventh Nat'l. Symp.breadboard circuit model. The technique effective- on Reliability and Quality Control, pp. 448-ly utilizes the formal procedures of parameter 454; 1961.variation, previously developed for use with adigital computer, to enable complete statistical [3] L. H. Stember, H. S. Scheffler, and J. J.and/or worst-case analysis results to be obtained. Duffy, "Circuit Analysis TechniquesSince the technique involves the use of a circuit Utilizing Digital Computers," Proc.breadboard rather than a specialized set of Seventh Nat'l. Symp. on Reliability andmathematical equations to represent circuit be- Quality Control, pp. 361-374; 1961.havior, it is more generally applicable and mayrequire less engineering time and expenditure [4] A. P. Lechler, D. G. Mark, and H. S. Schef-of funds than is the case with the purely mathe- fler, "Applying Statistical Techniques to thematical methods of circuit analysis. Analysis of Electronic Networks," Proc.

The parameter-variation analysis techniques 1962 Nat'l. Aerospace Electronics Conf.;represent a practical and low-cost solution to the pp. 168-172.problem of circuit design variability analysis.They can be applied with a minimum of investment [5] H. S. Scheffler, J. J. Duffy, and B. C.in equipment and also with a minimum of re- Spradlin, "Mandex--A Worst-Case Circuiteducation of the circuit designers, since these Analysis Computer Program," Proc. 1962engineers are accustomed to the informal use of Nat'l. Aerospace Electronics Conf.; pp. 38-breadboards to validate their designs. The 53.techniques represent a formalized approach todesign procedures that are already familiar to [6] B. J. Grinnell, "Analysis Testing for Im-the designer. By being formal, however, the proved Circuit Reliability," Proc. Eighthtechniques enable a much more complete analy- Nat'l. Symp. on Reliability and Quality Con-sis of circuit behavior to be conducted. Such trol, pp. 103-108; 1962.

40 IEEE TRANSACTIONS ON RELIABILITY March

[7] D. Harnett, and F. Vieweger, "Laboratory [8] H. L. Morgan, "Limit Testing ElectronicEvaluation of Circuit Reliability," Proc. Circuitry," Electronic Equipment Engi-Symp. on Guided Missile Reliability, pp. neering, pp. 45-50; March, 1962.45-51; December, 1958.