... . : . .. * .. i ** 11 f > . 1 OF T ORNL P 2723 T, . iebie be . op . , . . . de 1 *** . . ) . . 1. Mio. - . > . EEEFEEEE . ;. - S . li to MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS -1963 - - - - - --- --- - - -- -- -- --- -- ----.. . - ORNE 2725 CONF-670505-2 DEC 6 1966 I I.STI PRICKS 7 MASTER 1983.0 mn .65 . COMPUTER DESIGN OF EDDY-CURRENT TESTS* RELEASED FOR ANNOUNCEMENT IN NUCLEAR SCIENCE ABSTRACTS C. V. Dodd III and W. I, Deeds** Metals and Ceramics Division Oak Pidge National Laboratory Oak Ridge, Tennessee, USA The starting point of any eddy-current test is the coil which gener- ates the eddy currents. We have developed a method (sometimes called a relaxation or iteration process) of analyzing the physical properties of an eddy-current coil with the help of a large digital computer. In par- ticular, these physical properties include eddy-current coil impedance, actual eddy-current distribution in the sample (and hence the sensitivity to small defects), and the induced voltages in various "pickup" coils. Within the present restrictions of axial symmetry and sinusoidal driving currents, the technique can be applied to a probe coil of any size and shapa, with or without ferrite. The sample may also be any size and applied to several specific cases including a coil above a conducting plane, the same coil with & ferrite cup above a conducting plane, and a coil encircling a rod. The calculations begin with the differential equations for the vec- tor potential in a conductive isotropic, inhomogeneous medium using low frequencies (between 10 hertz and 10 megahertzł) V? A = *+ wo + wv» (x A) *Research sponsored by the U.S. Atomic Energy Commission under Contract with the ünina Carbide Corporatiori. *** Consultant from the University of Terinessee. PO 1 4 . LEGAL NOTICE This report wo prepared as 12 account of Osvorment sponsored work. Neitor tbe Ualled Stator, oor the Commission, wor any person acting on behall of the Commission: A. Mrken any warn -ty or representa uon, exprossed or implied, with respect to the accu- racy, completendus, or usefulose of the information contained in this roport, or that the uso of any informs.una, apparatus, method, or process dinciosodia this a spirt may not lalringe privately owned rights; or B. Aerumns any liabilities with roopact to the use of, or for damages resulting from the ''; use of any information, apparatus, method, or procon discloso in this roport. As used in the above, "person acting do behalf of the Commission" lncludin tyem- ployee or contractor of the Commission, or employer of such coatractor, to the ortoat that such employee or contractor of the Commission, or employce of such contractor people, dissemrutes, or provides access to, any information purnuant to bilo oraploymeat or contact with the Commission, or dig omployment with sixch contractor. : 2011!!! . We next make the assumption of axial symmetry and sinusoidal driving currents. Both these assumptions are valid for most eddy-current test problems. This allows us to reduce our vector equation to one scalar equation for the components around the axis. Also the time dependence can be represented by complex numbers. After making these assumptions and expanding the various vector quantities the equation becomes: This is the differential equation which we must soj.ve. It is a very difficult differential equation to colve in a closed form, and, even if we obtained a closed form solution, we would stiil need some computer - approximations to evaluate, our closed form functions. Therefore, we chose to use an approximate method to solve problems using this equation. We first lay out the problem on a two-dimensional lattice of points, in a plane extending outward from the axis of symmetry, as shown in Fig. 1. We then use finite difference calculus to approximate the various terms in our differential equation. From this the vector potential can be determined at a point in terms of the vector potentiel at the four nearest neighbors. The equation is: Mr,z (1+a A + A to a ZA r,za r, 2 r,2 sta,21 + + *74+8,2 r,z+a (26a/v) (time,) 4,40,2 + Axe, 2 A, 2+* As,eme * a*uin, , 2 + + 2 + + ja?whip. z Orz 2 Mr+a, z' x,z+a + a , a 2. riz riz The various values of Hope on , and Ir, for each point in the lattice are stored in the computer. The vector potential at the boundary of the mesh is set equal to zero. The computer then starts through the lattice, calculating the vector potential at each point in terms of the neighboring 34 *, pointe. After a number of runs tarough the entire mesh, the value of the vector potential at each point will converge to the value predicted by the differential equation. Typically, it takes a 70 x 70 mesh 400 iterations to converge within 1%, at, a computation cost of $200. Once the vector potential is known, any physically observable electro- magnetic phenomenon may be calculated from it., Figure 2 shows amplitude and phase contours of the vector potential for a square cross-section coil above a conducting plane. Note how the amplitude is attenuated and the phase is shifted by the presence of the metal. One of the properties that we can calculate from the vector potential is the induced voltage in the coil, given by: . i.. v = jw Ads, oi, in finite difference approximation: v = jw ) 21(119) Acz • i coli This is the voltage induced in the driving coil. However, this same method may be applied to calculate the voltage induced in any pickup coil. It is a good approximation as long as the current density in the coil is much less than the eddy-current density in the metal. Once we know the voltage induced in the driving coil, we can calculate its impedance. The driving coil impedance, normalized by dividing by the magnitude of driving coil 1.1pedance in air is: j coil 21(1/a) Ag, (with conductor present) {za (7/8) As,z (with coil in air) cbiz PARA ---- .. -.- . Hi, . The normalized imçedance has been calculated for a number of different frequencies for a coil above a metal surface. The spacing between the coil and the surface, or the liftoff, has also been varied. For experimental confirmation the Irapedance was measured at different frequencies and spacings for a family of four coils of different sizes, but having the 3tame relative dimensions. We can trade among the fre- quency, conductivity, and the square of the linear dimensions of the coil bect use they appear in our relaxation equation only as the product a fuko. For example doubling the frequency and halving the conductivity will still provide the same product. Figure 3 shows how the measured ană calculated impedances agree. The accuracy of the measured values is rather poor in the low frequency region and better in the higher fre- quency regions. The calculated values had some inaccuracy along the spacing direction (along lines of constant Ruutio). This is due to the fact that impedance is affected so strongly by liftoff and this method of calculation does not define the precise location of the coil and the metal. The error is always less than one lattice space. The agreement in values of Rawo is quite good for the higher frequencies. The coil inductance may be calculated by the relation: - 1- )* av { 2x(/8) Ag, 2 coil The values of the coil inductance calculated in this way were approxi- mately 2% lower than the measured values. This is probably due to the fact that the boundaries should extend to infinity instead of only two coil diameters. The close boundaries (where the vector potential is assumed to be zero) have the effect of reducing the vector potentiai ard hence the coil inciuctance. As an example of the use of this technique we shall show how it can be used to design an eddy-current coil for an impedance bridge instrument to measure the conductivity of aluminium. We shall stipulate that the coil diameter will be about 0.4 in. 80 that we can measure the conductivity or small parts. We assume that we want to sperate at a point on the impedance plene with a ratio of liftoff-to-coil radius equal to 0.0470 and with vario R equal to 8.66. The impedance at this point on the plot in Fig. 3 3.8 a sensitive function of conductivity and the parasitic errors (interwinding capacitance, direct-current resis- tance) encountered in the neasured values are small. The frequency may be calculated to be approximately 10 kilohertz. If we specify that the coll should have an impedance of about 100 ohms to be compatible with the bridge, we can calculate tile air inductance of the coil to be; 100 2 millihenries . air“ (0.7) 75 x 203 *** The coil would need about 390 turns of No. 40 wire to have 2 millinenries inductance, Figure 3 also shows some problems that arise in making the conduc- tivity measurements. We note at the chosen point on the impedance plane that the inpedance change is as great for only 10 mils of liftoff as it is for a factor of 4 change in the conductivity. Another property which we can calculate is the actual eddy-current distribution. The edwy-current density which is given by, * o : -0 =jw, . . is directly proportional to the vector potential and lags it by 90°. Thus in Fig. 2 the phase and amplitude contours of the vector potential in the metal are also phase and amplitude contours of the eddiy-current distribution. This, in turn, can be used to estimate the sensiiivity of a coil to a defect. A defect can be represented as a current equal in magnitude and flowing in the opposite direction to the induced eddy currents. The vector potential of a coil with a defect present is the SUM addition of the current of the defect to the eddy currents gives, of course, zero current flowing through the defect. Although the impedance change due to an actual defect is difficult to calculate, we may approx-" imate the impedance change due to a small, spherical deiect by:5 2= o vol we (). Or, in terms of parameters we have already calculated in our computer program, the normalized impedance is: % - 3 (2+uc) (9) Mais e o ( 23 1/2 m,:') case coil Thus, in Fig. 2 the normalized impedance change for a small, spherical defect whose radius is 1/40 the coil diameter is: 2 = 0.016 AR2 e2ją Thus the impedance change is proportional to the square of the relative vector potential with a phase shift of twice that of the vector potentiul at the defect, This technique ney also be applied to fields with ferrite and ferromagnetic materials present. For instance ferrites can be used to shape the eddy-current flow and increase the sensitivity to defects. Figure ho shows how the phase and. amplitude (and hence the eddy-current distribution) contours vary when we place 9 ferrite cup (relative perme" : ability of 1000) around the coil shown in Fig. 2. Notice how the ferrite concentrates the vector potential, and consequently the eddy currents, directly beneath the .coil. This increases the sensitivity to defects and also gives a sharper pulse when the coil noves across a defect, which allows eddy-current instruments with time dit'ferentiating defect enhancement c..rcuits to perform better. The normalized impedance change for a small, spherical defect 18: 2n = 0.054 A 6290. Comparison of this with the preceding equation shows how ferrite has increased the sensitivity to defects. We have also applied this technique to calculate the vector poten- tial for rods. Figure 5 shows the vector potential of a coil encircling a nonferromagnetic rod. Figure 6 shows how the vector potential varies if we assume the rod to have a relative permeability of 200, making it ferromagnetic. Notice how the vector potential is attracted by the rod. Also the eddy-current density is relatively constant over a large outer portiori of the rod and rapidly decreases toward the center of the rod. As the frequency is increased, the relatively constant current over the outer portion of the rod remains constant, but the rate of decrease toward the center of the rod becomes much steeper. This illustrates how this technique can sometimes uncover some unexpected results. - ..- .. * With H U ATTARIT Trtului . - SUMMARY This technique has been applied to a number of eddy-current prob- lems and can be applied to many others. The restrictions of our present program, axial symmetry and sinusoidal driving currents are not true limits. As computers grow larger and faster, we will be able to program and solve problems with three-dimensional sonsymmetric lattices. Sinus- oidal driving currents are only a convenience, and could be replaced by time-sequential currents, and a time-sequential relaxation performed. A nonlinear medium can also be handled in a time-sequential relaxation if the nonlinearity oî the medium is known. This technique gives further insight into eddy-current phenomena, allows the design of actual eddy-current systems, and takes the design of eddy-current coils from an empirical to a scientific basis. REFERENCES 2. Dodd, C. V., A Solution to Electromagnetic Induction Problems, ORNL-TM-1185 (1965). 2. Binns, Kenneth John, and P. J. Lawenson, Analysis and Computation of Electric and Magnetic Field Problems, Macmillan Company, Now York, 1963. .. 3. Tunstall, G. N. und C. V. Dodd, A Computer Program to Solve Eddy- Current Problems, ORNL- (to be published). . 4. Dodd, C. V., Design and Construction of Eddy-Current Coolant.. Channel Spacing Probes, "Mi-rotecnic' 5 and 6, Vol. 18 (1964). . 5. Burrows, Michael Leonard, A Theory of Eddy-Current Flaw Detection, University Microfilms, Inc., Ann Arbor, Mich. (1964). ---.... ..... 10......... LIST OF SYMBOLS Symbol Name MKS Units Vector potential Vecto: potential at a defect Vector potential relative to that of the coil in air weber meter weber meter weber meter meter Distance between lattice points Applied current density ampere meter ampere meter2 Induced current density Square root of -] Radius of a spherical defect meter his !! Inductance henry Number of lattice points in the coil Number of turns on the coil meter Mean coil radius Spacing between coil, and sphere in coil radii Distance meter Time second Voltage second Volume meter3 Impedance ohm Dielectric constant Permeability ſarad meter henry meter mno meter radian second Conductivity Angular frequency ORNL-DWG 65-6601 . Z ' .. COIL :. 1 . .: . 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R -T BLANK PAGE . - -- . ܙ ܙܫܝܐ ܙ ܪ ܙ . . ܕ . . ܂ ܕ * ܝܢ ܙܪ ܩ ܫ ܙܫ :; :. ܀ - . . ܂ ܀ . . ܟ ܕ ܫܕܪ . ܂܂ ܙܫ ܝ 1 . . . . . . . ܐܝܕܢܬܚܐܚܐ ܙܝܢܐ 11 . L . . O en beiden ...... . . . . . . . s1 . . . . . . . . * • - • - - ......... I 1 - - C C - - . LI .. . f C !. . 1 11 . . . . - - ... ::8. .. . . . . ... . . 7.. 1 . . - - - • - - - 1 - . . . . . . ...... 1 } LEGEND 23 27 Caik. Bouwprince FENCA Besonderes Cowokerbe.Koubacarias Laws and PAMELA CONSTANT PHASE 6 . . . 165772 . 1 11 - P ܕ ܫ . ܫ ܫܦ ܫ ܫ - ܕܐܲܐܵ܀܀ : ; : -;www22ܐ ܟܫܐ ; 200 -.܀ܐܝܼ. ܕܲ. ܕܚܚܲܝܚܼ. : ] . - - ENCIRATING A EiGORE 5 (e-week aa.-f -;-i ;-'? - *-- - " ܘܢ . ܂ . . .. ' 2:112; ; ; ; ;; ; . ' . . ܙܚܝ. .ܫ.ܫܝ . . ܐ ܫܕܘܗܚ ܢ ܃ ܃ ... ܃ C - -1 ܂ -- ܚ - - - ܂ . ...ܝܼܲ . ܂ ܝ.. .. ܀ - ܀ - ܫ 1 ܐ - ܕ - ܀ . . .. ܐ... 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