Novel Multiplying Circuits with Application to Electronic Wattmeters

PROCEEDINGS OF THE I.R.E.

Novel Multiplying Circuits with Application toElectronic Wattmeters *M. A. H. EL-SAIDt, SENIOR MEMBER, IRE

Summary-The primary relationship underlying this investiga-tion is the exponential law relating plate current to plate voltage in adiode when operated in the so-called "retarding-field" region. Byextending this classical mode of operation to multigrid tubes, thetube yielded further operating conditions in which plate current isaccurately proportional to the product of a linear function of platevoltage and an exponential function of grid voltage over wide ranges.

An analysis is given of various multiplying circuits using a singlemultigrid tube under this mode of operation. Two methods aredescribed for compensating accurately the inherent plate rectifica-tion due to the exponential grid curvature.

The circuits are most suited for electronic wattmeters havingexceptionally good features and predictable performance over thefrequency range from 20 cycles up to the neighborhood of 50 Mc.

I. INTRODUCTION

OR DIRECT measurement of power at the com-mon frequencies of power circuits, the electro-dynamometer type of wattmeter is satisfactory,

but for direct measurement of small amounts of power,or for the measurement of power at the high audio fre-quencies or at radio frequencies, such an instrumentcannot be used because of its appreciable insertion loss,inductive and capacitive qualities. For this reason,electronic wattmeters have been developed and used.With one exception,'-' these depend upon the use of twoaccurately matched tubes operating in the square-lawregime. These circuits have the disadvantage that it isdifficult to match the tubes sufficiently accurately andthat, even when matched, the square-law relation holdsover such a small range that voltage dividers are un-avoidable for voltages exceeding one or two volts. It isdifficult to correct the angles of these dividers at highfrequencies, and the power absorbed by the divider mayconstitute a serious loss. The insertion unit in these in-struments is a T or ir configuration, and it is difficult toprovide for a satisfactory ground system with these con-figurations.Another type of electronic wattmeter is described by

J. R. Pierce4 in which a multigrid tube of the hexodetype is used. In these instruments, the multigrid tube isarranged to operate in a regime so that its plate currentversus the voltage of the first grid for various values of

*Decimal classification: R245.3. Original manuscript receivedby the Institute, November 24, 1948; revised manuscript received,May 31, 1949. Presented, IRE West Coast Convention, Los Angeles,Calif., October 1, 1948; and URSI-IRE Meeting, October 9, 1948,Washington, D. C.

t Fouad University, Cairo, Egypt.1 E. Peterson, U. S. Patent No. 1,586,533.2 H. M. Turner and F. T. McNamara, "An electron tube watt-

meter and voltmeter and a phase-shifting bridge," PROC. I.R.E.,vol. 18, pp. 1743-1748; October, 1930.

3 E. Mallett, "A valve wattmeter," Jour. IEE, vol. 73, p. 295;1933.

4 J. R. Pierce, "A proposed wattmeter using multielectrode tubes,"PROC. l.R.E., vol. 24, pp. 577-584; April, 1936.

the voltage oIn the third grid is a family of straight lines,which when extended meet at one point; while at thesame time the plate current varies linearly against thevoltage of the third grid. Existing multigrid tubes havesuch a small range of linearity that the error introducedby rectification is serious enough to preclude the use ofone tube in a wattmeter. When two tubes are used in apush-pull arrangement, although the error due to volt-age deflection is considerably reduced, the deflectionsdue to current alone are still appreciable. It is difficultto match the tubes under these operating conditions,and even when matched, the useful voltage range is ex-ceedingly small.

This paper describes two types of electronic watt-meters for the measurement of ac power. The first typeis mostly designed for use in circuits in which there arealmost sinusoidal current and voltage relationships. Itsbasic multiplying circuit is of novel design and simpleconstruction using a single vacuum tube. The secondtype is essentially the first, but with an additional cir-cuit to make it generally applicable to voltages andcurrents having complex wave forms. The primary rela-tionship leading to the main design equations of the in-corporated multiplying circuit is the expcnential law re-lating plate current to plate voltage in a diode whenoperated in the so-called "retarding-field" region. By ex-tending this mode of operation to multigrid tubes, itwas found that the tube yields further current-voltagerelationships that are of importance in obtaining a pre-dictable nonlinear performance when more than onesignal is applied. These relationships are useful directlyfor a mathematical application to a specific type of amultiplying circuit.

II. "RETARDING-FIELD" CHARACTERISTICOF DIODES

It is known5 that in a diode the plate current lb is re-lated to the temperature-limited current Is by the ex-pression:

Ib = I8bEmE (1)where Em is the potential at the virtual cathode, b=e/kT= 11600/T volts-', where e is the charge of theelectron (1.602X10-1 coulomb), k is Boltzman's con-stant (1.380X10-23 watt/sec.), and T is the absolutetemperature in degrees Kelvin. For a given diode at afixed cathode temperature, the magnitude of Em is a

function of the plate voltage Eb. As Eb becomes suffi-ciently negative, Em approaches Eb, until at a certain

I Saul Dushman, "Thermionic emission," Rev. Mod. Phys.. vol.2, p. 381; 1930.

1949 1003S


plate voltage Eb', the potential minimum is equal toEb' and its position is at the plate. For all values of Ebmore negative than Eb' a region known as "retarding-field region," Eb=Em, and the lb- Eb characteristic isexponential and obeys the law:

Ib = ITEbEb. (2)The exponential portion of the diode characteristic

occurs at plate voltages more negative than those forwhich the three-halves power law holds, and only atplate currents determined by the Maxwellian theory ofinitial electron velocity distribution. The theoreticalvalue of b corresponding to a cathode temperature of10000 K is 11.6 volts.-1 This exponential relationship isclassical and has found practical applications; for in-stance, in analyzing' the behavior of the so-called peak-type of vacuum-tube voltmeter with small applied volt-ages. It is important both theoretically and practically,because it is essentially independent of variations intube construction and processing, and therefore givesaccurately reproducible results that do not demandcareful selection of tubes. Fig. 1 shows this exponential

Fig. 1 "Retarding-Field" characteristic of Type 954 con-nected as a diode: lb versus Eb.

characteristic for the Type 954 connected as a diodewith all the grids strapped to the plate. The slope b ofthe ln -Eb curve is 9.64 volts-'.

III. "RETARDING-FIELD" CHARACTERISTICSOF MULTIGRID TUBES

The development of the exponential mode of opera-tion of multigrid tubes will be presented here briefly;

6 C. B. Aiken, "Theory of the diode voltmeter," PROC. I.R.E.,vol. 26, pp. 859-877; July, 1938.

but some of the clarifying details are given in AppendixI. When the screen grid voltage of a tetrode or a pentodeis reduced to a value near cathode potential and the gridvoltage is sufficiently negative so that a virtual cathodeis formed at the grid, and both grid and plate currentswill vary exponentially against grid voltage. These lowcurrents are determined by initial velocity distributionand occur at grid voltages more negative than those forwhich the three-halves power law holds.

Fig. 2 shows these grid and plate currents for the

Fig. 2-"Retarding-Field" characteristics of Type 954: lband I versus Ec for various values of Eb.

Trype 954 plotted against grid voltage on a semilog chart.The curves are taken for plate voltages between 20 and400 volts with the screen grid voltage held constant at+3 volts. It will be seen that the curves are accuratelyexponential up to about 10 microamperes and that thegrid current L, is substantially independent of platevoltage. The slope bp of the ln Ib-E, curves is constant,being practically 9.13 volts-'. The slope bg of the ln10-E, curve is 10.86 volts-'. The curves show that,over the full plate voltage range, the grid and plate cur-rents, respectively, obey the relationships:

IC = 1,C8EbgEc (3)and

1b = I b8bpE, (4)

where 8,, and lb, are, respectively, the extrapolatedvalues of grid and plate currents at E, = 0. Furthermore,the plate current curves of Fig. 2 show that since bp isfixed, the value of lb, is determined by the plate voltageonly.

- -

1004 September

El-Said: Electronic Wattmeter Multiplying Circuits

Fig. 3 shows the same experimental data of Fig. 2,plate current versus plate voltage plotted on a linearchart for various constant values of grid voltage. Thesecurves show that, over the full plate voltage range from20 to 500 volts, the plate current varies accurately lin-early with plate voltage. Also, the lines determine a

Fig. 3-"Retarding-Field" characteristics of Type 954: Ib,

IC and Ic2 versus Eb for various values of EC.

common intersection on the plate voltage axis at Eb=-660 volts. Under these conditions, the value of lb,varies linearly with plate voltage, and (4) for plate cur-

rent can therefore be re-written as:

lb = (mEb + n)EbpEc, (5)

where Eb and E, are, respectively, the plate and gridvoltages,m is the slope of the extrapolated plate current-plate voltage characteristic for E, = 0, and n is the extra-polated plate current for Eb = E, = 0.

Referring to Fig. 2 and 3 it is evident that a multi-grid tube operated under initial velocity conditionsyields a plate current accurately proportional to theproduct of a linear function of plate voltage and an ex-

ponential function of grid voltage over wide ranges.

The grid current is an exponential function of gridvoltage, and substantially independent of plate voltage.

IV. MULTIPLYING CIRCUITSConsider the simplified circuit of Fig. 4 in which a

pentode V-1 is shown with its electrodes suitablypolarized to operate in the exponential regime with (3)and (5) valid. Two ac input voltages vp and v, are appliedin the plate and grid circuits, respectively. The source

impedances of vp and v, are not shown, since these can

be neglected compared to the corresponding high tubeimpedances in the operating region of Fig. 3. For thesame reason, grid rectification is neglected, and only (5)will be considered. With no alternating voltages applied,let the dc plate and grid voltages be so adjusted that

the direct current through the plate is given by:

lb = (mEb + n)ebpEc, (6)

where Eb and EC are the dc plate and grid voltages, re-

spectively.

V-1

Fig. 4-Simplified multiplving circuit incorporating a pentodeunder initial velocity conditions.

When vp and v0 are applied, the plate voltage becomes(Eb +vp) while the grid voltage becomes (E,+v,), andthe plate current becomes (Ib +ib) . Therefore:

ib = Ib(EbPvg - 1) + MOVE6bpvg, (7)

where mo = mEbpEc and equals the plate conductance at thequiescent operating point. Over an infinitesimal gridvoltage excursion, (7) reduces to:

i-=Ibp(v,0) + mo(v,) + mobp(vpvg). (8)

Equation (8) indicates that with infinitesimal grid volt-age excursion, the change in plate current, consequentupon application of alternating plate and grid voltages,comprises a term proportional to the product of thetwo voltages in addition to terms proportional to eachvoltage. The average change in plate current equals(ib)av. proportional to (v,v,),. provided that v, and v,have no dc components.

In order to obtain a useful amount of output by in-creasing the grid voltage excursion, plate rectificationoccurs due to the nonlinearity inherent in the exponen-

tial curvature. The multiplying circuit of Fig. 4 wouldbe ideal, had the plate current been proportional to theproduct of linear functions of plate and grid voltages.However, two methods will be described here for com-

pensating accurately the effect of plate rectification.One method compensates for the average value of platerectification, whereas the other compensates for theinstantaneous value.

A. Compensation by Average Grid RectificationSince plate current is exponential against grid volt-

age, plate rectification occurs and the average changein plate current consequent upon application of alter-nating plate and grid voltages contains a componentdetermined by grid voltage only. This component is in-dependent of plate voltage and can therefore be compen-sated by changing the average potential to the grid.Since, as previously indicated, the grid current is sub-stantially independent of plate voltage, it is thereforereasonable to see if the tube can bias itself automatically

954 PENTODE CONNECTIONEh 5.6 volts Ec' +3 volts 77FOREc-0638

Ib (1.978 Eb 1300) V9J3E J I9FOR EC,-0.6651Ic1 065 610 86E ,v.o. o 5

Erp. 0.503 e-9.13 c Meg.QS

9mgf9.13 It / 4 IbFOR Ec .-O7O2z

IbFOR Ee--0.665

,p, .s,3 (E +658)/ /

lb FOR Ec 0.8091

ILA---~~~~i, 'C2_IFORE --0.665"

|| ';- _ PLAsTE VOIL.TAG E Eb,volts8~~ ~8Q- - N

0)

1949 1005

PROCEEDINGS OF TIIE I.R.E.

to the proper operating point to compensate for averageplate rectification by using the grid current to produceaverage grid rectification.

Consider the circuit shown in Fig. 5 (a) which is simi-lar to that shown in Fig. 4, except that a grid leak re-sistor R, by-passed by a capacitor C, is connected inseries with the grid. Also, a plate load resistance R, by-passed by a capacitor C, is connected in series with theplate supply Ebb. Let the source impedances of the alter-nating voltages v, and v, be negligibly small comparedto the corresponding tube impedances in the operatingregion shown in Fig. 3. Let, also, the values of Cq andC, be such that all the alternating voltages v, and v, areeffectively applied respectively across the grid-cathodeand plate-cathode spaces of the tube.With no alternating voltages applied to the system,

let the dc potential at the plate be Eb, and the directcurrents flowing in the plate and grid be Ib and I re-

pV

V-i.

-aT C iYAEb(a)

Taking average values of both sides and solving for(ib)av. we get:

(E Pb)av.E-bpAE, - 1(ib),x. = Ib _bpv -

1 + moRp(EbPvo)av.E-bpAE,-bp/AE,

+ mo(vPEbPvg)av.1 + mIoRp(,bpv\)av.E-bpAE, (9)

Equation (9) shows that the average value of thechange in plate current consequent upon the applicationof v, and v, is composed of an average product term ofthe form (Vp,Zbpvg)av. and an additional voltage term de-pending upon the alternating grid voltage only. Sincethis additional voltage term is due to plate rectification,therefore, if by means of grid rectification the value ofAE is so adjusted that the additional voltage term in(9) vanishes identically, then:

(Ebpvg)av. - EbpAECI

i.e., the value of AE, must be given by

1AEC = - In (EbVg)av.,t(0

bp

which gives a condition for perfect grid bucking. Underthis condition, (9) reduces to:

MO (V=EbPio)av.

1 + moRn (Ebpvg) av.(11)

Particular interest is taken in the simple case wherevP= VP cos (wt+0) and v,== V9 cos wt (for other cases,see Appendix III). Substituting these values in (11) weget:

(b)Fig. 5 Multiplying circuit incorporating a pentode tinder initial

velocity conditions and using average grid rectification tocompensate average plate rectification.

spectively. The grid current L, will flow through R0,thus biasing the tube negatively; the bias dependingupon R, and its magnitude is E, =I1R,. Also, the platecurrent lb will flow through R', thus dropping the bat-tery voltage to Eb at the plate.When v, is applied, grid rectification occurs and the

current through R, increases. The total grid voltage be-comes (-E,+v,-AE,) where AE, is the average recti-fied grid voltage depending upon the magnitude of R,and the applied alternating grid voltage (see AppendixII).When the alternating voltages vp and v, are applied

simultaneously, the plate current will be (Ib-+ib); theaverage value changing from lb to (Ib+ (ib)av.). The totalplate voltage will be (Eb+Vp-(ib)av.Rp). Therefore,substituting these values in (5), the plate current con-sequent upon the application of alternating voltages tothe system is given by:

lb + ib = [mEb + mv, -M(ib)av Rp + nI(bE+b AE,

mO -jJ,(jbpVo)( b)av. m=- bV Cos 4,

1 + moRn Jo(jbpV g)(12)

where -j JI(jb, V0) is modified Bessel Function firstkind first order, and Jo(jbpVgJ) is modified Bessel Func-tion first kind zero order. Fig. 6 shows a plot of thequotient

-jJ,(jbpVg)Jo(jbpVg)

against bpV,,, and it will be seen that the deviation ofthis quotient from linearity is less than 3 per cent forvalues of bpV, up to 0.5. M1ultiplying both sides of (12)by Rp and considering only the linear portion of theBessel quotient, we obtain

mORP IAEb = (ib)av.Rp = m bRV,V1 cos 4,

1 + moRp 2(13)

which shows that upon the application of pure and co-herent alternating plate and grid voltages, the dc poten-tial at the plate changes by an amount proportional toV, V0 cos 4), provided that grid bucking is perfect. Thecircuit in this case can be used to measure the average

1006 September

(10)

VI


value of the product of two alternating voltages. Underthese conditions, when the peak ac grid voltage is lim-ited to less than 1/2 bp, the accuracy of measurement isbetter than + 1.5 per cent of full output. M\eanwhile,for bpV=0.5, the full output is 12.5 per cent of thepeak ac plate voltage at unity power-factor and withmoRp = 1; being only doubled for moRp>> 1.

quotient -Ji(j bpVg)anitpeFig. 6-A plot of the quotient Ji (j bp V5) - and its per

cent deviation from linearity versus bpV,.

As to grid bucking, it is possible (see Appendix II) toadjust the grid rectification efficiency so that the tubebiases itself automatically to such an operating pointthat almost perfect grid bucking is achieved. The con-dition for optimum grid bucking requires that the mag-nitude of Rg be given by:

1R9

I,--, or: bb,I,R, = 1bb Ic

(14)

wherebg

bp(see Appendix II).However, by examining (9), it will be apparent that

when vp=0, the condition for perfect grid bucking de-mands that (ib)av, should be zero independent of themagnitude of v.. Therefore, practically, the multiplyingcircuit of Fig. 5 can be adjusted for optimum gridbucking by applying appropriate ac voltages to the gridwith the plate input circuit short circuited, and adjust-ing R, for no change in the dc component of plate cur-

rent. The same result can be achieved by fixing thevalue of Rg and introducing a small adjustable grid biasto control the magnitude of 1, so as to fulfill the requi-site grid rectification efficiency for optimum grid buck-ing. These practical methods of adjustment require onlya fair knowledge of the parameters 6, b;, and I in (14)for the particular type of tube used. It will be necessaryto readjust the screen grid voltage in order to restore theplate current to its initial value before and during theadjustment.The circuit of Fig. 5(b) is similar to that in Fig. 5(a),

except that the alternating voltages v, and v, are fed inparallel with the plate and grid, respectively. In thisarrangement, since capacitive feed is used, any dc com-ponents in the applied voltages will not reach the tube.A resistance-capacitance filter Ro-Co is connectedacross the plate and cathode; the dc potential at theplate appearing across Co.

o lo 20 30 40 50 60 70 80 90 100GRID VOLTAGE QV9 mv rms

Fig. 7 Observed values of dc output voltage due to imperfectgrid bucking versuls rms grid voltage for various values of Roin the circuits of Fig. 5.

Fig. 7 shows results of measurements of the dc out-put voltage due to imperfect grid bucking in the circuitof Fig. 5(b) using the Type 954. The measurement ismade by a balanced degenerative dc voltmeter con-nected across Co. These curves show that for the par-ticular value of R,=0.423 megohm, the output due to

1949 1007


imperfect grid bucking is less than 4 millivolts over an

rms grid voltage range up to 40 millivolts. For smallervalues of R, the output is positive, whereas for largervalues the output displays a negative loop. The negativeloop curves show that the ac grid voltage range can beincreased to about 70 millivolts without having an ap-

preciable output voltage due to imperfect grid bucking,as shown by the curve for R, = 0.434 megohm.

be of the polystyrene dielectric type. With these cir-cuit components, and with Eb=250 volts, the full out-put at unity power factor is about 20 volts dc for 200volts peak ac plate voltage, and 70 millivolts peak ac

grid voltage.Circuit Performance as a Wattmeter: The circuit of Fig.5(b) is particularly suited for a simple electronic watt-

Fig. 8-Observed values of dc output voltage due to optimum gridbucking versus rms grid voltage for various values of the heatervoltage in the circuit of Fig. 5.

Fig. 8 shows results of similar measurements with thecircuit initially adjusted for optimum grid bucking at a

heater voltage about 5.7 volts. The curves show that thecircuit adjustment for optimum grid bucking is not ap-

preciably affected by a reasonable change in heatervoltage.The pentode Type 954 is particularly suited for a

large plate voltage swing without showing a serious er-

ror, due to imperfect linearity of plate current-platevoltage characteristics. For this Type, 4 microamperesis a suitable value for Ib, and 1.0 microampere for 1,. Theplate conductance at 4 microamperes is of the order of0.005 microampere per volt, corresponding to an in-ternal resistance of 200 megohms. The requisite valueof R, for optimum grid bucking is of the order of 0.5megohm, and a suitable value for Rp is 100 megohms.Since the dc voltage drop across Rp is 400 volts; theresistor should therefore be selected for minimum volt-age coefficient. This is necessary in order to minimizeresistor rectification due to dc polarization. Also, for thesame reason, capacitors Cp and C0 should preferably

Fig. 9 Basic circuit of electronic wattmeter incorporatingthe multiplying circuit of Fig. 5.

meter. Fig. 9 shows a diagram of the basic circuit ofwattmeter of this type in which V-1 is the basic measur-

ing tube, and V-2 is a twin triode connected as a conven-

tional degenerative dc voltmeter. The indicating meter,ia is connected across a reversing switch S. The voltagecomponent of power to be measured is applied to theplate of V-1, while all the current component practicallypasses through the series resistor r and develops a volt-age drop which is proportional to current and is appliedto the grid of V-1.

Fig. 10-Relative deflections of electronic wattmeter versus

dynamometer wattmeter at unity power factor.

Assuming that v = V cos (wt +4) and i = I cos wt, the

change in the dc voltage across R, consequent upon the

application of v and i is obtained by substitution in (13),hence:

f.0

EH 5.72V'25 ~~~~.Eh.s.~~~64~

.20 E.-E 58v

CIO~~~~~~~~~~~~~~~~~~~I

+15 6..

1I0 20 304050 78090100

GRID VOLTAGE mv rms

I-

1008 September

1I-Said: Electronic Wattmeter Multiplying Circuits

moRp 1

AEb = -R-2b(r (VI cos)1 +moRp 2

which is proportional to the mean ac power dissipatedin the load.The circuit was tested for wattmetric indication by

comparing its reading with that indicated by a dyna-mometer wattmeter at 60 cycles. Fig. 10 shows relativeresults obtained at unity power factor and Fig. 11 atvariable power factors. The values of power factor indi-cated in Fig. 11 are calculated from the reading of the

Fig. 11-Relative deflections of electronic wattmeter versusdynamometer wattmeter at various power factors.

dynamometer wattmeter, a voltmeter, and an am-

meter. The curves show that both wattmeters agree

closely. Fig. 12 shows that the deflection of the elec-

10 20 30 40 50 60 70 so

Fig. 12-Relative deflection of electronic wattmeter versusrms grid voltage with a fixed ac plate voltage.

tronic wattmeter with a constant ac plate voltage isvery closely proportional to the magnitude of the ac

grid voltage, as has been theoretically predicted andshown in Fig. 6.

Preliminary trials for determining the frequencycharacteristic of the circuit shown in Fig. 9 indicated a

reasonably flat frequency response up to about 10 mega-

cycles. Some errors were observed above this frequencyindicating the presence of feedback. The circuit was

then tested with the grid input circuit short circuitedand a variable frequency voltage of about 150 volts rmsapplied to the plate. In this test, a dc output of one percent of full scale was observed at about 5 Mc increasingapproximately as the square of frequency. Since, in thistype of operation, the grid-plate capacitance is insuf-ficient to cause appreciable error at this frequency, itwas found that the observed errors are due to feedbackthrough the cathode and screen-grid lead inductances.The effects of these two types of feedback, upon the

performance of the tube under such operating conditions,are in opposite directions. One type of feedback cantherefore be neutralized by means of adjusting theother. However, in the Type 954, an examination of theelectrode structure showed that there are some auxiliarymetallic parts, such as a top cap, a bottom ring, and a

getter support, all connected to the bottom ring to-gether with the cathode, rather than with the suppres-

sor, as is usually the modern policy of tube construction.The resulting direct capacitance between plate andcathode is of the order of 1.5 ,u,uf. In the megacycle re-

gion, and with 150 volts rms on the plate, an apprecia-ble radio-frequency current flows through the cathodelead inductance, thus developing an in-phase voltage inthe grid circuit of the order of few millivolts. In theType 954, the amount of feedback through the screen-

grid circuit is insufficient to compensate for this cathodefeedback. However, with the aid of a small adjustableneutralizing capacitor of the order of 1 ,u,uf connectedbetween the plate and the screen grid it was possible toincrease the feedback through the latter to an extentsufficient to compensate for the excessive cathode feed-back. This arrangement extended the frequency rangefor the Type 954 to 20 Mc. There is a good possibilityof exceeding this range by using tubes which do nothave such an appreciable plate to cathode capacitanceas in Type 954.The multiplying circuit of this wattmeter is highly

sensitive to supply voltage changes the power supplyshould be well regulated. A good degree of stability isachieved by the use of a bucking tube of the same typeas the wattmeter tube and connected to the other arm

of the bridged degenerative voltmeter circuit of Fig. 9.Nevertheless regulation, within 0.2 per cent is neces-

sary at the low alternating plate voltage ranges. Thecircuit stability may also be improved by using tubeshaving tungsten or tantalum filaments since these havea value of b = e/kT of the order of 3 to 4 volts-'. The use

of such tubes require a maximum ac grid voltage of theorder of 100 millivolts rms for bpV,=0.5. This has thefurther advantage of reducing effects of stray pickupand feedback at high frequencies.

In some fields of application of this wattmeter, as forinstance in the measurement of power at low power-fac-tors, the use of a two-tube push-pull arrangement maybe necessary, in order to reduce the errors caused by

I~~70

60

50

z /40O

0 -Fw

30o

20 ____.

/0 GRID VOLTAGE mv rms _

I

1949 1009

.1,


imperfect grid bucking and imperfect linearity of platecurrent-plate voltage characteristics. Fortunately, sucherrors in this type of operation can be determined andcontrolled.

This wattmeter has the advantages of a wide fre-quency range and simplicity, in addition to the fact thatit absorbs only a minute fraction of the measured power.It also enables the measurement of very small amountsof power.

B. Compensation by Instantaneous Grid RectificationIn this method of compensation, the instantaneous

values of rectified plate current due to the exponentialgrid curvature are compensated by varying the instan-taneous grid voltage on a proper logarithmic curvature.The system, in this case, obtains the product of the ap-plied alternating voltages as if the plate current wereproportional to the product of linear functions of plateand grid voltages.

Consider the circuit shown in Fig. 13 which is similar

Let the direct current through the plate be given by(6). The rectified plate current consequent upon appli-cation of alternating voltage v, will be given by:

lb + Aib = (mEb + n)E-bpE±+bpIbg In (1+vg!Ecc)

= IbJbplbg In (1+vglEcc)

Let in the ideal case bp = b,, then Aib=Ib(vg/Ec,) indi-cating that the output plate current is an exact dupli-cate of the input voltage v., and that plate rectificationis perfectly compensated. The system, in this case, be-haves as if the plate current were accurately propor-tional to the product of linear functions of plate andgrid voltages.When the alternating voltages vp and v, are applied to

the system, the plate voltage will be (Eb+Vp - (ib)av.Rp)and the grid voltage will be (-E-+e,). The plate cur-rent consequent upon application of alternating voltageswill be:

b+ ib =[mEb+mVp-m(ib)av.Rp+n]e-bpEc+bp/bg In (i+vgfEcc)

= [lb+mOvp-MO(ib)av.Rp] [1+1

and if bp= b, then:

v±ib = MOVp + E(I'b - MO(ib)av.Rp)

EccFig. 13-Multiplying circuit incorporating a pentode under initial

velocity conditions, and using instantaneous grid rectificationto compensate instantaneous plate rectification.

mO+ - vpv - MO(ib)av,Rp.

Ecc

to that in Fig. 5 except that the by-pass capacitor C0 isremoved and the alternating voltage v, is applied to thegrid through the grid leak resistor R. An external biasE,, is also shown.With no alternating voltages applied, the direct cur-

rent through the grid will bias the tube negatively; thebias will be E =IcR, -E, where =fI, E-bgEc. There-fore: Ec +E, =Ic8R, r-bgEc When the external bias E,,is much larger than Ec, the value of R, will be practically(Ecc/I0), and the relation between E, and E,, will begiven by:

1 E

ECC= IcsR0EboEc, i.e., -E = - ln--

indicating that the grid maintains a bias proportionalto the logarithm of the applied external bias.When the alternating voltage vg is applied, grid rec-

tification occurs, and the instantaneous voltage devel-oped at the grid will be (-E +e,), where eg is given by

1 Ece + v

-Ec + e =-In

bv IC8Rgi.e.,

1 /

e-=-ln 11+--. (15)

Equation (16) holds for all wave forms of the voltageVP and v. It will be seen from (15) and (16) that, whenthe instantaneous voltage across the grid-cathode space

of the tube is a proper logarithmic function of the ap-

plied voltage v0, the compensation of the exponentialcurvature is accurate. In this case, the circuit obtainsthe product of the two voltages v, and v, without pro-

ducing errors due to increasing the grid voltage excur-

sion.

Taking average values of both sides of (16) and as-

suming that vp and v, have no dc components, the changein the dc voltage across Rp is given by:

AEb = (ib)av.Rp = moRp 11 + moR -(VpVg)av.

1 + moRn Ec(17)

Although the circuit is extremely simple, yet it has thedisadvantage that the accuracy of obtaining the prod-uct (v,v,) depends upon the discrepancy in the ratiobp/b, from unity, in addition to a working frequencyrange limited by the frequency characteristics of R, andits associated capacities. The circuit can be arranged togive satisfactory performance up to high audio frequen-cies.

However, the system is greatly improved by the ad-dition of a radio-frequency logarithmic circuit. This ad-ditional circuit depends upon the fact that, with proper

(16)

1010 September


adjustments, the plate current of the pentode tube isaccurately logarithmic against plate voltage; the rela-tionship being:

Ib=A+[KlnEb, (18)

where A is a constant and K is the slope of the Ib-ln Ebcurve. Fig. 14 shows the circuit diagram and operatingcharacteristics of the Type 6AU6 for a logarithmic rela-tion over the plate voltage range from 0.5 to 2.0 volts.In this circuit the value of K is mainly controlled by thescreen grid voltage and the self-bias resistor. If the di-

lb R

wSEb 56AU6 >

p

1.0EL b.53 +0.245InEb_m a EE 2.3 lb-0.9

3.6 ___Oa

21 _ _ _ _ _ _ Ql

+ 2.8 2.0 Z a:~~~~~~~' 046

retoe2 tepaevotg agefo7. 1to 20vot.

cr -

z

0

rect curnvlae PLATE VOLTAGE Em iNu n I.tto~U,Dr-00 0 0 0 00i c 0 0 0 0000- N~ e)

Fig. 14 A pentode arrangement for a logarithmic plate cur-rent over the plate voltage range from 0.5 to 2.0 volts.

rect current voltage on the plate is adjusted to the mid--dle of the logarithmic range and a limited alternatingvoltage applied to the plate through a low impedance,the consequent change in plate current will be:

ib = K In (1 +E (19)

CP

R5V 'VVp g TTFig. 15 Multiplying circuit incorporating a pentode under initial

velocity conditions combined with a logarithmic circuit to com-pensate instantaneous plate rectification.

Consider the multiplying circuit shown in Fig. 15,which is similar to that in Fig. 13 except that an addi-tional logarithmic circuit is coupled to the grid circuit ofV-1 by means of the base resistor R. Thealternatingvoltage v, is applied between plate and cathode of V-3

through a coupling capacitor C3 and across a resistor R3.The voltage drop across R is injected in the grid circuitof V-1 through the cathode lead. If the plate current ofV-3 is much greater than the cathode current of V-1, asis actually the case, feedback from V-1 into V-3 isnegligibly small.With no alternating voltages applied, let the system

be adjusted that at any instant, the plate current of V-1is given by (5); whereas the plate current of V-3 isgiven by (18). The direct current through the plate ofV-3 will flow through R, whereas the sum of the screengrid and plate currents will flow through R3, droppingthe battery voltage Ebb3 to Eb3 at the plate.When the alternating voltage v, is applied, the conse-

quent voltage drop across R sill be given by KRln(1 + (vgjEb3)); the grid voltage of V-1 thus becomes

-Ec+ KR ln (1 +-).

When the alternating voltages v, and v, are appliedsimultaneously, the plate voltage of V-1 will be[Eb+vp- (ib)av. Rp], and the plate current will be:

lb+ ib= [mEb+mvp

-m(ib)av.Rp+n]Ebp [-Ec+KR In (+vglEb3)J- [Ib-I-movUp-mo(ib)5VR,p]EbpKR In (1+vglEb3)

If by a careful pre-adjustment bpKR = 1, then:

v vib= moVp + -- (lb - mO(ib)2V.Rp)

Eb3

mO+ --vpvg -MO(i)a.Rp

Eb3

which is similar to (16). The change in the dc plate volt-age consequent upon application of vp and vg is givenby:

moRp 1AEb = (ib)av.Rp = (VpVg)av.' (20)

1 + moRp Eb3

The requisite value of R can be calculated from1/bpK. The practical method of adjusting the system iscarried out experimentally by applying the alternatingvoltage vg with the plate input circuit of V-1 short cir-cuited and adjusting R so that the change in the dc out-put voltage is zero. The same result can be achieved byfixing the value of R and adjusting the value of K bychanging the screen grid voltage of V-3. Should bpKR= (1 ±A) where A is a small fraction, the dc outputvoltage due to imperfect compensation by the logarith-mic circuit is approximately ± (A/4) ( Vg/Eb3)2 IbRpwhere Vg is the peak value of v. This amounts to about30 millivolts for a value of A= 0.001, (Vg/Eb3) =0.6 andIbRp= 400 volts. The full output is 30 volts at unitypower factor and 200 volts peak on the plate formoRp= 1. However, provision can be easily made in thecircuit for a fine adjustment of the condition b,KR= 1.

19*49 toll

PROCEEDINGS OF THE 7.R.E.

In practice, values of K of the order of 1.0 milliam-pere are obtained from ordinary receiving tubes. Therequired value of R is about 100 ohms assuming bp= 10volts-'. With these values the frequency range of thecircuit is extended to the neighborhood of 30 Mc. How-ever, it is necessary to compensate for the unavoidableshunting capacities across R. These usually amount toabout 15 micromicrofarads and a small inductive com-ponent of R is desirable.

APPENDIX I

"Retarding-Field" Characteristics of Multigrid Tubes

The extension of the exponential mode of operation ofdiodes to multigrid tubes was accomplished by investi-gating-at first-the retarding-field characteristics oftriodes.

In a triode, when the plate voltage is reasonably low,and the grid voltage is sufficiently negative so that avirtual cathode is formed at the grid, the grid current-grid voltage characteristic becomes exponential with aslope b0 closely equal to b. Since the plate current con-stitutes electrons which pass through the grid mesh, theplate current-grid voltage characteristic is also expo-nential, but with a slope bp considerably less than b. Fig.16 shows these exponential curves of the Type 954 con-

0

I-~~~~~~~~~~~~~~~~~~~~~I

z

o.0 -

3.62

asatide0' ad Cvru EC for__ vaiou valesof

Eb 2O, bp529

Eb-b0, b.6 .66 Eb.20O,bg -IIO8

Et,. 5V b -6.81 Eb 1Ov bg 9.985

GRID VOLTAGE EC Volts

-4.0 -3.5 -3.0 -2.5 -2.0 -1i5 -10 -05

Fig. 16-"Retarding-Field" characteristics of Type 954 connectedas a triode: It, and 1h versus E, for various valuies of Eb.

nected as a triode with grids 2 and 3 strapped to theplate. The curves are taken for plate voltages between 5and 40 volts. The slope b, of the In10-E, curves is, inall cases, fairly close to that observed for the same tubeconnected as a diode, i.e., 9.64 volts-'; but the slope bp

of the ln Ib-E, curves is only 6.81 at Eb=5 volts, de-creasing to 3.62 at Eb=40 volts. This considerable de-crease in the observed value of bp is due to the fact thatthe field distribution between grid and cathode is appre-ciably affected by the plate voltage.7

It was inferred that the dependence of bp upon platevoltage can be avoided by shielding the plate from thegrid-cathode region. This was proved by taking meas-urements of bp for tetrodes and pentodes. Fig. 17 showsresults of the measured values of bp against Eb for theType 954 connected progressively as a triode, tetrode,and pentode. In these measurements the screen grid po-tential was held constant at a value just convenient topermit a few microamperes to flow in the plate circuit.From these curves it is evident that as the plate is pro-

10

954 PENTODE CONNECTION

8_8i s-,954 TETRODE CONNECTION (G-z STRAPPED TO PL bT _

6 4 TRIODE CONNECTION (GISTRAPPED TO G2 a GS STRAPPED TO PLATE)

'954 TRIODE CONNECTION (G2& G3 STRAPPED TO PLATE)

PLATE VOLTAGE Eb volts

0 50 100 150 200 250 300 350

Fig. 17-Measured values of lp versus Eb for the Type 954connected as a triode, tetrode, and pentode.

gressively more and more shielded from the cathode, thevalue of bp increases and becomes substantially inde-pendent of plate voltage.

XVith a fixed value of bp, the general shape of the re-tarding-field characteristics of multigrid tubes are asthose shown in Figs. 2 and 3 for the Type 954. Thesecharacteristics differ from those of triodes in that theslopes bp and b5, and the grid current are substantiallyindependent of plate voltage. Furthermore, the platecurrent varies linearly with plate voltage. In general,different types of multigrid tubes have different valuesof the constants m, n, and I,C in (3) and (5). The valuesof bg and bp mainly depend upon the operating tempera-ture of the cathode. For tungsten filaments, the value ofb is of the order of 3 to 4 volts-'. In all types, cathodestabilization is important in order to obtain consistentperformance, and tube ageing for at least 100 hours maytherefore be necessary.

Perhaps the most interesting feature of these charac-teristics is the wide range over which the plate current

E. G. James, G. R. Polgreen, and G. W. Warren, "Instrumentsincorporating thermionic valves and their characteristics," Jour.IEE, vol. 85, p. 242; August, 1939.

1012 .September

400 450 50011%


varies linearly with plate voltage. If, however, we con-sider the effect of Eb upon Em when all other electrodesare at a constant potential, it is conceivable that theplate current varies in accordance with the functionEbEbIl; where y is the amplification factor. When A ishigh, the value of the exponent bEb/,u is a small fractionof unity and the function becomes practically linearwith Eb. Furthermore, as the plate voltage is increased,the position of the potential minimum moves towardsthe cathode causing u to increase. This increase inguwith Eb tends to make the mode of variation of lb withEb more closely linear over a wider range of Eb. Thecurves of Fig. 3 show practically good linearity over thefull plate voltage range from 20 up to 500 volts. In tubeshaving a higher screening factor, this linear range ex-tends to 1,000 volts, but the internal resistance becomesexcessively high.

Fig. 18 shows the values of yu and rp for the Type 954as calculated from the data in Fig. 2. The curves showthat u increases accurately linearly with plate voltage,but is substantially independent of grid voltage. On theother hand, rp is independent of plate voltage, but variesexponentially with the grid voltage. The transconduct-ance can be obtained from Fig. 2 by multiplying thevalue of lb by the slope b,. All these relations are appar-ent from an examination of (5).

Fig. 18-Values of amplification factor and plate resistance versusplate voltage for the Type 954 under initial velocity conditions.

APPENDIX II

Comparison of Perfect and Practical Grid BuckingIn the description of the operation of the multiplying

circuit of Fig. 5, it was assumed that the tube can biasitself automatically to the proper operating point to com-pensate for average plate rectification by using thegrid current to produce average grid rectification. Un-der this assumption, the average rectified grid biasshould be given by (10) for perfect grid bucking. It willbe proved here that, at least for the cse where v,Vgcos wt, perfect grid bucking can almost be achieved by a

particular adjustment of the grid leak resistor. In thiscase the value of AE, from (10) should be given by:

1AEC = - In Jo(jbpVg).

bp(21)

Considering the grid circuit, and with no alternatingvoltage applied, let the direct current through the gridbe given by:

IC = Ies Cbgc

When vg is applied, the grid voltage becomes (-E^+vg-AE,), and the grid current will be:

IC + j, - IC bg(-Ec+VoAEc)

giving

ic ICI[fbgvg.-ebgAEc - 1.

Solving for SEC= (ic)av,Rg by taking average values ofboth sides and multiplying by Rg we get:

AEc = IcR, [(ebgvg) ebEC - j],

and when v = Vg cos wt, then:

AEC = IcR [Jo(jbqVg) *brAEc- I]. (22)

Equation (22) relates the average rectified grid volt-age to the applied ac grid voltage. This expression issimilar to that obtained by Aiken8 on the theory of thediode voltmeter, and has no explicit solution.

If, however, it is possible to adjust the magnitude ofthe grid leak resistance Rg so that the average rectifiedgrid voltage is equal to that required for perfect gridbucking, then R, should be determined by substitutingfrom (21) into (22) and solving for Rg. Hence:

1 [Jo(jbpVg)]1+5 ln [Jo(jbpVg)]bpIc [Jo(jbgVg)] - [Jo(jbpVg)]'-+ (

Equation (23) shows that for the case (5=0, i.e.,bp=b, the denominator vanishes, and Rg should be in-finite. But, for the case a positive, i.e., b,<bg which isusually true, the value of Rg becomes finite. In this lat-ter case, it is clear from (23) that the magnitude of Rgfor perfect grid bucking should depend upon the magni-tude of Vg. This indicates that it is not possible with asimple linear grid-leak resistor to achieve perfect gridbucking at all values of ac grid voltages.

Let therefore the adjustment of the grid leak resist-ance be such that (21) and (22) are satisfied only atsmall values of Vg. XVe have then:

right-hand side of (21)1- In (1 + lb42V02)p

and right-hand side of (22)

8 C. B. Aiken, 'Theory of the diode voltmeter," PROC. I.R.E..vol. 26, pp. 859-877; July, 1938.

1949 1()13


---'b2Vg2 (1 + 6)lbpVg2+ bglcRv b I

I

By equating both results, we get:

IR,= or 6b,I,R, = 1.

6bbjl

Z - 1IbRI

1 +mORgZ(25)

and the per cent error in the dc output due to imperfectgrid bucking is given by:

(14)

With the adjustment of the grid leak resistor as givenby (14), it is possible to calculate the dc output voltageof the circuit of Fig. S due to imperfect grid bucking. Re-arranging (22) we get:

AEC\AEC=- In (eb)gav.g - -lIn 1 + (24)

bg~ bg hCR

For small values of vg the rectification efficiency islow, and AE,/IcR, is small compared to unity, thus:

/ AEC \ AEcIn 1 + f

ICR g 1-,R g

Therefore,ln (eboVu)

AEc IcRgI

1 + blgcRofor a first approximation. Using successive approxima-

tion, and substituting from (14) into (24), we get:

E5PAEC = [('b60V0)fa. I/X

where

X = + 26 + 62 262 In (Ebgvg)

Li

U

-+300b /

lb 50 .. C..

rr i,j U Ol/V.

+5_0

_5

0 0.1 0.° °3 b 0'43

Fig. 19-Theoretical values of the dc output vgrid bucking versus b,V, for various valunity for the multiplying circuit of Fig. 5.

Let

Z - (ebpvg). * e-bpAEc = (jbpvg) [(ebgvg

lb Z - 1 (,bpvg)a_ 100.

MO Z (VPEbpvV)av.(26)

Equations (25) and (26) are general and apply for anywave form of the ac grid voltage. For the case v,= VIcos wt, Fig. 19 shows a plot of the dc output voltage dueto imperfect grid bucking as calculated from (25) forvarious values of bbjIR, close to unity. These curves

show that for the condition bb,0IcR, = 1, the output ispositive, increasing as b,V, is increased; but for valuesslightly greater than unity, the output shows a negativeloop. It is interesting to see that the curves in Fig. 19agree in shape and order of magnitude with those ob-tained experimentally in Fig. 7.

20 30 40 5o 60

GRID VoLt-AGE mi/l; volts rms5.

Fig. 20-Theoretical percentage error due to imperfect grid buckingversus rms grid voltage for various values of 15bbI1R, close tounity for the multiplying circuit of Fig. 5.

J \ / / Fig. 20 shows the per cent error in the dc output volt-age due to imperfect grid bucking as calculated from(26) for unity power factor. These curves show that it is

__/X possible to adjust the grid leak resistor such that forrms grid voltages less than 40 millivolts, the order of theerror is less than one per cent at unity power factor and

Q.~ 0.6 o 7 with only 15 volts rms on the plate.It should therefore be concluded from the above that

roltage due to imperfect if a linear grid-leak resistor is used, the adjustment forlues of t5h0IcR, close to optimum grid bucking corresponds to a value of bb,I,R,

slightly greater than unity. The requisite grid rectifica-tion efficiency for optimum grid bucking equals bs,V,,for values of b,V, less than 0.5, decreasing slightly for

-)]V lIXY higher values.

then (9) can generally be rewritten as:

Z - 1 moR2pZ (Vpebovo)av(?ib)av.Rp = IbRFp +

m __v____1 + moRpZ 1 + mORPZ (6bpvg)aV.

from which the absolute value of the dc output voltagedue to imperfect grid bucking is given by:

APPENDIX III

Accuracy of Measurement by the Multiplying Circuit ofFig. 5 under Complex Wave Forms

It was proved in the text that (ib)av. of (9) is propor-

tional to (V, Vg cos 0) in the simple case where vp,= Vcos (wt+o), v= Vs, cos wt and b,V0.0.5. If the alter-

I-I K

September,1()14

El-Said: Electronic Wattmeter illultiplying Circuits

nating voltages v, and v, have complex wave forms, theaccuracy of measurement of (VpVg)av. is subject to an er-ror caused by the fact that only average plate rectifica-tion is compensated. This type of error does not exist inthe multiplying circuit of Fig. 15.

Equation (9) gives the quantities generally involvedin determining the value of (ib)av. under any wave formof the alternating voltages vp and v/. Also, (26) gives theper cent error in the indicated reading. Estimation ofthis error under standard wave forms-expressed inFourrier Series-indicates that second harmonic termscontribute almost to the full value of the discrepancy inthe indicated reading from that of a perfect multiplyingcircuit. The error will therefore be considered due to sec-ond harmonic. Two cases will be illustrated here.

Case a: vp= Vp cos (wt+0), and vg= V, (cos wt+kk cos2 wt) where kg is the amplitude ratio of the second har-monic to the fundamental of the ac grid voltage. The fol-lowing expressions are used in the analysis:

(EbVg)av = Jo(jbVg) + [Jo(jbkVg) - 1]

± kv(bVg)3[1 + -kgbV,,(VpEbVg)av. = Vp Cos Of -jJi(jbVg)

+ 4 kv(bV )2[1 + 2kvbVg + 1(bVg)2]}.

Fig. 21 shows the calculated per cent error versus rmsvalue of the fundamental grid voltage for various values

m '

& -2A-

°16 10 2 05

/f'ag^ae*/ 68poD J3 mi~//i volts rnms

Fig. 21-Calculated error in the indicated reading of the multiplyingcircuit of Fig. 5 when the alternating grid voltage contains asecond harmonic.

of kg. These curves show that the error is of the order of20 per cent for 100 per cent second harmonic at bpVg=0.5; decreasing to about 3.5 per cent at bpVg=0.125.

Case b: v= V, cos wt, and vZ,= Vp [cos (wt+q$) +kpcos (2 wt+0))], where kp is the amplitude ratio of secondharmonic to fundamental of the ac plate voltage. Thefollowing expression is used:

(Vpebvg)av. = Vp cos 4)[-jJf(jbV0,) + kji-2J2(jb V,)]Fig. 22 shows the calculated per cent error versus the

rms value of the ac grid voltage for various values of

~~~ 0 ~ 1

9~~~-2

0 /0 2o 30 4o 50 6

&'ndam.,tit/ CA'IO VoL4rA6e mil//liol/4f r'ms

Fig. 22-Calculated error in the indicated reading of the multiplyingcircuit of Fig. 5 when the alternating plate voltage contains asecond harmonic.

k,. The curves show that the error is of the order of 14per cent for 100 per cent second harmonic at bp V, = 0.5;decreasing to about 2 per cent at bpV =0.125.

It will therefore be concluded that though the meas-urement of ac power by the wattmeter circuit of Fig. 9 issubject to a reasonably serious error when the currentand voltage are complex, advantage can be taken by re-ducing the voltage drop across the series resistor r. Theaccuracy is greatly improved by using the appropriateseries resistor which gives a value of bpVg about 0.125(quarter full scale). Under this condition, the error inthe measurement of the total ac power is of the order of+5 per cent when both current and voltage contain 100per cent second harmonic; decreasing to less than 3 percent for 50 per cent.

ACKNOWLEDGMENTS

The author wishes to extend his appreciation to Gen-eral Radio Company, Cambridge, Mass., whose co-op-eration made this development possible, with especialindebtedness to D. B. Sinclair, assistant chief engineer,whose constant interest has carried the investigation toa stage of development; and to R. A. Soderman, devel-opment engineer, for the able assistance in performingthe experimental tests and many helpful calculations.The author also wishes to express appreciation to

Fouad University, Cairo, Egypt, for Fellowship facili-ties for circuit development.

1949 1015

Documents

Novel Multiplying Circuits with Application to Electronic Wattmeters