Computaton of Low-Fequeny Eet Feds n Anayss of Eetomagnet Fed Exposue Žejo Šth, Bojan Tuja, Sead Bebeovć Fauty of Eeta Engneeng and Computng Unvesty of Zageb Unsa 3, Zageb, Coata phone:+385 69 865, fax: +385 69 66, e-ma: zejo.sth@fe.h, bojan.tuja@fe.h, sead.bebeov@fe.h Abstat A method fo numea omputaton of ow-fequeny eet fed nea powe appaatus and systems based upon the method of moments and the thn-we appoxmaton s desbed n ths pape. Man featue of the method s a smooth appoxmaton of unnown ne hage denstes, whh s enabed by appaton of ub spnes. The appabty of the method s ustated by the anayss of the oupatona and genea pub eet fed exposue n substatons and n the vnty of powe nes. Keywods Eet Fed, Intega Equatons, Method of Moments, Cub Spnes, Oupatona Exposue. Intoduton The anayss of the exposues to ow-fequeny eet fed s nowadays an ntega pat of desgn n the aea of powe appaatus and systems. Natona standads defne maxma vaues of eet fed stength fo oupatona and genea pub exposue to suh a fed and desgn souton must be heed wth espet to that tea. In the anayss of oupatona and genea pub eetomagnet fed exposue nea powe appaatus and systems we ae nteested n the steady-state tmehamon eetomagnet feds at the fequeny of 5-6 Hz. At these fequenes the eetomagnet fed s a quas-stat fed and effet of eet and magnet feds an be anayzed sepaatey []. We study feds n the aeas whee humans may stay fo onge peods (oupatona peod of stay s eght hous pe day and genea pub peod of stay s hous pe day). Thus, we ae nteested ony n feds that ae fa away fom soues and theefoe we may mode soues (hages) as beng dstbuted ove one-dmensona nes (thn-wes). We neget nsuatos beause they nfuene the eet fed ony n a nea egon. The most effetve souton method fo suh nea and unbounded pobems s the appaton of the method of moments that s based upon the bounday ntega fomuaton fo nown saa eet potenta of ondutve pats. Geomety of ondutve eements of powe appaatus and systems s appoxmated by staght nes and seond-ode uves. Unnown dstbuton of hage densty n the eet fed omputaton s appoxmated by ub spnes, and the oeffents of the dstbuton ae detemned fom nown potentas by pont mathng. The appaton of desbed omputatona appoah s ustated by the anayss of the oupatona eetomagnet fed exposue n a / V substaton and on the nteseton of V and V tansmsson nes at the fequeny of 5 Hz. Computed feds ae ompaed to the aowed vaues of eet and magnet feds that ae defned by the Coatan egsaton.. Numea auaton In omputaton of eet fed we dea ony wth ondutos at nown potentas. The eath s assumed to be a pefet onduto and we tae nto the onsdeaton the eath nfuene by the method of mages. The phaso of the saa eet potenta & ϕ( ) at any pont on the tota sufae of ondutos s eated to the phaso of ne hage densty & λ ( ) at any pont on a thn-we eements (ogna and mage ne hages) by the equaton [] & λ( ) & ϕ( ) = () π In ode to detemne the unnown funton & λ ( ), the method of moments s apped. The thn-wes ae dvded nto the fnte segments ( =,... NS) that may be staght nes and ua as. On the th fnte segment we expess & λ by N B bass funtons t as N B & λ = K t = ()
A ub dstbuton (N B = ) s assumed fo λ & on eah segment and the dependene of the bass funtons t upon the dmensoness paamete s ( s ) s j t ajs j= = ; =,,3, In ode to defne the bass funtons t we use the vaues of the ne hage densty and the devatves of the ne hage densty at the begnnng (s = ) and at the end (s = ) of the segment. The ne hage densty on eah segment s defned by & λ () s = + s + 3s + s () Thus, the vaues and the devatves at the begnnng and at the end ae: & λ ( ) = ; & λ ( ) = (5) & λ () = + + 3 + ; & λ () = + 3 + 3 The oeffents (=,,3,) ae 3 = & λ = 3 = λ ( ) ; = & λ ( ) & λ ( ) & λ ( ) + 3 & λ ( ) & λ ( ) & ( ) + & λ ( ) & λ ( ) + & λ ( ) = The substtuton of the oeffents (=,,3,) nto () esuts n & λ ( s) = & λ ( )( 3s + s ) + & λ ( )( s s + s ) + (7) + & λ ()( 3s s ) + & λ ()( s + s ) Aodngy, the oeffents a j n (3) ae ontaned n the matx 3 [ A] (8) 3 The oeffent K n () s the vaue of the ne hage densty at the begnnng of the -th segment, K s the devatve of the ne hage densty at the begnnng of the -th segment, K 3 s the vaue of the ne hage densty at the end of the -th segment, K s the devatve of the ne hage densty at the end of the -th segment. The substtuton of () nto () esuts n the nea ntega equaton t ( ) t( ) & ϕ( ) = { L + L } π (9) NS +, = = π = We deve a nea equaton system fo unnown ompex oeffents L by pont mathng at the ponts defned by s = and s = on eah segment. The ne 3 3 ntegas n (9) ae omputed numeay usng a gobay adaptve sheme based on Gauss-Konod ues. The souton of the system detemnes the dffeentabe appoxmaton of the ne hage & λ on eah segment. Aftewads, by the usage of the expesson (3) (6) & t ( )( ) E ( ) = { + + NS L 3 = = π ( )( ) L+, 3 = 3 π T e x y z T Ex,max + Ey,max + Ez,max t } E = E () t + E () t + E () t dt = () we may auate the veto-phaso of the eet fed stength at any pont. Suh vetos ae eptay poazed and the magntude vaes wth espet to tme. We use the effetve vaue of the magntude () = as an equvaent vaue of eptay poazed fed vetos n studes of eetomagnet fed exposue. 3. Anayss of Eet Fed Exposue The Intenatona Commsson on Non-Ionzng Radaton Poteton (ICNIRP) ssued the Gudenes [3] fo mtng the eetomagnet fed exposue up to 3 GHz. The anayss of the eet fed exposue may be dvded nto two pats: A. Anayss of the oupatona exposue The anayss of the oupatona exposue deas wth heathy aduts who ae awae of the s and who ae ey to be subjet to meda suveane. The eommended oupatona efeene eve fo the fequeny of 5 Hz s E ef = V/m. The aepted mt vaue fo oupatona exposue n Coata s E = 5 V/m. B. Anayss of the genea popuaton exposue The anayss of the oupatona exposue has to be based on boade onsdeatons suh as heath status, envonmenta ondtons, spea senstvtes, possbe effets on the ouse of vaous dseases, as we as mtatons n adaptaton to envonmenta ondtons and esponses to any nd of stess n od age. The nfuene of these onsdeatons s nsuffenty expoed, and the mts fo genea popuaton exposue have to be onsdeaby smae. The eommended genea pub efeene eve fo the fequeny of 5 Hz s E ef = 5 V/m. The aepted mt vaue fo genea pub exposue n Coata s E = V/m. As an exampe of the appaton of the desbed poedues we anayze the oupatona exposue n a V pat of a substaton that onssts of fve V nes and two powe tansfomes. The mode of the substaton s shown n Fg..
z (m) 3 y (m) x (m) Fgue. Mode of the V pat of a substaton The omputaton of the eet fed s defned by nown potentas of wes, whh ae equa to phase-to-gound votages of symmeta thee-phase system. As the mt vaue of oupatona exposue s defned n the ase of a homogeneous fed, we have to auate the mean vaue of the effetve vaue of the magntude of the eet fed between the gound and the heght of m. The esuts ae shown n Fg.. y (m) 5 C3 C C C C9 C8 C6 C5 C3 C C C 5 75 5 5-5 E (V/m) >. <. <. <. < 9. < 8. < 7. < 6. < 5. 5 5 5 3 Fgue. Aeas of oupatona stay
Anothe exampe taen nto onsdeaton s the nteseton of V and V tansmsson nes. Effetve vaue of eet fed s auated on etangua pane m above gound (Fg.3.) Resuts of the auaton ae shown n Fg.. Maxma vaues of eet fed stength s E eff,max = 33 V/m, theefoe anayss has to be done fo genea pub exposue. V tansmsson ne Hozonta pane m above gound V tansmsson ne z[m] 5 5 5 5 5 3-5 - -5 - Fgue 3. Inteseton of tansmsson nes E[V/m] 5 5-5 5 - -5-5 -5 Fgue. Eet fed stength m above gound
Fna esut of the anayss s the detemnaton of the aea n whh the genea pub exposue n duaton of hous pe day s fobdden. Ths s the aea whee effetve vaue of the eet fed stength exeeds V/m. Ths aea s shown n Fgue 5. 5 m. Conuson A numea method fo omputaton of ow-fequeny eet fed nea powe appaatus and systems that may be used fo anayss of oupatona and genea pub fed exposue has been poposed. The omputaton s based on the numea souton of the bounday ntega equatons by pont mathng tehnque and may be easy apped to any ompex thee-dmensona geomety. -5 - Refeenes [] Haznada, Z. and Šth, Ž. Eetomagnet Feds, Waves and Numea Methods, IOS Pess, Amstedam, pp 359-388 [] Šth, Ž. "Hgh votage nsuatng system desgn by appaton of eetode and nsuato ontou optmzaton IEEE Tans. on Eeta Insuaton, vo., pp 579-58, Aug. 986. [3] ICNIRP: Gudenes fo mtng exposue to tme-vayng eet, magnet, and eetomagnet feds (up to 3 Ghz), Heath Phys. 75(), pp 9-5, 998. -5-5 5 5 m Fgue 5. Aea n whh s genea pub exposue fobdden