EFFECT OF THE SIZE AND COORDINATE OF A FERROMAGNETIC SPHERE ON THE IMPEDANCE OF A LOOP

V. K. Zhukov

(Presented by the Scientific and Technical Seminar of the Department of Information and Measurement Technology)

If a ferromagnetic metallic sphere is placed in a through-type eddy-current transducer with a short winding, this will cause a change in the impedance of the transducer. The greatest change in impedance will occur when the sphere is in the plane of the winding. The perturbation introduced by the sphere depends on its size and electromagnetic parameters, as well as on the coordinate of the sphere in the plane of the winding.

In known works, problems concerning a sphere in a homogeneous field [1, 2] and a sphere on the axis of a loop or a short solenoid [3, 4] have been considered. The cited works do not answer the question of the influence of the sphere coordinate on the impedance of the transducer, which must be known when using an eddy-current transducer for selective detection of metallic objects.

In order to study the influence of the sphere coordinate on the impedance of the transducer, let us consider the problem of a small ferromagnetic sphere in the plane of a large loop. We assume the dimensions of the sphere \(2a\) (Fig. 1) to be much smaller than the dimensions of the loop \(2R_b\), which gives grounds for approximately considering the sphere as being in a homogeneous electromagnetic field.

Let us determine the vector magnetic potential of the loop with the sphere. For this purpose we solve the wave equation

\[ \nabla^2 \bar A + \kappa^2 \bar A = 0 \tag{1} \]

in the spherical coordinate system \(r, \Theta, \varphi\) (Fig. 1).

Fig. 1

By virtue of circular symmetry, the vector \(\bar A\) has only one component \(\dot A_\varphi = \dot A\); therefore equation (1) in the adopted coordinate system will have the form

\[ \frac{1}{r^2} \left\{ \frac{1}{dr} \left( r^2 \frac{\partial \dot A}{\partial r} \right) + \frac{1}{\sin \Theta} \frac{\partial}{\partial \Theta} \left( \frac{\partial \dot A}{\partial \Theta}\sin \Theta \right) + \frac{1}{\sin \Theta^2} \frac{\partial^2 \dot A}{\partial \varphi^2} \right\} + \kappa^2 \dot A = 0 . \tag{2} \]

Since \(\dot A\) does not depend on \(\varphi\), i.e., \(\partial \dot A/\partial \varphi = 0\), (2) is simplified:

\[ 2r \frac{\partial \dot A}{\partial r} + r^2 \frac{\partial^2 \dot A}{\partial r^2} + \frac{\partial^2 \dot A}{\partial \Theta^2} + \frac{\cos \Theta}{\sin \Theta}\frac{\partial \dot A}{\partial \Theta} + k^2 r^2 \dot A = 0. \tag{3} \]

For the conducting medium of the sphere, \(k^2 = \omega^2 \gamma \mu \mu_0\), where \(\omega\) is the angular frequency of the current flowing through the loop, \(\gamma\) is the specific electrical conductivity, and \(\mu\) is the relative magnetic permeability of the material of the sphere. For air, \(k^2 \approx 0\).

We shall solve equation (3) by the Fourier method; for this purpose we represent \(\dot A\) as the product

\[ \dot A = \dot A(r)\cdot \dot A(\Theta) \tag{4} \]

of two factors, one of which, \(\dot A(r)\), is a function only of the coordinate \(r\), while the other, \(\dot A(\Theta)\), is a function of the coordinate \(\Theta\).

After substitution of (4) into (3), the latter splits into two equations:

\[ \frac{\partial^2 \dot A(r)}{\partial r^2} + \frac{2}{r}\frac{\partial \dot A(r)}{\partial r} + (k^2 - \lambda/r^2)\dot A(r) = 0, \tag{5} \]

\[ \frac{\partial^2 \dot A(\Theta)}{\partial \Theta^2} + \frac{\cos \Theta}{\sin \Theta}\frac{\partial \dot A(\Theta)}{\partial \Theta} + \lambda \dot A(\Theta) = 0, \tag{6} \]

in which \(\lambda = n(n+1)\) is the separation parameter.

By the substitution \(\dot A(r) = y(r)/\sqrt r\), equation (5) is reduced to the Bessel equation

\[ r^2 y'' + r y' + \left[k^2 r^2 - \left(n+\frac{1}{2}\right)^2\right]y = 0, \tag{7} \]

whose general solution has the form

\[ y(r) = \dot C_n' Y_{n+\frac{1}{2}}(kr) + \dot D_n' H_{n+\frac{1}{2}}(kr). \tag{8} \]

In order to satisfy the condition that the solution be bounded at \(r=0\), one must set \(\dot D_n' = 0\), whence

\[ y(r) = \dot C_n' \cdot Y_{n+1/2}(kr), \tag{9} \]

\[ \dot A(r) = \frac{1}{\sqrt r}\dot C_n' Y_{n+1/2}(kr), \tag{10} \]

In (10), \(Y_{n+1/2}(kr)\) is a Bessel function of half-integer order with argument \(kr\), and \(\dot C_n'\) is a constant of integration.

The general solution of (6) is represented by the Legendre polynomial \(P_n(\cos \Theta)\) with argument \(\cos \Theta\):

\[ \dot A(\Theta) = \dot C_n'' P_n(\cos \Theta). \tag{11} \]

Substituting (10) and (11) into (4), we find the general solution of equation (3) for the conducting medium of the sphere:

\[ \dot A = \sum_{n=0}^{\infty} \frac{1}{\sqrt r}\dot C_n P_n(\cos \Theta)Y_{n+1/2}(kr). \tag{12} \]

In air \(k^2 = 0\), and therefore wave equation (1) is transformed into the Laplace equation

\[ \nabla^2 \dot A_{\mathrm{в}} = 0, \tag{13} \]

which, by means of the Fourier transform in spherical coordinates, can be represented by two equations

\[ \frac{\partial^2 \dot A_{\mathrm{v}}(r)}{\partial r^2} + \frac{2}{r}\frac{\partial \dot A_{\mathrm{v}}(r)}{\partial r} - \frac{\lambda}{r^2}\dot A_{\mathrm{v}}(r)=0, \tag{14} \]

\[ \frac{\partial^2 \dot A_{\mathrm{v}}(\Theta)}{\partial \Theta^2} + \frac{\cos\Theta}{\sin\Theta}\frac{\partial \dot A_{\mathrm{v}}(\Theta)}{\partial \Theta} + \lambda \dot A_{\mathrm{v}}(\Theta)=0. \tag{15} \]

The separation parameter is \(\lambda=n(n+1)\).

Equation (14) is Euler’s equation, having the general solution

\[ \dot A_{\mathrm{v}}(r)=\dot M_n r^n+\dot N_n\frac{1}{r^{n+1}}. \tag{16} \]

Equation (15), like (6), has Legendre polynomials as its roots,

\[ \dot A_{\mathrm{v}}(\Theta)=QP_n(\cos\Theta). \tag{17} \]

Combining (16) and (17), we find the expression for the vector potential in air,

\[ \dot A_{\mathrm{v}}=\sum_{n=0}^{\infty} P_n(\cos\Theta) \left[\dot M_n r^n+\dot N_n\frac{1}{r^{n+1}}\right]. \tag{18} \]

Above we agreed to regard the field of the loop in the space occupied by the sphere as homogeneous. The vector potential of this field is related to the magnetic-field intensity \(\dot H_0\) and to the coordinates of the center of the sphere \(r,\Theta\) by the relation

\[ \dot A_0=\frac{1}{2}\mu_0 \dot H_0 r\sin\Theta . \tag{19} \]

Far from the sphere, the vector potential \(\dot A_{\mathrm{v}}\) of the resultant field must tend to the potential \(\dot A_0\) of the loop field, i.e.,

\[ \sum_{n=0}^{\infty} P_n(\cos\Theta)\dot M_n r^n = \frac{1}{2}\mu_0 \dot H_0 r\sin\Theta . \tag{20} \]

From (20), equating the coefficients of \(r^n\) with identical exponents, we find

\[ \dot M_1=\frac{1}{2}\mu_0 \dot H_0 r\sin\Theta . \tag{21} \]

\[ \dot M_n=0 \quad \text{for } n\ne 1. \tag{22} \]

The constants of integration \(\dot N_n\) and \(\dot C_n\) are determined from the boundary conditions

\[ \dot A\big|_{r=a}=\dot A_{\mathrm{v}}\big|_{r=a}, \tag{23} \]

\[ \frac{1}{\mu}\frac{\partial}{\partial r}(r\dot A)\bigg|_{r=a} = \frac{\partial}{\partial r}(r\dot A_{\mathrm{v}})\bigg|_{r=a}. \tag{24} \]

After substitution and transformations, we find

\[ \dot N_1= \frac{1}{2}\mu_0 \dot H_0 \operatorname{tg}\Theta\, a^3\, \frac{(1+2\mu)Y_{3/2}(\kappa a)-\kappa a\,Y_{1/2}(\kappa a)} {(\mu-1)Y_{3/2}(\kappa a)-\kappa a\,Y_{1/2}(\kappa a)}, \tag{25} \]

\[ \dot N_n=0 \quad \text{for } n\ne 1. \tag{26} \]

Let us denote

\[ \frac{(1+2\mu)Y_{3/2}(\kappa a)-\kappa a\,Y_{1/2}(\kappa a)} {(\mu-1)Y_{3/2}(\kappa a)-\kappa a\,Y_{1/2}(\kappa a)} =\dot{\tau}. \tag{27} \]

Knowing the constants of integration, we write the final expression for the vector magnetic potential in air:

\[ \dot A_{\mathrm{v}}=\frac{1}{2}\mu_0 \dot H_0 r\sin\Theta \left(1+\frac{a^3}{r^3}\dot{\tau}\right). \tag{28} \]

The first term in (28) determines the field of the loop, and the second the disturbance introduced by the sphere.

The vector potential \(\dot A_0\) [5] and the intensity \(\dot H_0\) [6] of the magnetic field of the loop without the sphere are determined by the expressions

\[ \dot A_0=\frac{\mu_0 I}{\pi \xi}\sqrt{\frac{1}{\xi}} \left[\left(1-\frac{\xi^2}{2}\right)K(\xi)-E(\xi)\right], \tag{29} \]

\[ \dot H_0=-\frac{I\,E(\xi)}{\pi R_{\mathrm{v}}(1-\xi^2)}, \tag{30} \]

where \(\xi=r/R_{\mathrm{v}}\), \(I\) is the current through the loop, and \(K(\xi)\) and \(E(\xi)\) are complete elliptic integrals.

The magnetic flux of the loop without the sphere can be determined by integrating \(\dot A_0\) over the contour of the loop:

\[ \dot \Phi_0=\int_0^{2\pi}\dot A_0 R_{\mathrm{v}}\,d\varphi. \tag{31} \]

Substituting \(\dot A_0\) from (29) into (31) and carrying out the integration, we find

\[ \dot \Phi_0=2\mu_0 R_{\mathrm{v}} I \left[\frac{1}{2}K(\xi_{\mathrm{v}})-E(\xi_{\mathrm{v}})\right]. \tag{32} \]

Here

\[ \xi_{\mathrm{v}}=\frac{R_{\mathrm{v}}-r_{\mathrm{pr}}}{R_{\mathrm{v}}} \tag{33} \]

determines the normalized coordinate of the inner edge of the loop made of wire with radius \(r_{\mathrm{pr}}\).

The vector potential of the field of the sphere in the plane where \(\sin\Theta=1\) is equal to

\[ \dot A_{\mathrm{sh}}=\frac{1}{2}\mu_0\dot H_0 r\frac{a^3}{r^3}\dot{\tau}. \tag{34} \]

If the sphere is displaced relative to the center of the loop by a distance \(h\), then the vector potential of the field of the sphere at the point with coordinates \(R_{\mathrm{v}}, \pi/2, \varphi\) is determined from (34) after substituting

\[ r=\sqrt{h^2+R_{\mathrm{v}}^2+2hR_{\mathrm{v}}\cos\varphi}. \tag{35} \]

The magnetic flux of the sphere penetrating the plane of the loop is determined by integrating (34) over the contour of the loop, taking into account (30), (35), and the new notation \(\xi_{\mathrm{sh}}=h/R_{\mathrm{v}}\):

\[ \dot \Phi_{\mathrm{sh}}=\mu_0 R_{\mathrm{v}} I\, \frac{E(\xi_{\mathrm{sh}})}{(1-\xi_{\mathrm{sh}}^2)^2}\, \frac{a^3}{R_{\mathrm{v}}^3}\dot{\tau}. \tag{36} \]

The change in the magnetic flux linked with the loop, caused by introducing the sphere, is determined by the relation

\[ \frac{\dot{\Phi}_{\mathrm{ш}}}{\dot{\Phi}_{0}} = \frac{E(\xi_{\mathrm{ш}})} {(1-\xi_{\mathrm{ш}}^{2})^{2}\,[K(\xi_{\mathrm{в}})-2E(\xi_{\mathrm{в}})]} \frac{a^{3}}{R_{\mathrm{в}}^{3}}\,\tau \tag{37} \]

The changes in the impedance \(\dot{Z}\) and the emf of self-induction \(\dot{E}\) of the loop are related to the changes in magnetic flux by

\[ \frac{\Delta \dot{Z}}{\dot{Z}_{0}} = \frac{\Delta \dot{E}}{\dot{E}_{0}} = \frac{\dot{\Phi}_{\mathrm{ш}}}{\dot{\Phi}_{0}} \tag{38} \]

and, consequently, are described by expression (37).

In (37) the factor

\[ \kappa_{\xi} = \frac{E(\xi_{\mathrm{ш}})}{(1-\xi_{\mathrm{ш}}^{2})^{2}} \tag{39} \]

determines the influence of the coordinate of the sphere; the factor

\[ \kappa_{a} = \frac{a^{3}}{R_{\mathrm{в}}^{3}}\, \frac{1}{K(\xi_{\mathrm{в}})-2E(\xi_{\mathrm{в}})} \tag{40} \]

determines the influence of the dimensions of the sphere; and the factor \(\tau\) determines the influence of the electromagnetic parameters of the sphere and of the frequency of the exciting field on the impedance and the emf of self-induction of the loop.

In Fig. 2 the dependence of \(\kappa_{\xi}\) on the normalized coordinate of the sphere \(\xi_{\mathrm{ш}}=h/R_{\mathrm{в}}\) is presented graphically. It is seen from the graph that, when the sphere is moved from the center of the loop to the edge, the perturbation introduced by it increases by two orders of magnitude.

Fig. 2

The ratio \(1/[K(\xi_{\mathrm{в}})-2E(\xi_{\mathrm{в}})]\simeq 1\), therefore \(\kappa_a \simeq a^{3}/R_{\mathrm{в}}^{3}\). This means that the change in the emf and in the impedance of the loop caused by the sphere is proportional to the cube of the ratio of their radii.

At low frequency, for \(ka \to 0\), the Bessel functions of half-integer order are

\[ Y_{1/2}(ka) \simeq \sqrt{\frac{2}{\pi ka}} \left[ ka-\frac{(ka)^{3}}{6} \right], \]

\[ Y_{3/2}(ka) \simeq \sqrt{\frac{2}{\pi ka}}\, \frac{1}{3}(ka)^{2}, \]

as a result of which \(\tau\) tends to 2.

For \(ka>0\), the coefficient \(\tau\) is complex and, in modulus, less than two.

REFERENCES

1. M. A. Divelkovskii. The problem of a sphere placed in a uniform alternating magnetic field. ZhTF, vol. IX, no. 5, 1939.

2. M. N. Bobrova. The complex magnetic moment of the inhomogeneity of a sphere in a uniform alternating magnetic field. Proceedings of SFTI, no. 72, 1972.

3. A. A. Fomin. A sphere in the field of a current loop. ZhTF, vol. XXXIII, no. 9, 1963.

4. A. N. Nikitin and V. S. Sobolev. On the action of a conducting sphere on a coil carrying current. Defektoskopiya, no. 2, 1969.

5. V. S. Sobolev, Yu. M. Shkarlet. Surface and screen transducers. Novosibirsk, “Nauka,” 1967.

6. V. A. Govorkov. Electric and Magnetic Fields. Moscow, “Energiya,” 1968.