THEORY OF A THROUGH-TYPE SENSOR APPLIED TO DIAMETER CONTROL OF THIN COPPER WIRE BY THE EDDY CURRENT METHOD

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

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

The rapid development of the electrical-engineering industry, electronics, radio engineering, and communications requires further development of the cable industry, both quantitatively and qualitatively. A bottleneck in the cable industry is the high-quality manufacture of thin and ultrathin copper wire. One of the quality indicators is the exact correspondence of the actual wire diameter to the nominal one. At present, the manufacturing tolerance for the diameter of wire with nominal sizes from 0.05 mm to 0.19 mm is \(-2 \div +3\) microns. This imposes high requirements on control-and-measuring equipment and on the accuracy of die manufacture. However, the present state of these problems is far from meeting these requirements. Let us begin at least with the process of manufacture and the methods for checking dies. For the manufacture of thin and ultrathin wire, pobedit and diamond dies are used, up to 18 dies per drawing route. Their service life ranges from two to five weeks. They then have to be reground to larger diameters. Thus, frequent checking of a large number of dies is required. Moreover, when selecting a route, even deviations of die diameter from nominal values by \(\pm 1\) micron lead to significant disturbances in the correctness of the technological drawing process, as a result of which the quality of the wire noticeably deteriorates and the number of breaks increases. However, up to the present time most dies are checked with an ordinary micrometer with a scale division of 10 microns and a factory error of \(\pm 4\) microns, by measuring a wire pulled through the given die. Only the most critical—final—dies are checked with lever micrometers and special photo-optimeters with a scale division of 1 micron, since these instruments require very careful and skillful handling, are easily thrown out of adjustment, and in general are of little use under shop conditions. In addition, all these instruments are mechanical, contact instruments and require considerable contact force—up to 300 g—as a result of which the measured wire may be crushed, and hence the readings may be incorrect.

Hence there arises a need to create a highly sensitive and preferably contactless instrument for the precise measurement of wire diameter in static conditions with an error not exceeding \(\pm 0.5\) microns, which would be

be simple to operate, reliable, and suitable for shop-floor conditions. This same instrument can be used for inspection of finished products in the quality-control department, since here there is the same discrepancy between the tolerances for diameter and the means of measuring it.

In addition, because of the variable service life of dies, it is desirable to monitor the diameter of the wire during the drawing process in order to prevent rejects. Hence there arises the problem of controlling wire diameter dynamically. The accuracy required here is almost the same, while the operating conditions are much worse. Since the drawing speed reaches 30 m/sec, all contact methods are certainly ruled out here. Of the noncontact methods, the most promising for measuring the diameter of fine and ultrafine wire are the eddy-current method and the photoelectric method.

In the present article, as well as in our papers [4, 5] included in the same collection, the principal questions associated with controlling the diameter of fine copper wire by the eddy-current method under static and dynamic conditions are considered.

The eddy-current method, as applied to measuring the diameter of cylindrical bodies, consists in placing the given metallic body in the cavity of a coil fed by a high-frequency current. The eddy currents arising in this body under the action of the electromagnetic wave incident upon it form their own electromagnetic field—the reflected wave—which leads to a change in the resultant field inside the sensor. This, in turn, leads to a change in its active and reactive resistance, that is, in its inductance and quality factor. The magnitude of these changes depends on the diameter and specific electrical conductivity of the introduced body (and, if it is a ferromagnetic, also on its magnetic properties). We shall be interested in the case of monitoring a nonmagnetic cylindrical body, namely copper wire. It should be noted that, in addition to the single-winding sensor described, there is also a two-winding sensor design, in which inside the winding fed by alternating current there is another winding in which an emf is induced, the magnitude and phase of which depend on the parameters of the body being monitored. In the practice of nondestructive testing of products by the eddy-current method, this sensor design has become most widespread, because the energy losses in the sensor windings themselves here have almost no effect on the measurement results, which increases the stability of the sensor. However, a two-winding sensor gives good results only at relatively low frequencies, whereas in monitoring fine and ultrafine wire it is necessary to work at frequencies up to tens of megahertz. Here the use of single-winding sensors already makes it possible to achieve greater sensitivity and accuracy than in the case of using two-winding sensors, since at such frequencies generator measuring circuits are the most effective, that is, circuits in which the sensor is a component part of the oscillatory circuit of a self-oscillator. Indeed, using the frequency method, it is possible to measure changes in the sensor inductance with an error of 0.01% or less, whereas when measuring an emf of such frequency it is not possible to reduce the error below 1–2%. In addition, single-winding sensors are simpler in design, which plays an important role in monitoring fine wire, since in this case the winding diameter must be made very small.

However, up to the present time there has been no sufficiently rigorous theory of a single-winding through-type sensor. The existing theory of Dodd—Grabovetskii [1, 2] does not take into account the phenomenon of the occurrence of eddy-

... currents and the associated additional energy losses in the sensor winding. This did not lead to large errors at low frequencies, provided that the diameter of the test body was much greater than the diameter of the winding wire. However, at high frequencies, when monitoring the diameter of fine and ultrafine wires, when the diameters of the measured wire and of the sensor winding wire turn out to be approximately equal, these additional losses may exceed those energy losses that are taken into account by Grabovetskii’s formulas.

In the present work we shall use more rigorous analytical expressions relating the total resistance of the sensor to the parameters of the wire being measured; to obtain them we used the method employed in the theory of induction heating [3]. Here the losses of electromagnetic energy are already taken into account not only in the metal of the tested article (the wire being measured), but also in the metal of the sensor winding.

In deriving the formulas, the following simplifications were made:

1. Edge effects were not taken into account, i.e., an infinitely long system was considered. In practice this assumption is valid if the ratio of the winding length to its diameter is

\[ \frac{l}{D_v} > 5 . \]

1. A sensor with a single-layer winding was considered, since in this case it is possible to achieve maximum stability of the electrical parameters of the sensor.

2. The sensor winding was assumed to be surrounded on all sides by air. In practice, if another dielectric is used instead of air, this will not introduce noticeable changes into the total impedance of the sensor.

In order to take into account, as far as possible, all energy losses occurring in the sensor, it is considered as a system consisting of three parts: the measured wire, located along its geometric axis; the air gap between the sensor winding and this wire; and the sensor winding.

Using as the starting point the equations of the electromagnetic field for conducting and nonconducting media, the active and reactive powers absorbed in each of the named parts are calculated separately. In this procedure, the powers absorbed in the measured wire and in the gap are calculated by the usual method, based on consideration of the case of incidence of a cylindrical electromagnetic wave on the outer surface of a solid isotropic metallic cylinder. To take into account the power absorbed by the sensor winding itself, it is replaced by an equivalent metallic tube, and the case of incidence of a cylindrical electromagnetic wave on the inner surface of a hollow cylinder is considered. But since in reality the electromagnetic wave is radiated not by some extraneous source located inside the tube, as is the case in [3], but is excited by the current passing through the winding itself, it is necessary to introduce an additional term characterizing the power losses at direct current. Then the total powers absorbed by the whole system are calculated, and from this the total complex impedance of the sensor is found. The expressions obtained have the following form:

\[ Z_{\mathrm{total}} = 6.2 \cdot 10^{-4}\frac{w^{3}}{l} \left\{ \sqrt{2}\cdot 10^{-4} d_i^{2} f \psi_a + \frac{D_v}{k_{\mathrm{ukl}}} \left[ \sqrt{\rho_i f}\,F_{\mathrm{vshch}} + 0.645\cdot 10^{4}\frac{\rho_i}{d_i} \right] \right. \]

\[ \left. + j\left[ 10^{-4} d_i^{2} f(\sqrt{2}\psi_r-1) + 10^{-4}D_v^{2}f + D_v\sqrt{\rho_i f}\, \frac{G_{\mathrm{vshch}}}{k_{\mathrm{ukl}}} \right] \right\}\ [\mathrm{ohm}]; \]

\[ Z_{\mathrm{no\text{-}load}} = 6.2 \cdot 10^{-4} \frac{w^{2}}{l} \left\{ \frac{D_{\mathrm{in}}}{k_{\mathrm{lay}}} \left( \sqrt{\rho_{\mathrm{w}} f}\, F_{\mathrm{ext}} + 0.645 \cdot 10^{4} \frac{\rho_{\mathrm{w}}}{d_{\mathrm{w}}} \right) \right. \]
\[ \left. {} + j \left( 10^{-4} D_{\mathrm{in}}^{2} f + D_{\mathrm{in}} \sqrt{\rho_{\mathrm{w}} f}\, \frac{G_{\mathrm{ext}}}{k_{\mathrm{lay}}} \right) \right\}\; [\Omega], \]

where

\(w\) is the number of turns of the sensor;

\(l\) is the length of the sensor winding (in cm);

\(D_{\mathrm{in}}\) is the internal diameter of the sensor winding (in cm);

\[ k_{\mathrm{lay}} = \frac{d_{\mathrm{w}} w}{l} \]

is the winding packing factor of the sensor;

\(\rho_{\mathrm{w}}\) is the electrical resistivity of the sensor-winding wire
(in ohm·cm; for copper wire \(\rho_{\mathrm{w}} = 1.8 \cdot 10^{-6}\) ohm·cm);

\(f\) is the frequency of the electromagnetic field (Hz);

\(d_{\mathrm{w}}\) is the diameter of the sensor-winding wire (in cm);

\(\psi_{a}, \psi_{r}\) are the Weber and Fischer functions, formed from Bessel functions and dependent on the generalized argument

\[ p = \frac{r_{0}\sqrt{2}}{\Delta_{\mathrm{e}}} = \frac{d_{0}\sqrt{2}}{2 \cdot 5.03 \cdot 10^{3}\sqrt{\rho_{0}/f}} \simeq d_{0} 10^{-4} \sqrt{\frac{2f}{\rho_{0}}}; \]

\(\rho_{0}\) is the electrical resistivity of the measured wire
(in ohm·cm);

\(d_{0}\) is the diameter of the measured wire (in cm);

\(F_{\mathrm{ext}}, G_{\mathrm{ext}}\) are the Burch and Davis functions for the case of incidence of an electromagnetic wave on the inner surface of a hollow metallic cylinder.

For the analysis of such expressions, graphical diagrams constructed from them are usually used; these diagrams show the dependence of the relative increments of the sensor resistance on various parameters of the item being tested over a wide frequency range. In our case such parameters will be the diameter and the electrical resistivity of the wire being controlled. When generator-type measuring circuits are used—and under the given conditions they are the most effective, since bridge circuits are unsuitable for such frequencies, while resonant circuits have low sensitivity and accuracy—it is convenient to use a complex diagram in the axes

\[ \frac{\Delta X}{X_{\mathrm{no\text{-}load}}} = f\left( \frac{\Delta R}{X_{\mathrm{no\text{-}load}}} \right) \]

(Fig. 1). Indeed, generator-type measuring circuits respond to changes in the reactive resistance of the sensor (the frequency of the generated oscillations changes) and to its quality factor (the amplitude of the oscillations changes). This complex plane gives us precisely a visual representation of the relative changes in the reactive resistance of the sensor and of the reciprocal quality factor when the various parameters of the wire being controlled and the frequency change. On this diagram, at a constant diameter of the sensor winding, each point uniquely determines the resistivity of the material of the measured wire and its diameter. The solid arc-shaped lines show changes in the increments of the sensor resistance with a change in the argument

\[ p \]

at various values of the filling factor, equal to

\[ v = \frac{d_{0}^{2}}{D_{\mathrm{in}}^{2}}. \]

The change in the argument here is achieved by changing the frequency or

of the specific electrical resistance \(\rho_0\). The dashed lines show changes in these increments when the wire diameter is changed at different frequencies. Already from an analysis of this diagram the following conclusions can be drawn:

1) to increase the sensitivity of the sensor, one should strive for the greatest possible filling factor

\[ v=\frac{d_0^{\,2}}{D_v^{\,2}}; \]

2) since a change in \(\rho_0\) leads only to a change in the argument \(p\), the sensitivity of the sensor to a change in the specific electrical conductivity of the wire will be greatest at small values of the argument \((p=1\div4)\) and least at large values of the argument \((p>10)\);

3) the sensitivity of the sensor to diameter, on the other hand, will increase as the argument increases;

4) for diameter control at large values of the argument it is convenient to use the change in the reactive resistance of the sensor, which is done in oscillator frequency circuits. However, when measuring thin wire, large values of the argument can be obtained only by increasing the operating frequency. But even for a wire of diameter \(100\ \mu\), the value of the argument \(p=10\) corresponds to a frequency \(f=91\) Mc, i.e., the frequency already enters the meter-wave range. In this case the influence of various destabilizing factors increases sharply, in particular parasitic capacitances and inductances; it becomes difficult to use circuits with lumped parameters, that is, the use of an extremely high frequency is undesirable. Therefore it makes sense to derive expressions for the sensitivity of the sensor in reactive resistance (or, equivalently, in inductance) to the diameter of the wire being checked and to its specific resistance as a function of the argument, and to choose the operating frequency by carefully analyzing them. In addition, these expressions are necessary for the quantitative calculation of the sensitivity of the sensor to a change in the diameter of the measured wire and of the error due to a possible change in its specific electrical conductivity. These expressions are easily obtained from the general expression for the total reactive resistance of the sensor by differentiating it with respect to the corresponding parameter. Indeed, the absolute change in the reactive resistance when the wire diameter changes by \(\Delta d_0\) can be found as

\[ \Delta X_{\text{tot}}=\frac{\partial X_{\text{tot}}}{\partial d_0}\,\Delta d_0. \]

The relative change will be

\[ \frac{\Delta X_{\text{tot}}}{X_{\text{tot}}} = \frac{\partial X_{\text{tot}}}{\partial d_0}\cdot \frac{\Delta d_0}{X_{\text{tot}}}. \]

Fig. 1. Diagram of increments of the complex resistance of the sensor for a nonmagnetic metallic cylinder.

But

\[ \frac{\Delta X_{\text{total}}}{X_{\text{total}}} = S_{\text{rel}}\frac{\Delta d_0}{d_0}, \]

whence

\[ S_{d_0(X)} = \frac{\partial X_{\text{total}}}{\partial d_0} \cdot \frac{d_0}{X_{\text{total}}}. \]

Analogously to this,

\[ S_{\rho_0(X)} = \frac{\partial X_{\text{total}}}{\partial \rho_0} \cdot \frac{\rho_0}{X_{\text{total}}}. \]

Substituting the values of the derivatives, we obtain:

\[ S_{d_0(X)} = \frac{ 2d_0^2 \left[ \sqrt{2}\,\psi_r + 10^{-4}d_0 \sqrt{\frac{f}{\rho_0}} \frac{\partial \psi_r}{\partial p} - 1 \right] }{ d_0^2(\sqrt{2}\,\psi_r-1) + D_{\text{v}}^2 + 10^4 D_{\text{v}} \sqrt{\frac{\rho_{\text{i}}}{f}}\, \frac{G_{\text{out}}}{k_{\text{coupl}}} } \left[ \frac{\%\,X_{\text{total}}}{\%\,d_0} \right]; \]

\[ S_{\rho_0(X)} = - \frac{ 10^{-4}d_0^3 \sqrt{\frac{f}{\rho_0}} \frac{\partial \psi_r}{\partial p} }{ d_0^2(\sqrt{2}\,\psi_r-1) + D_{\text{v}}^2 + 10^4 D_{\text{v}} \sqrt{\frac{\rho_{\text{i}}}{f}}\, \frac{G_{\text{out}}}{k_{\text{coupl}}} } \left[ \frac{\%\,X_{\text{total}}}{\%\,\rho_0} \right]. \]

Figure 2 presents a combined plot of the sensor sensitivity to the wire diameter and to its specific resistance as a function of the argument \(p\) at a fill factor \(v=0.01\). It can be seen from it that the sensitivity of the sensor to the diameter rises sharply, while the sensitivity to the specific resistance falls sharply only when the argument increases from 0 to 6–8. With a further increase in the argument, the change in the sensitivities slows down more and more, and at arguments \(p>15\) the sensitivities remain practically constant. Therefore it makes no sense to seek an increase of the argument beyond 10.

Fig. 2. Dependence of the sensor sensitivity to the diameter and to the specific electrical conductivity of the monitored copper wire on the argument (frequency). Fill factor \(v=0.01\).

Very useful for the proper choice of the sensor winding diameter is the plot of the sensor sensitivity to the wire diameter as a function of the fill factor. Such a plot for several values of the argument is presented in Fig. 3.

We have already said that the choice of the operating frequency and of the sensor parameters must be made not only from an analysis of the sensor sensitivity,

but also taking into account the need to ensure the maximum stability of its parameters, for the accuracy of the instrument when a generator measuring circuit is used will depend on the stability of the generator frequency, which for \(L\)–\(C\) generators is determined chiefly by the stability of the parameters of the oscillatory circuit and, consequently, of the sensor. In order to ensure reliable self-excitation of the generator and to reduce the frequency correction, it is necessary to strive for the maximum quality factor of the sensor. In addition, technical and operational requirements must be taken into account, which often prove decisive. Thus, the question of choosing the parameters of the sensor and the operating frequency is solved by a comprehensive analysis of all the requirements imposed on it. In this connection, the analytical expressions given above and the graphs constructed from them prove very useful for the correct and well-founded selection of the optimal operating frequency and optimal sensor parameters for each specific case.

Fig. 3. Dependence of the sensitivity of the sensor to the diameter of the copper wire being monitored on the filling factor at different values of the argument.

REFERENCES

1. H. B. Dwight, M. M. Bagai. Calculation for Coreless Induction Furnacess, Trans. AIEE, v. 54, 1935.

2. V. P. Grabovetskii. A contactless method for measuring resistivity and geometric dimensions by means of eddy currents. Avtomatika i telemekhanika, vol. XX, No. 7, 1959.

3. A. M. Vainberg. Induction Melting Furnaces. Gosenergoizdat, 1960.

4. V. K. Zhukov, V. E. Dreizin, I. G. Leshchenko, M. S. Roitman. Measuring circuits for monitoring the diameter of fine wire by the eddy-current method. (The article is included in this collection.)

5. V. K. Zhukov, V. E. Dreizin, I. G. Leshchenko. Errors in measuring the diameter of fine wires by the eddy-current method. (The article is included in this collection.)