CONDUCTING ELLIPTICAL CYLINDER IN A SLOTTED EDDY-CURRENT TRANSDUCER

R. M. ZAKIROV, V. K. ZHUKOV

(Presented by the scientific-technical seminar of the Department of Information-Measuring Technology)

In [1] a solution is given to the problem of an elliptical conducting cylinder placed in a transverse homogeneous alternating magnetic field. In deriving the analytic expression for the vector potential of the resultant magnetic field, simplifications were made that reduced its accuracy.

In the present work a more rigorous solution of the indicated problem is given.

The wave equation for the vector potential \(\dot A\) of the magnetic field in the orthogonal system of elliptic cylindrical coordinates has the form:

\[ \frac{\partial^2 \dot A}{\partial \xi^2} + \frac{\partial^2 \dot A}{\partial \varphi^2} + 2q(\operatorname{ch}2\xi-\cos 2\varphi)\dot A =0. \tag{1} \]

Here

\[ q=\frac{1}{4}\kappa^2 h^2 = \frac{1}{4}\omega^2\left(\varepsilon_a-i\frac{\sigma}{\omega}\right)\mu_a(a^2-b^2), \tag{2} \]

where \(\omega\) is the angular frequency of the exciting field;
\(\varepsilon_a=\varepsilon_0\varepsilon\) is the dielectric permittivity of the medium;
\(\mu_a=\mu_0\mu\) is the magnetic permeability of the medium;
\(\sigma\) is the specific electrical conductivity of the medium;
\(a\) and \(b\) are the major and minor semiaxes of the ellipse formed by the normal cross section of the cylinder;
\(h=\sqrt{a^2-b^2}\) is half the distance between the foci of the ellipse.

For the conducting medium of the cylinder,

\[ q=q'=-\frac{1}{4}i\omega\sigma\mu_a h^2, \tag{3} \]

and for the nonconducting space surrounding the cylinder,

\[ q=q''=\frac{1}{4}\omega^2\varepsilon_a\mu_a h^2. \tag{4} \]

Let us denote the vector potential of the field in the core by \(\dot A'\), and in the space by \(\dot A''\). Then the wave equations of the field in the core and in the space will have the form:

\[ \frac{\partial^2 \dot A'}{\partial \xi^2} + \frac{\partial^2 \dot A'}{\partial \varphi^2} + 2q'(\operatorname{ch}2\xi-\cos 2\varphi)\dot A' =0, \tag{5} \]

\[ \frac{\partial^2 \dot A''}{\partial \xi^2} + \frac{\partial^2 \dot A''}{\partial \varphi^2} + 2q''(\operatorname{ch}2\xi-\cos 2\varphi)\dot A'' =0. \tag{6} \]

The solution of equation (5), as follows from the physical picture of the field, must be symmetric with respect to the minor axis of the ellipse

Fig. 1. Elliptical cylinder in a transverse magnetic field

and asymmetric with respect to the major axis (Fig. 1a). A solution satisfying this condition is expressed in terms of Mathieu functions [2]:

\[ \dot A'=\sum_{n=0}^{\infty} \left[ \dot C_{2n+1}\operatorname{Se}_{2n+1}(\xi,q')\operatorname{se}_{2n+1}(\varphi,q') + \dot D_{2n+1}\operatorname{Fey}_{2n+1}(\xi,q')\operatorname{se}_{2n+1}(\varphi,q') \right], \tag{7} \]

where \(\dot C_{2n+1}\) and \(\dot D_{2n+1}\) are integration constants, and \(\operatorname{Se}_{2n+1}(\xi,q')\), \(\operatorname{se}_{2n+1}(\varphi,q')\), and \(\operatorname{Fey}_{2n+1}(\xi,q')\) are Mathieu functions. Taking into account that, for physical reasons, \(\dot D_{2n+1}=0\), we obtain

\[ \dot A'=\sum_{n=0}^{\infty} \dot C_{2n+1}\operatorname{Se}_{2n+1}(\xi,q')\operatorname{se}_{2n+1}(\varphi,q'). \tag{8} \]

The vector potential \(\dot A''\) in space is the sum of the vector potential of the exciting field \(\dot A_0\) and the vector potential \(\dot A_1''\) of the field of eddy currents in the core:

\[ \dot A''=\dot A_0+\dot A_1''. \tag{9} \]

The vector \(\dot A_0\) is related to the magnetic-induction vector \(\dot{\mathbf B}_0\) of the exciting field by the equation \(\operatorname{rot}\dot{\mathbf A}_0=\dot{\mathbf B}_0\), from which, for the given coordinate system, taking into account that \(\dot{\mathbf B}_0=\tau_\xi \dot B_\xi+\tau_\varphi \dot B_\varphi\), \(\dot{\mathbf A}_0=\tau_z\dot A_0\), it follows that

\[ \dot A_0=\dot B_0 h\,\operatorname{sh}\xi \sin\varphi. \tag{10} \]

Let us proceed to the solution of equation (6). If the displacement currents in the space surrounding the cylinder are neglected, i.e., if \(q''=0\), and \(\dot A''\) is replaced according to (9) and (10), then (6) is transformed into the expression

\[ \frac{\partial^2 \dot A_1''}{\partial \xi^2} + \frac{\partial^2 \dot A_1''}{\partial \varphi^2} =0. \tag{11} \]

Equation (11) is solved by the method of separation of variables. The vector \(\dot A_0\) of the external field is a function that is odd with respect to the angle \(\varphi\), hence—

therefore, the function \(\dot A_1''\) must also be odd. In addition, as \(\xi \to \infty\) one must have \(\dot A_1'' \to 0\). Thus, the sought solution of equation (11), in the form of a superposition of all possible solutions, will have the form

\[ \dot A_1''=\sum_{m=0}^{\infty}\dot N_{2m+1}e^{-(2m+1)\xi}\sin(2m+1)\varphi, \tag{12} \]

where \(\dot N_{2m+1}\) is a constant of integration.

After substituting (10) and (12) into (9), the vector potential of the field in space will be determined by the expression

\[ \dot A''=\dot B_0 h\,\operatorname{sh}\xi \sin\varphi+ \sum_{m=0}^{\infty}\dot N_{2m+1}e^{-(2m+1)\xi}\sin(2m+1)\varphi. \tag{13} \]

The constants of integration \(\dot C_{2n+1}\) in (8) and \(\dot N_{2m+1}\) in (13), for the case of a nonmagnetic cylinder \(\mu=\mu_0\), are found from the boundary conditions at \(\xi=\xi_0\)

\[ \dot A'=\dot A'';\qquad \frac{\partial \dot A'}{\partial \xi}=\frac{\partial \dot A''}{\partial \xi}. \tag{14} \]

Substituting into (14) the values of \(\dot A'\) and \(\dot A''\) from (8) and (13), we obtain two equations:

\[ \sum_{n=0}^{\infty}\dot C_{2n+1}\operatorname{Se}_{2n+1}(\xi_0,q')\operatorname{se}_{2n+1}(\varphi,q') = \]

\[ =\dot B_0 h\,\operatorname{sh}\xi_0 \sin\varphi+ \sum_{m=0}^{\infty}\dot N_{2m+1}e^{-(2m+1)\xi_0}\sin(2m+1)\varphi; \tag{15} \]

\[ \sum_{n=0}^{\infty}\dot C_{2n+1}\frac{\partial}{\partial \xi} \left[\operatorname{Se}_{2n+1}(\xi_0,q')\right]\operatorname{se}_{2n+1}(\varphi,q') = \]

\[ =\dot B_0 h\,\operatorname{ch}\xi_0 \sin\varphi- \sum_{m=0}^{\infty}(2m+1)\dot N_{2m+1}e^{-(2m+1)\xi_0}\sin(2m+1)\varphi. \tag{16} \]

We shall use the orthogonality property and the normalization conditions of the Mathieu functions [2]:

\[ \int_0^{2\pi}\operatorname{se}_m(\varphi,q)\operatorname{se}_p(\varphi,q)= \begin{cases} 0, & m\ne p;\\ \pi, & m=p\text{ — for real }q;\\ L_{2n+1}, & m=p\text{ — for imaginary }q. \end{cases} \tag{17} \]

In addition, in the right-hand side of (15) we express the functions \(\sin\) in terms of Mathieu functions

\[ \sin(2m+1)\varphi=\sum_{n=0}^{\infty}P_{2m+1}^{(2n+1)}\operatorname{se}_{2n+1}(\varphi). \tag{18} \]

After this, we multiply both sides of expression (15) by \(\operatorname{se}_{2n+1}(\gamma,g')\) and integrate over the interval from \(0\) to \(2\pi\). Taking into account that in our case \(q\) is an imaginary quantity, we obtain

\[ \dot C_{2n+1}\operatorname{Se}_{2n+1}(\xi_0,q')L_{2n+1}= \]

\[ = \left[ \dot B_0 h\,\operatorname{sh}\xi_0 P_1^{(2n+1)} + \sum_{m=0}^{\infty}\dot N_{2m+1}e^{-(2m+1)\xi_0}P_{2m+1}^{(2n+1)} \right]\cdot L_{2n+1}, \tag{19} \]

The remaining integrals are equal to zero.

From (19) we find

\[ \dot C_{2n+1}=\frac{1}{\operatorname{Se}_{2n+1}(\xi_0,q')} \left[\dot B_0 h\,\operatorname{sh}\xi_0\,P_1^{2n+1} +\sum_{m=0}^{\infty}\dot N_{2m+1}e^{-(2m+1)\xi_0}P_{2m+1}^{(2n+1)}\right], \tag{20} \]

which we substitute into expression (16)

\[ \sum_{n=0}^{\infty} \frac{\dfrac{\partial}{\partial \xi}\left[\operatorname{Se}_{2n+1}(\xi_0,q')\right]} {\operatorname{Se}_{2n+1}(\xi_0,q')} \left[ \dot B_0 h\,\operatorname{sh}\xi_0\,P_1^{(2n+1)} + \sum_{m=0}^{\infty}\dot N_{2m+1}e^{-(2m+1)\xi_0}P_{2m+1}^{(2n+1)} \right]\operatorname{se}_{2n+1}(\varphi,q') = \dot B_0 h\,\operatorname{ch}\xi_0\sin\varphi - \sum_{m=0}^{\infty}\dot N_{2m+1}(2m+1)e^{-(2m+1)\xi_0}\sin(2m+1)\varphi . \tag{21} \]

Substitute into (21) \(\operatorname{se}_{2n+1}(\varphi,q')\) in the form [2]

\[ \operatorname{se}_{2n+1}(\varphi,q') = \sum_{m=0}^{\infty}P_{2m+1}^{(2n+1)}\sin(2m+1)\varphi \tag{22} \]

and equate the coefficients of the sines with identical arguments that appear in the left- and right-hand sides of expression (21). After this we obtain a system of \((m+1)\) equations with \((m+1)\) unknown constant coefficients \(\dot N_1,\dot N_3,\ldots,\dot N_{2m+1}\).

We group the terms containing the unknown coefficients on the left-hand sides, and transfer the free terms to the right-hand sides of the equations. After the indicated transformations, we obtain a system of equations which, in matrix notation, can be represented as

\[ \alpha\cdot K=\beta, \tag{23} \]

where \(\alpha\) is the matrix of coefficients of the unknowns;

\(K\) is the column matrix of the unknowns, whose elements have the form

\[ K_{2m+1}=\dot N_{2m+1}e^{-(2m+1)\xi_0}; \tag{24} \]

\(\beta\) is the column matrix of the free terms.

The matrix \(\alpha\) is square and symmetric. Its general element is determined from the expression

\[ \alpha_{2i+1,\,2k+1} = \alpha_{2k+1,\,2i+1} = \sum_{n=0}^{\infty} \frac{\dfrac{\partial}{\partial \xi}\left[\operatorname{Se}_{2n+1}(\xi_0,q')\right]} {\operatorname{Se}_{2n+1}(\xi_0,q')} P_{2i+1}^{(2n+1)}P_{2k+1}^{(2n+1)}, \tag{25} \]

for \(i\ne k\) and \(i=0,1,2,\ldots,m;\ k=0,1,2,\ldots,m\).

The elements of the main diagonal are found from the formula

\[ \alpha_{2i+1,\,2i+1} = (2i+1) + \sum_{n=0}^{\infty} \frac{\dfrac{\partial}{\partial \xi}\left[\operatorname{Se}_{2n+1}(\xi_0,q')\right]} {\operatorname{Se}_{2n+1}(\xi_0,q')} P_{2i+1}^{(2n+1)}P_{2i+1}^{(2n+1)}, \tag{26} \]

for \(i=0,1,2,\ldots,m\).

The elements of the matrix \(\beta\) are found in the form

\[ \beta_1 = \dot B_0 h \left\{ \operatorname{ch}\xi_0 - \operatorname{sh}\xi_0 \sum_{n=0}^{\infty} \frac{\dfrac{\partial}{\partial \xi}\left[\operatorname{Se}_{2n+1}(\xi_0,q')\right]} {\operatorname{Se}_{2n+1}(\xi_0,q')} P_1^{(2n+1)}P_1^{(2n+1)} \right\}, \tag{27} \]

\[ \beta_{2i+1}=-\dot{B}_0 h\,\operatorname{sh}\xi_0 \sum_{n=0}^{\infty} \frac{\dfrac{\partial}{\partial \xi}\left[\operatorname{Se}_{2n+1}(\xi_0,q')\right]} {\operatorname{Se}_{2n+1}(\xi_0,q')} P_i^{(2n+1)}P_{2i+1}^{(2n+1)}, \tag{28} \]

for \(i=1,2,3,\ldots,m\).

The solution of the system of equations (23) in matrix notation will be as follows:

\[ K=\alpha^{-1}\cdot\beta, \tag{29} \]

where \(\alpha^{-1}\) is the matrix inverse to the matrix \(\alpha\).

The constants \(\dot{N}_1,\dot{N}_3,\ldots,\dot{N}_{2m+1}\) are determined from the expression

\[ \dot{N}_{2i+1}=\dot{K}_{2i+1}e^{(2i+1)\xi_0}. \tag{30} \]

Let us determine an expression for the magnetic flux penetrating the rectangular measuring winding of the slotted transducer, whose dimensions are respectively equal to the transverse dimensions of the magnetic core. In doing so, the influence of the magnetic core on the character of the distribution of the magnetic flux in the slot is not taken into account.

\[ \dot{\Phi}=\int_S \bar{B}\,d\bar{S} =\int_S \operatorname{rot}\bar{A}\,d\bar{S} =\oint \bar{A}\,d\bar{l}, \tag{31} \]

where \(dl\) is an element of the length of the integration contour coinciding with the contour of the measuring winding.

Fig. 2. Elliptical cylinder in a slotted sensor

The vector potential \(\bar{A}\) coincides in direction with the \(z\)-axis and does not depend on the coordinate along the latter. Therefore, when integrating expression (31) along the sides parallel to the \(z\)-axis, \(\dot{A}\) may be taken outside the integral sign, while the integrals along the sides perpendicular to the \(z\)-axis are equal to zero. Substituting expression (13) into (31), we obtain

\[ \dot{\Phi}=\oint \bar{\dot{A}}\,d\bar{l} =\dot{A}\int_{(MN)} dl+\dot{A}\int_{(PQ)} dl =4\dot{A}l=4\dot{B}_{0}lh\,\operatorname{sh}\xi\sin\varphi+ \]
\[ +4l\sum_{m=0}^{\infty}\dot{N}_{2m+1}e^{-(2m+1)\xi}\sin(2m+1)\varphi =\dot{\Phi}_{0}+\Delta\dot{\Phi}, \tag{32} \]

where \(\dot{\Phi}_{0}=4\dot{B}_{0}lh\,\operatorname{sh}\xi\sin\varphi=4\dot{B}_{0}lc=\dot{B}_{0}S\) is the magnetic flux of the exciting field penetrating the measuring winding;

\[ \Delta\dot{\Phi} =4l\sum_{m=0}^{\infty}\dot{N}_{2m+1}e^{-(2m+1)\xi}\sin(2m+1)\varphi \]
is the increment of the magnetic flux due to the introduction of the elliptical cylinder into the sensor gap.

Let us rotate the core by \(90^\circ\) (Fig. 1b) and find, for this case, the vector potential in the space surrounding the cylinder.

As a result of the rotation of the coordinate system, the solutions of equations (5) and (6) must be symmetric with respect to the major axis of the ellipse:

\[ \dot{A}'=\sum_{n=0}^{\infty}\dot{C}_{2n+1}\operatorname{Ce}_{2n+1}(\xi,q')\operatorname{ce}_{2n+1}(\varphi,q'), \tag{33} \]

\[ \dot{A}''=\dot{B}_{0}h\,\operatorname{ch}\xi\cos\varphi+ \sum_{m=0}^{\infty}\dot{N}'_{2m+1}e^{-(2m+1)\xi}\cos(2m+1)\varphi. \tag{34} \]

The constants of integration, as in the preceding case, are found from the boundary conditions. The expression for \(\dot{C}_{2n+1}\) is found from the first boundary condition (14)

\[ \dot{C}_{2n+1} = \frac{1}{\operatorname{Ce}_{2n+1}(\xi,q')} \left[ \dot{B}_{0}h\,\operatorname{ch}\xi_{0}L_{1}^{(2n+1)} + \sum_{m=0}^{\infty}\dot{N}'_{2m+1}e^{-(2m+1)\xi_{0}}L_{2m+1}^{(2n+1)} \right]. \tag{35} \]

Substituting (35) into the second boundary condition (14) and equating the coefficients of the cosines with identical arguments appearing in the left- and right-hand sides, we obtain a system of \((m+1)\) equations with \((m+1)\) unknown coefficients \(N'_{1},N'_{3},\ldots,N'_{2m+1}\), which can be represented in matrix form as

\[ \varkappa\cdot K'=\lambda, \tag{36} \]

where \(\varkappa\) is a square symmetric matrix of the coefficients at the unknowns \(\dot{N}'_{2i+1}e^{-(2i+1)\xi_{0}}\);

\(K'\) is a column matrix of the unknowns, whose general term is \(\dot{N}'_{2i+1}e^{-(2i+1)\xi_{0}}\);

\(\lambda\) is the column matrix of the free terms.

The general term of the matrix \(\varkappa\) is determined by the expression

\[ \varkappa_{2i+1,\,2k+1} = \varkappa_{2k+1,\,2i+1} = \sum_{n=0}^{\infty} \frac{ \dfrac{\partial}{\partial \xi}\left[\operatorname{Ce}_{2n+1}(\xi_{0},q')\right] }{ \operatorname{Ce}_{2n+1}(\xi_{0},q') } L_{2i+1}^{(2n+1)}L_{2k+1}^{(2n+1)}, \tag{37} \]

for \(i\ne k\) and \(i=0,1,2,\ldots,m,\ k=0,1,2,\ldots,m\). The elements of the main diagonal are determined by the expression

\[ \varkappa_{2i+1,\,2i+1} = (2i+1)+ \sum_{n=0}^{\infty} \frac{ \dfrac{\partial}{\partial \xi}\left[\operatorname{Ce}_{2n+1}(\xi_{0},q')\right] }{ \operatorname{Ce}_{2n+1}(\xi_{0},q') } L_{2i+1}^{(2n+1)}L_{2i+1}^{(2n+1)}, \tag{38} \]

for \(i=k=0,1,2,\ldots,m\), and the elements of the matrix \(\lambda\) are

\[ \lambda_1=\dot B_0 h\left\{\operatorname{sh}\xi_0-\operatorname{ch}\xi_0 \sum_{n=0}^{\infty} \frac{\dfrac{\partial}{\partial \xi}\left[\operatorname{Ce}_{2n+1}(\xi_0,q')\right]} {\operatorname{Ce}_{2n+1}(\xi_0,q')} L_1^{(2n+1)}L_1^{(2n+1)} \right\}, \tag{39} \]

\[ \lambda_{2i+1}=-\dot B_0 h\,\operatorname{ch}\xi_0 \sum_{n=0}^{\infty} \frac{\dfrac{\partial}{\partial \xi}\left[\operatorname{Ce}_{2n+1}(\xi_0,q')\right]} {\operatorname{Ce}_{2n+1}(\xi_0,q')} L_1^{(2n+1)}L_{2i+1}^{(2n+1)}, \tag{40} \]

where \(i=1,2,\ldots,m\).

In expressions (38)–(40), \(L_{2i+1}^{(2n+1)}\) is the coefficient in the expansion of the Mathieu function

\[ \operatorname{ce}_{2n+1}(\varphi,q')=\sum_{m=0}^{\infty}L_{2m+1}^{(2n+1)}\cos(2m+1)\varphi . \]

The solution of the system of equations (36), in matrix notation, will be as follows:

\[ K'=\chi^{-1}\lambda, \tag{41} \]

where \(\chi^{-1}\) is the matrix inverse to the matrix \(\chi\).

Using an expression analogous to (30), one can determine the constants \(N'_1,N'_3,\ldots,N'_{2m+1}\).

Substituting (34) into (31), we find the expression for the magnetic flux

\[ \dot\Phi=\dot\Phi_0+\Delta\dot\Phi, \tag{42} \]

where \(\dot\Phi_0=4Bl\,\operatorname{ch}\xi\cos\varphi\) is the magnetic flux of the exciting field,

\[ \Delta\dot\Phi=4l\sum_{m=0}^{\infty}N'_{2m+1}e^{-(2m+1)c}\cos\varphi \]

is the increment of the magnetic flux caused by the insertion of the cylinder into the sensor gap.

The relative change in the emf of the measuring winding caused by inserting the cylinder into the transducer is equal to the relative change in the magnetic flux

\[ \frac{\Delta\dot E}{|\dot E_0|}=\frac{\Delta\dot\Phi}{\Phi_0}. \tag{43} \]

The results of calculations using the expressions obtained, making it possible to give a quantitative estimate of the influence of the cylinder’s noncircularity on the transducer parameters, will be published later.

REFERENCES

1. V. K. Zhukov and R. M. Zakirov, “Defectoscopy,” 1970, No. 5, p. 102.
2. McLachlan, Theory and Application of Mathieu Functions. Moscow, IL, 1953.
3. Andre Angot, Mathematics for Electrical and Radio Engineers. Moscow, “Nauka,” 1965.
4. N. S. Piskunov, Differential and Integral Calculus. Moscow, Fizmatgiz, 1962.