IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 47, NO. 3, JUNE 1998 801

Compensation of the Frequency DependentSystematic Error of a Time-Division

Watt Converter Using Two MultipliersLjubisa D. Jovanovic,Member, IEEE

Abstract—The paper presents an original method for thecompensation of the frequency-dependent systematic error oftime-division multipliers (TDM’s). The method is based on usingtwo identical multipliers working at different conversion frequen-cies. Full compensation of the error can be achieved using simplecalculations, taking into account only the values of the output sig-nals of the multipliers and the ratio of the conversion frequencies.It has been shown theoretically by computer simulation that theerrors in the method are less than+50 �W/W for input signalfrequencies up to one tenth of the lower conversion frequency.For practical realizations of the method, there is no need to usetwo complete TDM’s because some simplifications are possible.A prototype watt converter based on this method has beendeveloped, and results of measurements show that an uncertaintyof �50 �W/W is achieved in the input signal frequency rangefrom 50 Hz to 10 kHz at a lower conversion frequency of only100 kHz.

Index Terms—High-precision wattmeter, in-phase error, powermeasurements, quadrature error, systematic error of the TDM,time division multiplier (TDM), watt converter.

I. INTRODUCTION

T IME division multipliers (TDM’s) of the Goldberg type[1] (Fig. 1) are used nowadays, due to their good features,

in high-precision watt converters. The precision of thesemeasuring instruments is a few parts in 10[2]. A TDM ofthe Goldberg type, however, introduces a systematic amplitudeerror (in phase) depending on the square of the ratio of theinput signal frequency to the conversion frequency [3], [4],thus making this measuring method, and the measuring devicesbased on this method, applicable to signals of only a singlefrequency or to signals in only a very narrow bandwidth.Recently there are requirements for the precise measure-ment of the electric power not only at the line frequencyof the input signals, but at a significantly extended rangeof frequencies (up to and beyond 10 kHz). Because of theincreased widespread thyristor applications in industry thereare special requirements for precise power measurements whenboth voltage and current signals are distorted, i.e., in thecases when both signals contain harmonics. By increasingthe conversion frequency one can decrease the systematicerror of the wattmeter. However, the conversion frequency

Manuscript received May 19, 1997; revised June 26, 1998.The author is with the Institute Mihajlo Pupin, Beograd 11000 Yugoslavia.Publisher Item Identifier S 0018-9456(98)09868-4.

cannot be increased without limits because of imperfectionsof the components used in the measuring devices. In pre-vious published solutions [5] the conversion frequency hasbeen increased to the value limited by imperfections in thecomponents used. Alternatively, the designing of the converterof the multiplier has been directed toward solutions whichminimize the influence of the imperfections of the components[6]. In this manner the frequency-dependent systematic errorinherent in the TDM has been reduced but still remains finite.The first method [7], which compensates for the systematicerror of Goldberg TDM’s over a relatively wide frequencyrange, uses corrective frequency-dependent circuits on bothinputs of the multiplier which reduce the systematic errorof the multiplier at lower conversion frequencies where theimperfections of the components are insignificant. Using thismethod of compensation there is no requirement for theconversion frequency to be two or three orders of magnitudehigher than the input signal frequency, as is required by analternate solution [5]. In this case, high performance can beobtained even when the conversion frequency is just one orderthe frequency of the input signals. According to the principle ofsuperposition (due to the linearity of the TDM), the systematicerror is eliminated; this is also the case when the frequencyspectra of the input signals are not single frequencies, i.e.,when both voltage and current input signals are distorted.

This paper presents an additional original method for com-pensating the frequency-dependent systematic error which isbased on a different principle: two identical multipliers whichwork on two different conversion frequencies. All of theadvantages of the previously mentioned the method [7] arealso valid for the two multiplier method presented in this paper.Some additional advantages to the practical realizations of thisnew method are described.

II. THEORY OF THE SYSTEMATIC

ERROR COMPENSATION METHOD

It has been shown [3], [4] that the output of the classicaltime-division multiplier (Fig. 1) for sinusoidal input signals

(1)

and

(2)

0018–9456/98$10.00 1998 IEEE

802 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 47, NO. 3, JUNE 1998

Fig. 1. Block diagram of the TDM.

contains a frequency-dependent systematic amplitude error (inphase) which can be expressed with sufficient accuracy by

(3)

where represents the conversion frequency, constantde-pends on the TDM parameters, andrepresents the frequencyof the sinusoidal input signals. This error can be reduced byusing two identical TDM’s, TDM1 and TDM2, working atdifferent conversion frequencies, and (Fig. 2). Asshown in Fig. 2, input signals and are simultaneouslyapplied to the inputs of both multipliers. According to (3), theoutput signals of the multipliers could be expressed by

(4)

and

(5)

where and are the conversion frequencies of themultipliers TDM1 and TDM2, respectively, and is the out-put signal of an ideal frequency-independent TDM wattmeterdetermined by

(6)

where is a constant. The value of the output signalof an ideal wattmeter can be calculated from (4) and (5) andexpressed as

(7)

where is the ratio of the conversion frequencies expressedby

(8)

Thus, full compensation of the frequency-dependent sys-tematic error of the TDM can be achieved using simplecalculation, taking into account only output signals of the mul-tipliers and the ratio of the conversion frequencies,

Fig. 2. Basic principle of the compensating method using two TDM’s.

By using this method there is no need to know the value of theinput signal frequency. Due to the linearity of the TDM andthe principle of superposition, this calculation is also valid ifboth input signals are distorted, i.e., composed of harmonics.Theoretically viewed, this calculation could be done using onlyone TDM by alternating its conversion frequency betweenand if the input signals are adequately stable. Practically,this could be done in laboratory conditions using an existingTDM with slight adaptations.

The basic principle of the compensating method may beseen in the block diagram of Fig. 2, where dotted lines indicatethe part performing the operation defined by (7). The essenceof the method is in the fact that by amplifying the outputsignal of one TDM by the factor , the absolute value ofthe systematic error contained in this signal becomes equalto the absolute value of the systematic error contained inthe output signal of the second TDM. Then by subtractingthe amplified signal from other output signal one obtains thedifference, which represents a frequency-independent signalbecause the two errors mutually cancel. Further amplificationof this signal is not necessary for the systematic frequencyerror compensation. It can be noted that the factor canbe greater or less than unity, depending on the values of the

and Since a TDM does not introduce a systematicquadrature error [3], [4], this type of error is also absent inthe output signal

For the theoretical analysis of the method, it is important todetermine the conversion frequencies, especially the optimumvalue of the conversion frequencies ratioBecause all the previous analyses and derivations are basedon the assumption that systematic amplitude error can beexpressed by (3) with sufficient accuracy, it was necessary todetermine the accuracy of (3), which approximates the realrelation between the error and the frequency of the inputsignals, and up to which value of approximation (3)is acceptable. The choice of the conversion frequencies isa compromise. The upper limit for the higher conversionfrequency is determined by the performance capability of themultiplier and the required wattmeter accuracy. The lowerconversion frequency has to be sufficiently spaced from theupper conversion frequency, but not outside the range in whichthe TDM is linear and not out of the range in which

JOVANOVIC: COMPENSATION OF THE FREQUENCY DEPENDENT SYSTEMATIC ERROR 803

TABLE ITHEORETICAL VALUES FOR THE SYSTEMATIC ERRORS�(fc1 = 100kHz) AND

�2(fc2 = 141:421 kHz) AND THE ERROR OF THEMETHOD �1 � 2�2

approximation (3) is acceptable. In determining the optimalvalue of the ratio , one should also take intoaccount the amplitudes of the input signals at the summingblock (Fig. 2) as well as the results of the computer simulationof the error and some of the circumstances related to therealization of the technical solution described later. Takinginto account all the facts mentioned above, in developing aprototype watt converter based on this methodwas selected as an optimal conversion frequency ratio, thus

Errors obtained by computer simulationfor the actual values of the input signal frequencies andconversion frequencies are shown in Table I in order toillustrate the efficiency of the method. The first two rows ofTable I contain theoretical values of the systematic amplitudeerrors and as the results of the computer simulation forconversion frequencies kHz andkHz, respectively, and for the modulation factor of the pulseconverters (defined in the same way as in [4]). Thethird row in Table I contains the residual error(expressed in parts per 10 which represents the valuesof the frequency dependent systematic error which remainsuncompensated in the output signal , i.e., the error of themethod of compensation. From these results it can be seenthat the theoretical error obtained is less than 50 parts per 10for input frequencies up to 10 kHz, and that above 10 kHz theerror increases significantly.

III. PRACTICAL REALIZATION OF THE METHOD

This method of compensation can be applied in variousways to practical realizations of wideband TDM based wattconverters. First of all, a single TDM watt converter capableof changing its conversion frequency between two values,and , could be used in a laboratory if the input signals werestable enough between the readings of the output values fordifferent sets of conversion frequencies. The other possibilityis to directly apply the basic principle shown in Fig. 2 by usingtwo complete TDM’s. In that case all parts of the two TDM’swould be duplicated. However, without changing the essenceof the method it is possible to make some simplificationswithout duplicating all functional blocks of the TDM’s. Thereis no need to duplicate the reference voltage block, the outputfilter, and a part of the pulse amplitude modulator. Applyingthese simplifications, a prototype watt-converter based on thiscompensation method has been developed. The block diagramof this prototype is shown in Fig. 3. The input voltage signalis applied at two voltage to pulse converters working atconversion frequencies 100 kHz andkHz. Output signals of these two converters are applied atthe corresponding blocks of the analog multiplexer. The inputcurrent signal is applied at the primaries of two current

Fig. 3. Block diagram of the developed watt converter based on simplifiedbasic principle.

transformers. The secondaries of the current transformers areeach connected to the related block of the analog multiplexer.The number of turns in the secondaries of both transformersare the same (equal to), while one of the primaries has oneturn and the other has two turns. By using different numbersof turns in the primary coils of the current transformers therequired different factors of amplification (1 and )are achieved. The output signals of the two blocks of theanalog multiplexer are of the opposite polarity so they aresubtracted in the summing block preceding the lowpass filter.An advantage of this method over the compensation method[7] is the possibility of applying the current of the secondaryof the current transformer directly to the analog multiplexer ofthe pulse amplitude modulator, as is done in one of the mostaccurate TDM-based watt-converters [2].

In order to test and verify the systematic error compensationmethod, several measurements were performed. The first groupof measurements were made on the watt-converter at an inputsignal frequency of 50 Hz over the full range of amplitudes ofthe input signals (0–120 V; 0–5 A) for varying power factorsbetween one and zero (capacitive and inductive). The errorsobtained are within 50 W/W.

The second group of measurements were made on the wattconverter, with a modified input current circuit at higher inputfrequencies ranging from 50 Hz to 10 kHz. In order to avoidthe influence of the frequency-dependent error of the currenttransformers, these measurements were made at unity powerfactor by applying the same voltage signal at the voltageinput and the current inputs, which are modified by currenttransformers, but retaining the ratio of the input currents ofthe pulse amplitude modulators equal to 2 Theamplitude of this input voltage signal was maintained constant(110 V) while its frequency was varied from 50 Hz to 10kHz. By fine adjusting one of the conversion frequenciesit was possible to obtain equal values of output signals atthe limit values of this frequency range. Output signal errorsover the range of input signal frequencies were within50

W/W. Although these measurement results at higher inputfrequencies were not obtained under real conditions for apower converter with a real signal current of 5 A with acurrent transformer, they are important for proving the validityof the compensating method because the possible influence of

804 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 47, NO. 3, JUNE 1998

amplitude and phase characteristics of a current transformeris excluded.

IV. CONCLUSION

The method for the compensation of the frequency-dependent systematic error of the TDM’s presented in thispaper is based on using two identical multipliers, TDM1 andTDM2, working at two different conversion frequencies,and It has been shown that full compensation of the errorcan be achieved using simple calculation, taking into accountonly the output signals of the multipliers and the conversionfrequencies ratio By using this method thereis no need to know the input signal frequency. The methodis also valid when both input signals are distorted, i.e., arecomposed of harmonics. It has been shown theoretically, bycomputer simulation, that the error of the method is less than

50 parts per 10for input signal frequencies up to one tenthof the lower conversion frequency, so this method does notrequire the conversion frequency to be two or three orders ofmagnitude higher than the input signal frequency as in thesolution [5]. Theoretically viewed, this calculation could bedone using a single TDM with the possibility of alternatingits conversion frequencies between two values, andThis could be possible in a laboratory if the input signalsare adequately stable between readings of output valuesat different conversion frequencies using existing TDM’swith slight adaptations. It has been shown that for practicalrealization of the method there is no need to duplicate allfunctions of the two TDM’s, and some simplifications arepossible. Unlike the compensation method [7], this methodpermits the coupling of the secondary coil of the currenttransformer directly to the analog multiplexer of the pulseamplitude modulating circuit as in [2]. A watt converterbased on this method has been developed and the resultsof measurements show that at the unity power factor anuncertainty of 50 W/W was achieved in the input signal

frequency range from 50 Hz to 10 kHz when the lowerconversion frequency is only 100 kHz. Although these resultsof measurements at higher frequencies are not obtained underreal conditions for a power converter with a real signal currentof 5 A and with a current transformer, they are important forproving the validity of the compensating method because thepossible influence of the amplitude and phase characteristicsof the current transformer is excluded.

REFERENCES

[1] E. A. Goldberg, “A high-accuracy time-division multiplier,”RCA Rev.,vol. 23, pp. 265–274, Sept. 1952.

[2] P. Miljanic, B. Stojanovic, and P. Bosnjakovic, “The development ofa high precision power meter,” inProc. IEEE Conf. Precision Electro-magnetic Measurements Digest, The Netherlands, 1984, pp. 67–68.

[3] P. N. Miljanic, B. Stojanovic, and R. Bergeest, “Systematic error oftime-division wattmeters when voltage and/or current are distorted,”IEEE Trans. Instrum. Meas., vol. IM-36, pp. 357–361, June 1987.

[4] P. Filipski, “The systematic errors of a time-division power converterunder sinusoidal and nonsinusoidal conditions,”IEEE Trans. PowerDelivery, vol. PWRD-1, pp. 61–67, July 1986.

[5] , “A TDM wattmeter with 0.5 MHz carrier frequency,”IEEETrans. Instrum. Meas., vol. 39, pp. 15–18, Feb. 1990.

[6] L. D. Jovanovic and B. M. Stojanovic, “Improvement in performanceof conversion type multiplier,” inProc. XXXI Conf ETAN, vol. 2, Bled,Yugoslavia, June 1987, pp. 249–255.

[7] L. D. Jovanovic, “A Method of compensation of the frequency depen-dent systematic error of the time-division watt-converter,”IEEE Trans.Instrum. Meas., vol. 46, pp. 416–419, Apr. 1997.

Ljubisa D. Jovanovic (M’97) was born in Bel-grade, Yugoslavia, in 1956. He received the B.S.and M.S. degrees in electrical engineering fromBelgrade University, Belgrade, Yugoslavia, in 1982and 1989, respectively. He is currently pursuing thePh.D. degree.

Since 1982, he has been at the Institute MihajloPupin, Belgrade, Yugoslavia, working on researchand development of different instrumentation andequipment applied in industrial automatic control.His main research activities are related to the in-

strumentation for precise electrical power and energy measurements.