MICROWAVE EDDY-CURRENT TECHNIQUES FOR QUANTITATIVE NDE. A. J. Bahr Stanford Research Institute International Menlo Park, CA 94025

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1 MCROWAVE EDDY-CURRENT TECHNQUES FOR QUANTTATVE NDE A. J. Bahr Stanford Researh nstitute nternational Menlo Park, CA 9425 ABSTRACT The objetives of this work are to develop an eletromagneti sattering model that an be used to predit the sattering from a rak in a onduting surfae and to evaluate the feasibility of using this odel in onjuntion with mirowave-measurement tehniques to determine the dimensions of suh a rak. Suh a theory has been developed, and its derivation is presented. Theory and experiment are ompared for retangular slots measured at 1 GHz, and the agreement is found to be good. The neessary measurement protool for determining the dimensions of a rak is disussed, and an example of inverting the measured data to determine the dimensions of a retangular slot is presented. NTRODUCTON n the low-frequeny eddy-urrent testing of metals, urrents are aused to flow in a test speimen by plaing it in the magneti field of an indution oil. The flow of urrents is affeted by the eletrial properties and shape of the test speimen, and by the presene of disontiuities and defets. n turn, these urrents reat on the exiting oil and affet its impedane. Thus, the presene of a defet is determined by monitoring the test oil impedane. Suh eddy-urrent tests are typially onduted at frequenies of less than 1 MHz where indution fields predominate and the eletromagneti wavelength is greater than 3 m. However, in quantitative NDE, where it is desired to obtain the defet dimensions from an analysis of the measured data, the use of suh low frequenies does not provide the degree of sensitivity to hanges in defet dimensions that is neessary for obtaining an aurate determination of these dimensions from an inversion of the eddy-urrent data. The problem of obtaining suffiient auray beomes more diffiult as the flaws of interest beome smaller. This problem would be alleviated if.higher frequenies were to be used in eddy-urrent inspetion. Thus, the work reported here addresses the possibility of onduting eddy-urrent measurements in the mirowave*frequeny regime (1 GHz to 1 GHz). Previous work 1 2 using frequenies in the range 1 GHz to 3 GHz has shown that good sensitivity to small raks an be obtained, and that there is a lear orrelation between rak depth and the deteted signal. n using mirowave frequenies, the radiation fields assoiated with the sensors beome an important onsideration, and the physis involved is best desribed in terms of fields and waves. For example, a defet should be thought of as produing a hange in the sattering of eletromagneti waves from the metal surfae. t should also be noted that, sine the use of mirowave frequenies auses the urrents indued in the test objet to flow essentially on the surfae (i.e., the skin depth is typially less * Referenes are listed at the end of the paper. 12 than 1 mat 1 GHz), mirowave eddy-urrent tehniques are limited in metals to surfae inspetion, e.g., to detetion and haraterization of surfaebreaking raks. n order to invert the measured mirowave eddy urrent data to obtain rak dimensions, it is neessary to have a theoretial model that relates the eletromagneti sattering from a rak to the rak dimensions. The requisite theory should be variational so that approximate solutions for irregular rak geometries an be obtained. n addition, the theory should not be restrited to any partiular frequeny range so it an be used to larify any distintions between onventi nal (lowfrequeny) and mirowave (high-frequeny) eddyurrent tehniques. Finally, suh a model would be useful for establishing an optimum measurement protool. A suitable general theory has been developed, and its derivation is outlined below. Then, as an example, the theory is applied to ross-polarized baksattering of a plane wave from a retangular slot in an aluminum plate, and the measurement protool neessary to determine the slot dimensions is disussed. The results of this theoretial example are ompared with experimental results obtained at 1 GHz and are found to be in good agreement. Finally, graphial inversion of the theoretial eletromagneti sattering from the sloi is performed to illustrate the.proess of obtaining the slot dimensions from the measured data. A THEORETCAL MODEL FOR ELECTROMAGNETC SCATTERNG FROM SURFACE-BREAKNG CRACKS N METALS A general eletromagneti sattering-measuremer system is shown shematially in Fig. 1, whih illustrates the general bistati ase where the transmitter and reeiver are separated. The probes are arbitrary, but it is assumed that a single eletromagneti mode propagates at some point in the transmission line(s) [waveguide(s)] that onnet the probe(s) to the transmitter and reeiver. t is also assumed that the metal shields and test body exhibit finite ondutivity. The starting point of the theory is the Lorentz reiproity therem,3 whih involves an integral

2 / / SURFACE 1:-- AT NFiNTY FGURE / METAL BODY UNDER TEST ', ' ARBTRARY PROBES ' ' 1-;; s.. / / GENERAL ELECTROMAGNETC SCATTERNG-MEASUREMENT SYSTEM over the losed surfae, S, equal to Sa + Sb + Sa + S + Sm. f there are no soures enlosed within the volume defined by S, the theorem takes the following form: JJCF. x H' - x Hl ;; ds = o s (1 l where n is a unit vetor that points gutwarg from the enlosed volume. The quantities E and H are the eletri and magneti fields, respetively, that exist on the surfae S. The unprimed and primed fields are defined by the following onditions: Unprimed field--no rak is present, "a" is a reeiver, "b" is a transmitter. Primed field--a rak is present, "a" is a transmitter, "b" is a reeiver. n addition, if gyromagneti media exist within the losed volume, all de magneti fields within the volume must be reversed in the two ases. 4 The evaluation of the surfae integral in Eq. (1) requires a knowledge of the fields on the various parts of the surfaes. First, on all the metal surfaes the tangential eletri field an be related to the surfae magneti field by means of a surfae impedane, Zs: For the plane waves, Zs is related to the skin depth, as, by the well-known tormula l+j where a is the ondutivity of the metal. Seond, on the surfae at infinity one has the following radiation ondition; (2) (3) E jwn x (n x A) (4a) ti -j w (n x A) no (4b) where w is the radian frequenfy, no is the intrinsi impedane of free spae, and A is the vetor potential. Third, sine single modes are assumed to propagate in the transmission lines, it an be shown 5 that the integrals over the transmission-line ross setions redue to the following form:... X JJCF. H' - E' X H) n ds = -4PAra (Sa) sa Jj(E" X H' - E' X it). n ds 4PArb (5b) sb where P is the power available from the transmitter and r i the refletion oeffiient of the propagating mode in the transmission line. Finally, the unknown eletri field in the rak mouth an be expressed as a fititious magneti urrent, M', viz., By using Eqs. (2), (4), (5), and (6), one an onvert Eq. (1) into the following form: 4PA(rb-ra) = -Zs Jf(Ht H) ds + JJ(M' H) ds s s Equation (7) expresses the differene between the sattering oeffiients measured with probes "a" and "b" in terms of fields that exist in the rak mouth when it is either open or overed by a ondutor having surfae impedane Zs. Hene, to relate this theoretial result to an experimental measurement, the measurement system must be apable of measuring this differene in sattering oeffiients. n onventional eddy-urrent systems, one usually measures the hange in impedane, Z'-Z, of a probe as it passes over a rak, rather than the hange in sattering oeffiient. n the monostati ase, the distintion between "a." and "b" disappears, and one finds from transmission-line theory that the hange in sattering oeffiient and the hange in impedane are related by (6) (7) 4PA (r'-r) = ' (Z'-Z) (8) where and ' are the total urrents flowing in the transmission line without and with a rak present, respetively. However, the sattering oeffiients will be retained in the present development beause they are more fundamental to a wave analysis. Equation (7) is a linear integral equation that relates the unknown refletion oeffiient, rb, to the unknown tangen_tial magneti field, Ht', and magneti urrent, M', in the rak mouth. The quantity H is the magneti field that exists on the surfae of the metal test objet in the absene of a rak. One way of solving this equation is to use the method of moments. 6 Suh a solution possesses the variational harateristis 7 that are desired. 13

3 n the moment-method sol tion, one expands M' in a set of basis funtions, Mn: N M' = L Vn Mn (9) n=l Te only onditions on the Mn are that they be l1nearly 1ndpendent, and that their superposition approximate M' "reasonably well" (herein lies the "art" in the method of moments). One also needs to invoke the ondition that the tangential magneti field be ontinuous aross the rak mouth; i.e., ( 1) FLFFCO Yn t, s {', ff(am H) (aw ) :; no ( 14a) 4PA ff( H) ds s (14b) J"fM H)(aw ) ds [fa; (t ) A ds iff(a.m rt) (aw. H) ds s ds (14) where A 1 is the inident magneti field, (M') is the indaed+magnti field just outside the rak outh ad Hf (-M') is the ind ed magneti field Jgst 1ns1de the ra.k mouth. He is a funtion of -M' rather than of M' beause of the equivalene piniple. 6 Also, beause the Ht operators are lnear, Eq. (9) an be substituted into Eq. (1) to g lve the result N +"""" LJ V n. ( 11) t n=l Next, one hooses a set of. testing funtions, W m!hat are similar (but not neessarily eual) to the Mn. By taking the dot produt of eah Wm with Eq. (11) and integrating that produt over the rak mouth, one obtains the following result: -ffw Hi ds S m t Thus, Eq. (1) has been onverted into a set of salar inhomogeneous linear equations, whih thus an be solved for the unknown oeffiients, V. n ( 12) The result of eliminating the V in Eq. (9) and substituting the result into Eq. (7)nis best expressed in matrix form: +.t[yr + ] -1 ij} + [Y r + y J -1 C i J} ( 13) where the normalized oeffiients and matrix elements are given by 14 yr +r -Jfw (Mn) ds ( 14d) mn S m Ht yc + -Jfw (Mn) ds (14e) mn S m. Ht {f(wm Hi) ds i {', and (14f) m {f(a,. A) (aw +i) Rt ds en {f(mn H) ds ( 14g) f(am H) (aw. A)ds Here±aM and aw are unit vetors pointing along M and W, respetively. Equation (13) applies to all eletromagneti eddy-urrent measurements of raks. At this stage, no restritions have been made on the operating frequeny or the distane between the probe(s) and the sample. The first term in the equation--i.e., the one involving Zs--an be interpreted as the hange in sattering aused by removing a small volume (the rak) of metal that has finite ondutivity. The seond term an be interpreted as the hange in sattering aused by.energy being stored in the rak and being reradiated. The oeffiient FLFFo ontains lhe effets of hanging the distane between the probe and the sample surfae (lift off), and of the rak orientation. At low frequenies, one finds that the finite-ondutivity term dominates; at high frequenies, where the rak beomes resonant, the effets of energy storage beoma predominant. Thus, the theory provides a lear distintion between onventional and mirowave eddy-urrent tehniques. AN EXAf 1PLE To illustrate how Eq. (13) an be evaluated in a speifi ase, onsider the simple ase of perpendiular plane-wave exitation of a retangular slot ut in a perfetly-onduting plane. The geometry of suh a slot is shown in Fig. 2. Assuminq the reeiver is ross-polarized to the transmitter, ra = and Eq. (13) beomes

4 Fo sin 28 {19b) l )2 tan e _4_ k/ab sin 2 (a) and (19) i 4 1 )2 tan e k;ab sin2(k4a) (19d) FGURE 2 RECTANGULAR SLOT GEOMETRY ( 15) Here, the quantity C1C is a slot oupling fator that gives the frequeny dependene of the oupling between the slot and the inident field. The normalized slot admittane, n Y, is a slot parameter that 5 is independent of the ex1tation. t is important to note that Ys is the sum of a radiation admittane, Yr, that depends on the boundary onditions external to the slot, and a avity admittane, Y, that depends on the geometry inside the slot. Thus, if the geometry of the slot avity hanges, but not the geometry of the slot mouth, only the avity admittane needs to be realulated. To alulate the avity admittane for a retangular slot, one an expand Rf (M 1 ) in transverseeletri waveguide modes. Then Eq. {14e) beomes where the slot admittane is Y 5 Yr + Y and it has been assumed that only one bas1s funtion is needed to approximate the fields in the slot mouth. n this ase, one an take advantage of the knowledge that exists onerning the solution for the fields in the aperture of a narrow-slot antenna. 8 This solution should provide a good resonane-region approximation to the fields in the mouth of a slot that is deeper than it is wide. Hene, one hooses the basis funtion and test funtion to be -+ - Ml = ax b 1 Sln. [ k (a a )] X - 2 = Wl ( 16) where ax is a unit vetor along the x-axis, k is the wave number, and the slot dimensions and oordinate system are defined in Fig. 2. Use of this approximate funtion onverts the moment-method solution to a perturbation solution. Now, for a plane inident wave with magneti field Ho. one has ;;nd ax Hi t H os 8 ( 17) - it 2H sin ax = e (18) where e is the angle between H and the x-axis and it has been assumed that there are no refletions from the probe. Thus, using Eqs. (16), (17), and (18) the oeffiients of the normalized slot impedane (1/oYs) in Eq. (15) beome 2 H ab FLF = * The term resonane region refers to frequenies where the slot length is equal to or greater than one-half wavelength. (19a) 15 where {21) This result differs slightly from that obtained in Ref. 5 beause of a differene in the definition of slot voltage. The radiation admittane Eq. (14d) an be alulated by expanding the aperture fields in a plane-wave spetrum.9,lo Assuming kb<<l, one has where Re(Yr) = (l/rr ) m{yr) { Cin (ka) + {Cin (ka) - 1/2 Cin (2ka)] os ka - [Si (ka) - 1/2 Si (2ka)] sin ka} (l/rrn ) {si (ka) + [Si (ka) - 1/2 Si (2ka)] os ka + [Cin (ka) - l/2 Cin (2ka) ln 3 sin ka} X Cin (x) J 1 - os u du and Si (x) j si u du and (22a) (22b) (23a) (23b) Equations (19), (2), and (22) show expliitly how slot oupling and slot admittane depend on the frequeny and the dimensions of the slot. n order

5 to show expliitly how the liftoff fator [Eq. (l9a)] depends on these parameters, it is neessary to relate Ho and PA. For example, if the soure antenna were equivalent to a magneti dipole loated at a large distane, R, from the slot, one would have (24) Hene, if k is large (wavelength is small), the dominant effet of hanging R will be to hange the phase of the refletion oeffiient. Thus, the lous of the refletion oeffiient in the refletionoeffiient plane as R is hanged will be a nearly irular ar. This behavior an be used to disriminate between the signals produed by variations in liftoff and by a bonafide rak. Having obtained Eqs. (19), (2), and (22), it is now possible to alulate the ross-polarized sattered power given by (25) as a funtion of the frequeny and slot dimensions. t is onvenient, however, to normalize the sattered power first to suppress the dependene of the result on the harateristis of the probe and slot orientation, viz., (26) The right-hand side of this equation is plotted (in db) in Fig. 3 as a funtion of the produt of frequeny and slot length, with the ratios of slot width and depth to slot length as parameters. n this figure, ka/n l defines the resonane region where eletromagneti energy an propagate into the slot with low attenuation. n this region, the sattering is seen to be a strong funtion of slot depth, whih is a desirable harateristi from the standpoint of obtaining an aurate determination of depth from a sattering measurement. For frequenies below the resonane region, the fields inside the slot are evanesent, and so the sensitivity of the sattering to hanges in slot depth dereases rapidly as the slot approahes one slot length in depth. Thus, eddy-urrent measurements for determining slot depth quantitatively are best onduted in the resonane region. t should be noted, however, that more than one slot depth an give the same value of sattered power in this frequeny region, and so it may be neessary to ondut measurements at more than one frequeny to resolve this ambiguity. So far, this example has negleted the ontribution of the surfae-impedane term to the sattering. ndeed, in the ross-polarized ase, C = in Eq. (13), and the other term ontaining Zs is small for most metals. On te other had, in the o-polarized ase, C = l, and CO = 2 os e. At very low fre uenies, 1/Ys, and the quasi-stati sattered power, PQ, 1s determined entirely by the surfae impedane: 2 (.!) 1 2 a: w 1-1- od: u (/) w N a: od: _J.. J, -1 (/) a: u w N :::::i -2 od: ::E a: z a/b FGURE 3 NORMALZED FREQUENCY - ka/rr NORMALZED CROSS-POLARZED POWER SCATTERED FROM A SLOT AS A FUNCTON OF NORMALZED FREQUENCY Of ourse, as the frequeny is inreased from zero, the energy stored in the slot also ontributes to the sattering. For a deep slot.(d/a 1), the Rayleigh sattering term, PR, an be approximated by expanding Eq. (15) for small ka. The result is (28 These two normalized sattered powers are plotted as funtions of frequeny in Fig. 4. n making the omputations it was assumed that a = 2.5 mm and that the material was aluminum with Zs = 3.26 x lo-7 v'"f (l + j) ohms, where f is the frequeny. This figure learly shows the dominane of the surfaeimpedane term (quasi-stati term) at low frequenie and the dominane of the energy-storage term (Raylei term) at high frequenies. The rossover ours in this example at about 1 khz. t is important to note that neither of these low-frequeny approximations to the slot sattering ontains any depth information. Hene, one onludes from this example that eddy-urrent measurements of rak depth are best onduted at frequenies where the wavelength is ommensurate with the rak length. 16

6 al " -4 r: w w r: -1 w f- f- <X: -12 u en -14 w N...J -16 <X: :2: r: -18 z FGURE 4 AN FREQUENCY - Hz COMPARSON OF RAYLEGH AND QUAS-STATC SCATTERNG FOR A DEEP SLOT DEALZED MEASURH1ENT PROTOCOL Equation (13) provides the basis for defining an idealized eddy-urrent measurement protool for determining rak dimensions. This protool an be divided into four main steps: ( 1 ) Calibrate the system at eah measurement frequeny using a "standard rak" to determine the lift-off fator, (2) Detet the real rak while keeping the distane between the probe and the speimen the same as in the alibration, (3) Measure the rak in at least two different orientations to determine the orientation fator (assuming the rak length is muh larger than the rak width), (4) Collet suffiient data to permit unambig-. uous inversion using the model to obtain the rak dimensions. (deally, a minimum data set would onsist of amplitude and phase at two frequenies--the use of more frequenies may be required in the resonane region in order to resolve ambiguities). Sine the data will not be perfetly aurate and the rak geometry will not be known preisely, it is likely that statistial tehniques, adaptive learning tehniques, or both, will be required to obtain suffiient auray for rak dimensions determined from eddy-urrent measurements. n any ase, Eq. (13) should provide a useful basis for designing experiments. EXPERH1ENT The amplitude and phase of the ross-polarized baksattering from a series of retangular slots in an aluminum plate were measured using the mirowave system whose shemati diagram is shown in Fig. 5. This system uses an orthomode oupler to disriminate against o-polarired baksatter, and a homodyne detetion system to provide in-phase () and quadrature (Q) output signals. The sensitivity of this 17 system is urrently about -75 dbm; this sensitivity is determined by the degree to whih the transmitting and reeiving portions of the system an be isolated in the absene of a rak by the orthomode oupler. The antenna used in the system is a lens-foused horn with a beamwidth at its foal point of about 3.5 mm at the operating frequeny of 1 GHz. An aluminum plate with six slots of different sizes eletrodisharge-mahined into its surfae was prepared aording to the layout shown in Fig. 6.*. Slots 1, 2, and 3 have a ross setion (ax b) of 2.54 mm x.25 mm; slots 4, 5, and 6 have a ross setion of 1.27 mm x.25 mm. Thus, a/b = 1 for the first set of slots, and a/b = 5 for the seond set of slots. Also, at 1 GHz, ka/rr = 1.7 for the first set, and ka/rr =.85 for the seond set. Finally, slots 1 and 4 were speified to be.25 mm deep, slots 2 and 5,.5 mm deep, and slots 3 and 6, 1. mm deep. The measured in-phase and quadrature voltages obtained at 98 GHZ by translating the slots through the mirowave beam are shown in Fig. 7. The plate was aligned perpendiularly to the mirowave beam and was positioned so that a linear translation of the plate aused the enters of the slots to pass through the enter of the beam, thereby maximizing the peak signal obtained from eah slot. The slots were aligned with their lengths at an angle of abou 6 to the eletri polarization vetor, thus ensur1ng that some of the inident energy would be oupled into the ross-polarized mode by eah slot. The in-phase and quadrature voltages were ombined to form a polar display on a storage osillosope--ommon pratie in low-frequeny eddy-urrent work. The orresponding polar display for slots 1, 2, and 3 (the 2.5 mm-long slots) is shown in Fig.?(). This type of display learly shows the differenes in the amplitudes and phases of the sattered signals produed by the different-depth slots. n tis ase, the signal produed by slot 3 (1. mm deep) 1s very different from the signals produed by the other slots. However, all three signals are learly distinguishable. The approximate model (desribed previously that assumes a sinusoidal distribution of eletr1 field in the slot mouth an be used to alulate the theoretial slot response for the parameters used in the experiment. For a foused mirowave beam, the hange in exitation of the slot aused by moving the slot through the beam an be approximated by setting (29) where k is the wave number, x' is distane along the sanning diretion measured from the enter of the slot, Ro is the distane between the mirowave len and the aluminum plate, and J 1 is the esse fut1on of first kind and first order. Also, 1n th1s s1mple model, it is neessary to assume that the inident field is onstant over the slot mouth for eah position x'. *This test plate was prepared under the diretion of Dr.. Buk of the Rokwell nternational Siene Center, Thousand Oaks, California.

7 RF GENERATOR SOLATOR RF r------, A TTENUATOR r------, TUNER N-PHASE OR THO POWER MODE DVDER COUPLER PHASE SHFTER 3-dB QUADRATURE COUPLER 1 db COUPLER - 3 khz FERRTE MODULATOR SOLATOR N-PHASE POWER DVDER Transmit/Reeive Antenna (Dual Polarized) Y-Poaition Translation Sensor f::::::==-=:> X-Posilion Translation Sensor -----,.7 - Material Under Teat 1 qb COUPLER MXERS 3-kHz AMPLFERS Signal Proessing & Reord/Display FGURE 5 MCROWAVE SYSTEM FOR MEASURNG CROSS-POLARZED BACKSCATTER USNG HOMODYNE DETECTON ALUMNUM PLATE 5.4 em 15.9 em 7.94 ;;;;=;1---j---j_1,..4 mm x.25 mm t-3.94 em--t-3.94 m-j SLOT 1 SLOT 2 SLOT em 1 1 lslot 4 SLOT 5 SLOT 5 SLOTS mm x.25 mm +- ->-- _.. _1 SLOTS L _!T y SCAN AREA {7.62 em x 1.16 em) em FGURE 6 LAYOUT OF SLOTTED ALUMNUM PLATE (slots are aligned in the x diretion) 18

8 (a) N-PHASE VOLTAGE, V 1 (b) QUADRATURE VOLTAGE, VQ > > > >!)!) SLOT 3 SLOT 2 SLOT 1 SLOT 3 SLOT 2 SLOT 1 () POLAR DSPLAY FREQUENCY = 98 GHz vi- FGURE 7 MEASURED SLOT RESPONSE FOR SLOTS 1, 2, AND 3 The result of the alulation for slots 1, 2, and 3 is shown in Fig. 8. The absolute amplitude and phase is undetermined in this alulation; therefore, the theoretial plot has been normalized so that the peak response for slot 3 mathes the experimental value for that slot. The experimental peak values of eah slot response are indiated by Xs. A omparison of Figs. 7() and 8 shows remarkable agreement, onsidering the approximate nature of the model. The experimental results obtained for slots 4, 5, and 6 are shown in Fig. 9. n that measurement, the gain was inreased over that used for the larger slots and, as a result, liftoff effets beame notieable, as is evidened by the high bakground or lutter in Figs. 9(a) and (b). However, as expeted, the polar display [Fig. 9()] allows the slot signals to be learly distinguished from the liftoff signal beause the two types of signals are nearly orthogonal. n this ase, the length of the slots (1.27 mm) auses the operating frequeny of 98 GHz to lie below the resonane region, with the result that hanges in slot depth produe relatively little hange in the phase of the sattered signal. NVERSON n view of the good agreement between theory and experiment, it appears worthwhile to examine the measurement-error sensitivity of an inversion proess that is based on the simple model. Sine all the slots were loated at the same distane from the mirowave lens and had the same orientation relative to the polarization of the inident wave, it was simplest to use one of the slots (slot 3) as a referene slot for alibrating the system v ia the model. Slot 2 was hosen as the unknown slot whose dimensions were-being sought. Hene, the ratio of the measured omplex signal for slot 2 to that for slot 3 was ompared to the same ratio obtained from theory. deally, measurements at two frequenies that are far enough apart to produe measureable hanges in sattering are needed in order to determine all three dimensions of a slot, as was mentioned in the setion on measurement protool. However, the existing experimental system did not permit signifiant hanges in operating frequeny to be made; therefore it was neessary to assume that one of the slot dimensions was known. The slot length, a, was hosen for this dimension, as it is the most likely to be known. The orresponding theoretial response for these The amplitude and phase of the relative sattering from slot 2 at 99.9 GHz are shown in Fig. 11 smaller slots is shown in Fig. 1. n this ase, the theoretial plot was normalized to the experimental peak value for slot 6 after the lutter (lift parameter. t is assumed that a = 2.5 mm. Also as funtions of slot dpeth, with slot width as a off signal) had been subtrated. Again, agreement indiated in the figure are the estimated ranges for between theory and experiment is fairly good. 19

9 FGURE 8 THEORETCAL SLOT RESPONSE FOR SLOTS 1, 2, AND 3 (x indiates measured peak value) 2

10 (a) N-PHASE VOLTAGE, V 1 (b) QUADRATURE VOLTAGE, VQ > > : --> > LD LD d d SLOT 6 SLOT 5 SLOT 4 SLOT 6 SLOT 5 SLOT 4 () POLAR DSPLAY FREQUENCY = 98 GHz Vr - FGURE 9 MEASURED SLOT RESPONSE FOR SLOTS 4, 5, AND 6 21

11 FGURE 1 THEORETCAL SLOT RESPONSE FOR SLOTS 4, 5, AND 6 (x indiates measured peak value) 22

12 UJ :::> J a_ :2 UJ >...J UJ a: PHASE AMPLTUDE SLOT 2/SLOT 3 f = 99.9 GHz a = 2.5 mm b/a =.12.1 MEASURED.8 PHASE MEASURED ' 135 Ol Q) -o UJ 125 (f.) a_ 115 UJ >...J UJ a: D. dla (±11%) RELATVE DEPTH, d/a.25 FGURE 11 AMPLTUDE AND PHASE OF THE SCATTERNG FROM SLOT 2 (relative to slot 3) AS FUNCTONS OF SLOT DEPTH the measured data. The measurement errors orresponding to these ranges are: amplitude ±1 db; phase, ±4. One an see from the figure that, if the slot width an be estimated to within ±2. the measured data determines the slot depth to within ±11% (the ross-hathed area). t is interesting to note that, in this ase, the use of amplitude data alone would only inrease the unertainty in the depth determination by a small amount, namely to ±14%. The slot depth measured from a sanning eletron mirograph of a rubber replia of slot 2 is d/a =.21. This value lies approximately in the enter of the ross-hathed slot-depth range shown in Fig. 11. f the ross-setional dimensions of a rak annot be obtained by mirosopi examination or by some other means, all three of the rak dimensions must be obtained from the eddy-urrent measurement alone. The auray of the required inversion solution will depend on how sensitive the sattering is to hanges in eah rak dimension. Calulations using the model developed here show that this sensitivity depends on the produt of rak length and operating frequeny, with maximum sensitivity obtained in speifi portions of the resonane region. Thus, obtaining maximum auray in determining rak size will neessitate the use of a frequeny that is appropriate to the size range being measured. SUMMARY A general theory for the eletromagneti sattering from a surfae-breaking rak in a onduting material has been developed. The theory is valid for any frequeny, and provides a basis for defining a measurement protool for the purpose of determining rak dimensions from eddy-urrent measurements. The theory also shows that, at low frequenies, eddyurrent measurements of raks are dominated by effets of finite ondutivity while, at high frequenies, the measured signals are determined mostly by energy storage in the rak. Approximate numerial results have been obtained for the ase of a retangular slot. This example reveals all of the essential harateristis of the baksattering as a funtion of frequeny and slot dimensions, and gives insight into what an be expeted for the behavior of a signal sattered by a real rak. For example, when the slot length is greater than one-half wavelength, resonanes an our; these resonanes make the determination of slot depth from measured sattering aurate but introdue ambiguities. For smaller slot lengths, there is a one-to-one relation between slot depth and sattered signal, but the amplitude of the sattered energy is smaller and slot depths that are greater than one slot length are not well resolved. The effets of liftoff and slot orientation are also eluidated in the example. 23

13 Experimental results obtained at 1 GHz using eletrodisharge-mahined slots in an aluminum plate were found to be in good agreement with theory. The smallest available slot, whih was 1.27 mm long,.25 mm wide, and.25 mm deep, ould be distinguished from lutter (liftoff) by using phasesensitive detetion and a polar display. An example of using the measured data and the theoretial model to determine slot depth was given. One an onlude that it should be possible to obtain an aurate determination of the dimensions of a surfae-breaking rak from mirowave sattering measurements. Although further improvements in the theory are possible, questions onerning the pratial and eonomi realization of the tehnique should be addressed first. ACKNOWLEDGEMENT Tehnial disussions with Dr. B. A. Auld of Stanford University and Dr. R. B. Thompson of Rokwell nternational Siene Center were most helpful in this work. Thanks are also due to Dr. A. C. PhilTips of SR for designing the 3-kHz portion of the phase-sensitive detetion system used in the measurements. This work was sponsored by the Center for Advaned NDE operated by the Rokwell nternational Siene Center under Contrat F C-518. REFERENCES 1. R. J. Hruby and L. Feinstein, "A Novel Nondestrutive, Nonontating Method of Measuring the Depth of Thin Slits and Craks in t-1etals," The Review of Sientifi nstruments, Vol. 41, pp (May 197). 2. A. Hussain and E. A. Ash, "1irowave Sanning Mirosopy for Nondestrutive Testing," Proeedings of the 5th European Mirowave Conferene, Hamburg, Germany, pp (September 1975). 3. R. E. Collin, Field Theory of Guided Waves (New York: tgraw-hi 11, 196). 4. P. R. Misaa, "A General Reiproity Theorem," EEE Trans. 1irowave Theory Teh., Vol. MTT-27, pp (April 1979). 5. A. J. Bahr, "Using Eletromagneti Sattering to Estimate the Depth of a Retangular Slot," EEE Trans. Antennas Propagat., to be published. 6. R. F. Harrington and J. R. Mautz, "Computational Methods for Transmission of Waves Through Apertures," Eletroma neti Satterin, P.L.E. Uslenghi, Ed. New York: Aademi Press, 1978). 7. R. F. Harrington, Field Computation by Moment Methods (New York: Mamillan Company, 1968). 8. R.W.P. King and C. W. Harrison, Jr., Antennas and Waves: A Modern A}proah (The M..T. Press, Cambridge, Mass., D. R. Rhodes, "On a Fundamental Priniple in the Theory of Planar Antennas," Pro. EEE, Vol. 52, pp (Sept. 1964). 1. D. R. Rhodes, "On the Stored Energy of Planar Apertures," EEE Trans. Antennas Propagat., Vol. AP-14, pp (Nov. 1966). 24