Multiplier d.c. error specifications:are they to be measured or computed?

M.M. Stabrowski

Indexing terms: Errors, Multiplying circuits

Abstract: Definitions of d.c. error terms for analogue multipliers are introduced, and actual measurementmethods are described. An efficient computer program package permits error-term calculation in perfectagreement with the definitions. Sample segments of this program and calculation results are presented.Approximation by polynomials of two arguments is relevant to other aspects of component and circuitanalysis.

Streszezenie: Podano definicje biedow statycznych analogowych przetwornikow mnozqcych i opisano metody iclibezposredniego pomiaru. Btedy te niozna obliczac w sposob catkowicie zgodny z definicjami, korzystajqc zzaprezentowanego w artykule pakietu programow komputerowych. Przedstawiono przykladowe segmcnty programu iwyniki obliczeii. Aproksymacja wielomianami dwoch argumentow. stosowana w programie, jest uzyteczna i w innychdziedzinach analizy obwodow i ich elementow.

1 Basic specification of d.c. multiplier errors

The error specifications of a multiplier determine howwell it performs its basic function of multiplying twoinput voltages Ux and Uy under fixed external conditions.For an ideal device, the nominal output voltage Uzn isgiven by

Uxn = 0-1 (1)

The gain (scaling factor) is assumed to be 0-1, as is normalfor integrated analogue multipliers.

One of the simplest and most comprehensive ways ofspecifying multiplier errors is to measure the departureof the actual output voltage U2 at a number of points Ux,Uy from its nominal value Uzn. A contour map, showingthe interpolated errors at all points in the Ux-Uy plane,may be drawn for a given multiplier. Such a set of equalerror contours, called 'iso-vers', is presented in Fig. 1.

This method of d.c. error description, although usefulfor manufacturing control and unit classification, is fre-quently much too extensive for the user. An alternativeapproach assumes that multiplier inaccuracies result fromtwo basic sources, namely offsets and nonlinearities.Such a model can be described by

U, = ± Uzo + 0-lA(Ux ± Uxo) {Uy ± Uyo)

+ f(Ux,Uy) (2)

where A = gain error constant

Uxo, Uyo = input offset voltages (feedthroughs)

UZo = output offset

f(Ux,Uy) = nonlinearity.

Expansion and regrouping leads to the standard multiplierequation

Uz = 0-\AUxUy ±0-lAUxoUy ±0-lAUxUyo ± Uzo

+ f(Ux,Uy) (3)

Paper T236 E, first received 16th March and in revised form14th June 1978Dr. Stabrowski is with the Department of Electrical Engineering,Warsaw Technical University, ul. Ch/odna 11M 420, PL00891,Warszawa, Poland

Furthermore, the nonlinearity function f{Ux, Uy) may beapproximated as

f(Ux,Uy) = \Ux\ex + \Uv\e> (4)

where the constants e^, ey are labelled partial input non-linearities. These six error terms are treated as standardby many multiplier manufacturers and appear in the morecomprehensive catalogues.1'2

The question of multiplier a.c. specifications is farmore complex. They are of paramount interest in appli-cations such as modulation or detection, and a discussionof their practical aspects may be found elsewhere.1'2

In the case of integrated multipliers, however, d.c. errorspredominate up to several hundred hertz and are sufficientin many cases to describe the operation of the device.1

2 Measurement of error terms

Despite widespread agreement on the definition of thed.c.-error terms introduced in eqns. 3 and 4, in practice,

10

Fig. 1 Contours of equal error (iso-vers), in millivolts, for anintegrated multiplier

ELECTRONIC CIRCUITS AND SYSTEMS, NOVEMBER 1978, Vol. 2, No. 6 173

0308-6984/78/0601 73 + 04 $01-50/0

the specifications depart significantly from these de-finitions.

In the direct-measurement method,1'2 the outputoffset voltage Uzo is measured with both inputs at zerovoltage.

Feedthroughs are measured as the peak-to-peak out-put when a ± 10 V low-frequency sawtooth signal is appliedto one input (the nominal maximum input voltage), andzero voltage to the other. The output offset is first trimmedto zero.

The scaling factor (gain) is obtained as the average ofits measured values in either two or four quadrants, withthe input voltages at their nominal maximum values of±10V. Output offset is first trimmed to zero, and thefeedthroughs are adjusted to their minimum values.

Partial nonlinearities ex and ey are measured byapplying the nominal maximum voltage to one input anda± 10 V low-frequency sawtooth to the other. Nonlinear-ities are thus determined for the boundaries of the multi-plier Ux—Uy plane (Fig. 1).

The direct measurement of d.c.-error terms involvesa compromise between the exact application of the de-finitions and the time or effort required for the measure-ments. It is clear that the measured error terms describethe multiplier performance in isolated regions of theUx—Uy plane. To make the situation even worse, eachterm is measured in a different region.

3 Computer-aided determination of the error

specifications

Perfect conformity with the error definitions, and trulymeaningful results, may be obtained through computer-aided analysis of the measured data. Conceptually, com-putation of the error terms is quite simple; the measure-ments that were used to construct the iso-vers map areapproximated by a polynomial of two variables Ux andUy comprising the four right-hand terms of eqn. 3. How-ever, even large computer-program libraries, aboundingin approximation programs for one-variable problems^do not contain universal programs for two-variableproblems.

To solve these difficulties, an efficient and versatileFortran program package was written, using Forsytheorthogonal polynomials.3'4 The proper approximationbasis is formed by the Forsythe-polynomial productsFkx{x)-Fly{y). Analysis of multiplier errors is carriedout in the square defined by

0 <k <kmax

0 < / < kmax

(5)

Forsythe polynomials are generated in the program packageby subroutine POLYGEN. Its simplified version, omittingthe generation of generalised Forsythe polynomials, islisted in Appendix 6.1. The variables MMAX and KMAXare equal to the number of Xi[Ux]lyi[Uy] nodes and tothe number of Forsythe polynomials kmax (eqn. 5) inthe approximation basis. Forsythe-polynomial coefficientsare stored in array COEFF.

Degenerated normal equation systems with diagonalcoefficient matrices may be written concisely in terms oftensor products as

m^kmax,x J L^kmax,xm

kmax,ym

1

1S =

r Fox i

"kmax.x

®

• ^ O y "

Fly

Prkmax,ym

r Zl i

zimax _

(6)

Matrix elements Fix, Fiy and zt are themselves row matricesof Forsythe-polynomial coefficients, or the correspondingvalues of z (i.e. Uz) for constant y values. The matrix scontains the coefficients of the Forsythe-polynomialproducts; its elements are computed during the solutionof the degenerated normal equation system.

Although eqn. 6 looks repellent and confusing, itsFortran realisation is rather elegant. The solution is ob-tained through simple division of left- and right-hand co-efficients.

Another segment of the program package may be ofgeneral interest. Function DOUVAL, listed in Appendix6-2, computes values for a polynomial of two variablesfor a given point x,y. For polynomials in a single variable,Homer's algorithm or more refined methods4 are used.The algorithm presented here for two variables makesno claims to ultimate efficiency, but it reduces the com-putation time very distinctly. It should be noted thatthe final coefficients of the approximating polynomial(the sum of the Forsythe-polynomial products), stored inarray FUNA, are arranged in the same order as in a Newton(Pascal) triangle.

This medium-size Fortran program package of about400 statements was run on a Control Data Cyber 73 com-puter. Its memory requirements were standard, i.e. 22500central memory words. Execution for 121 z values re-quired less than 2 s of central-processor time. This programmay be easily implemented on any modern minicomputer.

Table 1 : Computer-calculated and catalogue d.c.-error specificationsof an integrated multiplier

Gain error.Uxo.Uyo.Uzo.€x>

%mVmVmV%%

Cataloguespecifications

0-803030

00-8000-300

Calculatedspecifications

0-721115

7-0003

0-125

4 Results and conclusions

The program package described here was used to computethe d.c.-error specifications for a multiplier having the iso-vers map shown in Fig. 1. Table 1 compares the computederror terms with the catalogue specifications. The multi-plier output was first trimmed to zero with both inputsat zero voltage (Fig. 1). The calculated output offset

174 ELECTRONIC CIRCUITS AND SYSTEMS, NOVEMBER 1978, Vol. 2, No. 6

Uzo differs from zero, since it is defined as the fourthright-hand term of eqn. 3, and not as the output voltagewhen both inputs are at zero.

Fig. 2, which shows the contour map of nonlinearity,is of particular interest. Nonlinearities in the investigatedunit are particularly prominent. It is clear that eqn. 4cannot adequately characterise the multiplier which wastested, since the nonlinearities are highly irregular.

Such a computer-aided analysis of multiplier errorsmay be easily carried out in an online mode. In routineproduction tests, essentially all the data needed for theuse of the program package are measured.

Approximations involving polynomials of two argumentsare useful in many areas of components and circuit analysis.A number of modern circuit-analysis programs assumepolynomial description of the nonlinear components. Themethod described here may be used directly to complementSPICE or NAP2 programs for circuit analysis, or in connec-tion with other similar programs.

Fig. 2 Contours of equal nonlinearity error in mV for an in-tegrated multiplier

5 References

1 'A primer on analog multiplier specifications'. Burr-Brown appli-cation note AN-51, Dec. 1972

2 BURWEN R.S.: 'A complete multiplier/divider on a singlechip'. Analog Dialogue, 1971, 5, p. 3-15

3 FORSYTHE G.E.: 'Generation and use of orthogonal poly-nomials for data-fitting on a digital computer'. J. Soc. Industr.Appl. Math., 1957, 5, p. 74-88

4 RALSTON A.: 'A first course in numerical analysis' (Me Gra'v-Hill, New York, 1965)

6 Appendixes

6.1 Listing of subroutine POLYGEN for generationof Forsythe polynomials

SUBROUTINE POLYGEN(X, MMAX, KMAX,COEFF, POLY,

$POLSQ)

1213

1003233

300

REAL MAGAZ(4)DIMENSION COEFF(10,10), X(20), POLY(20,10),

$POLSQ(10)SIGMP = 0DO 1001 = l.MMAXSIGMP = SIGMP V* X(I)POLY(I, 1)= 1CONTINUEMAGAZ(4) = SIGMPMAGAZ(3) = MMAX* 1 $ MAGAZ(2) = 0ALPHA = 0 $ BETA = 0DO300J= l.KMAXDO 3001= 1,KMAXCOEFF(I,J) = 0CONTINUEPOLSQ(1) = MMAXDO 490 K = 2, KMAXKRED = K - 1 $MAGAZ(l) = MAGAZ(2) $

$ COEFF(1,1)=1

4041

410

43

43944

46420

490

KREDUC = K - 2MAGAZ(2) =

$ MAGAZ(3)ALPHA = -MAGAZ(4)/MAGAZ(2)IF(K.GT.2)GOTO40KREDUC=1 $ GO TO 41BETA = -MAGAZ(2)/MAGAZ(1)DO410J = 2,KN = J - 1COEFF(K, J) O COEFF(KRED, N) + ALPHA*

$COEFF(KRED,J)$ + BETA*COEFF(KREDUC, J)

CONTINUECOEFF(K, 1) = ALPHA* COEFF(DRED, 1) +

$BETA*COEFF(KREDUC, 1)MAGAZ(4) = 0 $ MAGAZ(3) = 0DO 420 I = 1, MMAXPOLY(I, K) = COEFF(K, K)DO439N= l.KREDPOLY(I, K) = POLY(I, K)*X(I) + COEFF(K,

$(K-N))CONTINUEPOLSQ = POLY(I, K)*POLY(I, K)MAGAZ(3) = MAGAZ(3) + POLYSQMAGAZ(4) = MAGAZ(4) + X(I)*POLYSQCONTINUEPOLSQ(K) = MAGAZ(3)CONTINUERETURNEND

6.2 Listing of function DOUVAL for computation oftwo variables polynomial value

FUNCTION DOUVAL(KMAX, FUNA, X, Y)DIMENSION FUNA (19,19), YEXP(19)NCOLMX = KMAX*2 - 1NROWMX = NCOLMXYEXP(1)= 1DO 100 I = 2, NCOLMXYEXP(I) = YEXP(I - 1)*Y

100 CONTINUEDOUVAL = 0DO 210 NCOL = 1, NCOLMXIMAX = NCOLMX - NCOLIF(IMAX)21,22

ELECTRONIC CIRCUITS AND SYSTEMS, NOVEMBER 1978, Vol. 2, No. 6 175

21 XSUM = FUNA(NROWMX, NCOL)DO 220 I = 1, IMAXXSUM = XSUM*X + FUNA(NROWMX - 1 ,

$NCOL)220 CONTINUE

GO TO 25

22 XSUM = FUNA(NROWMX, NCOLMX)25 DOUVAL = DOUVAL + XSUM*YEXP(NCOL)

210 CONTINUERETURNEND

M. M. Stabrowski was born on 3rdApril 1942 in Lublin. In 1964 hereceived the M.S. degree in electricalengineering from Warsaw TechnicalUniversity. Just after graduation hebegan work at Warsaw TechnicalUniversity in the field of electronicmeasurement instrumentation. In 1973he obtained the Dr. E.E. degree alsofrom Warsaw Technical University. Hisrecent areas of scientific activity

include computer-aided design and microcomputer-basedinstruments and measurement systems.

John V. Hanson was born in Bradford,England in 1932. He received theB.A.Sc. degree in electrical engineeringfrom the University of Toronto in1955, after which he spent two yearsat Imperial College, London, as anAthlone Fellow, receiving the D.I.C.and M.Sc. (Eng.). He received thePh.D. degree from the University ofLondon in 1968. In 1964 he joinedthe Electrical Engineering Department

of the University of Waterloo where he is presently activein modulation systems, receiver design and single-sidebandsystems.

R. Datta received his B.Sc. in electricalengineering from London UniveristyEngland, and his M.A.Sc. from theUniversity of Waterloo in 1977. Hejoined Raytheon of Canada in 1969,where he worked in the design ofV.O.R. and DME navaid systems. Atpresent he is working at Communi-cation Research Centre, a branch ofthe Canadian Department ofcommunication, as a project leader,

where he is responsible for the design of small (1 and 2 m)transmitting and receiving satellite ground stations.

176 ELECTRONIC CIRCUITS AND SYSTEMS, NOVEMBER 1978, Vol. 2, No. 6