Analytical Tnvest iga t ions of Electroma!:netir Fields in Mine Environments 

Ilnlimited, U.S. Cov't. work not protected by [J.S. Copyright. 

SIIPF'I~EMENTAR Y NOTE LSPONSORTflG ACT TVITY 

The views and conclusions con- U.S. Eureau of Mines, 

tained in this doc:ument are those Pittsburgh Mining and Safety 

of the authors and shotlld not he Research Center 

interpreted as necessarily repre- 1J.S. BOM Technical Project Officer, 

senting the official policies or Dr. H.K. Sacks 

recommendations of the Departments 

of Interior and Commerce. 

This rcport contains a summary of analytical research on electromagnetic 

methods of nondestructive testing of cylindrical conductors. The principal 

oh-jective is to providc theoretical insight to the operation of electromagnetic 

sensing of the internal properties of wire ropes that are used extensively in 

mine hoists. An overall review of past work is also included. 

KEY WORT),? 

Nondestructive testing, nondestructive evaluation, electromagnetic probing, 

mine safety, cable testing, electromagnetic induction. 

Section 1 

Here we present an overall summary of the principal achievements during the course of the investigations carried out under this phase of the 

researc,h contract. 

In Section 2 we give an overall summary of the Non-Destructive Methods 

that utilize the interaction of electromagnetic fields with the sample. 

Some specific topics are: Some Early NUT Work in England, The 1mportar.t 

Efforts by Semmelink in South Africa, More Recent Applications and Practices, 

Early Theoretical Investigations, Related Investigations in Geophysical 

Prospecting, More Recent Analytical Studies, An Alternative Formulation for 

Solenoid Excitation, The Dual Problem, and finally we introduce The Prolate 

Spheroidal Void Model of Hill and Wait that is covered in more detail in 

later sect ions. 

In Sectlon 3 we deal with the electromagnetic theory of an infinitely 

long cylinder for the case when the electrical conductivity and the magnetic 

permeability arc uni-axial tensors. This is an idealized yet relevant model for a stranded wire rope or cable that is to be excited by an external 

alternating-current source. The general and special forms of the solution 

are discussed in the context of non-destructive testing (NDT) of the rope. 

The method of deducing the electromagnetic response for a finite soJrce is 

described in the special case where azimuthal symmetry prevails. 

In Section 4 a remarkably simple and novel solution is obtained for 

the fields induced in an anisotropic cylindrical shell that is located coaxially within a long solenoid. This could be the basis of a non-destructive 

measuring scheme for stranded wire rope when we have actually accounted for 

the winding geometry in a relatively crude fashion. 

In Section 5 we extend a previous analysis for the series impedance of 

a long solenoid to allow for the cylindrical layering of the encircled conductor. The results are discussed in the context of non-destructive testing 

of stecl ropes that may 1lC-ive external or internal corrosion. It is shown 

that even an internal layer of reduced conductivity and permeability will 

be detectable if the frequency is sufficiently low to permit penetration of 

the primary field. 

In Section 6 we present a self-contained analysis for the impedance of 

a solenoid that encircles a conducting cylinder that has an internal flaw 

or anomaly that is also cylindrical in form. A perturbation method is used 

to obtain an expression for the fractional change of the impedance as a 

function of the size and location of the anomaly. 

In Section 7 we consider the effect of a thin prolate spheroidal void 

in an infinite conducting circular cylinder. This is used as model for a 

broken strand in a wire rope. The rope is excited by an azimuthal magnetic 

line current which is a model for a thin toroidal coil. The anomalous 
external fields are computed from the induced electric and magnetic dipole 

moments of the void. 

In Section 8 we consider again the thin prolate spheroidal void in an 

infinite conducting circular cylinder. But now the rope is excited by an 

electric ring current which is a model for a thin solenoid or multi-turn 

wire loop. The anomalous external fields are computed from the induced 

electric and magnetic dipole moments of the void, For this type of excitation, the induced axial magnetic* dipole moment is the dominant contributor 

to the scattered field. 

In Section 9 a stranded wire rope is again idealized as a homogeneous 

conducting and permeable cylinder of circr~lar cross section and of infinite 

length. The rope is excited by a coaxial solenoid or finite length multiturn coil that carries an azimuthally directed alternating current. The 

novel feature here is that the rope and the enclosing solenoid may have a 

uniform velocity relative to each other. Using a non-relativistic analysis, 

the nature of this dynamic interaction is examined and numerical results 

are presented for parameter values that are relevant to both static and dynamic conditions in non-destructive testing of such cylindrical conductors. 

It is shown that the dynamic interaction with the rope specimen is not appreciably modified from that for the static condition unless the motional 

velocity is somewhat greater than about 10 m/s. 
INTRODUCTION 

Wire ropes are used extensively in many life sustaining situations. 

Elevator and mine hoist cables are two notable examples but we might also 

mention the support cable for aerial t'ramways, ski chairlifts and gondolas, 

helicopter and suspension cables. In this review, we will restrict attention to wire ropes used in mine hoists. There is an obvious need to perform tests of the integrity of such ropes without in any way impairing 

their function. Apart from careful visual examination and measurements of 

the external diameter, the non-destructive test methods available utilize 

electromagnetic fields, x-rays or mechanical waves. Here we will review 

progress in the electromagnetic methods, 

The early history of the subject will be described briefly since this 

provides a remarkably good introduction to the working principles. We will 

then progress rather quickly in time up to the currently used techniques 

and operating procedures. Next we drop back in time to summarize some of 

the basic papers that deal with the basic concepts and the techniques for 

dealing with the testing of cylindrical conductors by both electric and 

magnetic methods. At this juncture, we call attention to the extensive 

related work on electromagnetic. probing of geophysical targets such as ore 

bodies and other sub-surface conductors. Finally, we turn to the various 

investigations, primarily of a theoretical nature, that have been carried 

out; wz include here the most recent work, in the later sections of this 

ill(erirn stlnuniir-tr report we tltlal wit11 sl,ecific aspe(\ts in more detail. 
SOME EARLY NDT WORK IN ENGLAND 

Wall [I] gives an early account of the design and operation of electromagnetic rope testers. The test results were given for specially constructed ropes. Using basically an A.C. technique, his method makes it 

possible to detect broken wires within a rope that represent less than 5% 

reduction in the total cross-sectional area. As the author indicates, the 

investigation was motivated by testing of wire ropes used in collieries in 

England. But the author also indicates that other important applications 

are to tram cables, airship mcoring ropes, and to suspension bridge ropes. 

Wall is an early proponent of the A.C. method as opposed to D.C. magnetic testing favored in more recent times. He lists some of the advantages 

as Follows: a) the rope need not continuously move through the magnetic 

system, b) the A.C. signal is easy to detect and c) any remnant magnetism 

due, say, to a previous test is wiped out. 

I!c is interested in testing colliery ropes of the locked coil type up 

to 2% inch diameter. In such locked ropes, the outer and one or more of the 

inner layers are locked together to form flexible sheaths. This constrains 

the rope to maintain its circular form as it passes over the winding drum. 

He then quotes a classical skin depth formula that gives a guide to the 

penetration of the A.C. currents into the core of the rope. He points out 

cogently that, once the A.C. energy penetrates through the outer locked 

coil sheaths, it will be "practically" uniformly distributed through the 

remainder of the cross-section since the wires are so effectively stranded. 

The tests were made by means of a laminated iron yoke that has projecting limbs with machined tunnelled holes as indicated in Fig. la. The 

sample length of the rope was fixed concentrically in the holes. A search 
coil of 20 turns was wound around the center portion of the rope that is 

midway between the projecting limbs. The relationship between the flux 

density in the rope and the peak value of the magnetizing excitation at 

20 and 50 Hz was measured. Wall was able to ascertain that the flux penetration was almost complete at 20 Hz but only partially so at 50 Hz. 

In the same pioneering paper, Wall studies the effect of mechanical 

strain on the magnetic permeability with special reference to A.C. excitation. He also examines the change in the reluctance of the air space due 

to the eccentricity of the rope in the magnet system. 

He concludes that "whilst a flaw of about 3% of the total cross section 

is detectable, a flaw of about 5% produces a pronounced effect, whilst a 

flaw of about 18% gives a very striking disturbance of the record". Some 

of Wall's results are summarized in Fig. lb. 

In a follow-up study, Wall and Hainsworth [2] investigate the way in 

which the flux is distributed within the rope. As they point out, this 

information is relevant to the estimation of the depth of a flaw in a locked 

coil or similar rope as used in collieries. They feel a mathematical approach 

to the problem is intractable. Instead, they build up a special sample or 

physical model with embedded search coils. 

The experimental configuration was chosen to be a replica of a locked 

coil rope except that the layers of wires were not so close as in actual 

ropes. Straight mild steel wire rods of 118" diameter were employed; each 

rod was coated with insulating enamel before assembly. A first layer or 

central group of 7 wires was formed and a search coil (A) of 200 turns was 

arranged to embrace this group. A second layer of 20 wires was then added 

and the search coil (B) embraced this layer. Next came the third layer 

with a further embracing search coil (C). Finally, a fourth layer of 44 
wires was formed and the search coil (D) embraced it. Each of the four 

search coils had 200 turns. Then, by connecting pairs of consecutive 

search coils in opposition, it was possible to measure the flux magnitude 

associated with each layer. 

The differences between the waveform of the induced EMF'S in the 

various layers are very striking as indicated in Fig. 2. Also, as indicated in Pig. 3, the B-H response curves are shown for central group and the 

surface layer. Here the screening effect for the central group is very 

apparent. For the surface layer, the situation is somewhat different since 

apparently the A.C. excitation has the effect of increasing the apparent 

permeability. To some extent this tends to counteract the eddy current 

screening. 

THE IMPORTANT EFFORTS BY SEMMELINK IN SOUTH AFRICA 

Semmelink 131 gives an extremely interesting account of the early 

history of electro~nagnetic testing of wire ropes in South Africa. He 

indicated that in a paper read to the Transvaal Institute of Mechanical 

Engineering in 1906, Mr. C. McCann proposed an "electrical apparatus" for 

ascertaining the cross sectional area of wire ropes. In this scheme, a 

detachable coil around the rope was supplied through a stepdown transformer 

from tile mains supl~ly. A lami-nated yoke completed the magnetic circuit and 

an ammeter was connected in series with the coil. It was claimed that 

f I the current falls exactly in proportion to the size of the rope". 

Semmelink was also well aware of T.F. Wall's [I] work in England that we 

described above. He points out that the disadvantage of Wall's method is 

that the rope speed must be very slow due to the very low frequencies 

being adopted. Furthermore, the locked type ropes and the use of high 
A.C. flux densities causes severe heating of the rope. 

The first experiments described by Semmelink were conducted in 1946. 

A magnetizing coil of twenty turns of welding cable was used carrying a 

current of 100 amps at 50 Hz together with a co-axial search coil of 

several thousand turns. The rope was passed through the centers of both 

coils. It was found that all visible corrosion and broken wires caused 

voltage variations in the search coil that were measured by a peak reading 

voltmeter and an oscilloscope^ Semmelink [3] then tested about fifty main 

winding (hoist) ropes. The tensile strength of the wire was either 1231134 

tons (2000 lb.) per square inch or 128/140 tons per square inch. The ropes 

have six strands of approximately thirty wires laid on a sisal core which 

is impregnated with lubricant. The diameter of the ropes varied from la 

inches to 2 inches. The ropes were similar to the type shown in Fig. 4. 

The steel was ascertained to have the following magnetic properties: relative permeability = 44 for a magnetizing force less than 1 oersted, maximum 

permeability = 320 for a magnetizing force of 25 oersteds, maximum flux 

density = 14000 gauss for a magnetizing force of 200 oersteds, retentivity 

= 9500 gauss, coercive force = 20 oersted, hysteresis loss = 0.35 watts 

per c.c. at 50 Hz. He found, what is now generally known, that an increase 

of tensile stress in the wire causes an increase of permeability. For 

example, a stress of 65 tons per square inch causes an associated increase 

of permeability of 14% even for a magnetizing force of less than 1 oersted. 

He also noted that twisting of the wire caused a decrease of 8% of permeability for 180" of twist per foot of wire. Semrnelink [3] also measured 

the chililge of D.C. resistance of the wire rope for stresses varying from 

zero to 70 tonslsquare inch. Only 1.1% increase was noted over this range 

that was in accord with the percentage increase of length due to elasticity. 
the reactive component E and the resistive component ER can be balanced X 

out in an appropriate adjustment of the potentiometers. Thus, the "output" 

for a given rope can be indicated on a double pen recorder. 

The rope speed during a test depends on the response of the recorder 

and of the detector circuit. Furthermore, due to the choice of a low 

operating frequency (e-g. 80 Hz), the response of the detectors can not be 

fast and Semmelink chooses a time constant of 0.1 second. Thus, if the 

shortest variation to be detected along the rope is 10 cm, the rope speed 

should not exceed 10 cm in 0.1 second (i-e. the rope speed should be less 

that 200 feet per minute). 

The coils, wound on a bakeli.te former, are 6 inches long, 5 inches 

outer diameter and 3 inches inner diameter. The two magnetizing coils have 

10 turns tvich. For n magnetizing current of 1 amps, this gives an exciting 
field at the center of 0.43 oersted. The pick up or search coil also has 

ten turns. The whole assembly is hinged with spring loaded contacts to 

maintain the electrical integrity needed when the two halves are clamped 

around the rope. 

A number of interesting results have been obtained by Semmelink [4]. 

For example, a decrease of eddy currents causes an increase of flux with a 

resulting increase on the E trace and a corresponding decrease on the 

X 

ER trace. This often is associated with decreased contact between the 

strands. The reverse situation has also been found when there is increased 

contact between the strands. Sllch stressful situations occur at the crossover points where ropes pass from one layer to another on winding drums. 

Slight corrosion in the rope leads to a deposit of non-conducting 

material between the wires and to a reduction of eddy currents. This 

usually means a decrease of both E and ER traces. Semmelink also 

X 

indicates that internal corrosion appears to occur usually over relatively 

short lengths often at intervals corresponding to the circumference of the 

cl rum. 

External corrosion manifests itself in a large reduction of the E 

X 

trace but only a small reduction in the ER trace. Such corrosion can 

occur over 1-ong lengths of the rope (e.g. 1000 feet). 

Semmelink [4] found that inadvertent J3.C. magnetization of the rope 

(e.g. from the earth's magnetic field) could lead to vi-olent transients in 

the E trace when the rope moves at high speed through the coil assembly. X 

In one case, the rope had remained at rest for long periods with one section 

of the rope extending from the drum to the headgear sheave in a northerly 

direction. 
a sharp reduction presumably duch to n decrease in the axial magnetic flux. 

In a number of cases, sections of rope with apparent missing wires were 

found. This was believed to indicate a wire separation at a faulty spot 

weld. It is unfortunate that the numerous test results published by Lang 

are not accompanied by "ground truth" information on the actual state of 

the rope. He does show a few intcresting comparisions between the measured 

breaking tensile strcngth and the "K" and "X" readings at 10 Hz for locked 

coil ropcAs. Sucli data, howcver, wcrc not provided for the 30 and 80 Hz 

tests on the stranded ropes. 

Larsen et a1 [7] describes various devices for both D.C. and A.C. rope 

testing that are currently available. They mention that the Rotesco device 

as described by Lang [6] has been successful in 1970 in predicting tensile 
reductions of about 5%, although the extensive comparisons were not "quanti tat ive" . 

Morgan [8] presents a general review of electromagnetic non-destructive 

test methods applied to wire ropes. The discussion is entirely qualitative. 

He makes a number of recommendations for further development in Australia. 

Morgan and Symes [9] then describe the activities that led to the creation 

of a government sponsored project at the University of New South Wales. 

They give a largely qualitative description of their current research on 

both methods of testing wire ropes. They also describe some tentative ideas 

about an A.C. device that is to operate at a frequency of 1 kHz. Surprisingly, they make no mention of many A.C. devices that hitherto have been 

used elsewhere. It is possible they will find that 1 kHz is rather too 

high a frequency for effective penetration to the core of most mine hoist 

ropes. 

Stachurski [lo] gives a very useful summary of the physical concepts 

employed in D.C. magnetic testing for flaws in wire ropes. A number of 

interpretative schemes are outlined in a qualitative manner. Also, he uses 

prescribed forms of flaws, breaks, and cracks to calibrate the device. Much 

riseful data on the design and implementation of the defectograph device are 

also given. It is evident that this Polish group have highly developed the 

D. C. magnetic technique. 

Egen and Benson [ll] describe some interesting tests on a special prepared rope with prescribed types of imperfections. There were seven types 

of faults: 1) splice, 2) 2 to 3 wires filed half-way through, over a length 

of about $ inch, 3) an added No. 18 AWG wire laid into groove of core, 4) 

spike inserted into core and wires were spread apart, 5) 3 wires of 2 inch 
length were removed from the core, 6) one wire in core was cut and 7) a 

two inch length of core was removed. Using three D.C. devices (the Polish 

MD-8 Def ectoscope, the Swiss PMK-75 Kundig device and the Canadian Rotesco 

device). None of these devices detected fault No. 6 and the Fundig device 

did not detect fault No. 2. Otherwise, all other faults were detected by 

all three instruments. The results indicated that, for the rope tested 

(i.e. 7/8 inch diameter, 6 x 25, FW, RLL, FC, XIPS), each instrument observed 

about the same magnetic field variations. The very similar performance of 

these three instruments is probably a consequence of the basic similarity 

of the operating principle of each of these D.C. devices. Tt is a pity 

that the A.C. Kotesco device was not tested on the same rope. 

Rergander [I21 International, Inc. describes the Polish D.C. magnetic 

devicc for NDT of wire ropes known as the Defectograph MD-8 that he developed 

with Dr. J. Stachurski. The rope being tested moves through the permanent 

magnet wl~icli is magnetized to its saturation point. He claims that a 0.2% 

change of the rope cross-section can be detected via a measurement of the 

clxternal magnetic leakage field. The c.m.f. induced in the search coil 

can aiso be compensated for changes in the rope speed. IJsi~q an anpirical 

approach the probe coil response is related to such parameters as the loss 

of cross section of the rope, the length over which the loss occurs and the 

radial location of the internal flaw. 

EARLY THEORETICAL INVESTIGATIONS 

~ochschild [13] gives a review of papers hy ~Erster [14-181 and presents sornc useful plots of the current density in conducting cylinders for 

A.C. excitation by a solenoidal coil. He points out that no matter how 

carefully the test coil system is designed, small defects will go undetected 
unless the response time of the instrument is less than the time taken from 

the defect to pass through the effective region of the coils. For example, 

3 localized defect passing through at 0.03 inch wide differential test coil 

at 100 feet per minute will not be detected unless the response extends upwards to at least 200 Hz. A common limitation on bandwidth is the ink-pen 

recording devices, whose frequency response seldom exceeds 200 Hz. 

McClurg [19] also exhibits some of the results of ~orster and colleagues 

in graphical form. In particular, he shows that the impedance variations of 

solenoids (i.e. feed through coils) encircling cylindrical conductors that 

have surface and sub-surface cracks. The empirical data used are from 

~orster's papers. McClurg discusses the application of these eddy current 

techniques to metal cutting operations where such things as uniformness of 

the hardness is desirable. 

Graneau [20] introduces the interesting concept that induced currents 

in a conductor, flowing along closed curves, can be represented by a number 

of filamentary circuits. These currents and the energizing current can 

ttlen be deduced in principle from a system of circuit equations with self 

and mutual inductances that are postulated from the physics of the problem. 

lJsing somewhat heuristic reasoning, Graneau obtains an expression for the 

current anywhere in the metallic object by infinite series of increasing 

powers of the energizing frequency. The coefficients are undetermined 

functions of the filament circuit resistance, and mutual inductances between 

the filaments. He concludes that there is a clear di-vision between quantities depending on geometry and properties, As a consequence, the induced 

currents can be expressed as an explicit function of frequency. This is 

really quite strange since exact solutions of idealized forms such as layered 

cylinders with external dipole excitation do not exhibit this behavior r21-251. 
1 RELATED INVESTIGATIONS IN GEOPHYSICAL PROSPECTING 

Electromagnetic methods of non-destructive testing of solid conductors 

are closely akin to techniques that are now used in geophysical prospecting 

for metallic ore bodies [21-341. Tt is unfortunate that these two groups 

have had little interaction. This writer was involved in the theoretical 

developments in multi-frequency and transient electromagnetic methods in 

geophysical exploration. Tn fact, as long ago as 1950 it was proposed that 

conductivity and permeability of a spherical ore body could be ascertained 

from its electromagnetic response in either the frequency or its time domain. 

A similar analysis was carried out for a cylindrical ore body with a spccif ied condr~c tivity and permeability. The basic concept was that geometr ical 

configuration of the source and recei,~er coils with respect to the target 

could be arranged to have a negligible effect of the determination of ore 

body size, conductivity and permeability, The subject has advanced considerably since those "early days"' and modern accourlts are published regu1ari.y 

in Geophysics, the journal. of tlie (U. S .) Society of Exploration Geophysicists 

and Geophysical Prospecting, the journal of the European Society of Geophvsic-al Exl)loration. 

The work that is relevant to the electromagnetic probing of wire ropes 

is the analysis of '3 homogeneous conductive and permeable cylinder of infinite 

length. In an "early" paper 1221, a general solution was given for the total 

ficllds produced when a line source or current-carrying cable was located 

parallel LC] t0c c*yl inder. AT\ ox,~c.t ~wo-cl irnc2ns ional s0111t ion was obtained 

using a wave impedance approach. Tlie 'low frequency version of the general 

sol~~tion was expressed in a quasi-static form and numerical results for the 

induced dipole term were given. Some examples are shown in Figs. 7 and 8 
for the induced magnetic dipole responses of spheres and cylinders. In 

each case the ordinates are proportional to the induced dipole moment. 

Strictly speaking, the monopole term should also be considered when dealing 

with cylinders that are effectively infinite in length. This type of analysis 

was later extended to dipolar excitation of the infinite cylinder when again 

the monopole or azimuthally independent induced current was not considered 

in the numerical examples [22]. 

Further work on this subject was carried out by David A. Hill and the 

writer [30] where more realistic situations were treated such as the excitation of a conductive cylinder of finite length by an external magnetic dipole 

where all significant induced monopole, dipole, and multipole contributions 

were retained in the calculations. The analytical and numerical techniques 

used in these papers would seem to be applicable in a quantitative analyses 

of electromagnetic non-destructive testing of solid conductors. 

MORE RECENT ANALYTICAL STUDIES 

Vein [35] points out, what is usually accepted, that the mutual impedance between two closed circuits is dependent on conducting solids in the 

immediate environments. He promotes the concept of transfer impedance but 

feels ill at ease in relating this to the reciprocity theorem for generally 

contin~ious media. With this motivation, he works through the analytical 

details of a number of classical problems such as the mutual impedance between coaxial circular loops in the presence of planar, cylindrical, and 

spherical conductors. He assumes, without really providing a justification, 

that azimuthal symmetry prevails in each case. No real harm is done, however, 

since no numerical results of any kind are provided. The derivations seem 

to be unnecessarily complicated and, even then, reliance is made on formulas 
quoted from exercises in W.K. my the's [36] classic text. 

Burrows [37], in a significant thesis, exploits the reciprocity theorem 

1 in eddy-current testing and the subsequent development of a f law-detection 

theory. He is very quick to recognize the limitations of quasi-static theory 

such as assumed by Vein [35]. Rut it should be stressed immediately that 

Burrows actually assumes azimuthal symmetry of his detection coils even 

1 though the "flaws" may be asymmetrical. This is a valid procedure but it 

1 appears that some fundamental information is lost when both the source and I 

the probe coil fully encircle the cylindrical sample. Burrows represents 

internal Flaws in terms of induced electric and magnetic dipoles that, in 

turn, produces tlle secondary inf hence. He also provides some very useful 

tabulations of the infinite integrals that describe the internal fields 

within both solid cylindrical and tubular samples of circular cross sections 

for azimutllally symmetric excitation. These same integrals can be used to 

predict the response in a corresponding azimuthally symmetric detector coil 

due to an internal (small) flaw. Actually, this same approach has been 

followed up very recently by Hill and Wait [38] who did not restrict the 

results to azimuthal symmetry of the probe coil. 

1 Dodd, Deeds and Luquire [39] have obtained integral solutions for the 

vector potential produced by a circular coil for a number of different geometrical configurations. The solutions are limited to axial symmetry. In 

calculating the exciting fields of a rectangular cross-sec tion circular 

1 coil, they assume that a straight-f orward superposition of current over the I 

cross-section is valid. This is a quasi-static assumption that probably 

needs to be investigated, particularly when the coil of finite width and 

length encircles a highly conducting cylinder. 
Dodds et a1 cinotc Burrows' [37] formulation for deducing the secondarv 

induced voltages due to a defect or flaw in an adjacent conductor. Here 

they write down a formula for the defect-produced voltage, induced in coil 

2, by a current in coil 1 in terms of the electric and magnetic scattering 

matrices for an electrically small defect. 

In the same paper, Dodds et a1 give explicit solutions in integral 

form for the rectangular cross-section coil located over a two-layer planar 

conductor. Actually, the form of these solutions are very similar to 

earlier investigations of electromagnetic induction in layered models of 

the earth's crust. The NDT community is apparently not aware of this extcnsive literature>. An example is the book by Keller and Frischknecht [40] 

that reviews the current status of such problems at least up to 1965. Of 

course, Dodds et a1 give the explicit form of the fields of a circular coil 

with rectangular cross-section, while the geophysicists restrict their 

attention to small loops. Dodds et a1 also gives solutions for various 

combinations of pick-up coils and the corresponding secondary effects due 

to embedded defects that can be characterized by the polarizability matrices 

mentioned above. 

Cheng, Dodds and Deeds [41] have presented a general formulation for 

the time-harmonic eddy currents, produced by a circular coil of rectangular 

cross-section, for a underlying planar conductor of any number of layers. 

The integral solutions obtained in a straight-forward fashion yield algebraically complicated results. A number of these results could have been 

obtained by using impedance methods based on analogies with transmission 

line theory (see for example [42] and [43]). 
Dodd, Cheng and Deeds [44] give a formal vector potential solution to 

a coil coaxial with any number of cylindrical conductors. While they state 

the derivation is quite general, the solution is only carried through for 

complete symmetry about the common axis. 

Dodd and Deeds [45] give the same solutiorls for a uniform coil with 

uniform A.C. current excitation over a two layer planar conductor. They 

then repeat the solution for the coil encircling a two-layer cylindrical 

conductor of infinite length. A single numerical example is given for the 

normalized impedance of the coil as over the two-layer planar structure 

that exhibits the effect of the thickness of the upper layer. 

Kahn et a1 [46] present an interesting analysis of how eddy currents 

in a solid conductor are diverted around a surface crack. One of the basic 

assumptions is that the magnetic field tangent to the surface is a constant 

even in the presence of the crack. They also present solutions for diffraction by a semi-infinite crack (i .e, a half-plane) in an otherwise infinite 

medium. Neither of these solutions are "rigor~us'~ as claimed by the authors 

but the results do provide considerable insight into how defects, of other 

than infinitesimal size, will modify to external fields. Kahn and Spa1 

[47] have also presented some results for the calculations of eddy currents 

in a long cylinder with a radial crack at the surface. Details of the 

analytical method are not yet available but presumably the approach is 

similar to that used in treating the surface crack in the planar conductor. 

It is appropriate to call attention to the close similarity of such problems 

to earlier theoretical studies in geophysics where one is interested in 

the perturbation of time-varying geomagnetic fields near coastlines [48] 

and other laterally varying features in the earth's crust [49]. 
AN ALTERNATIVE FORYULATION FOR SOLENOID EXCITATION 

As we have indicated, a commmon method of non-destructive testing 

(NDT) of metal rods and tubes is to induce eddy currents by means of an 

encircling solenoid carrying an alternating current. The impedance of the 

solenoid is related to the cross-sectional area and the electrical properties of the sample. A formula for this impedance was obtained by ~'drster 

and Stambke [16] on the assumption that end effects could be ignored. Also, 

they assumed that the cylindrical sample was centrally located within the 

solenoid. The same derivation was essentially repeated by Hochschild [13] 

and Libby [51]. 

A feature of the ~Erster-~tambke derivation is that the effect of 

the air gap is introduced in a somewhat heuristic fashion wherein 

the field in this concentric region is assumed to be the same as the one 

for the empty solenoid. We feel it is worthwhile to provide a more general 

derivation of the impedance formula. We also show it applies to the case 

of a non-concentric air gap. Finally, we mention the relevance of the 4 

current analysis to the dual problem where the cylindrical sample is excited by a toroidal coil: 

To simplify the discussion, we consider first the concentric air gap 

model with a homogeneous cylindrical sample of radius a with conductivity 

a and magnetic permeability y. The situation is indicated in Fig. 9 where 

the solenoid of radius b encloses the sample, both of which are assumed to 

be infinite in length. Our objective is to find an expression for the 

impedance of the solenoid per unit length since this is the basis of the 

NDT eddy current methods that are commonly used. 
In terms of cylindrical coordinates (p,,), the only component of 

the magnetic field is HZ since the exciting current in the solenoid is 

uniform in both the axial and in the azimuthal direction. Within the 

sample, HZ satisfies the Helmholtz equation 

where y2 = iwu and where we have adopted a time factor exp(iwt). Here 

u is the angular frequency that is sufficiently low that displacement currents in the sample can be neglected. If not, we merely replace a by 

a + i&w where E is the pentiittivity. Also, it goes without saying that 

the field amplitude is sufficiently small that non-linear effects can be 

ignored. 

For the highly idealized situation described, we can immediately write 

for p < a where I is a modified Bessel function of argument yp and 

0 

where A is a constant. From axw well's equations the azimuthal component 

of the electric field is 

also for p < a. Now we can immediately form an expression for the 

"impedance" Zc of the cylinder: 

where = y/'~ = (iL1w/o)' is the intrinsic or wave impedance of the sample 

material. 

Now, for the air gap region a < p < b , we write corresponding field 

expressions 
and 

E = -BooI1 + CooK1(y0~) o@ (6) 

$ where B and C are constants and where 0, = ~,/(i~~~) = P 12m 

in terms of the permittivity c and permeability p of the air region. 0 0 

Here and in the above, the Bessel function identities 310(x)/3x = Il(x) 

and aKo(x) /ax = -Kl(x) have been employed. 

Compatible with the requirement that tangential fields must be continuous at p = a we can write 

+ZH ] =O 

CEO@ coz 

p= a 

This immediately tells us that 

In the external region p > b , the field exprcssions must clearly 

have the form: 

where D is another constant. 

Now the solenoid current is idealized as a continuous current distribution j amps/m in the azimuthal direction defined such that 

0 

Application of these conditions immediately leads to 

D = c - I1(y0b)B/K1(yob) 

and 
Among other things, this tells us that the magnetic field external to the 

solenoid (i-e. p > b) has the form 

H 02 = €-B[I~(Y,~) /K~(Y~~)I + c~K~(Y~P) (15) 

The quantity of immediate interest is the impedance Z of the solenoid 

itself. Clearly, within the limits of our basic assumptions, 

Z = const. x E (p=b)/jo 00 

The corresponding impedance of the empty solenoid is denoted Z . Thus, 

0 

it follows that 

which is explicit since C/B is given by (8). 

We now can simplify the impedance ratio formula if we invoke the small 

argument approximations for Bessel functions of order y a and yob. That 

0 

is, we use Io(x) = 1, I1(x) ' x/2, Ko(x) " -1ogx and K (x) l/x. This 1 

exercise leads to 

where no restriction has been placed on the magnitude of ya. Here R and 

X denote the resistance and reactance, respectively. 

The fornlula for z/z') given by (18) is in agreement with ~6rster and 

Stnmbke [16] (if one remembers they used the o3.d German designations Jo 

and J1 for modified Ressel functions). ~'drster and Stambke [16], Hochschild 

[13] and Libby [51] present extensive numerical data for this quasi-static 

approximation to %/z in Argand diagrams in the complex plane for various 

0 

values of lya 1 and . Two examples, using dimensionless parameters, 
are shown in Figs. 10 and 11 when the ordinates and abscissae are normalized 

hy Xo which is the reactance of the empty solenoid. That is, we assume 

% iXo corresponding to negligible ohmic losses in the solenoid itself. 0 

I/ 

The real parameter a is defined by a = ya exp(-i*/4) = (o~u) 2a. In Fig. 

10 the sample radius b is assumed to be the same as the sample radius a 

(i.e. no air gap). Different values of the magnetic permeability are 

shown. Not surprisingly, when a is small, R vanishes and x/Xo 

tends to p/pO. However, in general, the eddy-currents have the effect 

of reducing X/X~ , which is the effective flux, and to introduce a resistive portion R/X~ In Fig. 11, the relative permeability of the sample 

~/~O. = 1 but the filling factor a2/b2 assumes different values. The 

results indicate that the presence of the air gap reduces the sensitivity 

of the device for probing the conductivity but the effect is predictable. 

Actually, if a is sufficiently small (i.e. lyal << l), (18) reduces 

to 

Z/Z~ = X/X~ = 1 + (a2/b2)[u/uo - 11 (19) 

which is consistent with the curves in Figs. 10 and 11. In this D. C. limit 

the results only depend on the magnetic permeability of the sample. 

The formal extension of the theory to the case where the excitiy, 

solenoid is no longer concentric with the cylindrical sample can also be 

dealt with. The situation is indicated in Fig. 12~ AS before, cylindrical 

t-oordinatrs (p,$,z) are chosen co-axial with the sample. But now, the 

st~iftctl ~.onr(li~~ntrs (pl,@',z) are chosen to be coaxial with the exciting 

sol eno i ti . 't'he shift is 

po as indicated in Fig. 12 where we do impose the 

rather ohviorls physical restriction that b > po + a. We may show that the 

field in the non-concentric air gap has the form [50] 
This is valid in the non-concentric air gap region (i.e. p > a and p' < b). 

The needed azimuthal component is obtained from 

The relevant quantity for the impedance calculation is the "average" fie Ld 

- 

E 00' at the solenoid. Then it Eollows that the impedance Z per unit 

length of the solenoid with the sample divided by the impedance Z of the 

0 

empty solenoid is given by 

where E = 1 and E = 2 for m f 0 and where C,/B is known [50]. 0 m 

Not surprisingly, (22) reduces to (17) for the centrally located sample, 

i . e. Im(yopo) = 0 for po +O when m#O. 

We again may invoke the small argument approximations for Bessel 

functions of order yea, yoh, and yop0. Lo and behold, these show that 

2/Zo reduces again to the formula given by (18). This confirms the conjecture of ~Erster and Stambke [16] who seemed to be gifted with keen 

physical insight into such problems. Of course, we do not expect the 

result to hold in any sense when the dimensions of the solenoid become 

comparable with the free-space wavelength. In that case, many other complications arise sctch as the assumed uniformity of the solenoid current. 

THE DUAL PROBLEM 

There is an extremely interesting dual to the problem we have discussed. That is, rather than exciting the cylindrical sample with an 

azhthaZ electric current, we employ an azimuthaZ magnetic current. This 
is an idealized representation for a thin toroidal coil but, again, it 

effectively is of infinite length in the z or axial direction. The assumed 

source discontinuity is now in the electric field at p = b which has only 

a z component. The much more ccmplicated case of the toroidal coil of 

finite axial extent was analyzed recently [53]. 

Under the present assumption of axial uniformity, the admittance Y 

per unit length of the toroid is the dual of the impedance Z of the solenoid 

discussed above. Thus, all the earlier equations apply if we make the 

following transformations: ipw + a, llo -b c0, n + TI-1 

no + no1, HZ + EZ, 

EO + -HQ, HoZ 'f EoZ, and E -+ -H 04 00- Then the dual of (18) is the ratio 

of the admittance Y of the toroidal coil with the sample to the admittance 

Y without the sample. It is written explicitly 0 

for the case where lyobl 1. That is, the radius of the toroidal coil 

should be much smaller than the free-space wavelength. Also, in full analogy to the earlier discussion, the quasi-static result holds for any location of the cylindrical sample within the toroid. Furthermore, in the low 

frequency limit where lyal = a << 1, we see that 

which depends only on the conductivity of the sample. Thus this type of 

excitation should be preferred with probing the effective conductivity in 

the axial direction in the sample. 
THE PROLATE SPHEROIDAL VOID MODEL OF HILL AND WAIT 

As we have indicated, an appropriate source for excitation of axial 

electric currents is a toroidal coil which encircles the rope, and a magnetic current model for the toroidal source coil has been analyzed [53]. 

Expressions for both the interior and exterior fields were derived for a 

homogeneous wire rope. As we shall outline here, the presence of a small 

slender void within the rope can be treated. Such a void can be corisidered 

a model for a broken strand. The void is also allowed to be oriented at 

any arbitrary angle to the rope axis to account for the winding geometry 

of the wire rope. The induced electric and magnetic dipole moments are 

computed from the primary fields and the electric and magnetic polarizabilities of the void. We then obtain expresssions for the external fields of 

electric and magnetic dipoles of arbitrary orientation. Of course, it is 

these external fields which are the observable quantities in any EM nondestructive testing method. The particular expressions, derived for the 

external fields of internal el-ectric and magnetic dipoles, should be useful 

in future analyses of other types of small imperfections in wire ropes. 

Of colirse, in such cases, the electric and magnetic polarizabilities would 

be different. 

In a previous analysis [53], we analyzed an azimuthal current sheet 

source which encircled the rope, The current sheet was allowed to have 

arbitrary width in the z Qr axial direction and arbitrary azimuthal extent. 

This source results in fairly complicated expressions for the electric and 

magnetic fields. Since, in this paper, we are primarily concerned with the 

fields scattered by the void, we take the simpler special case for the source 

shown in Fig. 13, Specifically, a magnetic current ring of strength K is 
located at a radius b in the plane z = 0. This is a model for a thin 

toroidal coil which completely encircles the rope. The rope is assumed 

to be infinitely long and has radius a. It has conductivity per- W ' 

mittivity E and permeability p . The surrounding free space has per- W ' W 

mittivity E and permeability p . For now, we defer discussion of the 

0 0 

void properties and consider only the homogeneous rope. 

Because of the symmetry of the source and the rope, the fields are TM 

(Transverse Magnetic) and independent of . The magnetic field has only 

a nonzero azimuthal component H~~ , and the electric field has only non- @ 

zero axial and radial components lZPr and The superscript pr dez P 

notes the fields in the absence of the void. These primary fields are 

derived elsewhere [53], and the explicit forms both inside and outside the 

rope are given in Appendix A of a recent report [38]. We note that EPr 

P 

and H~~ are zero for p = 0. On the other hand, EPr is nearly inde,- 0 z 

pendent of inside the rope for sufficiently low frequencies. 

We now select a thin prolate spheroid of conductivity o permitti- v' 

vity E and permeability p inorder tomodelabrokenstrand. The v' v 

prolate spheroidal shape is a convenient one because its electric and magnetic polarizabilities are known, However, we would not expect a significant difference for a thin circular cylinder of the same length and volume. 

To account for the winding geometry of the rope, we allow a rotation of the 

major axis of the spheroid about the p' axis by an angle a. Thus, the 

A 

major axis is oriented at an angle a to the unit vector z' and an angle 

n/2 + a to the unit vector ' as indicated in Fig. lc. 

Since the void has a contrast in both the electric and magnetic properties, both electric and magnetic dipole moments will be induced [36]. The 

electric polarizabilities for the incident electric field applied along the 
e major axis, a , or along the minor axis, ae , are given by ma j m in 

e 

a = -V(u -0 ) maj wv 

and 

-2V (n -ov) - - W 

a min 1 + CJ 10 9 

where V is the volume of the thin prolate spheroid. Since we anticipate 

the use of very low frequencies, we have neglected displacement currents. 

To include them, n would be replaced by n + iwe and nv would be 

W W W 

replaced by 0 + iwev in (25) and (26). The magnetic polarizabilities v 

for the incident magnetic field applied along the major axis, am 

ma j 9 or 

m along the minor axis, a , are similarly given by min 

and 

-2Viw (ljw-pv) 

a" - - min 1 + pv/pw (28) 

In order to compute the induced dipole moments, it is first necessary 

to resolve the incident electric and magnetic fields into components along 

the major and minor axes. The resultant induced dipole moments can then 

be resolved into the more convenient , , and z components. When thi-s 

is done, the induced electric dipole moments are found to be 

P= €2 = E~ (amaj cos2a + ae sin2a) , min 

Pr e e (Ids)()= EZ (amin - a maj ) sinacosa , (30) 

(1~1s)~ = E~~ ae 

p min 

Pr In (29)-(31), the primary electric field components E- and lZPr are 

z P 

evaluated at p', Q', z'. Similarly, the induced magnetic dipole moments 

are found to be 
Note that the units of the induced electric dipole aomcnts in (29)-(31) are 

ampere meters, and the units of the induced magnetic dipole moments in (32) 

through (34) are volt-meters. These induced dipole moments are the sources 

of the scattered field. 

The scattered field external to the rope is the observable quantity 

in any NDT system. Since we anticipate the use of coils for sensors, we 

SC have considered the scattered magnetic field components Hp (p,@,z), 

HiC(p,$,z), and HSC(p,@,z) [38]. Each of these components can be 

Z 

written as a superposition of the contributions fror,, each of the six induced dipole sources. (Actually, there are only five nonzero sources since 

(Kdk) is zero for the specific configuration considered here). Explicit 

expressions for the three scattered field components are given elsewhere 

[381. 

The quantities of most interest in NDT are the external (p>a) magnetic field components which are observed with the sensing coils. The 

primary magnetic field has only a @ component HPr but the scattered @ 

magnetic field has all three components. For the numerical results, the 

following parametric values remain fixed: a = 1 cm, o = 1.1 x 106mho/m, 

W 

- = 200 llo, b = 2 cm, frequency = 10 Hz, p = 2 cm, o = 0, and pv - po. v 

For this low frequency, the conduction currents dominate, and the permittivities E and cV are unimportant. The above values of 

W Ow and Pw 
are roughly representative of stainless steel, but the permeability of steel 

is quite variable [54]. For the above parameters, the radius a is approximately one skin depth. For convenience here, we also choose z = z' = 0 

and 0' = 0. 

In Figs. 14, 15, and 16, we show the magnitude of the scattered magnetic field components normalized by VH" as a function of . The rea- 0 

son for showing the dependence is that, although the primary field is 

independent of $ , the scattered field varies considerably in . This 

variation might dictate the use of multiple sensing coils spaced in azimuth 

around the rope. The curves are normalized by the primary field because the 

anomalous scattered field is measured in the presence of the primary field 

and their ratio is thus of interest. We also normalize the results to the 

volume V of the void in order to make the curves more general. However, a I 

typical value for V might be on the order of 10-'m3 (=I cm x 1 mm x 1 mm) . 

As indicated in Figs. 14 and 15, HSC and HSC are of approximately the 

P 0 

same level, but HSC is odd in $ while HSC 0 is even. We also may note 

P 

the decrease in scattered field as the void is moved from the outer region 

(p'/a = 0.9) toward the center (p' /a = 0.1) of the rope. Because a = 0' 

in Figs. 14 and 15, only the axial and radial electric dipole moments and the 

azimuthal magnetic dipole are excited. Also, the radial electric dipole is 

very small because its exciting field E~~ is small. The calculations 

P 

reveal that (Ids) and (KdR)@ contribute approximately equally to the Z 

scattered field. In Fig. 16, we have z = z' = 0 and p'/a = 0.5, but 

we allow a to vary from 0' to 30'. Nonzero values of a allow two additional dipole moments (the azimuthal electric and the axial magnetic dipoles) 

SC 

to be induced. The result is a decrease in the level of H and HSC 0 P 
and. an increase in the level of H". These new results are described 

z 

elsewhere [38] . 

The phase of the scattered field was also computed, but was found to 

be of less interest. Also, the individual dipole contributions were computed, and they are of individual interest since they will be excited 

differently for other sources or other void configurations. 

Although we have generated numerical results for the specific case of 

a thin prolate spheroidal void, the formulation given here actually yields 

the external fields for arbitrary induced electric and magnetic dipole 

moments. Thus the formulation is useful for any type of rope imperfection 

that can be characterized by induced electric and magnetic dipole moments. 

This requires only that,the rope imperfection be small in terms of the rope 

radius and the rope skin depth. Larger imperfections should have a similar 

qualitative behavior, but could probably be rigorously analyzed only by 

solving an integral equation for the fields in the imperfection. Also, 

it would be a simple matter to perform similar calculations for a solenoidal 

coil of the type used in present NDT systems. The scattered field calculation remains unchanged, but the primary field would be TE(E'~ = 0) 

z 

rather than TM(H~~ = 0). Some calculations for this case have been carried 

z 

out by Burrows [37] for the special case where both exciting and sensing 

coils are coaxial with the tubular specimen. 

CONCLUDING REMARKS 

We have attempted to give an overall view of NDT electromagnetic 

methods. As indicated, the investigations have proceeded along different 

lines. The very practical and perhaps most useful work to date has been 

empirical in nature. There has been no attempt made to deduce the magni- 
tude of the secondary fields due to the internal flaw; instead a large 

number of actual ropes are tested and the results presented. On the 

other hand, the analytical approaches have considered only highly idealized 

situations. Nevertheless, some important principles are disclosed that may 

have an important bearing on operating procedures. For example, the 

toroidal coil exciter could be utilized in practical schemes in co~junction with localized sensor coils that can characterize the three-dimensional 

field configuration in an adequate fashion. 

A basic theoretical aspect of the NDT problem that we have not really 

addressed is to account for the anisotropic nature of the effective conductivity and the permeability of the wire rope. The spiral construction 

of stranded wire ropes sometimes described as a "twisted bunch of spaghetti" 

defies any simple microscopic description. However, from a macroscopic 

point, we might describe the effective anisotropy by conductivity and 

permeability tensors. Some preliminary efforts in this direction have 

begun [55]- It is obvious that much remains to be done and we have addressed some of thcsc topics in the subsequent sections. 
REFERENCES 

T.F. Wall, "Electromagnetic testing for mechanical flans in steel wire 

ropest', J. Inst. Elect. Engrs., vol. 67, pp. 899-911, 1929. 

T.F. Wall and C. H. Hainsworth, he penetration of alternating magnetic 

flux in wire ropes", J. Inst. Elect. Engrs., vol. 71, pp. 374-379, 1932. 

A. Semmelink, "Electro-magnetic testing of winding ropest1, Trans. of the 

South African Institute of Electrical Engrs., vol. 43, no. 5, pp. 113-129, 

May 1953. 

A. Semmelink, "Electromagnetic testing of winding ropes", Trans. of the 

South African Institute of Electrical Engrs., vol. 47, No. 8, pp. 206-244, 

Aug. 1956. 

R.H. Hiltbrunner, "Le Controle ~a~ngti~ue des cgbles avec le d6fectoscope 

P 

integra", Econmie et Technique des Transports, no. 119, June 1957. 

J.G. Lang, "The principle and practice of electromagnetic wire rope 

testing", Canadian Mining and Metallurgical Bulletin, pp. 415-424, 

April 1969. 

C.H. Larsen, P.A. Egen, R.D. Jones, and H.A. Cross, "Wire rope applications and practices associated with underground coal mining in the U. S.", 

Final Report (from Eattelle Laboratorl:es) on U.S. Bureau of Mines Contrcxc-t No. H0107747, 22 June 1971. 

J.P. Morgan, "Investigations on wire ropes in mine hoisting system", hoe. 

Australian Inst. of Min. and Metal, no. 215, pp. 59-85, 1965. 

I I J. P. Morgan and H. E. J. Symes, Non-destructive testing of wire ropes", 

Bulletin No. 6, Australian Mineral Industries Research Assoc. Ltd. 

(National Library of Australia Card No. ISSN 0313-6973), 1976. 

J. Stachurski, "Magnetic testing of steel wire ropes", Report from 

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Cracow, Poland, 1976. 
methods 11. Practical results and industrial applications of methods 

using test coils", Zcitschrift MetaZZkunde, vol. 43, no. 5, pp. 172-180, 

1952 (in German). 

[16] F. Forster and K. Stambke, "Theoretical and experimental foundation of 

non-destructive testing using eddy currents 111. Quantitative methods 

of non-destructive testing using test object encircling coils", Zeitschrift 

Metczllkziru2t., vol. 45, no. 4, pp. 166-179, 1954 (in German). 

[ 171 F. ~&-stc-!r, "'I'11eort.t ical and experimental foundation of non-destructive 

testing using eddy r-urrcnts TV. Practical apparatus for non-destructive 

testing using encircling coils for eddy currents", Zeitschrift MetaZZkunde, 

vol. 45, 110. 4, pp. 180-187, 1954 (in German). 
F. FErster and H. Breitfeld, "Theoretical and experimental foundation 

of non-destructive testing using eddy currencts V. Quantitative testing 

for cracks in metallic qbj ects using test object encircling coilg", 

Zeitschrift MetaZZkunde, vol. 45, no. 4, pp. 188-199, 1954 (in German). 

G .O. McClurg, "Non destructive eddy current testing", IRE Trans., vol. 

IE-11, no. 1, pp. 20-26, 1959. 

P. Graneau, "Coupled circuit theory for electromagnetic testing", 

Progress in Non-~estructive Testing (ed. by E.G. Stanford and J.H. 

Fearson), vol. 2, pp. 163-188, ,1961. 

J.R. Wait, "A conducting sphere in a time varying magnetic field", 

Geophys., vol. XVI, pp. 666-672, Oct. 1951. 

J.R. Wait, he cylindrical ore body in the presence of a cable carrying an oscillating current", Geophys., vol. XVII, pp. 378-386, April 

1952. [In eqn. (13), replace / by np2/u1 both in the numerator 

and denominator, similarly, in (14), replace (kfn) by n(kk1). Also, 

the summations in (66), (67), etc. shotild include the n = 0 terms] . 

J.R. Wait, "A conducting permeable sphere in the presence of a coil 

carrying an oscillating current", Can. J. Phys., vol. 31, pp. 670-678, 

May 1953. 

J.R. Wait, "Some solutions for electromagnetic problems involving 

spheroidal, spherical and cylindrical bodies", J. Res. NBS, vol. 64B, 

pp. 15-32, Jan./~ar. 1960. 

S.H. Ward and D.C. Fraser, "A conducting permeable sphere and cylinder 

in an elliptically polarized alternating magnetic field", J. Geomag. 

and Gcoelectricity, vol. 18, no. 1, pp. 23-41, 1966. 
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no. 4, pp. 245-266, 1972. 

J.R. Wait, "On the electromagnetic induction in elongated ore bodies", 

Geophysics, vol. 38, no. 5, pp. 984-985, 1973. Correction: vol. 39, 

no. 2, p. 235, 1974. 

W. Kertz, "~eitungsfahiger Zylinder im transversalen magnetishen 

Wechself eld, Cerl. B~itr. (;cophys., vol. 69, pp. 4-28, 1960. 

S.H. Ward, "unique determination of conductivity susceptibility, size 

and depth in multi-frequency electromagnetic exploration", Geophysics, 

vol. 24, no. 3, pp. 531-546, July 1959. 

J.R. Wait, "Electromagnetic coupling between a circular loop and a 

conducting sphere", Geophysics, vol. 18, pp. 970-971, Oct. 1953. 
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[40] C.V. Keller and F.C. Frischknecht, Electrical Methods in GeophysicaZ 

Prospsctiq. NewYork: Pergamon Press, 1966. 

[41] C.C. Chang, C.V. Dodd, and W.E. Deeds, "~eneral analysis of probe coils 

near stratified conductors", Int'Z. J. of DT, vol. 3, pp. 109-130, 

197111972. 

[42] J.K. Wait, EZt?ctromagnetic Waves in Stratiffed Media. Pergamon Press, 

Chap. 2, 1st Ed. 1962 and 2nd Ed. 1970. 

[43] J.R. Wait, "Fields of a horizontal dipole over a stratified half-space", 

IEEE Trans. Antennas Propagat., vol. AP-14, no. 6, pp. 790-792, Nov. 1966. 

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arbitrary number of cylindrical conductors", J. AppZ. Phys., vol. 45, 

no. 2, pp. 638-647, Feb. 1974. 
1501 J.R. Wait, "The electromagnetic basis for non-destructive testing of 

c ylindr-cal conductors", J17':EE: ?'rans. Instrmevz~~ation & Measurement, vol. 

IM-27, No. 3, 235-238, September 1978. 

151 1 H .I,. I,ibby, In lroduction 1 o E7cctrma~nct:ic FJondsstructive I'est Methods. 

New York: John Wiley & Sons. 1971, Sec. 5.2, pp. 135-150. 

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pp. 4893-4897, 1977. 
. ---------. 

4.5 cm 

--------- I ---------. -3 

. ------- ---. ----------- ,--------- 

20 TURN 

SEARCH COIL 

(after Wall 1929) 

Fig. la. Sketch of Wall's device (1929). the B-H curve, and 

the resulting induced EMF in the search coil. 
Induced EMF 

in Search Coil 

AMP-TURNS 

at 20 HZ. 

AMPERE TURNS (PEAK VALUES) 

Distance from end in feet 

(After Wall 1929) 

b% ; lb. Son~c. ti:~t,l froc- wall's test of a locked coil rope. (EMF Waveform 

h1p1 it udc~ not given) . 
Model Wire Rope 

Insulated individual 

Wire Rods of ''8 in. d 

location of embedded 

search coil A 

/r 

EMF in A (Flux \ / 

in central group) u 

(after Wall and 

Hainsworth 1932) 

EMF in C-D 

(flux in surface 

(flux in 3rd 

lay0 r ) 

Current 

Waveform, I = 5.4 amps 

"ig. 2. Special model of a wire rope used by /\Tall and Hainsworth 

(1932) to study internal fields. 
Peak value of excitation curre 

(after Wall and 

Hainsworth, 1932) 

AMPS. 

n t 

5 10 15 AM 

Peak value of A.C: - 

Direct Current -+ 

Fig. 3. Some measured data for the special wire 

rope model. 
after 

Semmeli n k 
SCHEMATIC 

DIAGRAM 

(of ter Semmelink, 1956) 

Fig. 6. Lrnproved model use by Semmelink (1956). 
Values of 

p/pO shown 

(from Wait 1953) 

Sphere Dipole Response 

Fig. 7. Induced magnetic dipole for a conducting 

permeable sphere as a function of its 

conductivity a, permeability p, angular 

frequency w, and radius a. 
Values of p/pO shown 

0.1 I 10 lo2 lo3 lo4 

Cylinder Di pole Response 

RI = PI+ LQI 

(from Wait 1952,1960) 

Fig. 8. Induced (line) magnetic dipole for a conducting 

permeable cylinder. 
SAMP 

Fig. 9. Cross-sectional view of 

cylindrical sample located 

centrally. 

Fig. 12. Cross-section view of the nonconcentrlcally located sample. 
Values of 

filling factor 

2 2 a /b shown 

Fig. 11. Argand plot 

t>F the impedarce 

7. = R + iX normalized 

to the reactance ?, 

0 of tlic empty bolenoid, 

for 1, = 11 . 0 
(infinite length) Fig. 13. Thin toroidal coil surrounding 

a metal rope of infinite length 

a) Perspective view, b) Top view, 

c) Prolate spheroidal void. 

Void 

Fig. 15. The normalized magnitude 

of the scattered azimuthal magnetic field for various void 

locat icns p' . 
17 - illj:. i h. 'i ~lorrn~il izccl nlngni tutlci of the scat tcirod 

rnci i;lI 1n;ignct i L- i ielcl for varic-)us void 

~>rIc.nt;lt idns (1.. 
py 

the polarization current model suggested by Unz and correctly 

interpreted by Epstein, Rell, Smith, and Brice is also pointed 

out. ) 

Theresa M. I.avcl'le, Micarvlr~nvc..? ij7 :'or i!r*:: 1 rzrcti7)c 'I'crntiny;, Yaterials --- - -- -- Eval. -- 

uation, Vol. 25, No. 11, pp. 254-258, Nov., 1967. 

(The use of microwave radiation in nondestructive testing of nonmetals is investigated. Techniques involving reflection, transmission and scattering are discussed and applied to general 

nondestructive tests. The paper also discusses the results of 

a laboratory test program in which the following measurements 

arc considered: 

1. Dimensional Measurements - Determination of range and accuracy for thickness variations. 

2. Flaw Detection - Determination of types of flaws and specinien configuration. 

3. Material Property - Determination of types of physical and 

chemical properties that can be detected and possible degree of resolution. 

The general cest procedure is to irradiate the sample with microwave energy and to monitor the transmitted or reflected energy 

for phase or amplitude change. The tests are conducted at X-band 
gg) 

V .V. Klyuev, G .(; . Kapel in, !?:-!.,i:r.~.i'vi;!~: !.I;? alzd 7'7~:r)~e !.%cn!. 1irsearc.h %Q ,'.'c~cl/i- 

'1'hrn:~~jh I/"r~~zqr,.ll!c.i~,,:: fc?r Mc,)rl i l:o?ai 8 :; :::O,'ii?!7 nfi;ic:c f-s, - Def'ekt - oskopiya - , Vol . 6, pp. 701-, 1969. 

V.V. Klyuev, PI.1,. Faingoi z, and (; .(:. Kapel in, .'!~~~crcrnt of a Currentf1oncj!cr*l ',1;1 ;'/(I tc > il,rb.clZ lc 7 ,!(i l !?r /'/,-it:,- of a ?r,:vy-'r!/~)~c Traansducer, 

- Defektoskopiya, Val. 6, pp. 425-432, .July/August 1970. -- 

(The emf of a square striding-type transducer located over a parallel moving plate is determined. The voltage hodographs obtained analytically are confirmed experimentally. Recommendations are given for tuning out the effect of the travel speed of 

the objcct being inspected.) 

(A general expression is analyzed for the insertion cmf of a pmetrating pickup with a bimetallic ferromagnetic: cylinder whose 

surface layer has a thickness equal to 0.01-0.06 of the outer 

radius. Simplified equations are formd for designing pickups. 

Cases are considered in which there are various relations 
exciting an applied transducer powered by current in the form 

of a single step. The resultant formula is used to compute the 

relative density of eddy currents during a contin~~ous change of 

the time of the transient process for certain fixed distances 

from the surface into the depth of a metallic medium. The distribution of current density is computed with respect to the 

depth of a metallic half-space for various instants of time.) 

M .M. Shel, lim~nonic Structure of the Secondary EMF of a Deposited Sensor, 

Defektoskopiya, Vol. 7, No. 4, pp. 378-382, 1971. 

(An approximate expression is obtained which can be used to calculate the higher harmonics of the secondary emf of a deposited 

sensor in checking ferromagnetic parts. The relationship between 

the harmonic structure and the shape of the hysteresis loop of 

the specimen being investigated is determined. Experimental results are cited which confirm the correctness of the basic theoretical premises. ) 

V.V. Vlasov and V.A. Komarov, The Magnetic Fi~ld of Eddy Currants Above 

A Surface Crack in MetaZ With Excitation of nzem by an Applied Inductor, 

Defektoskopiya, Vol 7, No. 6, pp. 665-675, 1971. 
V . A. Sandovskii, Fie Zd of t ir~ V~ct,~r-i)ot~nfri iv n Cond7~ct-ive Scmispacc 

I~i~uliny Its Movement Rc.ht.l:7?e to a Wall fkving cx C~rrr~nt, Def ektoskopiya, 

Vol. 8, No. 1, pp. 75-80, January/February 1972. 

(The problem of the steady-state field of the vector-potential 

in a conductive semispace is solved for its movement relative 

to a wall having a current. The rigorous solution with generally accepted assi~mptions for problems of such a class is presented 

in qundratures. This solution is analyzed for the case of a 

moving semispace of magnetic and nonmagnetic materials and is extended to any sollrce of a constant magnetic field.) 

V.V. Vlasnv and 13.2. Volkov, il,np/Csd Edd!r-Current S3stem Hnz?inp a Hall 

Element ns th~ :;en~~~~Jrrr rrntl' i:cwcl in? to thc Tan?entini! Compovent of the 

nc-.fert i7ir/d, Defektoskopiya, -- Vol. 8, No. 1, pp. 84-89, January/February, 

1972. 

V.V. Klyuev and M. I.. Faingoiz, N(?~zcfcst~~catiuc Tc..;ting of Movirq Cu~~rv-nl- 

(:nductir~~ .4r Lic2o.s 713 ~Yearzs of Pass-Snhr*qu,:rh Conucr fers flninp the Co?ls t (In l - 

fl'?'~Zd Afel hod, Defektoskoyiya, Vol. 8, No. 2, pp. 27-31, March/April, 1972. 

(A determination is made of the induced vector potential of outer 

and inner cylindrical pass-through converters excited by a cons tan t current in the monitoring of moving current-conduc ting 
pp. 1824, JanuarylFebruary, 1974. 

V.V. Klyuev and M.L. Faingoiz, Inspection of Movin(g, Ferromagnetic Components IJ~ th Through Bdd;! Currant Converters, Defektoskopiya, Vol . 10, 

No. 2, pp. 106-110, MarchIApril, 1974. 

P ? 

G .A. Bur tsev and E. E. Fedorishcheva, ,7imp7e ~,pproxhation for th& i!apnctostntic Fiel:!~ c f Surface Defects and Inhomo.~cncities, Def ektoskopiya, 

Vol. 10, No. 2, pp. 111-118, March/April, 1974. 

V .V. Klyuev and ?I.L. Faingoiz, Inspection of a Moving Metal ;?trip with an 

?j'f-Axb[7 Sh?:cZ&ci Co~ltact Transducer, Defektoskopiya, Vol. 10, No. 3, 

pp. 24-29, ~ay/June, 1974. 

V.F. Avduevskii, CaZcuZating the Response of Eddy-Current Transducers, 

Defektoskopiya, Vol. 11, No. 5, pp. 27-30, September/October, 1975. 

V . P . Zuev and V .N. Novikov, I?Zcc*f~ostat?:c i)s?bfcctoscopy o-? f'j! Lirdrieaz 

S[,tl'?imcns uith Axioi! Duct.7, Defektoskopiya, Vol. 11, No. 5, pp. 31-36, 

September/October, 1975. 

V.P. Kurozaev, Yu. I. Steblev, and V.E. Shaternikov, Influencr I)$ Part 

Geometry on the Results of Their Inspection b?y Eddy-Current Tranaducerr;, 
(

dipole with a density of surface magnetic charges which falls 

off exponentially with depth into the sample. The theoretical calculations are compared with experimental data.) 

V.A. Sandovskii, CaZcuZtzti~g the Resistance Introduced by Cracks in an 

Applied Transducer, Vol 12, No. 2, pp. 95-101, MarchIApril 

1976. 

(The impedance injected into an applied transducer by a crack 

is found by approximation in the form of the product of 

three functions depending on a generalized eddy-current parameter. 11 holograph is plotted for small and large openings 

of cracks.) 

B.I. Kolodii and A. Ya. Teterko, Determination of the Transverse Magnetos tatic Fie Id of a Cylinder with an Eccentric C!j lindrical Tnc Z7~sion, 

Defektoskopiya, Vol. 12, No. 3, pp. 44-50, May/June 1976. 

(Using the method of the vector potential in bipolar coordinates, the magnetostatic field of an infinite cylinder with 

an eccentric cylindrical inclusion situated in a transverse 

field is determined. Special attention is paid to the field 

of a semi-inf inite solid with a cylindrical i~lclusion paralell to the surface. For the case of an eccentric cavity 
(The problem of calculating the electromagnetic field of a coil, 

carrying an alternating current, that encloses a metallic infinitely long cylinder arranged noncoaxially with respect to 

the coil is considered. The problem is solved by using scalar 

functions that enable separation of variables in cylindrical 

functions. The problem is reduced to finite expressions defining the induced emf of a feedthrough transducer. The effect 

of displacement p on the induced emf is analyzed. The author 

0 shows that the amplitude-phase method can be used to compensate 

for the influence of po when monitoring any of the parameters 

u, 0, or R. The attenuation of the p is not measured in the 

0 differential method.) 

V.E. Shcherbinin and M.L. Shur, Calculating the Effect of the Roundar.l:~s 

of a Product on the Field of a Cylindrical Defect, rjefektoskopiya, - Vol. 

12, No. 6, pp. 30-35, November/December, 1976. 

(Calculations are made of the defect field in the form of an 

infinite cylinder, taking magnetic reflections from the spatial boundaries and the boundaries of the defect into consideration. The authors show that taking the last factor into 

account leads to a quantitative and qualitative change in the 

calculated values of the defect field.) 
pg 

ial in automatic plant being a practical consequence. It is 

important to know the magnetic history of some materials in 

order to work at a suitable magnetization level. For the general case it has been suggested that one may be able to characterize surface opening cracks, but not sub-surface flaws, 

by an equivalent depth width and angle. Experimentally improvements in tapes, microprobes, methods of magnetization, 

lift off control and use of electronic tailoring have led to 

imporved signal-to-noise, sensitivity and resolution.) 

Il .A. Hill and J .R , Wait, il~~n%!j::is oJ A lternatirrg Currcnf; F:xc<takion of n 

Wfre Rope by iz 'Fcro-i&'l ('c~if, Journal of &plied Physics, -- Vol. 48, No. 12, 

pp. 489 3-4897, December, 19 77. 

(An idealized magnetic sheet current model for a toroidal coil 

which encircles a conducting ferromagnetic rope is analyzed. 

This configuration is suitable for the nondestructive testing 

of wire ropes and cables. Numerical results for the axial 

electric current density induced in th? rope reveal that low 

frequencies on the order of 10 Hz are required to produce a 

uniform current in a typical rope. For a toroidal coil which 

does not completely encircle the rope, the azimuthal symmetry 

is lost and harmonics in @ are produced. These harmonics are 

large but decay rapidly away from the source.) 
ear motion of an arbitrary current distribution past materials 

of constant permeability is presented. The application of the 

Green's function technique to the field equations yields integral representations of the induction, eddy currents, and electromagnetic forces. Due to interface coupling of the boundary 

conditions along the surface of the conductor, Green's functions 

are shown to satisfy integral equations. In the case of a conducting slab, explicit solutions for the Green's functions are 

derived. Application to magnetic levitation and the calculations of forces on moving coils are developed. Results are 

compared with experimental drag measurements.) 

D.A. Hill and J.R. Wait, EZectromagnetC~ Pi~%d Pcrtmhntion by nn Intern07 

Void in a L'orduct;ng CyZinder Pkcited by a Wire Loop, -. Preliminary Report 

to U.S. Bureau of Vines on Contract No. H0155008, - pp. 1-55, 14 ~ui~, 1978. 

(A thin prolate spheroidal void in an infinite conducting circular cylinder is used to model an internal flaw in a wire rope. 

The rope is excited by an electric ring current which is a model 

for a thin solenoid or multi-turn wire loop. The anomalous external fields are computed from the induced electric and magnetic dipole moments of the void. Computer plots of the scattered 

fields are generated to illustrate the effects of various 
netic method of nondestructive testing (n.d.t.) is used in 

which the rope is magnetized by means of either an electromagnet or permanent magnet and magnetic sensors are used to 

detect anomaious magnetic fields due to wear, corrosion or 

broken wires in the rope. A prototype instrument has been 

produced for testing stranded haulage ropes used in mines 

and the technique can, in principle, be extended to test 

other types of wire ropes, for example general stranded ropes 

of any diameter, locked coil hoisting ropes, and large mooring cables for offshore structures.) 

James R. Wait, The Electromagnetic Basis for Nondestructive Testing of 

CgZindricaZ Conductors, IEEE Transactions on Instrumentation and Measurement, Vol. IM-27, No. 3, pp. 235-238, September 1978. 

(Using an idealized model, we deduce the impedance per unit 

length of long solenoid of many turns that contains a cylindrical sample. The sample with a specified conductivity and 

magnetic permeability need not be centrally located within 

the solenoid provided all transverse dimensions are small compared with the free-space wavelength. The derivation is relatively straightforward and it provides a justification for 

earlier use of the impedhnce formula. The dual problem, where 

the solenoid is replaced by a toroidal coil is also considered. 
(ypp

solenoids that are coaxial with a conducting cylinder in relative motion. The formulation is based on the first order Lorentz 

transformation and the results obtained are sufficiently general to encompass any such situation that could arise in nondestructive testing schemes. A numerical example, relevant to steel wire 

ropes of 2 cm radius used in mine hoists indicates that, even with 

relative velocities as high as 10 m/s, the mutual impedance at 10 

Hz differs little from that calculated for zero velocity. However, 

the non-reciprocal effects could be significant for higher velocities and/or for more highly conducting and larger ropes.) 
--- 

SECTION 3 

.JAMES K. WAIT 

Abstract-The electromagnetic theory of an infinitely lonq 

cylinder is presented for the case when the electrical conductivity and the magnetic permeabil ity are uni-axial tensors. 

This is an idealized yet relevant model for a stranded wire 

rope or cable that is to be excited by an external alternatingcurrent source. The general and special forms of the solution 

are discussed in the context of non-destructive testing (NDT) 

of the rope. The method of deducing the electromagnetic response for a finite source is described in the special case 

where azimuthal symmetry prevails. 

INTRODUCTION 

There is a rapidly growing interest in the non-destructive testing 

of materials using electromagnetic fields [I]. The basic idea is to 

induce currents into the target and then observe the secondary response. 

The material properties are then deduced from the measured data. At 

least this is the objective which is seldom attained. 

A par titularly important example of non-destruc tive testing deals 

with the examination of metallic cables or stranded wire ropes [Z]. Both 

the electric and magnetic properties may be an indicator of the mechanical condition of the material. For example, internal breaks in the 
strands will decrease the longitudinal component of both the effective 

conductivity and permeability of the cable. Also, internal corrosion 

will tend to inhibit azimuthal and radial current flow so the effective 

transverse component of the conductivity would be mainly affected. 

STATEMENT OF PROBLEM 

Our purpose here is to set forth a general analysis for the electromagnetic fields that can be induced in an anisotropic cylindrical structure of infinite length with a circular cross section. The procedure to 

be employed bears some similarity to the analyses of isotropic cylindrical structures [3]. We simplify the problem to some extent by choosing 

uni-axial forms for the conductivity and permeability tensors with their 

principal axes to be taken parallel to the axis of the cylinder. In 

spite of the seemingly simplified description, the resulting field solution does not seem to be available. 

FORMULATION 

To be specific, we chose a cylindrical coordinate system (p,@,z) 

such that the surface of the cylinder is p = a where a is the radius. 

The external region p > a will contain the sources but for the time 

being we will restrict our attention to the internal region p < a. 

Maxwell's equations for the internal region, for a time factor 

exp (iwt) , are 

-+ -+ 

fa)E = curl H (1) 

+ -+ -iw(p)H = curl E (2) 

-+ -+ where E and H are the vector electric and magnetic fields respectively, 

and (0) and (y) are the tensor conductivity and permeability, respective1.y. In view of our stated assumptions, we may write 
and 

La 

Here it should be noted that op, a@ and a are scalar complex conZ 

ductivities; thus, for example, a = gP + WE: where 

g~ 

and E are 

P P P 

the real conductivity and the real permittivity, respectively, in the p 

direction. Similarly, we could write 

l1 Y, - 1 - i where p and pi 

P 

are both positive real. However, for virtually all applications in NDT, 

the elements of (a) and (p) can be regarded as real since the imaginary parts are entirely negligible. A more important limitation is that 

we restrict attention to time harmonic fields of sufficiently small magnitude that the elements of (a) and () do not vary with the field 

magnitude (i.e. we are within the linear regime). 

Equations (1) and (2), expressed in cylindrical coordinates, are 

Leading up to later developments, we now assume that the field components 

vary according to exp (-im@)exp (-ihz) . For single-valuedness, m is restricted to positive or negative integers including zero. The parameter 
A, as yet, is unrestricted. Maxwell's equations are now simplified to 

To proceed further, we can now deconipose or decouple the above set 

of equations into two cases. In the first case, we set H = 0 whence 

Z 

it is a simple matter to deduce that E satisfies 

Z 

The other case is when we'set E = 0 whence H is found to satisfy Z Z 

Now clearly the total field is the superposition of these two cases. 

In fact, it is not difficult to show that 
GENERAL SOLUTION AND ITS PROPERTIES 

Solutions of (17) and (1.8) are modified Bessel functions. In fact, 

for fields that are finite at r, = 0, we can verify that 

where 

and where f (A) and f (A) are unspecified functions of X and ym h ,m 

Without actually solving anything, we can at this stage draw a number of important inferences about the eddy current testing of wire ropes 

and cables. When we impress an axial electric current, we expect, of 

course, that the axial electric field in the cable will be dominant. In 

fact, if the axial variation of the field is small (i. e. A is small), the 

forms of the solutions simplify. Then 

1 .. 

where fi =(io 1~ and 6 = m / ) Furthermore, since 20 P 

HZ is negligible in such cases, E and E4 are also small. The induced currents 

P 

and the secondary external f i e l d s a r e thus predominantly a function of the 

axial electrical conductivity oz and the azimuthal magnetic permeability 
p@ . However, the higher-order harmonics (i. e. m > 0) are influenced to 

some extent by the radial magnetic permeability p . P 

Another comparable situation is when the impressed magnetic field is 

axial and uniform. Then 

1, . k.; 

where ; ~(ip o w) and I = m(n /o ) -. Now the significant induced curz(P 0 P 

rents are azimuthal and driven by the azimuthal electric field 

Clearly, in this case, the secondary fields are mainly a function of the 

axial magnetic permeability p and the azimuthal electric conductivity z 

Analogously, the higher order harmonics are influenced by the radial 

electric conductivity o '. 

P 

The two cases considered above can be described as the E-field or 

the H-field method of excitation. Actually, it is the latter H-field 

configuration that is the basis for nearly all existing methods of eddy 

current testing of cables and wire ropes. In the case of the so-called 

DC technique, the external excitation is a solenoid that carries a large 

azimuthal current. The secondary H-field is then heavily influenced by 

the axial magnetic permeability. On the other hand, in the so-called AC 

method, the secondary field results from the induced eddy currents in the 

cable. These are primarily a function of the azimuthal conductivity O@ 

in addition to the axial magnetic permeability p . 

z 

The E-field configuration would arise if the cable were excited by 

a toroidal or doughnut shaped coil [4]. The resulting induced currents 

now flow principally in the axial direction. In general, in this case, 

the finite value of X needs to be considered. Nevertheless, the 
secondary fields are now mainly a function of the axial conductivity c~ Z 

and the azimuthal permeability . 

The situation generally becomes quite complicated when the effects 

of the non-zero values of X and In are considered. The resulting 

fields then become hybrid [3] bein2 neither purely E or H. In such cases, 

the secondary f i e l d s w i l l depend, in general, on all tensor elements of 

the conductivity and permeability. Then it appears that quantitative 

calculations must be resorted to in order to gather further insight. 

CASE OF AXIAL SY-WqETRY 

In the following discussion we will restrict attention to azimuthally 

uniform excitation such that all the harmonics for m $ 0 may be discarded. We will consider, however, fields that have a significant variation in the axial direction. Now the field components are decomposed into 

two sets. For the E-field type, 

and 

E = (iX/o )H F P 4) 

For the H-field type, we would have 

-iX z 

I3 z = fh(A)lo(v~)e 

and 

-imp vf (A) 

E=- I1(vp)e -iAz 

x2 + imp 4 

F 4) 
Two important physical parameters are the axial series impedance 

Z (A) and the axial series admittance Ys(A) of the cable. These are 

S 

defined and given as follows: 

E (A' + iwci kr )I (ua) 

z (A) = ---- - - P Lo 

s 2naH 2nao uIl(ua) 

P 

and 

(A2 + imp )I (va) 

Ys(A) = - - - - PO 0 

2n aE 2naiuv vI (va) P 1 

Two limiting cases follow immediately: (I), lual and lva/ << 1, 

then we obtain the static forms 

and 

z (A) 1/(na20 ) S Z 

Y (A) l/(na2iwpz) S 

(2) , 1 ua 1 and lva 1 >> 1, then we obtain the asymptotic high frequency 

and 

These limiting cases have the expected dependencies on the electric 

properties. Not surprisingly, the static impedance (i.e. resistance) 

depends only on the axial conductivity. The analogue admittance depends 

in the same on the axial permeability. 
THE EXCITATION PROBLEM 

We now consider the excitation problem. The external sousce is 

taken to be a solenoid of radius b of finite axial extent that is concentric with the cable. The situation is illustrated in Fig. 1. The 

surface electric current density in the solenoid, in the present idealization, has both an axial and an azimuthal current density defined by 

and 

j (z) = j sin$ \ Z 0 

R for -- R 

<Z<- 2 2 

j$(z) = j cos$ 

0 

where is a pitch angle of the current flow. Here one should note that 

$ = 0 corresponds to purely azimuthal current flow which is the condition 

approached in a solenoid of many turns. In general, for a pitched winding, 

the axial component of the current flow should also be accounted for. The 

corresponding boundary or initial conditions for the tangential magnetic 

fields, according to Ampere's law, must be 

-Iwhere b-- = limit of b t n when the positive quantity n tends to zero. 

For the same excitation, we also have continuity of the tangential 

electric field. Thus 
We now use the Fourier integral representation 

where it follows that 

The desired representations are thus 

and 

In the free space region p > a, it is convenient to derive the 

fields from electric and magnetic Hertz vectors [3] that have only z 

components U and V, respectively. The fields are thus obtained from 

where k2 = E IJ u2. 

0 0 0 

To be compatible with the selected source fields, we assume the 

respective forms, for a < p < b, 
where R2 = h2 - k2 and Ae and Ah are function of X yet to be 

0 

determined. Now the functions 6 and 6h can be determined immediately e 

by imposing the impedance and admittance conditions at p = a that are 

specified by (38) and (39). Then, using (56), (57), (58), and (59), it 

is easy to show that 

and 

Now in the outer region p > b the appropriate forms for the Hertz 

potentials are 

+a2 

and 

because the fields must be non-inf inite as p + m. 

The source conditions specified by (46) and (47) can now be applied 

to yield 

and 

where 

h -(~jo/nA)cosl(l sin(A,t/2) 

A (A) = 

P'[' 0 +(fill-Rh)Kol 
h R = c~/A~ = '(I1 - 6hK1)/K1 (69) 

In the above four equations, the indicated modified Bessel functions are 

all functions of Pb. 

CONCLUDING REMARKS 

In principle, we have now solved the problem since the resultant 

fields are now given in terms of the source current. The next step would 

be to deduce the pick up voltage in a sensor such as a small loop or electric probe. This is a straight forward process once we have specified 

the source and sensor configuration. 

The method of obtaining the field expressions for the concentric 

current sheet is illustrative of the procedure to use in more general 

situations. For example, if the exciting current distribution is no 

longer azimuthally symmetric, we would require that higher harmonics 

of order m be included. This would tend to complicate the calculation 

for anisotropic cylinders because the modified Bessel functions are not 

of integer order. Never theless, this would be wor thwl-iile. 

A particularly important extension would be to consider the uniaxial tensors (3) and (4) to be directed along equi-angular spirals. 

This would correspond to the basic structure pf stranded wire ropes. 

Analytically, this does not seem to be a trivial extension, since the 

axial E and H fields would then be solutions of coupled second order 

equations. 

Further work on this subject is underway. The present preliminary 

report is intended as a guide to the analytical procedure that will be 

needed. 

'ER 

OR CABLE 

Fig. 1 ANISOTROPIC WIRE ROPE OR 

CABLE EXCITED BY A CONCENTRIC 

CURRENT SHEET OF FINITE AXIAL 

EXTENT. 
- 

Section 

- -. . - . - -- - 

4 

A remarkably simple and novel solution is obtained for the 

fields induced in an anisotropic cylindrical shell that is 

located coaxially within a long solenoid. This could be 

the basis of a non-destructive measuring scheme for stranded 

wire rope. 

The non-destructive testing of wire ropes and cables exploits the electromagnetic response characteristics of cylindrical conductors. T1 the 

operating frequency is sufficiently low, the primary field can penetrate 

effectively to the interior of the sample and respond to internal imperfections such as fissures and voids. A straight-forward scheme to achieve 

this objective is to insert the sample into a long solenoid and measure 

the series impedance of the latter at a number of frequencies1,'. The 

electromagnetic basis of this method was discussed in an earlier paper" 

where the sample was idealized as a homogeneous cylinder with'a specified 

conductivity and permeability. 

It could be argued that a stranded wire rope is a cylindrical conductor but, because of the coinplicated spiral structure, it woi~ld not be 

isotropic. To shed some light on this problem, we examined the rc>sponsc 

of cylindrical conductors that were characterized by uni-axial conductivity and permeability tensors4. To facilitate the boundary value solution, 

the axcbs of these tensors were taken to be coaxial with the cylinder. A 
rigorous solution for general anisotropy would be worthwhile but it 

appears this leads to couplet1 second order equations for the wave functions that are not immediately solvable by analytical methods5. As an 

interim approach, we adopt here a very simple model that consists of a 

thin shell whose conductivity along orthogonal spirals is different. 

The remarkable simplicity of the quasi-static form of the solution is 

justification for studying this problem in its own right. 

We consider a thin cylindrical shell of thickness d and radius a 

that is encircled by a solenoid of radius b. The shell is characterized 

by a conductivity a along spirals with a pitch angle of I); the conductiS 

vity in the transverse direction is a The nature of the idealizations t' 

will be evident in the formulation of the problem. The objective is to 

obtain an expression for the series impedance of the solenoid. 

To be more specific, a cylindrical coordinate system (p,@,z) is 

chosen and the shell of assumed infinite length is defined by p = a. 

The encircling solenoid of negligible thickness is at p = b. The impressed azimuthal current density in the solenoid is j amps./m. The 

region inside and outside the cylindrical shell is homogeneous with 

intrinsic propagation constant y and intrinsic characteristic impedance 

112 . For free space conditions, of course, 'y = i(~ p )ll'w and I') = (pO/c0) 0 0 

for a harmonic time factor exp(iwt). 

Leaving aside many elementary details, we can now write down appropriate expressions for the axial fields and circumferential fields in 

the various regions: 
anisotropic shell. A direct application of Ampere's law requires 

L11;i t 

H (a+Oj - H - (a-0) = -n dl< (a) S b t t 

an tl 

Ht (a-tO) - Ht(a-0) = (J-~E (a) s :; 

wtle~t~ the slrbscrillt s o:. t ctesignates that fi.eJ.d component is in the 

direction of the spiral or transverse to it. Also, for this model, r 

% 

and E are continuous thro~gh the shell so we do not need to disti1lg:rish t 

between rrlle two vnlues on each sicle. Kcw we note that 

H (a?-0) = H Jato) -C - H (atO)*S z s n t o (6) 

and 

I1 (n.0) = H (a+O) *So + H (at-0) *Ct, @ S t 

C 7) 
where C = cos$ and S = sin$ in terms of the pitch angle $. It is now 

0 0 

a simple matter to rewrite (4) and (5) in the form 

HZ(a+O) - HZ(a-0) = otd(EZSo-E Co)Co - ~sd(~ZCo+~ So)So 0 0 

and 

H (a+O) 0 - HO(a-O) z0 0 z0 0 = btd(E S -E Co) So + osd(E C +E So)co 

where it is understood that the E and the E components are evaluated Q, Z 

at p = a. 

We can apply (8) and (9) to general forms given by (I), (2), and 

(3). Then, bearing in mind that E (a+O) = E (a-0) and EZ(a+O) = EZ(a-0), 0 0 

we can deduce that 

where all the Bessel functions in the elements of the determinants have 

common argument ya. 

The series impedance Z per unit length of the solenoid can be obtained from its basic definition Z = E$(p=b)/j. But what is most meaningful is the ratio Z/Z~ where Z is the corresponding series impedance 0 

of the solenoid in the absence of the core sample. It easily follows 

that 

z/zo = 1 - (B~/C*)K,(~~)/I,(~~) 

Algebraically, this seems rather complicated but a great simplification 

ensues if we invoke the q~~asi-static approximation where 1 yb 1 << 1. Then, 

using the small argument approximation for all the modified Ressel 
Co. I E LII~ spiral 1s wow~d :.n that the wires are nearly axial (i .e. 

S = sin$ << 1) , we see that the effective coiiduc tance of the shell -is 

o 

~+d. Conversely, if the pitch angle 1) is near 90" (i.e. wires almost 

circumferential so that C = cosq << 1). the effective conductance is 

O 

0 d. 

S 

More complicated rnodels could be constructed by employing several 

concentric she7 13 possibly in combination with intermediate regions of 

finite isoLropic properties. This co~lld ?e.id LO .I more flexible yet 

still tractable ~oodel to desc-ribe the salient cllrrent flow partern ir: 

a stranded wi rc. rope. Work or1 the subjcc t continues. 
REFERENCES 

1 LIBBY, H.L.: 'Introduction to electromagnetic non-destructive test 

methods', (John Wiley & Sons, 1971), Sec. 5.2, pp. 135-150). 

2 FORSTER, I?., AND STAMBKE, K.:'~heoretische und experimentelle 

Grundlagen der zerstGrungfreien Werkstoffprufung mit Wirbelstromverfahren', 

Pt. III., Zcits. MetaZZkde, 1954, 45, pp. 166-179. 

3 WAIT, J.R.: 'Electromagnetic basis for non-destructive testing of 

cylindrical conductors', IEEE Trans., 1978, IM-27, pp. 235-238. 

4 WAIT, J.R.: '~lectromagnetic response of an anisotropic conducting 

cylinder to an external source', Radio Sci., 1978, 13, 

5 WAIT, J.R.: 'Electromagnetics and plasmas', (Holt, Rinehart and 

Winston, 1968) . 
:NTRODU(:TION AND FCRMULATION 

In an earlier paper [I], we had described the electromagnetic basis 

for the non--destructive testing of c;rlinclrical conductor..;. The model vas 

higlily idealized; it consisted of a long solenoid that encircled the honogeneous cylindrical sample. Here we w~sh to generalize the theory to accounl 

for concentric layering within the sample. This is a possible model LL) account 

for the effect of internal and external corrosion that can occur in cabies 

or wire ropes. 

The derivation is given in outline only since the method is well <1ocumented elsewhere [ 2, 31. IJe deal specifically wlth a cvPi ndrical ~arny.1~2 tila: 

has an outer radius a and a core of radius a There is an intermedidte 1 3 - 

annular region bonndrd by cylindrical surfaces of radii a and d 2 3- The three 
regions have electrical properties a E and p where j = 1'2'3. For 

j' j' j 

example, the outer layer has conductivity a permittivity e and permeabil- 1' 1 

ity pl, the intermediate layer has properties 0 2 ' e2 and p2 and the core has 

properties a3, E* and p 3' 

The sample is located within a solenoid of radius b but it need not be 

centrally located provided the free-space wavelength is much greater than b. 

Then, proceeding in the same manner as in the previous paper [I], we can 

derive an expression for the series impedance Z of the solenoid per unit 

length in the form 

where Z is the series impedance when the solenoid is free of the sample. 0 

Here Z is the inwards looking wave impedance at the sample. For example, C 

if the latter were homogeneous with electrical properties a E and p 1' 1' 1' 

we would have 

where y = [iplw(0 +iclw)11'2 is the intrinsic propagation constant of the I 1 

sample. Here I and I are modified Bessel functions of the first kind. 

0 1 

This special case was derived in detail in the earlier paper [I]. 

THE GENERAL WAVE IMPEDANCE 

To account for the concentric layering of the sample, we need to employ 

the appropriate expression for the radial wave impedance Z . This can be 

C 

obtained in a rather prosaic manner by setting up wave function in each of the 

homogeneous regions and determining the unknown coefficients by matching 

tangential fields at the concentric interfaces. A more physically meaningful approach is to employ non-uniform transmission line theary as so elegantly 
of radiu~ 3, is given by an analogous form: L 

where 

and 

where y = ip.w(n.+ir.~)~~~~ and n = iU.w/y_ for j = 1,2,3. i [ :I 1 1 j 3 1 

Act\is?ly, subjccc to such assumptions as axial and azimuthal uniEormity, 

the expressions for Z :and Z are exact. For computation, however, we can C C 
1 

and u1 and radius a This, of course, corresponds to the general case 1' 

under the condition a - 2 a3 = 0. As indicated in the inset in Fig. 1, the 

sample is also located centrally within the solenoid of radius b. To be 

definite, we also choose the sample radius a to be 2 cm which is kept fixed 1 

i.n all that follows. Also, the conductivity of the (uncorroded) sample is 

taken to be 1.1 x lo6 mhoslm and its relative permeability 11 = pl/uo = 200. re1 

These are typical values for steel. The corresponding series impedance 

Z = R + iX of the solenoid is shown plotted in Fig. 1 in an Argand diagram. 

The abscissa and ordinate are actually R/Xoprel and X/Xourel, respectively, 

where we have assumed that the impedance of the same empty solenoid is 

Z = iX being purely inductive. Various fill factors F = a2/b2 are shown and 

0 1 

the operating frequencies are indicated. As expected, at sufficiently low 
frequencies, R tends to zero and X approaches X x yrel X F. Not surprisingly, 0 

the series impedance of the solenoid is sensitive to the diameter of the 

sample. 

We now consider an external rust or corrosion layer of thickness R at 

the outside of the sample as indicated in the inset in Fig. 2. This is a 

special case of the general analysis when a = 0. The uncorroded region of 3 

radius a has the same properties as before (i.e. o2 = 1.1 x lo6 mhos/m and 2 

- 're1 - p2/y0 = 200) but the external layer now assumes the values o = 1 1 

mfio/m and = yo. 1 The results are, shown in Fig. 2 where we have taken the 

fill factor F to be 1 (i.e. al = b). Different values of R, the thickness 

of the rust layer, are shown. Not surprisingly, the effect of an external 

rust layer is to reduce the effective radius of the cylindrical steel sample. 

Finally, we consider an internal c.orrosion layer. As indicated in Fig. 

3, this corresponds to the case where the full formula for Z as given by c' 

(3), is needed but we set o = tr = 1.1 mhos/m and 11 /p = y3/p0 - -. 

1 3 10 're1 

200 which are the "uncorroded" regions. In the central annular "corroded" 

region, we choose p = y and o = 1.0 mho/m. For the results shown in Fig. 2 0 2 

3, we have also chosen the radius a of the core to be 0.3a but the value 3 1 

of R ranges from being zero (i.e. no rust layer) to a value of 0.3a As 1' 

seen in Fig. 3, the series impedance of the solenoid is still markedly 

affected by the presence of the corrosion layer, even though it would not 

be visible to an outside observer. Of course, the frequency must be sufficiently low so that the primary fields of the solenoid can penetrate to the 

interior of the sample without significant dissipation. 
CONCLUDING REMARKS 

The present calculations, while based on a highly idealized model, 

indicate the potential of non-destructive schemes using electromagnetic 

waves [5]. More sophisticated models based on the layered cylinder formulations can easily be implemented. Actually, three dimensional models that 

allow for the finite extent of the primary field and the internal anomaly 

have been examined elsewhere [6,71. 

Certainly, the sensitivity of such devices to the electrical, magnetic, 

and geometrical properties needs further study. Here we have only indicated 

a few examples that might promote further developments in this area. 
a wire rope by a toroidal coil", J. AppZ. Phys., Vo1. 48, No. 12, pp. 

4893-4897, 1977. 

[7] D.A. Hill and J.R. Wait, "Scattering by a slender void in a homogeneous 

conducting wire rope", Appl. Phys. (Springer-Verlag) , Vol. 16, pp. 791- 

398, 1978. 
FIGURE CAPTIONS 

Fig. 1. Normalized series impedance of solenoid of radius b for a concentric 

homogeneous cylindrical sample of radius a for various fill factors 1 

Fig. 2. Normalized series impedance of solenoid of radius b = a for a 1 

sample with an external. rust layer of thickness R. 

Fig. 3. Normalized series impedance of solenoid of radius b = a for a 1 

sample of an interior rust layer of thickness R. 
RESISTANCE .. R 

Fig. 1. Normalized series impedance of solenoid of radius b 

for a concentric homogeneous cylindrical sample of 

radius a for various fill factors F = avb'. 1 I 
0.5 1.0 

RESISTANCE R 

Xo Prel 

Fig. 2. Normalized series impedance of solenoid of radius b = a 1 for a sample with an external rust layer of thickness R. 
RESISTANCE . . 

R 

Fig. 3. Normalized series impedance of solenoid of radius t = :i 

for a sample of an interior rust layer of thickness Y. 
Section 6 

JAMES R. WAIT 

and 

Robert I,. Gardner 

~bstract-A self contained analysi s i s presented for the 

impedance of a solenoid that encircles a conducting cyl inder 

that has an internal flaw or anomaly that is also cylindrical 

in form. A perturbation method is used to obtain an expression 

for the fractional change of the impedance as a function of the 

size and location of the anomaly. 

INTRODUCTI ON 

There is a need to evaluate the internal structure of wire ropes and 

similar cylindrical conductors. This has led to an extensive technology 

called non-destructuve testing (NDT) . A good survey of the electromagnetic 

methods are given by Libby [I]. One of the basic configurations is a solenoidal coil that tightly encircles the wire rope specimen. As indicated in 

a two-dimensional analysis [2], the impedance of the solenoid can be related 

uniquely to the conductivity and permeability of the wire rope if it can be 

assumed homogeneous. It is the purpose of the present analysis to examine 

the effect of an internal flaw. While more complicated cases [3] can be 

treated , we consider here an idealized two-dimensional anomaly or cylindrical flaw of radius c that is contained within the otherwise homogeneous 

cylinder of radius a. The situation is illustrated in Fig. 1 which we will 

describe in more detail below. 
of the encircling solenoid . 

A preliminary problem that we need to consider is the excitatiori oi a 

cylindrical flaw by zr known form oi primary field. In this manner, the 

polarizabilities or strengrhs of the induced monopole and dipole can be 

ascertained. We the11 use these results in the composite problem where t.he 

exciting field is produced by the encircling solenoid. Finally, some nurnrriL-a1 results are presencecl to illustrate the fractional change of the solenoid impedance as a function of the dimensions and location of the anomaly 

PROTOTYPE PROBLEM 

As indicat-ed above, it is useful to consider the following prototype 

solution. A circular cylinder of conductivity 0 and permeability 11 with 

a a 

radius c is excited by an electric line source of strength M at a radial 
where the modified Bessel functions on the right hand side are of order m. 

It is then a simple matter [5] to show that the resultant field external to 

the> cylinder is 

where 

Il = iW/y , va = illaw/ya and ya = (ipaoaw) 4 . 
and 

or 

COMPOSITE PROBLEM 

Now in the composite problem indicated in Fig. 1, we have 

whicl~ wnul~! bc the fields i~i:iiclt\ tile l~omogenrous cylinder (i.e. p<a) if there 

wiis no internal anom;-llp t)r €Law. These applied fields excite or induce the 

n~i)nol~olc., tlLl~olc, ant1 rnultipolcs within the cylindrical flaw. These in turn 

produce secondary fields that arc. observable outside the main cylindrical 

conductor. In what follows, we actually assume that a - p, >> C. 
In order to deal with the interaction of the above secondary fields 

with the wall boundary at p = a , we need to translate back to the (p,@, z) 

coordinates. First of all, we may employ the same addition theorem as used 

in the prototype problem: 

-P 

?'he other addition needed is obtained straight-forwardly by differentiating 

bothsidesof (15) withrespect top Thus 1' 

wilere we note that 

Tt is now a simple matter to deal with the wall interaction. This 

nomints to replacing Kn(yp) in both (15) and (17) by Zn(yp) where 

where 

a 11ci 

'I'l~is soll~t ion is ciltirely analogous to the earlier result but now the equiv;llc.nt 1 inc. source is inside the cylinder. Here z is the radial wave 

an 

inil~eda~~~c looking outwards at p = a into the external free space region. 

Now we can write down the desired form for the secondary field. 

Clearly it is given by 
If lyal << 1 , this reduces to the very simple result that 

A = (L: -ll)c2/(bln2) 

a (26) 

which is compatible wiLh physical intuition. Here the contribution From the 

induced electric dipole term is of second order. 

NUMERICAL RESULTS AND DISCUSSION 

Some ~~umerical results for the fractional impedance change /A / of the 

solenoid or sensor were ohtained from (25) for the case where the wire rope 

conductivity cr = 10%ho/m and the relative permeability p/p = 44. The 

0 

wire, ro[rt. r~tiliils ;I was 2. 54~111 (i.e. 2 inch diameter rope). The flaw or 

anomaly i tst'lf was ;tss~imed to be an air-filled void with free space properties (i.c. L' 0 

= IT x 10-'henries/m, = 0 and E~ = 8.854 x 10-12farada/m) 
Most of the curves shown in Figs. 3, 4, 5, and 6, on a log-log plot, 

differ imperceptibly from straight lines with a slope of two. In fact, for 

2 the I1.C. limit, (26) tells us that A (c/a) . However, for the higher 

frequencies, the dipole term has a non-negligible contribution and the result is that the curves tend to be somewhat steeper. 

F1'6, 1: The composite problem 

1, 

I'rototype problem 
Ratlio of Arlomaly D.iaineter to Cylinder Diameter 

I Figure 3 : Fractional Inipec.lance (:tinnge for Sensor as a Function 

of Anomaly Size. The Arloinaly is in the Center of 

the Cylinder. 
10-3 10 -L 

Ratio of Anom:lly Diameter to Cylinder Diameter 

Fi.gurcl4 : Fractional Impedance Change for Sensor as a Function 

of Arlomaly Size. The Anomaly is Slightly Offset from 

the Center of the Cylinder (f', = .la). 
'! 

r'! 5 

,; :,1 6; * *. 

! 

.? ., . ' ;.,... 11, i:ii ;;!\c.r,~;~:Ly Diameter to Cyl.'incicr Diameter 

' ' : Fr.2~ ion31 Impedailce Change for a h~nsor as a k'uncl ioii 

of ,\ilc.~ns L y Size. The Ano~nal y is Of Esei one--11al.r t lic: 

(.v 7 i citicbr X,~dius (q = .5a) . 
Rn t i o of Anoml y Diameter to Cylinder Diameter 

L , !:ractionnl Impedance Change for a Sensor as a 17unction 

of hnonialy Size. The Anomaly is Near the Cylinder 

Sur Face. ( e, = .8a) 
Sc3c.t ion 7 

1)AVIT) A. HILL 

and 

JAMES R. WAIT 

Abstract. A thin prolate spheroidal void in an infinite conducting circular cylinder is used to model a broken strand in a 

wire rope. The rope is excited by an azimuthal magnetic line 

current which is a model for a thin toroidal coil. The anomal.ous 

external fields are computed from the induced electric and magnetic dipole rnments of the void. The results have applications 

to nondestructive testing of wire ropes. 

Electromagnetic nondestructive testing is widely used in the metals 

industry to inspect and evaluate materials [I]. Both dc (direct current) and 

ac (alternating current) methods [2,3] are currently used for inspection of 

wfre ropes [4 which are made of electrically conducting f errmagnetic material. 

In both met'lmods, the magnetic. field is applied along the axis of the rope, and 

rope irregularities produce magnetic field changes which induce voltages in 

the sensing coi1.s. 

An alternative ac method would excite the rope with an axial electric field 

rather rllarl ;.ill axial niagrletic f ield. Since axial electric currents would be 

fntluc~*tl, this rncrhotl might be nlore scnsitivc to broken strands than the present 

ac me cllod wh iclk irlcluccs 0111 y az ilnr~ t-I~il currents which tend to flow normally to 

the strarlds. S11c:h a rriethod could also he used on electrical conductors which 

are not ferrmagnetic. 
111 

An appropriate source for excitiltion of axial electric currents is a toridal 

coil which encircles the rope, and a magnetic current model for the toroidal 

source coil has been analyzed [5]. Expressions for both the interior and exterior 

fields were derived for a homogeneous wire rope. 

In this paper we wish to allow for the presence of a small slender void 

within the rope. Such a void can be considered a model for a broken strand. 

The void is also allowed to be oriented at any arbitrary angle to the rope axis 

to account for the winding geometry of the wire rope. The induced electric and 

magnetic di-pole moments are compr~ted from the primary fields and the electric 

and magnetic polarjzabilities of tlle void. We then obtain expressions for the 

external ficlds of electric and magnetic dipoles of arbitrary orientation. It 

- 

is these external fields which are the observable quantities in any Dl nondestructive testing method. The particular expressions, derived for the external fields 

of internal electric and magnetic dipoles, should be useful in future analyses 

of other types of small imperfections in wire ropes. Of course, in such cases, 

th~ electric and mapnetic nolarizahil i ties wor~ld he different. 

Primary Field Excitation 

In a previous analysis [5], we analyzed an azimuthal current sheet source 

which encircled the rope. The current sheet was allowed to have arbitrary width 

in the I. or axial direction and arbitrary azimuthal extent. This source results 

in fairly complicated expressions for the electric and magnetic fields. Since, 

in this paper, we are primarily concerned with the fields scattered by the void, 

we take the simpler special case for the source shown in Fig. 1. Specifically, 

a magnetic current ring of strength K is located at a radius b in the plane 

z 0. This is a model for a thin toroidal coil which completely encircles the 

rope. The adopted time dependence is exp (iwt) for all source and field quantities. 
pennenbili~y \J in order to model n broken strand. The ~jrolate spheroidal shape v 

is a convenient one because its electric and magnetic polarizabilities are known. 

However, we would not expect a significant difference for a thin circular cylindel of the same ler~gtli and volume. To account for the winding geometry of the 

rope, we a3low a rotaLion of the major axis of the spheroid about the p' axis by 

A 

an angle a. Th11r; the rnajor axis is oriented at an angle a to the unit vector z' 

and an angle n/2 -t a to the unit vector ;' as indicated in Fig. lc. 

Since the void has a cvntrast in both the electric and magnetic properties, 

both electric and magnetic dipole moments will be induced 161. The electric poe larixabi l it ies for the iticiJent electric field applied along the major axis, ci maj' 

e 

or along the minor axis, a are given by min ' 
e 

a = -v(oW-ov) maj 

and 

- 2v (ow-ov) 

a - min l+o;/aw 9 

where V is the volume of the thin prolate spheroid. Since we anticipate the 

use of very low frequencies, we have neglected displacement currents. To include 

them, ow would be replaced by a + iw~ and oV would be replaced by W W 

0 + WE in (1) and (2). The magnetic polarizabilities for the incident mag- v V 

m m netic field applied along the major axis, a or along the minor axis, maj' amin 7 

are similarly given by 

o;" = -Via (pw-pv) maj (3) 

and 

a--- - min 1 + pVhw 

In order to compute the induced dipole moments, it is first necessary to 

resolve the incident electric and magnetic fields into components along the 

major and minor axes. The resultant induced dipole momen'ts can then be resolved 

into the more convenient , , and z components. When this is done, the 

induced electric dipole moments are found to be 

(Ids) = pr 

e 

- a )sinacosa , E~(amin maj (6) 

(Ids)p = E~~ ae 

p min 

In (5)-(7), the primary electric field components Epr and EPr are evaluated 

Z P 

at p' ,@' ,z' . Similarly, the induced magnetic dipole moments are found to be 
P - 

consj~derctl here). 

The till-ee scattered field components can be compactly written in the fo.Llow- 
In the first 3 by 3 matrix. G~ is the p component of H at (p.4,~) pro- PP 

duced by a unit p-directed electric dipole at p 4) ,) , ce is the p com- 

~4) 

ponent of H produced by a unit $-directed electric dipole at (p',@',zl), etc. 

The second 3 by 3 matrix is similar except that the G~ elements correspond to 

unit magnetic dipo1e.s. We choose the above matrix notatiorl rather than the elegant dyadic Green's function notation [7] because we are interested in the relative irrlportance of each source and G term. Also, for the cylindrical geometry 

encountered here, all of the matrix elements can be derived in a systematic 

manner f rain z components of electric and magnetic Hertz vectors [B]. The specific 

expressions for the matrix elements are derived in Appendix B. 

Numerical Results 

The quantities of most interest in NDT are the external (p>a) magnetic 

field components which are observed with the sensing coils. The primary magnetic 

field has only a 4) component H~~ which is given by (A-7) for a < p < b or 

$ 

by (A-1.1) for p > b. The scattered magnetic field has all three components as 

given by (11). A computer program has been wri-tten to compute the external prinlary and scattered magnetic field components. 

There are too many parameters to present a thorough parametric study here. 

For all nrimel-ical results shown the following parametric values remain fixed: 

a - 1 cm, a = 1.1 x 106mho/m, pw = 200 po, b = 2 cm, frequency = 10 Hz, p = 2 W 

cm, o = 0, and pv - 

- 1. For this low frequency, the conduction currents 

v 

dominate, and the permittivities E and cV are unimportant. The above values 

W 

of Ow and p are roughly representative of stainless steel [9], but the perme- W 

ability of steel is quite variable [lo]. For the above parameters, the radius 

a is approximately one skin depth. 
Ti1 Figs. 4, 5, and 6, we retain z = 5a and a = 0" but allow z' to vary. 

Also, we set p' /a = 0.5. Note that the level of H" and H"' increases as 

P - cb 

z ' appl-oac11e.c: Z. o ow ever, liSC is very small for z' = z, and its peak level 

z 

is ai: about- z' /a = 3.75. Since a - 0" in Figs. 2-6, only the axial and radial. 

electric dipole moments, (ids) and (Tds) and the azimuthal magnetic dipole, z P' 

(KdR) are induced. Also, (Ids) is very small because its exciting field 9 ' P 

E"' is soull. The cnlct~lrlf ions reveal that (Ids)Z and (Kd&)+ contribute 

P 

approximately equally to the scattered field. 

In Fib;s. 7-9, we have z = z' = 521 and p'/a = 0.5, but we allow (3 to 

vary from O0 to 30". Nonzero values of a allow two nddltional dipole mornects, 

(Ids) and (KdY!)z to be induced. The result is a decrease in the level OF 9 

H~~ and H: and an increase in the level of H~'. @ z 
The phase of the scattered field was also computed, but was found to be of 

less interest. Also, the individual dipole contributions were computed, and 

they are of individual interest since they will be excited differently for other 

sources on other void configurations. 

Concluding Remarks 

An ideal-ized model for a wire break in an infinite conducting rope has been 

analyzed. The thin prolate spheroidal void has been shown to produce a significant anomaly in the external magnetic field when the rope is excited by a thin 

torol-dal coil. In general, all three components of the scattered magnetic field 

are nonzero, but the azimuthal dependence is fairly complicated and contains 

some deep nulls. This suggests that lnultiple sensing coils of various locations 

and orientations might bc worthwhile. 

Altfio~~gl~ we have generated numerical results for the specific case of a 

thin prolate spheroidal void, the formulation give:i here actually yields the 

external fields for arbitrary induced electric and magnetic dipole nioments. 

Thus the formulation is useful for any type of rope imperfection that can be 

characterizcd by induced electric and magnetic dipole moments. This requires - 

only that the rope i-mperfection be small in terms of the rope radius and the 

rope skin depth. Larger imperfections should have a similar qualitative behavior, but could probably be rigorously analyzed only by solvir~g an intclgrnl 

ecluat ion for the fields in the imperfection. 

Also, it is a simple matter to perform similar calculations for a solenoidal 

coil of the type used in present NDT systems [3]. The scattered field calculation 

remains unchanged, but the primary field would be TE(E'~ = 0) rather than 

z 

TM(H:~ = 0). Some calculations for this case have been carried out by Eurrows 

[11] for the special case where both exciting and sensing coils arc coaxial with 

the tubular specimen. 
K .I,-; iht. :;t.rcngch of I Ile n1it~:nt5cic current ri~il:, KO is n zero order modified - 

ilt.:i:.rai fu~rt L J otl* iil~(l 131- i~tlcb ' CICI~C>~~\>~ tl-Lf 1 c~ cnLiaLion with rcspcrt to the argument. 

dr; ti.e region bet.ween the. rope and the source (a < p < b), we find 
-00 

where 

Outside the source (p > b), we find 

Epr = - z 

u2hoKo (up) exp (-ihz)dh , 

-0 

lip' = / ~WC~UA~K~ (up) enp (-ihz)dh , 4) -w 

where 
APPENDIX I3 - Dipole Fields 

'l'lrt? tlt+r lv'it ion of t 1 it~lci:; [)rod\rc:ild by a clipol ch source: wi tl~in :I c ircular 

wire rope i :i al)alogoc1.~ to tlliit for ;1 d ipo1.c source in a c: ircular tunnel [8,9], 

and some of the dctcz-i.1~ are excluded here. The fields within the wire rope for 

p' < p < a can be obtained from Uw and V which are the z components of 

W 

electrdc and magnetic Hertz vectors. 

where 

,[ -, = p c [ ,e-im($-mt 1,-'"("-"'I dX 03-31 

-CO -03 

A,(h) and H (A) are functions of the source dipole and will later be given for 

111 

the six d:lpol.e types. Pm and \ are determined from the boundary conditions 

at p = a and are given by 

h iwr 

m 

w wa 

wher c-1 
-iwc K;(ua) 

- y 0- ------ 

m u Km(ua) 

'I'hc nrgi~rnen t of tilth nrod i I I cbd 13cbr;sel func ti 011:; IIn and Km is wa unless 

indicated otherwise. 

The external fields (p > a) can be obtained from 

U = r[um] and V = r[vm] 

where 

and 

vm = Tm(X)Km(up) 

Continuity of E and HZ at p = a yields Z 

and 

The external magnetic field components can be written 

H = I'[R ] , 1 = T[H~~] , and HZ = r[~~~] P pm @ 

where 

OE m 0 =-iXuT K' (up) + m m P SmK,(u~) , 

- - - -mX T R (up) - iwe US K' (up) , P mm o mm 

HZm = -u2vm = -u2~ K (up) . m m 
Similar exprcasione are available for the electric field components. 

I:quntions (8-1)- (11-12) yielrl the c-xtcrnnl. mngne t ic f ield components in 

terms of Am and I3 whi.clr are functions of the source dipole. We now tabulate 

m 

Am and Bm for the six dlpole types [8]. 

For a radial electric dipole (Ids) we have 

P ' 

and 

and 

and 

and 

-iX A = ('Ids) -- n1 I;(wP') 4rr2iw6w 

W 

For an azimuthal electric dipole (Ids) , we have 4 

For an axial e1.ectrl.c dipole (Ids) we have 

1,' 

B =O. 1n 

For a radial magnetic dipole (KdR) we have 

P' 

For an azimut.hal magnetic dipole (KdK)$ , we have 
-1 A = (KdR) ' ---- m I;(wP') @ 4rr2w2 

and 

and 

For an axial mangetic dipole (KdR)z , we have 

We are now able to define the matrix elements required in (11) in terms of 

H , , and HZ given in (3-9). For each of the six dipole types, we can P 

define the required three elements 

For a radial electric dipole 

in a column matrix 

(Ids) P' 

we have 

For an azimuthal electric dipole (Ids)$ , we have 

as follows. 

For an axial electric dipole (Ids)Z , we have 
For u etad la 1 rnngtlet f r- d lpolr? (Kri K) wc Itiivr? 

I' ' 

For an azimuthal magnetic dipole (Kdll)$ , we have 

have 

In order to numerically evaluate the G elements, we encounter the infinite 

summation and integration of the 1' operator as indicated by (B-9). However, 

all of the G elements are either even or odd in m and , and the summation 

and integration can be taken over positive values of m and 1 to reduce the 

computation time by a factor of 4. Also, all G elements are calculated in 

paral.le1 which means that the required modified Bessel are calculated only once 

resulting in a largc t.ime savings. In cvaluatillg the X integration, a variable 

step size intcgrat -ion is used to deal effectively with tt.lrt rapid variation for 

small A and tile slower variation for largc A. Thus the subroutine which 

comp~ites all 18 element:; ninlics use of all these time saving features and is 

- 

quite fast. 
References 

11. I,. Libby: In Lr~~duction to E:Lec:Lrmagnetic flodestructive Test- Methods 

(Wiley Interscience, New York 1971). [Gives numerous references]. 

C.H. Larsen, R. A. Egen, R.D. Jones, H.A. Cross: Report to the U. S. Bureau of 

Mines on Contract No. H0101741, June (1971). 

J.G. Lang: The Canadian Mining and Metallurgical (CIM) Bulletin. 415 (1969). 

E.R. Hoskins: Report to the U.S. Bureau of Mines on Contract No. H0346193 

(South Dakota School of Mines & Technology) July (1975). 

D.A. Hill, J.R. Wait: J. Appl. Phys 48, 4893 (1977). 

W. R. Smythe: Static and Dynamic Electricity , Sec. 5.28, p. 171, 3rd ed. 

(McCraw-Hil.1 Hook Co. , New York 1968). 

C. T. Tai: 1)tjndic Grcerz 's Functions in Elcctrorr~qnetic Theory (Scranton, 

Intext 1971). 

J.R. Wait, D.A. Hill: Appl. Phys. 11, 351 (1976). 

D. R. J. White: Electrmugne tic Shie M<ng Material and Performance, Sec. 2.1 

(Don White Consultants, Inc., Germantown, MD 1975). 

A.G. Kandoian: HeSercnce Datn for Radio lhginecrs, pp. 4-32 (ITT Corp. , 

New York 1968). 

M.L. Burrows: Ph.D Thesis (Dept. of Electrical Engrg., University of 

Michigan, Ann Arbor 1964). 
-7. 1 . , 7'!w 11ormn ? i ::cd mngn i t~idc, I'ig . 3. Tl~e normalized nagnj tudc 

cs f t l~c. sc;l t t c,rcd rsd i nl of the scattered azimuthal 

magrlet ic f f ld For var i 011s a1ap;netic f ie1.d for various 

void locations p'. void locations p'. 
CUT'N 

m -l $3rOGV) - m,. c 

E-c '-4-0 

- 2": u-d l-d T uu +a0 

0 c-l 

e2 

.d 

Lr, 

oum 

GU3 

rd 2Ub 

+mo 

E+ cec 6 C, 
Section 8 

DAVID A. HILL 

and 

JAMES R. WAIT 

Abstract. A thin prolate spheroidal void in an infinite 

conducting circular cylinder is used to model an internal flaw 

in a wire rope. The rope is excited by an electric ring current 

which is a model for a thin solenoid or multi-turn wire loop. 

The anomalous external f i e l d s a r e computed from the induced 

electric and magnetic dipole moments of the void. For this type 

of excitation, the induced axial magnetic dipole moment is the 

dominant contri.butor to the scattered field. The rcslllts have 

application to nondestructive testing of wire ropes. 

Introduction 

Electromagnetic NUT (Nondestructive Testing) is widely used for inspection and evaluation of metals [I], and electromagnetic methods in NDT of 

wire ropes 121 have recently been reviewed 131. Both DC and AC methods 

[4,5] are currently i.n use. In both methods, the magnetic field is applied 

along the axis of the rope, and rope irregularities produce magnetic field 

changes which induce voltages in the sensing coils. A two-dimensional model 

of solenoidal (electric current ring) excitation of a homogeneous wire rope 

has now been analyzed fully [6]. The toroidal coil (magnetic current ring) 
131 

excitation was also treated by invoking the principle of duality. A further 

analysis deals with a two-dimensional model for a wire rope with an internal 

anomaly [7], and the resulting change in the input impedance of the surrounding sol.enoid . 

Here we consider a three-dimensional model in which an electric current ring is used to model a thin solenoid or multi-turn wire loop. The 

analogous problem of a magnetic current ring has already been solved [8] ; 

thus the needed primary field expressions can be obtained from the magnetic 

ring current solution by duality, We allow for the presence of a small 

slender void within the otherwise homogeneous rope. Such a void can be 

considered a model for a broken wire within the rope or siailar internal 

flaw, The void is allowed to be oriented at any arbitrary angle to the 

rope axis to account for the winding geometry. Following a previous 

analysis (91, we compute the induced electric and magnetic dipole moments 

from the primary fields and the electric and magnetic polarizabilities of 

the void. The external fields of the induced dipole moments are then calculated using the previous formulation for dipoles of arbitrary orientation 

[9]. Thesc external fields are the observable quantities in any nondestructive testing method.. 

The most common methods of wire rope testing detect the average or 

integrated value of the circumferential electric field with a receiving 

solenoid which completely encircles the rope. Usually this is done by 

making and measuring the change in the mutual impeds~~ce oftwo coaxial 

loops or similar arrangement. But, in addition, one should be able to 

probe the detailed structure of the external magnetic field components 

with small sensing coils'. Because this possibility has not been well 

explored, we present here, in graphical form, the azimuthal dependence 
of the three components of the external magnetic field for a significant 

range of the relevant parameters. These plots are intended to be an aid 

to the design of an effective configuration of sensing coils. Although 

the analysis is carried out for the AC case, DC excitation is included as 

a special case. 

Primary Field Excitation 

The geometry of the rope and the source are shown in Fig. 1. The rope 

is infinitely long and has radius a, conductivity 0 permittivity E and 

w ' w' 

(magnetic) permeability p The surrounding free space has permittivity w* 

and permeability 11 . For now, we defer discussion of the void proper- 0 0 

ties and consider only the homogeneoiis rope. 

The soilrce Is an electric current ring of strength I located at p = b 

in the plane a = 0. This is a model for a thin and narrow solenoid which 

completely encircles the rope. This source is the dual. of the magnetic 

current ring or thin toroidal coil which was considered previously [9]. 

The time dependence is exp(iwt) for all source and field quantities. 

Due to the symmetry of the source and the rope, the fields are TE to z 

(Transverse Electric) and independent of 4. The electric field has only 

a nonzero azimuthal component E'~, and the magnetic field has only nonzero @ 

Pr axial and radial components H" and 11 . The superscript pr denotes the 

2 P 

ficlclk; .in the absznce of thc void. Sirlce the dual problem of the magnetic 

ring source has already been solved [9], the fields of the electric ring 

source can be obtained by duality [lo]. The resultant expressions for the 

primary fields both inside and outside the rope are given in Appendix A. 

The axial magnetic field H~~ is nearly independent of p inside the rope z 

for sufficiently low Frequencies. On the other hand, H~~ and E~~ are zero 

P 0 

at the rope center p = 0. For the D.C. limit (wO), E~~ is zero everywhere. 0 
Induced Dipole Mornen ts 

To allow comparison between the toroid 191 and solenoid excitation, we 

again select a thin prolate spheroid in order to model a broken strand. As 

indicated in Fig. lc, the spheroid has conductivity a permittivity E v' v ' 

and permeability pv. We also allow a rotation of the major axis of the 

spheroid about the p' axis by an angle a in order to account for the winding 

b* geometry of the rope. Consequentlfi the major axis is oriented at an angle 

h h 

a to the unir vector z' and an angle ~r/2 + a to the unit vector Q'. 

The prolate spheroidal void model is a convenient one because the electric and magnetic polarizabilities are known [ll]. The electric polarizabile ities for the incident electric field applied along the major axis, a or maj ' 

e along the minor axis, a are given by min' 

and 

-2v (ow-a,) 

a - min 1 + ov/ow ' 

where V is the volume of the thin prolate spheroid. Displacement currents 

have been neglected in (1) and (2), but they can be included merely by replacing o by aw + iwe and o hy fi + iw~ . The magnetic polarizabiliti~s W W v v v 

m for the incident magnetic field applied along the major axis, a or along maj' 

m the minor axis, a are given by min' 

and 

-2~im(p~-~~) 

am - - min 1 + pVhw 

Although the polarizabilities in (1)-(4) are derived for a thin prolate 

spheroid, we would not expect a significant difference for a thin circular 

cylinder of the same length and volume. 
In order to compute the induced dipole moments, it is necessary to 

first resolve the incident electric and magnetic fields into components 

along the major and minor axes. The resultant dipole moments can then be 

resolved into themore convenient p, @, and z components. When this is 

done, the induced electric dipole moments are found to be 

Pr e e (Ids)= = -a )sina cosa , E@ (amin maj 

The induced magnetic dipole moments are found to be 

(Kdll)z =: pr 

m Hz (amaj ?os2a + a min sin2a) , 

- Pr m rn (~dll)~ - tlz (amin - a )sina cosa , ma j (9) 

In (5)-(7), the primary field components E~~ HPr, and H~~ are evaluated $' 2 P 

at p', $', 2'. 

External Scattered Field 

The total scattered field can be written as a superposition of the 

contributions from each of the six induced dipole sources given in (5)-(10). 

(Actually, there are only five nonzero sources since (Ids) is zero for the P 

specific: configuration considered here.) The scattered magnetic field components IlSC (p, Q, z) , (p,$, z) , and H:' (P,Q, z) are of particular interest P 

when small coils are used as sensors. However, the scattered electric 

field components E"(~,$,Z), E;'(~,Q,Z), and E"(~,$, z) can also be cornP z 

puted with little additional effort. The $-averaged value of 
interest when a solenoid which completely encircles the rope is used as a 

sensor. 

Both the magnetic and electric scattered fields can be compactly 

written in the following matrix form: 

and 

In the first 3 by 3 matrix, G~ is the p component of H at (p,$,z) produced PP 

by a unit p-directed electric dipole at (pV,$',z'), Ge is the p component PO 

of H produced by a unit $-directed electric dipole at (p',$',zV), etc. The 

second 3 by 3 matrix is similar except that the Gm elements correspond to 

unit magnetic dipole sources. For the cylindrical geometry encountered 

here, all the matrix elements can be derived from z components of electric 

and magnetic Hertz vectors [12], and the specific expressions for the G elements 

have been given previously [9]. The expressions for the matrix elements 

in (12) are derived in Appendix B. The $-averaged value of E" which is $ 

of interest for solenoid reception is defined as 

The specific form of E"" is also given in Appendix B. Of course, instead 0 - 

of (13), one could also obtain E~~ by integrating the flux over the area of 

0 

the wire loop, but the present approach is simpler and more convenient. 
136 

Numerical Results 

Numerically efficient computer programs have been written to evaluate 

the scattered fields as given by (11) and (12) and the primary fields as 

given by (A-5)-(A-7) for a < p < b and by (A-9)-(A-11) for p > b. 

In Figs. 2, 3, and 4 the magnitudes of the three scattered magnetic 

field components H~~ SC , and HZ are shown plotted as a function of 4 for 

a selected set of the parameters. The vertical scale is in per cent of the 

primary axial magnetic field IIir running from to 1%. The volume of 

the small slender void V is taken to be lo-* rn3 (= lmm x lmrn x lcm) and the 

void itself is oriented in the axial direction (i.e. a=OO). Since the scattered fields are proportional to V, the curves are easily scaled to other 

values of V. Also, the constitutive parameters of the void are taken to be 

those of free space: E = co, ov = 0, and 11 = po. In these cases, 4' is 

v v 

taken to be zero, but for 4' not equal to zero, the horizontal axis can 

simply be replaced by 4 - $'. Other selected values of the parameters are: 

a = 1 cm, o = 1.1 x lo6 mho/m, pW/pO = 200, b/a = 2, p'/a = 0.5, z'/a = W 

3.75, p/a = 2, z/a = 5, f = 10 Hz. 

The magnitudes of the individual dipole contributions to the total 

scattered field are also shown for each component of H in Figs. 2, 3, and 4. 

Since a is zero, there are only three induced dipole moments (KdR) (Kdll)p, 2' 

and (Ids) Asexpected, (Kdf,) is themajor contributor. (Kdf,)pissmall 4. Z 

because H~~ is very small inside the rope. 0 

SC Actually, for solenoid reception, the value of E is of primary impor- 4 

tance. In Fig. 5, we also show the @ dependence of E" for three values of 0 

p'/a for z'/a = 5. Of course, the greatest 4 variation occurs for the void 

near the surface (p'/a = 0.9). In Table I the normalized averaged scattered 

- 

azimuthal electric field IE~~/E~~~ is shown for the same parameters which were 4 4 
used in Figs. 2-4. This is the quantity which indicates the sensitivity of 

- 

the solenoid sensor used in present systems. The values of IESC/EPr 1 are typi- @ @ 

cally rather small (2 .01%). Note in Fig. 5 that IE~~/E~~ 1 has a comparable 0 0 

level for pl/a = 0.1, but a much larger level for p'/a = 0.5 or 0.9. This - 

is due to the higher 0 harmonics in ESC which do not contribute to E". @ 0 

It is also the case that the normalized scattered magnetic field components 

have a higher level because of the higher order harmonics in 0 for p'/a $ 0. 

Concluding Remarks 

Excitation of a wire rope containing a slender void has been analyzed 

for a thin solenoid source. The prolate spheroidal void has been shown to 

prod~~ce ananomaly in the external fields which is typically larger than 

that produced for the case of toroidal excitation [9]. Again, the external 

scattered magnetic field components (not shown here) have a rather complicated azimuthal dependence containing some peaks and nulls. This suggests 

that multiple sensing coils of various locations and orientations rni-ght be 

worthwhile. 

- 

SC The @-averaged value of the scattered azimuthal electric field E @ 

has also been examined. This is the quantity which is sensed by a solenoidal 

sensor which encircles the rope, and Burrows [15] has also examined this 

quantity. This scattered field is typically quite small compared to the 

primary field as indicated in Table 1. Of course, the solenoid sensor does 

not make any use of the higher harmonics in @ which can be considerably 

larger than the $-averaged value. However, a possible advantage of the 

solenoid sensor is that ESC is not highly dependent on the void location. @ 

Thus the anomaly is related more to the void's volume than location. A 

system which senses the scattered field directly without any @ averaging 
will detect a greater anomaly ;;hen the voicl is located near the surface of 

the rope. 

On the basis of calculations at other frequencies, one general conclusion is that 10 Hz is a good choice. Higher frequencies suffer from 

poor penetration of the rope. Also, the exciting solenoid and the sensors 

should be located as close to the rope as possible to maximize the relative 

scattered field. 

The individual contributions of the various induced dipole moments to 

the scattered field are illustrated in Figs. 2-4. As expected for solenoidal 

excitation, the induced axial magnetic dipole moment is the dominant contributor to the scattered field. Actually, this dipole moment requires 

that the magnetic permeability of the rope he different from that of the 

void. In fact, calculations show that if the rope is nonmagnetic, the 

effect of the anomaly is quite small because only an electric dipole moment 

is induced. 

A further worthwhile study would be to examine the response as a function of void location for a fixed geometry of the sensor. Some additional 

calc:illations, in a co~nputcr print-out format, that can give such information, 

may be obtained from the authors on request. 

Although we have only examined the thin prolate spheroidal void, we 

have derived expressions which are useful for other shapes. The only requirement is that the imperfection be sufficiently small that its scattered 

fields can be represented by induced electric and magnetic dipole moments. 

Another convenient imperfection to treat would be an oblate spheroidal imperfection which could model a region of corrosion. Also, it might be 

possible to treat a rope model where the conductivity and permeability are 

tensors. This anisotropic feature of ropes is expected because of the 

winding geometry; the subject has been examined by Wait [16]. 
Appendix A - Primary Fields 

The primary fields for cxcitaticrr~ by a 1nagrIeti.c current ring of 

strength K have been derived previously [9]. By duality [lo], the previous results hold for electric current ring excitation if we make the 

following transformations: K -t I, H~~ -+ E;~, E~~ -+ -H'~, E~~ -+ -H Pr -F 

4 z Z P p ' 

Eo, c0 + pO. iupw-+ Ow + iocw, and a + iw~~ -+ iiwpw. The resultant primary W 

field expressions take different forms in the three regions, p < a, a < p 

< b, and p > b. 

Inside the rope (p < a), we find 

w 

H~~ = - I ~XW~~I~ (wp) exp(-ihz)dX , P -00 

00 

EP~ = iwp, 4 waoI1 (wp) exp (-iXz)dX , -00 

where w = (X2+y;)li = ~w~~(o~+~wE~) and I. and I1 are zero and first Yw 

order modified Bessel functions. The factor a. is given by 

where 

-1 

-YoBo 

a = 0 w2alo (wa) K~ (ua) 

K is a zero order modif:ied Bessel function, and prime ' denotes differen0 

tiation with respect to the argument. (In the previous paper [9], the 
expression for B should actually read Bo = -~b~;(ub)/(Zru)). 0 

In the region between the rope and the source (a < p < b) , we find 

where 

where 

Outside the source (p > b), we find 

E~~ = iw,Jo 4 / uAoKl (up) exp (-iXz) , -00 
Appendix B - Scattered Electric Fields 

Tlie derivation for the electric fields produced by a dipole source 

within a wire rope is a direct extension of the derivation of the magnetic 

field derivation given previously [9], and some of the details are excluded 

here. The external fields can be obtained from U and V which are the z 

components of electric and magnetic Hertz vectors 

U=r[u] and V=r[vm] , m 

where 

and 00 00 

rEl= [ 1. -im($-$' ) .-iX (z-z ' ) dX . 0-3) -m -00 

The expressions for S and Tm have been given for each of the six dipole m 

types [91. 

The electric field components can be written 

where 

won 'i , 

= -ihuS 

m 

K' 

m (up) - - P TmK,(uP) 9 

arid 
Erc 

0 

k 
For an axial magnetic dipole (KdR) we have 

z ' 

The $-averaged value of E as given by (B-4) can be written 4 

(B- 12) 

(B- 13) 

where 

and 

E = iop T K1(up) . 40 0 0 0 

As expected, only the m = 0 term of the electric Hertz vector contributes 

to 5. 
References 

H. I,. Libby: Introduction to lilondestructivc Test Methods (Wiley Interscience, New York, 1971). 

E.K. Hoskins: Report to the U.S. Bureau of Mines on Contract No. H0346193, 

South Dakota School of Mines and Technology, (July 1975). 

J .K. Wait: Review of electromagnetic methods in non-destructive testing 

of wire ropes, Preliminary Report to the U.S. Bureau of Mines on Contract 

No. R0155008, (March 1978). 

C.H. Larsen, R.A. Egen, R.D. Jones, H.A. Cross: Report to the U.S. Bureau 

of Mines on Contract No. H0101741, (June 1971). 

J.G. Lang: The Canadian Mining and Metalurgical (CIM) Bulletin, 415 (1969). 

J.R. Wait: The electromagnetic basis for nondestructive testing of cylindrical conductors, IEEE Transactions on Instrumentation and Measurement, 

111-27, 235 (September 1378). 

J.R. Wait. and R.L. Gardner: Non-destructive testing of a cylindrical 

conductor with an internal anomaly - A two-dimensional model, Preliminary 

Report to the U.S. Bureau of Mines on Contract No. H0155008, (May 1978). 

D.A. Hill and J.R. Wait: J. Appl. Phys. 48, 4893 (December 1977). 

D.A. Hill and J.R. Wait: Scattering by a slender void in a homogeneous 

conducting wire rope, Appl. Phys. , 16, 391 (1978). 

R. F. Harring ton: Time-Hamonic EZectrmcrgnetic Fields (McGraw-Hill, 

New York, Sec. 3.2, 1961). 

W.R. Smythe: Stat<c and Dynamic Electricity (McCraw-Hill, Sec . 5.28, 

1968). 

J.K. Wait and LA. tiill: Appl.. F'hys. 11, 351 (1976). 

A.C. Kandoinn: Reference Data for Radio Engineers (ITT Corp., New York, 

pp. 4-32, 1968). 
14. D. R. J. White: Electrmagnctic SizieZding Material and Performance 

(Don White Consultants, Inc., Germantown, MD, Sec. 2.1, 1975). 

15. M.L. Burrows: Ph.D. Thesis, Department of Electrical Engineering, 

University of Michigan, Ann Arbor (1964). 

16. J.R. Wait: Electromagnetic induction in an anisotropic cylinder, Radio 

Science, 13, (september/October 1975). 
- 

Table 1 - Normalized E' @ 

Parameters* 

Same as Fig. 5 

p'/a = 0.1 

* 

Parameters for each entry are the same as Fig. 5 except where noted . 
Void location 

p=a (€09 PO) 

p=b 

(infinite length) 

Fig. la 

Perspective View 

Fig. Ib 

Top View 

(a;, E,,, ,XV) 

FIG, 1c 

Prolate spheroidal void 

Fig. 1. Thin solenoid surrounding a metal rope of infinite length. 
Fig. 2. Contribution of the individual 

dipole moments to H;' for an 

axially oriented void. 

Fig. 4. Contribution of the individual 

dipole moments to H~~. 

z 
Fig. 5. Normalized magnitude of the 

scattered azimuthal electric field 

for various values of the void 

location pV/a for zV/a = 5. 
-- Section 9 

DAVID A. HILL 

and 

JAMES R. WAIT 

Abaaact-A stranded wire rope is idealized as a homogeneous 

conducting and permeable cyl inder of circular cross section and 

of infinite length. The rope is excited by a coaxial solenoid or 

finite length mu1 ti-turn coil that carries an azimuthally directed a1 ternating current. The rope and the enclosing solenoid may 

have a uniform velocity relative to each other. Using a nonrelativistic analysis, the nature of this dynamic interaction is 

examined and numerical results are presented for parameter values 

that are relevant to both static and dynamic conditions in nondestructive testing of such cyl indrical conductors. It is shown 

that even for motional velocities v as high as 10 m/s the dynamic interaction with the rope specimen is not appreciably modified 

from that for the static condition (i.e., for v = 0). 

INTRODUCTION 

The nondestructive testing of metallic structures often utilizes the interaction of electromagnetic fields with the specimen [1,2]. A particularly 

good example is a steel stranded wire rope such as i.s used in mine hoists. In 

this case it may be desirable to test the rope under dynamic conditions such 
I that the sensor and the rope have a uniform velocity relative to each other. 

11t is the purpose of the present psper to examine the effect of the motional 1 

(velocity on the nature of the electromagnetic fields excited by a coaxial 1 

Jsolenoid or finite lenath coil that carries an alternating current. We also 

(consider the mutual impedance be tween two solenoids that are both coaxial with 

1 the rope. Such a configuration is quite common in testing both solid and tu1 bular conductors. 

FORMULATION 

I We consider a simple model; the rope is treated as a uniform cylindrical 

I structure of radius a. It is excited by a current-carrying concentric sole- 

( noid of radius b as indicated in Fig. 1. Cylindrical coordinates (p,$, z) 

fixed within the rope are chosen such that the surface of the cylindrical conI ductor is p = a while the solenoid is idealized as an azimuthal current sheet of 

I axial length located at p = b. The solenoid is taken to be moving with a uni- I 

I form velocity v in the positi-ve z direction relative to the rope frame. 

I (p,$,z). Azimuthal variations of the fields are assumed to be negligible I 

1 which is the case when the azimuthal current is constant and the rope is lo- I 

1 cated concentrically within the solenoid. 

The solenoid current surface density in amperes/meter at p = b is specified to be 

jg(z, t) = Real part of (~I/e)exp(iwt) 

for (l/2) + vt > z > (-l/2) + vt and zero outside this interval. Here N is 

the total number of turns that are wound uniformly throughout the axial length 

4 of the solenoid. I is the magnitude of the current. The angular frequency 

is w and t is the time. Now it is not difficult to show that 
where it is understood here and in what follows that we take the real part 

of the indicated complex quantity. 

We are now interested in the interaction of the fields from this primary 

current with the conducting cylinder. While the analysis can be carried 

through quite rigorously by using the full-blown Lorentz transformation [3,5], 

here we will use a nonrelativistic approach [6] that is justified when terms 

containing v2/c2 are negligibly small. This will certainly be the case in 

any practically realizable remote sensing scheme [e.g., even for a motional 

velocity v = 100 m/s, v2/c2 2 10-l3 where c = 3 x lo8 m/s]. Also we will 

assume that the fields in the region external to the rope are quasi-static 

in nature. This is justified when the significant dimensions of the sensor 

and the rope diameter are small compared with the operating free-space wave7 length Ao[e.g.,at a typical operating frequency of 10 Hz the A = 3 x 10 m]. 

0 

-1 In this context it is worth pointing out that 1 h 1 , where A is the axial 

wave number, should also be small compared with A. over the significant range 

of the integration variable A. 

EXTERNAL FIELDS 

In view of our assumptions the fields in the region p > a can be obtained from a magnetic type Hertz vector that has only a z component 

Il (p, z, t) which satisfies ~aplace's equation 0 

Taking a cue from (2) ve construct the following appropriate integral form 

for TI as: 
and 

H = a2n /az 2 

0 2 0 

The corresponding integral representations are 

where 
1 and 

IJe are now in the position to apply the source conditions 

Ho,(b - 0) - Ho,(b + 0) = je(z.t) 

and the impedance condition 

where Z(X) is discussed below. Thus we deduce that 

B(X) = NIb sin(XR/2) 

- -[ - (h%/2) ~<~(lXlb) 

1 and 

D(X) = c(X) - B(X)I,(IX~~)/K~(IXI~) 

Here we have made use of the Wronskian relation 

SOLUTION FOR INTERNAL FIELD 

I Now we must specify the internal structure of the cylinder in order to I 

( obtain an explicit expression for z(X). For present purposes we will assume I 

homogeneity with a conductivity 0 and magnetic permeability 1-1. Displacement currents in the cylinder a r e a l s o neglected. 

The fields within the cylinder (i.e. p < a) can also be obtained from 

an azimuthally independent magnetic Hertz vector that has only an axial cornIponent TI. In the rope frame (p,$, z) , it satisfies the time dependent wave I 
The corresponding integral form is 

h 

where fl satisfies 

where 

2 

w = [X -1- iw(w -t- Xv)] 3 

The field components in the region p < a are obtained from 

2 H = a wapaz P 

2 E = pa rr/atap @ 

and 

The corresponding expressions for the field components are then 

'==A [:! ] { ['l:::] e -iXz e i(w + Xv) t dA 

-03 

HZ (A) 

where 

The appropriate solution of (24) that is finite at p = 0 is clearly 

where A(A) is determined below. 
Thus, we readily deduce that 

This completes the specification of the ratio R(X)/C(X) as given by (19). 

A 

Also since fI and A are continuous at p = ape ascertain from (4) and (31) 02 z 

that 

The fields everywhere in the (p,@,z) frame are now given in terms of the 

known parameters of the proble,m. 

To express the fields and f;:, in the (p,) coordinates in the frame 

of the solenoid we need to apply the transformation law 

and 

-+ 

where i is a unit vector in the positive z direction. These forms are valid 

z 

for the external region p > a; the corresponding forms for p < a are s'milar. 

The important point is that the axial magnetic field is not changed as a result of the uniform axial motion. 

RESULTS FOR INTERNAL FIELD 

In any non-destructive testing scheme the internal field of the sample 

interacts with the material properties and, in turn, produces a secondary 

field that is detected in the external region 173. Thus, it is of interest 

to exarnirle the interrlal axial magnetic I1 at some fixed distance z' from the 

z 

solenoid as indicated in Fig. 1. 

The field in question is given by 
159 

Now z = z' + vt if the solenoid is located at z = 0 at t = 0. Thus we have 

simply that 

The v dependence is contained entirely in A(X) that is defined by (34) where 

B(X) and C(X) are given explicitly by (18) and (19). 

The integration over X indicated by (38) is performed numerically using 

a variable step size in order to deal effectively with the relatively rapid 

variation for small values of IX I. In general, at least for the case v $ 0, 

the integrand is not an even function of X so the j-ntegration must be carried 

out over the full range of X from -rn to +. Kere we rraintain the requirement 

that Re. w > 0 or that w tends to IX I as X tends to + OJ. Some results for 

I plotted as a function of z ' are shown in Figs. 2 through 9 where the curves 

are normalized by choosing NI = I. ampere as the total current in the solenoid. Positive values of z' correspond to points that are ahead of the solenoid in the sense that the solenoid is moving with a uniform velocity v in 

the positive z direction ----- relative to the rope. Of course the results 

apply to the spectal case when the solenoid is fixed and the rope moves with 

a relative velocity v in the negative z direction. 

In each of the curves for IH I shown in Figs. 2 through 9, the rope radius 

Z 

- - 2 cm, the solenoid radius b = 2.5 cm,and the solenoid length R = 2 cm. 

We then choose various values and combinations of the following parameters: 

operating frequency w/2n, rope conductivity U, rope magnetic permeability 

(relative to free space) v/p and the velocity v. 

0 

In Fig. 2 we consider the case of a solenoid that is stationary with 

respect to the rope (i.e, v = 0) and illustrate the effect of frequency on 

t11c field IH- I at the surface of the rope, p = a. Here p/po = 200 and L 
160 

6 U = 10 mho/m. As expected the curves are synunetricnl about z' = 0. In 

each there is a pronounced peak within the region of the solenoid which again 

is not surprising. IIowever, there is an enhancement of the peak for the A.C. 

cases (i-e,, f = 10 and 100 Hz) over that for the D.C, case (i.e., f = 0). 

At 100 Hz this is particularly noticeable since the magnetic flux is being 

excludcd from the interior oL the rope. In Fig. 3 the corresponding curves 

are shown for IB 1 at the center of the rope, p - 0. Here we can see that, z 

particularly at 100 Hz, the field magnitude is much reduced which is consistent with our statement above. 

In Fig. 4 we now illustrate the influence of tlre motional velocity on the 

field H at the surface 9 = a for frequency f - w/2n == 10 Hz, rope conductiz 

6 vity 0 = 10 mho/m and a rel-ative permeability p/p - 200. The three cases 

0 

shown are for rope velocities v - 0, 10 and 50 m/s. The corresponding results 

are showm in Fig. 5 far the internal field at the rope center p = 0 where 

therc is a decided "delayu of the buildup of the field magnitude; this effect 

becomes quite appreci-able at v = 50 mi::. 

In Fig. 6 we show the influence of the relative permeability p/po £or 

6 the case p = 0, f - 10 Hz, v = 0, and for a = 10 mho/m. The curves are 

particularly peaked when p/p0 = 1 correspanding to a nonmagnetic metal but 

the maxima became smoothed out as u/p increases. Corresponding results are 

o 

shown in Fig. 7 for v = 50 m/s. Again these show the pronounced "delay" of 

the maximum which we sight: describe as a hy&omagn~tic drug. 

In Fig. 8 we illustrate the influence of the rope conductivity CJ on the 

internal Eield for the case where v = 0, f = 10 Hz and p/p, = 200. Not surprisingly there is a progressive decrease of the field magnitude as u in5 6 7 

creases from 10 , to 10 , to 10 mho/m. In the latter case the attenuation 

is quite severe. The corresponiling curves are shown in Fig. 9 for a mot-iunal 
161 

velocity v = 50 m/s. Again we have a good illustration of the hydromagnetic 

6 

drag effect that is particularly evident for = 10 mho/m. 

MUTUAL IMPEDANCE FOR COAXIAL SOLENOIDS 

In various nondestructive testing schemes for wire ropes and similar 

tubular metal specimens the mutual impedance between two coaxial solenoids 

is measured. Such a configuration is depicted in Fig. 10 where we can specify that the bottom solenoid is excited by a current I and the induced 

voltage V' is measured in the upper solenoid. Both solenoids have a fixed 

separation s and they move in the upward direction with a velocity v relative to the cylindrical rope. For convenience we select both solenoids to 

have N turns uniformly spaced over a length R. Clearly the voltage V' is 

to be obtained from 

where 

Then if we define the mutual impedance Z' according to 

m 

iwt) Z' = V1/(1e m 

it follows readily that 

+cx, 

sin (hR/2) z1 m = ipow3N2 J [ -iis 

-03 

[~~(Iilb) - (~/~)~~(Ihlb)ld~ 

where C/B is given explicitly by (19). In the limiting case where the rope 

is absent we have the simpler formula 

2 

m 0 

1.' = iXo = 2ip wb N K (ib) 1 (ib) coshs dh (Xk/2) 1 
162 

Here Xo is the free-space mutual reactance between the two solenoids. 

The quantity we consider for the final numerical examples is the nor- 

--- 

- rnalized response defined by the complex ratio (2' - iXo) /Xo. In Fig. 11 the 

m 

real and imaginary parts are plotted as a function of frequency from 1 to 

1000 Hz for the foll-owing parameters: v = 0, a rope radius a = 2 cm, ra6 dius of both solenoids b = 2.5 cm, rope conductivity CI = 10 mho/m, rope 

permeability p/p = 200. The two sets of curvesin Fig. 11 correspond to 

0 

s = 0 and s = 4 cm. Xn the first case of zero separation we are dealing with 

the self impedance of one solenoid whereas in the second case the solenoids 

are distinctly separated. In both situations it is evident that the imaginary 

part considerably exceeds the real part at the lowermost frequencies. The 

corresponding results for a motional velocity of v = 50 m/s are shown in 

Fig. 12. Now we must distinguish between positive and negative values of s 

according to whether the :"receivingv solenoid is above or below the "transmitting" solenoid, respectively. The marked changes between the curves in 

Figs-11 and 12 are entirely due to the difference between motional velocity 

of zero and 50 m/s. It is particularly significant that in the latter case 

the reciprocity theorern is violated but such an effect is really not unexpected 

because of the lack of symrnetry in the problem. 

Finally, in Fig. 13, the norvalized response at 10 Hz is plotted as a 

function of the motional velocity v from 1 to 1000 m/s for the three conditions 

s = 0, +4 and -4 cm. Otherwise the parameters are the same as before. These 

curves show very clearly that, for v somewhat less than 10 m/s,the influence 

of the relative motion is negligible. In fact if v = 1 m/s the curves have 

reached the v = 0 asymptotes. 
The electromagnetic fields induced in the cylindrical specimen or rope 

are seen to be influenced by the relative motion of the sensor. However, 

the effect is quite small unless the motional velocity is on the order of 

10 m/s or higher. The same conclusion would apply to more complicated rope 

models if the average properties and geometrical dimensions are of the same 

order. 
REFERENCES 

[I] W.L. Libby, Lntroduction to Electromagnetic Non Destructive Test Methods, 

New York: Wiley Interscience, 1971. 

[2] V.V. Klyuev and M.L. Faingoiz, o on destructive testing of zwving current 

conducting articles by means of pass-thro~igh converters using the constant field method," Defcktoskopiyn, Vol 8, No. 2, pp. 27-31, ~archlhpril 

1972. 

[3] H.N. Kritikos, K.S.H. Lee and C.H. Papas, "~lectromagnetic reflectivity 

of nonuniform jet streams," Radio Science, Vol. 2, No. 9, pp. 991-995, 

Sept. 1967. 

[4] R.H. Ott and G. Hufford, "Scattering by an arbitrarily shaped conductor 

in uniform motion relative to the source of an incident spherical wave," 

Radio Science, Vol. 3, No. 8 pp. 857-861, Aug. 1968. 

[5] J.R. Wait and D.A. Hill, "~lectromagnetic interaction between a conducting cylinder and a solenoid in relative motion," Preliminary Report -to 

U.S. Bureau of Mines mz Contract No. HOZ55008, 9 January 1979. 

[ 61 L .D. Landau and E.M. Lif shitz, EZectrodynamios of Continuov.~ Media, 

Oxford: Pergamon Press, Sec. 4.9, pg. 205, 1960. 

[7] D.A. Hill and J.R. Wait, "Scattering by a slender void in a homogeneous 

conducting wire rope," Applied Physics, Vol. 16, pp. 391-398, 1978. 
Fig. 1. Cylindrical Conductor of radius a excited 

by a concentric solenoid of radius b that 

has a velocity v relative to conductor. In 

examples that follow, a = 2 cm, b = 2.5 cm 

and R = 2 cm. 

Z' (cm) 

Fig. 3. Axial magnetic field at axis, p = 0, for same conditions as in Fig. 2. 
Fig. 4. Field at surface = a, for velocities v = 0, iO 

itrid 50 m/s, for f = 10 Hz rnd other conditions as 

abovth . 
Fig. 5. Field at axis p = 0, for conditions as in Fig. 4. 


Fig. 8. Field at axis p = 0 for v = 0, f = 10 Hz, 

p/pO = 200 and CJ = lo5, lo6 and 107 mho/m. 

Fig. 10. Coaxial solenoid arrangement that 

moves with velocity v (upwards 

in figure) relative to cylindrical 

conductor or rope. 

Fig. 12. Normalized response for v = 50 m/s and ot:~er conditions as in Fig. 11. 
Fig. 13. Normalized response for coaxial sensor a function of v for 

f = 10 Hz, o = lo6 rnho/m and p/po = 200, s = 0, -4 and +4 cm.