Measurement of the Sinewave RMS Value in Noncoherent Sampling Mode. Daniel Belega 1, Dominique Dallet 2

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1 Measurement o the Sewave RMS Value oncoherent Samplg Mode Daniel Belega, Domique Dallet 2 Politehnica University o imişoara, Faculty o Electronics and elecommunications, Bv. V. Pârvan, r. 2, 3223, imişoara, Romania, phone: ( , ax: ( , daniel.belega@etc.upt.ro 2 Laboratoire IMS - Université de Bordeaux - ESEIRB UMR CRS 528, 35 Cours de la Libération, Bâtiment A3, 3345, alence Cedex, France, phone: ( , ax: ( , domique.dallet@ims-bordeaux.r Abstract- In this paper, a method to estimate the value o a noncoherent sampled sewave by means o the ormula used an AC analog electronic voltmeter is presented. Addg some modications on the algorithm, the accuracy obtaed this case was improved. his method is well suited or real-time applications which the measurement o the sewave value with relative high accuracy is suiciently. he perormances o the method proposed are proved by means o computer simulation as well as experimental results. I. Introduction One o the most important parameter o a sewave is its root mean square ( value because it relates directly to the sewave power. he value o a sewave voltage can be measured by an AC analog electronic voltmeter []. For this purpose the AC analog electronic voltmeter contas a mean value converter [], [2], volvg two basic operations: irst the put is applied to a rectier and secondly the mean value o the rectier output is determed. o obta the sewave value the mean value converter output signal is multiplied by a scale actor which depends on the rectier type. In an AC analog electronic voltmeter which contas a mean value converter with a ull-wave rectier, the value o the put sewave x(t characterized by its amplitude A and requency ( is given by: m 2 2 x(dt t where: is the value o x(t (the ideal sewave value is ideal A 2 ; x(t is the modulus o x(t, obtaed at the output o the ull-wave rectier; m is the mean value o x(t, obtaed at the mean value converter output (the ideal value is m ideal 2A ; K is the scale actor ( K ideal m ideal 2 2. In this paper the measurement o the sewave value by means o a discrete-time system noncoherent sampled mode is vestigated. he theoretical expression o the value o a noncoherently sampled sewave obtaed by the ormula ( is derived. Based on this irst task, a new method is proposed to crease the estimator accuracy and its perormances are studied. II. Expression o the sewave value the noncoherent samplg mode Let us consider a sewave o amplitude A, requency ( and phase ϕ ( t As( 2 t ϕ. x When the signal x(t is digitized by mean o a digital waveorm recorder the discrete-time signal x(n, n,,2,... is obtaed. he relationship between the put requency, and the samplg requency, s is given by: ( (2

2 s (3 where is the number o recorded samples, is the number o complete recorder cycles and is the ractional part o the recorded cycles ( <. For the samplg process is considered as coherent te o requencies [3]. he theoretical value o x(t considerg the recordg terval is: ( ( As ( 2 t ϕ which K 2 2. So, the value o x(n can be estimated by: dt (4 ˆ and the relationship between and is given by (see Appendix ideal ideal ideal s 2 ˆ is n As 2 n ϕ lim ˆ ( s( ϕ cos( cos( ϕ s( s( ϕ. <.5 ϕ.5 ϕ ϕ ( 2 ( 2 < ϕ ( 2 ( 2 < (5 (6 (7 which ϕ ϕ, ϕ, ϕ <. From the above expression it can be established that or ϕ < 2.5, then ideal. Fig. shows (calculated by (7 and ˆ (calculated by (5 as a unction o the number o recorded cycles (i.e.. he sewave obtaed by simulation is characterized by its amplitude A 2 and its phase ϕ 23 rad. he number o recorded cycles varies the range [4, 45] with an crement o.. he number o recorded samples is 248. Fig.. (solid le and ˆ (dotted le as a unction o the number o recorded cycles. As it was observed Fig. the derences between and ˆ are relatively small. In order to be more precise, usg (7 the relative error o the value measurement was established as:

3 ε s 2 ( s( ϕ, cos( cos( ϕ s( s( ϕ,, <.5 ϕ.5 ϕ ϕ ( 2 ( 2 < ϕ ( 2 ( 2 < From the above expression it has been demonstrated that the maximum relative error ε,was given by: ε max max max s 2 (, ( cos(, 2 cos(, ( 2 s(, <.5.5 < Fig. 2 shows the maximum o the relative error modulus o ε % as unction o the number o recorded cycles obtaed durg ϕ scan (ϕ varies the range [, 2 rad. with an crement o 5 rad. he number o recorded cycles varies the range [4, 45] with an crement o.. he sewave signal is the one used to obta the results presented Fig.. (8 (9 Fig. 2. he maximum o the relative error modulus o the value measurement as unction o the number o recorded cycles obtaed durg ϕ scan the range [, 2 rad. As it can be observed rom Fig. 2 or 4 the maximum o the relative error modulus is smaller than.26%. III. A new method to improve the value estimation It is obvious that the bias o the value measurement is determed by the part o signal period at the end o (. A method to reduce the bias o the value measurement is to multiply a priori the signal x(t by a wdow w(t. hus, the signal x w (t x(t w(t is obtaed. Ones o the most used wdows are the cose wdows [4], deed by : w ( t H ah h cos 2ht ( where H is the wdow order and a h are the wdow coeicients. he theoretical value o x w (t durg the recordg terval is: w ( ( As ( 2 t ϕ w( t In case o coherent samplg, ater some calculus, it can be established that dt. ( (

4 a w ideal. (2 Based on the above expression the value o x(n can be estimated by: a n x a ( n w( n ˆ w where w(n is the discrete-time H-term cose wdow. It should be noted that or a discrete-time H- term cose wdow, a is equal to the normalized wdow peak signal ga PSG n ( PSG w( n. Fig. 3 shows the maximum o relative errors modulus o the value measurement obtaed usg (5 or ˆ and (3 or as unction o the recorded cycles number durg ϕ scan. When the value is estimated by, the Hann wdow and 4-term Blackman Harris wdow were used. he amplitude o sewave was A 2. he phase o the sewave ϕ varies the range [, 2 rad. with an crement o 5 rad. he number o cycles recorded varies the range [4, 45] with an crement o.. he number o the recorded samples is 248. he same sewave as Fig. is used. (3 (a (b Fig. 3. he maximum o the relative errors modulus o the value as unction o the recorded cycles number obtaed durg ϕ scan the range [, 2rad. he value is estimated by: a ˆ ( x mark and dotted le and b with Hann wdow (star and dotted le and 4-term Blackman-Harris (circle and dotted le. From Fig. 3 it is clearly evident that the value is more accurately estimated by than ˆ. When the value is estimated by the maximum o the relative error modulus is smaller than.6% (more than our times smaller than when the value is estimated by ˆ. IV. Experimental results he perormances o the method proposed are also veried by means o experimental results. For this purpose several acquisitions are made at derent sewave requencies between Hz. he amplitudes o the sewaves are equal to 2 V. he sewaves are obtaed rom the HM83 signal generator. he MS32C5x board is used as the acquisition system. he samplg requency is 4877 Hz. For each requency a number o 25 records are collected. Each record contas 24 samples. he sewave value is estimated also by means o the Interpolated Discrete Fourier ransorm (IpDF method with Hann wdow [5]. he IpDF method provides very high accurate estimate o the amplitude o a sewave (and also o the sewave value. Fig. 4 shows the modulus o the derence between the average o the sewave values estimated by the proposed method and the one o the values estimated by means o the IpDF method as a unction o

5 requency. In the proposed method the Hann wdow and 4-term Blackman-Harris wdow are employed. Fig. 4. he modulus o the derence between the average o the sewave values estimated by the proposed method and the one o the values estimated by means o the IpDF method as a unction o requency. In the proposed method the Hann wdow (star and dotted le and the 4-term Blackman-Harris wdow (circle and dotted le are used. he results obtaed by the proposed method der rom the ones obtaed by means o the IpDF method begng to the ourth digit ater the decimal pot. hus, the sewave values are high accurately estimated by the proposed method. V. Conclusion he expression o the value o noncoherently sampled sewave calculated by the ormula based on an AC analog voltmeter works is derived. o creases the value estimation accuracy obtaed by usg the derived expression a method is proposed. By appropriate choice o and values the value can be relative high accurately estimated by the proposed method. he perormances o this method have been proven by computer simulation and also by experimental results. he ma advantage o the proposed method is that this is very simple to implement. hus, the proposed method is well suited or real-time measurement o the value o a discrete-time sewave. Appendix Calculation o he value o the part o the sewave x(t (given by (2 that was discretized is ( As ( As ( 2 t ϕ ( 2 t ϕ dt ideal dt K ( ( As ( 2 t ϕ As ( 2 t ϕ dt. dt (A. For ϕ < < t, where t (-ϕ(2, it can be established hus, becomes As ( 2 t ϕ dt As( 2 t ϕ dt s( s( ϕ. A (A.2

6 s ideal t < t 2, where t 2 (2-ϕ(2, it can be established ( s( ϕ. (A.3 As t ( 2 t ϕ dt As( 2 t ϕ dt As( 2 t ϕ dt cos( cos( ϕ. t A A (A.4 hus, becomes t 2 <, it can be established ideal cos ( cos( ϕ. (A.5 As t ( 2 t ϕ dt As( 2 t ϕ dt As( 2 t ϕ dt As( 2 t ϕ hus, becomes 2A A s t2 t ( s( ϕ. ideal 2 s t2 ( s( ϕ For ϕ < 2, the same procedure is used and the same expression or is obtaed, but with ϕ replaced by ϕ-. hus, expression o is given by ideal ideal ideal s 2 ( s( ϕ cos( cos( ϕ s( s( ϕ <.5 ϕ.5 ϕ ϕ. ( 2 ( 2 < ϕ ( 2 ( 2 < dt (A.6 (A.7 (A.8 ϕ which ϕ, ϕ, ϕ <. ϕ < 2 Reerences [] he Measurement, Instrumentation, and Sensors Handbook, CRC Press and IEEE Press, 999. [2].. Leng, Electronique des Systèmes de Mesures mise en oeuvre des procédés analogiques et numériques, Edition Masson, 992. [3] A. Ferrero, R. Ottoboni, High-Accuracy Fourier Analysis Based on Synchronous Samplg echniques, IEEE rans. Instrum. Meas., vol. 4, no. 6, pp , Dec [4] A. H. utall, Some wdows with very good sidelobe behavior, IEEE Acoust. Speech Signal Processg, vol. ASSP-29, pp. 84-9, Feb. 98. [5]. Grandke, Interpolation Algorithms or Discrete Fourier ranso o Weighted Signals, IEEE rans. Instrum. Meas., vol. IM-32, no. 2, pp , une 983. Acknowledgments he work presented this paper is perormed under the type A project code 35, supported by CCSIS Romania. his project volves collaboration between the Department Measurements and Optical Electronics o the Faculty o Electronics and elecommunications imişoara and the Laboratoire IMS-ESEIRB - Université de Bordeaux.