Simulation and practical realization of the new high precise digital multimeter based on use of dual-slope ADC

EUROPEAN TRANSACTIONS ON ELECTRICAL POWER Eum. Trans. Elecrr: Power 2004; 145-19 (DOI: 10.1002/etep.2)

Simulation and practical realization of the new high precise digital multimeter based on use of dual-slope ADC

Predrag Petrovic*.t

Cucak College of Engineering, Sverog Save 65, 32000 Cacak, Yugoslavia

SUMMARY

The measurement method described in this paper can be applied to the accurate determination of AC signals during the period of one second. This paper presents a novel design for a high accuracy multimeter based on the use of very slow, high-precision and cheap dual-slope A/D converters. Measurements of voltage and current are made in successive periods by a stroboscopic technique (synchronous undersampling). The assumed stationarity of the electric utilities is validated by measurements with an experimental set-up consisting of a fast high- precision sigma-delta ADC (1EE Proceedings, Electric Power Applications 1999; 4: 383-390). The necessary synchronization is reached by software measurements of the frequency of the measured signal. Assuming stationarity of the observed system (electric utilities) in the stated one-second interval, it is proved that precise digital processing can be achieved without using a sample-and-hold circuit (IEEE Transactions on Instrumenta- tion and Measurements 2000; 6: 1245-1 248). The suggested measuring system was simulated, practically realized and tested, and the obtained results confirmed completely the starting postulates. Processing precision of 0.01 % was achieved. Copyright 0 2004 John Wiley & Sons, Ltd.

KEY WORDS: adaptable algorithm; dual-slope ADC; software synchronization; synchronous undersampling; Nyquist criteria; high accuracy; electric utilities; digital multimeter

1. INTRODUCTION

Many applications involve digital processing of periodic signals. For example, both voltage and current in electric utilities are periodic signals containing harmonic components. The measuring method proposed in this paper is based on selecting samples (the original assumption) of the input variable in a large number of periods in which the system (in this case, electric utilities) is considered to be stationary. The stationary condition can be proved by values obtained from measurements of RMS values of system voltage [ 11. From that finding a mathematical presentation was concluded, i.e. confirmation that the band width of that system is very limited. That fact enabled application of the Nyquist criterion. By stationarity of the system, we understand that slowly changing quantities like current and voltage, and their harmonic content, are constant within the measuring interval. In this case, undersampling contrary to the Nyquist criteria is possible. The measuring time is about one

*Correspondence to: F'redrag Petrovic, Cacak College of Engineering, Svetog Save 65, 32000 Cacak, Yugoslavia. 'E-mail: pegi 1 @yu 1 .net

Copyright 0 2004 John Wiley & Sons, Ltd.

Received 29 March 2001 Revised 6 November 2001

Accepted 19 November 2001

6 P. PETROVIC

second. The current makes this system non-linear due to the fact that one cannot predict in advance the type of load which will be used and when it will be connected to the investigated system. However, after a certain number of periods, the current can be considered as a slowly changing variable during processing. This is why very slow, low-cost, but very accurate A/D converters [2], such as the dual- slope type, were used in the proposed measuring system. The voltage and current from real electric utilities were used as input variables. The sampling procedure is initiated arbitrarily. The distance between two consecutive samples is given by:

&lay = N . T + At ( 1 )

where N is the number of periods between sampling, T is the period of the input voltage, and At is the delay determined by the delay of elements in the processing circuit. At depends on the harmonic content of the input signal [I] . All conditions, which have to satisfy both Nand At to get an accurate result of measuring, can be derived [ 11. For that reason they cannot be arbitrary. Calculation by the proposed method is carried out on the basis of the following relations:

where W is the number of samples needed for accurate processing of observed values (W = 40, At = 0.5 x s). The reason for taking 40 samples is that even the most complex harmonic content processed by the proposed method demands more than 40 voltage and current samples [ 11. Index k provides the jump from period-to-period of the measured voltage and current and delay At for samples at every next processing.

According to the suggested measuring method it is assumed that, during the measuring interval, the harmonics content of the measured signal does not change very much. The algorithm deals with an arbitrary harmonics content. In the case of current with high harmonic content, as in a system supplying phase-switched loads (such as thyristor controlled), the accuracy of the method is preserved provided the system is stationary within the measuring interval. The possible non-linear distortions in transition processes do not last very long, so they can be avoided when the functioning of this multimeter designed for measuring periodic variables is considered.

The obtained conditions for the value of delay X = At [I], is completely equal to the Nyquist condition. In other words, as was described in the Introduction [ 11, we are not able to carry out real spectral reconstruction of the processed signal by the proposed method of measurement because of the extremely low speed of sampling. It is only possible to carry out ‘virtual’ (delay in time) spectral reconstruction. As the delay X = At is responsible for the movement forward of the moment of sampling from period-to-period (or more periods depending on the parameter N value) the delay must satisfy the Nyquist criterion, i.e. it must be in accordance with a basic postulate of synchronous sampling, where the here-proposed measurement method conditionally belongs. If the current and

Copyright 0 2004 John Wiley & Sons, Ltd. Eum. Trans. Electl: Power 2004; 1 4 5 1 9

NEW HIGH-PRECISION DIGITAL MULTIMETER 7

voltage signals had different harmonic content, the conditions for delay X = At would determine a signal with ‘richer’ harmonic content (which is usually the current signal because of its greater dynamics), (on the basis of equations (38) in the Appendix of Reference [l]) i.e.:

4 n, 1 5 V r , s 5 M1,M2

= kn, k E N(natura1 number)

( r + s)wX

n (r+s)wXW

# O = . 2 ( r + s)wX

2 sin

=2 wx 4 M i + M 2 ’ 2

where MI is the number of the highest harmonic of the voltage, and M2 is the number of the highest harmonic of the current signal. It is necessary to take the samples equidistantly on the interval of one period and in that way, on the basis of established limits in processing [ 11, make it possible to calculate accurately the observed electrical values in electric utilities.

One of the basic principles (postulates) in the processing of the continual signals by the dual-slope A/D converters is that at its input there has to be a sample-and-hold circuit, which is a possible source of systematic errors. In Reference [3] it is shown that use of this circuit is not necessary.

2. SIMULATION OF THE SUGGESTED MEASURING METHOD

Additional testing of the suggested digital measuring system was carried out by simulation of its work in the program package Matlab (version 5.1), module Simulink. Simulink is a module specialized for the simulation of system dynamics in a graphic environment.

In Figure 1 a block-diagram of the suggested digital measuring system is shown, whose construction consisted of ready-made Simulink models. The special advantage of such a program environment (surrounding) is that we are able to give an arbitrary input signal (in the sense of its harmonic content, white noise presence, different irregularities which may occur and simulate jump functions, changeable phase position between processed voltage and current signals) which is further processed. Thus, both voltage and current signals are introduced into simulation, while a completely arbitrary phase relation and amplitude value were given to them, as well as absolutely arbitrary noise power which was superimposed on them. After forming a complex harmonic input signal by superpositioning (adding), the signal was taken into the circuit for sample-and-hold (unit delay) which is located in front of real ADC. The signal was taken from the output sample-and-hold circuit into the D flip-flop as a delay element, which is clocked from the special signal generator (rectangular series of impulses) for which the arbitrary duty ratio can be given. In this way, a continual signal was measured and that sample was held constant up to the moment of the next measuring (sampling) which was given on the basis of the previously proposed form. Every next sample was taken from one of the next periods of the input signal, which can be adjusted by the choice of parameters of the simulation model. Voltage and current signals were multiplied and then integrated in time thus obtaining data of momentary circuit active power value.

All model parameters are adjusted to real dual-slope ADC and its speed and conversion precision. Because of all this we can be sure (claim) that this kind of simulation completely presents real conditions of exploitation, where the only difference is that by simulating its work was tested in more extreme conditions than the ones which may be expected in practice. The obtained results confirmed fully all suppositions and conclusions that were made before.

Copyright (g 2004 John Wiley & Sons, Ltd. Eum. Trans. Elect,: Power 2004; 145-19

8 P. PETROVIC

m~ie mtn Oenenlor

Sine Ubnl

Sine Wan4

P u l r O e n e m o r l

Sine Wn3

C o n u n t P+

8and.Limilad w I W i l e Noisel

Sine w W a r 6

Figure 1. Block-diagram of simulation model for measuring active power, based on the suggested measuring concept with dual-slope ADC and sample-and-hold circuit.

In the simulation model constructed (designed) in this way, ideal synchronization in processing with the frequency of the processed signal was achieved. That is very difficult in practice, but together with planned hardware resources (precise comparator and microprocessor with 16-bit architecture) we will be able to determine accurately the frequency of the processed signals. Simulink gives the possibility (it has such input block) for introducing a sinus signal whose frequency varies completely arbitrarily, so this block was also used during the simulation, but the results were still beyond our expectations (to the third decimal).

In order to estimate by simulation the power of digital processing without using the circuit for sample-and-hold, a model in Simulink has been constructed which corresponds to the imagined measurements method.

On the above model (Figure 2) using a special selector of analogue signals (Switches 1 and 2), passing through of complex periodic signals of current and voltage is applied to the circuit, which carries out their further processing. This selector passes through signals in the intervals, which corresponds the work speed of dual-slope ADC. A pulse generator is used to which duty-ratio and period can be given. The selector passes through the signal; from the first input during the time when the signal from the second input is bigger than the given threshold, otherwise it would pass through (let out) a signal from the third input. From Figures 1 and 2 it can be noticed that voltagekurrent signals with 3 harmonics (beside the basic one, third, fifth, either even or odd) were used because they are the most dominant harmonics in practice. However, it is not a limitation factor in such simulation. In other

Copyright 6 2004 John Wiley & Sons, Ltd. Eum. Trans. Electr: Power 2004; 145-19


Whit. N o h

Sine w Wave2

Oaneralor

comtrn11

~ I

Puba Oanarrlorl

Sin. Wn.3

Comtrnl 4

I B a n z i t a d W h L Noise1

Figure 2. Circuit models, which carry out processing of active power without using the sample-and-hold circuit.

words we are able to introduce higher harmonics, but the conclusions presented above are still valid. All this was worked out in detail in Reference [ 11 when arbitrary harmonic content was accepted. After mathematical analyses for that case certain conclusions were made about the required necessary number of samples and delay time during processing [l], in order to get absolutely accurate (theoretical) results of measuring. Simulink gives the possibility (it has such input block) for introducing a sinus signal whose frequency varies completely arbitrarily, so this block was also used during the simulation, but the results were still beyond our expectations (to the third decimal).

In the above-described processing, signals of small amplitude were used because such amplitudes can be expected on the plate after scaling and introduction of signals into the ADC and because of clearance. The results obtained when calculating active power were followed, when the product of the two compound periodical signals polluted by very strong noise is processed, which is most critical from the digital processing point of view.

Processing results obtained by this model have been compared with the ones given by the model in Figure 1, where the power of noise which is superpenetrated into an input complex periodical signal has considerably been amplified. Figure 3 is an illustration of the signal obtained in this way.

In Table I the results of processing over three models of digital processing are given. In the real environment such a strong noise can rarely be met, and because of this the model with a circuit for sampling should be accepted as very accurate. The model without a sample-and-hold circuit gives much better results at great distortion because processing includes those distortions as well, which the sample-and-hold circuit does not have to catch. In practical realizations this circuit brings in a significant error when processing very dynamic signals.


P. PETROVIC

-2

o 5 10 15 20 25 30 35 40 45 50 time [mrl

Figure 3. Voltage signal to which an extremely powerful noise signal is superpenetrated (with power of 0.004) or which is used for testing the processing concept with and without a sample-and-hold circuit.

Comparing the results it can be noticed that the processing method without the sample-and-hold circuit is immune to a very strong noise, while when processing with this circuit where there is a strong noise present it comes to a considerable deviation from the value of active power, which was calculated by the definition formula. In the real system it will be difficult to meet such a strong noise, which can be prevented by additional filtering, which has already been explained in Reference [ 11.

An additional test of the proposed method for measuring electric values was carried out on the basis of checking possibilities for more precise reconstruction of the input voltage signal according to the known values of voltage samples obtained over experimental set-up [ 11 (Appendix A).

Based on the suggested concept of processing, the complete realization of the dual-slope A D converter can be achieved through the resources of the microprocessor itself (only the most modern type of processor is satisfactory here, however, with a considerably high performance), as well as through its counting resources (16-bit architecture). All these elements will lead to reduced costs of the suggested solution, without any changes to the basic conclusion [I] , dealing with the adaptability and the necessary number of recorded measures. It is assumed that, during these observations, the system remains stationary (electric utilities), and that the harmonious content is preserved, within the time interval necessary to conduct the measuring according to the suggested method. In the same way we can process other slowly changing values such as pressure, temperature, etc.

Table I. The values of active power when processing signals possessed by basic harmonic with amplitude of 1 V, third harmonic with amplitude of 0.3 V and fifth harmonic with amplitude of 0.1 V, for different

methods of power processing.

Active power Active power Active power Number of Noise calculated by processed with processed without measurement power definition formula sample-and-hold circuit sample-and-hold circuit

0 . m 1 0.5454 o.oO01 0.5465 0.0002 0.5476 0.0005 0.5508 0.001 0.5561 0.001 5 0.5613

0.5464 0.5444 0.5418 0.5274 0.5239 0.5222

0.5458 0.5477 0.5484 0.5509 0.5555 0.5602

Copyright c) 2004 John Wiley & Sons, Ltd. Eum. Trans. E l e m Power 2004; 14:5-19


(A

(A P c)

3

The suggested method gives very accurate results when measuring time-average power, regardless of the harmonic content of the input signal. After detection of the signal harmonic content, the necessary number of samples can be specified in the form of an adaptive algorithm [ 11. The suggested method of processing AC values in electric utilities theoretically gives an absolutely accurate result of measuring, if the observed system can be considered stationary for a sufficiently long period of time [l] . The proposed approach is suitable for real-time processing and is characterized by a low computational burden in comparison to the described algorithms in References [4,5]. This gives an opportunity to develop a measuring system with very simple and inexpensive hardware, in contrast to the highly sophisticated and expensive hardware described in Reference [5].

+ b

synchronization

3. PRACTICAL REALIZATION OF THE PROPOSED MULTIMETER

DISPLAY

The block-diagram of the proposed multimeter is shown in Figure 4. After being adjusted to the measuring range of a converter, both voltage and current signals are

brought into the acquisition board. The voltage signal has been adopted from a precise resistance network (in the range W O O V). The current signal from an accurate current transformer has been taken to the input of the dual-slope A/D converter (Linear Technology ADC TC530) (in the range 0-lOA).

scaled voltage current

MICROCONTROLLER

?--l control 7 7 4

I A/DCONVERTER I

I (DUAL SLOPE) 16 BITS

ZERP CROSSING DETECTOR

Figure 4. Block scheme of the digital multimeter.

Copyright I() 2004 John Wiley & Sons, Ltd. Eum. Trans. Electr Power 2004; 145-19

12 P. PETROVIC

The current transformer has a secondary circuit operation amplifier which provides practically zero resistance, and thus much better linearity of the transfer function of all transformers.

The TC530 are serial analog data acquisition sub-systems ideal for high-precision measurements (up to 17 bits plus sign). The TC530 consists of a dual slope integrating A/D converter, negative power supply generator and 3-wire serial interface port. Key A/D converter operating parameters (Auto Zero and Integration time) are programmable, allowing the user to trade-off conversion time for resolution. Data conversion is initiated when the RESET input is brought low. After conversion, data is loaded into the output shift register and EOC is asserted indicating new data is available. The converted data (plus Over-range and polarity bits) is held in the output shift register until read by the processor, or until the next conversion is completed allowing the user to access data at any time. The TC530 requires a single 5 V power supply and features a -5 V, 10 mA output which can be used to supply negative bias to other components in the system. We use two ADC, one for voltage and one for current signal. The voltage and current signals must adopt a range of f 2 V to activate maximum linearity of used ADC, and over special resistant circuit specify reference signal on 1.025 V.

As a sample-and-hold circuit we used Analog Devices circuit AD684. The AD684 is ideal for high- performance, multichannel data acquisition systems. Each SHA channel can acquire a signal in less than 1 ms and retain the held value with a droop rate of less than 0.01 mV/ms. Excellent linearity and AC performance make the AD684 an ideal front end for high-speed 12- and 14-bit ADCs. The AD684 has a self-correcting architecture that minimizes hold mode errors and ensures accuracy over temperature changes. Each channel of the AD684 is capable of sourcing 5mA and incorporates output short circuit protection. Low droop (0.01 mV/ms) and internally compensated hold mode error results in superior system accuracy. Independent inputs, outputs and sample-and-hold controls allow user flexibility in system architecture. Fast acquisition time (1 ms) and low aperture jitter (75 ps) make the AD684 the best choice for multiple channel data acquisition systems.

A special circuit detects the passing of the signal through zero with a comparator, in this way achieving synchronization of the measuring cycle with the electric utility frequency. The microcontroller generates sampling intervals and is able to perform necessary calculations, based on sampling values of the measured signals. The analyses of operation of the zero-crossing detector proved that satisfactory accuracy could be achieved. Even more accurate detection can be attained in this procedure by algorithms reported in References [4,6]. The number of zero crossings of the signal is evaluated by a test on the sign of consecutive samples. The program corrects for multiple transitions due to noise and checks that the distance between zero crossings is approximately compatible with the frequency expected by the program. An odd integer number of valid zero-crossings are taken, and the frequency of the signal is calculated from the period of time between the first and the last, divided by the number of periods. For better accuracy the values of the samples around these two zero-crossings are interpolated by means of a least squares procedure.

In the existing electric utilities the system frequency swings in the range of 49.06-50.02Hz (allowed by existing regulations). This can certainly influence the accuracy of the proposed measuring concept, due to the error made when determining the sampling period by zero crossing detection which can be as high as 2%. When the period is read using an internal counter in the microprocessor, a zero crossing is required. The least expensive comparators have a detection rate of 50 V/ps. As the system voltage is scaled to about 2 Von the board (ratio of 1: 150), this comparator triggers at about 2.5 mV, so the error is about 20ns. This error can be ignored, since there is no accumulation.

Complete control of the measuring process and all necessary calculations are done by the microcontroller (Motorola 68HC 1 1). This microcontroller works by the program, which is stored in EEPROM, and complete calculation was realized using Equations (1) and (2), with specified N = 6,

Copyright 0 2004 John Wiley & Sons, Ltd. Eum. Trans. Electr Power 2004; 145-19

NEW HIGH-PRECISION DIGITAL MULTMETER 13

Table 11. The results of measuring at the National Institute for Measurements in Belgrade with the proposed digital multimeter, in Part (a) we used 40 samples to calculate the observed values, and in

Part (b) we used 80 samples.

Number of Measured RMS Measured RMS Measured value of value of current [A] value of voltage [V] active power [W]

2 3 4 5 6 7 8 9 10

(b) I 2 3 4 5 6 7 8 9 10

120.0459 120.056 120.0158 120.0352 120.0963 120.0862 120.0353 120.0359 120.0862 120.0657

120.0162 120.0 165 120.027 1 120.01 66 120.0267 120.0266 120.0169 1 20.0 169 120.0269 120.0267

4.0035 3.9891 4.002 3.9956 4.002 4.001 3.9936 3.9946 3.9926 3.997 1

3.9992 3.9987 3.9982 3.9977 3.9902 3.9987 3.9987 3.9977 3.9979 3.9982

479.58 1 479.93 1 480.23 1 479.03 1 480.23 1 480.28 1 480.181 479.731 479.43 1 479.581

479.703 479.755 479.807 479.805 479.706 479.755 479.755 479.807 479.755 479.757

W = 40, At = 0.5 x 10-3s. Thus, obtained results are sent to a display so that the process of measuring can be followed visually, too. An interface circuit is designed to enable a connection between meter and PC, with a possibility to choose the number of samples needed to calculate the observed AC values. Over the installed display we can visually follow results of calculating (RMS values of voltage, current, active power and frequency of the basic harmonic of voltage).

The installed keyboard, which is used to set-up the measuring range and the start of measuring, is not shown in Figure 4. Beside this, we can choose the type of measurement, i.e. this multimeter can be used for measuring the RMS values of voltage and current, as well as the average and reactive power.

The multimeter was checked at the National Institute for Measurements in Belgrade (the official laboratory for this purpose). The obtained results completely proved our assumptions [ 1,3] (Table 11). The instruments for calibration in the Institute during the measurements with the proposed multimeter show that RMS values of voltage, current and active power were Urns = 120.0083 V, IRMs = 3.99977 A, and Paverage = 479.984 W. These parallel measurements showed the same results up to the third decimal. All this provides verification for using the suggested measuring concept in extremely precise reference and laboratory measurements. As we can see from Table 11, if we use more samples then the results of the calculation will be better.

The multimeter can be used as a watthour meter in real electric utilities, where it is essentially monitoring the basic AC values (voltage, current and frequency). For this reason we can use ADC with low resolution (up to 12 bits) for realization.

Copyright 2004 John Wiley & Sons, Ltd. Eum. Trans. Elecrr Power 2004; 145-19

14 P. PETROVIC

The results show that the accuracy of the proposed algorithm is acceptable under all conditions. It is independent of the harmonic content of the voltage and current signals and has good accuracy even if the frequency deviates from the nominal value.

4. CONCLUSION

A new design of a digital multimeter that performs measurements of basic parameters of electric utilities (voltage, current, power, energy, and frequency) has been described. The algorithm, without the high computational burden as in References [4,5], is suitable for on-line measurements. A dual- slope A/D converter with very simple and inexpensive hardware, in contrast to the highly sophisticated and expensive hardware described in Reference [ 5 ] , meets all the price and accuracy requirements for the design of the measuring system. This reduces the price of the entire device, while the high level of accuracy in processing AC values is preserved. The necessary synchronization is achieved by software measurements of the frequency of the measured signal. The algorithm is of an adaptable type and depends on the harmonic content of the input signal and network frequency (50 or 60 Hz) [ 11. Voltage measurements on the system show that it is inertial [l], so the error made when sampling during several periods is below the value required for class 0.1 instruments. By eliminating the sample-and- hold circuit from the final design of the measuring system [3], a possible source of systematic errors is eliminated and hardware requirements are significantly simplified. All this reduces the price of the entire device, while the high level of accuracy in processing AC values is preserved. The multimeter has been simulated, built and tested, proving an accuracy of 0.01%.

5 . LIST OF SYMBOLS AND ABBREVIATIONS

The A/D or ADC integrating (dual-slope) A/D converter uses an integrator connected to a reference voltage to generate an analogue value which is compared with the input analogue value by comparator. The time taken for the output ramp of the integrator to reach the input-signal level than gives the binary solutions.

RMS T N At W MI M2 EOC SHA

root mean square value period of the input voltage signal (50 Hz in Europe) number of periods between sampling delay determined by the delay of elements in the processing circuit number of samples needed for accurate processing of observed values number of the highest harmonic of the voltage number of the highest harmonic of the current signal end of conversion sample-and-hold amplifier

APPENDIX A

The files with data about net voltage measuring values were software processed [ l ] in order to be transformed from the original format (second complement) into a decimal record. In that way we came

Copyright ((1 2004 John Wiley & Sons, Ltd. Eum. Trans. Electr: Power 2004; 14:5-19


to a one-dimensional series of great length (with over 16000 members, which corresponds to over 60 periods of net voltage). These series are input files for calculations carried out afterwards in the program package Matlab. According to these real data about the voltage value in the electric utility, we are able to estimate the current value, which corresponds to some random load. Besides, we can ever carry out submersion of some fast changeable signal (like Heaviside function of determined amplitude and length or noise signal of determined strength), by which we simulate different irregularities which can be expected on the real utility. Determination of the current value for the known type of loading and starting conditions are carried out in a discrete domain, by solving differential equations which correspond to a specific load type. Thus we come to a series of current samples of the same length as the voltage series of samples.

Knowing both the voltage and current series, we can calculate the RMS value of voltage and current signals, as well as the active power for two cases: using the definition form and on the basis of the suggested concept of processing, which is described in this paper. According to the shape of input files, it can be seen that there are 260 samples on the length of one period (20ms) i.e. each sample at a distance of 7.7 x lop5 s. Calculation of the RMS values of voltage and current, as well as active power will be carried out on the basis of the following relations:

where u( i ) and i ( i ) are the voltage and current samples. These equations can apply from any member of the series, at the length of 260 samples, which are in one period. The obtained values are used for comparing with the values obtained by applying the suggested measuring method. Calculation by the proposed method is carried out on the basis of the following relations:

1 40 p* =-Cu(269.k) i (269 .k)

k= I 40

The reason for taking 40 samples is that even the most complex harmonic content processed by the proposed method demands more than 40 voltage and current samples [ 11. Index 269 provides a jump from period-to-period of the measured voltage and current and delay for 9 samples at every next processing.

Copyright C) 2004 John Wiley & Sons, Ltd. Euro. Trans. Elect,: Power 2004; 145-19

16 P. PETROVIC

In order to achieve the expression, which defines the current for specific loading the following equations are given in the space condition of the following shape:

ai q Ri + L- + - = u ( t )

at c bi 4 L- = -Ri - - + u( t ) dt C

a i R 1 1 at L LC L - = - - i - - - q + - u ( t )

i = l . i + O . q + O . u ( t ) q = O . i + l . q + O . u ( t )

On the basis of the above-presented equations in the space condition we have written the program

Calculating the current program for optional loading on the basis of state condition equations: within the program package Matlab.

clear loadu452.mat kkk=size(u452); vel=kkk( 1,Z) ; t=0:1.27/(vel-1):1.27; f=50; ur452=~452.*310./30000; eps=1OA(-80);

XL=l.O; XC=O.l; ILO=O; uco=o ;

R=l.;

L=XL./(2.*pi.*f); C=l./(XC.*2.*pi.*f+eps); L=L+eps ; C=C+eps ; R=R+eps ;

B=[l./L;O];

D=[ 0; 01 ; s=lsim(A,B,CC,D,ur452,t,[ILOC.*UCO]); zbir=O ; zbirl=O; zbirZ=O; zbir3=0; zbir4=0; zbir5=0; figure ( 1 )

A=[-R./L-l./(L.*C);IO];

CC=[10;01];

Copyright 0 2004 John Wiley & Sons, Ltd. Eum. Trans. Electr Power 2004; 145-19


plot(t,ur452,t,s(:,l),’r’); grid figure ( 2 ) p1ot(t(1:1000),ur452(1:1000),t(1:1000),~(1:1000,1),’r’); grid forii=261:520 zbir=zbir+s(ii,l).*s(ii,l); zbirl=zbirl+ur452(ii).*ur452(ii); zbir2=zbir2+s(iifl).*ur452(ii);

end forjj=1:40 zbir3=zbir3+s(jj*7,1).*s(jj*7,1); zbir4=zbir4+ur452(jj*7).*ur452(jj*7); zbir5=zbir5+s(jj*7,1).*ur452(jj*7);

end ieff=sqrt(zbir./260) ueff=sqrt(zbirl./260) p=zbir2./260 ief=sqrt(zbir3./40) uef=sqrt(zbir4./40) pf=zbir5./40

The ability of this program is to calculate (work out) current samples series for arbitrary loading according to input samples series of voltage from the electric utility, which is recorded by the described experimental set-up [l]. After presenting the results graphically, this program is used to calculate the value of the active power, both on the basis of definition form and on the suggested measuring method. We can conclude from the program that 40 samples are taken on the basis of the suggested method because it meets (satisfies) conditions of even the most complex harmonic content which can be expected. Special attention should be paid to the transition processes which occur with certain types of loading, when circuit should let calm down in other words, the usual regime of the circuit work should be re-established, and only after that measuring of determined (planned) electric values on the basis of described procedure can be done. Namely, during transition processes an error may occur when reading the instrument, which works on the basis of, suggested measuring method where dual-slope A/D converter is used. That error is about 2%.

Using software developed in this way we are able to check different types of loading and this is presented Table AI. Beside active power, RMS values of voltage and current were checked.

From the results given in Table AI it is obvious that the suggested measuring method gives high precision when calculating basic electric values in electric utilities, if the stationary condition of the observed system is satisfied. As the whole calculation is based on objective input data about the measured voltage samples in the electric utility, these results can be accepted as highly credible for the suggested measuring concept.

In order to check the possibility of processing slowly changing signals such as AC signals, we can start from the supposition that the shape of a voltage signal, which can be expected in the real electric utilities, is known in advance [ 11. These samples can be integrated into a certain time interval, which is suitable for the concept of processing without using the sample-and-hold circuit, which is proved to be one possible type of digital processing of such stationary signals [3]. In this way, a possible noise which exists in the system is taken into account when processing, as well as other non-linearities and fast changes which may occur in the shape (form of input signal which is processed). Assuming that we know the values of integrals in different time intervals, using this system of equations we can calculate unknown amplitudes of harmonics and thus reconstruct the input signal.


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Using this system over the program packet Mathematica, unknown values of harmonics amplitudes were calculated and later used for reconstruction of the input signal. On the basis of this harmonic content, the values of amplitudes of harmonics are determined, and consequently RMS values of voltage.

A general remark when solving such problems is in placing the points, which are used as known, when solving the above equations, and i.e. defining the time interval of integration, on which a real series of data is integrated (sample of input voltage signal). They should be equally placed on the interval of the semi-period of the basic signal while in other cases this procedure of interpolation causes great error. In this way interpolation procedure can describe the input signal in the most optimal manner. As the results show, this assumption has brought about much better results which confirms the possibility of reconstruction of a complex periodical signal on the basis of digital processing without using a special circuit for sample-and-hold. The reason for that was using integral values of input functions on the defined time interval as known values.

REFERENCES

I . Petrovic P, Marjanovic S, Stevanovic M. Digital method for power frequency measurement using synchronous sampling.

2. Hnatek ER. A User's Handbook of D/A and A/D Converters, 1st edn, Wiley: New York, 1976. 3. Petrovic P, Marjanovic S, Stevanovic M. Measuring of slowly changing AC signals without sample and hold circuit. IEEE

4. Xi J, Chicharo JF. A new algorithm for improving the accuracy of periodic signal analysis. IEEE Transactions on Instru-

5 . Toivonen L, Morsky J. Digital multirate algorithms for measurement of voltage, current, power and flicker. IEEE Transac-

6. Tarach D, Trenkler G. A noise-adaptive digital null detector. lEEE Transactions on Instrumenfation and Measurement

IEE Pmceedings, Electric Power Applications 1999; 4:383-390.

Transactions on Instrumentation and Measurement 2000; 6: 1245-1 248.

mentation and Meusurement 1996; 4:827-83 1.

tions on Power Deliver?, 1995; 1:l 16126.

1997; 2:435438.

AUTHOR'S BIOGRAPHY

Predrag Petrovic (born in 1967). (member of the IEEE), received the B.S.E.E. and M.Sc. degrees in electrical engineering from the University of Belgrade Yugoslavia, in 1991 and 1994, respectively, and is pursuing a Ph.D. degree in the field of digital signal processing at the University of Novi Sad. Since 1991, he has been a Teaching and Research Assistant at the University of Kragujevac, Cacak College of Engineering, where his main interests are digital signal processing, microcontroller programming, AD conversion, mathematics, and cryptology.

Copyright i:' 2004 John Wiley & Sons, Ltd. Euro. Trans. Electr Power 2004; 145-19

Documents

Simulation and practical realization of the new high precise digital multimeter based on use of dual-slope ADC