Impedance of a single-turn coil due to a double-layered sphere with varying properties A. A. Kolyshkin Rémi Vaillancourt CRM-293 June 994 Department of Applied Mathematics, Riga Technical University, Riga, Latvia, LV 00. Departments of Mathematics and Computer Science, The University of Ottawa, Ottawa, ON, Canada KN 6N5. Department of Applied Mathematics, Riga Technical University, Riga, Latvia, LV 00. Departments of Mathematics and Computer Science, The University of Ottawa, Ottawa, ON, Canada KN 6N5.
Abstract This paper presents an analytical solution for the change in impedance in a coil due to the presence of a conducting double-layered sphere symmetrically situated with respect to the coil. In the case the relative magnetic permeability, µ(r) = r α, of the outer sphere is a function of the distance r from the center of the sphere, α is an arbitrary real number, the solution is expressed in the form of a series containing Bessel functions and can be used to compute the change of impedance. Numerical results for the case α = 2 are presented. Résumé On présente une solution analytique du changement d impédance d une bobine induit par une sphère à deux parois en position symétrique par rapport à la bobine. La perméabilité magnétique relative de la paroi extrérieure µ(r) = r α est fonction de la distance radiale du centre, où α est une constante réelle. On exprime la solution sous forme d une série qui contient des fonctions de Bessel. On présente des résultats numériques pour le cas où α = 2.
I INTRODUCTION The method of eddy currents is widely used to control the quality of materials of different shapes. In practice, controlled objects often have a spherical form []. However, there are only a few known analytical solutions to eddy current problems in spherical geometry. An analytical formula for the change of impedance in a single-turn coil situated above a conducting spherical layer is presented in [2] while in [3] a similar problem for a multilayered sphere is considered. The paper [4] is devoted to the interaction of a coil with a nonmagnetic ball. An analytical solution to a similar problem is also given in [5], Chapter 9. All the above cited solutions are obtained in the case all the parameters of a medium are assumed to be constant. If the conductivity and/or magnetic permeability of the sphere vary with the spatial coordinates, as it often happens in practice, no analytical solutions are known to the authors. (There exist analytical solutions for the cases the conducting medium is a halfspace, a double layer, or an infinitely long cylindrical tube, [6] [8].) In the general case, the medium is usually divided into a large number of thin layers it is assumed that all the parameters of the medium are constant within a given layer, and the technique for a multilayer sphere is used [3]. The number of layers depends on the properties of the medium and in some cases may be large. For example, up to 50 layers are used for the solution of a similar problem in a horizontal multilayer medium [9]. Therefore there is a need to construct an analytical solution to the spherical problem considered here in the case the parameters of the medium are not constant. This paper presents an analytical solution to a problem of a double-layered sphere in the case the relative permeability of the medium µ is a function of the radial coordinate of the system, more precisely, µ(r) = r α, and α is an arbitrary real number. The change in impedance in the coil is found in the form of an infinite series. Numerical computations are obtained for the case α = 2. II FORMULATION OF THE PROBLEM Consider a coil of radius ρ c horizontally situated above two concentric spheres of radii ρ and ρ, respectively, ρ < ρ, and the axis of the coil passes through the center of the spheres (see Fig. ). Let the density of the external current in the coil be described, in the spherical coordinates (ρ, θ, ϕ), with center at O, by the formula I = I ϕ e jωt e ϕ, e ϕ is a unit vector in the azimuthal ϕ-direction. We denote by R 0, R and R 2 the following three regions: (a) the empty space R 0 : r > ρ, 0 θ π, 0 ϕ 2π, containing air; (b) the outer spherical shell R : ρ < r < ρ, 0 θ π, 0 ϕ 2π, which is a conducting medium with constant conductivity σ and variable relative magnetic permeability µ(r); (c) the inner ball R 2 : 0 r ρ, 0 θ π, 0 ϕ 2π, which is a conducting medium with constant conductivity σ 2 and constant relative magnetic permeability µ 2. The electromagnetic field in a region with variable relative magnetic permeability, µ(r), is described by the following subset of the full set of Maxwell s equations: 3
Figure : Single-turn coil above double-layered sphere. curl E = B t, () curl H = I + I e, (2) I = σe, B = µ 0 µ(r)h, (3) E and H are the electric and magnetic field strengths, respectively, B is the magnetic induction vector, I is the current density, I e is the external current density, σ is the conductivity, µ 0 and µ are, respectively, the magnetic constant and the relative magnetic permeability of the medium. In (2) the displacement current is neglected, as it is usual in problems of eddy current testing. Introducing the vector potential curl A = B, (4) from () and (4) we have ( curl E + A ) = 0. t Thus we can introduce the scalar potential ψ by the equation E + A t = grad ψ. (5) Then, using (2) (5), we obtain [ ] curl µ 0 µ(r) curl A = σ grad ψ σ A t + Ie. (6) 4
Because of axial symmetry, the vector potential A has only one nonzero component, in the azimuthal direction, A = (0, 0, A(r, θ)). (7) By means of basic operations of vector analysis, (6) and (7) become ( grad div A µ 0 µ(r) µ 0 µ(r) A + dµ A µ 0 µ 2 (r) dr r + A ) e ϕ r = σ grad ψ σ A t + Ie. (8) Using the gauge and the expression grad div A = σ grad ψ µ 0 µ(r) A = A(r, θ) e jωt e ϕ, and projecting (8) on the ϕ-axis, we obtain the equation 2 A r + 2 A 2 r r + cot θ A r 2 θ + 2 A r 2 θ A 2 r 2 sin 2 θ ( dµ A µ(r) dr r + A ) r jωσµ 0 µ(r)a = µ 0 µ(r)i ϕ, (9) I ϕ is the projection of the vector I e on the ϕ-axis. III MATHEMATICAL ANALYSIS Since an analytical solution to the general equation (9) cannot be found, we suppose that µ(r) is of the form ( ) α r µ(r) =, (0) α is an arbitrary real number. Substituting (0) into (9) and using the dimensionless radial coordinate r d = r/ρ c we obtain the following system of equations in each of the regions R 0, R and R 2 (the subscript d to r d is omitted and all the geometric quantities are measured in units of ρ c ): ρ c 2 A r 2 2 A 0 r 2 + 2 α r 2 A 2 r 2 A 0 r + cot θ A 0 r 2 θ + A 0 r 2 θ A 0 2 r 2 sin 2 θ = µ 0I ϕ, () A r 2 θ + 2 A r 2 θ A ( 2 r 2 sin 2 θ jβr 2 α + α ) A r 2 = 0, (2) + 2 r A r + cot θ + 2 r A 2 r + cot θ r 2 A 2 θ + r 2 2 A 2 θ 2 A 2 r 2 sin 2 θ jβ2 2A 2 = 0, (3) β = ρ c ωσ µ 0, β 2 = ρ c ωσ2 µ 0 µ 2, and A i (r, θ) denotes the vector potential in the region R i, for i = 0,, 2. 5
The boundary conditions have the form A 0 r=ρ = A r=ρ, A r=ρ = A 2 r=ρ, A 0 = A, (4) r r=ρ µ r r=ρ A = A 2, (5) µ 2 r r=ρ µ 2 r r=ρ µ = µ(ρ ), µ 2 = µ(ρ). The current, I ϕ, in the coil is represented in the form I ϕ = Iδ(r r )δ(θ θ ), (6) I is the amplitude of the current and δ(x) is the Dirac delta function. Introducing the new variable ξ = cos θ, we express the solution of problem () (6) by the following integral transform: à i (r, n) = D n  i (r, ξ)p () n (ξ) dξ, i = 0,, 2, (7) Âi(r, ξ) = A i (r, θ), P n () (ξ) is an associated Legendre function of the first kind and D n = By (7) problem () (6) reduces to [ P () n (ξ) ] 2 dξ = 2n(n + ) 2n +. d 2 à 0 dr 2 + 2 r dã0 dr n(n + ) r 2 à 0 = µ 0 I 2n + 2n(n + ) P () n (cos θ ) sin θ δ(r r ), (8) d 2 à dr 2 + 2 α r dã dr n(n + ) ( à r 2 jβr 2 α + α ) à = 0, (9) r 2 d 2 à 2 dr + 2 dã2 n(n + ) à 2 r dr r 2 2 jβ2ã2 2 = 0, (20) à 0 = r=ρ à dã0, r=ρ dr = dã r=ρ µ dr, r=ρ (2) à = r=ρ Ã2 dã, r=ρ µ 2 dr = dã2 r=ρ µ 2 dr. r=ρ (22) It is convenient to consider the solution of equation (8) in the two subregions R 0 and R 02 of R 0, corresponding to the intervals ρ < r < r and r > r, respectively. We denote the solution in R 0 and R 02 by Ã0 and Ã02, respectively. The bounded general solutions of (8) in R 0 and R 02 are à 0 (r, n) = C r n + C 2 r n, à 02 (r, n) = C 3 r n, (23) respectively. Since the vector potential is continuous at r = r, then à r=r 0 = Ã02 r=r. (24) 6
Multiplying equation (8) by r 2, integrating from r = r ε to r = r + ε, and considering the limit as ε +0, we obtain dã02 dr dã0 r=r dr = µ 0 I 2n + r=r 2n(n + ) P n () (cos θ ) sin θ. (25) The general solution to equation (9) can be expressed in terms of Bessel functions (see [0], p.46): γ = α + 2 2 à (r, n) = C 4 r a J p (βr γ ) + C 5 r a Y p (βr γ ), (26), a = α, β = β j, p = 2 γ (α + )2 + 4n(n + ). α + 2 The solution to equation (20) which remains bounded as r 0, has the form à 2 (r, n) = C 6 r J n+/2 (k 2 r), (27) k 2 = β 2 j. Using (23), (26) and (27), and determining the constants C to C 6 from the conditions (2), (22), (24) and (25), we obtain C = µ 0 I 2n(n + ) P n () (cos θ ) sin θ r n+, C 2 = C ρ 2n+ c /c 2, C 3 = C 2 + C r 2n+, C 4 = dc 5, C ρ n + C 2 ρ n C 5 = ρ a [dj p (βρ γ ) + Y p (βρ γ )], ρ[c4 ρ a J p (βρ γ ) + C 5 ρ a Y p (βρ γ )] C 6 =, J n+/2 (k 2 ρ) d = d d 2, c = [dj p (βρ γ ) + Y p (βρ γ )](nµ a) βγρ γ [dj p (βρ γ ) + Y p (βρ γ )], c 2 = [dj p (βρ γ ) + Y p (βρ γ )][a + µ (n + )] + βγρ γ [dj p (βρ γ ) + Y p (βρ γ )], d = ρ a µ 2 [ J n+/2 (k 2 ρ) + 2k 2 ρj n+/2(k 2 ρ)]y p (βρ γ ) + 2ρµ 2 J n+/2 (k 2 ρ)[aρ a Y p (βρ γ ) + βγρ a+γ Y p (βρ γ )], d 2 = ρ a µ 2 [ J n+/2 (k 2 ρ) + 2k 2 ρj n+/2(k 2 ρ)]j p (βρ γ ) 2ρµ 2 J n+/2 (k 2 ρ)[aρ a J p (βρ γ ) + βγρ a+γ J p (βρ γ )] and denotes the derivative with respect to the whole argument. Inverting the integral transform (7) we obtain the series A i (r, θ) = n= à i (r, n)p () n (cos θ), i = 0,, 2. (28) 7
Formula (28) gives the complete solution to problem () (6). The change of impedance in the coil caused by a two-layered conducting sphere is given by the integral Z = jω A 0 (r, θ) dl, (29) I L is the contour of the coil, L A 0 (r, θ) = A ind 0 (r, θ)e ϕ, and A ind 0 (r, θ) is the induced part of the vector potential. More precisely, A ind 0 (r, θ) is described by formula (28) Ã0(r, n) is replaced with à ind 0 (r, n) = C 2 r n. Substituting (28) into (29) we obtain the change in impedance in the form Z = jπωµ 0 ρ 2 sin 2 θ n= n(n + ) ( ρ r ) 2n [P () n (cos θ )] 2 F F 2, (30) F = [dj p (βρ γ ) + Y p (βρ γ )](µ n a) βγρ γ [dj p (βρ γ ) + Y p (βρ γ )], F 2 = [dj p (βρ γ ) + Y p (βρ γ )][a + µ (n + )] + βγρ γ [dj p (βρ γ ) + Y p (βρ γ )], For some particular values of α, formula (30) can be simplified. For example, if α = 2, equation (9) becomes Euler s equation, whose general solution is à (r, n) = C 4 r γ + C 5 r γ 2, (3) γ,2 = 3 ± + 4n(n + ) + 4jβ 2. 2 Repeating the derivation of the change of impedance, Z, in the case formula (9) is replaced with (3) for α = 2, we obtain Z = ωπµ 0 ρ 2 sin 2 θ Z 0, and with Z 0 = j n= ( ρ r ) 2n P () n (cos θ )] 2 n(n + ) nµ (gρ γ + ρ γ 2 ) gγ ρ γ γ 2 ρ γ 2 gγ ρ γ + γ 2 ρ γ 2 + (n + )µ (gρ γ + ρ γ 2 ), (32) g = g g 2, g = µ 2 ρ γ 2 [ J n+/2 (k 2 ρ) + 2k 2 ρj n+/2(k 2 ρ)] 2µ 2 γ 2 ρ γ 2 J n+/2 (k 2 ρ), g 2 = µ 2 ρ γ [ J n+/2 (k 2 ρ) + 2k 2 ρj n+/2(k 2 ρ)] 2µ 2 γ ρ γ J n+/2 (k 2 ρ). 8
Figure 2: The change in impedance, Z 0, as a function of β, for r =.6,.8, 2.0, Figure 3: The change in impedance, Z 0, as a function of β, for ρ =.,.0, 0.9. IV NUMERICAL RESULTS Formula (32) was used to compute the change of impedance, Z 0 = X + jy, in the case the relative magnetic permeability, µ(r), of the media R and R 2 are r 2 and the constant µ 2, respectively. The computational results are presented in Figs. 2 4 for increasing values of β, as indicated by the arrow. In Fig. 2, Z 0 is shown as a function β for three values of the distance, r =.6,.8, 2.0, of the coil to the origin. The remaining parameters were set at: ρ =.0, ρ =., µ 2 =, µ =, µ ρ 2 2 = ρ, β 2 2 =. The unit of length is the radius, ρ c, of the coil. It is seen from the figure that Z 0 decreases as r increases. Moreover, Y changes more rapidly than X. In Fig. 3, Z 0 is shown as a function of β for three values of the radius, ρ = 0.9,.0,., of the inner sphere with variable magnetic permeability. The remaining parameters were set at: r =.5, ρ =.3, µ 2 =, µ =, µ ρ 2 2 = ρ, β 2 2 =. 9
Figure 4: The change in impedance, Z 0, as a function of β, for β 2 =, 2, 3. It is seen that a small change in the radius of the internal sphere has a large influence on Z 0. This fact is used for controlling the thickness of coverings with variable magnetic permeability µ. In Fig. 4, Z 0 is shown as a function of β for β 2 =, 2, 3. The remaining parameters were set at: r =.5, ρ =.0, ρ =.2, µ 2 =, µ =, µ ρ 2 2 = ρ. 2 It is seen that a change of conductivity of the inner sphere has a significant influence on the output of the eddy current probe. This fact is used in eddy current testing of multilayer products. Note that the technique developed in this paper can be used for more general problems. For example, one can take a multilayer sphere the magnetic permeability of the ith layer is given by µ i (r) = r α i, (33) and α i are distinct constants. Moreover, an analytical solution can also be found in the case the conductivity of the ith layer, σ i (r), has the form σ i (r) = r δ i, (34) δ i are distinct constants. Finally, one can find an analytical solution for a multilayer sphere in the case both µ i (r) and σ i (r) are given by (33) and (34), the constants α i and δ i can be distinct. V CONCLUSION In this paper, a formula is obtained for the change of impedance in a single-turn coil due to the presence of a conducting double-layered sphere with a variable magnetic permeability, µ(r), of the outer layer of the form µ(r) = r α, α is an arbitrary real number. The technique developed here can be used to find solution of a similar problem for a multilayered sphere the magnetic permeability and conductivity of each layer are given by (33) and (34), respectively. ACKNOWLEDGMENT This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant A 769 and the Centre de recherches mathématiques of the Université de 0
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