ACCOUNTING FOR THE INFLUENCE OF THE MAGNETIC-CORE POLES 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)

In a slotted eddy-current transducer, significant discrepancies are observed between theory and experiment [1]. One of the main causes of the discrepancy is the distortion of the eddy-current field of the cylinder caused by the proximity of the magnetic-core poles. To verify this assertion, the following experiment was performed. The cylinder, together with the measuring windings, was placed in a homogeneous transverse field produced by Helmholtz coils having radius \(R = 50\) cm. Two measuring windings, located on opposite sides of the cylinder, each had 50 turns. The geometrical dimensions of the latter were \(24 \times 24\) mm. A copper rod of diameter 13.9 mm with electrical conductivity \(47.5 \cdot 10^6\) S/m was used as the cylinder. With this arrangement of the experiment, the conducting cylinder was in a homogeneous transverse magnetic field created without the aid of a magnetic core.

The discrepancies between the values of the induced emf determined experimentally and those calculated from the analytical dependence [1]

\[ \frac{\Delta \dot{E}}{-|\dot{E}_0|} = i \eta \dot{F}(\kappa_c r_0), \]

did not exceed \(5 \div 7\%\) over the entire frequency range of the magnetizing current.

The experimental data confirm the assumption stated above concerning the influence of the magnetic-core poles on the sensitivity of the transducer.

Let us attempt to take into account the influence of the magnetic-core poles when deriving an analytical dependence between the parameters of the cylinder and those of the slotted transducer.

Owing to the proximity of the magnetic-core poles, the topography of the magnetic field created by the eddy currents of the cylinder changes both in the air gap and in the space occupied by the magnetic core, as compared with the field pattern without a magnetic core.

It is shown in [1] that the increment of the vector potential in the air space surrounding the cylinder, in a rectangular coordinate system, has the form

\[ \Delta \dot{A} = \dot{B}_0 y \frac{r_0^2}{x^2 + y^2} \dot{F}(\kappa_c r_0), \tag{1} \]

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

Having determined the components of the increment of the induction vector, \(\Delta \dot{B}_x=\dfrac{\partial \dot{A}}{\partial y}\) and \(\Delta \dot{B}_y=-\dfrac{\partial \dot{A}}{\partial x}\), we find its modulus

\[ \Delta \dot{B}=\dot{B}_0\frac{r_0^2}{x^2+y^2}\dot{F}(\varkappa_c r_0). \tag{2} \]

Taking into account the assumption stated above, the vector is distorted in the vicinity of the ferromagnet and enters normally into the poles of the magnetic core. Consequently, it fully participates in creating the magnetic flux penetrating the measuring windings near the poles of the magnetic core:

\[ \Delta \dot{\Phi} = \int_S \Delta \dot{B}\,dS = \int_{-c}^{c}\int_{-l}^{l} \dot{B}_0 r_0^2 \dot{F}(\varkappa_c r_0) \frac{dy\,dz}{x^2+y^2} = \]

\[ = \dot{B}_0 r_0^2 \dot{F}(\varkappa_c r_0)\cdot 2l\cdot \frac{2}{d}\arctg\frac{c}{d} = \dot{\Phi}_0\dot{F}(\varkappa_c r_0)\frac{r_0^2}{dc}\arctg\frac{c}{d}, \tag{3} \]

where \(\dot{\Phi}_0=\dot{B}_0\cdot 2l\cdot 2c\) is the magnetic flux of the primary field,
\(2l\) is the size of the winding along the \(z\) axis,
\(2c\) is the size of the winding along the \(y\) axis,
\(d\) is the distance from the axis of the specimen to the winding.

After simple transformations of (3), we obtain an expression for the relative increment of the emf of the measuring winding:

\[ \frac{\Delta \dot{E}}{-|\dot{E}_0|} = i\eta_1\dot{F}(\varkappa_c r_0), \tag{4} \]

where

\[ \eta_1=\frac{r_0^2}{dc}\arctg\frac{c}{d}. \]

Expression (4) was calculated for values of the quantities \(d\), \(c\), and \(r_0\) corresponding to the dimensions of the transducer manufactured for conducting the experiment.

Figure 1 (curve 2) shows the dependence of the ratio of the normalized introduced emfs, calculated from expression (4) and found experimentally, on the value of the normalized height of the measuring winding.

Fig. 1

The presented dependence shows that allowance for the influence of the poles, under the assumptions made, reduces the discrepancy between

theory and experiment. However, the difference still remains substantial. This is apparently explained by the fact that, in deriving expression (4), the distortion of the field in the transducer gap was not taken into account.

To take into account the distortion of the cylinder’s own field in the transducer gap, we shall use the method of images to derive the analytical dependence. The essence of the method consists in replacing the influence of the ferromagnetic boundary on the applied field of a linear current by a simple system of currents located behind the boundary surface. Usually, fields having either one boundary surface or several (no more than four) are amenable to analysis, but the angles must be integral multiples of $\pi$ [2]. Since a pole of the magnetic core has 3 boundary surfaces and the angle between the planes is $\frac{3}{2}\pi$, such a field cannot be analyzed by the method of images. Therefore we simplify the problem by replacing the surface of the pole end by an infinite surface (Fig. 2). The field in air, where a linear current $i$ flows, located at a distance $h$ from an infinite plane with relative magnetic permeability $\mu$, is composed of the fields of the current $i$ and of its mirror image $i(\mu-1)/(\mu+1)$. The field created by the eddy currents in the cylinder may be regarded as a superposition of the fields of linear currents. Therefore one may write

Fig. 2

\[ \Delta \dot A=\Delta \dot A_1+\Delta \dot A_2, \tag{5} \]

where $\Delta \dot A_1$ is the vector potential due to the eddy currents in the cylinder and determined from expression (1), and $\Delta \dot A_2$ is the vector potential due to the mirror images of the eddy currents. Here the quantity $(\mu-1)/(\mu+1)$ is taken equal to 1, which will lead to an insignificant error for $\mu \gg 1$.

The flux linking the measuring frame is

\[ \Delta \dot \Phi=\oint_L \Delta \dot A\,dl =\dot \Phi_0 r_0^2 \frac{2(c^2+d^2)+4(h-d)} {(c^2+d^2)\left[(c^2+d^2)+4h(h-d)\right]} \dot F(k_c r_0), \tag{6} \]

where $l$, $c$, and $d$ are the same as in (3).

The expression for the relative increment of the emf of the measuring winding is

\[ \frac{\Delta \dot E}{-|\dot E_0|} =i\eta_2 \dot F(k_c r_0), \tag{7} \]

where

\[ \eta_2=r_0^2 \frac{2(c^2+d^2)+4(h-d)} {(c^2+d^2)\left[(c^2+d^2)+4h(h-d)\right]}. \]

Expression (7) was calculated for a real transducer. Curve 3 (Fig. 1) shows the dependence of the ratio of the normalized introduced emfs, calculated from expression (7) and found experimentally, on the value of the normalized height of the measuring winding.

Comparison of the results of theory and experiment makes it possible to draw the following conclusions.

1. When the height of the measuring winding is considerably smaller than the height of the magnetic core, the discrepancies between the theoretical and experimental data do not exceed the experimental error.

1. With measuring windings whose height is close to the height of the magnetic core, the difference between the theoretical and experimental results reaches 23%. This is explained by the assumptions that were made in deriving expression (7), when the pole of the magnetic core was replaced by a ferromagnetic half-space.

Thus, taking into account the influence of the poles of the magnetic core on the topology of the magnetic field makes it possible to increase the accuracy of the analytical expression relating the parameters of the cylinder and the transducer.

REFERENCES

1. V. K. Zhukov, R. M. Zakirov, Ferromagnetic cylinder in a slotted eddy-current transducer. In the present collection.
2. K. Binns, P. Lawrenson. Analysis and Computation of Electric and Magnetic Fields. Moscow, “Energiya,” 1970.