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11TECHNICAL SCIENCES
БАГАТОПАРАМЕТРОВИЙ ЕЛЕКТРОМАГН1ТНИЙ МЕТОД КОНТРОЛЮ ЦИЛ1НДРИЧНИХ
СТРУМОПРОВОД1В
Горкунов Б.М.
доктор технгчних наук, професор кафедри ¡нформацшно-вимгрювальних технологш i систем На^онального техтчного yHiверситету «Хартвський полiтехнiчний iHcmumym»
Львов С.Г.
кандидат технiчних наук, професор кафедри iнформацiйно-вимiрювальних технологш i систем Нацюнального техтчного yнiверситетy «Хартвський полiтехнiчний iнститyт»
Борисенко €.А. кандидат технiчних наук, доцент кафедри iнформацiйно-вимiрювальних технологш i систем На^онального техтчного yнiверситетy «Хартвський полiтехнiчний iнститyт»
Тамер Ш. астрант
кафедри iнформацiйно-вимiрювальних технологш i систем На^онального техтчного yнiверситетy «Хартвський полiтехнiчний iнститyт»
MULTI-PARAMETER ELECTROMAGNETIC METHOD OF TESTING CYLINDRICAL
CONDUCTORS
Gorkunov B.
Doctor of technical sciences, professor of the department of Information-measuring technologies and systems, National technical university "Kharkiv polytechnic institute "
Lvov S.
Candidate of technical sciences, professor of the department of Information-measuring technologies and systems, National technical university "Kharkivpolytechnic institute"
Borysenko Y.
Candidate of technical sciences, associate professor of the department of Information-measuring technologies and systems, National technical university "Kharkivpolytechnic institute"
Tamer S.
Postgraduate student of the department of information-measuring technologies and systems, National technical university "Kharkivpolytechnic institute"
Анотащя
У робот розглянуто принцип побудови електромагттного перетворювача з просторово-перюдичним полем для контролю магттних, електричних i геометричних параметрiв цилшдричних протяжних металевих виробiв. Отримано основт математичт вирази, що описують розподш напруженосп електромагттного поля всередиш виробу, мгж виробом i обмотками перетворювача, а також за проввдниками перетворювача. Бiбл. 7, рис. 1. Abstract
The paper observes the approach by which the authors developed electromagnetic transducer with spatially-periodic field structure. This transducer is for estimating magnetic, electrical and geometrical parameters of metal objects of cylindrical form. Authors derived key mathematical equations, describing distribution of magnetic field strength inside and outside the object. Applying these equations allow authors to calculate parameters of spatial harmonic, such as amplitude and phase. The authors show, that using the mathematical apparatus after results of measurement it is possible to estimate electrical and geometrical parameters of object under study.
Ключовi слова: електромагнiтний перетворювач, напруженють магттного поля, магнина проникшсть, електропроввднють.
Keywords: eddy-current electromagnetic transducer, spatially-periodic field structure, magnetic permeability, electric conductivity.
The task of quality monitoring for production manufactured in energy and machinery industries together with embedded diagnostics of operating equipment, requires increasing accuracy, sensitivity and reliability of NDT means, being used. In order to enhance
functionality of NDT means considering growth of nomenclature of controlled parameters a task of development of multi-parameter means for monitoring is actual. Electromagnetic methods are considered to be such in particular.
Currently, the development of multiparameter measurement methods is of particular practical interest. These methods allow us to obtain the fullest information concerning object under study (OUS). In this regard eddy-current methods and devices with output signals depending upon large number of parameters of the OUS and transducer have undisputable advantages.
For simultaneous determination of OUS's two parameters the most suitable approach is to apply single frequency eddy-current transducers. In this case two parameters of the transducer's output signal need to be measured. Afterwards, it is supposed to solve a system of two equations, connecting these parameters with OUS's certain properties [1, 2].
If we want to define three or more OUS's parameters, it is obligatory to involve more input quantities, which gives us extra independent equations. We can define these quantities using eddy-current transducer which operates on several fixed frequencies. In that case, OUS is penetrated with the multi-frequency field, and then we extract amplitude and phase for harmonics of each frequency [3].
Nevertheless, utilizing this approach is difficult in practice, due to complexity of phase synchronization for sinusoidal fields and extraction of output parameters out from the eddy-current transducer's output signal.
The same downside of complexity of the output signal processing, which lies in time-domain analysis with extracting parameters of the harmonics, is inherent for eddy-current transducers based on pulsing excitation field [4].
It appears, as it is shown below, that multi-parameter simultaneous measurements can be simplified by utilizing single-frequency sinusoidal spatially-periodic magnetic fields.
This paper utilizing this approach studies aspects of eddy-current transducers' theory applied in multi-parameter measurements with magnetic fields having spatially-periodic structure.
Consider the problem of determining the distribution of the alternating magnetic field in the wire with a current located at a distance d from the center of the ferromagnetic cylinder of radius a. (fig. 1)
Fig 1 The collocation of the object of control, exciting and measuring wires of the electromagnetic transducer
The important condition is that the lengths of the wire and cylinder are significantly bigger than the transverse dimensions of the last one (a and d respectively).
Using Maxwell's equations and Ohm's law [5-7] we obtain mathematical description of the magnetic field in the conductive media. Providing the condition of constancy of the magnetic permeability | and electric conductivity ct in the cylindrical object this equation is following:
AH =
dH
dt
(1)
Solving equation (1) together with the boundary conditions we define the regularities of field distribution in a cylindrical object:
on the surface of a cylinder with radius r = a
n = (B(i) - B(m))= 0
n = (h(i) - H(m) )= 0
where H - magnetic field strength, |o - magnetic constant.
We assume, that the electromagnetic field is quasi-stationary, with a wavelength exceeding transverse dimensions of the wire and the cylinder. We also considering the following system: a threadlike wire with current I produce field in continuous cylindrical object with longitudinal Z-axis in cylindrical coordinates (fig. 1) and this current flows in the positive direction of Z-axis. Produced magnetic field contain only transverse
components H{Hr, H9, H0 ), where H and H9 are r-th h 9-th components of magnetic field strength.
on the surface of a cylinder with radius r = d
n = {b(l) -B(m) )= 0 -( Û (!) Û (m) )-
n = (H(l) - H
)= J
(2)
(3)
(4)
(5)
where upper indexes in brackets correspond to the fields of different areas:
(i) - inside cylindrical object (0 < r < a); (m) - between surfaces of cylindrical object and virtual cylinder of radius d (a < r < d);
(l) - behind the surface of this cylinder (d < r < »); j - current density vector; I, n - vector normal to the surface of a cylindrical object.
As so as functions, which describes field distribution are periodical, initial conditions can be arbitrary.
Conditions (2) - (4) establish regularity of the normal components of the induction vector B and tangential components of the vector of the magnetic field strength under transition from given media to another, and (5) describes a leap of field strength in case of crossing cylinder surface. There is a convenient way to represent flowing current, which has the component along Z-axis, using Fourier transformation:
jz (9, t ) =
itot
er-L 5(cp) = d
Jo + Z Jn cosM
n=1
(6)
where n - spatial harmonic number, t - time, ra -cyclic frequency, 8(9) - delta-function.
Hr (r, p, t ) = e/rotJ Z fn (r )sin(np)
(7)
Hv(r, 9, t) = eiratj X gn (r )cos(n9) (8)
n
where fn(r) and gn(r) - functions undefined yet. Since studied magnetic field has two components so the coordinate representation of the equation (1) could be rewritten as the following system:
1 d f df„ ^ n +1 _ 2n . .
I--^T fn gn =
r dr I dr ) r r
(9)
1 d L dgn Y «in
r dr t dr
2n
gn +fn =GMi'®g„. (10)
2 on 2 Jn r r
This system of equations considering boundary conditions (2)-(5) has solution:
f!P (r, to) =
Jn f a
Dn (iro) t d
g? (r, **) = Jn f °
Dn (iro) t d
n
\ n—1
n—1
—1(y4i ) — /„+1(y4i )_ [in—1 (y4i )+l„+1(^41 )
f(m)(r,/®)=—Jrl a I 1)/n—1M)—(n+1)l„+1 i
Dn (/ro) t d
gnm) <r»-—J f a
v n—1
[fc — 1)/n—1(xVi )—+ 1)/n+1^ )
f( ) =— g<l ) (r, -) =
Jn
2Dn (/ro) t r
n+1 I
1
2n
n—1 / \n+1
r ) f a
d
\n—1 / \ n+1
r ] +f£
a ) t r
1—1+(,+1)f a
2n
Ln+M
M )[
(11)
(12) ; (13) ; (14) (15)
n
r
where generalized parameters x = a^JMqMO® , y = r^jMoM°® , discriminant
Dn (ira) = [(m + l)/n_! (x4i )-(M- l)ln+i (xVi )J,
In+i, In_i - modified Bessel function of the corresponding orders.
Thus, provided we knew m, o, a and d of the object and values of current I and valid radius r, using formulas (1)-(15) we can calculate functions f„(r, 9, t) and gn(r, 9, t) for any spatial harmonic, and by these functions reconstruct the structure of the magnetic field distribution in any area (inside the object, between the object and a wire with a current, behind this wire).
Formulas (11)-(15) describe magnetic field of the threadlike wires which transversal dimensions infinitely small. But practically we use wire with some finite width and radial height. The latter is conveniently realized either in a form of threadlike wire or as pole having some number of thin wires with currents codi-rected alongside cylinder's generatrix (e.g. alongside Z-axis). In the latter case currents of individual wires create total current.
To consider the radial thickness of the ribbon-like wire or pole with total field producing current we have to substitute r value for some effective radius, the value
of which depends on the shape of the pole, in the formulas for field's strength.
To account finite width of the pole with total current we substitute 9 for (9 - a) in formulas (7) and (8) and integrate these formulas of a on a range from -y go y (where y - angular half-width of the pole). As a result, we get:
Hr (r, p, t ) = em jZ
sln(ny)
ny
fn (r )sln(np) (16)
H9(r, 9, t) = eiratjX ^ gn (r)cos(n9) (17)
n ny
It is convenient to build electromagnetic transducers based on spatially-periodical field structure using transformer approach. For such transducers measuring wires should be situated on a circle with radius r ~ a which is less than d. Using several measuring wires with different angular coordinates 9 allows us to obtain several primary prevailing harmonics since other harmonics fade, like (a / d)n-1.
The translational symmetry let us use rather narrow frame-like measuring wires, placed alongside the object in parallel with excitation wires, in the way shown in fig. 1.
n
The components of the magnetic field H or Hp penetrate into these measuring wires. In order to perceive Hr measuring wire need to be perpendicular to coordinate r, and to perceive Hp - perpendicular to p.
For instance, if we place measuring wire between the object and the excitation wire (a < an < d), then to
calculate the r-th component of the transducer's EMF for the mode of operation "transducer with the object" based on formulas (7) - (15) we get formulas for calculation of module and phase:
E(n) =-mWa O= -iano eia —H Wa S x
2nd
v aH y
+ h
(n )
f V-1
a
v aH y
a
v an y
2n
sin (ny)
cos(n
ny
M
(18)
tg 0(n > = Im h (n V Re h
(19)
In these formulas ®r - instantaneous value of the magnetic flux (r-th component), which induces EMF Er in the measuring wire; S - area of the frame-like measuring winding; Wh h Wh - number of turns of the excitation and measuring windings; complex parameter hr(n) describes object's reaction to the excitation field (n-th harmo nic), which is defined as:
= Ah1Bh1 + Ah2 Bh2 i Ah2Bh1 - Ah1Bh2
Bh1 + Bh2
Bh1 + Bh2
Ah1 = W - l)bern-iX + (^r + l)bern+iX Ah2 = (^r - l)bein-ix + (^r + l)bein+ix Bh1 = (^r + l)bern-lX + ((V - l)bern+lX
Bh2 = W + l)bein-1x + W - l)bei
n+1X
(20)
(21) (22)
(23)
(24)
where x - generalized parameter.
Formulas (21) - (24) are based on (11) - (15) using modified Bessel ber„ x, bei„ x and Kelvin functions.
Conclusions.
Analyzing equations (16) and (17) we can conclude, that provided the quantity y is finite we have sin(ny)/ny < 1, that results in decrease of an amplitude of n-th spatial harmonic of the field, produced by either ribbon-like wire or pole with a current comparing to the amplitude of the same harmonic of the field, produced by thread-like wire. What is more, as the order of harmonic increases, the decrease of amplitude, mentioned
above, become substantial. It should be noted that in the limit y ^ 0 (16) and (17) become (7) and (8) respectively. As a result of the study authors obtained equations for estimating amplitude and phase for any harmonic of the output signal of the transducer with spatially-periodic model for representation magnetic field inside and outside of metal cylindrical object.
References
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3. Gorkunov B.M., Tyupa I.V. Eddy current two-parameter control of ferromagnetic cylindrical products // Vesnik NTU "HPI", Kharkov. - № 5. - 2004. - P. 9399. (Rus.)
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