I IEO World Congress Fundmentl nd pplied etrology September 611, 29, Lisbon, Portugl FST ND CCURTE ESUREENT OF THE RS VLUE OF NONCOHERENT SPLED SINE-WVE Dniel Beleg 1, Dominique Dllet 2 1 Fculty o Electronics nd Telecommunictions, Politenic University o Timişor, Timişor, Romni, emil: dniel.beleg@etc.upt.ro 2 University o Bordeux - ENSEIRB, IS Lbortory, 3345 Tlence Cedex, Frnce, emil: dominique.dllet@ims-bordeux.r bstrct In tis pper new estimtor or te mesurement o te rms vlue o noncoerent smpled e-wve is proposed. Te ey eture o tis estimtor is tt te e-wve oset, priori estimted, is removed rom te e-wve. ormul used to estimte te ewve oset is lso given in te pper. It s been proved by mens o computer simultions nd experimentl results tt by te proposed estimtor ccurte rms mesurements re obtined i te totl rmonic distortion (THD) o te ewve is smller tn or equl to 3 db. eywords: e-wve rms estimtion, noncoerent smpling mode, windowing. 1. INTRODUCTION In mny engineering pplictions, it is very importnt to now wit ig ccurcy te rms vlue o e-wve becuse it relted directly to its power. In digitl mesurements, te e-wve is oten smpled in noncoerent wy (noncoerent smpling mode). Dierent metods ve been proposed in scientiic literture to mesure te rms vlue o e-wve in tis cse. Tese metods cn be clssiied eiter in time-domin metods [1]-[2] or in requency-domin metods [3]-[8]. In [2] te rms vlue o noncoerent smpled e-wve is clculted by te ormul used in n C nlog electronic voltmeter. For tis purpose irst te rectiied men vlue o te ewve is clculted nd ten te result obtined is multiplied by te ctor orm or e-wve. To increse te mesurement ccurcy te e-wve is priori multiplied by te sequence o coe window. Te estimtor proposed in [2] is bsed on low complexity lgoritm nd it is te most simple to implement nd so, te ster. In [2] it s been proved by mens o computer simultions nd experimentl results tt or ree oset nd ig spectrl purity e-wve te mesurements obtined ug tis estimtor re reltively ig ccurte. Te e-wves oten encountered in prctice ve n oset nd spectrl purity not so ig. Unortuntely, tese situtions ve not been nlyzed in [2]. In tis pper irst te inluence o te oset nd rmonic components o e-wve on te ccurcy o te rms vlue mesurement obtined ug te estimtor proposed in [2] is nlyzed. From tis nlysis new estimtor or te mesurement o te rms vlue o ewve wit oset nd rmonic components is proposed. Te eiciency o tis estimtor is nlyzed by mens o computer simultions nd experimentl results s well. 2. PROPOSED ESTITOR FOR RS VLUE ESUREENT Let us consider e-wve x(t) o mplitude, requency (equl to 1/T), pse, nd oset d, wit rmonic components. Te t rmonic component is crcterized by teir mplitude, requency (equl to ), nd pse. It is ssumed tt te noise o x(t) is smller tn te oset nd rmonic components. Te signl x(t) is smpled t te requency s (equl to 1/T s ) nd smples re cquired. Tus, te ollowing discrete-time signl is obtined: xm 2 s m d 2 Te rtio between te requencies nd s is: s 2 m, s m,1,, 1 (1) J (2) were J nd re respectively te integer prt nd te rctionl prt o te number o cquired e-wve cycles, nd.5 <.5. Wen = te smpling process is coerent wit te input e-wve (coerent smpling mode). Conversely, te noncoerent smpling mode is crcterized by. Te ltter mode is very common in prcticl pplictions. Te estimtor proposed in [2] to mesure te rms vlue o x( ), rms, is given by: 1 1 m wm x m x w ˆ m m rms (3) NPSG NPSG
were w( ) is window sequence, is te orm ctor ( / 2 2 ), nd NPSG is te window normlized pe. signl gin NPSG w m / m 1 Te e-wve x( ) is multiplied by w( ) in order to reduce te bis o te rms vlue determined by te prt o signl period T t te end o (J + )T. By tis multipliction te signl x w (m) = x(m) w(m) is obtined. Te coe windows re used in te estimtor ˆ. It sould be noted tt or tese windows NPSG is equl to te irst window coeicient,. In [2] it s been proved by mens o computer simultions nd experimentl results tt or d = nd smll totl rmonic distortion (THD), rms is estimted wit reltive ig ccurcy by ˆ rms. In te ollowing te cse in wic d nd THD s n importnt vlue is investigted. Since te expression o ˆ rms contins te modulus o x w ( ) te reltive error o rms,, is smller tn upper limit, lim. For = tis upper limit is given by (see (.13) rom ppendix): 2 rms d lim (4) were is te teoreticl vlue o rms ( / 2 ) nd is te teoreticl vlue o rms vlue o te t rmonic component ( / 2 ). For te expression o te upper limit it is very diicult to obtin nd it will tereore not be derived. From (4) it is obvious tt te ccurcy o te rms mesurement increses s d nd ( = 2, 3,..., ) decrese. Tus, te best solution to obtin n ccurte rms mesurement is to remove te oset nd te rmonic components rom x( ). In order to remove te rmonic components it is necessrily to estimte ec rmonic component prmeters. Tis ts is too complex nd increses te complexity o te mesurement o rms nd tereore, it will not be mde. On te oter nd, d cn be estimted by: 3. SIULTION ND EPERIENTL RESULTS Te im o tis section is to determine te eiciency o te estimtor ~ rms by mens o computer simultion nd experimentl results. 3.1. Simultion results Fig. 1 sows te mximum o te modulus o, mx, s unction o nd d (Fig. 1), nd THD (Fig. 1b), nd d nd THD (Fig. 1c). () 1 xw m dˆ w m (5) NPSG NPSG were w () is te irst component (DC component) o te discrete Fourier trnsorm (DFT) o x w ( ). Tus, to increses te ccurcy o te rms mesurement, dˆ is priori removed rom x( ) nd rms is estimted by: ~ rms 1 xm dˆ wm m. (6) NPSG (c) Fig. 1. mx s unction o: () nd d, nd THD, (c) d nd THD.
Te modelled x( ) is crcterized by: = 1 V, J = 11 nd = 124. Te e-wve s te 2 nd nd 3 rd rmonic components o mplitude 2 nd 3 = 2 /2. Te pses o te e-wve nd rmonic components re uniormly distributed on [, 2) rd. Te Hnn window is used. In Figs. 1 nd 1b, vries in te rnge [.5,.5) wit step o 1/4 nd in Fig. 1c, =.2. In Figs. 1b nd 1c, THD vries in te rnge [8, 2] db wit step o 5 db nd in Fig. 1, THD = 5 db. In Figs. 1 nd 1c, d vries in te rnge [, 1] mv wit step o 5 mv, nd in Fig. 1b, d = 5 mv. Te signl is pplied to n idel digitizer wit 14-bit bipolr nlog-to-digitl converter (DC). Te ullscle rnge (FSR) o te DC is equl to 5 V. Tus, te quntiztion noise o te DC is te only noise wic ects te e-wve. Tis is modelled by uniorm dditive noise. For ec nd d (Fig. 1), nd THD (Fig. 1b), nd d nd THD (Fig. 1c), mx occurring during te pses vrition is retined. Ec time 1 runs re done. s expected, te vrition o d does not ect very muc te estimtion o rms - te greter mx obtined rom te results sown in Fig. 1 is equl to.8%, wic is smll error. On te oter nd, mx increses s te THD increses. Tus, te worst cse is obtined or THD = 2 db, wen mx is equl to 1.84% (in bot Figs. 1b nd 1c), wic is n cceptble error. However, it sould be noted tt or smller THD more ccurte mesurements o rms re obtined or exmple or THD = 3 db, te greter mx is equl to.3% (Fig. 1b), wic is reltive smll error. ny oter simultions were perormed or dierent vlues o (smller tn FSR/2) nd J (iger tn 5) nd in ll cses beviour lie te one depicted in Fig. 1 ws lwys observed. 3.2. Experimentl results Te results obtined ug te estimtor ~ rms re compred wit tose obtined ug te interpolted DFT (IpDFT) metod [5]-[8]. Te IpDFT metod provides very ccurte mesurements o te mplitude o e-wve (nd lso o te rms vlue). In bot cses te Hnn window is employed. For tis purpose grpicl interce s been lso implemented ug TLB. In tis grpicl interce te cquired e-wve nd its spectrum re sown. To compute te spectrum te 4-term minimum error energy window is employed [4]. Te rms vlues obtined ug te estimtor ~ rms nd te IpDFT metod re lso given. Te e-wves re obtined rom dierent signl genertors. Te cquisition system s 14-bit bipolr DC wit FSR equl to 6 V. Te smpling requency is equl to 48.77 Hz. Te e-wves re irst obtined rom te gilent 3322 signl genertor. Te e-wves prmeters re: mplitude 1.5 V, requency 1.2 Hz nd oset 1 mv. Te user s mnul speciies THD smller tn.4% t tis requency. number o 25 records re collected o = 124 smples ec. Fig. 2() sows, or ec record, te results obtined ug te estimtor ~ rms nd te IpDFT metod. Fig. 2 sows by mens o te grpicl interce te results obtined or te 13 t cquired e-wve. () Fig. 2. For te e-wves obtined rom gilent 3322 signl genertor: () Te vlues o te rms mesurements obtined ug te estimtor ~ rms nd te IpDFT metod or ec record, Te results obtined or te 13 t cquired e-wve. For ec record, te results obtined ug te estimtor ~ rms diers rom tose obtined ug te IpDFT metod beginning to te ourt digit ter te deciml point. Tus, rms is ccurtely mesured ug te estimtor ~ rms. Tis beviour is cieved becuse te cquired e-wves ve ig spectrl purity (see Fig. 2). oreover, te ccurcy o te proposed estimtor is investigted or e-wves wit importnt rmonic components. For tis purpose symmetric e-wves re cquired rom te TG315 signl genertor. Te mplitude o te signls is 1 V, te requency is 1 Hz nd te oset is 1 mv. Te symmetry ensures ig THD. number o 25 records re collected o = 124 smples ec. Fig. 3() sows, or ec record, te results obtined ug te estimtor ~ rms nd te IpDFT metod. Fig. 3 sows by
mens o te grpicl interce te results obtined or te 15 t cquired signl. oreover, te proposed estimtor is very simple to implement. Tereore, tis is well suited or rel-time mesurement o te rms vlue o discrete-time e-wve. PPENDI Determintion te limit o te reltive error o rms, lim Let us ssume tt te e-wve is coerently smpled, i.e. =. Te teoreticl rms vlue o te continul-time signl x w (t) is given by: () x t w t 2t 2t d wt. 2 (.1) ssuming tt w( ) is H-term coe window wit te coeicients i, i =, 1,, H 1. In te time-domin w( ) is deined s: H 1 2t wt cos 2. (.2) It sould be noticed tt or coe window w(t) = w(t) nd NPSG =. From (.1) te ollowing inequlity cn be estblised: 2 We ve: 2t wt 2t wt d wt. (.3) Fig. 3. For symmetric e-wves obtined rom te TG315 signl genertor: () Te vlues o te rms mesurements obtined ug te estimtor ~ rms nd te IpDFT metod or ec record, Te results obtined or te 15 t cquired signl. From te spectrum o te 15 t cquired signl presented in Fig. 3 it ollows tt ter te undmentl te igest spectrl line is equl to 21.7 db, wic is ig vlue. Even in tis cse te proposed estimtor provides reltive ccurte rms mesurements ce tey diers rom tose obtined ug te IpDFT metod beginning to te tird digit ter te deciml point. 4. CONCLUSION In tis pper n estimtor or mesurement te rms vlue o noncoerent smpled e-wve is proposed. Te signl used in te estimtion is te e-wve rom wic te oset, priori estimted, is removed. Te perormed simultions nd experimentl results conirm tt te proposed estimtor s reltive ig eiciency or ewve wit THD smller tn or equl to 3 db. H 1 H 1 1 2t wt 2t H 1 2t 2t 2t cos 2 2t cos 2 2t 2t cos 2. (.4) Te irst integrl rom te lst expression o (.4) is given by:
2. T / 2 2t 2J 2t (.5) For < te second integrl rom te lst expression o (.4) is given by: H 1 1 H 1 1 2J 1 ti1 i1 t2j ti t1 2t i 1 2t 2t 2t 2t cos 2 2t cos 2 2t cos 2 i were ti T, i 1, 2,, 2J. 2 ter some lgebr we obtin: H 1 1 T 1 1 2 2 2 J J 2t 2t cos 2 2t cos 2 2J 1 i cos 2 J i. (.6) (.7) It sould be noticed tt te sme result is cieved or < 2. Tus, rom (.4) nd (.7) it cn be estblised: 2 2 2t wt 2 NPSG. 2 2 (.8) By similr demonstrtion te ollowing equlity is cieved: 2 2 2t wt NPSG. (.9) Te lst integrl rom (.3) is given by: w t H 1 H 1 1 H 1 t cos 2 t cos 2 t cos 2 Ug (.8) - (.1), (.3) becomes: d 2 NPSG. NPSG (.1) (.11) were is te teoreticl vlue o rms ( / 2 ) nd is te teoreticl vlue o rms vlue o te t rmonic component ( / 2 ). From te bove expression it ollows tt te limit o te, lim, is given by: d lim NPSG (.12) 2 Bsed on (.12) te limit o te reltive error o ms, lim, cn be determined: lim d lim NPSG. (.13) 2 REFERENCES [1]. Novotny nd. Sedlce, RS vlue mesurement bsed on clssicl nd modiied digitl signl procesg lgoritms, esurement, vol. 41, no. 3, pp. 236-25, pril 28. [2] D. Beleg nd D. Dllet, esurement o te ewve rms vlue in noncoerent smpling mode, IEO-TC4 Symposium, vol. I, pp. 251-257, Isi, Romni, Sept. 27. [3] D. Petri, Frequency-domin testing o wveorm digitizers, IEEE Trns. Instrum. nd es., vol. 51, no. 3, pp. 445-453, June 22. [4] C. Oelli nd D. Petri, requency-domin procedure or ccurte rel-time signl prmeter mesurement, IEEE Trns. Instrum. nd es., vol. 39, pp. 363-368, pril 199. [5] D. C. Rie nd G.. Vincent, Use o te discrete Fourier trnsorm in te mesurement o requencies nd levels o tones, Bell Syst. Tec. J., vol. 49, pp. 197-228, 197. [6] T. Grnde, Interpoltion lgoritms or discrete Fourier trnsorms o weigted signls, IEEE Trns. Instrum. es., vol. I-32, no. 2, pp. 35 355, June 1983. [7] G. ndri,. Svino, nd. Trott, Windows nd interpoltion lgoritm to improve electricl mesurement ccurcy, IEEE Trns. Instrum. es., vol. 38, no. 4, pp. 856-863, ugust 1989. [8] D. Beleg nd D. Dllet, ultirequency signl nlysis by interpolted DFT metod wit mximum side lobe decy windows, esurement, vol. 42, no. 3, pp. 423-426, pril 29.