On Separate Control of the Parameters of Cylindrical Nonmagnetic Products by the Eddy-Current Method

V. K. Zhukov

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

One of the reasons limiting the wide application of the eddy-current method for monitoring production parameters is the difficulty of separating the effects exerted on the characteristics of an eddy-current sensor by different parameters of the product. A cylindrical nonmagnetic product has two parameters that directly affect the sensor: the diameter of the cylinder and the electrical conductivity of the material from which it is made. Nevertheless, even in such a simple case it is sometimes difficult to separate the effect of diameter from the effect of electrical conductivity. To suppress the sensitivity of the sensor to the interfering parameter, it is recommended that monitoring be carried out by the reactive resistance or the imaginary component of the emf of the measuring winding, and at a definite frequency. This frequency must be sufficiently high if the dimension is being monitored, and relatively low if the electrical conductivity is being monitored. The most effective tuning-out is obtained for the influence of electrical conductivity. However, when monitoring the size of products of small diameter, it is difficult to ensure supply of the sensor with a current whose frequency would correspond to obtaining a sufficiently low sensitivity to electrical conductivity, since the frequency turns out to be very high. For this reason it is of interest to investigate other possible output parameters of the sensor from the standpoint of using them for separate control.

The stated problem is not posed for the first time; it has been considered in a number of works, among which the works of N. M. Rodigin and N. E. Korobeinikova [1, 2, 3] and B. V. Goncharov [4] should first of all be noted. In the cited works the authors analyze the sensitivity of the active \(R\) and reactive \(X\) resistances of a single-winding sensor, as well as the real and imaginary components of the emf of a two-winding sensor, to the parameters of the product. In doing so they operate with normalized increments of the output quantity. However, technical devices operating with eddy-current sensors usually respond to the relative change of its output parameter. Therefore it is desirable to investigate the sensitivity of the relative increments of the sensor output quantity to the relative change in the parameters of the cylindrical product that caused them.

In addition to those listed above, the following may be used as output quantities: the quality factor \(Q = X/R\) and the total impedance \(z = \sqrt{X^{2}+R^{2}}\) of a single-winding sensor; the amplitude \(E_m = \sqrt{E_a^{2}+E_p^{2}}\)

and the phase \(\varphi=\operatorname{arctg} E_p/E_a\) of the emf of the measuring winding of a two-winding sensor.

In the present work, expressions are derived for the relative sensitivities of \(X\), \(R\), \(z\), \(Q\), and \(\varphi\) to the diameter and electrical conductivity of a cylindrical nonmagnetic product, and they are analyzed from the standpoint of their use for separate control.

The changes in the components of the resistance and emf of the measuring winding, caused by introducing the product into the sensor, may be represented in normalized form as

\[ \Delta R/X_0=\Delta E_a/E_0=\operatorname{Re} j\eta(\dot{\mu}_{\mathrm{eff}}-1) =\operatorname{Re} j\eta \dot{\tau}_k, \tag{1} \]

\[ \Delta X/X_0=\Delta E_p/E_0=\operatorname{Im} j\eta(\dot{\mu}_{\mathrm{eff}}-1) =\operatorname{Im} j\eta \dot{\tau}_k, \]

where \(\eta\) is the filling factor, equal to the ratio of the cross-sectional areas of the product and the sensor, and

\[ \dot{\mu}_{\mathrm{eff}}= \frac{2I_1\!\left(\sqrt{j}\,\kappa a\right)} {\sqrt{j}\,\kappa a\, I_0\!\left(\sqrt{j}\,\kappa a\right)} \]

is the effective magnetic permeability introduced by F. Förster, which is expressed through the modified Bessel functions \(I_0(\sqrt{j}\,\kappa a)\) and \(I_1(\sqrt{j}\,\kappa a)\) of the complex argument \(\kappa a\). The quantity \(\kappa a=\sqrt{a\omega\gamma\mu}\), determined by the radius of the product \(a\), the frequency \(\omega\) of the current feeding the sensor, the electrical conductivity \(\gamma\), and the magnetic permeability \(\mu\) of the product, is called the generalized parameter.

For brevity, in what follows we shall denote

\[ \dot{\mu}_{\mathrm{eff}}-1=\dot{\tau}_k. \tag{2} \]

Let us represent the reactive resistance of the sensor as the sum \(X=X_0+\Delta X\), in which \(X_0\) is its resistance without the product and \(\Delta X\) is the resistance introduced by the product. On the basis of expressions (1), this sum will be equal to

\[ X=X_0(1+\operatorname{Im}j\eta\dot{\tau}_k). \tag{3} \]

We shall denote the relative sensitivity of \(X\) to a change in the radius of the product by \(S_a^x\) and define it as

\[ S_a^x=\lim_{\Delta a\to 0}\frac{\Delta X/X}{\Delta a/a} =\frac{\partial X}{\partial a}\cdot\frac{a}{X}, \tag{4} \]

where \(\Delta X/X\) is the relative change in the reactive resistance caused by a small relative change in the product radius \(\Delta a/a\). Having determined the derivative \(\partial X/\partial a\), after transformation we find

\[ S_a^x=\frac{\eta}{1+\operatorname{Im}j\eta\dot{\tau}_k}\,\alpha_x, \tag{5} \]

where

\[ \alpha_x=2\operatorname{Im} \frac{I_1'\!\left(\sqrt{j}\,\kappa a\right)} {I_1\!\left(\sqrt{j}\,\kappa a\right)}. \tag{6} \]

As is evident from the expression obtained, the sensitivity of the reactive resistance to the radius of the product depends on the filling factor \(\eta\) and on the generalized parameter \(\kappa a\). By arguments and transformations analogous to those that were made in find-

substituting \(S_a^x\), for the relative sensitivity of the reactive resistance to the electrical conductivity of the product we obtain:

\[ S_\gamma^x=\frac{\eta}{1+\operatorname{Im} j\eta\tau_k}\cdot \beta_x , \tag{7} \]

where

\[ \beta_x=\operatorname{Im}\left[ 1-\frac{2I_1(\sqrt{j}\,\kappa a)}{\sqrt{j}\,\kappa a\, I_0(\sqrt{j}\,\kappa a)} -\frac{I_1'(\sqrt{j}\,\kappa a)}{I_0'(\sqrt{j}\,\kappa a)} \right]. \tag{8} \]

The sensitivity \(S_\gamma^x\), like \(S_a^x\), depends on \(\eta\) and \(\kappa a\). From formulas (5) and (7) we see that the ratio of the sensitivities

\[ \chi=S_\gamma^x/S_a^x=\beta_x/\alpha_x \tag{9} \]

is a function only of \(\kappa a\) and does not depend on \(\eta\). The dependence of \(S_a^x\) and \(\chi_x\) on \(\kappa a\) is shown graphically in Figs. 1 and 2. The sensitivity of the reactive resistance to the electrical conductivity of the product, as is seen from the graph (Fig. 2), decreases with increasing \(\kappa a\). Already at \(\kappa a=10\) it is less than 0.05 of the value of the sensitivity to the radius. Hence it follows that, in order to monitor the size of a product by the change in reactive resistance, the value \(\kappa a\) must be chosen large. From the same graph it is evident that the sensitivity to electrical conductivity always remains below the sensitivity to radius; therefore, when monitoring electrical conductivity by the change in \(X\), it is impossible to eliminate the influence of changes in radius for any \(\kappa a\).

Fig. 1. Sensitivity of the reactive resistance of the transducer to the radius of the product.

Fig. 2. Dependence of the ratio of the sensitivities of the reactive resistance on the generalized parameter.

Let us represent the active resistance of the transducer winding, similarly to (3), in the form

\[ R=R_0(1+Q_0\operatorname{Re} j\eta\tau_k), \tag{10} \]

where \(R_0\) and \(Q_0\) are the active resistance and quality factor of the transducer winding without the product, and \(\operatorname{Re} j\eta\tau_k\) is the increment of the active resistance caused by insertion of the product. The sensitivity of the active resistance to the radius of the product

\[ S_a^R=\lim_{\Delta a\to 0}\frac{\Delta R/R}{\Delta a/a} =\frac{\partial R}{\partial a}\cdot\frac{a}{R}, \tag{11} \]

and to the electrical conductivity of the product

\[ S_\gamma^R=\lim_{\Delta\gamma\to 0}\frac{\Delta R/R}{\Delta\gamma/\gamma} =\frac{\partial R}{\partial\gamma}\cdot\frac{\gamma}{R} \tag{12} \]

we find, after determining the derivatives \(\partial R/\partial a\) and \(\partial R/\partial \gamma\):

\[ S_a^R=-\frac{\tau_k Q_0}{1+Q_0 \operatorname{Re} j\eta\cdot \tau_k}\,\alpha_R, \]

\[ S_\gamma^R=\frac{\eta Q_0}{1+Q_0 \operatorname{Re} j\eta\cdot \tau_k}\,\beta_R. \]

Here

\[ \alpha_R=2\operatorname{Re}\frac{I_1^2(\sqrt{jka})}{I_0^2(\sqrt{jka})}, \tag{13} \]

\[ \beta_R=\operatorname{Re}\left[ 1-\frac{2I_1(\sqrt{jka})}{\sqrt{jka}\,I_0(\sqrt{jka})} -\frac{I_1^2(\sqrt{j}\cdot ka)}{I_0^2(\sqrt{j}\cdot ka)} \right]. \tag{14} \]

The ratio of the sensitivities of the parameter under consideration

\[ \chi_R=\beta_R\alpha_R \tag{15} \]

is a function only of the generalized parameter \(ka\). The character of this function can be judged from the graph in Fig. 4. Analysis of the graph leads to the conclusion that the parameter \(R\) can be used for tolerance inspection of size if \(ka=2.5\) is ensured. The magnitude of the sensitivity \(S_a^R\) in this case can be found from the graphs shown in Fig. 3.

Fig. 3. Sensitivity of the active resistance of the measuring winding of the transducer to the radius of the product.

Fig. 4. Dependence of the ratio of sensitivities of the active resistance on the generalized parameter.

For a two-winding transducer the principal output parameters are the real and imaginary components of the emf. As follows from (1), and as the theoretical analysis shows, the ratio of the sensitivities of the real component of the emf is described by expression (15); and the ratio of the sensitivities of the imaginary component of the emf by expression (9). Consequently, the principal parameters of the transducer in this case also can be used only for size inspection.

One of the derived parameters of the transducer with a clear physical meaning is its quality factor

\[ Q=\frac{X}{R}=Q_0\frac{1+\operatorname{Im}j\eta\tau_k}{1+Q_0\operatorname{Re}j\eta\tau_k}. \tag{16} \]

By a method analogous to those applied above, one can find the sensitivity of the Q-factor to the radius \(S_a^Q\) and to the electrical conductivity \(S_\gamma^Q\) of the product, as well as their ratio, which is of interest to us:

\[ S_a^Q = \eta\, \frac{(1/Q_0+Re\,j\eta\tau_k)\alpha_x+(1+Im\,j\eta\tau_k)\alpha_R} {(1/Q_0+Re\,j\eta\tau_k)(1+Im\,j\eta\tau_k)}, \tag{17} \]

\[ S_\gamma^Q = \frac{(1/Q_0+Re\,j\eta\tau_k)\beta_x-(1+Im\,j\eta\tau_k)\beta_R} {(1/Q_0+Re\,j\eta\tau_k)(1+Im\,j\eta\tau_k)}, \]

\[ z_Q = \frac{(1/Q_0+Re\,j\eta\tau_k)\beta_x-(1+Im\,j\eta\tau_k)\beta_R} {(1/Q_0+Re\,j\eta\tau_k)\alpha_x-(1+Im\,j\eta\tau_k)\alpha_R}. \]

The ratio of the sensitivities of the Q-factor, in contrast to \(X\) and \(R\), already depends on three parameters: \(\eta\), \(Q_0\), and \(\kappa a\). Figures 5 and 6 show the dependences of \(S_a^Q\) and \(z_Q\) on \(\kappa a\) for several combinations of \(\eta\) and \(Q_0\). From

Fig. 5. Sensitivity of the Q-factor of the sensor coil to the product radius.

Fig. 6. Dependence of the ratio of the Q-factor sensitivities on the generalized parameter.

the graph, and also from formula (17), it can be seen that for certain combinations of \(\eta\), \(Q_0\), and \(\kappa a\) the ratio \(z_Q\) is equal to zero, i.e., the sensor proves to be insensitive to changes in electrical conductivity. As follows from the formula, these combinations must satisfy the condition

\[ (1+Q_0 Re\,j\eta\tau_k)\beta_x - Q_0(1+Im\,j\eta\tau_k)\beta_R = 0, \]

whence it follows that, in order to obtain \(z_Q=0\), it is necessary to have:

\[ \eta = \frac{1/Q_0\beta_x-\beta_R} {\beta_R\cdot Im\,j\eta\tau_k-\beta_x\cdot Re\,j\eta\tau_k}, \tag{18} \]

\[ Q = \frac{\beta_x} {(1+Im\,j\eta\tau_k)\beta_R-\beta_x\cdot Re\,j\eta\tau_k}. \tag{19} \]

With the aid of the last expressions, by specifying two quantities, \(\kappa a\) and \(\eta\), or \(\kappa a\) and \(Q_0\), one can find the third, corresponding to zero sensitivity to the electrical conductivity of the product.

The sensitivity of the Q-factor to the product radius, for any combinations of \(\kappa a\), \(\eta\), and \(Q_0\), remains higher than the sensitivity to electrical conductivity; therefore this derived parameter of the sensor can be recommended only for use in tolerance control of dimensions.

It can be shown that the ratio of the sensitivities with respect to another derived parameter of the transducer—the phase angle \(\varphi = \arctan E_p/E_a\)—is equal to the ratio of the sensitivities with respect to the \(Q\)-factor and is determined by expression (17).

The total impedance of the measuring winding, in accordance with expressions (3) and (10), will be equal to

\[ z=\sqrt{(1/Q_0+Re\,j\eta\tau_\kappa)^2+(1+Im\,j\eta\tau_\kappa)^2} \tag{20} \]

The sensitivity of \(z\) to the radius

\[ S_a^z=\lim \frac{\Delta z/z}{\Delta a/a}=\frac{\partial z}{\partial a}\cdot \frac{a}{z} \]

and to the electrical conductivity of the product

\[ S_\gamma^z=\lim \frac{\Delta z/z}{\Delta \gamma/\gamma}=\frac{\partial z}{\partial \gamma}\cdot \frac{\gamma}{z} \]

will be obtained by determining the derivatives of the corresponding functions:

\[ S_a^z=\eta \frac{(1/Q_0+Re\,j\eta\tau_\kappa)\alpha_R+(1+Im\,j\eta\tau_\kappa)\alpha_x} {(1/Q_0+Re\,j\eta\tau_\kappa)^2+(1+Im\,j\eta\tau_\kappa)^2}, \tag{21} \]

\[ S_\gamma^z=\eta \frac{(1/Q_0+Re\,j\eta\tau_\kappa)\beta_R+(1+Im\,j\eta\tau_\kappa)\beta_x} {(1/Q_0+Re\,j\eta\tau_\kappa)^2+(1+Im\,j\eta\tau_\kappa)^2}. \tag{22} \]

The dependence of \(S_a^z\) and \(S_\gamma^z\) on \(\kappa a\) for four values of \(Q_0\) and \(\eta=1\) is shown in Fig. 7. From the graphs of this figure it can be seen that, for any \(Q_0\), there are such values of \(\kappa a\) at which \(S_a^z=0\) or \(S_\gamma^z=0\).

The sensitivity of the total impedance of the winding to the radius of the product will be equal to zero

Fig. 7. Dependence of the sensitivity of the total impedance of the transducer winding on the generalized parameter:
a) to the radius of the product, b) to the electrical conductivity of the product.

when the numerator of expression (21) becomes zero:

\[ (1+Q_0 Re\,j\eta\tau_\kappa)\alpha_R+(1+Im\,j\eta\tau_\kappa)Q_0\alpha_x=0. \tag{23} \]

It follows from this that, in order to obtain \(S_a^z=0\) for given \(\kappa a\) and \(\eta\), it is necessary to ensure

\[ Q_0=-\frac{\alpha_R}{\alpha_R Re\,j\eta\tau_\kappa+\alpha_x(1+Im\,j\eta\tau_\kappa)}. \tag{24} \]

Since \(\alpha_x\) is negative, \(Q_0>0\).

The total impedance of the measuring winding will be insensitive to electrical conductivity under the condition that the numerator of expression (22) is equal to zero,

\[ (1/Q_0 + \operatorname{Re} j\eta\tau_k)\beta_R + (1 + \operatorname{Im} j\eta\tau_k)\beta_x = 0. \tag{25} \]

From this condition we determine the values of \(\kappa a\), \(\eta\), and \(Q_0\) corresponding to \(S_i^z = 0\). In doing so, two quantities are prescribed, and the third is found.

For a two-winding transducer, an arbitrary parameter analogous to the parameter \(z\) of a single-winding transducer is the emf amplitude of the measuring winding \(E_m\). The sensitivity of \(E_m\) to the parameters of the product is equal to the sensitivity of \(z\) when \(Q_0 \to \infty\), i.e., when the resistance of the measuring winding to direct current is zero.

The expressions obtained in this work and their analysis have shown that most of the possible output parameters of the transducer, with an appropriate choice of current frequency, filling factor, and quality factor, provide zero or nearly zero sensitivity to the electrical conductivity of the product; therefore these parameters can be used in tolerance inspection of dimensions.

Measurement of electrical conductivity with suppression of sensitivity to small fluctuations of diameter can be ensured only by the total impedance of the measuring winding of the transducer; consequently, it can be recommended for use as the output parameter of the transducer in tolerance inspection of electrical conductivity.

REFERENCES

1. N. M. Rodigin, I. E. Korobeinikova. Use of the “zero” components of the emf increment of a coil for detuning from interfering parameters in inspection by the eddy-current method, Proceedings of the Institute of Metal Physics, issue 24, 1965.

2. I. E. Korobeinikova, N. M. Rodigin. On separate measurement of the parameters of a nonferromagnetic tube. Proceedings of the Institute of Metal Physics, issue 24, 1965.

3. N. M. Rodigin, I. E. Korobeinikova. Quality control of products by the eddy-current method, Mashgiz, 1958.

4. B. V. Goncharov. Proceedings of TsNIITMASH No. 62, Moscow, 1966.