EXPERIMENTING WITH MAPLE TO OBTAIN SUMS OF BESSEL SERIES

Transcription

1 EXPERIMENTING WITH MAPLE TO OBTAIN SUMS OF BESSEL SERIES Walter R Bloom Murdoch Uiversity Perth, Wester Australia bloom@murdoch.edu.au Abstract I the study of ulse-width modulatio withi electrical egieerig may authors develo a series reresetatio of the modulated wave usig a double Fourier series based o roerties of the carrier ad referece waveforms. Assumig a itegral frequecy ratio we ca also comute the sigle Fourier series directly, ad equatig these we obtai two equivalet reresetatios, oe ivolvig the usual trigoometric fuctios, ad the other ivolvig these together with Bessel fuctios of the first kid. Comarig coefficiets we are led to a rage of series such as J + J = ad these results ca be cofirmed usig MAPLE. Geeralizig the above series we ca the use MAPLE to study the likely sum of each of the series ad J + s J + s J s J s for, s =,,..., all of which coverge. I this aer we ivestigate the atters that arise, ad show that i most cases cosidered the sums ca be give quite secifically ad have very simle forms. This aroach is useful as both a research tool ad also for itroducig studets to the rather difficult area of Bessel series. Itroductio The advet of owerful symbolic maiulatio ackages like MATHEMATICA ad MAPLE has had a rofoud effect o the availability of tools for mathematical research ad teachig for the latter,

2 see [] ad [] for examle. I this aer we take a more exerimetal aroach, which we illustrate with a examle arisig from the study of ulse-width modulatio withi electrical egieerig where certai Bessel series together with their sums are derived, ad the further oes cojectured through use of MAPLE. The ricial aim will be to idicate both the geeral aroach ad how MAPLE ca be used to obtai likely results exerimetally, while at the same time avoidig comlicatios i resetatio. The format of the aer is as follows. I the secod sectio we derive two seemigly differet series reresetatios of a simle eriodic fuctio, ad i the third sectio we suggest some derived series that could be usefully ivestigated ad use MAPLE to obtai their likely sums. While these results are develoed from just a sigle examle, it is easy to see how the above methods ca be adated to roduce a rage of other Bessel series. Sigle ad double Fourier series aroaches We cosider the referece waveform v ref t = si t ad carrier waveform give by t, 0 < t < 3 v car t = t +, < t < 3 3 t, < t < 3 which have the followig grahs: t - -3 Fig. It is easily checked that the itersectio oits are, i icreasig order, 0,,, 3,. I the study of ulse-width modulatio PWM it is usual to cosider the modulated outut wave give by, v car < v ref v t = 0, v car > v ref We emhasize that while the referece ad carrier waves are quite ideedet, the outut wave deeds comletely o the geometric relatioshi of these two. We examie two methods for comutig

3 a series reresetatio of the outut wave, the first beig the sigle Fourier series of the eriodic fuctio v, ad the secod derived from a associated double Fourier series. For the above examle, usig the first method we are just comutig the sigle Fourier series of the fuctio, 0 < t < v t = 0, < t <, < t < 3 3 0, < t < the values at the itersectio oits lay o rôle here, which is give by + si t The two-variable aroach ivolves evaluatig the double Fourier series of the doubly eriodic two-variable fuctio, si t < y < + si t F t, y = 0, otherwise ad the restrictig F to the lie y = t see Fig. below. The eveloig siusoids are evaluated by solvig the equatios v ref = v car, from which we obtai si t, 0 < t < 3 t = si t, < t < si t, < t < 3 We are lookig at the lie y = t ad bad 0 to. I comutig the double Fourier series of the doubly eriodic fuctio F we ca take the itegratio betwee ad 5 show by dashed lies i Fig., ad withi this bad the boudaries of the regio where F 0 are from si t to + si t t Fig.

4 The required itegral is + 0 si t si t e imt+y dydt which after some maiulatio ca be show to be equal to, m, = 0 i i i i 3 i, m = ±, = 0 6 0, m 0, ±, = 0, m eve, 0, eve +, m odd, 0, eve +, m eve, 0, odd, m odd, 0, odd where we aeal to [3], Chater II for elemetary roerties of the Bessel fuctios J of the first kid. The substitutig y = t we obtai m,= c m e im+t = + 3 si t + + m=,m eve, eve m=,m odd, eve m=,m eve, odd m=,m odd, odd, eve, odd J 0 J 0 cos mt si t+ + si mt cos t + cos mt si t+ si mt cos t+ J 0 si t + J 0 si t

5 If we comare this with the sigle Fourier series the the coefficiet of si t will be = , ad is give after some rearragemet of the terms by J + J J + J which, usig MAPLE, evaluates to o takig 000 terms. 3 Some derived series For coveiece we cosider the series without the term, ad take differet values of as follows: J + J J + J 3 If we comare this with the sie series the it follows easily that 3 takes the value for =, 6, 0,...At this stage it is iterestig to see how the series will evaluate for the itermediate values of. Now i view of the form of the above aalysis will ot give the sums for values of other tha ad r where r =,,... However we ca comute these usig MAPLE, which we do to resectively 00, 500 ad 000 terms, to obtai the table below. For examle, the MAPLE code that roduces the fourth colum other tha r for r =, 3,... is Sum BesselJ+,Pi*//, =..500 Sum BesselJ-,Pi*//, =..500 Sum BesselJ+,Pi*//, =..500+Sum BesselJ-,Pi*//, =..500; for from 3 by to 5 do rit,evalf% od; / 00 terms 500 terms 000 terms

6 It turs out that the covergece of these series is rather slow which makes the comutatios a little tedious. O examiatio of the above table of values ad icludig the results already obtaied aalytically it is aaret that the atter is J + J, = J + J =, = r, r =, 3,..., = r, r =,,... 0, = r Now by cosiderig a examle with a differet choice of referece ad carrier waveforms, resectively v ref t = si t ad the triagular wave t, 0 < t < v car t = t, < t < 3 3 t +, < t < t - -3 Fig. 3 we obtai the outut wave J + J =, = +, 5 If we traslate the carrier by to the right the the grah becomes

7 t - -3 Fig. ad this gives the outut wave J + J =, = 3, = r, r =,,..., = r, = r, r =, 3,... 6 Agai some of the values have bee obtaied aalytically, ad the remaiig oes either usig MAPLE or comarig with those obtaied earlier. We ca cosider the followig series related to that i : J + J + J + 3, = 3, =, = r = / eve =, = r 0, odd 0, = r If we subtract from 7 we have J + J =, = J 7, Now cosider a alteratig versio of the origial series the we exect from other aalysis

8 to obtai 0 for the eve terms + J + J + J + J =, = 0, 9 Thus the origial series leads to a variety of results ivolvig Bessel series. We have carried out the calculatios to 500 terms The loss of accuracy comared with takig 000 terms is offset by the cosiderable differece i comutatioal time aroud miutes comared with over hours for each value. Note that from 5 ad 9 + J + J = For this to work we must have comarig ad 0, odd J + which gives J +r ad the cosiderig the eve terms J +r J = +, =,, odd 0, eve J r = 0, r =,,... J r =, r =,,... r After scalig the argumets of the Bessel fuctios we also obtai J +r 3. Related MAPLE calculatios Let us write B, s = As we have idicated above, J + s J r =, r =,,... r B r, = 0, r =,,... J s 0

9 We carry out some MAPLE calculatios o B r, s to obtai 0, eve, s =, 5,... B, s =, odd, s =, 5,... ad the to evaluate B, s, s =,, 3 we use MAPLE takig 500 terms to obtai the secod colum i the table below s =. The third ad fourth colums are obtaied by takig resectively s = ad s = 3. Agai, the MAPLE code that roduces the first twety terms of the third colum is just Sum BesselJ*-+,Pi**-//*-, =..500 Sum BesselJ*--,Pi**-//*-, =..500; for from by to 0 do rit,evalf% od; s = s = s =

10 The atter for B, is clear, ad i fact we have obtaied exerimetally, = r, r =,,..., s = B, s = 0, r, r =,,..., s = 0, = r, r =,,..., s = 3 0, eve, s =, 5,..., odd, s =, 5,... It is ot at all obvious from the above table what the geeral exressios for B, ad B, 3 would be but, for examle, we ca show that for odd B, 3 = cos 5 6 which gives B, 3 = 3, B 3, 3 = 0, B 5, 3 = 3, B 7, 3 = 3, B 9, 3 = 0,... 0 Coclusio The above aroach leads to a variety of Bessel series ad their sums, may of which have bee verified directly by comarig sigle ad double Fourier series. Refereces [] Pleym, Harald, Samles of aimatios from teachig egieerig mathematics with MAPLE V R5, i Yag, W-C et al eds, Proceedigs of the Fourth Asia Techology Coferece i Mathematics ATCM 999, ATCM Ic, USA, 7-7, ATCM, 999. [] Pleym, Harald, Educatioal use of MAPLE i egieerig mathematics, i Yag, W-C et al eds, Proceedigs of the Sixth Asia Techology Coferece i Mathematics ATCM 00, ATCM Ic, USA, , ATCM, 00. [3] Watso, G. N., A treatise o the theory of Bessel fuctios, d ed., Cambridge Uiversity Press, 9.