ANALYTICAL SOLUTIONS TO THE PROBLEM OF EDDY CURRENT PROBES
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1 ANALYTICAL SOLUTIONS TO THE PROBLEM OF EDDY CURRENT PROBES CONSISTING OF LONG PARALLEL CONDUCTORS B. de Halleux, O. Lesage, C. Mertes and A. Ptchelintsev Mechanical Engineering Department Cathlic University f Luvain Belgium, B-1348 INTRODUCTION Eddy current testing is currently used t determine cnductive specimen physical characteristics and t detect defects by measurements f electrical impedance f an eddy current prbe. In general, reliable quantitative NDE requires accurate measurements and a thery t interpret them. In sme NDE applicatins related t cated metals inspectin eddy current prbes which induce planar r linear currents in a sample are used fr flaw detectin, thickness and cnductivity measurements [1]. Fr prbes f this type, e.g. rectangular r meander like cils, there still exists the need f theretical mdeling. In this study we develp a theretical apprach fr a prbe cnsisting f tw infinite length straight parallel cnductrs. The vectr ptential expressin is btained in the case when the cnductrs are parallel t and equidistant frm the plane bundary surface f the cnductive bject. The cnductrs underg equal currents with ppsite signs. Curves f electrical impedance per unit length f the prbe placed ver a layered half space are calculated numerically fr ferrmagnetic and diamagnetic materials. The electrical impedance experimental values btained n tw finite length prbes are in gd agreement with presented theretical curves. The mdel makes accurate eddy current measurements pssible using this type f prbe. Theretical predictins fr eddy current prbes cmpsed f several lng straight cnductrs parallel t but nn equidistant frm plane bundary surface f cnductive bject als becme pssible. THEORETICAL RELATIONS The gemetry f the prblem is shwn in Figure 1. Tw parallel infinite length straight wires are placed parallel t and equidistantly frm a layered cnductive half space. The wires are separated by the distance 2A. underg equal harmnic currents I f ppsite signs. The lift-ff f the system f wires is I, the cnductive layer thickness is m. The electrical prperties f regins 3 and 4 are 113,03 and 114,04 respectively, cnductivity 01 =02 is zer and relative permeability III = 112 is unity. All media are istrpic and hmgeneus. It is f interest t btain the vectr ptential expressin A(x,y,z,t) and Review f Prgress in Quantitative Nndestructive Evaluatin, Vl. 15 Edited by D.O. Thmpsn and D.E. Chimenti. Plenum Press, New Yrk,
2 -/ / regin I 1-'.0'.::0 I-' (J = ~*--r ~--~----~ 2 2 ~~~~~~~~~~~~~~~ ~O' Figure I. Tw infinite parallel wires placed ver cated cnductive half space. cnsequently detennine the electrical impedance per unit length. The diffusin equatin fr the regins 1,2 is given as fllws. a 2 A(y,z) + a 2 A(y,z) = 0 ay2 az2 (1.1) Taking int accunt the independence A(x,y,z) alng the x c-rdinate and separating y, z variables A(x,y)=Y(y)Z(z), (1.1) can be rewritten 1_a 2 Y(y)+_I_&Z(z)=0 Y(y) ay2 Z(z) az2 (1.2) The general slutin f the last equatin can be represented by the infmite integrals, AI (y,z) = J ~ sin(ay)e-"'da ~(y,z) = J (c 2 e'" + d 2 e-"')sin(ay)da (1.3) (1.4) where the vectr ptential indices are related t the regin f interest, ~,c 2,d 2 are cnstants, a is the integratin parameter. The expressin under the integral sign is the particular slutin f the equatin (1.2). In the regins 3, 4 cnductivity is different frm zer, and the diffusin equatin is given by (1.5) with the k-parameter being k = ~J.10'0), j being H. Applying the same methd we btain the fnnulae, 370
3 ~(y,z) = J (c 3 e"'z + d 3 e-",z)sin(ay)da A4 (y,z) = J 04e"'Z sin(ay)da (1.6) ( 1.7) 2 2 a 3 = a + J1l3(J 3(i) a~ = a 2 + j1l4(j 4(i) (1.8) The bundary cnditins between the different regins are : Al (y,o) = A2(Y'0), A 2 (y,-1) = A3(y,-I), A3 (y,-i- m) = A4 (y,-i- m) 0.9) a~(y,z)i - 8~(Y'Z)i = Il/((y- A)-O(y + A» 8z z= az t~o 1 8A2(Y,Z)i = ~ 8A3 (Y,Z)i 0.10) 112 8z z~-i 113 8z z~-i :3 aa3i:'z)iz~_i_m = :4 8A:;:'zl~_J_m where (y) is the Dirac functin. With the bundary surface cnditins the slutin can be expressed in tenns f the integral 1 J sin(aa) sin(ay) A(y,z) = -- da (1.11) 1t 0 a which is nt always cnvergent, especially when y = A, which relates t the case f tw parallel lines f current. T bypass the prblem let us cnsider, instead f lines f current, tw parallel strips f current r the fllwing linear distributins f current in the plane z=o (Figure 2) I i(y) = A -A' Al <y<a2 2 I i = 0, elsewhere (1.12) where i( y) is the linear density f current, Al ' 1..2 are spatial limits f the strip f current. The bundary cnditins give us the fllwing system t find unknwn cnstants : OJ = c 2 + d i(y) 01 + C2 - d2 = - sin(a.a.) a1t c e-w + d ew - c e-a,1 + d ea,l (1.13) cx(c2e-a1-d2ew) = J..l j C 3 e-u,m + d 3 eu,m = a 4 e-u m J..l CX3(C3e-UJ1 - d 3 eu,/) CX 3 ( C3e- u,m - d 3 ea,m) = ~: cx, a4 e u m 371
4 z regi n I 1.1(1 =0 I I r-~-r~ ~~~~--~-, ~202=O ~~~~~~~~~~~~~~~ ~O J _j~404 Figure 2. Tw strips f current ver layered cnductive half space. Slving the system (1.13) t fmd ~,substituting the result in (1.3), taking the integral ftrst ver y within the limits [1.. 1,1.. 2 ] and after numerical integratin by a we btain the vectr ptential value and cnsequently can determine the electrical impedance per unit length. If mre than tw cnductrs cmpse the prbe, summatin ver separate pairs must be perfrmed. RESULTS Using the abve methd the electrical impedance curves were cmputed fr the infmite length prbe including fur strips with varius distances 21.. between them. Strip' width W was taken t be abut 4 mm. In the case when the gap 21.. between strips is large in cmparisn with the prbe width the induced eddy currents are signiftcant just belw the strips and almst vanish in the area between them. When decreasing the distance 21.., the magnitude f the currents in the sample under the strips als diminishes, and additinal maxima arise due t the interactin between the induced eddy currents. The relative eddy current density n sample surface fr very clsely placed strips (W=2A.=4 mm) as a functin f y c-rdinate is shwn in Figure 3. Thus that type f the prbe is differential and can be used t detect mechanical and structural defects. cq!pqplate thickness = 4 mm 2I..=4mm W=4mm f=5hz a=48ms/m Iift-ff= lj(y,o)1 n(l/m) Figure 3. Eddy currents relative density distributin at the surface f cpper plate. 372
5 21.. = 50 nun W=4nun Iift-ff= 0 f= 500Hz 1J1Il(l/m) Cpper 300 ",/~ Y(m) -,~.-~c<~~inum Figure 4. Eddy current density vs. depth distributin. Lnrm. cpper plate R nrm. Figure 5. Electrical impedance curves fr cpper plate and half space. Lnrm 2 half space steel steel layer 2 mm n cpper half space 1.6 cpper layer 2 mm n steel half space 1.2 cpper half space Rnrm Figure 6. Electrical impedance curves fr steel and cpper half space cvered by cnductive layers. 373
6 Table 1. Cil and measurement parameters fr the prbes A and B. Parameter prbe A prbe B Number f strips 2 18 Prbe length (mm) Distance between strips (mm) Lift-ff (prtective slide) (mm) 9 5 Inductance in air* (JlH) Resistance in air* (Ohm) *-measured at 50 khz. The density f eddy currents vs. depth dependence fr cpper, brass and aluminum samples is shwn in Figure 4. The electrical impedance curves fr diamagnetic half space and 8 mm cpper plate are shwn in Figure 5. Fr frequencies exceeding 1 khz the curves are indistinguishable. Sme cases f layered half spaces fr dia and ferrmagnetic materials are illustrated in Figure 6. EXPERIMENT In rder t verify the validity f the theretical mdel, experimental wrk n cpper plates has been perfrmed. All impedance measurements were taken at rm temperature with a digital bridge GR1693 in the frequency range khz. All measured impedance values are averaged using 10 measurements. Tw prbes inducing planar eddy currents were manufactured. Actual dimensins and electrical prperties f the prbes A, B are given in Table 1. The gemetry f the prbes examined in this paper is illustrated schematically in Figure 7. The present design allws t diminish the influence f prbe ends by making them perpendicular t the sample surface. Impedance measurements were perfrmed n a simple cpper plate and cpper plate cvered by brass fil f 75 Jl111. Prbes were applied against the sample surface withut additinal lift-ff, s the ttal lift-ff was determined by the prbe prtective slide. The results are presented n the theretical impedance diagram, shwn in Figures 8, 9. There exist several discrepancies between the cmputed and measured data fr the prbe B, hwever fr the prbe A the experimental results match theretical predictins very well. Needless t say that the mdel is nt an exact ne and naturally the prbes' length is the mst imprtant factr which influences the mdel accuracy. Figure 7. Gemetry f the prbe placed ver layered cnductive sample. 374
7 0.6 x thery experiment Rnrm Figure 8. Experimental impedance pints (prbes A,B placed ver 3 mm cpper plate). Lnrm 0.6 thery experiment R 0.16 nrm Figure 9. Experimental impedance pints (prbes A,B placed ver 3 mm cpper plate cvered by 75 J.IIIl brass fil). CONCLUSION It has been shwn that the apprximate mdel f a pair f lng straight cnductrs, wires r strips, can be used t determine theretically the electrical impedance f a prbe cnsisting f these elements, that gives the pssibility f accurate eddy current measurements using this type f prbe. The accuracy f the mdel becme higher with increasingly high prbe length t width rati. Fr the shrt prbes the effects f extremities lead t significant errrs. REFERENCES H. Rse, S. M; Nair, Inverse Prblems 7, L31 (1991). 375
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