M. SZMAJDA 1, K. GÓRECKI 1, J. MROCZKA 2, J. BORKOWSKI 2

Transcription

1 M. SZMAJDA 1, K. GÓRECKI 1, J. MROCZKA 2, J. BORKOWSKI 2 1 Technical University o Opole, Poland 2 Technical University o Wroclaw, Poland szmajda@po.opole.pl, janusz.mroczka@pwr.wroc.pl ANTIALIASING FILTERS IN POWER QUALITY DIGITAL MEASUREMENT SYSTEMS * Digital signal processing exists in the systems which perorm the processing o power waveorms. Each measurement equipment using digital signal processing and computing its discrete spectrum should be equipped with an antialiasing ilter which unortunately creates errors in magnitude and phase spectrum responses. This paper discusses the spectrum distortion analysis o the power signals caused by antialiasing ilter implementation in the analog-digital signal processing system paths. The inluence o the aliasing phenomenon and a real type magnitude response o the antialiasing ilter on the measurement uncertainty o the power signal amplitude characteristic was speciied. The values o the minimal sampling requencies or the power signal spectrum computations were proposed. Use o these requencies assures the maximum error level at the speciied level (0,1%, 1% and 10%). The research includes low-pass realization o the ilter magnitude responses approximated by polynomials: Butterworth, Bessel, Chebyschev and elliptic. Keywords: Antialiasing, ilters, digital signal processing system, power quality 1. INTRODUCTION The current state o technology makes it possible or electrical and non-electrical quantities to be measured. Analog measuring instruments were substituted by digital systems. These systems should be equipped with antialiasing ilters. A special example o these systems are devices which process voltage power signals, compute their requency spectrum and power quality parameters. Analog-digital conversion generates so-called aliasing errors. In case o wrong selection o the analog-digital conversion parameters, there is a possibility o signal spectrum deormation by the aliasing phenomenon. The inluence o this phenomenon on results o research o the spectrum is limited by applying antialiasing ilters beore the analog-digital converter (ADC). The ilter is characterised by limited steepness o slopes, non-zero width o the transition band and real type o spectrum characteristics (Butterworth, Chebyshev, Bessel, elliptic). Establishing the inluence o these parameters on computation accuracy o the measured signal spectrum and proposing the optimal choice o them is a very interesting subject. 2. THE ALIASING PHENOMENON, THE DIGITAL SIGNAL PROCESSING SYSTEMS PATH The spectrum o a sampled signal is presented by the ollowing Eqs. (1), (2): S p ( 1 ) = T n= S n, (1) T * Article published with the support o the Foundation or Polish Science.

2 where: T - sampling period, S() - spectrum o the continuous signal beore the sampling operation, n - natural number. Equation (1) presents the relation between a continuous-signal magnitude spectrum and a sampled-signal magnitude spectrum. The special eature o the equation is the appearance o duplicated magnitude spectrum (aliases) around the total multiple o the sampling requency 1/T (Fig. 1b). A S() A) B) S() p S() p C) A/T A/T g n si a Ali g n si a A li B/2 B/2 -B/2 s s +B/2 s 2s -B/2 s B/2 s 2-B/2 s +B/2 s 2 s Fig. 1. The principle o creation o aliasing. A) spectrum o continuous signal, B) duplicated spectrum o discrete signal, C) creation o the aliasing phenomenon. Sig na l Conditioning Antialiasing Filter Analog Input Analog-digital Converter Digital Processing Fig. 2. The structure o a typical analog-digital signal processing path. A consequence o duplication o the magnitude spectrum is that it is impossible to speciy the signal requency only on the basis o the time samples because one can lead to an ininite number o harmonics through the discrete-time points. The second phenomenon closely related to spectrum duplication is aliasing. It includes overlapping o duplicated aliases in the requency domain, when the highest harmonic o a signal is higher than Nyquist requency (Fig. 1c). Furthermore, it is impossible to perorm an inverse Fourier transormation without distorting the analog output signal in the time domain. To avoid this kind o ambiguity, low-pass antialiasing ilters are applied between the conditioning circuit and analog-digital converter (Fig. 2) [1,2]. 3. THE SAMPLING FREQUENCY AND ANTIALIASING FILTER CUT-OFF FREQUENCY SELECTION CRITERIA IN POWER SIGNALS PROCESSING SYSTEMS The selection o digital signal processing parameters (especially DFT) starts by speciying the signal bandwidth and the number o considered signal periods (width o time window). The sampling requency has to meet the Nyquist condition [1,2] in which the sampling requency must be at least twice higher than the maximum measured signal requency. The power signal harmonics controlling systems, which in most cases perorm computation o THD actor (Total Harmonic Distortion), should apply a sampling requency higher than 4 khz. (THD actor takes into account the irst 40 harmonics o the power signal [3]). Thus, the considered minimal power signal bandwidth, which is needed to compute the THD actor, equals 2 khz. In practice, the sampling requency is several times higher than 2 khz. Other

3 systems, which may be sensitive to random short-time duration voltage changes, have to take into consideration a much greater bandwidth (100 khz). The bandwidth is relevant to the duration o distortion. The character o the investigated distortions also speciies the number o power signal periods which should be considered in computations. For the sake o averaging eatures o DFT, the spectrum o the short-time duration distortion should be evaluated, considering the minimal number o periods in contrast to the slow-variable distortion. In such cases, the measuring time-window width should include the maximum number o periods, what simultaneously leads to the increase o the spectrum resolution. The relation between the spectrum resolution, sampling requency and number o samples is presented in the ollowing equation: s res=, (2) N where: s - sampling requency, N - number o samples. To avoid a spectrum leakage phenomenon in DFT computation, the total multiple o signal periods should be taken into account (in case o power signals, the total multiple o 20 ms) [3]. Several ways to eliminate the errors generated by the phenomenon are used in practice. The irst solution is the choice o the time window dierent rom the rectangular shape on the investigated signal [4,5]. In the case o power signals the Hanning window or a rectangular one is used. The application o the rectangular shape window orces synchronization o the sampling requency with the measured signal requency (more rigorous than or the Hanning window). Obviously the synchronization should respect the total number o signal periods. The second solution is the extension o the measuring time window [6,7]. This solution takes into consideration a larger number o power signal periods (i.e. 4-30). This situation results in averaging o short-time duration disturbances on all analysed periods. The accurate location and interpretation o the disturbances may be diicult when such situation occurs. The selection o the sampling requency depends on the kind o inormation which is to be obtained as the result o DFT computations. It is signiicant whether the system should provide data concerning temporary disturbances in power signal or long-term inormation determining components o the signal spectrum. The sampling requency which may be applied in the digital power signal processing systems depending on the number o ull periods and number o time samples are shown in Table 1. The table includes only requencies which are a total multiple o 1 Hz. Table 1. The sampling requency (in khz) used in digital power signal processing systems depending on the number o ull periods and time samples. Amount o samples AMOUNT OF POWER SIGNAL PERIODS N THE INPUT SIGNAL MODEL

4 The voltage power signal is presented by a sinusoidal waveorm with a rms value o 230 V and requency o 50 Hz. Testing o disturbances in the signal, particularly power quality parameters, is simple. The ast and expensive analog-digital converters and wide pass-band conditioning circuits are not required in the systems. There are many types o disturbances in electrical networks beginning with low requency transients (lickers) to broad-band impulses (electrical discharges, commutation o loads, etc.) in real power distribution networks. The problem o generating the disturbances is described in literature [8,9,10]. The occurrence o impulse-type disturbances in the signal widens its magnitude spectrum. The width o the magnitude spectrum depends on the impulse width, its steepness and amplitude. The increasing number o impulses causes also an increase o the participation o higher harmonics in the signal. In the critical case the signal magnitude spectrum may stay equal in all bands. However, this case occurs very rarely but it allows to check the maximal error level. The input signal model guarantees that the measurement uncertainty will be smaller than computed, even in the most distorted networks. 5. THE MAGNITUDE SPECTRUM MEASUREMENT ERRORS - RESEARCH METHODOLOGY The measurement uncertainty o the magnitude spectrum in digital measurement systems depends on two actors. The irst one is the dierence in the pass-band range between shapes o the ideal ilter magnitude response (Bode approximation) and real ilter magnitude response (approximated by i.e. Butterworth or Bessel polynomials). The second one is the inluence o a duplicated spectrum (alias), which was created by the sampling process (aliasing phenomenon). It is necessary to speciy the participation o both phenomena in the inal uncertainty o the signal magnitude spectrum measurement. The inluence o ilter real magnitude response shape. Filters o real amplitude spectral characteristics are described by some basic parameters: - type o applied approximation polynomial, - width o pass-band, - width o transition-band, - attenuation in stop-band, - cut-o requency, - ilter order. For the sake o the papers the additional parameter called the width o the utilitarian-band was deined. The parameter deines a part o the pass-band where the error caused by the dierence between the ideal and real ilter magnitude response is maintained at an assumed level (i.e. 0.1%). The our most important characteristic types o the ilter magnitude responses: Butterworth, Bessel, Chebyshev and elliptic (Cauer) are taken into account in this paper. These characteristics, in most cases, appear in ilters included in the digital signal processing systems. The ilters are generally available as monolithic circuits realizing one or more types o characteristics. Sotware using the discussed characteristics types applied in PC computers and in special digital signal processors (DSP) is also popular. The type o characteristics is closely related to the ollowing parameters: - pass-band ripple, - stop-band ripple, - steepness o the characteristic slopes (related to the width o the transition-band).

5 Figure 3 presents low-pass realizations o the individual characteristics. The values placed on the requency axis were normalized in relation to the ilter cut-o requency (cut). Each o the characteristics has its special unique properties. Butterworth ilter characteristics. This ilter has the lattest possible pass-band magnitude response and moderate steepness o slopes above the cut-o requency. In the pulse response o the Butterworth ilter there occur oscillations rising with the increase o the ilter order. Chebyshev ilter characteristics. Filters o this type have steeper attenuation above the cuto requency than Butterworth ilters. This advantage comes at the cost o the amplitude variation (ripple) in the pass-band. The Chebyshev has even more ringing in its pulse response than Butterworth. The research includes Chebyshev characteristics with the utilitarian-band ripple o db. This value corresponds to a deviation o the real and ideal characteristics in the utilitarian-band range o about 0.1%. Bessel ilter characteristics. Due to its linear phase response, the ilter has excellent pulse response (minimal overshoot and ringing). The magnitude response is not as lat and steep as in the Butterworth ilters. Elliptic (Cauer) ilter characteristics. The magnitude response has steeper slopes or a given number o ilter order. The phase response o this ilter eatures the worst linearity. The magnitude response in the pass-band and stop-band sections has even ripple. The research includes magnitude response with utilitarian-band ripple equal to db and 10 db, 20 db, 30 db, 40 db, 50 db, 60 db in stop-band ripple. Fig. 3. The ilter magnitude responses o the ollowing types: Butterworth, Bessel, Chebyshev and elliptic 20 db. Fig. 4. The Butterworth ilter magnitude characteristics o 4th and 8th order beore (cut = 1Hz) and ater cut-o requency correction (cut = 2.2Hz or 4th order and cut = 1.5Hz or 8th order). To perorm the design o an antialiasing ilter and speciy the cut-o requency cut it is necessary to take into account the act o 3dB attenuation o the magnitude response. The

6 higher harmonics in an input signal with a bandwidth reaching the cut-o requency may be attenuated. The error inserted by an antialiasing ilter will then equal about 30% or the cuto requency. To avoid this distortion it is necessary to correct the ilter cut-o requency. The new value should ensure minimal attenuation in the whole utilitarian-band and keep an error level in the assumed range. For urther research, the error level was established as 0.1%. The ilters which meet this condition will then not insert a larger error to signal spectrum. The characteristics in Fig. 4 present hypothetical magnitude responses o 4th and 8th Butterworth ilter orders beore and ater cut-o (cut) requency correction. In case o using a 4th-order Butterworth ilter, which has to transmit a 1Hz bandwidth signal with a maximum error o 0.1%, a cut-o requency o about 2.2 Hz should be used. The cut-o requency increases more then twoold. There is also the inluence o the ilter order in this relation. For ilters o the same type but o the 8th order, the cut-o requency should be 1.5 Hz. The relation between the cut-o requency ater and beore correction may be deined as the cut-o requency multiplier (CFM). CFM cut ater correction =. (3) cut beore correction The inluence o the aliasing phenomenon The second actor which has the inluence on the measurement accuracy o the input signal magnitude spectrum is the aliasing phenomenon. The inluence o the phenomenon was determined considering the described input signal model. The magnitude spectrum o the signal is constant in the whole band up to a sampling requency value. The principle o estimation o the inluence o the aliasing phenomenon consists in speciying the proper sampling requency. The use o the computed value ensures keeping the error level within the assumed range or the given ilter type characteristics, ilter order and bandwidth. The principle was presented with the help o a Chebyshev ilter o the 4th order, with a bandwidth o 2 khz and maximum aliasing error equal to 10% (Fig. 5). Fig. 5. The principle o perorming computation or a 4th-order Chebyshev ilter with an utilitarian-band o 2 khz (cut = 3 khz) at assumed maximum aliasing error o 10%. a) comparison o the main band and the duplicated band (alias); b) expanded part o Fig. a; c) comparison o main band, alias and inal sum o characteristics; d) the percentage values o errors introduced to the signal magnitude spectrum in the utilitarianband range ( Hz) by the aliasing phenomenon. The previously mentioned input signal is applied to the input o an antialiasing ilter. The act is that the input signal has a constant magnitude spectrum in the whole bandwidth. As a result o iltration we obtain a signal which eatures the magnitude spectrum exactly itted to

7 the magnitude response o the ilter (Fig. 5a). The cut-o requency was chosen considering the CFM value (about 3 khz) in a way to ensure that in the whole bandwidth the maximum dierence between the ideal and real magnitude spectrum equals 0.1%. Figure 5b shows the character o signal amplitude spectrum changes in the range o band not exceeded by value 0.1% (there are also visible pass-band ripples which are characteristic or signals iltered by Chebyshev ilters). The grey lines in Fig. 5a and 5c represent a basic amplitude characteristic (alias) duplicated around the sampling requency. As a result o the sum o these spectra in the pass-band range, an expansion o the signal amplitude spectrum (called the aliasing phenomenon) appears. For the assumed characteristic type, ilter order and ilter pass-band, the inluence o the aliasing phenomenon may be limited by shiting spectra until the error disappears. The operation results in a speciication o the sampling requency - about 7 khz. Percentage representation o the error or the sampling requency was shown in Fig. 5d. The maximum error value in the whole utilitarian-band or this case equals 9%. 6. THE RESULTS OF RESEARCH The inluence o the ilter magnitude response shape The research on the inluence o the ilter magnitude response shape on the signal spectrum aimed at speciying a correction actor, called the cut-o requency multiplier (CFM). To obtain a requency cut-o which guarantees limitation o the error level caused by the dierences between the ideal and real ilter magnitude response below an assumed value, it is necessary to multiply the signal bandwidth by CFM. The error was established at the 0.1% level as a value which has the minimum inluence on the inal computation accuracy. The CFM value is a unction o the ilter magnitude response characteristics type and ilter order. The research was perormed in order rom 1 to 20 and considered the ollowing ilter types: - Butterworth, - Bessel - Chebyshev: utilitarian-band ripple db (0.1%), - elliptic: utilitarian-band ripple db (0.1%), stop-band ripple 10 db, 20 db, 30 db, 40 db, 50 db and 60 db. The CFM values are presented in Table 2 and in Fig. 6, Fig CFM Butterworth Bessel Chebyshev Elliptic 10dB Elliptic 20dB Elliptic 30dB Elliptic 40dB Elliptic 50dB Elliptic 60dB Filter order Fig. 6. The CFM dependence on ilter order or dierent characteristics types. Table 2. Filter cut-o requency multiplier values (CFM) as a unction o the ilter order and types o characteristics. r Filter type

8 Butterworth Bessel Chebyshev Elliptic 10dB Elliptic 20dB Elliptic 30dB Elliptic 40Db Elliptic 50dB Elliptic 60dB ,5 CFM 3 2,5 2 1,5 Butterw orth Bessel Chebyshev Elliptic 10dB Elliptic 20dB Elliptic 30dB Elliptic 40dB Elliptic 50dB Elliptic 60dB 1 0, Filter order Fig. 7. The CFM dependence on ilter order or dierent characteristics types (the expanded part o Fig. 6). The characteristics shown above present a decrease o CFM with an increase o the ilter order until about the 6th order. A urther increase o the ilter order does not cause a signiicant decrease o CFM and stays saturated at the 1 value. An exception is the Bessel ilter, where or orders 2 to 20, CFM varies between values 18 and 19. As a consequence, it is necessary to use wide-band ilters or a narrow-band, but lower accurate characteristics ilter in the band range (1% or more). The narrow-band iltering circuits which implement this type o magnitude response, will insert relatively high distortion or higher harmonics. Thus, these circuits ind application in systems where maximum linear phase response is required and accuracy o magnitude characteristic is not so signiicant. The elliptic and Chebyshev ilters reach a CFM actor which equals 1, the astest. The ilters show stronger non-linearity o the phase response and consequently ripples occur in the

9 time domain o the output signal. A compromise between linear phase response and relatively low CFM value is the Butterworth ilter. The inluence o the aliasing phenomenon. The aim o the perormed research is to speciy a minimum sampling requency which should be applied in the system to ensure the maximum aliasing phenomenon error below an assumed value. The requency is dependent on the ilter order and utilitarian-band width. The computations concern cut-o requency correction with the help o the CFM actor. Thus, all the characteristics insert an error o a maximum value o 0.1% to the ilter pass-band. The computation was perormed or the ollowing parameters: - error values: 0.1%, 1% and 10%; - width o utilitarian-band: 2 khz, 2.5 khz, 10 khz, 25 khz and 50 khz; - ilter orders: types o ilter magnitude responses: - Butterworth, - Bessel, - Chebyshev - utilitarian-band ripple value db (0.1%), - Elliptic - utilitarian-band ripple value db (0.1%) and stop-band ripple value: 30 db, 40 db, 50 db and 60 db. The utilitarian-band values were selected to make the implementation in the systems o measuring slow- and ast-variable power signals possible. The 2 khz and 2.5 khz utilitarianbands are used to measure the THD actor in the power signal, which requires considering the irst 40 harmonics [3]. The measurements o the spectrum content are very oten made up to the 50th harmonic, inclusive. Thus, the research considers a 2.5 khz utilitarian-band. The remaining ranges o bands may be used to measure the wide-band signals or disturbances like over-voltages. The analysis o errors introduced by the aliasing phenomenon includes all ilter types considered during the CFM actor testing, with the exception o 10 db and 20 db elliptic ilters. According to the deinition, ilters o this type do not have large stop-band attenuation and can insert signiicant errors to the magnitude characteristic o signals. Figures 9 to 14 show the results o analyses. Figure 8 shows the manner o data representation in the diagrams.

10 Butterworth Bessel Chebyshev Elliptic 30dB Elliptic 40dB Elliptic 50dB Elliptic 60dB Butterworth Bessel Chebyshev Elliptic 30dB Elliptic 40dB Elliptic 50dB Elliptic 60dB Band 50 khz Sampling requency Band 25 khz Band 10 khz Band 2,5 khz Band 2,0 khz order n order n+1 Sampling requency exceeds 1MHz Fig. 8. The manner o data representation in the diagrams. Each o the ilter orders is represented by a group o computations or 7 characteristic types. Proper columns in the groups were assigned to the individual characteristic types. The irst column shows data concerning the Butterworth ilter, the second a Bessel ilter, the third a Chebyshev ilter, elliptic 30 db, 40 db, 50 db ilters and the inal one presents data proper or the elliptic 60 db ilter. The columns are divided into 5 parts which correspond to proper ilter pass-bands. Lack o a column or the given ilter order means that the sampling requency exceeds 1 MHz or cannot be speciied (i.e. in case when the ilter characteristic is unable to ensure an assumed attenuation). Sampling requencies marked rom diagrams visibly decrease with an increase o the ilter order.

11 Fig. 9 The dependence o the sampling requency selection on ilter order and utilitarian-band or maximum errors introduced by the aliasing phenomenon equal to 0.1% (orders 1-10). Fig. 10 The dependence o the sampling requency selection on ilter order and utilitarian-band or maximum errors introduced by the aliasing phenomenon equal to 0.1% (orders 11-20). Fig. 11 The dependence o the sampling requency selection on ilter order and utilitarian-band or maximum errors introduced by the aliasing phenomenon equal to 1% (orders 1-10). Fig. 12 The dependence o the sampling requency selection on ilter order and utilitarian-band or maximum errors introduced by the aliasing phenomenon equal to 1% (orders 11-20). Fig. 13 The dependence o the sampling requency selection on ilter order and utilitarian-band or maximum errors introduced by the aliasing phenomenon equal to 10% (orders 1-10). Fig. 14 The dependence o the sampling requency selection on ilter order and utilitarian-band or maximum errors introduced by the aliasing phenomenon equal to 10% (orders 11-20).

12 This decrease is visible in the diagrams assigned or the error levels o 1% and 10%. This trend maintains until approximately the 6th order. Above this value the sampling requency changes or each band are minimal. The use o ilters higher than 7th order seems to be groundless. The remaining elliptic ilters (30 db and 40 db) do not have such exactly monotonically decreasing character. Sampling requencies which should be applied in the system are alternately this higher and lower and this can be seen in diagrams assigned to the error levels o 0.1% and 1%. It is closely related to the stop-band elliptic magnitude response shape, where a ripple characteristic exists. The ripples have special eect at low attenuation levels in stop-bands, which can be particularly seen in the elliptic 30 db ilter. According to the deinition, these ilters have low stop-band attenuation. Thus, there are no specic sampling requencies or the error level o 0.1% in the diagrams (lack o appropriate columns). On the average, the highest sampling requencies are obtained or the Bessel ilter (even above 400 khz or 1% error), and are the result o mild transition rom a pass-band to a stopband and the high CFM values. The elliptic 60 db ilter presents the lowest sampling requencies, but rom the 8th order it has values comparable to Butterworth and Chebyshev ilters. 7. CONCLUSION The paper presents the estimation o the inluence o errors inserted to the magnitude spectrum o the transmitted signal by an antialiasing ilter. The errors are generated by ilter amplitude response approximation with the help o real characteristic (Butterworth, Chebyshev, Bessel, elliptic) and creation o the aliasing phenomeneon ater perorming a discretization process o the input analog signal. The impact o the irst type o error may be reduced by using a correction actor (CFM). The ilter utilitarian-band width should be multiplied by the CFM actor. The resulting value is a ilter cut-o requency which ensures the maintenance o error at the level below 0.1%. The second error, ater considering the CFM actor, was established at levels o 0.1%, 1% and 10%. For such values, the minimal sampling requencies were established and they should be applied to the system as a unction o the ilter order, characteristic type and width o the ilter utilitarian-band. Assuming the width o the ilter utilitarian-band, aliasing phenomenon error and the dynamics o signal changes (represented by the ilter order), a minimal sampling requency is obtained while designing the system, as a result o computations. The sampling requency values should consider synchronization with the input signal. A system designed or transmitting ast variable signals should apply a low-order antialiasing ilter with a maximally linear phase response. The Bessel ilter has such characteristics. To minimize nonlinearity o the amplitude characteristics in the pass-band in ilters o this type, the cut-o requency must be relatively high. A better solution is the use o a Chebyshev ilter (with small pass-band ripple), which does not require such a high cut-o requency (on the average 6 to 7 times lower than or a Bessel ilter). The phase response o the Chebyshev ilter is more nonlinear than in the case o a Bessel ilter. A compromise solution is the use o low-order Butterworth ilters having better pass-band magnitude response, linearity and phase response nonlinearity than the Chebyshev ilter. The sampling requency o the system with the application o a 4th-order Butterworth ilter and 10 khz pass-band equals 80kHz. The cut-o requency is then 22 khz.

13 In applications measuring low-variation signals, practically all ilter types (i.e. THD actor measurement) may be used. The best results can be obtained with the elliptic 60 db ilter. To obtain a magnitude response measurement error at the 0.1% level and to perorm measurement in the range o 2 khz, a 7th-order ilter may be used. The ilter cut-o requency should be 2.2 khz, at the minimal sampling requency o 5 khz. Finally, considering requency synchronization, the sampling requency may be chosen at 5.12 khz or 6.4 khz. The computations speciy the maximum error level which may appear in the input signal magnitude spectrum, ater iltration by a real magnitude response ilter. Errors have a boundary character and are computed assuming the worst case condition - that is where accumulation o individual distorting phenomena and the input signal magnitude spectrum are equal in the whole band. The results o computation may give satisying eects in the strongly- as well as lightlydistorted environments. REFERENCES 1. Oppenheim A., Shaer R.: Discrete time signal processing, New Jersey: Prentice Hall, Madisetti V., Williams D.: Digital Signal Processing Handbook, Boca Raton: CRC Press LLC, EN 50160: 2000: Voltage characteristics o electricity supplied by public distribution systems. 4. Harris F.: On the Use o Windows or Harmonic Analysis with the Discrete Fourier Transorm, Proceedings o the IEEE, vol. 66, pp , January Zhang F., Geng Z., Yuan W.: The Algorithm o Interpolating Windowed FFT or Harmonic Analysis o Electric Power System, IEEE Transactions On Power Delivery, vol. 16, no. 2, pp , April Rong-Ching Wu, Ta-Peng Tsao, Chia-Ching Ning: The Optimization o Spectral Analysis or Signal Harmonics, IEEE Trans. On Power Delivery, vol. 16, no. 2, pp ,April Heydt G.T., Fjeld P.S., Liu C.C., Pierce D.,Tu L., Hensley G.: Application o the Windowed FFT to Electric Power Quality Assessment, IEEE Transactions On Power Delivery, vol. 14, no. 4, pp , October Pomilio J.A., Deckmann S.M.: Flicker Produced by Harmonics Modultaion, 8th International Conerence on Harmonics and Quality o Power, October 1998, Athens, Greece, pp Dan A.M., Czira Zs.: Identiication o Harmonic Sources, 8th International Conerence on Harmonics and Quality o Power, October 1998, Athens, Greece, pp Dan A.M.: Identiication o licker sources, 8th International Conerence on Harmonics and Quality o Power, October 1998, Athens, Greece, pp FILTRY ANTYALIASINGOWE W CYFROWYCH SYSTEMACH POMIARÓW JAKOŚCI ENERGII ELEKTRYCZNEJ Streszczenie Artykuł przedstawia oszacowanie wpływu błędów wnoszonych przez iltr antyaliasingowy do charakterystyki amplitudowej przenoszonego sygnału w systemach cyrowego przetwarzania sygnałów elektroenergetycznych, a w szczególności systemów pomiarowych parametrów jakości energii elektrycznej. Błędy te generowane są w wyniku aproksymacji charakterystyki amplitudowej iltru za pomocą charakterystyki rzeczywistej (Butterwortha, Czebyszewa, Bessla, eliptycznej) oraz powstawania zjawiska aliasingu po przeprowadzeniu procesu dyskretyzacji wejściowego sygnału analogowego. Zaproponowano minimalne częstotliwości próbkowania dla zastosowań obliczania widma sygnałów elelktroenergetycznych zawierających zakłócenia wolno- i szybkozmienne. Stosowanie wyznaczonych wartości zapewnia utrzymanie poziomu błędu na określonym poziomie (0.1%, 1% oraz 10%). Badane błędy mają charakter graniczny. Są one wyznaczone przy założeniu najgorszego przypadku, tj. gdy występuje sumowanie się błędów generowanych przez poszczególne zjawiska zniekształcające oraz przy założeniu stałości widma amplitudowego sygnału wejściowego w całej szerokości pasma. Wyniki obliczeń mogą zatem z powodzeniem dawać zadowalające eekty zarówno w środowiskach słabo jak i silnie zakłóconych.