THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS

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1 R775 Philips Res. Repts 26, , 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated expressios are derived for the systematic ad the radom error i etwork fuctios due to elemet toleraces. A relatio is derived betwee the radom error ad the summed-squared sesitivity provided that the elemets have equal probability distributios. These expressios have bee foud useful i the aalysis of filter etworks. 1. Itroductio The elemets used i electrical etworks deviate from their face value. That is why it is ot possible to get the specified etwork fuctio but oly a approximatio. It is importat to have a measure of how large the deviatio may be expected to be. Schoeffler 1) has chose the sum of the magitudes-squared of the first-order sesitivity as a measure for the sesitivity of the etwork fuctio (e.g. voltage-trasfer fuctio, iput impedace) due to toleraces of the compoets. We shall use a statistical approach 2). Compoets ca be cosidered as stochastical variables with a mea (equal to the omial value of the compoet) ad a stadard deviatio. I geeral, there will be a systematic error ad a radom oe (the systematic error caused by e.g. temperature deviatios or agig will ot be take ito accout there). A approximated expressio is derived for both the systematic error of a etwork fuctio ad the radom error (sec. 3). The systematic error is show to be proportioal to the secod-order sesitivity while the radom error is proportioal to the first-order sesitivity. Therefore, the systematic error is a order of magitude smaller tha the radom error if the toleraces are small. The ifluece of the probability distributio of the elemets upo both errors is ivestigated. A relatio is derived betwee the radom error ad the sum of the magitudessquared of the sesitivities of a etwork characteristic to chages i each of the elemets (sec. 4). This aalysis also reveals the idividual cotributio of each elemet to the summed error. The tolerace of a specific elemet ca be chose i terms of the apparet sesitivity of that elemet.. The results are applied to a RC-active filter realizig a Chebyshev characteristic (sec. 5). 2. Defiitios Let y deote the etwork fuctio of iterest. The values of the compoets

2 ERRORS DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS 415 of the etwork which realizes this fuctio are deoted as XI for i = 1 up, to. We ca write ' y mayalso be a fuctio of frequecy, time, etc. The fuctio y represets e.g. the modulus of the voltage-trasfer ratio of the etwork cofiguratio. The compoets Xl do ot have a fixed value, they are supposed to be stochastical variables. The omial value of compoet Xl which is assumed to be idetical with the mea is deoted as!-ll. We assume that the distributio of the compoet values will be symmetrical aroud Xl =!-ll. We ow expad y i a -dimesioal Taylor series aroud the value (Pl>!-l2'...,!-lh...,!-l): " All derivatives are take at the poit (PI'!-l2'...,!-lh...,!-lIl). F,urther, we assume that the stochastical variables Xl> X2', X are idepedet. (A etwork cosists i geeralof differet types of compoets. Elemet values that represet the same kid of compoets ca have equal elemet values or differet oes. All these facts ifluece the justificatio of the assumptio that the compoets are statistically idepedet.) The, we have " (1) Equatio (1) will be the startig poit for our derivatios. 3. The systematic ad the radom error 3.1. The systematic error The systematic error is defied as the differece of the mea value of the

3 416 H. W. HANNEMAN etwork fuctio 4) ad the exact value of the etwork fuctio /(!1-1,!1-2,...,!1-,,)' Let E(y) qeote the expected value of y. Usig eq. (1), the expectatio of y ca the be foud as follows: " E(y) = E(f(!1-t.!1-2,...,!1-I>...,!1-,,)) + E (2: (XI- fl-i) ::,) + " C deotes the sum of the terms cotaiig covariaces. For symmetrical distributios it ca easily be show that all terms cotaiig odd powers of (XI -!1-1). do ot cotribute to the sum 2). Thus, 11 For the systematic error s defied as the differece betwee the mea of the etwork fuctio (E(y)) ad the etwork fuctio itself we obtai i a first approximatio 11 where s = ~"\' a? b2~, 2.f.. ; öx, (2) 3.2. The radom error The calculatio of the radom error is essetially the calculatio of the variace. As i sec. 2 we assume that the compoets are stochastical idepedet variables which are. symmetrically distributed aroud the omial value. We apply the operator variace to eq. (1). This leads to the result

4 ERRORS DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS 417 which equals where C' deotes the sum of the terms cotaiig covariaces ad al 2 = var (XI)' Neglectig terms of a order higher tha oe yields for the stadard deviatio of y, ay, (3) The stadard deviatio of each compoet al ca be calculated from the probability distributio fuctio of the particular compoet XI' Most likely, the probability fuctio will be gaussia. Because of fiite toleraces the p.d.f. will be trucated at both tolerace limits (example 2). A drastically trucated gaussia distributio is the uiform distributio 3) which is used i example 1. Example 1 The compoet value XI is uiform-distributed betwee the tolerace limits ± a% (fig. I). The value of XI ca be represeted as 2 a III XI = III +-- (R- 0,5), 100 p(xi) (1-a)/J; f!.i (I+a)f!.i ---Xi Fig. 1. Statistical distributio of the compoet values of elemet XI'

5 418 H. W. HANNEMAN where R deotes a umber betwee 0 ad 1, chose at radom (pseudo-radom). The distributio fuctio P(X,) ca be deoted as ad X, > (l + a)!-ti or X, < (1 - a) Il" Now, ad (1 +a)~1 E(x, 2 ) = f P(X,) x? dx, = t (3 + a 2 ). (l-a)~1 Thus, the variace of the uiform distributio will be (4) If all compoets x, of the etwork fuctio f have a uiform distributio the systematic error ca be expressed as (5) while the variace of the etwork fuctio ay 2 becomes (:~,Y a/ = t'al L!-t? (6) Example 2 The probability desity fuctio of compoet value x, is a trucated ormal ~e. The degree of trucatio is 0: (the percetage of compoets with a compoet value less tha the lower tolerace limit (1 - a)!-ti or larger tha the

6 ERRORS DUE TO ELEMENT TOLERANCE!! OF ELECTRICAL NETWORKS 419 Fig. 2. Statistical distributio of the compoet values of elemet X,; P(XI): ormal distributio, h(x,): trucated distributio. upper tolerace limit (1 + a) #1) (fig. 2). The ormal distributio P(X,) is give by 1 {I (X'- fl,)2} p(x,) = -a-,(-2-)-1-/2 exp - 2' -a-, -. The doubly trucated distributio fuctio hex,) is equal to p(x,) Now, 1- a E(x,) = #,. The variace of the trucated ormal distributio is calculated as The fuctio cp(a) is give i table J, for some values of the degree of trucatio. TABLE I (7) cp(a) It ca be see from table I (as expected from example 1) that the value of cp(a) approaches t for large degrees of trucatio. 4. Relatio to the sesitivity The first-order sesitivity is writte as #1 "öf S(/'#I) = --, f "ö#1 accordig to Bode's defiitio.

7 420 H. W. HANNEMAN The sesitivity of a etwork with compoets /hl' /h2'..., /hl>..., /h" is defied by Schoeffler 1) as the sum of the magitudes-squared of the first-order sesitivities, Thus, Comparig ". 1 ~,( "DJ)2 ~ IS (f, /h,)1 2 = J2 L..._; /h/ "D/hl. (8), this result with the expressio we foud for the variace of y (eq. (3» we observe that i the two cases previously discussed (examples 1 ad 2) ~ ISI 2 is proportioal to ay 2 Thus we obtai i case of a trucated ormal distributio for the quotiet of var (y) ad ~ ISI 2 the importat result ((J( ct) a 2 / 2 (/hl' /h2'..., /hl>..., /h). A similar relatio ca be derived from the compariso sesitivity which is equal to of the secod-order ad the systematic error. 5. Applicatio to filter etworks We have derived expressios for the summed systematic ad radom errors. The values of the compoets ca be see as weight fuctios for both errors. It is importat, however, to kow also the value of each of the derivatives (firstorder ad secod-order) of the etwork fuctio. This eables the desiger to choose the tolerace of a specific elemet accordig to the relative importace of the first- ad secod-order derivative ad the compoet value itself. Observe that the systematic error is proportioal to the square of the tolerace (a), while the stadard deviatio is proportioal to a. This meas that for small tolerace values the systematic error will be less importat compared with the, I stadard deviatio tha it will be for high tolerace values. Taylor 4) has calculated the compoet tolerace of each elemet so that' the etwork characteristic will remai withi the tolerace limits for each choice of elemet value. This is ofte impractical, however, sice very arrow toleraces o the compoets would be required. A sesitivity aalysis is v~ry importat to the desiger of filter etworks. I order to show the usefuless of the method here preseted we will ow apply the method to a example.

8 ERRORS DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS 421 Example 3 The secod-order low-pass etwork cofiguratio we shall use is show i fig. 3 ad was first give by Salle ad Key 5). The port voltages are Vi ad Vout' The etwork characteristic of iterest (y) will be the modulus of the trasfer ratio Wout/Vll. With this filter a characteristic of the Chebyshev type (ripple 0 5 db) is realized. Vi Vout Fig. 3. Secod-order low-pass RC-active etwork. The etwork fuctio y is give by We have calculated the systematic error (s) ad the radom error (s.d. of y), deoted by r as a fuctio of frequecy (w). All elemet values have tolerace limits of 5 % ad the probability desity fuctio is uiform. This case is deoted as A i table Il. The systematic error is a order of magitude smaller tha the radom error. For differet toleraces, e.g. a %, table Il A ca still be used. The systematic error the becomes (a/5)2 times a large, while the radom error has to be multiplied by a/5. The percetages of the radom ad systematic errors are take to the value of IVout/Vll at w = O. We have also calculated the magitudes ofthe first- ad secod-order derivatives. By comparig the derivatives of y to K with the derivatives of y to the other compoets it ca be see that the first are relatively large. Thereore we have calculated the systematic ad the radom error for a etwork ith a tolerace limit of 1 % for K ad 5 % for the other compoets. The esults are give i table Il B. A oticeable reductio ca be obtaied. The ext-iarger derivatives belog to Cl' Agai the systematic ad the radom rrors are calculated for a tolerace limit of 1 % for K ad Cl ad 5 % for the th er compoet values (Rl, R2' R 3 ). The results are show i table Il C. Usig the relatioships here derived a aswer ca be obtaied to the questio f what elemet tolerace has to be chose i order to assure that a give peretage of the circuits has a etwork characteristic which deviates ot more ha le (IF (k > 0) from the expected characteristic (fig. 4).

9 422 H. W. HANNEMAN TABLE II The systematic ad the radom errors of the modulus of the voltage-trasfer fuctio of a secod-order active RC filter (Chebyshev) for various tolerace limits. The compoet values are: - R, = Rz = 1 Q; Cl = Cz = F; K = 1'841, givig a ripple of 0 5 db (see fig. 3) elemets uiform- oe elemet two elemets distributed with (K) (Cl' K) tolerace 5 % tolerace 1% tolerace 1% A B C freq. s (%) r (%) s (%) r (%) s (%) r (%) '0' ruc _Freq. -- with the correct elemet values ---_ characteristic if a systematic error appears I the stadard deviatio ai some frequecies Fig. 4. The modulus of trasfer fuctio of a Chebyshev filter (fifth-order).

10 ERRORS DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS Coclusio. The sesitivity of a etwork characteristic to deviatios of the elemet values has bee described i a statistical maer, which has resulted i approximated. expressios for the systematic ad. the radom error. A relatio has bee éstablished betwee the sum of the magitudes-squared of the first-order sesitivities ad the radom error. The method has bee applied to a example to demostrate its usefuless. Ackowledgemet I wish to thak Prof. Dr K. M. Adams of the Techological Uiversity of Delft, Ir H. N. Lisse of the Techological Uiversity of Eidhove ad my colleague Ir J. O. Voorma for may valuable suggestios. Eidhove, July 1971 REFERENCES 1) J. Schoeffler, IEEE Tras. Circuit Theory CT-tl, , ) J. Madel, The statistical aalysis of experimetal data, Wiley, New York, 1967, p ) P. W. Broome ad F. J. Youg, IRE Tras. Circuit Theory CT-9, 18-23, ) N. H. Taylor, Proc. IRE 38, , ) R. P. Salle ad E. L. Key, IRE Tras. Circuit Theory CT-2, 74-85, 1955.

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