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2—oo a ssoo icanisont= _a iee -_ ~——|2 __. | :ooi_--_.ai 2a oo . - i :_Sea |oo |eSpOCUMENTetDANCOPon i.rt — = ——seleato— _ (o eea 8_ \4ee |. a:oe 2. Sfrr a;beeeooteua-om ee etian_ iges aenn 88.Ft |- — _ee a a.oe_—_—.. a. a-aeLooeaoaoe. AG _eg aeSintrun23a aaiTEES__ete . cooe——oe a oea.__.a ae . .oo _4 aaa'ce .| a.oooeoea aa. —a ee . Contract; No. W-74CJ5 -eng-26 I%TALS 1zT\sD CERAMICS DIVISION C, V. Dodd W. E. Deeds 5. W. Luquire W. G. Spoeri APRIL 1969 OAK RIDGE NATIONAL L4BGRATDRY Oak Ridge, Tennessee 09 erat ed by WION CARBIDE CORPORATION U. S. ATOMIC ENERGY CCMMISSIOfl for the 3 445b 0266736 0 iii CONTENTS Page Abstract ............................. 1 1 . Introduction ......................... 1 I1 . General Theory ........................ 4 A . Differential Equation for the Vector Potential ...... 4 5 . Superposition of Vector Potentials ............ 4 C . Calculation of Various Electromagnetic Phenomena ..... 5 1 . Induced Voltage ................... G 2 . Mutual Inductance .................. '7 3 . Self Inductance ................... 8 4 . Coil Impedance .................... 8 6 . Induced Eddy Currents ................ 12 '7 . Dissipated Power ................... 13 E? . Electromagnetic Forces ................ 13 5 . Defects ....................... 9 9 . Fourier Analyses of Nonsinusoidal Driving Currents . . l.5 I11 . Some Important Special Cases ................. 18 A . Coil above a Two-Conductor Plane ............. 18 1 . General Derivation .................. 18 2 . Impedance of a Coil above a Two-Conductor Plane e . . 27 3 . Inductance of a Coil in Air ............. 24 4 . Defects in the Cladding Material ........... 29 5 . Defects in the Base Material ............. 30 6 . Eddy-Current Force .................. 31 1 . Voltage and Phase of a Reflection-Type Coil ..... 33 Current Instrument ................. 36 3 . Defects in the Cladding Material ........... 37 4 . Defects in the Base Material ............. 38 1 . Voltage and Phase of a Through-Transmission Coil ... 39 I3 . Reflection Coil above a Two-Conductor Plane ....... 33 2 . Lift-off Compensation in Yne Phase-Sensitive Edd;r- C . Through Transmission ................... 33 2 . Defects in a Plate .................. 40 iv Page D . Coil between ‘Jho Conducting Plates ............ .................. I. . General Derivation 2 . lnipedance of a Coil between Two Conducting Plates . . E . Coil Encircling a Two-Conductor Hod ............ 1. General Derivation .................. 2 . Impedance of a Coil Encircl-ing a Two-Conductor Rod . . 3 . Defects in the Gu.‘ier Mxterl’.al ............ L+ . Defects in the Inner Ma.teri.al 5 . Differentia.1. Coi.1 System ............... E’. Coil . j-nside a Two-Conductor ‘Tube ............. 1 .. General Deriv-a*t,i.on .................. 2 . ... ............ lnipedance of a Coil inside a Two-Conducto r. Tube 3 . Defects i.n the Inner Material ............ L.. . Defects in the Outer Material ............ N . Calcul.a.ted Results for Some Inipai-tant A_nplica’ij.ons ...... V . Fqeri.mentaI Verlfication .................. VI . Accuracy of Calculations ................... A . Axial Symmetry ...................... 9 . Current Sheet Approximation ............... C . iii.gh Frequency Effects .................. 1 . The Skin Effect ................... 2 . Interwinding Capacitance ............... 4= Displacemen-t Current Effect ............. 5 . Assumption of Linear Media. .............. VIT . Conclusions ......................... VI1 1 . Acknowl.ed.,gnen ‘is ....................... Appendix ............................. 3 . C o i . 1- t o - . S amp le C apa c it an c e .............. 41 it 1 46 47 &1 56 56 57 58 61- 61 67 67 68 68 ‘13 78 ‘I G 78 ‘79 79 80 81. 81 132 82 82 85 SOME EDDY-CURRENT FROBLENS AJTD THEIR I?TTEGR!-L SOLIPTIONS C. V. Dodd W. E. Deeds' J. W. Luquire' W. G. Spoeri Abstract We have obtained integral representations for the vector potential produced by a circular coil for a number of different geometrical configurations. From the vector potential we can calculate any observable electromagnetic induction phenomenon. Our present solutions are limited to cases with axial symmetry and linear media. are in the form of multiple integrals of ordinary- and rnodi- fied Bessel functions. The configurations that we have already solved include coils with rectangular cross sections in the followilzg loca- tions: above or on each side of a twO-ConduCtGr sheet bounded by parallel planes, between two parallel-plane conductors, encircling a tube or a rod consisting of one metal clad on another, and inside a tube. The electromagnetic phenomena that we have calculated include the coil impedance, the phase and amplitude of the voltage induced in pickup coils, the effects of small spheroidal defects on the impedance and. phase, and the forces generated in a metal. by the eddy cw- rents. is excellent in most cases and within experimental error in all cases. The solutions The agreement between calculated and lneasured pherromena I. Introduction One of the most important factors in any eddy-current test and one of the most difficult to analyze is the coil- that generates the eddy currents. flowing in a circular coil in the presence of' a conducting material is generally quite complicated. times in the past by use of various physical models. The electromagnetic field produced by an alternating current The problem has been analyzed a nimber of The solutions have 'Consultant from the University of Tennessee, Knoxville. L grad.i.m.l_lry become more acclrak and 'ihe models more sophisticated as the actual problem is bettei. approaillied. very compli.cated solutions, some of which are presented heye I 'Todhy we have very accurate but An early seri-es of paper by Forster2-" between 19.52 and 1754 dealt with both .the t'neory and practical aspects of eddy-cwt-z-ent testing. Iie arialyzed a coil above a conducting pl.meJ a:;swning the coil to be a magnetic d.?::pole, and a?? infinite coil encircling an i:nfini-Le rod. I-io~hschild~ in 1959 al.so gave an analysis of an infinite coil and Lnchxied- some eddy-current di-strtbiitions in the m-etal. Waidelich and Renkt.n6 in 1956 analyzed coil i.rrped.a,nce by an iiimge aI3prosch. theore tical. results agreed wel.1 with experiment for relatively high frequencies. coil. was a transformel- with a network tied to the secondary. ne1;work representation gave good results when coqa-red to experi.rr1en.t The diffusion of eddy-current pulses (Atwood and, Libby, Their T,i-?7by7 in 1.959 pre:;enf;ed a theory in which he assumed -the 'This 1963) can be rep:ce,i;ei_ited in this manner. In 1962, Russell, Schuster, aad Waidklich, 9 assuming that the flux was entirely coupled into the conductor, analyzed a ciij?-core coil. The semieiiqirical results agreed fa3.rI.y TAre7.l with the 'F.L.ied.riclz FErstes-, ---_- Z, Meta1.1.k. & - 163-.1.?1. (1952) . 3Fried~ich Fijrster and Kurt Stanibke, Z. Metallk. 45 (4), 166-179 (1954). '*Friedrich Forster, Z. Metallk. 45 (L;-) , 197-199 (1954). 511. I-IocbschJ.l.d, "Electroni~g:l_etic Met'ilods of Testing Metals, " - Progmss in Nondestructive 'lkstirig9 --I- Vol. 1, Macmillan Company, New York, 1959. 6D. L. Waidelich and C. J. Redcen, Proc. Natl. Electron Conf. 12, 186-196 (1956) . 9T. J. Russell, V. E. Schuster, and D. L, Waidelich, J. Electroz-. Control 13, 232-237 (1962). --___I = 3 experimental measurements. Burrows'' in 1964 gave treatments based on delta fbnction coils, and Burrows continued with the development of an eddy-em-rent flaw theory Dodd and Deeds13 in 1363 and DoddL4 in 1965 gave 3 relaxation theory to calculate the vector potential of a coil with a finite cross secti-on. While the relaxation theory is ve~j versatile and quite accurate for a very fine lattice, it, requires a large digital computer and is very expensive to use. Dodd" in 1967 and Dodd and Deedslb in 1968 devel- oped a set of integral equations that can be evaluated accurately on a small computer. Un€ortunately, the derivation of these equations is complicated and must be repeated for various coil and conductor configu- rations .) Vein'' in 1962, Chengl' in 1364, and This report presents the results for six cases of practical importance with some experimerkal verification of the calculations. For the computer programs used in evaluating the various formulas, refer to G'ne work by Luquire, Dodd, Deeds, and Spoeri."' lop. R. Vein, J. El-ectron. Control - 13, 471-4'34 (1962). "David H.S. Cheng, "The Reflected Impedance of a Circular Coil in the Boxi~ty of a Semi-lnfinite J!/iediwn," Ph.D. Ei.sserta-tion, University of Missouri, 196C;. University Microfilms, Inc., Ann Arbor, 'Michigan, 19G:L+. 'by a Relaxation Method," pp. 3C)0-%1~+ in Proceedings of the Symposium on Physics and Nondestructive Testing, Southwest Research Institute, San Antonio, Texas, 196.3. 12Ni.chael Leonard Burrows, A Theory of Eddy Cl.rrrent Flaw Detect?%, 13C. V. Dodd arid W. E:. Deeds, "Eddy Cixlrrent Impedance Calcu1a.ted I4C. V. Dodd, A- Solution to Electromagnetic Induction Problems, ORNL-TM-llE?5 (1%5] and M. S. Thesis, the University of Tennessee, l%5. 15Cs V. Dodd, Solutions to Electromagnetic Induction Problems, OIWL- ,T!M-1€Vv2 (19157) and Ph.D. Dissertation, the University of Tennessee, 146'1. "CC. V. Dodd and W, E. Deeds, J. Appl. Pkljrs. _I 3'9, 2i)29-2838 (1966). - '17J. W. Luqurire, C. V. Dodd, W. E. Deeds, and W. G* Epoeri, Csmputer Programs for Some Eddy Current Problems, ORNLTM-2502 (in preparm 11. General Theory - For the cases considered here, the vector potential adequately and convenient.ly represents the electromagnetic field. In this section we shall gi.ve the differential equation for the vector potential of a single loop of wire, show how a number of simi-lar vector potentials c3.n he superimposed to obtain the vector potential for coils of rectangular cross secti-on, and then derive equations for various physical. phenomena in terms of the vector potential. A. Differential Equation for tile Vector Potential -_m ._.__- We shall assume that space is divided irito regions of linear, iso- tropic, and homogeneous media, one of which contains an infinitely thin coil caryying curwmt, r. sinusoidal currcnt in the coi 1, whjch i s located at, (ro, za) in cylindri- cal coordinates. We shall also assume axial syrmetry and 3 'The differential eqimtioii for the spatial part of the vector poten- tial, A, is15 The soluti.on of this dj-fferential equ-ation is Q boundary value problem, and it wi1J. be effected for varri oils geonetr:i.cal configiratioiis in later sections. u1a.r coilductor configuration has been solved, any number of delta fuiction coils can then be superimposed to build up sny desired sliape of coil. Once this linear differential equation for a partic- B. Superposition of Vector ..-.__.I_ Potentials ... . ____-- We can write for "c'ne to-Lal vector po-Lential produced at (r,z) by the su.perposition of n coaxi.al d.elta function coils located at 1" and z, i 1 n 5 Tnis equation is valid for coils of any cross section. If we let the current, I, in the delta function coils approach a continuous cur- rent distribution of density io(',~), then the vector potential due to this current will also approach a continuous distribution. density of this vector potential distriblAion by plo( r,z,ro,zo), we can mite the integral form of Eq. (2.2): Denoting the since Ao(r,z,~D,z,,) is proportional to io(r,z), it will prove useful- for cases in which the current in each loop has the same magnitude and phase to express Eq. In such a ease, (2.3) in terms of the current density. nT io(r,z) = = eonstant , (coi.1 area) where I is now the current in each loop of wire. MultLplying the integrand of Eq. (2.3) by io(r,z)/io(r,z) and making use of Eq. (2.4) yields (2 .I> where we have a-ssumed the coil to have a rectangular cross section, as shown in Fig. 1. partj-cular coil and conductor configuration, the total vector potential produced by a coil. with rectangular cross section may be obtained simply by allowing the delta function current to approach the current density, t,,(r,z), and nfilring use of Eq. (2.5). Once the vector potential has been calculated for a C. Calculation of Various Electromagnetic Phenomena Orice the vector potential has been determined, any electromagnetic induction phenomenon can be calculated *om it. In this section we 6 ORNL-DWG 58-fO309 Fi.g. 1. A Rectangular Cross-Section Coil.. shall give the equations for the piienomens tha’i are of particular inLerest to the nolidestructive tester. 1. Induced Voltaae We have, for the voltage induced in a length of wire, For an axially symmetric coil wit‘n 3. single loop of raditis r, Eq. (2.5) becomes The to.tal. voltage induced in. a coil of ?I turns is then n ~~ ~. v = j2mjLri~(ri,zi) . i:-- 1 (2.8) 7 We can 3pproxima.te the above summation by an integral over a turn density of N turns per unit cross-sectional area: V -2 j2nwJ rA(r,z) Ndrdz . coil cross section For coils with a constant number of turns per unit cross-sectional area rA(r,z) drdz . JJ j2m.m v= coil cross section coil cross section If we specify a rectangular cross-section co 1 of length (4;- sad of inner and outer radii r; and r;, respectively, we can wvrike Ey inserting the vector potential from Eq. (2.5) into Fq. (2.11), we obtain the voltage induced in a coil (with primed parameters) by a current, I, flowing in a, coaxial coil: (2.19) (2.11) For the specj.a,l case of the self-ind.uced voltage, we have v= 2. li .e,, Mutual Indilc t anc e (2.13) The voltage generated in a '*pickup" coil with d-imensions r;, by a current, I, flowing in a "driver" coil with dimensions 12 a, is 8 or (2.14) Using Eq. (2.12) to calculate the voltage we have (2.15) This is the mutual inductance between the driver coil. and the pickup coil. By the reciproci-Ly t711e0~ern, this is equal. to the muti~.al. inductance betweei? the pickup coil. and the driver coil.. 3. Self Ind.ucta~ice 'The coI.1.'~ self indue-Lance is a special case of the mutual. i.nduc- .Lance. We shall let; the ~WC? coils become one and the same and drop the prFmes in Eq. (2.15), obtaining 4. Cot1 Impedance From tile self-induced voltage, we can calculate the coi.1 impedance: or V z=- I (2.17 1 Substituting from Eq. (2.11) for -the self-indmed voltage gives The coil impedance in the presence of a conductor is usually normalized by dividing it by the nagnitude of the coil impedance in air. 'c onduc t os /'air ZZ I n Q1 r, z= n 5. Defects (2.19 ;, Once we know the vector potential in a metal, we cm determine the effect of a defect. defect by the sum of a "ciurent defect" and a "magnetric: defect." current defect is caused b$ an abrupt change in the me+,al's conductivity, and the magnetic defect is the result of an abrupt change in its magnetic permeability. rials. The defects must be smll enough for the incident field to be essentially uniform over the defect, and the defect ctimensions must be small compared wlth the distance from the defect to the nearest material boundary. According to Bwrows,l8 we can represent a snlall The The latter is of importance only in ferroInagaetic rnate- We can modif4r BUTI'OW'S Eq. (5.12) t,o obtain the defect-induced voltage in terms of the vector potential: "Michael Leonard Burrows, A Theorj of Eddy Current Flaw Detection, University Microfilms, Inc., Ann Arbor, Michigan, 1'3GI+. (2.20) induc3d in coil 2 by a current v2 d.' 'i'liis is :;he defect-produced voltage, ll €l.o~hii_ng in coil 1. ---* Here, AI is the vector potenti-a1 at, the defect produced by a current 11 flowi.?ig in coil I., axid A2 + is the 'v-ect,or poten- tial that would be produced at the def'eel; if a. cwrent I, were appli-ed to coil 2; a and B are the current and^ magiietic scat-tering matrices, respecttvely. shape current defect a.nd a spheri.cal magnetic def'ec-t; is --f + --f --L For two a.xial.ly sy-timetric coils the vol~tage &le t;o any We shal-l. first coi?sid.~r current; defects. If we take the spec-ial. case of a :;piserori.dsi dei"zct, as showrl in Fig. 2, we can write for the current; shape and orientation factor - a cos'@ + a sin 7 Q , L T where (2.23) (2.22) II ORNL-GWG 68-7316‘ .-. ................ - I l- COIL AXIS SPHEROIDAL DEFECT 02 05 4 2 5 40 b/a Fig. 2. The Shape and Orientation Factor of a Spheroidal Defect. and -1 /2 6 ll-(b/a)’ I = asymmetry parameter, a = symmetry semiaxis, b I semiaxis perpendicillar to symmetry axis, o = conductivity of metal, (r - conductivity of defect. d Figure 2 shows a plot of the current shape md orientastion factor when the defect eonduc‘civity is taken to be zero. The orientation angle, 0, is the angle between Lhe theta (0) axis and the synnrietry axjs of the spheroid. spheroid becomes a flat disk. and the shape and orientation factor becomes infinite in such ip. w3.y that the product approaches If we let the ratio of b/a approach infinity, the Its volume (4~b*a/3) approaches zero, Vol. a22 = ( 2.2'/ ) where b is the radjiis of the disk. It is worth noting that we can divide the current defect equation into -two independent factors: (2.28) The factor in the fli~st square brackets dkpends only on the problem pararrxeters, such as coi.1 size and shape, frequency, and conductivity. It has both Teal and imaginary parts and is cal.J.=d. the "defect sensitiv- ity factor." the size, shape, and orientatton of the defect. If we allow the two c0il.s to become one and. the same; Eq. change due 'io a ciulrent defect. The factor in the second. s~rqu-are brackets depends only on (2.28) represents the impedance For a spheri-ea1 magneti-c defect, we have (2.29) where /I and 11 adre the permeahil.ities of the metal and of the defect, respectively. From Eq. (2.21) we have for spherical. magnetic defects d Thus we can calculate the effects of a dzfecl; from the vector. poten- Lial produced thcro - 6. Iniiuced. EMy Currents -, We have, from Ohmit s 1.a~ -+ -? 32 --f J = & := --.. 0- -- - -jwmA . a-i; - Dlie to axial. sylrxnet;-~y, Ey. (2.31) 'uecmes J = -joiiA(r,z) . (2.31) (2.32) 7. Dissipated Power From the vector potiential, the dissipated power density due to the eddy currents can be caleulated: where A is the root-mean-sqmre vector potential. The negative sign denotes a power loss from the field. 8. El ectrormgnetic Forces We can also calculate the electromagnetic forces i.n any conductor' that mnay be present. The force density is gi-ven by Stz.att,c;n'.9 as (2.34) This is the force exerted by an electromagnetic field on a unit volume of isotropic matter, neglecting electro- and magnetostrictive forces. These latter forces can justifiably be neglected, since they produce deformation of the Inaterial but no net Torce. The first -berm vanishes when the chazge density, p, is swnmed. over the electrons and ions. The third term is also taken to be zero for the interior of a metal. Th.e last term is due to the light presswe and is negligibly small. 'I'h'lis, the force density reduces to ( 2.35 ) We shall first consider only no.wnagnetic materials, which require only the first term in Eq. (2.35). ~cp. (2.31) gives Substituting for the current from '"5. A. Stratton, Electromagnetic Theory, McGraw-Hill Book Company, New York, 19)+1, 1.4 Expanding the --f curl of A in cy.l.indrical coordinates, yields By perforxfl-ing t'ne vector operations, we find that (2.37) (2.38) If the vector potentLal. consists of the sum of a time-harmonic part and a steady-state part A=A, +Ao, (2.40) Ey. (2.39) becomes We shall now consider only the z component of the force density, although the treatment of the r component is similar* harmonic vector potential, z4u, and its derivative with respect to z vary- with respect to the according to The time- %,) cos(wt f 4)') . (2.4-3) Thus we have &A, Expaading in terms of the real and imaginary parks of A, and -- gives a2 (2.45) This is the z component of the force density. To get the total force on the metal, we must integrate Eq. (2*45) over the metal. terms are the only ones that give a net force on the metal when the force is averaged over one cycle. Due to the cylindrical symmetry, there will be no net r component of force. The last two Zet 1.1s now consider magnetic materials. The first term in Eq. (2.35) gives the Lorentz force density, which we have aLready calculated; the second tern is due to magnetic materials: Substituting the (2.46) (2.47) ( 2 ” 4‘8 ) Tais force density must be integrated over the entire metal to obtain the total force. However, the derivative with respect to l/p is usually taken to be zero except on the boimdaries. Again, due to symmetry there is no net r component of force. ‘The total force is the sum of the eddy-current forces and the magnetic forces. ‘3. Fourier Analyses of Nonsinusoidal Driving Currents We have assumed that the applied current was sjnusoidal, while in most practical cases it is not. The current waveform in practice may 16 rang? from a pure sinusoi.ci plus a smal-3.. amoimt of hs.rmonlc3 distorti.on to pillses. However, we can represent any current as a Fourier srrtes of pure sinusoida,l ciirrents : (2.50) T -I the period of repetition. For the c0j.l voltage pi-orluced by the Fourier conlponents of the c1mren.t we have whex the fi-equency mbiIs the k-pedancei wh-ich has been normal-ized by dividi.-w by UL. produced by the pulse is and 0 represent the magnitude and phase of the impedance at *'m rn This irqedance can be read- ilirect.ly fi-om curves of The vol.tage m- 1 For aa example we shall take a square wavc pulse to be the applied cu~ent, as show in Fig. 3(a) - The current; can be mitten as ORNL-DWG 68--40341 ... . .. .. . U.. Fig. 3. Current Naveforms. (a) A square wave current pulse. (b) A delta fimetion current pulse. and thus the voltage in terns of the normalized impedmce becomes (2.54) For a second example we shall assume a unit impulse of current as shown in Fig. 3(b). For the current we have 18 (2.56) and thus the vol.'G'c;nge in terms of tile nor'm3.ri.zed impedaace is Now, as ilie frequency approaches infin-i-ly, the vol-tages for both the step function and the urit; impulse function would theoretically become infi- 1ij.tely large. For actual physical test cases, howevw, the series in botk Eqs. (2.55) and (2.57) me terminated due to t'ne Limited bandwidth of the systex's [email protected]. They can netther generate the infinite fre- qiienci.es required to produce the dk-iving currents nor anqlify the inPinite freqmncies of the voltages which Wo-dd be produced. - 1-11. Soin:: 1rIpxt;mt SLecial Cases _I_ .- __l_^_____ll We have seen how various electromagnetic phenomena can be calculated from the vec.l;or potential. In this section we shall presen'c the vector potentisl. for som special cases and use it. to calculate sone of the quantities of prac Lical tnterest. A. Coil above a Two-Cond.iictoT Plane --~- 1. General Derivation The coi.1. above a two-conductor pl.ane is shown in Fig. 4. The space is d.fvi.ded into four regri.ons, in each of which the properties are homo- geneous. The differential equation in each region is a specisl. case of Eq. (2.1). In air (regi-ons 1 and TI) we have The diTI'erentfia1 equation in a conduc-l;or (regions 111 and. Tv) is ORNL-DWG 67-2522 Fig. 4. PL Delta E'imction Coil above a !bo-Condu-c-tor ?lane. WP stml.3. now elloose a separation "constant," ~2. TIE positive sign makes the equations easier to emluate for boundaries ruming in the 20 r directton. We mite for the z depend-ence (3.4) or We define Equa1,i.on (3.3) i;iien becomes This is a first-order Bessel equat-i.oi1 and has the solu-tioi? ( 3 . '7 ) (3.8) Combining the solutions we have We now need to deLer-inriiie the constants A, By Cy and D. They are functions of the sepa-i-ation "constant" and are usually dirfereni; for each value of a. ?Aid-ividual solutions, if' 2 were a dri.scre.te variable; but, since a j.s a continuous variabl.e, the conrplete soJ.i.ation is an 3.n-t;eg-al over the entire rmge of cx. Our complete soluti.ori would be a sum of' all the Thus, '&e general sol.ut.:ion is 21 t We must take A(a) = 0 in region I, where z goes to plus infinity. Due to the divergence of Y, at the origin, D(a) = 0 in all regions. region IV, where z goes to minus infinity, B(a) must vanish. tions in each region then become In The solu- A(')(r,z) = $n,(a)e"OZ J,(ar) da (3.11) The boundary conditions between the different regions are Z=& z=Q z=-c z=-c Equation. (3. fi ) gives (3.u) (3 Il6) (3.17) (3.18) 22 We can sirriplify Rq. (3.22) by use of the Fourier-Bcssel equation, which is Equation (3.22) then b, '=Tomes (3.24) We can evalua,l;e the other integral eqiiations in a similar mamier. We shall drop the primes on the CY and nlak the fol.l.owing de-finit:i.onr Applying the reminder of the bouuidary condllioiis gives (3.26,) (3. Z?) 23 P (3.28) ( 3 * 29 ) (3.30) We now have six equations t7ith six unknowns. Their solution is (3.31) ( 3.32 ) (3.33) (3.34) (2.35) The expressions for the vector potential in each reglon thus become (3.36) 24 2 BingfF Big ADay2)=FSfas(ory)5(or)eto! a ¢,$082)(B1-B2) +(a9-81)(B 1482)eA x@fess:eee a(3.3) Q%(99-8 ,)(8,-B2) *(aig*B,)(B,3,) ee| (2) uTry jf ont AN(1,2) =5[Card (or)&P0 $ aze,|(0g*81)(Bi-R2) +(ao-81)(B1 482)e®¢ a] .xo{enorpapechiesi |e771ax(3.38) A% L(ao-B1)(B1-B2) +(Go+B1)(Br4B2) @%re| - 0(n,2)=a,[blarodavton) et! 1(By4Bg)e®EOY&(81-85)81 xate Sn ener |GY(3.39)(9-81 )(B1-B2) (Gio481)(By482) ee) . 4 fr .Osate,folerodsston) eto! L pop,el@atIe ax , xGabOrb) bla Oeeames (3.40) Maig-B,)(By+Bg) +(XpBy)(BBpeM. ‘hese are theequations forthevector potential ofadelta function coil above atwo-conductor plane. leshall nowuseEq.(2.5) tosuper- impose thedelta function coils toform acoil ofrectangular cross section asshown inFig. 5.Substituting Bq.(3.37) into Bq.(2.5) and reversing the order ofintegration gives w@2pfy. ) fF? fava,: APor,2)=|JfSeFiore)aslar)P9800?ena? . 2.((00+B1)(Bi-B2) +(a9-81)(B1+B2) 21 x&fortesa eae aoaroat,(3.41)0 (29-8, )(Bi-B,) +(Ag+,)(B,+82)eAYJ 25 .. ORNL- DWG 67- 2523R I ................. ... ..... - - ...__ - .. .. - -. . - - ............ Fig. 5. A Coil of' Rectangular Cross Seetion above a Two-Conductor Plane. 26 Lbon applyi nq Eqs. (3.42) and (3.43) the equations for kh~ veclor potential in the various regions TCJY a coil of I=e-tangill?r moss section be c Oine 27 Equation (3.44) for A' is -ralid in the region abo-ve the coil, and Eq. (3.45) for A'2) is valid for the region below the eofl. to give special treatment to region 1-11, between the top and bottom of the coil. for A(' ) (r,z) for the portion of the coil from z d.om to 8-r and the equation for ~I(~)(r,z) for the porkion of the coil from z ';$ to az. If we substitute Q, = z i.n Eq. (3.41+) and :l1 = z in Eq. (3.45) and add the two equations, we get We have For a point (r,z) in region I-LI, we can use the equation m We have completed the (3.W general derivation of the vector pot;enf;j.al of a coil of rectangular cross section above a two-conductor pl-sae. shall now use this vector potential to calculate 5 mmber of e1ect;r.o- magnetic induction phenomena for this configu.re"tion. We 2. Irrrpedance of a Coil above Two-Conductor Plane To ealci-fiate the coil impedance, PTI? substitute t'ne equation for -i;'ne vector potential at the coil, A(1~2)(r,z)y into Eq. (2.18) and. perform the integration over the dimensions of the coil.. The I-esu1.t 3.s 28 All I-engljhs are divided by ?,and all. a's and B's are ixlJlkipl?ed by F. After the dimensions are normalized, Eq. (3.49) brtCojnt:s NOMT all lengths are scale4 in term.: of i', ami we have Tor thd a's and E's II J. .L (3.53) (3.55) 29 A computer program (CLADT) designed to calculate the normalized coil impedance for this case is given by Luquire -- et al."l computer programs, CX0 is taken equal -Lo Q. that there is any appreciable difference between them. difference could have been neglected in Eq. (3.4) but has been carried through to retain generality. In all the It is only in rare cases In fact, the 3. Inductance of a Coil in Air From Eq. (2.16) the inductance of a coil in air is (3.55) A program (AIRCO5) designed to carry out this integration is presented elsewhere. 20 4. Defects in the Cladding Material We can use Eq. (2.28), with the driving coil and the receiving coil being the same coil, to obtain the impedance change due to a s~x%ll current defect: (3.57) For defects in the cladding material, A is the vector potential h. region 111. normalized dimensions, is The impedance change due to a small defect, in terms of 2oJ. W. Luquire, C. V. Dodd, W. E. Deeds, and W. G. Spoeri, Computer Programs for Some Eddy Current Problem, ORNL-TM-2501 (in pepam- 30 This impedance change may be mde dfmensionlt?ss b~, nomna!.ir,;.rig the impedance by dLviding by the magn'ltude of tiie eo51 irxpedaulce in air: 'To obte.iiz the d.c€'ecL sensitivity factor, we divide Eq. (3.59) by Vol 022. sjngle, ~inaXi-~ cwrenl; defect in 'die clai?ding material is giveii by Luquire et al. 20 ca!.culat,i.ng -the defect, sensiiivitjl rCac.i;or for a defect at a point on a lattj ce in Yne clad.6i.n.g mteri.al.. A prograril (DEFEC5) designed to calcul.a.te this fac~l;or for a In the same work. t'nere is another progra.m (DEUT) Tor I_ Wc, hav? dPsitr;fied 3 prnhrafu (DEE'EICB) io calculak the defect factor for a small current d?fect iii the b%se material. 20 do. . (3.60) sensri tivity 31 6 I Ed&$-Current Force The eddy-current force density at a point in ,EL conductor. can be calculated by substitutirg the integral equation for the vector poten- tial into Eq. (2.45). We will now compute %he net force of a coil above a single conductor. The vector potential In a single conductor reduces to ( 3.61.) and the only net force density, when averaged over time, is (3.62) BY inspectton we can write Eq. (3.6%) as (3.63) where the asteri-sk represents the complex conjugate. To get the total force on the metal, we integrate the force density over the vo1w.e of‘ the metal: Substitutlng Yne vector potentia,l from Eq. (3.61) into this equation and setting ct0 a gives 32 We c3,n re-verse the order of integrat.ion and peyforrn the integr,%tion ovw 1.- fimt. We shall use Lhs fact that al- J,(c*r! J-,(cr I\ r) d.i- = r 9\a-ufj I . r r-o (3.4 CX- x1 + jy, . Equa,.tion (3.65) then becomes (3.67) 33 This is the net eddy-current force produced on the cor?duetor. A computer program (FORCES) designed to eale-date this force is given by ixquire et ax. 2o -- B. Reflection Coil above a Two-Conductor Plane A reflection coil, as sliown in Fig. 6, is usually used in t'ne phase- sensitive eddy-current instrument. 21j 22 This configuration is actually 8 combination of special cases of' the single coil above a two-conductor plane. 1. Voltage and Phase of a Reflection-Type Coil We shall obtain the difference between the voltages induced in the two small pickup coils by a current in the large driver coil. up coils are connected in opposition so that, in air, the voltage cii-f- ference vanishes. However, when t'ne coils are placed r1ea.r a metal., there exists a voltage difference caused by- a "reflected wave" coming back from the metal. The pick- The net voltage induced in the pickup coils Is v=v - pu2 vpu, ' ( 3 . '70 ) where the first pickup is nearer the metal than the second pickup. According to Eq. (2.11) the voltage in terms of the vector potenti-al 5.s Due to the symmetry of the pickup coils, J,-J, is equal to ,t,-e,, and the coils are recessed an equal distance from khe ends of the driver. *IC. V. Dodd, Mater. Evaluation -- - 22(6), 26C-263 (1964). 22C. V. Dodd, Mater. Evaluation __ 26(3), 33-36 (1968). VATa 1 7 ~ . VA. Sg oa YWUps: fete1 Substituting the vector potential f'rom Eq. (3.48) into (3.71), per- forming the integration o-rrer r and z, and normalizing the dimensions gives We can make the following d.efinitions: LZ = length of the driver coil., Lt, = length of the pickup coil, L5 ::: distance the pickup coils are recessed, L = spacing between &river coil and metal or "lift-off." Substituting these into Eq. (3 (1 72) yields (3.73) A. computer. program (RF'XT) designed -to eal.cdate .LIE phase and anrplitude of this voltage may t;e found in the I-iteratweaZ3 335. W. Thquire, C. V. Dodd, W. E. Deeds, and PJ. G. Spoe~i, Computer Progrms for Some Eddy Current Problems, OKNL-'T'M-2501 (in preparm 36 ORNL-DW3 68-10310 (MAXIMUM L;FT ZERO LIFTOFF) . . . . . . . . . . ut V, - A sin (Q,. 1 Si! . (3.76) i 37 To determine the phase shift from @1 due to a change in lift-off, thickness 01" conductivity, we subtract Q from @I to get i (3.77) Luquire _I- et a1.23 give a program (DISC) designed to calculate this phase shift. Jk addition they give anot'ner program (RFLCYC) clesigned to csl- culate the phase shift for an incremental thickness change. It first calculates t'ne ptiase and amplitude of the voltage at zero and rmximum lift;-off' for the nominal. thickness value, sets the discriminator to trigger at the proper voltage VI, and then calculates the phase shift for small variations TYom the nominal thickness. The program will perform these calculations for a number of different values of r2hyrj. 3. Defects in the Cladding Material. We can write the equation for the voltage induced in the pickup coils by the presence of a current defect when a current, I, is fl-owing in the driver coil. simply by substituting the vector potential for region I11 in Eq. (2.28) to obtain we sha1.l now normlize t~ie dimemions in Eq. (3.78) anc~ again write the equation in terms of the dimensions of the driver coil, the pickup coil, the pickup eo51 recess, and the Iifi-off: 38 4- Defects in the Base Ivlatei-ial- - .......___I - .............. 39 A program (WDFTB) presented elsewhere23 is designed to calculate the d-cfect sensitivity factor for a srnnll curl-ent rlefect in the base material. C. Through TrmismLssion A through-transmission coil arrangement is sho~m 2.n FLg. 3. I-Iere we have a signal, transmitted by a large driTrer coil through a metal plate, received by a pickup coil. reflection case except that the pickup coil is now Zccated i_n region IV, This case is i2lmOSt identical. to the 1. Voltage and Phase of a Through-Transmissinn Coil We shall obtain the phase and amplitiide of the voltage induced in Subsbitutisng the plickup coil by a current J“lowing in the driver coil. the vector potential in region lYJ as gjven by Eq. (3.4’7)) with ri :::O into ~q. (2.11) gives 2 where 4 and T (4’ and TI’) are the lifi-off and length of thc driver (picklq) coil, respectively. performing t‘ne integration yie~ci:; We have presented elsewhere23 a computer progrm (WnU5) designed -to calmlate the phase and amplitude of the voltage for a. throu,@- transmission coil. I I 40 CRNL-DWG 68-7313 Y Pig. a. A 'l'l?rcJugh-Transm-j_ssion Coil. Amangement . 2. Defects in a Plate - \.le can eastly w-?'.te Llie equation for the voltage in the picklip coil. j-nd-ixed by Liie lpresence of n cul-rent defect when a current flows i.r-l t'ne driver coil- This i.s the same as the reffj.eckion case, except that now the pickup coil is on the opposite side of the metal. The vector potenkia.1- produced at the defect by a unit current fl.ow-i-og i.n the pickup is si.ven by the same expression. as that for the driver, 4 I except that the distance of the defect beneath t'ne metal now is set equal to c-z. If we norrr;tlize the dimensions by r and let, r2=O, we hwe which we divide by I, Wol in the work of' Luquire -- et alS2' there is a progrm (THRUDF) designed to calculate 1;h.e defect serisitivity factor for .f;'his case. tm obtain the defect sensktiv'i-Ly factor. D. Coil between 13x0 Conducting Plates I This configuration is used f~r spacing ineasxmments,2" and is shown in Fig. 9. 1. General Derivation 'The differeri$ial equation for. this case is Eq. (2.1jl and the general solution is given in Eq. (3. Uj) . equations for the vector potential in each regioxl we obtain I-Iowever, when we write the (3.84) 2'T. V. Dodd and R. W. McClwig, Fuel Element Coolaat Channel and Other Spacing Measurements by Eddy-Current Techniques, ORTJL-TM-129 ( 196%) . 42 k’iz. 3. A Dclts r’niction Coil between Two Condocti ag Plates. (3.85) and (3.86) (3.sri) (3.88) L* 3 (3.89) (3.91) z= e z=e ( 3 .93 ) We can substitute Eqs. (3.84) through (3.87) into Eqs. (3.88) m through (3.931, multiply by d J,(a'r) rdry reverse the order of integra,tion, and use the Fourier-Bessel integral to simplim them. then get We P2 C, - B, = - C4 . a (3 * 94.) (3.95) (3.96) (3.97) (3.913) (3.99) We have six equations and six unki?owlls. If we soJ.ve for each constant and substitute the resii.l.ts into Eqs . (3.8)+) thro~h (3.87) we get for the vector potential in each region m M and. J da J 1 I da. (3.1.02) ( 3.103 ) We now have Yne equa'iions for the vector pote?'i,ia,? due -to the currcerrt in a delta flrrrction co?.l- 'nelmeen two conductiilg planes. use Eq. (2.5) to superirqmse the vectoi- poten$ial-.s of delta function coil-s to find. the vector yoteiitial. produced by a coil of rectanglular cross secti.or? as shown in Fig. 10. Si_n.c-e we a:m concei-iiec? only with the Tegion cont&ii.:cig the coil, region TI-ilIT? we have We can. 45 OWL-DWG 68-7344 Fig. 10. A Coil of Rectangular Cross Section between Tsio Conducting Plates. Carrying out this integration yields 46 This is the eqmtion for the vectijr potential- of a coil with rectangular cross section in region TI-TIT. 2. Impedance of a Coil between Two Conducting ?late5 .......... .... .-. To calculate the coil impedance, we subs-kitute the equation for tile vector potential in region 11-1~ inix ~g. (2.1.8) and iiltegi-ate over -Yne dlmens~ons of t'ne coil. In addttion, we normalize the dimen- $ions i.i? terms of %. We then have I where f(a) is deflmec! by 'i"nis bpedance may be normalized- by dividi-ng by the magnitude of the air impedance, given in Eq. (3.55). culate the iiormalized impedance for this case i.s gi.ven by Luquire et A computer prograx. (Fill'NCO) riesi.gned to cal- -- ........... ..-- 25J-. W. Liqukre, C. V. Uodd, W. E. Deeds, and W. G. Spoerri, Computer Progi-ams for Some Eddy Curi-eilt -_ Problem, ON\.L-TM-2501 ( in p:rep~~riw E. Coil Encircling a Two-Conductor Rod We shall assume a delta f'unction cot1 encixcling an infinitely long, two-conductor rod, as shown in Fig. 11. 1.. General Derivation The general differential eyuaticn is t'ne same as Eq. (3.3) for a coil above a conducting plane. term, we have If we neglect the displacement current (3.108) Now, however, we shall assume the seyaration constant to be negative: 1 SZ(2) ,, = constant" = - ci2 . z(z) az2 Then z(z) F sim( ) + G cosrx( z-zo ) and Eq. (3.108) becomes ( 3.109 ) (3.110) (3.111) The solutj-on to Eq. (3.111) in terms of modified Bessel functions is We can now write the vector potential in each region. We shall use the fact that it is synmetric (with respect to z-z,) to eliminate the sine terms and the fact that KI (0 ) and Il(a) both diverge to 0 RN L-- DW G 6 7.- 25 24 I 111 I I .-i Fig. 11. Delta Function Coil Encircli.ng a Two-Conductor Rod. IV we have The boundary condi.tions between the different regions %re ( 3 .I13 ) ( 3.114) ( 3 - 115 ) ( 3.116 ) (3.117) ( 3.11G (3.119) (3.120) (3. Y21) ( 3 -122 ) 50 11’ we niulti.p3.y botil sides of Eq. (3.1.1.7) by cosa’ (z-,zo) and i-ntegi-ak from zero to infinity, we obtain We can reverse the order of integration and use the oi-t‘nogonal.-i.’iy pl-o-perties of the cosine in.tegral or use !;lie Fourier integral theorem Thus, we can solve tile ii1tegra.l. Eqs. (3. IJ-7) through (3.1.22) ~ We sbal.1. uf: a. to designate (a2 + jqm. 1‘ and ,R use primes to designate deri.u-s.tives with respect to the a.rgulilent. get fYom -the integral Eqs. (3.13.7) t‘nrough (3.122) 1 - to designate ”0 a.. We shall 1. 1. i cli 1 We 51 We now have six equations with six unknown constants. equa"cons may be solved to give the constants. following definitions: The We shall make the and The constants can then be written as P-p, (ar, 1 c, = ablsD (c~) 7 (3.133) (3.134) (3.135) (3.130) (3.13'7) ( 3 I 138 ) 52 Me can now write for the vector potenti-al. i-n each region ( 3.139 ) 'These are the equations for a delta fumction coil encircling a two- con&.i.cltor rod. We shall now superimpose these sohlt5on.s to form a coil with rectangular cross section as shobm in Fig. 12. Tnis i-nvolves sub.. s.Lil;uting Eqs. (3.139) t'nrough (3.142) into Eq. (2.5) and pwforrning Lhe intepation over the dimensions of the coil.. defj-nitions : L We make {;he folI.ovin,g (3.1.43) and. similarly 53 ORNL-DWG 67.- 2525R r---- b ---+ ' II rl ---A I I '2 ------ ___..._I__ Fig. 12. A Coil of Rectangular Cross Section Encircling a Two- Conductor Rod. We then get for the vector potential produced in the different regions by a coil of rectangular cross section (3.147) (3.743) Ti?. t'ne region of the coil, between regions TIL and IV, we have (3.149) or 55 It is theoretically possible to evaluate A ('' ' ) and its integrals over r and z. However, the fntegrals would require a very difficult numerical evaluation, and it is possible to obtain these in terms of a previously defined function. have shown that, Eason et d..z6 We ininst have z-zg > 0 and rO > r in the rtbove equation if it is to remttin finite. To do this we break the solution into two regions, using the ones that remain finite in their respective regions. We then multiply by rodrodzo and integrate over the cross section of the coil. We get By using "ihlis result in Eq. (3.l.50) we can write Thus, we have obtained Y'ne vector potential. for a coil of rectangular cross section i.n te.rrrs of an integral. of somewhat simpler fimc'cions. 56 2. Impedance ol” a Coil Encri-rcling a Two-Condi~-cl;or Rod ..._lll__ .-II_ 111_ ~ l_l^ To calcul.a,te the coil rimpedance, we substitute the equation for t‘ne vector potentia?.. at the coi.l., A(3’4)(r,z) into Eq. (2.1.8) and perfom the integratj-on over the dimensions of the coil; Nomml.i.zali.on with respect to r gives A compixter pl-.ogram (ENCCOS ) performs this rather difficult j_ni.,egration. 2 7 3. Defect:; in the Outer Material. 1- .... -__ll_-__l_. We car. mite the eqimtion for the impedance change d.ue to a current defect in regri.on IT by puttjng the vector potentisl for iOef;_ ‘-1 on 11 ii2ko Eq. (2.28). the ais irrped.ance, then If we normalize the dimenstoris by F and the impedance by 27J. W. Luquh-e, C. V. Dod.d., W. E. Deeds, and W. G. Spoeri, Computer ORPTL-TM-2501 ( in pyqjarati.oiir-- __I.. Z’rograms for Some __II_ Eddy Current Problemc; _____l_l_ _._^_ 2-9 57 from which we easily obtain the defect sensitivity factor since ( 3.157) Luqui.re et al. 27 give a program (LDllFT5) designed to calcula-Le the defect sensitivity factor due to a single defect in the outer c.onductor. the Same work is a program (ENDFTL) designed to calculate the d.efect sensitivity factor for a defect I.ocnted at successive poirrtls of' a LzLtice of posi.tions in the outer conductor. In 4. Defects in the brier Plateris1 bubstitutiotl of Ey. (3.145) into Fq. (2.28) yields the aonna1ized irrrpedance chmge due t~ a defect in region 1: A proarm (ENDFBS) designed to calculate the rlei"f2c-b sensitivity Factor for Ynis case is also given by Luquire -- et al. 27 58 5 . Dtfferentia,l. Coil Svstem A differentia!. coil system, as shorn in Fig. 13, is frequently used YnTs coil arrangemexlt is us& both with t'ne in the inspection of tubes. coils encircling a rod or tihe and with the coils positi.oned inside a tube. irr Fig. 14. driven by currmts 1,- and I,. The coi1.s am usually CoiTflected. i.n a bridge arrangement, as shorn We shall assume tha-t; the twc, cori.1~ are iden-tical and are The tokl.. voltage induced -in coil 1. i.s Where Z,, is the self-impedamce in the presence OS metal but not the ck-fect, M,, is the mutual coupli.ng between the two coil.s, ZId is the defect iinpedaiice diie 'io a current in coil I., Mlzd is the rflutiial. coupling between c0l.l.s 1. and 2 due to the flaw. &or current defects atid spbt:rical. magnetic defects, we have \$e also have a similar equation for v,( total) : (3" 1.62) 59 ORNL- DWG 67- 2525R2 I S tern. 60 OANL-DWG BR -10308 ~ ...... ~.. I COZlPENSAl RES1 STOI Fig. ll+. A Bridge An-angement Used with Differential Cotls. NOW MI, = M,, and = M2,d in general, and, since we clefi.ned tine two coils to be identical, Zll -: z22. Therefore, the voltage d.ifPere11ce is The circifit 3.s generally operated near null, so t'ria.1; 11-12 is very small. Slnce i.s also small, this term can usually be neglected. The voltage difference due to a defec'i then becomes Now if we take 6 as .tile cente:r-to-center :;pacine; between. the coils, we have Fwt'nermore, i.f we have only current defects FTP can write for the voltage difference az2 Vol I, . (3.166) The term in the braces can be called the differential defect sensi- tivity factor and is obtained by subtracttrig one defect sensitivity factor from the other. designed to calculate the differential defect sensitivity factor for a defect at points on a lattice from a lattice of values of the defect sensitivity factor. We have presented elsewhere27 a program (REflDIN) F. Coil inside a Two-Conductor Tube A delta function coil inside a two-conductor tube is shown in Fig. 15. 1. General Derivation We can start with the same general equation, Eq. (3.3.12), in term of modified Bessel functions that we used for a coil encircling a two- conductor tube and write the vector potential in each region. can drop the sine term and Ylze coefficients of K3(o) and I1(w). then have for the vector potential in each region Again, we We (3.167) ( 3.1.68) (3.1.69) (3.170) The boundary conditions for a delta function coil at (ro,zc) are 62 Fig” 15. A Delta Function Coil inside a Two-Conductor Tube. - r-a r-=a (3.1.73) (3. lrlL+) ( 3 . 1-75 ) 63 We shall first substitute Eqs. (3.167) through co Eqs. (3.1.70) into these six eqilations, again milltiply both sides by [ casu'( z-zo) d( z-zo), reverse the order of integration, and use the orthogonality properties of the cosine integral. 1 We then obtain We naw have six equations with six ~mErulovn constants. We shall mke the following d.efinitions: ( 3.186 ) (3.187) (3.188) (3.189) (3.190) W= shall now substitute these constants into thej r rpapectivc equations to obtain the vector potential in each region due to a del%& functioii coil.: 65 ( 3.194) Trrese izre the equations fox a delta function coil i.nside a Lwo- conductor tube. We shall now superimpose them to cbtain the results for a coil of rectangular cross section, as shown in Fig. 1.6. We do this by substituting Fqs. (3.191) through Eqs. (3-1-94) into Eq. (2.5) and integrating over the dimensions of’ the coil. We get for the vector poten- tial in the different, regions and 66 Fig. 16. A Coil of Kectangular Cros Tube. In the regj-on of the coil., between regions I and TI, we have We have now obtained the vector potential in each region. 2. &medance of a Coil inside a 'Two-Conductor Tube To calculate the coil impedance, we substitute the equation for the vector potential in region 1-11 into Eq. (2.18) and integrate. normalizing the dimensions with respect to the mean (foil radius we get After A computer program (I?srJCOS) has been desi.gned to calculate the norm1i.zed. imped-ance for this case.28 3. Defects in the Inner Material We can write the equation for tne impedance change due to a, current, defect in region 111 by substituting the vector potential in regton III into Eq. (2.28). If we normalize the dimensions by I- and -the impedance by the air value, we obtain - "J. W. Luquire, C. V. Dodd, W. E. keds, and W, G. Spoeri, Corpnter Program for Some Eddy Current ProbLems, 0?3C\JL-TM-2501 (in preparation). 68 l2 J x [sina(z-&,) sin~(z-~,)I d.a from which we can readily find the defect semi-tivity factor, since this is just VOI., n There is a prograrfl (TIVDFTS) desigried -to cal.cii~late the defecl; sensi- tivity factor due to a small cir~e-:i.t defect in the inner conductoy.28 in addi-tion, there is 8 prograin (INDFTL) designed to calcil.l-ate t‘ne defect sensii;ivi.ty factor at al.1 poi.nts in a lattice in the hner conductor. 4. Def’ects i.n the Outer Material. ~ .I... Substitution of Eq. (3.1.98) into Flq. (2.28) gives the no-maltzed impedance chairge clue to a defect in reg:ion ZV: ‘lhere is a program (iNUm5) designed to calculate the defect sensi- tivity Pac Lor for’ a cuwent defect in Liie out~r condiictor . 28 IV. Calculated H<?sults for Some Important Applicati.ons ................... 111111 WP have derived Lhe equalions for some physical properties for the We six riifSerent, coi 1 and conrjuctol- configurations show ia FL~. 17. shall now present computer evaluaiions OF some of these properli-es, 69 ORNL-DWG 68-7311 (01 CASE 1: COIL ABOVE TW-CONDUCTOR PLANE. IC) CAS€ 3: -1HROUGH-TSANSMISSION COiLS (AS USED IN PHASE-SENSITIVE INSTRUMENT. (bl CASE 2: REFILECTION-TYPE COIL (AS USED IN PHASE-SENSITIVE INSTRUMENT I AEOJE A TWO-CONDUCTOR PLANE. (dl CASE 4: COiL BElWEEN TWO-CONOUCIING PLANES IFOR SPACING MEASUREMENTS. le1 CASE 5: COIL ENURCLlNG TWO CONDUCTOR ROO. (fl CASE 6: COiL INSIM TWO CMIDUCTOR TUBE. Fig. 17. Eddy- Curreiit Problems Treated Here. 70 The first quzntity j.s the iiiiped.ance of a coil above a two-conductor plane. ness is varied froin zero to iiifini-Ly. There are two different curves corresponding to two differeint base nna-Lerials of different, conductivities. These curves i-ndicate t'ne accuracy of measuring chdding thickness and the effect of a change in conductivity of -tine base matei"ial. Figure 18 shows how the impedance vari.es as the eI.addi.ng thick- 0 73 0 72 0 71 0 70 F z ? 069 5 0 B 055 W k 0 rY - > 2 067 0 66 0 65 0 64 0 63 ORNl - DPG 67- E5 ........ I --------- I v =CONDUCTIVITY C =CL.nODING THICKNESS/i -0.0476 ILIFT OFF J,= ~F I I I U 009 010 Ot1 012 Oi3 044 0.15 Of6 RESISTIVE COMPONENT P'it;. 18. Irnpzdancr of a Coil above a TkTo-Conductor Planp. As a further ayp1-7 catton, we have calculated the defect sensitivity factor for a coil above a conducting plate. We have calculated the frzctoi- at poin-Ls -in a lattice and plotted contours of constant defect sensitivity. Figure 19 shows Line defect sensi-tivity factor contours superimposed on a scale drawing of the coil and conductor. We can we these contours to calculate the fmpedance change due Lo e, cui-rent defect at any locati~on in the metal. 01- due to movement of the deyect past the coi-I.. The normalized impedance change 7.s the product of the defect sensitivity facbor, the volume of the defeci;, anti the shape and. orientation factor. Wy eeANGULAR FREGUENCY ousESSE —————— FOZ osme SS SSE LL Yoan S == g “cab SS ee ZZ. me SSS LEE , mS Zo a MEDS SEE ge ae teaheONS _ LOPEZ i 1 |pe - Fig.19.Contours oftheDefect Sensitivity Factor foraCoilabove @Conducting Plate. '7 2 - Another application involves the ref]-ec Lion- type coil. We are attenptimg to determine tlie optimum operating conditions for inaximurn sensitivity to an incremmta!. tiiickmss change from a certain nomilial thliclulegs. varies as a finctTon of tlie operating conditions for four different nominal 'ihicknesses . Froni this graph, one can- choose the irptlirrwn fre- quency for a ps.r..ticulaii coil, m-Lerhl. ~ and thickness range. Figiinie 20 shows how the phase shift per jncremental t,hrichess (0- ANGULAR T:iCQUENCY /I- PERMEABll IIY 0.0 1 0.1 1 10 100 io00 T2wpo- Fig. 20. The Phase Sh.ri.P.L for ilO% Thickness Vai-iatioii fror~~ Nomi.na1 Thi.ckncss. For a coil encircling a two-conductor rod, Fig. 21 shows the normalized impedance ax a functi-on of the radFus of .l;b= inner (:onductor. The conductivi.ty of t'ne inner conductor differs for Yne -Lwo cwves, but a1.l other pa.srameters are the S~E - Thri.s ex3nipl.e demonstrates the range snd sensitivity of clad thi r_kj?css nieasurements arid the prob1.em.s rcsultirg from a change in the coiiduc::;ivity of the inner conductas ~ 7 3 ORNL-DNG 58-7318 c 0.64 0.6? 0.60 t 1055 2 :: L 056 W -2 - a W 0 54 0.5: 0 50 07 0.8 09 i.0 I4 12 4.3 4.1 15 1.8 RESISTIVE COMPONENT Fig. 21. Impedance Variation as a Function of Inner Radius. V. Experimental Verification While we do not yet; have experimental measimements for every case derived., we do have results for einough cases tu verify the general tech- niques. It is very difficult to cont~ol the coil dimensions precisely enough to obtain good agreement between calcul.atec1 and. measiired msults. Qlr fi.rst experimental measurements are of the indiuetance of a coi.1 in air. They are compared with calculated values in Table 1. The rneasiu?ements were made on a bridge wi.th a reported accziracy of +0.2'$. coil wind.ings. We feel that most of the ermr is due to small variatious i.n the We have constructed a fLmily of four coils of different sizes. but with %he same relative dimensions. coil at six different frequencies and at four different spacings above We rneaswred the impedance of each 74 Table 1. CalculaLed and Measured Coil ind-uctance I - ............ Coil Heas IJY e d. Calculated Error NUlIlber (d ( rrt ) ( %I ...............- 1_1 ^_ __- 1 11.38 11.1688 1.8 2 14-595 1.Lk.3438 1-72 3 134.17 135.216 0.38 4 3.157 3.16425 0.23 . 1111 _I_ a large, thick al.urni.num plate. in E’5.g. 22. the four different coils. la.t;ed valiies is excellent in ’ihe regions of the plot repi-esenting the The results weye normalized and plotted Each cxperirnental. pOri.iIt represents the average of values for The agreement b?i;-ween experimental and calcu- 05 t I 0 0476 ‘429 8 L ..... e.- C4LCULATED ”, V4LUCS a EXPERIMENTAL VALUES ~ ?=MEAN COIL RADIUS i w!l or ‘=CONSTANT 1, = LIFT -OFF/? 0 0 05 01 0 15 01 RESISTIVE COMPONENT b’j g . 22. Varia Lion OF Expcrirtien tal and Calcul a,t,nd Values of Normal ized Coil hpedance with Frequency 2nd Lift-Off. 75 higher frequencies. ments are relatively inaccurate. experimental error in all cases. At the lower frequencies, the experimental measure- The agreement is within the limits of We have obtained some experimental results for a reflection coil We measured the amplitude and phase of’ the above a conducting plate. voltage in the pickup coil as a function of lift-off‘ and plate thickness. Figure 23 compares the measured voltage with the calculated voltage, and Fig* 24 shows the measured and calculated phase shifts. In addition, for the through-transmission case, we measwed the phase and amplitude of the voltage of the receiver coil as a function of lift-off and metal thickness. I 2.4 - 2.2 - 2 .o 1.8, -V- CALCULATED VALUES 0 EXPERIMFNTAI- VALUES P2wpm-.1 i311 F= FREQUENCY T=CURRENT IN DRIVER COIL N=NO TURNS IN DRIVER COIL N’ZNO TURNS IN PICK-IJP COIL P=MEAN COIL HADIUS=O 2825 In <“=ANGULAR FRE OUENCY p-PERMkABILITY 0.92 O.Y4 0.96 0.95 4.00 1.02 1.04 1.06 THICKNESS/i Fig. 23. Vartation of Experimental and Ca.1culated Values of the Magnitude of the Reflected Voltage with Thich-ess and LiYt-Off. -r-” - ?=MEAN COIL RADIUS-0 2825 in “=ANGULAR FREQUENCY p =DE RM EA6 I Ll TY v=CONDUCTIVITY f=THICKNESS/i Fig. 24. Variation of Experimental and Calcsulated Values of’ the Phase Shift of the Reflected Voltage with Thickness and LifL-Off. Fiyiire 25 shows how the magnitude of the measured voltage compares wi-th the calculated values, and Fig. 26 shows the measured and calculal:,ed phase shifts. The agreement is fa-irly good and within the limits of experimental errar. We have measured and calculated the eddy-current force exerted on a large, thick aluminum plate by an alternating current flowi.izg in a coil.. The force was measined for two coils of different sizes but .the same relative d.imenaions. It was then di-vided by the squa,re of the number of ampere-turns and multiplied by lo7. Table 2. The averaged results are given in '77 ORNL-DWG 68-9862 ~ '.oo ~ -CdLCUL.ATED VALUES EXPERIMENTAL VALUES r= THICKNESS/? F= FR EQ U E tCY N=NO. TURNS IN DRIVER COIL N'=NO. TIJHNS iN PICK-UP COIL P=MEAN COIL RCOIUS w=ANGULAR FREQUENCY u-PERMEABILITY LT =CONDUCTIVITY 0.98 . I=CURRENT IN DRIVER COIL ........... ~~ ..... o,~ 0.94 izwp0=5.5372 i~ ...... L-. ~ ............ !.-- 0.4 0.5 0.6 ( THICKNESS/? 7 Pig. 25. Variation of Experimental and Calculated Values of the Magnttude of the Transmitted Voltage with T2iiehess and Lift-Off . ORNL-OWG 68-9860 I _- - f = THICKNESS/? _- - 05 06 07 -028 04 THICKNESS/i Fig. 26. Experimental and Calculated Values of the Phase Stni.ft of the Transmitted Voltage with Thickaess and Lift-Off. 78 Table 2. Calculated and. Messwed Force ..-. _l_l .._......_ _._. I-^._ -. . . . . __ ._.. . Measured. FoYce/r2 I2 Calculated Force Error (X newton/amp2) (X J.0-7 newtmn/amp*) (3.) r2 pLlj0 3. 082 8.628 2,. 65 77.05 329.8 862.8 3 . 082 8.622 24.65 w.05 329. Z 802.8 1,053 2.6'63 4.834 7.148 9.326 10 e 54 1.l2146 2.80859 5.11168 'i.52462 9.82756 10.7935 I,i.ft- Off,& = 0.0952 0.9414 2.316 4.149 5.988 7.692 8.575 0.987604 2.41421 4.29660 6.1.7 143. 7 . 8 5 3'1 8 8.53364 6.1 5.0 5.4 5.0 2.i 2.3 .+.7 4. I 3.6 3.0 2.1 -0.5 ~ .___II VT. Acciir-acv of Calculations This technicyip, like most otheiis used in engineering, is "exart; except for a f2w assuqtions we have io make in ord~r to work the probleril." We will now disciiss thc probable erpors in some of these assurflptjons. A. Axial Symmetry 1-11^-_1--._..-- This is a very good assmp.tion, but we cannot easily wind coils tha:t This error will vary with tile winding tech- have perfect axial symmetry. n-ique and will decresse as I;he number of turns on the coi.1 and tile coil- Lo-conduc-tor spacing increases. This error wiI.1. be effectively rwh~cecl when nornElized inipedance ris calculaLed. less than 0.01-%. L For a typical coil it should be B. Current Sheet Auproximtion 'I'his error arises because we have assumed a. current sheet, wiliJe we act1ia3.l.y haw a coi 1 wound bcith round, insulatPd wire. Some correction formulas are given by Rosa and Gi-overZ9 for the inductance of' a coil i.n air I From Eq. (93) by Rosa and Grover we have .m] . The s;ymbols D and d are the wire di.meters w-itki end wj-thout ins~Lliztion, respectively. This correction is positive and is usua1.l.y a small frac- tion or" the total inductance. TJsing the approxim&te Eg. (8'7) from Rosa and. Grover for the inductance we can wi.te for %he frzctfonal inductance change (6.2) where a11 dimensions are norm3lized by -the mean coi.1 radius. For a typical coil writh 3-00 turns, .tile change in indmhrice is G,l9$. In pracf;ic:e most, coils are not wound in precise I-ayei-s of wrire as assumed by Rosa and Grcwer, Instead, the windings are randomly piled on. each other until a C{~LI form is filled. T?i?s effect should be a very sho.rl;- range one., It wl.11 'nave a nri~c'il s~al-ler eff'ec-t on some ot'ner pheaomena, such 8,s normalized coil. imped.ance, which ilepend on the metal. C. High r'requency Effects -- There ,are a number of' high Treqiiency cffects, and they are probably the most serioius sources of error in this calculation -technique. 1. The Skin Effect; As the frequency i.ncreases, {;he current density ceases to be imi- i"cjrd.y distTPi.buted over the cross section of Yne wire but, bcevmes concen- trated near the su.rface. The resistance of the coil. increases, and the i.nductaice ilecreases As a first approxirrt8,tFon the resktance, R, of a circular cross section of' straight wire is'" 29Ee B. Rosa and 3'. W. Grover, Nat. Bur. Std. (IJ*SaI9 Tech, Netrs ___ 31111. -- 8( I), 1-237 ( 1912) " - Frederick W. Grover, Inductance Calculaticnns Working Forrnuhs a9d 30 'Fables, Dover, New York, 19W. 80 (6.3) Browin and Sharp Ereqii~ncy (hertz) _g_-_ I I Wire Gage ........ Nimber ._l___l. _____ ........... 20 G.C x 10' 30 67 x J03 4 0 680 x IO3 r7 1.1 x 1c6 'Yhe self-inductance decreases sI.i.ghLly at higher frequ.enc-ies du.;? to bhe concmtratioa of curr-eiit on the surface. wi-:re, A, the self-inductance chailges fron For a 1.eiigth of stm.ig1i.t .E 2 as the frequency goes fi-om ze~o to jtif'irliky, or a net change of .... X X-' h. Th3.s rep~esenis a net change of less than 1.3 pplil oil a typical coil. skin effect for t'ne lease of a s brai,ghL wire is mod.i.fi.ed. i.onsiderab1.y Wili.il the wire is wound. into a ~03.1.. or d er - of- mag it uii e e3 .t iilB t es . The IIowever, these equatrions my be used for 2. Interwi-nding ......... Capaci Lance The interwinding capax i.t,a.ace is -the distributed capaci-tance bets.r?en the inrii.vi.d.ua! 'Lurns of wire, and it decreases as tile -t;hlickness of t'ne wire insula-Lion is increased. acts much as a lurnped. capacitance in paral-lel wi.th a sei-ies induc.iance and. resis.t;ame. W'ni1.e it is a distributed capaci tanoej it; This prod.u.ces the resonant frequency 1 ".3=-. rn (6.4) B 1 ,4t frequencies much below t11e resonant i”requency ([A <: [Ai OJ ) the resistamce arid inductance vary as follows: 3o (6.6) Near resonance, the reactance changes rapidly from mxximwn inductive reactance through zero to maximum capacit7.ve reactance. being capacitively coupled between the turns, tends to flow across the loops of wire rather than through them. below -their resonant frequency. This effect also is short range and. tends to cancel oixt, in normalized ind.uctance ea.l.culations. The cwrent, In general, coils are used well. 4. Displacement Current Effect, In the actu%l crzleulation of the results, we have neglee1,etl %he ?is- placement, curyen t; tgrrns # We made the following approximtion: (6.8) real. s.nd imaginary parts of bhe irrrped.ance wa? +59 ppm and i11..3 ppm, respectively. For the coils am? conduc-iors norm3,1.1.y use& in ed.dy-cui~en'L work, t'ne eff ::et is The err*or in the coil i.ndiictar~ce was .--LAO ppm- comp le-L e iy negl.-ig:i.l, 1.e . We b.a.ve assumed that. the media are ].inear - that i.s, B : pH and D - GE, whei-e 1-1 ana c are constants. This is a good assmption i.f tile 1nateria1.s are nonfesrcma,gn.ztic or if al.3. [;he mgn.etic domaS.ns in a fermi>- magnetic mterial. are saturated. For f erroriiagneti.c materials, the rm:diu;l i~s no+-, only nonlinear, buL it fi-equer!t.i_gr has a large arnount of hysterresj.s. i*ihile the effects are fa3.rl.y small for low currents and the ferrite mate- rial-s experiment%!. measurements indicate that tjiplcal. calculated. vd-ues may be in error by abo-ct 10% for ferrormgne-iic mater7.al.s. VIZ. Concli~s ions These i-ntegral. solutions ofrer an acciii-ate way to calculate I;he observed e-ffec'is of actual. eddy-current, tests. The agi-eement bebween eqerimentsl and mlculatcd effects i.s good, althoug?~ Lhe nimber of experimental rneasiirements i.s s'cill somewlhat I.i.rtiited. the abili.ty to calculate these ed.dy-current pehnornei.3 axcurately will. lead! to eddy-current instriiments -which cam r 3 direct measurernen-ts of the physical properties of a specimen without. c&!.i.brabloa stand.a.x*ds. It is hoped that The authom wish to expi-ess their appreci.atioii 'Lo W. A. Sixpson and 1). D. Chitwcod for performring t,he experimental. measurements and to D. P. Godsey and L. D. Chi.twoud for consti-u.ctliig the coi~ls. Iix3;hcr thanks are expressed to E'ra.:flces Scarbom o:C Uie Meta1.s .atid Ceramics Di.vision Repoyts Office for t'ne typ.Lng arid preparation of t'nis repor.t. APPENDIX List of Symbols - In the first column the symbol used Is given and in the sqcond coliimri the name. units are given. of mass (MI, length (L), time (T), and electric charge (Q). In the third column the meter.-kilogram-scco.rld (MKS) In the last column the dimensions are given in term Symbol 14KS Units Name I- weber meter vector potential _.- magnetic induction weber meter2 meter clad thickness electric displacement coulomb meter2 electric intensity volt meter magnetic interis ity aqere meter applied current ampere app I.i ud current dens i ty r1 ciulr ent dens ity square root of minus one ampere meter‘ -I_ Dimens ions TQ L Q L’ T2 Q - Mi, - Q - ampere --- rne t e r TL~ 86 MKS Units Dimensions __ ^I_- Name .__.~_...__I S ymb o 1 L inductance henry F distance from metal to top of the coil met er L a2 Jl distance from metal Lo bottom of the coil meter 11 M N n t T v z mutual ind-uc tance turns per unit area Limber of turns coil inner radius coil outer radius mean coil. radius time period voltage henry 1. T., - tU-Cll meter 2 tur.11 meter L meter L meter 1, second T second T VO 1.t ohm ZO distance from metal to delta function coil meter L normalized i mpedamce 'n 0: separation constant a cinrent; scat tw ing ma,tr%x 4 -+ 1 L - 1 meter- 3. L - meter- A 2 8 magnetic s catterinz matrix Symbol Name MKS Units c d i e1.e c t I- i e c- ons t ant I-L permeah ility meter - farad meter henrv meter mho meter (T conduct ivi L y 0 angular f r equ-ency second Dimensions 1 T - ORI\JZ- 4,384 TJC-25 - Metals, Cerartiics, and Materials 1-2. 3. 4-23. 2,. 2'5 * 26. 27. 28. 2'3. 30. 81. 82. 83. 84. 3140 INTERNAL DI S TR IBUTIGK 1'33 , J.04, 105. KG--115. 11.6 . 117. llt3. 119. 1.20 . 121. Centra 1 Res eareh Library GRI!IL Y- 12 Technical Library Docmen 1; Refer exie e Sect ion Wooratory Records Depa.rtment Laboratory Records, OLNJ-RC: ORNL Patent 0:Cfi.e e 12. M. Adamson, Jr. G. E. Boyd w. 1). Brown G. W. Clark J. E. Cunningham C. V. Dodd J. 3 Frye, Jr. W P"ul.ker s on J. L. Gregg W. 0. Harm 8 547. 82 * 89. go. 91_. 92. 93. 34. * 95 * 96. 9'7 . 93. 97. 100. 101. 102. 122. 123. 124. 125-126. 1.27-128. 129. 130. 131, 132. 133-32'7- EXTERNAL DISTRIBUTION M. R. Hill C. E, Larson 11. G. i!~2cPherson D, L,. €Jason, Y-12 R. W. McClimng W. L. Moore H. 14. SmLYn K, E. Spear N. G. Spoeri D. A. SLmiYbeere; A. M, Weinberg V . 13. e I?. Vpplllu-ri c. M. ~ciarns, ,rr, (~onsl~itant) Leo Brewer. (consulhnt) 1,. S. Darken (consultant) J. A. Krl-uninansl (consultant) S . Aveyard, AERIE-Hxrwell, Didcot, Berkshire, Ehgland R. L. Brown, Jr., GE-Kanford D. F. Cope, RDT, SSR, AEC, Gak. Ridge IVati.ona.1 Laboratory W. E. Deeds, 'The TJriiversity of Tennessee Robert C e c.rubTriskas, Department of the Army, &rr!y Materials Donald 2. Green, ADTL, Hanford W. J. hrlrin, AN!, Oak Ridge Operations X L. Libby, (;E-Hanf or8 J. W. Lu-quire, The Tiniversi-ty of Tennessee M. C. McIlwain, K-QTJAL-AKA, C:. C. ivhrsha1.1. Space Fl.ight Center, 5. F. Pferce, 'L'h.e IJniversity of Tennessee C. J. Renken, Prgonne National Laboratory Roy S. Sharpe, AERE-Han-ell., Didcot, Berkshire, England. J. M. Simmons, AX, Washlrinqton E. E. Stambury, Tne gnriversity of Tennessee lj. K. Steveiis, AEC, Nashington J. A. Swartout, Union Carbide Corpomtion, New York D. L. WaideI-ich, Depa.rtn1en-k of ElectricEtl. Engineering, Iaborstory and University Divi-sion, AEC, Oak Ridge Operations Given distribution as sliown in ~~i1-45Oo under Net,als Cerarni-cs , arid Materials category and Mechanics Research Center", Watertown, Mass .) 0%2.'72 blarshall. Center, 4-l.a. 35812 University of .Missou.ri, Colrmbia, Mo. 6520.1