PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Volume 155                                                                                                                1968

SLOTTED INDUCTIVE SENSOR

V. K. ZHUKOV, V. E. DREIZIN, I. G. LESHCHENKO

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

One of the methods used for contactless measurement of the geometric dimensions of cylindrical products (wire, pipes) is the eddy-current method. In such cases, the sensitive element of the instruments is a high-frequency inductive sensor with a through coil [1], which, alongside its advantages, has a number of drawbacks. First of all, this is the dependence of the sensor parameters on the temperature of the inspected product, if it differs from the ambient temperature. In addition, when the dimensions of the inspected product are small, the inconvenience of inserting the product into the sensor begins to be felt, and the technological difficulties of manufacturing the latter increase sharply. These drawbacks can be eliminated if the inspected product is moved away from the sensor winding, which can be achieved by using a magnetic core that makes it possible to carry the magnetic flux, in the form of a concentrated beam, beyond the coil cavity.

In this case we arrive at a slotted inductive sensor, the design of which is shown in Fig. 1. The winding 2 of the sensor is wound on an annular magnetic core 1, which may have either one slot (Fig. 1,a) or two slots (Fig. 1,b). The inspected product 3 is placed in the slot. That part of the magnetic flux created by the winding current which closes through the magnetic core induces eddy currents in the product (in the region located between the poles), the magnitude of which depends on the geometric dimensions and electrical conductivity of the product. The magnetic flux of the eddy currents is directed opposite to the flux that produced them; therefore the resulting magnetic flux penetrating the winding will change. Consequently, from the change in the magnetic flux of the sensor one can judge the parameters of the specimen. In the single-slot sensor design, compared with the two-slot design, a larger part of the magnetic flux created by the winding closes through the magnetic core. Consequently, all other conditions being equal, the sensitivity of the single-slot design will be higher than that of the two-slot design. On the other hand, for a single-slot sensor it is undesirable for the ends of the product to be closed, since the short-circuited turn thereby formed will significantly affect the magnitude of the magnetic flux of the magnetic core, thus leading to measurement error. In the two-slot design this influence will be completely absent in the case when the position of the product relative to the magnetic core is the same in both slots.

Therefore preference should be given to the two-slot version of the sensor design.

Depending on the number of windings, the sensor may be single-winding or two-winding (by analogy with a through-type sensor). In a single-winding sensor, changes in the parameters of the article can be judged from the value of the total impedance of the winding, while in a two-winding sensor they are judged from the value of the emf induced in the measuring winding.

To make effective use of the sensor’s capabilities, let us find the analytical relationship connecting the values of the active and reactive added winding resistances with the parameters of a cylindrical specimen of circular cross section, for a single-slot sensor design.

Fig. 1

The expression obtained will also be valid for the two-slot design. In doing so, we assume:

1. The magnetic flux produced by the winding closes entirely through the magnetic core and, consequently, penetrates the slot.
2. The magnetic flux is uniformly distributed over the entire cross section of the magnetic core.
3. In the slot, the lines of magnetic induction (in the absence of the article) are parallel to the axis of the magnetic core, i.e., outside the slot the field is absent.
4. The dimension \(c\) of the magnetic core is considerably larger than the dimension \(h\); therefore it may be assumed that in the article the field does not depend on the coordinate \(z\) (Fig. 2).

The assumptions made do not affect the qualitative aspect of the conclusions obtained below, but lead only to some quantitative inaccuracy.

The vector potential \(\vec A''\) of the resultant field in the slot is determined by the sum of the vector potential \(\vec A_0\) of the exciting field and the vector potential \(\vec A_1''\) of the eddy currents induced in the article.

\[ \vec A'' = \vec A_0 + \vec A_1''. \tag{1} \]

Similarly, the vector potential \(\bar{A}'\) of the resultant field in the article will be determined as

\[ \bar{A}'=\bar{A}_0+\bar{A}_1'. \tag{2} \]

To determine the magnetic field in the slot, we shall find a joint solution of the wave equations for the vector potential in the article and in the slot.

In general form, the wave equation for the vector potential is

\[ \nabla^2 \bar{A}+\kappa^2 \bar{A}=0. \tag{3} \]

For the conducting medium of the article,

\[ \kappa^2=-j\omega\cdot\gamma\cdot\mu, \tag{4} \]

and for the slot,

\[ \kappa_0^2=\omega^2\cdot\varepsilon_0\cdot\mu_0. \tag{5} \]

Here \(\omega\) is the angular frequency of the current feeding the sensor; \(\gamma\) is the conductivity of the article material; \(\mu\) is the magnetic permeability of the article material; \(\mu_0\) is the magnetic permeability of free space and \(\varepsilon_0\) is the dielectric permittivity of free space. The displacement currents in the slot may be neglected, since they are considerably smaller than the conduction currents of the article and therefore will have little effect on the magnitude of \(\bar{A}''\). For this reason we shall take \(\kappa_0^2=0\).

Fig. 2

The vector \(\bar{A}'\) (as also the vector \(\bar{A}''\)) has only one component \(A_z\), which we shall denote by \(\dot{A}'\); therefore the wave equation in cylindrical coordinates \(r,\varphi,z\) for the conducting medium will have the form

\[ \frac{\partial^2 \dot{A}'}{\partial r^2} +\frac{1}{r}\frac{\partial \dot{A}'}{\partial r} +\frac{1}{r^2}\frac{\partial \dot{A}'}{\partial \varphi^2} +\kappa^2\dot{A}'=0. \tag{6} \]

The solution of equation (6) is found by the Fourier method. Let us represent the general integral of this equation in the form

\[ \dot{A}'=\dot{R}(r)\cdot\dot{\Phi}(\varphi) \tag{7} \]

and substitute (7) into (6). After substitution, equation (6) decomposes into two independent equations:

\[ \frac{\partial^2 \dot{R}(r)}{\partial r^2} +\frac{1}{r}\frac{\partial \dot{R}(r)}{\partial r} +\left(\kappa^2-\frac{n^2}{r^2}\right)\dot{R}(r)=0, \tag{8} \]

\[ \frac{\partial^2 \dot{\Phi}(\varphi)}{\partial \varphi^2} +n^2\dot{\Phi}(\varphi)=0. \tag{9} \]

The general solution of equation (9) is

\[ \dot{\Phi}(\varphi)=(\dot{C}'_{\varphi})_n\cdot\cos n\varphi +(\dot{C}''_{\varphi})_n\cdot\sin n\varphi. \]

Since \(\dot A'\) is an odd function of \(\varphi\) and its period is \(2\pi\), one should set \((\dot C'_{\varphi})_n=0\) and \(n=1\), after which

\[ \dot \Phi(\varphi)=\dot C''_{\varphi}\cdot \sin \varphi . \tag{10} \]

The general solution of equation (8) will be Bessel functions of the first and second kind of order \(n\) with argument \(\kappa r\):

\[ \dot R(r)=(\dot C'_r)_n\cdot I_n(\kappa r)+(\dot C''_r)_n\cdot Y_n(\kappa r). \tag{11} \]

There is no reflected wave in the cylinder; therefore \((\dot C''_2)=0\), and \(n=1\) for the reason noted above. Taking this into account,

\[ \dot R(r)=\dot C'_r I_1(\kappa r). \tag{12} \]

Substituting (12) and (10) into (7), we find the general solution of the wave equation (6):

\[ \dot A'=\dot C\cdot I_1(\kappa r)\cdot \sin \varphi . \tag{13} \]

The integration constant \(\dot C=\dot C'_r\cdot \dot C''_{\varphi}\) is to be determined from the boundary conditions.

The wave equation of the vector potential \(\dot A''_1\) of the eddy-current field in the slot, taking into account \(\kappa_0^2=0\), will be

\[ \frac{\partial^2 \dot A''_1}{\partial r^2} +\frac{1}{r}\frac{\partial \dot A''_1}{\partial r} +\frac{1}{r^2}\frac{\partial^2 \dot A''_1}{\partial \varphi^2}=0. \]

After substituting \(\dot A''_1=\dot R(r)\cdot \dot \Phi(\varphi)\), it splits into two equations:

\[ \frac{\partial^2 \dot R(r)}{\partial r^2} +\frac{1}{r}\frac{\partial \dot R(r)}{\partial r} +\frac{n^2}{r^2}\dot R(r)=0, \]

\[ \frac{\partial^2 \dot \Phi(\varphi)}{\partial \varphi^2} +n^2\dot \Phi(\varphi)=0. \tag{16} \]

The solution of the latter equation will be

\[ \dot \Phi(\varphi)=(\dot N'_{\varphi})_n\cdot \cos n\cdot \varphi +(\dot N''_{\varphi})_n\cdot \sin n\cdot \varphi . \]

Since \(\dot A''_1\) is an odd function of \(\varphi\) and its period is \(2\pi\), \((\dot N'_{\varphi})_n=0\) and \(n=1\), and

\[ \dot \Phi(\varphi)=\dot N''_{\varphi}\cdot \sin \varphi . \tag{17} \]

Equation (15) is nothing other than Euler’s equation, the general solution of which has the form

\[ \dot R(r)=(\dot N'_r)_n\cdot r^n+(\dot N''_r)_n\cdot r^{-n}. \]

This solution must satisfy the condition at infinity: \(\dot R(r)=0\) as \(r\to\infty\), which will hold when \((\dot N'_r)_n=0\). Taking into account that \(n=1\),

\[ \dot R(r)=\dot N''_r\cdot r^{-1}. \tag{18} \]

Thus the solution of the wave equation (14) will be

\[ \dot A''_1=\dot N\cdot r^{-1}\cdot \sin \varphi . \tag{19} \]

We find the vector potential \(\dot A''\) of the resultant field in the slot by substituting (18) into (1).

\[ \dot A''=\dot A_0+\dot N\cdot r^{-1}\cdot \sin \varphi . \tag{20} \]

We find the constants $\dot C$ and $\dot N$ from the boundary conditions. For $r=r_0$, i.e., on the surface of the cylinder, the conditions must be satisfied

\[ \dot A'=\dot A'', \tag{21} \]

\[ \frac{\partial \dot A'}{\partial r}=\frac{\partial \dot A''}{\partial r}. \tag{22} \]

The boundary conditions (21) and (23) give the following equations for finding the constants of integration:

\[ \dot C\cdot I_1(kr_0)=\dot N\cdot r_0^{-1}+\dot B_0\cdot r_0 \tag{23} \]

and

\[ \dot C\cdot k\cdot I_1'(kr_0)=-\dot N r_0^{-2}+\dot B_0, \tag{24} \]

in deriving which the vector potential of the exciting field $\dot A_0$ was expressed in terms of the induction of the exciting field $\dot B_0$ in the gap, according to expression (25)

\[ \dot A_0=\dot B_0\cdot r\cdot \sin\varphi . \tag{25} \]

Solving the obtained equations with respect to $\dot N$, we find that

\[ \dot N=\dot B_0\cdot r_0^2 \frac{I_1(kr_0)-kr_0 I_1'(kr_0)} {I_1(kr_0)+kr_0 I_1'(kr_0)} . \]

After substituting

\[ I_1'(kr_0)=I_0(kr_0)-\frac{I_1(kr_0)}{kr_0} \]

and replacing the unmodified Bessel functions by modified ones, the expression for $\dot N$ is transformed into the form:

\[ \dot N=\dot B_0\cdot r_0^2 \left[ \frac{2I_1(\sqrt{j}\cdot m)} {\sqrt{j}\cdot m\cdot I_0(\sqrt{j}m)} -1 \right], \]

where

\[ m=r_0\sqrt{\omega\cdot\gamma\cdot\mu}. \]

Let us denote the contents of the square bracket by $x$; then

\[ \dot N=\dot B_0\cdot r_0^2\cdot x . \tag{26} \]

Now we can write the final expression for the vector potential of the resultant field in the slot:

\[ \dot A''=\dot B_0\cdot r\cdot \sin\varphi \left(1+\frac{r_0^2}{r^2}x\right). \tag{27} \]

Knowing the vector potential of the field, it is not difficult to find the vector of magnetic induction of the resultant field. For this we use the expression

\[ \bar B''=\operatorname{rot}\bar A'' . \]

Since the vector $\bar A''$ has only one component $\dot A_z''=\dot A''$, determined by expression (27), then

\[ \bar B'=\bar r\,\frac{1}{r}\frac{\partial \dot A''}{\partial\varphi} -\bar\varphi\,\frac{\partial \dot A''}{\partial r}, \tag{28} \]

where $\bar r$ and $\bar\varphi$ are unit vectors.

From (28) we find the radial \(\dot B_r''\) and tangential \(\dot B_\varphi''\) components of the magnetic induction vector:

\[ \dot B_r''=\frac{1}{r}\frac{\partial \dot A''}{\partial \varphi} \quad \text{and} \quad \dot B_\varphi''=\frac{\partial \dot A''}{\partial r}, \]

and after substituting \(\dot A''\) from (27),

\[ \dot B_r''=\dot B_0\left(1+\frac{r_0^2}{r^2}\dot \chi\right)\cdot \cos \varphi, \tag{29} \]

\[ \dot B_\varphi''=\dot B_0\left(1-\frac{r_0^2}{r^2}\dot \chi\right)\sin \varphi. \tag{30} \]

We assume that the magnetic flux in the magnetic core is produced by the radial component \(\dot B_r''\) of the magnetic induction vector in the slot, i.e.

\[ \dot \Phi=\int_S \dot B_r''\, dS. \]

Representing the elementary area \(dS\) as \(c\cdot r_n\cdot d\varphi\) and substituting \(\dot B_r''\), determined by expression (30), we obtain:

\[ \dot \Phi=\int_{-\varphi_0}^{\varphi_0} \dot B_0\cdot c\cdot r_n\cdot \cos \varphi \left(1+\frac{r_0^2}{r^2}\dot \chi\right)d\varphi. \]

After taking the integral and making transformations, we find:

\[ \dot \Phi=\dot B_0\cdot c\cdot h\cdot \left(1+\frac{r_0^2}{r_h^2}\dot \chi\right). \tag{31} \]

The induction \(\dot B_0\) of the exciting field may be determined from the law of total current:

\[ \dot B_0=\frac{\mu_0\cdot \mu_m\cdot \dot I\cdot w}{l\cdot \xi}, \]

where \(\mu_m\) is the relative magnetic permeability of the magnetic-core material;

\(\dot I\) is the current in the sensor winding;

\(w\) is the number of turns of the winding; \(l\) is the slot width; \(\xi=1+\dfrac{l_m}{l\cdot \mu_m}\), and

\(l_m\) is the length of the mean line of the magnetic core.

Using the known relation

\[ \dot E=-w\frac{\partial \dot \Phi}{\partial t} =-j\omega\cdot w\cdot \dot \Phi \tag{32} \]

we find the self-induction emf \(\dot E\) in the sensor winding:

\[ \dot E=j\frac{\omega\cdot w^2\cdot c\cdot h\cdot \mu_0\cdot \mu_m}{l\cdot \xi} \left(1+\frac{r_0^2}{r_h^2}\dot \chi\right). \]

The power of losses in the sensor winding may be represented as

\[ \dot P=\dot E\cdot \dot I \tag{33} \]

or

\[ \dot P=\dot I^2\cdot \dot Z. \tag{34} \]

Substituting into (33) the value of \(E\) from (32) and equating the right-hand sides of (33) and (34), we find the total impedance \(\dot{Z}\) of the sensor

\[ \dot{Z}=j\,\frac{\omega w^{2} c h \mu_{0}\mu_{m}}{l\xi}\left(1+\frac{r_{0}^{2}}{r_{h}^{2}}\chi\right). \tag{35} \]

In the absence of a product in the slot, \(r_{0}=0\), and the impedance of the sensor becomes equal to

\[ \dot{Z}_{0}=j\,\frac{\omega w^{2} c h \mu_{0}\mu_{m}}{l\xi}. \tag{36} \]

The introduced impedance \(\Delta \dot{Z}=\dot{Z}-\dot{Z}_{0}\) is readily determined from (35) and (36). In order to make \(\Delta \dot{Z}\) independent of the coil parameters, we normalize it by dividing by \(\dot{Z}_{0}\):

\[ \frac{\Delta \dot{Z}}{\dot{Z}_{0}}=1+\frac{r_{0}^{2}}{r_{h}^{2}}\chi. \tag{37} \]

Equation (37) is the desired equation for determining the introduced reactive \(\Delta X\) and active \(\Delta R\) impedances. The real component of the right-hand side of (37) determines the introduced reactive resistance \(\dfrac{\Delta X}{X_{0}}\) (normalized), and the imaginary component determines the active \(\dfrac{\Delta R}{X_{0}}\).

Fig. 3

In Fig. 3 the dependence (37) is presented graphically. The indicated graph was constructed under the assumption that the relative magnetic permeability \(\mu_{m}=1\); therefore it applies only to non-ferromagnetic products. From consideration of the graph it is seen that it coincides with the analogous graph for a sensor with a through coil [3], only here the role of the filling factor is played by the ratio \(r_{0}^{2}/r_{h}^{2}\). Hence it follows that all conclusions concerning the influence of the product parameters and the current frequency on the impedance of a sensor with a through coil will also be valid for the slotted sensor.

LITERATURE

1. B. Z. Mikhlin. High-Frequency Capacitive and Inductive Sensors, Gosenergoizdat, 1960.

2. N. M. Shilov. Dissertation for the degree of Candidate of Physical and Mathematical Sciences, Tomsk, 1939.

3. V. P. Grabovetskii. Contactless method for measuring resistivity and geometrical dimensions by means of eddy currents. Avtomatika i telemekhanika, vol. XX, No. 7, 1959.