CC BY
22Journal of Siberian Federal University. Engineering & Technologies, 2018, 11(5), 512-527
yflK 621.3.019.34
Monte Carlo Numerical Method
in the Problem of Temperature Stability Analysis
of Electronic Devices
Denis V. Ozerkin*a and Sergey A. Rusanovskiyb
aTomsk State University of Control Systems and Radioelectronics
40 Lenin, Tomsk, 634050, Russia bPolus, JSC 56 v Kirov, Tomsk, 634050, Russia
Received 04.10.2017, received in revised form 02.01.2018, accepted 06.05.2018
The problem of ensuring high temperature stability ofparameters by reason of the characteristic features inherent in the integral performance becomes especially urgent for microminiature electronic devices. For the mathematical description of the electronic devices' temperature error it is proposed to use the method of experiment's statistical planning in combination with regression analysis. There are classes of electrical circuits in which the output parameter depends mainly on one-parameter electrical radio elements. In the article it is shown that the problem of obtaining the temperature error equation for such electrical circuits can be reduced to the problem of effective finding of the one-parameter electro radio elements' influence coefficients. A modification of the Monte Carlo statistical method with the computational factor experiment's scenario to find the temperature error equation is considered. Approbation of the proposed modification is carried out using the example of the electric circuit of the generator with the Wien bridge.
Keywords: electronic device, electrical radio elements, temperature stability, circuit simulator, SPICE model, factor experiment, regression analysis, Monte Carlo statistical method, temperature error equation.
Citation: Ozerkin D.V., Rusanovskiy S.A. Monte Carlo numerical method in the problem of temperature stability analysis of electronic devices, J. Sib. Fed. Univ. Eng. technol., 2018, 11(5), 512-527. DOI: 10.17516/1999-494X-0050.
© Siberian Federal University. All rights reserved
Corresponding author E-mail address: [email protected], [email protected]
Численный метод Монте-Карло в задаче анализа температурной стабильности электронных средств
Д.В. Озеркина, С.А. Русановскийб
"Томский государственный университет систем управления
и радиоэлектроники Россия, 634050, Томск, пр. Ленина, 40 бАО «НПЦ «Полюс» Россия, 634050, Томск, пр. Кирова, 56 в
Для микроминиатюрных электронных средств особенно актуальной становится задача обеспечения высокой температурной стабильности параметров в связи с характерными особенностями, присущими интегральному исполнению. Для математического описания температурной погрешности электронных средств предлагается использование метода статистического планирования эксперимента в сочетании с регрессионным анализом. Существуют классы электрических схем, в которых выходной параметр зависит в основном от однопараметрических электрорадиоизделий. В статье показано, что задачу получения уравнения температурной погрешности для таких электрических схем можно свести к задаче эффективного нахождения коэффициентов влияния а1 однопараметрических электрорадиоизделий. Рассмотрена модификация статистического метода Монте-Карло по сценарию вычислительного факторного эксперимента для нахождения уравнения температурной погрешности. Проведена апробация предложенной модификации на примере электрической схемы генератора с мостом Вина.
Ключевые слова: электронное средство, электрорадиоизделия, температурная стабильность, схемотехнический симулятор, SPICE-модель, факторный эксперимент, статистический метод Монте-Карло, регрессионный анализ, уравнение температурной погрешности.
Introduction
A significant place in the modern electronic devices' (ED) design is the task of ensuring the temperature stability of ED parameters under both external (environment) and internal (heat generation in electrical radio elements) thermal effects. The new element base and constructive material use, the new technological operation implementation lead to an essential reduction in the ED mass and volume. In general it affects the operational, design, technological and economic indicators positively. At the same time, for microminiature ED, the problem of ensuring parameters' high temperature stability becomes especially urgent, due to the characteristic features inherent in the integral design: the increase in the specific dissipated power of electrical radio elements (ERE), the mutual parameter correlation, the heat transfer complex mechanism, etc.
The earliest domestic publication on the ED thermal stability is work [1]. The book provides a detailed error analysis arising in the ED production process. Based on analysis, the book authors propose a technique for calculating the radioelectronic equipment tolerances. The technique involves a combine use of the electrical tolerance theory, probability theory and mathematical statistics. As an
approbation of the proposed methodology, the; book authorsgive estimatesof tolerancesfor several typknai cltotriean esscufta opet;sadyna; yn t eiatinuous and pulsedooode.The toeci cepranCekctrical circuiee are oonsidoreil orpaoately.
in a lorer punco e2i tir^ cnleulaeinn erlpoanse ee cdlani.tfue eotzi jioss^cp in i 1] rss^e found tv ire a logical ecicdeton. Tde eechnkiunpecvtD ko f(SPItr the ED mechnnical aited el(ccdaepar toeecance elteory. Tlee nlbefssaa mathamoticai mof ah in dicia rhcony ce dhr aelalivo earna enuntiro of N EDautput parameter:
AN = h
u =jo
/=0
= j(qo,q2,-,qn) a.
dq/ P/(qo,q2v•• ^n) '
AT, (1)
where a<p(q1,—2'''"' q"(-—-= B i - are the influence coefficients; a, - is the temperature
dqe C2 (qn,q2,...,q n(
coeikitieni nO /'-th ERE paraeeter; AT - is the atfference hetween Che ERE operating temperature antl Oie amitem ipmpcratneei ri,sst, ..., «n« -are ER— paramoleos; OtCb q2, ..., qn) - is the analytical dependenceof N ED outputparameter versusthe EREparameters.
Further development of the ED electric tolerance theory with reference to external and internal lon^ersdnie mflutnees was fvdtfl ik [3]. Tne auihor proceed an eafe^tts'-mrta(^(ifop nnding the li. ir^flit^n^ea(^(^eiOicipiite (if ihe tempeiaiure ne^i^t^i" onuatioa (t). kho nssenae of -kemelhod is the use of exeerivp^itat stotiiticaU pHanning in wililr regnessrnn analyst. M tiiir ense the global
mathematicnl modeO Ss ttio reyreeeion equdllon:
k k k 2
n = b0 + X biti + X bijtil j + X biil( + - ' (2)
i=1 i< j +=1
where n - statistical evaluation of N EDoutput parameter; b0, b,, btj, b u - are theempiricalcoefficients oftheregression equation.
The regression equation (2) makes it possible to estimate both linear and nonlinear interactions, depending onthe experimentalplan.
In the same paper [3] it was shown that for multiparameter ERE it is expedient to make a variation nsst by individual parameters but dirergp affect /'-th element by temperatuie. If during statietical date froceislng we additinnaUy ii^rn^^uiz(; the nagressio ncneffitionts in(2), then weobtain the temperature erzor equation:
-Vnut J AT; " J AT; jj
—= Trai—+5121 aia i — +•••, (3)
Nnut £ ; T * ; j T Tj
b; -Tq
where T; =--is the influence coefficient of the /'-th ERE; T0 - is the nominal temperature (zero
AT ■ bQ
AT;
vbribtlon level); TT - is -he temeerature variation intaerve.^; —- relative change in the ERE operating temperature. 1
The practical use of the temperature error equation (3) was demonstrated in [4, 5]. In particular, a regression analysis of the electronic circuit temperature stability was developed using computer circuit simulators,suchasCadenceOrCADandSpectrumSoftwareMicroCAP.
Problem statement
According to the invettigatien oesults in [P, 51 it was establiehed that tPevariation intheERE operating Semprratcse in 1he facton experimenl isnat always jvstified. Thora are ele ntrieaS c^ii^^u^^ t^la^ es 5n whixb TO ED rePotd parametes dtpends cr ane-serapeier ERE i^^^^s^issx b^pec^^te^^m inducOers) csa[nsy. Thosefose0 wgen nurrymg oug ¡a factor ecporiment is is ioffinient to gory hyone parameter for taoh suols EREo There is no oeod tea tats SiiSn noeoant tlie Ibmpeoatsiro depentttnne cf ige interrelated Ccrii"esctiteiS) paeamater complex within one ERE. Based on this, equation (1) takes the form:
N
i_l
dPiTi ) \i (x de= p i(ei ) 1
XT. (4)
The difference in the experiment planning course lies in anothermechanism for obtaining a normalizationfacto r:
or-r^- o(5)
q -n0\
whe re A=,-= - <7,Bf - id tlie varia]it) n int(2)nial(^l^e^paof z'-ih ERE nanometer.
AppSying tte sltiOi^seiitiatic^at ojieir^tin^n to (4) for nasli q, taotor, we obteinche timporatureniror equationancording to [2]:
AN
N
k k k /V)
T Aiaqi + raAijaq1a qj + EAnlaq- X ' i=1 i < j i=1
AT, (6)
dN q-Q d^N eiOe ¡0 ^li^M^e A- =--— - is the influence coeffieicnt oJF ^in^^jatr term a ; As =---i s the m ixed
N nq j dqfiqj N0
cecned-ordor influence f-clor cCcracterizing trie (t-p) paireactor inieractten on the outpot parameter; An -ce —M—A0 n secondt-rdea rnfluenco nooffieieni.
-r N
)n rhefinoieorm,ihc iemporatnre craor nquation Cor one-porameter ERE ts:
AN n n n
~NSdnL= ZapiAdTi + % Zaya.ATjATj + ..., (7)
Ndni i=1 i=l j==1
where a - is the influence coefficient of i-th ERE thermal dependent parameter; b, - regression
coefficient; a, - maximum temperature coefficient value of the variable parameter; AT,- - working
temperaturechanging of ,-th ERE.
Thus, the problem of obtaining equation (7) can be reduced to the problem of efficiently finding
the a coefficients for electronic devices, at that temperature function of the N output parameter depends
primarily onone-parameter electroradioelements.
Research theorypart
The aim of the research is to improve the system design method of the thermostable ED with the numerical Monte Carlo method.
A common method of electrical circuits' statistical analysis is the Monte Carlo method (the statistical test method) [6]. At the same time, the use of this method as applied to the study of the ED temperature stability generates two main problems:
1. Search or synthesis of ERE mathematical models by criterion of adequacy of parameters' temperature dependences to their real characteristics.
2. Monte Carlo method realization according to the factor experiment scenario in order to justify minimization of the statistical test number.
The complexity of first problem solving is due to the fact that there are practically no adequate ERE mathematical models with parameters' temperature dependences applied to the domestic element base. Separate attempts to create such models have been undertaken in [7, 8]. This article focuses on solving the second problem - the implementation of the numerical Monte Carlo method in the problem of analyzing the ED's temperature stability.
The calculation block diagram using the Monte Carlo method (Fig. 1) includes the basic procedures:
1. Random Q vector realization, i.e. generation of qt component parameters' random values in accordance with their distribution laws.
2. A single-variant analysis of the electrical circuit with the obtained random Q vector realization.
3. Calculation of the target V function's value in order to establish the ED output parameter.
4. Pre-set repetition of procedures No. 1, 2, 3, corresponding to the total test's number.
5. Statistical processing of the all tests' results.
Approaches in the implementation of No. 1, 3, 5 procedures are specific for the numerical Monte Carlo method in the problem of analyzing the ED's temperature stability. Let us consider this implementation in more detail.
It is known [9] that for single-parameter ERE (resistors, capacitors, inductors), the temperature coefficient characterizes reversible changes in the q parameter with a change in temperature. The parameter's temperature coefficient (PTC) aq - is its relative change with temperature change by a °C:
Wtth a parameCer'r liaear Cemparaturg depenelence,which is csuclly observedin a narrow range oa amaeent temperatures, tier Pi h c akulates as:
where qtT), aliei - ir tha pasameter's vclua, reeparsivaly, al an increcsrd (denneased) operating iemnaraSuee T act at a normal temperature T0.
if era hP) rependenor ii nenhpeat, ihonthe parametee'r temparoture sthbiliiy car be characterized byaralativcchange tn^e 5p param^er:
1 dq
a
error equation
Let ,-th ERE of the electrical circuit have a functional qtj parameters' set (Set,- vector): Set,- = {q,-1, q,2, q,3, ...}.
From the position of the temperature stability investigation of z'-th ERE, each functional parameter is a temperature function. Consequently, Set is also a vector function versus temperature:
Seti(T) = {qn(T), qfl(T), qfl(T), ...}.
In the special case of one-parameter ERE (elements with one pronounced functional parameter), the vector function degenerates into a one parameter's function versus temperature:
- 517 -
Set,(r) = {={)}-)( (8)
Tne sr enmj^^i^ne^ao , inhere nim /-rhonns^ramater ERE undenbothexteenaland internal thermal effeuto, is probabilttyvalue.buppose than ^beoj^roli^l^i-tyPTvnlue for d-ah ERE ur rubaeci to the normal distrfbuiionlaw:
f (-) =
1
V2n
nxp
CT-
(———0 )2
2ct2
where f(T) - is the probability density function of the random T value; T0 - is the normal temperature; c - is thestandard deviationoftherandom T value.
Define a certain range of operating temperatures for i-th ERE: [Tmin, Tmax]. According to the factor experiment scenario [5], the operating T temperature of i-th ERE can be equiprobably near the point T0min and T0max estimates. Then the sum of the probability density functions of a random T value is abimodal distributio n'io densityfunntion:
bi(T) _ f (Tmin) + f (Tmax) _ _
1
1
erms/2ji I c
-exp
(mmin d min)
2c
1r
-exp
CTmax TQnri.x)
2cm
whereronw-isthe normallzlagioeffirient; Tmm and dmax - random vailues of-merating timprraturns at the grcen range'- ttoundaries; cmin and amam - are ^lie; correspondmg amm and standamd deviations.
Obviously, the computationalfnctorexperimenh, in contrast to this real rxpeoiment, allom^^o^^ to specify the variation levels' values with high accuracy in each experiment realization. Consequently, under ege cnnditions ree iho eoaaiirulie^tio aial factoeM napetiment: Ocominst owd ^ C. nrus offiret Os groflucann sloowa on Fig. 2. TUe blmgdal distrihotion'r density funcUioncnntewrilten moce simply:
bOP)=-
1
norm • m
rJC/J0
exo
(pmig-
2n2
+ exjr
(yaicx ~ ffiF
2n2
The limiting case, when c = 0, allows us to introduce the discrete random T value's notion, i.e. operating temperature of i-th ERE, taking only the T0min and T0max values. The discrete random T value is determined by the Bi(T) distribution function analytically:
+
c
£ t« a o
£
.2 0,5
3
x>
tm
Temperature, °C
o 0,5
3
J3
/ tin' Tmax
Temperature, °C
Fi-. 2. The nimodol Oislrlbulion's densarfpnctionoftOe randomt vrlue: e -tcn^O) case, b - (a -
• 0) c ase
0 ; Tmin ] ;
Bi (t ) =
0,5 6 (Tmin ; Tmax ] lm(Tmax ;+co ) .
Thegraphicalformof the Bi(T) distribution function isshown in Fig. 3.
As a result, for /'-th ERE function (8), the definition domain consists of only two values:
(t )=
IT
q i i
min
T
maxl
Tl^e <7,)T)funcnon is, in its turn, one of the parametersaor the ED's relative error equation (4). Therefore, tie applied Of) eaaation 04), Üie aditsissibln hitiirpr^^titeis far i-th ERE are:
Qi =
The computational experimentfactors' area is the set of all q:
Q =
&=•
02 = Qn =
qi
qm
q2mm q2mlX
qmm
q«tax
where n - is the factor number,i.e. the one-parameterERE number arevaried in the experiment.
Under the conditions of realization of the computational factor experiment, a V vector will be obtained. The V vector's element is represented by the 0,- function of the ED's one-variant test, which containsaunique(non-repetitive) qt value's combination:
Fig.3.Thedistribution function of adiscrete random Ovalue
V =
Vm = miQiQli-miQ»^ Qm{l{
max max max I
q2 In J
wliere m = 2".
Statisticot on the computational experiment is aimed hit finding the coefficients'
regression. We introduce en auxiliary K matrix w ith m x (cml) f mnntisn csootiiintng^]^^ codevfiuss ofvariab (afacto rs' levels:
k =
1 -1 ... -1 1 -1 ... 1
11
1
2hg noro malrix cnlumm (listf^iaieoiit) contains uoit makst s j^iii^ iss inteodef to oaltutaiethe free term oftheregression equation:
b0 =
yT . K<0>
m
(9)
Tge tisrn^^ntintm ooiumns tfihe K mctrin |from I tc t] f eove io t OLcuma thg aocfficionOs' regression of ida cenrrspofhigg factor:
b =
vT- k<R>
bn =
V T. K <o>
(10)
m m
h'.KS factorr' tan^incac icttir^c;tirn inrire nompftatlena0 experi mttt^t cait be taken into account by fdding soiutnnt \oith the code reat(zg)irn of t)e <7,17, q,qtqe ... favors. Ii it oisvious that the nonlinear intoraoeion siil^TLiili: win lent So on tftreasf itt ilie d^mieti^^(tiii or" rlon tt motrix ta:
m x (w+H+X
wtero /t-is tixa numbfi oqfactoet' nonlinear interactions.
To hnt toe temperature error oicir^t^on ItX is ir saiy tr recaiculate the b, coefficients' eagpeeslon lo tic it,- inUugucq creffiaiente itii tho ttefmttly repaidem paramelor of /'-thERE:
0
sib )i Aqfs 0 '
(11)
wgere h;0 - the zero level of the variable parameter of i-th ERE; Aq - the variation interval el ihe l value.
Research experiment part
The object of experimental research is the electric circuit of the harmonic oscillations' generator with the Wien bridge (Fig. 4). The generator uses VD1 and VD2 zener diodes to limit the output signal.
- 520 -
ai =
Fig. 4. Wien bridge circuit
The circuit is characterized by a harmonic distortion coefficient in the range 1 ... 5 % [10]. To generate oscillations in the goneeahor, tha Barichnus en criarrton must 0a rompTiecO: tee phase nhifl of 180 e sncurs in the feedbnck irof; t8o total gain in the loop ie nsi lc ac Olnass one.
Ie t^tie cireiut R2 = R0h = and CI = C2 h C. Conseqnontlo, the ei^^c^hee^cal qoacitiesonance fresueneyis:
f = —— =-f-K = [59C3 Hz.
7 chrc en • fe4 •fe-a9
Using the frequency correction circuit (DA1, C3), the oscillation frequency is reduced to 10 kHz. Theaim nftne rerearcCexperimenO part ie to traC ihe trmpeealoce error ec[oation (7) for a generator
with aWdenbridne.ThsED'couttcleparametsr im^ie rciatineinsCabality -f- of the generation frequency.
One-parameter ERE,most influencingthe output parameter, are:C1, C2, C3, R2, R4.
The generator element base with the Wienbridge consists of four ERE types and four corresponding SPICE models (Table 1). For carrying out the computational experiment, the following ERE types are specified:
- for C1, C2 apply
0CK10-17B-M47-1H®±5 % B 0^0.460.107 Ty;
- for C3 apply
0CK10-17B-M47-63n®±5 % B 0^0.460.107 Ty;
- for R2, R4 apply
0CM P1-8Mn-0,5-10K0M±0,1 %-0,5-M-A-0m467.164 Ty
According to [11, 12], the resistance temperature coefficient of the resistors' types is aR = ±100i0-6 C~';the capacitance temperature coefficient of the capacitors' types is aC = -47i0-6 C-1.
- 521 -
Tablel.SPICEmodel listused in the Wien bridge modeling
No. SPICE model name Origin source
1 Thin film chip resistor P1-8Mn "SPICE-model development of Russian-made electronic component base's library" report
2 Chip capacitor for surface mounting K10-17B "SPICE-model development of Russian-made electronic component base's library" report
1 SiliconzenerKC156A r-diod.lib library, part of the Spectrum Sofware MicroCAP simulator (Russian localized version) [13]
4 Medium accuracy operational amplifier cmn^ r-opamp2.lib library part of the Spectrum Sofware MicroCAPsimulator (Russian localizedversion)[13]
In the nominal mode (no parameters' variation), the simulation of the generator's circuit with the Wien bridge was carried out in the OrCAD PSpice simulator [14]. The result were harmonic oscillations of f =10kHzfrequency and Um = 3V amplitude (Fig. 5).
To accurately fix the f and Um numerical values the PSpice Probe graphic postprocessor two target functions are formed:
1/Period(V(DA1:OUT));
Max(V(DA1:OUT)),
where Period - is the target function template for finding the oscillation period [14]; Max - is the target function template for finding the oscillation amplitude [14]; V (DA1: OUT) - is the target functions' argument, denoting the potential at the OUT output of the DA1 operational amplifier (and the whole circuit).
The main characteristics of the computational experiment plan are given in Tables 2, 3, 4. Note that the capacitor types used have a linear capacitance temperature coefficient (M47) over the entire operating temperature range [12]. The resistor type has a non-linear resistance temperature coefficient (M) [11], however, due to the small temperature range in the computational experiment (±10 °C), it is permissible to use a linear dependence. Thus, during computational experiment planning the parameter's temperature dependence was used for all variable ERE:
MT) = is0(l± TCN-AT),
where N0 - is the nomonalparameter value; TON - is l=epar=mete='st=mpe=aturecoefficient; AT = ±se°C- aae0et)on(n0ae=a(tamperature;
Based on the results of the computational experiment planning, the relative deviation (DEV) of the ERE's scale parameter is found necessary for the Monte Carlo statistical tests (the penultimate column):
.MODEL K10-17 CAP (C=1 DEV=0.047 %); .MODEL R1-8 RES (R=l DEV= 0.1 %).
Table2. Variation levels forCl, C2 (xl, x2 factors)
Capacitor operating temperatu re 9, °C Abselute c aparlty value BSuc, nF Temperature vaiiixitiito inpe°val l\T,°V Capacity absolgte tlnvmtion AC,nC Cpjppcity vmopeanm° dtnwtion vc VC -100% CABS C scalefactor 2aviation gorlh ai]PiC c model
Toplevel 37 1,00047 +10 +0,00047 +0,047 1,00047
Zero level 17 1 a 0 0 1
Bottom level 17 0,999^:3 -to -0,00047 -0,047 0,99953
TableC. VariationlevelsforC3 (vCfactac)
Capticitor ojctreting tantptiature C C Abstlute c apailly value CAls, nF "ftmperatae oariatien inieov al AT, °C Cajgatity ateohrte dCTmtion AC,nF Ccpgai1^il relative daniction Aa • t22% aabs C scalefactor deviation for the SPICE model
Toplevel 37 63,02961 +10 +0,02961 +0,047 1,00047
Zerolevel 27 63 0 0 0 1
Bottom level 17 62,97039 -10 -0,02961 -0,047 0,99953
TableC. Variationlevel sforR2,R4(x4, x5 factov)
Resscto r ogtrsti ng tantptiatu re °C 0C Abiolnte aesi^tamc e vaiue ASabs, Ohm "ftaiperatae oaciatien inier Aal KT, °C Reaisiance aXcolule iletittion AA ,OOm Resistance relctine daniction AA • t22% AAbs R sc aie tcctor devraiioo for tacsprc E model
Top leoal 37 10 010 +10 +10 +0,1 1,001
Zerolenel 27 10 000 0 0 0 1
Botlnmle vel 17 9 990 -10 2 0 - 2,1 0,999
Multivariate analysis of the electrical circuit (Fig. 4) using the Monte Carlo method is carried out in the OrCAD PSpice simulator. The circuitsi mulationparameters (Fig. 6, 7) are:
- run to time: At = 4 ms;
- start saving data after: tbeg = 3 ms;
askip the ditect current analysis;
- octpunvasimble - rutput vvlnrua of the op-dmp: aaour = a(Di-l:tC Ofem
- testnumber Vy theMonne Carlo method: MC = 100;
- name of the random variables' distribution law: BiModal;
-random cumOpe seed:TVo= S;
-upordinaies oflleo Poc-noCioned bimodal distribution law: (-1,1) (-0.99,1) (-1,1) (-0.99,1) (-0.99,0) (0.99,0) (0.9t,t) (t,l).
The sneutt of ^l^e muhivkriate aetlynis is She family oCUiaemonic oscillations originally. However, suchagcaphicsl reyieusnUattonis meonved-entfor perception and quantitative analysis. The noted deficiency tseusiiyeCimihateO: TCefunfiioeaCity of tiiePSpicrProbe graphical postprocessor makes it possibleto^e oept thvvahies of Retarget function (12) in a table form in each of the statistical tests (H&SC.
Run to time: j4m
Start saving data after: |3m Transient options
seconds (TSTOP) seconds
Maximum step size: |0.1u seconds
J* Skip the initial transient bias point calculation (SKIPBP)
(* Monte Carlo P Enable PSpice M support for legacy C Worst-case/Sensitivity Output variable: |V(DA1:0UT)
Monte Carlo options-
Number of runs: |l 00
Use distribution: |BiModal Distributions...
Random number seed: jl [1..32767] Save data from ¡All
a b c
Fig. 6. Modellngparametersio- timedomain anaSysisparemetarst -Monte Carlo analysis parameters, c - bimodal dirOsibution parameters
Ö a> T3
-2 0,5
-10 1 Variation levels
Fig. 7. Thenormalized density ofthe bimodal distribution
Evaluate Measurement » I ÎS 19 i 90 91 j 92 93 94 9
► I_ 1iPeriod(V(DA1:OUT)) ifc.0279S; m.ffimsk: ■100143«; 11.02698k: 999753k; 9.99426k 1002441k ieii2ik
i ... i
Fig. 8. Table of Monte Carlo test results
Each of the statistical test's results (Fig. 8) should be compared with a specific combination of variable factors. The essence of the comparison lies in the comparative analysis of the output text file (Fig. 9, a) generated by OrCAD PSpice, and the information presented in the implementation matrix of the computational experiment (Fig. 9, b, c).
For example, in a text file, a fragment with test No. 87 was found, with the following combination of ERE parameters:
C1 = 9,9953E-01; C2 = 9,9953E-01; C3 = 9,9953E-01;
R2 = 9,9900E-01; R4 = 9,9900E-01.
According to Table 2, 3, 4 it is easy to establish that the code values of the factor combination looks like:
C1 = -1; C2 = -1; C3 = -1; R2 = -1; R4 = -1.
Therefore, the test No. 87 and the target function value f = 10,02792 kHz, found from the table in Fig. 8, will correspond to the combination (-1 -1 -1 -1 -1) in the computational experiment implementation matrix.
The orthogonality of the implementation matrix's columns (Fig. 9, c) allows us to determine the regression coefficients according to (9) and (10):
b0 = 10,014T03; b = 0,00146i03 (C1); b2 = -0,00587i03 (C2);
b3 = -0,00046i03 (C3); b4 = -0,01248i03 (R2); b5 = 0,00312i03 (R4).
Consequently, the linear polynomial has the form:
f = (10,014+0,00146^1- 0,00587q2 -0,00046q3 -0,01248q4 +0,00312q5)i03. (12)
Using the obtained linear polynomial (12), we calculate the theoretical value of the output fT parameter in each experiment, and then find the su m of the difference squares between the experimental and theoreticalvalues ofthe output parameter.Thetotal sumof differencesquaresof values for a linear polynomM lSiA/2=if,9°-t0iS p03. Takmg into uccount tlie smaU value o°A/2,we can consider (12) to bean adeooatere geeuuion modet.
According to (11), the coefficients of the temperature error equation are determined:
= 0,3di sCd); Uu = -3,24/^0);
a3 = -0,09U (C3);a4 = -1,2U7 (R2=; a, = 0,312 (R4).
The temperature error equation in accordance with (7) will be:
Af- = 4,46-43-5ATC1 -5,86-43"5ATC2 -4,55-43"6ATC3 -f C4 C 2 (13)
- 4,25 - 43" 4 ATR2 + 3,42 - 43" 5 ATR4.
Research results
Analysis (13)allows us tostate:
- the temperature error of the generation frequency mainly depends on the temperature instability ofthefourERE:C1 andC2capacitors, R2 and R4 resistors;
- 525 -
- the linear polynomial (12) is recognized as an adequate regression model of the investigated process;
- to ensure a given temperature stability of the generator, three solutions are possible: the use of highly stable C1, C2, C3, R2, R4; partial temperature compensation of C1-C2 and R2-R4 pairs; thermostating R4.
Conclusions
1. A modification of the numerical Monte Carlo method for analyzing the temperature stability of electronic circuits is proposed.
2. The problem of finding the influence coefficients of the temperature error equation for the one-parameter ERE's case was solved.
3. The obtained temperature error equation of the generator with the Wien bridge (13) allows quantitatively and qualitatively to formulate the requirements for ensuring a given temperature stability of the device both at the stage of circuitry and at the stage of topological design.
References
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