Comparison of RMS Value Measurement Algorithms of Non-coherent Sampled Signals

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1 0.478/v MEAUREMET CIECE REVIEW, Volume, o. 3, 0 Comparison of RM Value Measurement Algorithms of on-coherent ampled ignals H. Hegeduš, P. Mostarac, R. Malarić Department of Electrical Engineering Basics and Measurement, Faculty of Electrical Engineering and Computing, University in Zagreb, Unska3, 0000 Zagreb, Croatia, hrvoje.hegedus@fer.hr Uncertainty and bias of RM measurement of digitally non-coherent sampled signal is dependent on the algorithm used. This paper presents the new Averaging two subsets method for RM value bias correction of non-coherent sampled signal. Methods for estimating RM values in the time domain are also compared. Keywords: RM measurement methods, estimation algorithms, non-coherent sampling T. ITRODUCTIO ODAY, high resolution analog to digital converters (ADC are often used for precision measurements of voltage and current. The measurement uncertainty of the RM value of the analog sinusoidal signal depends on the quality of the ADC and on the algorithms used for RM estimation. This paper will focus only on the algorithms for RM estimation of the sinusoidal signal. The comparison of classical algorithms and windowing based algorithms has been described in [, ]. RM value of the sinusoidal signals with more than 5 periods can be estimated with several windowing based algorithms with the uncertainty better then measurement uncertainty of present ADC systems. The goal of this paper is to present a new method for estimating RM value of the sinusoidal signal with less than 5 periods and to compare it with other RM estimation algorithms. Comparison is done by calculating RM value of the simulated signal of an exactly known RM value, by random signal phase to simulate non-coherent sampling and with similar sampling rates to common ADCs.. THE AALYI OF THE RM VALUE MEAUREMET A. RM estimation of sinusoidal signals By definition, RM estimation of analog signal x(t is based on the relation [4]: M T RM x ( t dt, ( T M where T M is time of measurement. Analog sinusoidal signal can be expressed as: x ( t ( ω t ϕ m 0 sin ( where m is signal amplitude, ω sig is signal angular frequency and ϕ is signal phase relative to start of the measurement. RM value of analog sinusoidal signal ( can be calculated as: sig RM ( ω T ϕ sin( ϕ m sig m ωsigt m. (3 Time of measurement can be estimated by the number of signal periods: T m ( M T sig, M Z,,, (4 where M is integer number of signal periods and is decimal part of noninteger period. RM value of the sinusoidal signal with ( M periods can be calculated using (5 or (6: ( 4π ϕ sin( ϕ 4π ( M RM, (5 ( π cos( π ϕ π ( M RM. (6 B. RM estimation of coherently sampled sinusoidal signals When sinusoidal signal is coherently sampled, an integer number of sinusoidal periods M is sampled that can be described by equation: T MT, M Z (7 0 m sig The RM value of sinusoidal signal ( with integer number of periods can be calculated using (8: m RM. (8 Equation (8 shows that the RM value of sinusoidal signal with integer number of periods is depended only on signal amplitude m. 79

2 MEAUREMET CIECE REVIEW, Volume, o. 3, 0 C. RM estimation of non-coherently sampled sinusoidal signal In real measurements coherent sampling is often hard to carry out. In this case, the measured signal consist of noninteger number of periods ( M where M Z and 0. The expression for RM value for non-coherently sampled signal can be estimated by using (5 or (6. It is not dependent only on signal amplitude but also on signal phase φ, integer number of sampled signal periods M and decimal part of the last sampled period. The difference between RM value of non-coherently sampled sinusoidal signal RM (5 or (6 and RM value of coherently sampled sinusoidal signal RM (8 is the bias of the RM measurement. The relative bias of the RM value of noncoherent sinusoidal signal can be estimated using (0 or (: RM RM RM RM RM, (9 RM ( 4π ϕ sin( ϕ 4π ( M ( π cos( π ϕ π ( M, (0. ( 3. THE AALYI OF METHOD FOR RM VALUE BIA CORRECTIO A. Reducing RM value bias by minimizing the decimal part of sinusoidal signal period The bias of the RM value can be reduced by minimizing the decimal part of period to the range defined by the number of samples per period of non-coherently sampled signal. Higher sampling rate ensures more samples per period and lower value. When the analog sinusoidal signal is sampled with number of samples per period, desired signal phase can be obtained by choosing between two samples. Range of signal phase error caused by non-coherent sampling φ can be estimated as (: π π ϕ,, ( where is number of samples per period of the sampled analog sinusoidal signal. Exact range of the decimal part of period can be estimated as (3:,, (3 where is number of samples per period of the sampled analog sinusoidal signal. When the decimal part of period is in range defined by (3, the maximum expected bias of the RM value can be calculated by finding maximum of the relation (. ( πs cos( πs ( ϕ ϕs π ( M RM MA, (4 s where φ is desired signal phase relative to start of the measurement, φ is phase error caused by non-coherent sampling, M is integer number of periods and is decimal part of period. Equation (4 will have its maximum when the cosine part is equal to : ( ( ϕ ϕ cos π, (5 s s ( ϕ ϕ 0 π. (6 s s For calculating approximated maximum expected bias of the RM value, approximations for square root (7 and sine function (8 is used: 0, (7 ( 0 sin. (8 After using approximations (7 and (8 on (4, maximum bias of the RM value can be approximately calculated as: s RM MA (9 ( M s Maximum bias of the RM value ( will be for maximum value of (0 from the range defined by (3: s ±, (0 RM MA. ( ( M Maximum expected bias of the RM value expressed in ppm is calculated in Table I using ( for some common parameters M and. TABLE. Maximum expected rms value bias (ppm Integer umber of samples per period number of periods M Maximum expected RM value bias calculated for some common parameters M and using equation (. 80

3 MEAUREMET CIECE REVIEW, Volume, o. 3, 0 B. Reducing RM value bias by using the ingle subset method The bias of the RM value of sinusoidal signal ( is highly dependent on the signal phase that is shown in Fig.. MA ϕ (6 M Maximum expected bias of RM value (5 and (6 is for parameters (7 and (8: (7 π ϕ (8 Approximated value of the maximum expected bias of the RM value for the ingle subset method is: π MA (9 ( M Fig.. Relative bias of the RM value of the sinusoidal signal using equation ( with parameters M 5, 000 and as a function of signal phase 0 < φ < 360 The bias of the RM value can be reduced by extracting single subset from main sampled signal with certain signal phases φ where the influence of the RM value bias is minimal. These phases can be calculated by equaling relation ( with zero: ( π cos( π ϕ π ( M RM 0 ( ϕ 0 ( cos π (3 π π ϕ k π, k Ζ (3 4 { 45, 35, 5, 35,...}, 0 ϕ (4 The main disadvantage of this method is that minimal variation of signal phase from values defined in (3 is causing significant increase of the RM value bias because the first deviation of function RM ( ϕ defined by equation ( has its extremes for these phases. Maximum expected bias of the RM value estimated by the ingle subset method can be calculated as (5: MA ( π cos( π ( ϕ ϕ π ( M (5 After using approximations for square root (7 and sine function (8, approximated value of maximum expected bias of the RM value can be calculated as (6: Maximum expected bias of the RM value (9 estimated by the ingle subset method for some common parameters M and is calculated in Table. Values show significant reduction of RM value bias calculated by the ingle subset method in comparison to values in Table. Table. Maximum expected RM value bias (ppm for ingle subset method Integer umber of samples per period number of periods M E0 3.E00 3.E-0.6E0.6E00.6E E0 6.3E-0 6.3E E0 3.E-0 3.E-03 Maximum expected RM value bias for ingle subset method calculated for few common parameters M and using equation (9. C. Reducing RM value bias by using the Averaging two subsets method The idea of the Averaging two subsets method is to extract two signal subsets of the same length from the main sampled signal with certain signal phases to get maximal and minimal bias of RM value of each subset. By averaging RM values of these two subsets, RM value bias can be significantly minimized. Minimum and maximum of function RM ( ϕ defined by ( can be calculated as: d dϕ RM π 0 ϕ k π, k Ζ { 0, 90,80, 70,...}, 0 (30 ϕ (3 To extract two signal subsets of one whole signal period from the main signal with 90 phase difference between the 8

4 MEAUREMET CIECE REVIEW, Volume, o. 3, 0 subsets, the sampled signal must contain at least one and half signal period. The example of sampled signal and two signal subsets are shown in Fig.. RM ( π cos( ϕ ( π M m (40 Different sign of sine and cosine part under square root in (39 and (40 is caused by 90 phase offset of the second signal subset relative to the first signal subset. Arithmetic mean of RM values of the first and second signal subset can be estimated as: AT RM RM (4 After using (39 and (40 on (4, RM value estimated by the Averaging two subset method can be calculated as (4 Fig.. ignals ubset and ubset extracted from the main sampled signal. ubset should have its cosine part equal to one and ubset equal to minus one: AT m ( π cos( ϕ π ( M ( π ( ( cos ϕ π M (4 ( ϕ cos π (3 ( π ϕ cos (33 ϕ kπ π k Ζ (34 π ϕ kπ π k Ζ (35 RM values of ubset and ubset can be calculated as (36 and (37: ( π cos( π ( ϕ ϕ ( π M RM (36 ( π cos( π ( ϕ ϕ ( π M RM (37 where range of the decimal part of period is defined in (3 and range of the phase offsets φ and φ is defined by the number of samples per period in equation (38: { ϕ, ϕ } π π, (38 By using approximations for square root (7, RM value can be approximately calculated as: AT ( π [ cos( ϕ cos( ϕ ] ( 8π M m (43 Both phase offsets φ and φ are rather small and have similar values so the difference between them is also a rather small number: ϕ ϕ ϕ (44 After using assumption (44 on equation (43, RM value estimated by the Averaging two subset method is approximately equal to RM value of sinusoidal signal with integer number of periods with zero bias: m AT RM (45 Relative bias of RM values calculated by the Averaging two subsets method can be calculated by equations (46 and (47: AT RM AT, (46 RM After using (34 and (35 on (36 and (37 it can be calculated: ( π cos( ϕ ( π M RM (39 AT ( π cos( ϕ π ( M ( π cos( ϕ π ( M. (47 8

5 MEAUREMET CIECE REVIEW, Volume, o. 3, 0 Maximum expected bias can be calculated by using equation (48 with ranges of, defined by (3 and ranges of φ and φ defined by (38. AT MA ( π cos( ϕ π ( M ( π cos( ϕ π ( M (48 To calculate approximated maximum expected bias (5, the following three approximations based on Taylor series are used: (49 8 AT MA 6 sin ( x x (50 cos ( x x (5! ( ϕ ϕ M ( ϕ ( ϕ ( M. (5 Maximum expected bias of RM value (48 and (5 is for parameters (53 and (54: (53 π ϕ ϕ (54 Table 3. Maximum expected RM value bias (ppm for Averaging two subsets method Integer umber of samples per period number of periods M E0.3E-0.3E-03 3.E00 3.E-0 3.E E-0 5.0E E-05 0.E-0.3E-03.3E-05 Maximum expected RM value bias for the Averaging two subsets method (3 calculated for few common parameters M and. Maximum expected bias of RM value estimated by the Averaging two subsets method for some common parameters M and is calculated using (5 with parameters (53 and (54 and presented in Table 3. Values exhibit significant reduction of RM value bias calculated by the Averaging two subsets method in comparison to values calculated by the ingle subset method in Table and values calculated by minimizing decimal part of period in Table. 4. THE IMULATIO OF RM ETIMATIO ALGORITHM A. Realization of methods in I LabVIEW The ingle subset method and the Averaging two subsets method are developed and tested in I LabVIEW development system [5]. For measurement of sampled signal phase and frequency both methods use built in functions like Extract single tone information that analyzes whole sampled signal to achieve the best result. ignal phase and frequency can be also measured with other algorithms [6, 7]. This step is very important because accuracy of both methods depends on the accuracy of the signal phase and frequency measurement. After that step, beginning and length of signal subsets can be calculated and signal subset extracted to calculate RM value. Extract ubset RM Main signal Measure φ and f Calculate φ and φ Average RM Extract ubset RM Fig.3. Block diagram of the Averaged two subset method algorithm. After ubset and ubset extraction, RM values of both ubset and ubset are calculated using the classical method with Rectangular window. Finally, RM is calculated by averaging RM values of ubset and ubset. B. Testing methodology Testing is realized by simulating non-coherent sinusoidal signal of known RM value that can be compared to the results of tested methods for RM measurement. The results are presented in graphs depending on the number of signal periods (Fig.4. - graph x-axis. Methods were tested on sinusoidal signals with number of samples per period 000 and number of periods.5 < M < 8 (in steps of 0.0 periods. For each step 500 measurements have been performed on different signals with random signal phase 0 φ < 360 and random signal frequency from f 50±0.5 Hz. In Fig.4 the value of 83

6 MEAUREMET CIECE REVIEW, Volume, o. 3, 0 maximum relative bias of these 500 RM measurements is shown. The arithmetic mean of these 500 measurements could show about 0 times better results of the bias of RM value but in this paper only the maximum bias of RM value (the worst case is presented. C. imulation results The results of the simulation of RM value measurement with the ingle subset method and the Averaging two subsets method is shown in Fig.4. The bias of the RM values estimated with both methods is lower than maximum expected RM value bias of these methods calculated by equations (9 and (48. Fig.4. Comparison of simulated RM value measurement with the ingle subset method and the Averaging two subsets method with maximum expected RM value bias estimated by equations (9 and (48 based on simulated sinusoidal signal 000, f 50±0.5 Hz, 0 φ < 360. Fig.5. Comparison of RM value measurement based on the ingle subset method and the Averaging two subsets method with windowing algorithms: Rectangular (no window, Hanning, 4 Term B-Harris, 7 Term B-Harris based on simulated sinusoidal signal 000, f 50±0.5 Hz, 0 φ < 360. The results of RM value measurement with the ingle subset method and the Averaging two subsets method are compared with Rectangular, Hanning, 4 Term B-Harris and 7 Term B-Harris windowing algorithms in Fig.5. Averaging two subsets method reduces RM value bias of noncoherently sampled sinusoidal signals better than any known windowing algorithms for signals from M.5 to M 5 periods. 5. COCLUIO The simulation results of the ingle subset method and the Averaging two subsets method showed that RM value bias is considerably reduced. These two methods are superior to all known windowing algorithms for signals with low period number. The new proposed Averaging two subsets method can be used in applications where both high precision and speed of RM value measurement is important [8, 9]. In future work, the method will be tested for signals with higher harmonics, presence of noise, and the influence of analog to digital converters with different conversion resolution. REFERECE [] ovotny, M., edlacek, M. (004. Measurement of RM values of non-coherently sampled signals. In Proceedings of the 3 th International ymposium on Measurements for Research and Industry Applications, IMEKO TC-4. Athens, Greece, [] ovotny, M., edlacek, M. (008. RM value measurement based on classical and modified digital signal processing algorithms. Measurement, 4 (3, [3] Hegeduš, H., Mostarac, P., Malarić, R. (00. Precision RM value measurement of non-coherent sampled signals. In 00 Conference of Precision Electromagnetic Measurement Digest. Daejeon, Republic of Korea, [4] Hegeduš, H., Malarić, R., Mostarac, P. (009. Analysis of methods for RM measurement of AC signals. In Proceedings I ational Conference ETAI 009, 39. [5] ational Instruments Corporation. I LabVIEW. Retrieved from [6] Malarić, K., Malarić, R., Hegeduš, H. (00. A nonlinear least-squares graphical tool ( Gaussfit for educational purposes. International Journal of Electrical Engineering Education, 47 (, -. [7] Mostarac, P., Malarić, R., Hegeduš, H. (00. ovel frequency measurement method with low sampling time. In 00 Conference of Precision Electromagnetic Measurement Digest. Daejeon, Republic of Korea, [8] Ptak, P., Kurkowski, M., Biernacki, Z., Zloto, T. (00. Digital measuring system for recording deformed functions. Measurement cience Review, (, [9] Espel, P., Poletaeff, A., dilimabaka, H. (00. Traceability of voltage measurements for nonsinusoidal waveforms. Measurement cience Review, 0 (6, Received May 6, 0. Accepted August 3, 0. 84