Available online at I-SEEC Proceeding - Science and Engineering (2013)

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1 Available online at I-SEEC 0 Proceeding - Science and Engineering (03) Proceeding Science and Engineering Science and Engineering Symposium 4 th International Science, Social Science, Engineering and Energy Conerence 0 Root mean square calculations o periodic unctions by sampling method S. Poomjan a,*,. aengtang a, P. Buranasiri a a Applied Physics Program, Faculty o Science, King Mongkut s Institute o echnology Ladkrabang, Bangkok, 050, hailand Abstract his paper proposes a technique o sampling method to calculate root mean square (RMS) o periodic unctions with error analysis. his proposed method will be quite useul or computational technique which is done by running a sequential time domain with exact resolutions to generate discrete values o periodic unctions. Error analysis rom arbitrary data collecting o sinusoidal unctions will be standardized to apply or any unction. In the end, the errors rom this method have been shown that they will be approached to the result computed rom the calculus integration technique when the number o sampling has been increased. From the experiment, we can ind the ormula o error analysis to calculate correct RMS by using low sampling size. 03 he Authors. Published by Kasem Bundit University. Selection and/or peer-review under responsibility o Faculty o Science and echnology, Kasem Bundit University, Bangkok. Keywords: Root mean square, periodic unction, sampling method. Introduction he root mean square mathematical operation is widely used in power engineering.[], as an example, it has been mentioned to be equivalent to direct current signal. RMS has been used as the representative or an alternative current signal, which luctuates with time domain. his paper proposes a technique o samplings methods to calculate root mean square (RMS) o a periodic unction with error estimation ormula, it will be better than a complicate calculus integration technique which has never mentioned about error estimation ormula. he interesting point o the paper is the discovery o error estimation ormula or sinusoidal unction, which may be used as the basic o all periodic unctions when the Fourier s series analysis is applied to distribute the periodic unctions. * Corresponding author. address: pattrix00@hotmail.com

2 55 S. Poomjan et al. / Proceeding - Science and Engineering (03) Fig.. he igure shows us that RMS equals Generally calculus integration technique is a amiliar technique to calculate RMS value o any periodic unction according to equation () as deinition o RMS[]. or RMS RMS where RMS is the root mean square value o unction (t) and is its period. () () Fig.. A periodic unction (t) has a period o. he root mean square value will be calculated between and +

3 S. Poomjan et al. / Proceeding - Science and Engineering (03) Reer to the deinition o expected value or mean o unction, it can be shown as ollowing. (3) hereore, the expected value o ( t ) is From relation (), RMS can be rewritten as. (4) RMS. (5) From relations o Eq. (3) and (5), the expected values o some sinusoidal unctions can be calculated (with the same period o ) as shown in able. able. he expected, RMS and RMS values o some sinusoidal unctions. (It have been ounded that the RMS value always equals to 0.50 ( ) exactly.) Sometime, calculated periodic unctions are too complicate to use integration techniques such as sinusoidal unctions multiplied by polynomial or exponential terms, or some unctions, which will not be convenient to be integrated, include some unction that cannot be integrated. hen the sampling method with easy root mean square technique may be used as another option to do.. Methodology Normally, the purposed sampling method is a systematic sampling[3] to collect data rom generating o periodic unctions, then the easy mean square o all such generated data will be used to calculate RMS in the next step. he Fig. shows the lowchart o the procedure to calculate RMS by using these methods.

4 554 S. Poomjan et al. / Proceeding - Science and Engineering (03) Fig. 3. he lowchart o the procedure o RMS calculation In our methods, the experiment introduces some ormulas to get acceptable errors, especially or sinusoidal unctions, which are undamental unctions to comprise with any periodic unction. First, sinusoidal unctions are primitive unctions to comprise any periodic unctions according to Fourier series concept, (t) = sin(t) was chosen to be the st unction in our experiment to study RMS calculation. hen, in the case o other periodic unctions, which are not sinusoidal orms, the Fourier series concept can be used to generate sinusoidal harmonic terms according to ollowing ormula and each term can be analyzed to calculate exact value o RMS[]. a 0 k ( a k kt cos b k kt sin ) (6) 3. Results and Discussion he experimental results o sampling size n = 4 and n = 0 are shown in able and 3, respectively. However, regarding to the experiments has been conducted in many times until we have ound the relation o the error versus sampling size. Figure 4 shows the variation o sampling size only or n = 4. he results o error values or sampling size n=4 to n=50 are plotted as shown in Fig. 5.

5 S. Poomjan et al. / Proceeding - Science and Engineering (03) Fig. 4. Data were collected rom (t) = sin(t) (Calculated RMS : (Average) 0.5, PPM Error : (Standard RMS Calculated RMS) x / Standard RMS) able. he sampling size (n) = 4, period () = , resolution o time ( r ) = /(n-) = , Using 4 values o t which provide 4 values o (t) to calculate RMS able 3. he sampling size (n) = 0, period () = , resolution o time ( r ) = /(n-) = , Using 0 values o t will provide 0 values o (t) to calculate RMS

6 556 S. Poomjan et al. / Proceeding - Science and Engineering (03) Conclusion Fig. 5. he graph o error versus sampling size (n) o a sinusoidal unction From the number o experiments, this method may be a new option to calculate root mean square. In addition, the error rom this method or sinusoidal unction has been related to sampling size (n) according to the ollowing equation. Error n (7) Ater the error ormula have been ound, exact value o RMS could be given as RMS = RMS stat + Error (8) RMS stat is RMS rom sampling method. For urther application o this error ormula, we may use it or general periodic unctions, which are summarized rom orthogonal terms o Fourier series.

7 S. Poomjan et al. / Proceeding - Science and Engineering (03) Reerences [] Mihaela Albu, G.. Hey. On the Use o RMS Values in Power Quality Assessment. IEEE RANSACIONS ON POWER DELIVERY, VOL. 8, NO. 4, OCOBER 003, p [] K. Ashoka Reddy, Boby George, V. Jagadeesh Kumar. Use o Fourier Series Analysis or Motion Artiact Reduction and Data Compression o Photoplethysmographic Signals. IEEE RANSACIONS ON INSRUMENAION AND MEASUREMEN, VOL. 58, NO. 5, MAY 009, p [3] Joan Joseph Castillo (009). Systematic Sampling. Retrieved 5 Nov. 0 rom Explorable :