Ferromagnetic Cylinder in a Slotted Eddy-Current Transducer

V. K. Zhukov, R. M. Zakirov

(Presented by the scientific and technical seminar of the Department of Information and Measurement Technology)

The article considers the response of the measuring winding of a slotted eddy-current transducer when a round ferromagnetic cylinder is introduced into the gap of the latter. In solving this problem, the following assumptions are made: the field in the gap is considered uniform; the influence of the edge of the transducer along the axis of the cylinder is not taken into account; the magnetic permeability of the cylinder is constant and does not depend on the magnetic-field strength. Let us proceed to the analytical determination of the vector potential of the magnetic field in the air space surrounding the cylinder, which is located in a transverse alternating magnetic field, taking into account the above assumptions (Fig. 1). The homogeneous Helmholtz equation for the vector potential \(\vec A\) of an alternating magnetic field has the form

Fig. 1. Cylinder in a transverse magnetic field

\[ \nabla^2 \vec A + k^2 \vec A = 0, \tag{1} \]

where \(k^2 = \omega^2 \mu_a \varepsilon;\ \omega = 2\pi f\) is the angular frequency of the magnetic field; \(\mu_a = \mu \cdot \mu_0\) is the absolute magnetic permeability of the medium; \(\mu\) is the relative magnetic permeability of the medium; \(\mu_0 = 4\pi \cdot 10^{-7}\ \text{H/m}\) is the magnetic constant; \(\varepsilon = \varepsilon_a - i\delta/\omega\) is the complex dielectric permittivity of the medium; \(\varepsilon_a = \varepsilon \cdot \varepsilon_0\) is the absolute dielectric permittivity of the medium; \(\varepsilon\) is the relative dielectric permittivity of the medium; \(\varepsilon_0 = 8.85 \cdot 10^{-12}\ \text{F/m}\) is the electric constant; \(\delta\) is the specific conductivity, \(\text{S/m}\).

We shall solve equation (1) in a cylindrical coordinate system, directing the \(z\)-axis along the axis of the cylinder. Under the assumptions made above, the vector \(\vec A\) has only one component, \(\vec A = \vec i_z A\), which does not depend on the coordinate \(z\). We denote the vector potential in the conducting medium of the ferromagnetic cylinder by \(A_c\), and in the air—

space—through $\dot A_b$. Since in the medium of the cylinder $\omega \mu_a \delta \gg \omega^2 \mu_a \varepsilon_a$, one may take $k^2=k_c^2=-i\omega\mu_a\delta$, and, correspondingly, in the air space $\omega\mu_0\delta \ll \omega^2\mu_0\varepsilon_0$, while $k^2=k_b^2=\omega^2\mu_0\varepsilon_0$. If the influence of displacement currents on the nature of the electromagnetic-field distribution is neglected, one may take $k_b=0$.

Expanding in (1) the Laplace operator $\nabla^2$ in a cylindrical coordinate system, we obtain two equations, respectively for the cylinder medium and for the air medium,

\[ \frac{\partial^2 \dot A_c}{\partial r^2} +\frac{1}{r}\frac{\partial \dot A_c}{\partial r} +\frac{1}{r^2}\cdot\frac{\partial^2 \dot A_c}{\partial \varphi^2} +k_c^2 \dot A_c=0, \tag{2} \]

\[ \frac{\partial^2 \dot A_b}{\partial r^2} +\frac{1}{r}\frac{\partial \dot A_b}{\partial r} +\frac{1}{r^2}\cdot\frac{\partial^2 \dot A_b}{\partial \varphi^2}=0. \tag{3} \]

Both equations are solved by the method of separation of variables. Taking into account the physical picture of the field, the solution of equation (2) will have the form

\[ \dot A_c=\sum_{m=1}^{\infty}\dot C_m J_m(k_c r)\sin m\varphi, \tag{4} \]

where $\dot C_m$ is a constant of integration; $J_m(k_c r_0)$ is the Bessel function of the first kind of order $m$.

The vector potential in the air space surrounding the cylinder is the sum of the vector potential of the exciting field $\dot A_0$ and the vector potential $\dot A_{b_1}$ of the eddy-current field in the cylinder,

\[ \dot A_b=\dot A_0+\dot A_{b_1}. \tag{5} \]

Substituting (5) into (3), we obtain

\[ \frac{\partial^2 \dot A_{b_1}}{\partial r^2} +\frac{1}{r}\frac{\partial \dot A_{b_1}}{\partial r} +\frac{1}{r^2}\frac{\partial^2 \dot A_{b_1}}{\partial \varphi^2}=0. \tag{6} \]

The solution of equation (6) must satisfy the following requirements: it must be odd with respect to the angle $\varphi$ and tend to zero as $r\to\infty$. The following function satisfies this:

\[ \dot A_{b_1}=\sum_{m=1}^{\infty}\dot N_m r^{-m}\sin m\varphi, \tag{7} \]

where $\dot N_m$ is a constant of integration.

Since the vector $\dot A$ is determined by the equation $\dot B=\operatorname{rot}\dot A$ to within the gradient of an arbitrary scalar function, the latter may be taken equal to zero.

Let us express $\dot A_0$ through the induction $\dot B_0$:

\[ \dot A_0=\dot B_0 r\sin\varphi. \tag{8} \]

Substituting (7) and (8) into (5), we obtain an expression for the vector potential in air,

\[ \dot A_b=\dot B_0 r\sin\varphi+\sum_{m=1}^{\infty}\dot N_m r^{-m}\sin m\varphi. \tag{9} \]

The constants $\dot C_m$ in (4) and $\dot N_m$ in (9) are determined from the boundary conditions for the magnetic field, which in the given coordinate system have the form

\[ \dot A_c=\dot A_b;\qquad \frac{1}{\mu_a}\frac{\partial \dot A_c}{\partial r} = \frac{1}{\mu_0}\frac{\partial \dot A_b}{\partial r}, \quad \text{at } r=r_0, \tag{10} \]

where \(r_0\) is the radius of the cylinder.

Substituting (4) and (9) into equalities (10), we obtain

\[ \sum_{m=1}^{\infty} \dot C_m J_m(k_c r_0)\sin m\varphi = \dot B_0 r_0 \sin\varphi + \sum_{m=1}^{\infty} \dot N_m r_0^{-m}\sin m\varphi, \tag{11} \]

\[ \frac{1}{2\mu_a}\sum_{m=1}^{\infty}\dot C_m \left[J_{m-1}(k_c r_0)-J_{m+1}(k_c r_0)\right]\sin m\varphi = \]

\[ = \frac{1}{\mu_0}\dot B_0\sin\varphi + \frac{1}{\mu_0}\sum_{m=1}^{\infty}(-m)\dot N_m r_0^{-(m+1)}\sin m\varphi. \tag{12} \]

Transforming the left-hand side of equality (11) in accordance with the recurrence relation [1]:

\[ J_\nu(x)=\frac{x}{2\nu}\left[J_{\nu-1}(x)+J_{\nu+1}(x)\right], \tag{13} \]

and equating the factors at the sines with the same argument separately in equalities (11) and (12), we arrive at the conclusion that \(\dot C_m=\dot N_m=0\) for \(m>1\), and

\[ \dot C_1= \frac{4\mu_a\dot B_0} {k_c\left[(\mu_a+\mu_0)J_0(k_c r_0)+(\mu_a-\mu_0)J_2(k_c r_0)\right]}, \tag{14} \]

\[ \dot N_1 = r_0^2\dot B_0 \frac{(\mu_a-\mu_0)J_0(k_c r_0)+(\mu_a+\mu_0)J_2(k_c r_0)} {(\mu_a+\mu_0)J_0(k_c r_0)+(\mu_a-\mu_0)J_2(k_c r_0)}. \tag{15} \]

Divide the numerator and denominator by the constant \(\mu_0\) and transform (15), using (13). Then

\[ \dot N_1 = r_0^2\dot B_0 \left[ \frac{2\mu J_1(k_c r_0)} {(\mu-1)J_1(k_c r_0)+k_c r_0J_0(k_c r_0)} -1 \right]. \tag{16} \]

Thus, expression (9) for the vector potential in the air space surrounding the cylinder will take the form

\[ \dot A_b = \dot B_0 r\sin\varphi \left[ 1+\frac{r_0^2}{r^2}\dot F(k_c r_0) \right], \tag{17} \]

where

\[ \dot F(k_c r_0) = \frac{2\mu J_1(k_c r_0)} {(\mu-1)J_1(k_c r_0)+k_c r_0J_0(k_c r_0)} -1. \tag{18} \]

Let us determine the resultant magnetic flux linking a rectangular measuring winding, which has the following geometrical parameters (Fig. 2): the sides \(DC\) and \(MN\) are equal to \(2l\) and are parallel to the \(z\)-axis; the sides \(DN\) and \(CM\) are equal to \(2c\) and are parallel to the \(y\)-axis.

\[ \dot\Phi = \int_S \dot{\vec B}\,d\vec S = \int_S \operatorname{rot}\dot{\vec A}_b\,d\vec S = \oint_L \dot{\vec A}\,d\vec l, \tag{19} \]

where \(L\) is the contour of the measuring winding.

The vector \(\dot{\vec A}_b\) is parallel to \(DC\) and \(MN\) and perpendicular to \(DN\) and \(CM\). In addition, it does not depend on the coordinate \(z\). Substituting (17) into (19), we obtain

\[ \dot\Phi = 4\dot B_0lr\sin\varphi \left[ 1+\frac{r_0^2}{r^2}\dot F(k_c r_0) \right]. \tag{20} \]

Here, by \(r\) and \(\varphi\) are meant the coordinates of the sides of the winding parallel to the \(z\)-axis. Therefore \(r\cdot\sin\varphi=c\), and the flux is

\[ \dot{\Phi}=4B_{0}lc\left[1+\frac{r_{0}^{2}}{r^{2}}\dot{F}(k_{c}r_{0})\right] =\dot{\Phi}_{0}+\Delta\dot{\Phi}, \tag{21} \]

where \(\dot{\Phi}_{0}=\dot{B}_{0}\cdot 2l\cdot 2c=\dot{B}_{0}S\) is the magnetic flux of the external exciting field penetrating the winding; \(\Delta\dot{\Phi}=\dot{\Phi}_{0}\dfrac{r_{0}^{2}}{r^{2}}\dot{F}(k_{c}r_{0})\) is the magnetic flux produced by eddy currents in the cylinder.

Fig. 2

In the expression for the total magnetic flux of a solenoid with a round product, a fill factor is introduced, equal to the square of the ratio of the product radius to the solenoid radius [2]. By analogy, the concept of a fill factor can also be introduced in the case under consideration; it should be understood as the ratio

\[ \eta=\frac{r_{0}^{2}}{r^{2}}. \tag{22} \]

The emf induced in the measuring winding by the flux \(\Phi_{0}\) is equal to

\[ \dot{E}_{0}=-w\frac{d\dot{\Phi}_{0}}{dt}=-i\omega w\dot{\Phi}_{0}, \tag{23} \]

where \(w\) is the number of turns of the winding.

Accordingly,

\[ \Delta\dot{E}=-i\omega w\Delta\dot{\Phi} =-i\omega w\dot{\Phi}_{0}\eta\dot{F}(k_{c}r_{0}). \tag{24} \]

For convenience of analysis, let us represent the emf increment due to eddy currents in the cylinder in normalized form:

\[ \frac{\Delta\dot{E}}{-|\dot{E}_{0}|} =i\eta\dot{F}(k_{c}r_{0}). \tag{25} \]

Fig. 3. Change in the effective permeability \(\mu_{\mathrm{eff}}\) as a function of the generalized parameter \(m\) and the relative magnetic permeability \(\mu\).

In the theory of eddy-current transducers, in order to estimate the influence of the specimen on the nature of the field distribution, the concept of the effective magnetic permeability of the specimen \(\dot{\mu}_{\mathrm{eff}}\), which is a complex quantity, is introduced [2]. Let us introduce the concept of effective magnetic permeability into expression (25)

\[ \frac{\Delta \dot{E}}{-|\dot{E}_0|}=i\eta(\dot{\mu}_{\mathrm{eff}}-1), \tag{26} \]

where \(\dot{\mu}_{\mathrm{eff}}-1=\dot{F}(\kappa_c r_0)\),

\[ \dot{\mu}_{\mathrm{eff}}= \frac{2\mu J_1(\kappa_c r_0)} {(\mu-1)J_1(\kappa_c r_0)+\kappa_c r_0 J_0(\kappa_c r_0)}. \tag{27} \]

The effective permeability (27) is the principal characteristic in testing by the eddy-current method. It is determined by the Bessel functions \(J_1(\kappa_c r_0)\) and \(J_0(\kappa_c r_0)\) of the complex argument \(\kappa_c r_0=i^{3/2}r_0\sqrt{\omega\mu_a\sigma}\). The quantity \(kr_0=r_0\sqrt{\omega\mu_a\sigma}=m\) is called the generalized parameter and reflects the basic electrophysical properties of the cylinder.

Expressions (26) and (27) for various \(m\) were analyzed with the aid of the “Nairi-2” electronic digital computer. On the basis of the calculation results, curves were constructed in the complex plane of the effective magnetic permeability (Fig. 3) and in the complex plane of the relative increment of the emf (Fig. 4).

It follows from expression (27) and Fig. 3 that, in the case of a nonmagnetic cylinder \((\mu=1)\), the hodograph of the effective magnetic permeability for the slotted transducer is analogous to the hodograph of \(\mu_{\mathrm{eff}}\) for a through-type transducer. However, in the case of a ferromagnetic cylinder \((\mu>1)\), the hodograph for the slotted transducer differs significantly from the hodograph for the through-type transducer. If for the through-type transducer, as \(m\to0\), \(\mu_{\mathrm{eff}}\to\mu\), then for the slotted transducer, at large \(\mu\), in the analogous case \(\mu_{\mathrm{eff}}\to2\).

Fig. 4. Complex plane of the relative increment of the emf for various relative magnetic permeabilities

In order to confirm the results of the theoretical investigations, experimental studies were carried out with a slotted transducer (Fig. 2). The dimensions of the poles of the magnetic core were \(120\times60\) mm. Four measuring windings of 20 turns each were glued to the ends of the poles. The geometrical dimensions, respectively, were: I — \(30\times20\) mm; II — \(50\times30\) mm; III — \(70\times40\) mm; IV — \(90\times50\) mm. Here the per-

the first digit is the dimension along the \(z\) axis, the second along the \(y\) axis. A copper cylinder with a diameter of 13.9 mm and electrical conductivity \(47.5 \cdot 10^6 \dfrac{\text{S}\cdot\text{m}}{\text{m}}\) was used as the test specimen. From the experimental results, hodographs of the normalized reduced emf were constructed (Fig. 4), where the solid lines are the result of the analytical calculation and the dashed lines are the result of the experimental data. Analysis of these curves makes it possible to conclude that, qualitatively, the experimental data confirm the results of the analytical calculation; quantitatively, however, a significant discrepancy is observed, and the greater the height of the measuring winding, the greater it is. This is explained by the fact that, in the analytical derivation, the influence of the poles of the magnetic core on the nature of the distribution of the eddy-current field created in the cylinder was not taken into account.

REFERENCES

1. André Angot. Mathematics for Electrical and Radio Engineers. Moscow, “Nauka,” 1965.

2. Nondestructive Testing, ed. by R. McMaster. Book II, Moscow, “Energiya,” 1965.