MASKING OF FERROMAGNETIC ELLIPTICAL SHELL IN TRANSVERSE MAGNETIC FIELD
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1 POZNAN UNVE RSTY OF TE HNOLOGY AADE M JOURNALS No 7 Electricl Engineering Kzimierz JAKUUK* Mirosł WOŁOSZYN* Peł ZMNY* MASKNG OF FERROMAGNET ELLPTAL SHELL N TRANSVERSE MAGNET FELD A ferromgnetic oject, locted in the Erth s mgnetic field, chnges its distriution. Through mesuring such disturnces, one cn locte the oject nd destroy it. To concel the oject, specil inding is plced inside its ferromgnetic shell, nd its tsk is to eliminte the disturnces in the distriution of the Erth s mgnetic field. A thin lled ellipticl shell, mde of ferromgnetic mteril, is emined s the oject model. There re coils, plced inside the shell, nd their tsk is to generte mgnetic field, hich is eliminting the effect, the shell is mking on the distriution of the Erth s mgnetic field in the surrounding re. Such procedure is clled mgnetic msking nd the inding used for this purpose is clled the msking inding. The possiility of uilding the msking indings for the ferromgnetic ellipticl shell, situted in trnsverse mgnetic field respectively to its mjor is, is lso emined. The solution of Mell s equtions, hich re descriing the mgnetic field distriution cused y the ferromgnetic shell presence in the Erth s mgnetic field, is found. Furthermore, the ility of selecting coils, hich re eliminting the perturtions of the mgnetic field outside the shell completely, is proven.. NTRODUTON Usully, ship hulls re constructed of ferromgnetic steel. The ferromgnetic hull cuses significnt disturnces in the Erth s mgnetic field distriution. These disturnces re detected t considerle distnce outside the ship nd they llo to recognize it. n order to void the influence of the ferromgnetic hull on the Erth s mgnetic field distriution, specilly shped indings re plced inside the hull. The mgnetic field, generted y these indings, reduces the chnges in the mgnetic field distriution outside the ship significntly. n this pper, one considers the ship hull model s n elongted ellipsoid shell of circulr cross section in the plne z const nd the shell thickness δ (Fig. ). Due to the fct tht the Erth s mgnetic field flu does not eceed vlue of out 5μT, it is ssumed tht the shell mteril hs the liner mgnetiztion chrcteristics. The doption of the liner mgnetiztion chrcteristics llos * Gdńsk University of Technology.
2 Kzimierz Jkuiuk, Mirosł Wołoszyn, Peł Zimny considering the msking inding for ech of the three components of the Erth s mgnetic flu seprtely. The clcultion of the msking coil for the component in the z is (Fig. ) is shon in pper []. This pper presents clcultion of msking indings for the field directed long the is (Fig. ).. MATHEMATAL MODEL One uses system of ellipticl coordintes, θ, φ, hich re relted to the rectngulr coordintes, y, z ssocited ith the ship (Fig. ) ith the folloing dependencies: sinθcos y sinθsin () z cosθ Fig.. The ship model nd its on rectngulr coordintes system, y, z nd the ellipticl ones θ, φ here the constnt is given y the formul: L H () nd L nd H re the length nd idth of the ship model respectively. The outer shell surfce is defined y the constnt ellipticl coordinte : L H.5ln () L H Mintining constnt shell thickness δ is not possile in the sme elliptic coordintes s re ssumed for the outer surfce. The shell thickness s ssumed s δ in the middle of the ship hull length for the coordinte, hich defines the loction of the inner shell surfce: Hδ Hδ ln () Assuming, tht the Erth s mgnetic flu in the spce surrounding the ship is uniform field directed long the is of the ship s on rectngulr coordinte, the Erth s flu hs folloing components in the ellipticl coordinte system:
3 Mking of ferromgnetic ellipticl shell in trnsverse mgnetic field θ cos sinθ sin θ coscosθ sin θ sin The mgnetic flu n in ech of the res is presented s the sum of the Erth s mgnetic flu nd fn, hereinfter referred s the perturtion mgnetic flu, hich is cused y the presence of the ferromgnetic shell: (6) n fn The components of the mgnetic flu in the ellipticl coordintes re determined y the reltionships (5). The mgnetic perturtion flu, hile ssuming liner mgnetiztion of the ellipticl ferromgnetic shell, fulfils the Mell s equtions in ech of the three sures: fn fn (7) here n the re inside the ellipsoid, n the inside of the ferromgnetic ll of the ellipsoid nd n the re outside the ellipsoid. The continuity of the norml mgnetic flu components occurs on the orders eteen prticulr res: (8) f f f f Wheres, in the cse of the condition for the continuity of the tngentil component of the mgnetic field, it is ssumed tht inside the ellipsoid is thin lyer of to msking indings represented s liner currents θ nd φ. f θ f f θ f fθ f f θ f ( ) ( ) θ ( ) ( ) coscosθ sin θ sin cos cosθ sin sin θ here μ the reltive ferromgnetic shell permeility, nd μ π -7 H/m the mgnetic permeility of vcuum. The inding represented y the liner current θ (θ,φ) is directed long the is θ, nd the second one, ith the liner current φ (θ,φ), is directed long the is φ. The presence of the msking indings cuses discontinuity of the mgnetic field tngentil components. Tking into ccount the liner mgnetiztion (5) (9)
4 Kzimierz Jkuiuk, Mirosł Wołoszyn, Peł Zimny chrcteristics, one otins the folloing oundry conditions for the tngentil components (9). Tking into ccount conditions (9), one ssumes the liner currents distriutions in the form of: cos cosθ θ θ sin () sin θ After tking into ccount equtions (), the oundry conditions for the tngentil components tke the form of: [ ( ) ] cos cosθ f θ fθ sin θ f f θ f f f θ f [ ( ) ] θ ( ) ( ) cos cosθ sin θ sin sin Due to the oundry conditions (), the mgnetic flu components re ssumed s follos: f k k(, θ) cos fθk θk( θ, ) cos f k(, θ) sin () here k,,. Sustituting the mgnetic flu given y the formuls () to the rottion equtions (7) recorded for the is, θ ellipticl coordinte system respectively, e otin: ( ) () sinθ k k sin θ () ( sinθ) θ k k sin θ θ () The mgnetic flu components epressed y (), () stisfy the third rottion eqution for φ is. Sustituting () nd () into the divergence eqution (7), e otin the eqution for the flu component φk in the form: ( sin θ) sin θ ( ) sinθ ( sinθ) θ (5) k k k Assuming the liner current θ in the form: θ (6) nd fter tking into ccount (), it cn e proven tht the oundry conditions () for the tngentil components of θ nd φ es re identicl nd they cn e replced y the oundry conditions for the surfce nd respectively: θ
5 Mking of ferromgnetic ellipticl shell in trnsverse mgnetic field 5 (7) Tking into ccount the oundry conditions (7) nd the limited vlues of mgnetic flu for i, the solution is dopted in the folloing form: > << < for for for (8) here: ctgh ln - the sphericl function of the generl type of the second kind [, ]. onstnts,, nd, y the oundry conditions (8) nd (7) nd tking into ccount (), is determined from the system of equtions: (9) here:,,, d d. d d Solving the system of eqution (9), e otin: [ ] () () here:. The condition, tht perturtion flu equls zero outside the ellipsoid, mens tht nd, hence, the liner current φ is: ()
6 6 Kzimierz Jkuiuk, Mirosł Wołoszyn, Peł Zimny nd the liner current θ from (6) tkes the vlue: ( ) ( ) θ (). RESULTS ANALYSS n order to investigte the prcticl fesiility of mking the inding msking ship, the size of the necessry liner currents defined y the reltions () nd () is defined. The vlue of Erth s mgnetic flu is ssumed s 5 μt nd the oject of moderte size, ith the length of L m nd the idth of H 6 m, is emined. n Figure one shos the clculted dependence of the line current on the ferromgnetic ship ll thickness, nd in Fig. - the line current dependence on the reltive permeility of the steel hull. The presented clcultions sho tht the mgnitude of the liner current is, prcticlly, liner function of oth: the shell thickness δ nd the reltive mgnetic permeility μ. The mplitude of the current θ, determined y the reltion (6), ill e greter ecuse is less thn one. Hoever, the required line current densities re in the rnge of ka/m for φ nd ka/m for θ. t is orth mentioning, tht the increse of the oject s geometric dimensions does not result in significnt mnner in the increse of the msking currents density. Fig.. The line current φ in A/m s function of the shell thickness δ in mi () nd s function of the reltive mgnetic permeility μ () REFERENES [] Jkuiuk K., Zimny P., Wołoszyn M.: Mskonie oiektu ksztłcie elipsoidy ziemskim polu mgnetycznym. nterntionl onference on Fundmentls of Electrotechnics nd ircuit Theory. Gliice-Ustroń, pp.5-6,. [] Leiedie N.N.: Funkcje specjlne i ich zstosoni. PWN, Wrsz 957. [] Hoson E.W.: The Theory of Sphericl nd Ellipsoidl Hrmonics. mridge t the University Press 9.
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