Adv. Sudies Theor. Phys., Vol. 7, 3, no., 3-33 HIKARI Ld, www.m-hikari.com hp://dx.doi.org/.988/asp.3.3999 Accurae RS Calculaions for Periodic Signals by Trapezoidal Rule wih he Leas Daa Amoun Sompop Poomjan, Thammara Taengang, Keerayoo Srinuanjan, Surachar Kamoldilok and Prahan Buranasiri Deparmen of Physics, Faculy of Science King ongku s Insiue of Technology Ladkrabang, Chalongkrung Rd. Ladkrabang, Bangkok Thailand 5 parix@yahoo.com Copyrigh 3 Sompop Poomjan e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac The purpose of his research is o presen a simple mehod using he leas daa amoun o calculae roo mean square (RS) values of periodic signals (consruced from Fourier series) which he highes harmonic order is known or able o be esimaed. In he research, rapezoidal rule wih he leas daa amoun was proved and ried o calculae definie inegral values of sinusoidal signals from a lo of rial runs unil a significan propery was found. Then he propery would be deployed o calculae RS values. Resuls of he research can provide no only more convenien processes of simple rapezoidal rule bu also more accurae resuls han he paened Simpson s rule in he same leas daa amoun. Keywords: roo mean square, RS calculaion, periodic signal, rapezoidal rule, Simpson s rule Inroducion For several decades, roo mean square (RS) calculaions for alernaing signals have been done by many mehods. The popularly rouine mehod is Simpson s rule based on definie inegraion which was paened more han years ago[]. Alhough he mehod of Simpson s rule has been more widely acceped by engineers and physiciss han rapezoidal rule, here has sill been no comparison
4 Sompop Poomjan e al. repor found o suppor Simpson s rule as he mos appropriae mehod o calculae RS values of periodic signals. In his research, rapezoidal rule mehod will be suggesed o replace Simpson s rule o calculae RS values of periodic signals because i is more accurae in he resul and no oo complicae in calculaion.. RS Calculaions Alhough here are many mehods for calculaing RS values, he sandard mehod is sill Calculus inegraion echnique for coninuous daa saed as : + T frs f ( ) d (..), T where f RS is he RS of f () beween he domain inerval of o, + T where T is he period of f () [].. Periodic signals A periodic signal is a signal consruced from many uniformly sinusoidal signals which can be expanded as he following Fourier series : + ) m ( am cos( m ) + bmsin( m ) f ( ) a ω ω (..), where f () is a periodic signal wih period of T, a is he DC signal of f (), ω is he basic angular frequency, a m and b m are Fourier coefficiens of T harmonic orders run from m,, 3,,, [3], where is he highes harmonic order of he sudied signal (Acually can be esimaed from previously experienced daa.). In order o simplify calculaion, T will be used as he period of all sudied signals. If he RS value of f () is required o be known, all harmonic erms will be expanded o calculae definie inegral values. Pracically f (), f ( ) and T can be known from collecing signal daa of general digial oscilloscopes, bu he highes harmonic order ( ) may be esimaed from rouine daa. Refer o equaion (..), f ( ) can be expanded as : f ( ) [( a) + ( a cos) + ( a cos ) +... + ( a cos( )) + ( b sin ) + ( b sin ) +... + ( b sin( )) + ( aa cos +... + ab cos( )sin( ) +... + b b sin(( ) )sin( ))] (..) From equaion (..), if π f ( ) d comes ou, he RS value can be obained
Accurae RS calculaions for periodic signals 5 successfully. In his research, all expanded erms will be inegraed by using rapezoidal rule wih he leas daa amoun. Because f ( ) is comprised of expanded harmonic erms, if all he erms are accuraely calculaed by rapezoidal rule, f ( ) can be accuraely calculaed by rapezoidal rule also..3 Simpson s Rule In order o esimae he definie inegral value of a funcion F () beween he ime inerval of o + T, Simpson s rule is always chosen, i can be formulaed as : + T 3T F ( ) d F( ) + 3F( ) + 3F( 3) + F( 4) +... + 3F( n ) + 3F( n ) + F( n) 8( n ) (.3.) [].4 Trapezoidal Rule [ ] A simple mehod for he definie inegral calculaion of a funcion F () beween he domain inerval of o + T is rapezoidal rule which a lo of researchers are raher no ineresed o use i because hey do no rus ha he linear inerpolaion of rapezoidal rule can calculae accurae RS values of periodic signals. Generally rapezoidal rule can be formulaed as : + T T F ( ) d [ F( ) + F( ) + F( 3) +... + F( n ) + F( n) ] (.4.), ( n ) where n is he daa amoun from collecing daa of F ) where ( i i + T ( i ) /( n ) is he domain beween and + T by running i from,, 3,, n, n [4]. If rapezoidal rule is deployed o calculae RS values, daa amoun can be eiher odd or even number. In his research, he creaed linear operaor of n nt ( F( )) will be he represenaive of long summaion of rapezoidal rule erms as :,
6 Sompop Poomjan e al. n F( )) [ F( ) + F( ) + F( 3) +... + F( n ) F( )] (.4.), ( n ), n ( nt + n where n is he daa amoun size from collecing daa of F ) ( i, where i + T ( i ) /( n ) is he domain beween and n by running i from,, 3,, n, n. By using a mahemaical esimaion, when daa amoun is greaer, definie inegral values calculaed by rapezoidal rule will approach closely o coninuous definie inegral values as : n lim ( nt ( F( ))) F( ) d, (.4.3) n n Experimens for Checking Accuracy of Definie inegral π Calculaion of sin ( m ) d by Using Trapezoidal Rule π In order o discuss he resul of sin ( m ) d, m will be inegers run from,, 3, for consideraion of he h m harmonic componen signal of each periodic signal beween is period. The resul of his inegral value can be saed according o basic Calculus as : π π cos(m) m π (..), sin ( ) d d for m will be inegers run from,, 3, Le π π mπ, nt (sin ( m)) sin () + sin ( ) +... + sin ( n ) n m( n ) π ( ) + sin (m n π )
Accurae RS calculaions for periodic signals 7 π be he inegral value of sin of n.. Discussion on m π The inegral value of sin ( m) d by using rapezoidal rule wih daa amoun ( ) d calculaed by rapezoidal rule wih saring he leas daa amoun of n 4 is used (If n or 3 were seleced, he summaion resul would be equal o zero) for he experimen, resuls are saed as : π,4t (sin π,5t (sin π π 4π 6π ( )) sin () + sin ( ) + sin ( ) + sin ( ) (4 ) 3 3 3 π ( )) sin (5 ) () + sin π ( ) + sin 4 4π ( ) + sin 4 6π ( ) + sin 4 8π ( ) 4 π π 4π 6π 8π π T(sin ( )) sin () + sin ( ) + sin ( ) + sin ( ) + sin ( ) + sin ( ) (6 ) 5 5 5 5 5 π,6 Afer running compuer programs (such as icrosof Excel 7, ATLAB ec.) π o calculae (sin ( )) by using T, n n 4, resuls of he calculaions can sill be equal o π for all of n 4. Therefore, a simple equaion of a rapezoidal rule propery can be saed as: π T, n π (sin ( )) sin ( n ) (..), for n 4 () + sin π ( ) +... + sin n ( n ) π ( ) + sin n ( n ) π ( ) n
8 Sompop Poomjan e al.. Discussion on m π The inegral value of sin leas daa amoun of n 4, resuls are saed as : π,4t (sin () d calculaed by rapezoidal rule wih saring he π 4π 8π π ( )) sin () + sin ( ) + sin ( ) + sin ( ) (4 ) 3 3 3 π π 4π 8π π 6π,5T (sin ()) sin () sin ( ) sin ( ) sin ( ) sin ( ) (5 ) + + + + 4 4 4 4 π π 4π 8π π 6π π,6t (sin ( )) sin () + sin ( ) + sin ( ) + sin ( ) + sin ( ) + sin ( ) (6 ) 5 5 5 5 5 Afer using compuer programs (such as icrosof Excel 7, ATLAB ec.) o π calculae (sin ( )) by using T, n n 4, resuls of he calculaions canno be consan as π for all n 4, bu hey will become o π again when n 6. Therefore, he simple equaion of a rapezoidal rule can be saed as : π T π, n (sin ()) sin ( n ) (..), for n 6..3 Discussion on m > () + sin 4π ( ) +... + sin n 4( n ) π ( ) + sin n Afer he propery of rapezoidal rule had been discovered for m and, a big lo of rial runs were carried on o verify his propery by many compuer programs for m > unil i was found ha he leas daa amoun ( n ) would depend on m according o he following equaion (.3.). n m + (.3.) 4( n ) π ( ) n
Accurae RS calculaions for periodic signals 9 Therefore, equaion (..) can be improved o be equaion (.3.) as : π ( m)) sin ( n ) (.3.), mπ ( ) +... + sin n m( n ) π ( ) + sin n π T, n (sin () + sin (m for n m +, or π + (sin,m T ( m)) (.3.3) π ) 3 Experimens for Checking Accuracy of Definie inegral π Calculaion of cos ( m ) d π, sin( m ) d π, cos( m ) d π, sin( m )sin( p) d, π cos( m )cos( p) d π, sin( m)cos( p) d π and cos( m )sin( p) d by Using Trapezoidal Rule π As same as he rial runs of sin ( m ) d, he definie inegral values of π π cos ( m ) d, sin( m ) d, cos( m ) d, sin( m )sin( p) d, π π π cos( m )cos( p) d, sin( m)cos( p) d π π and cos( m)sin( p) d are equal o inegral values calculaed by he mehod of Calculus when he appropriae daa amoun is also n m + and m > p.
3 Sompop Poomjan e al. 4 A Use of Trapezoidal Rule o Calculae RS Values of Periodic Signals Consruced from Fourier Series From he equaion (..), if he RS value of f () is required o be known, he inegral value of f ( ) beween is period will be finished successfully firs. See he calculaion process below. π f ( ) d π +... + ( b +... + b [( a ) sin( )) b + ( a cos) + ( a cos ) + ( a a cos +... + a sin(( ) )sin( ))] d +... + ( a b cos( )) cos( )sin( ) + ( b sin ) + ( b sin ) Because all expanded erms are arranged o be able o calculae he definie inegral values by using rapezoidal rule according o he previous experimen, herefore he definie inegral sign can be changed o rapezoidal rule as: π f ( ) d π, + +... + ( b +... + b T ([( a ) sin( )) b + ( a cos) + ( a a + ( a cos +... + a sin(( ) )sin( ))] cos ) +... + ( a b cos( )) cos( )sin( ) + ( b sin ) + ( b sin ) Thus, he definie inegral of f ( ) can be saed as : π f π ( ) d, + T ( f ( )) (4..) If he scale and he period are changed o any general periodic funcion, finally he RS value of equaion (..) can be formulaed as :
Accurae RS calculaions for periodic signals 3 f RS + T + T T ( f ( )), + f ( ) d T (4..) T 5 A Demonsraion of Using Trapezoidal Rule for RS Value A creaed periodic signal of g( ) 3.5 +.3sin + 3.48cos 7 +.5sin will be demonsraed o show accuracy of rapezoidal rule. See signal g () in he fig.. Fig. shows signal g() wih period of T. From he Calculus inegraion echnique, he RS value of g () is equal o 4.5396. If n ( ) + is used for calculaing RS values by rapezoidal rule and Simpson s rule, errors from rapezoidal rule and from Simpsons rule will be. and -.446 respecively as in able.
3 Sompop Poomjan e al. ehod Trapezoidal Rule Simpson's Rule Daa Amoun (n) RS Value 4.5396 4.9946 Error. -.446 Error Percenage (%). -.736767 Table. shows performances of rapezoidal rule which can calculae RS value of g( ) 3.5 +.3sin + 3.48cos 7 +.5sin more accuraely han Simpson s rule. 7 Conclusion An accurae RS value of a periodic signal comprised of Fourier series can be calculaed by using he simple mehod of rapezoidal rule wih he leas daa amoun ( n ) which can be deermined from he highes harmonic order ( ) of he periodic signal according o n +. From he experimen, rapezoidal rule was deployed o calculae accurae RS values of periodic signals. The resul of he experimen indicaes ha rapezoidal rule can provide more accurae RS values han paened Simpson s rule for periodic signals consruced from Fourier series.
Accurae RS calculaions for periodic signals 33 References [] Seven A. Lombardi, ETHOD AND APPARATUS FOR PROCESSING A SAPLED WAVEFOR, Unied Saes Paen, Paen No. 5,88,983, 998. [] ihaela Albu and G. T. Heyd, On he Use of RS Values in Power Qualiy Assessmen, IEEE Transacions on Power Delivery, 8(3), 586-587. [3] Andrzej Konsany uciek, A ehod for Precise RS easuremens of Periodic Signals by Reconsrucion Technique Wih Correcion, IEEE TRANSACTIONS ON INSTRUENTATION AND EASUREENT, 56(7), 53-56. [4] N. Dharma Rao,.E., and Prof. H. N. Ramachandra Rao,.Sc., Soluion of ransien-sabiliy problems hrough he number-series approach, The Proceedings of he Insiuion of Elecrical Engineers, (964), 775-788. Received: Sepember 9, 3