ATEE 4 ENERGY DSTORTON COEFFCENTS OF CRCULAR CROSS SECTON BMETALLC CONDUCTORS UNDER ERODC NON SNUSODAL CONDTONS Soin. ANTONU Electicl Engineeing Det. OLTEHNCA Univesity Buchest The enegy distotion coefficients of cicul coss section bietllic conductos e couted unde eiodic non sinusoidl conditions. The oeties of the distotion coefficients long with thei nge of vition e discussed in eltion with the conducto diensions.. NTRODUCTON The enegy distotion coefficients (THD Totl Distotion Coefficients of owe) s defined by A. Ţugule [] give globl diensionless evlution of the weight of honic coonents ( ) nd diect cuent coonent in the ctive owe loss in line solid conductos oeting unde eiodic non sinusoidl conditions with esect to eithe the totl ctive owe (THD ) o the fundentl coonent of the ctive owe (THD ) THD THD. () The ssocited foule e exessed in tes of the honic esistnce incese coefficients of solid conductos R nd the sque of the honic distotion coefficients δ of the eiodic cuent whee THD δ THD THD R δ () THD R ( ω) R δ R... (3) R n the cse whee the fist honic coonent of the cuent is bsent (double ltennce ectified cuents fo instnce) the enegy distotion coefficients e coesondingly edefined [] tking s efeence the diect cuent coonent of the ctive owe nd cuent esectively
ATEE 4 δ R THD THD THD THD (4) whee... 3 δ (5). HARMONC RESSTANCE NCREASE COEFFCENTS OF THE CRCULAR CROSS SECTON BMETALLC CONDUCTORS The odel unde considetion is vey long cylindicl solid conducto of dius suounded by coxil hollow cylinde of inne dius nd oute dius b lced in n insulting ediu. The coesonding conducting doins D nd D nd the insulting doin D 3 e suosed to be line nd hoogeneous with constitutive etes s indicted in fig.. Fig.. Coss section of the bietllic conducto The consideed conducto cies eiodic non sinusoidl cuent of intensity i(t) with etun th lced t n infinite distnce. The solution of the electognetic field diffusion oble in the conducto [] yields the exessions of the honic esistnce incese coefficients s ( ) N M k R Re (6) whee λ λ N M (7)
ATEE 4 k λ b σ µ λ σ µ σ σ µ µ. (8) n the bove foule n nd n e the odified Bessel functions of the fist nd second kind of intege ode nd colex guent the colex ogtion constnt in the oute conducto coesonding to the th honic of the cuent is ω µ σ j ω µ σ ( j) α ( j) (9) α is the eciocl of the coesonding enettion deth fo the fist honic of the cuent. 3. ENERGY DSTORTON COEFFCENTS OF BMETALLC CONDUCTORS Enegy distotion coefficients e clculted fo steel luiniu bietllic conductos of tye A/SA [3] used in the ovehed lines fo electic owe tnsot. Thei coe is de of one o oe concentic stnded steel wies nd thei oute lye is de of siil concentic stnded luinu wies. The theoeticl odel esented bove cn be used fo the study of such stnded conductos within the liits of the following oxitions: () the constitutive etes e tken s sufce vege vlues µ Sv d NS S µ S µ Sv d NS S σ S σ Al v NAl 4 d Al σ Al ( b ) whee N S nd N Al e the nube of nd d S nd d Al e the dietes of the steel esectively luinu wies nd µ S µ µ Al µ σ S 7 6 S/ σ Al 34 6 S/; (b) the steel coe is gneticlly line s the couttion of the cuent distibution in the coe confis the fct tht it eins unde 3% of the totl cuent so tht the level of the gnetic field stength is lced in the line t of the steel gnetiztion cuve. The enegy distotion coefficients e couted fo ll the 37 tyes of steel luiniu conductos in the selected set tking into ccount honic coonents of six tyicl non sinusoidl cuent wvefos syetic ltenting ectngul cuent wvefo (DS) ulsting ectngul cuent wvefo (D) syetic ltenting tingul cuent wvefo (TS) ulsting tingul cuent wvefo (T) single ltennce ectified sinusoidl cuent wvefo (RM) double ltennce ectified sinusoidl cuent wvefo (RB) nd two exeientl ltenting cuent wvefos E nd E []. The vition nge of the enegy distotion coefficients of the consideed set of bietllic conductos e esented in Tble fo the consideed cuent wvefos long with tht of the cuent distotion coefficients (THD Totl Honic Distotion)
ATEE 4 THD THD () edefined by A. Tugule [4] fo the cse whee thee cuent wvefo hs no fundentl coonent s THD THD. () Cuent Cuent distotions Rnge of enegy distotion coefficients wvefo THD THD THDin THDx THDin THDx DS 43536 48346 7498 749 986 377447 D 7778 36 58799 675 458 547 TS 73 53 444 8763 488 3656 T 868 7489 73364 7533 75434 34879 RM 777 4995 499636 9648 998544 RB* 43536 48346 9938 4348 35999 3734 E 3546 333 947 35 3774 5684 E 7977 443 538 63489 973 7389 * - coefficients couted ccoding to eltions () (4) Tble. Vition nge of THD nd THD coefficients 4. CONCLUSONS An nlysis of the couted vlues of the enegy distotion coefficients of bietllic conductos cying eiodic non-sinusoidl cuents esults in the following eks: () The vlues of the THD coefficients deend on the conducto stuctue nd diensions nd the cuent wvefo. (b) The onotonous deendence of the THD nd THD coefficients on the couttion ete α α Al b is etubed by sll oscilltions. This cn be exlined by the fct tht the thickness of the luinu in conducting lye b ( ) b ( ) is not in constnt tio with the oute dius b of the bietllic conducto. (c) The deendence of the THD coefficient on the couttion ete α is oe onounced when the honic coonents of the eiodic ltenting cuent e invesely ootionl to thei ode (the cse with DS wvefos) ~ thn when these honic coonents e invesely ootionl to the sque of thei ode (the cse with TS wvefos) ~. n tun the deendence of the THD coefficient on the couttion ete α is less onounced in the cse of invese ootionlity of honic coonents on thei
ATEE 4 ode thn in the cse of invese squed deendence on thei ode when diect cuent coonent is esent (D vesus T). This tend is illustted in figs. nd 3 fo the eltive vlues THD ( α ) THD ( α ) THD (.5) THD (.5) whee the bietllic conducto with couttion ete α.5 ws tken s efeence. The discussion does not ly to the exeientl cuent wvefos E nd E which lck coison efeence no to the double ltennce cuent wvefo RB fo which diffeent foule e used. Fig.. Reltive vition of coefficients THD fo diffeent cuent wvefos Fig. 3. Reltive vition of coefficients THD fo diffeent cuent wvefos
ATEE 4 (d) The esence of diect cuent coonent s it is the cse with the ulsting D T RM wvefos is ssocited with less onounced deendence of the THD nd THD coefficients on the couttion ete α s coed with tht esented by the ltenting DS TS E E wvefos (see lso figs nd 3). (e) The lst two eks e vlid s well fo othe stuctues of solid conductos [567]. (f) The vition nges of the enegy distotion coefficients suggest distinct vlues fo these coefficients nd even significnt detues fo the coesonding distotion coefficients fo the se tye of cuent wvefo. This obsevtion sustins the conclusion tht the enegy distotion coefficients THD do indeed bing dditionl infotion on the chcteiztion of the enegy tnsfe unde eiodic non-sinusoidl conditions which would not be vilble if the globl distotion coefficients THD only would be used. (g) t is hoed tht the cuent use of the globl distotion coefficient THD fo cuents nd voltges will be bndoned in fvo of o t lest coleented by the use of enegy distotion coefficient THD which is ble to exess oe dequtely the weight of the honic coonents of cuent nd voltge in the ctive owe loss s coed to the coesonding fundentl honic ctive owe loss. ACNOWLEDGEMENTS The logistic suot offeed by the Nueicl Methods Lbotoy te of the Electicl Engineeing Detent of the olytechnic Univesity Buchest nely of. C.D. on ssoc. of. Gbiel Ciuin ssist. of. M. ie nd coute ssistnt G. on is gtefully cknowledged. REFERENCES. A. Ţugule Ae totl honic distotion fcto the ight ete fo the enegetic effects estition unde non-sinusoidl conditions? Rev. Rou. Sci. Techn. Électotechn. et Éneg. 43 4. 49 496 (998).. S. Antoniu Fenoene defonte în conductoe sive h.d. Thesis olytechnic Univesity Buchest 4. 3. SR CE 89 RS Stndd oân: Conductoe entu linii eiene cu sâe otunde cblte în sttui concentice 4. A Ţugule Este fctoul de distosiune un fcto otivit entu estie efectelo enegetice în egi defont? Annul Confeence of the Electicl Engineeing Detent olytechnic Univesity Buchest 9 Ail 996. 5. S. Antoniu Enegy distotion coefficients fo ectngul coss section busbs cying eiodic non sinusoidl cuents Theoeticl Electicl Engineeing Ntionl Woksho 3 Octobe 4 olytechnic Univesity of Buchest. 6. S. Antoniu Enegy distotion coefficients fo cicul coss section conductos cying eiodic non sinusoidl cuents Theoeticl Electicl Engineeing Ntionl Woksho 3 Octobe 4 olytechnic Univesity of Buchest. 7. 7. S. Antoniu Enegy Distotion Coefficients of Coxil Cbles Unde eiodic Non sinusoidl Conditions ATEE 4 olytechnic Univesity of Buchest Novebe 4.