Wide-Band AnalogFunction Multiplier

JOSEPH A, MILLER,Minneapulis-H oneywell

Minneapolis, ilIinn.

ByAARON S. SOLTES

Air ForceCambhdge Resea"ch Cente,'

Cambhdge, Mass.

and RONALD E. SCOTTResearch Labol'ato,'y oj Elect"onics

MITCamb"idge, Mass.

Beam-deflection tubes perform nonlinear squaring operations that are the basis of thisanalog multiplication method. Speed and accuracy are high. Performance is primarilylimited by associated circuitry rather than the tubes

DEVELOPED on the quarter-squareprinciple, a simple analog

function multiplier takes advantageof the characteristics of recentlydeveloped beam-deflection square-law tubes, such as type QK-329.These can provide full parabolicsquare-law action to an accuracybetter than 1 percent of full scale,over'a' frequency range from d-c tothe vhf region.

This particular multiplier wasbuilt to explore the possibility ofusing these square-law tubes forthis application. Commerciallyavailable plug-in amplifiers wereemployed in the associated circuits.Results obtained showed that per-formance of this relatively crudemodel was, on the whole, limited bythe associated circuitry rather thanthe square-law tubes. Neverthelessa combination of accuracy and speedof response had been achieved thatexceeded any other known methodof analog multiplication.

A quarter-square multiplier isinstrumented around the identity

xy = Hx + y)2 - (x - y)2 (1)

The left-hand term is the desiredproduct and requires the perform-

160

ance on the right-hand side of theoperations of addition, subtraction,multiplication by a constant, andsquaring. All operations but squar-ing are linear and are readily ac-complished using conventional tech-niques. The particular methodselected for achieving the two non-linear squaring functions, however,presents a design problem and isprincipally responsible for the char-acteristics that distinguish onequarter-square multiplier fromanother.

BASECURRENT2.10ma ,

,

50 40 30

FIG. I-Comparison 01 curved portion01 static characteristic 01 square.lawtube with parabola

The beam-deflection square-lawtubes used as the starting point inthe present design are capable ofproviding accurate reproduciblefull-parabolic transfer characteris-tics in a noncritical manner. Overtheir range of operation, speed ofresponse' and accuracy capabilitiesare essentially independent of eachother. That is, the same accuracyobtainable under d-c conditions isachievable at the upper end of itsfrequency range.

Square-Law Circuit ElementsThe principle employed to pro-

duce a square-law characteristic inthese tubes is that of deflecting aflat sheet of electrons across atarget electrode containing para-bolic apertures.

The discussed tube is essen-tially a copy, with only minor modi-fications of the simplest of the QK-256 series of experimental beam-deflection square-law tubes.' Moreprecise methods of measuring andplotting the nonlinear static charac-teristics have indicated that theaccuracies achievable are betterthan was previously stated.

Figure 1 is a plot of a typical

February, 1955 - ELECTRONICS

Complete analog multiplier employs amplifiers that provide a single-ended low-impedance output. at an open-loop doc gain of over 10.000. Present frequencyresponse is limited by amplifier bandwidths

Type QK-329 beam-deflection tube pro-vides full parabolic square-law actionwith I percent accuracy at full scale

static characteristic made on anautomatic precision (0.1 percent)plotting board; Within an inputrange about the origin of approxi-mately ±35 volts the error is toosmall to measure by such means andremains less than 1 percent up to±40 volts. Within these limits thestatic characteristic may be ideal-ized to a close approximation as aparabola

iou' = io + k(eo + ein)2 amp (2)

with its vertex displaced from theorigin by amounts eo and· to.

CurrentScale factor Ie is expressed in mhos

per volt and is essentially a constantfor a given tube over a wide rangeof variations in cathode to anodevoltage, when operated with itsaverage deflection-plate potential,Eb,." maintained at a fixed frac-tion of Eilu The voltage requiredto center the parabola on the verti-cal axis, -eo, is generally small.Its magnitude may differ from tubeto tube, but remains constant withtime for a given tube and is notsensitive to changes in operatingpotentials. Self-centering schemes

ELECTRONICS - February. 1955

are, therefore unnecessary to holdeo constant.

The current, io, is a function oftotal current and subject to changewhen any of the operating param-eters that affect total current arevaried, such as heater voltage, Eb •••

or E fl,. Normal precautions appro-priate to d-c amplifier design aretherefore advisable to keep io stable.

As with most beam-deflection de-vices, best operation of the QK-329is obtained with a balanced input.The input conductance between de-flection plates is small enough toignore under most conditions.Where precise operation at d-c isrequired, account should be taken ofthe possible presence of diode cur-rents of about 10 microamperes be-tween the cathode and the posi-tively biased deflection plates. Thiscurrent is not an inherent propertyof the tube type, but rather a con-sequence of the fact that its effecthad not been noticed in earlierapplications.

The MultiplierEquation 1 can be instrumented

in a variety of ways. The blockdiagram of Fig. 2 illustrates the

arrangement of functional compo-nents employed. No effort wasmade to obtain any particular over-all multiplier scale factor. Threeidentical plug-in operational ampli-fier units are used in standard feed-back computer configurations ofunity gain to perform both the in-verting and subtracting functions.These amplifiers provide a high-impedance differential pair of in-puts and a single-ended low-imped-ance output at an open-loop d-c gainof over 10,000. Both amplifier in-puts handle signals in the sub-tractor stage, whereas one of theinputs in each of the two inverterstages is only used for zeroing pur-poses.

The output or product of a multi-plier with identical inputs may con-tain frequency components up totwice as high as those present ineither input. Consequently, the sub-tractor must be capable of operat-ing up to a maximum frequency ofdouble that which must be handledby the inverters.

In order to further extend thefrequency range of the subtractorit was found necessary to reduce theimpedance level of its associated ex-

161

A procedure has been devised forsystematically eliminating theseerror terms in a convergent man-ner. Some method of observing theform of the multiplier output as afunction of an input signal is re-quired. Error terms are theneliminated in the order shown inEq. 4 by the following adjustmentroutine:

Equating At and A2 cancels thesquare-law error, Eq. 4A. This isdone by setting y to zero and ob-serving the form of the output as afunction of x as A is varied. Whenthe plot is a straight line A, and A2

are equal.The next two steps eliminate the

21 = «0)1 + B1 + B, (3A)

z, = (eO)2 + BI - B, (3B)

The term C is the (zero signal)off-zero term at the outpu't of themultiplier. It includes the subtrac-tor zero and also any d-c unbalancepresent in the cathode followers orbetween the outputs of the square-law stages.

Equation 3 may be expanded intothe desired product and three typesof error terms by carrying out theindicated operations

Kxy + Ll = 2(A1 + A2)xy

(A) + (A, - A,) (x' + y2)(B) + 2(A,2, - A,z,)x + 2(Alz1 + A,z,)y(C) + A,Z,2 - A2z22 + C

tion 1 may be rewritten to includethese terms and their adjustments,in the form

Kxy + Ll = A1(x + y + Z,)2 -

A2(x - y + Z2)2 + C (3)

where, in the above equation factorK is the over-all scale factor of themultiplier and t. is the total errorat the output caused by misalign-ments.

Scale factors of the squared sum-and-difference channels, At and A2,

include the square-law, cathode-follower and subtractor stages.

The off-center terms at the inputsto the square-law stages, Zt and z'"include the respective square-Ia'\\stage centering voltages, eo, and theinverter zero adjustments, B, usedto set them to zero

errorterms

square-law error (A) Jlinear error (B)constant error (C)

analog function multiplier appearsin Fig. 3.

Instantaneous output accuracywithin ±0.5 percent of maximumproduct was consistently achievedwithin the input operating rangesof ±25 volts after alignment of thefunction multiplier.

A dynamic range of approxi-mately 30 db at either input and60 db at the output was obtained.

Overall amplitude response wasflat for either or both input fre-quencies from d-c to 90 kc (outputflat to 180 kc) with a gradual 1'011-off at higher frequencies.

The overall phase response at 9('kc was 65 deg and decreased almostlinearly with frequency. Phase re-sponse was measured with one inputa constant to make input and outputfrequencies identical.

Long-term drift from all causesincluding adjustments was within 1percent of maximum product afteran initial settling period of about 3hours. The output zero requiredthe longest settling time, whileother adjustments reached stabilitymore rapidly. Conventional regu-lated power supplies fed by a 2-per-cent a-c line regulator were used topower the multiplier during the sta-bility measurements.

Multiplier AdjustmentCircuits employed to instrument

the basic multiplier equation pro-duce a nominal over-all scale factorother than one-to-one. No efforthas been made to achieve a unityscale factor. The circuits, unlesscompensated, may introduce a num-ber of extraneous terms that arisefrom misalignments.

Magnitudes of errors produced bysome of the potential sources of ex-traneous terms, such as deviationsof the effective gains of the invert-ers from unity or the sum-and-dif-ference network from equality,depend upon the accuracy and sta-bility of passive resistive compo-nents. Errors contributed by theseparts of the multiplier may be mini-mized during construction by use ofaccurate and stable resistors andare therefore not considered.

Other errors, more subject tovariation with time (those depend-ent upon the stability of active orreplaceable components) are besteliminated by adjustments. Equa-

FIG. 2-Multiplier system using iden-tical amplifier units for inverting andsubtracling

tel'l13J computing resistance belowthat used in the inverters.

Ground potential is used as anabsolute" zero reference for the mul-tiplier input and output signals.The plug-in amplifier circuits whenzeroed do not introduce any shiftin reference level between their in-puts and outputs. However, the in-put and output of a square-lawstage normally operate at differentpotentials. The deflection plates ofthe beam-deflection tube ratherthan its output, operate near groundpotential. This avoids complicatingthe driving circuits to the square-law stages, that are already bur-·dened with provisions for formingbalanced sum-and-difference sig-nals.

The potential differences between-squ~r_e-Jawstage outputs and sub-tractor inputs are eliminatedthrough the use of conventionalvoltage-divider step-down arrange-ments. To prevent excessive signalattenuation, cathode followers areinserted between the high-imped-ance voltage-divider taps and thelower impedance inputs to the sub-tractor.

Push-pull sum-and-difference sig-nals for the square-law stage in-puts, ± (x + y) and ± (x - y), areformed with respect to ground po-tential in a symmetrical passivesumming network. To do this, thenetwork is supplied with balancedversions of the multiplier input sig-nals, ±x and ±y. Signals +x and+y are derived directly from theinput terminals of the multiplier,while the inverters provide theirnegative counterparts. A schematicdiagram of the complete wide-band

162 February, 1955 - ELECTRONICS

and

FIG. 4-Plot of multiplier characteris-tics. Z-KXY. using square-law tubes

StabilityMultiplier drift is primarily zero

drift of the output circuits. If themaximum-output signal level wereincreased to make full use ofthe capabilities of the differentialamplifier, the percent drift wouldbe improved. Use of drift-stabilizedamplifiers would also help.

The square-law stages of thepresent multiplier contribute only12 percent of the observed overalllong-term drift at the multiplieroutput or approximately 0.12 per-cent of maximum output. The dif-ferential output of the multipliercircuit provides a degree of inher-ent discrimination against theeffects of drift that could resultfrom changes in total current in thesquare-law tubes.

Sizeable increases in bandwidth,accuracy and zero stability areachievable with existing square-lawtubes by improving the associatedcircuitry. Square-law tubes of thegeneral type employed, therefore,provide a nucleus around which con-siderable forward progress in ana-log multipliers can be made.

The work described here wassupported in part by the ArmySignal Corps, the Navy Dept.(O.N.R.) and the Air Force(A.M.C.) under contracts DA-36-039sc100 and 3-99-10-022 with theResearch Laboratory of Electron-ics, M.LT.

The authors wish to acknowledgethe contributions of J. Gradijan, G.Fine and A. Moccia of the Air ForceCambridge Research Center.

spect to the cathode. The IR dropsproduced in the sum-and-differencenetwork by these changes in cur-rent are equivalent to shifts of thecentering voltage adjustments fromtheir original settings and intro-duce corresponding errors.

The ideal remedy is to modify thetube design to minimize the deflec-tion currents. It is possible thatsome of the other developmentalsquare-law tubes already built pos-sess improved characteristics inthis respect. Short of this, moreaccurate multiplier performance isattainable with the same square-lawtubes by finding operating condi-tions with lower deflection currents.Reducing the impedance levels atthe inputs to the square-law stages(sum-and-difference network) alsoimproves performance. Inverterswith higher output power ratingswould then be required to maintainthe original amount of drive.

Frequency response is at presentlimited by the bandwidths of theplug-in amplifiers. The QK-329square-law tubes have been success-fully operated from d-c to vhf. Theycontributed no measurable phaseshift to the over-all multiplier phaseshift mentioned in the previous sec-tion on performance. Increases inmultiplier frequency response of100-to-1 over the present modelcould thus be made before thesquare-law tubes offered a directobstacle to such improvements.For efficient design, a multiplierwith two identical inputs shouldhave a differential amplifier at itsoutput with twice the bandwidth ofthe input.

QK- 329

Ko X 1.000

INVERTER

+300V

B2 = - (eO), - (eo)d2 (5B)

The order in which B, and Bo areadjusted is of no consequence.

The term B, is adjusted to thecriterion of Eq. 5A by setting y tozero and observing the form of theoutput as a function of x as B, isvaried. The observed output, 6.,will be a constant for the proper ad-justment of B,.

The teI;m B2 is adjusted in a simi-lar manner by setting x to zero andobserving the form of the output,6., as a function of y, while varyingB2• The output will again be a con-stant for the proper adjustment ofB2 to the criterion of Eq. 5B.

The final step is to adjust C sothat the multiplier output is at zeropotential with respect to groundwhen the input terms are zero.

Performance LimitationsAccuracy of the present multi-

plier stays within close limits forinput signals up to a certain leveland gradually deteriorates as theinputs get larger in a mannersimilar to gradual overloading. Theerror appears to be produced prin-cipally by variable current flow be-tween deflection plates and cathodeof the square-law tubes, that occurswhen the deflection plates exceed acertain positive potential with re-

linear error terms, Eq. 4B, by set-ting B, and Bo so that Z, = Z2 = O.The conditions for this are foundfrom simultaneous solution of Eq.3A and 3B to be

FIG. 3-Schematic of complete multiplier whose performance is primarily limitedby circuitry rather than the square.law tubes

REFERENCE(1) Aaron S. Soltes, Beam Deflection

Nonlinear Element, li]LECTRONICS, P 122,Aug. 1950.

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