MAGNETIC FIELD AROUND TWO SEPARATED MAGNETIZING COILS

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1 The 8 th Intenational Confeence of the Slovenian Society fo Non-Destuctive Testing»pplication of Contempoay Non-Destuctive Testing in Engineeing«Septembe 1-3, 5, Potoož, Slovenia, pp MGNETIC FIELD OUND TWO SEPTED MGNETIING COILS B. Buda Beg 146, 474 Žiovnica, Slovenia, dako.buda@pompt.si BSTCT If the aangement of coils with the eddy cuents testing is not a standad one it is vey impotant to undestand the physics of the new aangement to be able to intepet the esults popely. vey useful tool is the mathematical teatment of field equations. The numeical methods ae often applied, since Maxwell equations ae athe complex patial diffeential equations and thee ae usually eal and imaginay components of the field to be taken into account. In the aticle thee ae basic algoithms given that ae used with the method of finite diffeences. Thee ae only main ideas given how to solve moe complicated poblems. The actual concete esults ae pat of a boade poject and will be published late. Keywods: Numeical solution of patial diffeential equations 1. Intoduction In ode to be able to detect longitudinal as well as pependicula suface cacks in feomagnetic bas of cicula coss-section it is necessay to make a special constuction of magnetizing and seconday coils [1]. Fig. 1: Pimay and seconday coils. 17

2 The magnetizing coil is made of two equal pats being sepaated longitudinally. The seconday coil is detecting adial magnetic flux and is placed in the middle of the pimay coil accoding to Fig. 1. It is not possible to speak about homogenous magnetic field. It is vey impotant to define the pope optimal distance (W) between both pats in ode to make the whole system sensitive to both types of cacks. The magnetic field in the neighbohood of all thee coils can be calculated assuming some idealization of coils. Compute simulation of diffeent positions and dimensions helps a lot with constuction of actual aangement. It is shown in due text how it is possible to assess the necessay sepaation in the pimay coil and how it is possible to assess the induced voltage in the seconday coil influenced by moving of a defective ba though the whole aangement.. Maxwell equations fo the magnetic field Fo the case of a feomagnetic ba with the cicula coss-section it is vey convenient to stat with the calculation of vecto potential in cylindical coodinate system []. V v v V V σ. µ.µ. µ. µ. ε. ε. = (1) t t v v i t By intoducing new vaiable: V = ω W. e, we can have the eal and the imaginay component of v v v the vecto potential : W = + i.. It is possible to wite two sepaated equations fo the eal and fo the imaginay component. v v + ω. µ. µ. σ. = () v ω µ. µ. σ. v = (3). The vectos and have geneally thee components in space and so we have a system of 6 patial diffeential equations to solve, fo each space component and fo the eal and imaginay pat. To illustate the pocedue let us limit to the system of two dimensions. Namely the magnetizing coil has the fom of a cylinde and if we choose the souce of the coodinate system in the axis of the coil, the poblem can be much simplified. If the poblem is otationally symmetical, only one component = is diffeent fom zeo. ϕ The following pai of equations is to be solved in cylindical coodinate system assuming that the poblem is otational symmetical: F. 1 = (4) F. = (5) whee F = ω. µ. µ. σ a.. 18

3 On the othe hand it is possible to wite the coesponding expession fo the two components of the magnetic field density B : B = B = B z 1 =. (. ) 1 B = z.(. ).1 Bounday conditions Fo the mesh points that ae lying on the bounday between the ai and the mateial, the basic bounday condition must be fulfilled. One must keep in mind that when cossing the bounday the nomal component of the magnetic field density ( B ) must be peseved. On the othe hand the tangential component of the magnetic field stength ( H ) must be peseved as well. It is pactically impossible to solve the system of equations geneally. Fo some special simplified cases it is possible to find maybe even analytic solution, but much moe often it is necessay to use some numeical methods. Thee ae seveal algoithms available but it depends on the expeience of the eseach woke which method should be used. It is not necessay to calculate the vecto potential to some geat pecision. moe o less ough assessment is usually good enough.. Method of finite diffeences We have solved this poblem by the method of finite diffeences. The coodinate system was chosen as shown on Fig.. Instead of looking fo the geneal solution fo the unknown vecto potential we wish to find the solution in discete mesh points as shown in Fig.. J I W Fig. : The mesh points whee the vecto potential will be calculated. Fo each mesh point ( a linea numeical expession coesponding to the patial deivatives fom Equation 4 o Equation 5 can be witten. Fo example: Instead of Equation 1 and Equation the following linea expession can be witten fo the point ( : 19

4 ( ( J ) + ( J + 1) 1 ( J + 1) ( + ( ( ( I 1, ( + ( I + 1, F = ( ( ) ( ( J ) + ( J + 1) 1 ( J + 1) ( + ( ( J ) ( ( ) ( I 1, J ) ( J ) + ( I + 1, J ) + F = (6) (7) In Equations 6 and 7 means the mesh distance in adial diection and in longitudinal diection. If thee ae IKON mesh points chosen in the diection I and JKON mesh points in the diection J, it is necessay to find the solution to IKONJKON linea equations with the same numbe of unknowns. Fom the solutions in discete mesh points it is also possible to calculate the values of the eal and the imaginay component of magnetic field density. The poblems how to wite the coesponding numeical diffeence equation in the cones and on the lines of symmety can be avoided by application of commonly used algoithms in methods of finite diffeences [3]. It is also convenient to use unequal spaced mesh. Fa fom the coils whee nothing is being changed any moe, the logaithmic mesh is vey often applied. lso the mesh points inside the mateial ae sometimes dense close to inteesting spots. ll these modifications of the mesh epesent some mino additional difficulty and some mathematical expeience is needed..3 Explanations of symbols used in Equations 1-7, eal and imaginay components of the amplitude of vecto potential a adius of the ba B, eal components of the magnetic field density B z, Bz B imaginay components of the magnetic field density F dimensionless fequency adius t time V vecto potential W amplitude of vecto potential z coodinate z σ electic conductivity µ pemeability of empty space µ elative pemeability ω = π f fequency 3. adial field between the two pats of the pimay coil The aangement of coils was simulated accoding to Fig and the distance W between two pats was vaied. Thee wee two equal pats of magnetizing coil simulated with a feomagnetic od in the middle. The adial component of the magnetic field stongly depends on this distance and on the fequency and on the gap between the seconday coil and the suface of the ba. The elative pemeability of the feomagnetic ba in the middle is also of decisive impotance. ll

5 this data togethe ae giving the necessay infomation to calculate the distibution of the magnetic field in the vicinity of defective spot. Fom the compute calculations also the adial magnetic flux could be evaluated. On basis of these simulations we could constuct a vey sensitive appaatus fo detection of suface cacks of both kinds on a feomagnetic ba. The most impotant issue is that the sepaation of the pimay coils must not be too small. It must be big enough to bing the field fom inside of the ba acoss the suface to the outside whee the adial seconday coil can catch the flow lines emeging fom the inteio. Since both pats of the pimay coil ae as equal as possible the seconday coil acts as the diffeential aangement of a pai of seconday coils. The only diffeence is that in this case thee ae not two induced voltages subtacted but the two pats of magnetic field ae flowing in opposite diection. It is inteesting to simulate diffeent geomety and diffeent physical popeties on the distibution of the magnetic field at diffeent fequencies. The esults ae pat of a poject whee so called BUD egion aound coils mentioned above will be investigated thooughly. 3.1 adial magnetic field at the points J=JTUL between two pats of magnetizing coil We solved a simple case and hee only main final conclusions ae given. Only one fouth of the whole coss-section is taken into account due to the symmety accoding to Fig.. The length of pat of magnetizing coil is chosen is 6. The adius of the od a= 6, the adius of the coil is 9, whee and can be chosen delibeately. The numbe of points in adial diection JKON= 3, the numbe of points in longitudinal diection IKON= 3, the points on the suface of the od in adial diection JK= 7, the points on the suface of the coil in adial diection JTUL=1. Pactically we chose the diamete of the ba 7 mm, the length of one pat ot the pimay coil was 5 mm long and the seconday coil had mm diamete. t the fequency of 5-7 khz the aangement of coils was extemely sensitive fo the longitudinal and pependicula suface cacks of the ba. The calculations of the adial component of the magnetic field wee done also fo the moving od though the coils beaing an unifom adial longitudinal suface cack with a good defined stat and good defined end. ll these calculations ae pat of a moe complex poject that is not yet complete. 4. Conclusions The mathematical methods fo solving Maxwell equations ae an excellent tool to veify new ideas and when looking fo new possibilities. The emeging field fom the inside of the nonfeomagnetic ba can help detecting pependicula cacks since the eddy cuents cannot flow so easy if the field tuns to adial diection. In this case namely a pependicula cack epesents some hindance to eddy cuents flowing in axial diection.. Using a pesonal compute it is possible to simulate vaious cases of cicula symmety also tubes, combinations of feomagnetic and non-feomagnetic mateials. ll this gives much bette undestanding of NDT method itself and epesents a new tool fo futhe investigations. 5. efeences [1] Buda B.: Magnetic Leakage Fields as Indicatos of Eddy Cuent Testing, The 8 th Intenational Confeence of the Slovenian Society fo Non-Destuctive Testing, Septembe 1-3, 5, Potoož, Slovenia. [] Buda B.: The Electomagnetic Field in the Neighbohood of a Defect in a Mateial, eseach Techniques in NDT, Vol. V, cademic Pess, London, 198, [3] Smith G. D.: Numeical Solution of Patial Diffeential Equations, Oxfod Univesity Pess, Oxfod,