SEPARATE INSPECTION OF VARIOUS PARAMETERS OF PRODUCTS MADE OF FERROMAGNETIC MATERIALS USING THE EDDY CURRENT METHOD

V. E. Dreizin, V. K. Zhukov, I. G. Leshchenko

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

At present, three main types of nondestructive testing are distinguished:

1) measurement of product parameters over a wide range of their variations;

2) tolerance inspection of these parameters, i.e., measurement of their small deviations from certain nominal values;

3) indication of defects such as discontinuities, foreign inclusions, or local inhomogeneities.

We shall confine ourselves to considering the first two types of testing, since it is precisely here that the difficulties of separate inspection and of suppressing the influence of interfering factors are most clearly manifested. In defect indication these difficulties, in most cases, are overcome comparatively easily by using differential probes with self-compensation, where suppression of the influence of fluctuations in electromagnetic and geometrical parameters is achieved by comparing two adjacent points of the part being inspected, as well as by dynamic analysis of the signal when scanning the entire surface of the inspected product; this analysis consists in separating abrupt signal changes caused by the presence of a defect from smooth changes determined by some variation in electromagnetic or geometrical parameters from one region to another.

Of the two remaining types of nondestructive testing, the more general is the first, since tolerance inspection may be regarded only as a special case of measuring product parameters, when their values fluctuate within some small, previously known limits around certain nominal values. Tolerance inspection is singled out as a separate type because the approach to solving problems of separate inspection in the two cases proves to be somewhat different. Naturally, no sharp boundary can be drawn between these two types of testing. The criterion that makes it possible to assign each specific case to one type or the other should be the possibility of linearizing the dependences of probe parameters on product parameters within a specified range of their variation.

In a general consideration of multiparameter separate inspection, three main points can be identified.

First, the necessity of a preliminary study of the character of the influence of each product parameter on the probe, i.e., finding the func-

functional relationships between the directly measured parameters of the transducer and the unknown parameters of the product.

Second, the need to obtain multidimensional information from the transducer in accordance with the number of unknown parameters of the product.

Third, the need to decode the information obtained.

Let us briefly consider each of these. The first point reduces to finding functional relationships between the directly measured parameters of the transducer and the unknown parameters of the product.

In general form these functional relationships may be written as

\[ X_i = F_i\left(Y_1, Y_2, \ldots, Y_n\right), \tag{1} \]

where \(X_i\) is one of the directly measured parameters of the transducer (or its increment),

\(Y_1, Y_2, \ldots, Y_n\) are the unknown parameters of the product (or their increments).

These relationships may be found either experimentally or analytically. The experimental approach is most convenient in the study of nonmagnetic specimens, since here only one electromagnetic parameter remains: electrical conductivity. This makes it possible to obtain easily, for study, a set of specimens with one varying parameter (when studying the effect of electrical conductivity, identical specimens are made from different materials; when studying the effect of geometrical dimensions, specimens of different sizes are made from one and the same material).

However, when studying ferromagnetic materials, obtaining sets of specimens with one varying parameter becomes very difficult, since here, in addition to electrical conductivity, at least two more electromagnetic parameters are added: the components of the complex magnetic permeability of the substance. They have to be selected from a large number of specimens with different values of the electromagnetic parameters. For this, naturally, it is necessary to have equipment for accurately measuring these parameters. Such equipment has not yet been created at the present time.

Moreover, it is not enough merely to obtain the necessary relationships; they must also be presented in a form convenient for analysis.

At present, the generally accepted practice is the graphical representation of functions in the form of normalized complex planes proposed by Foerster:

\[ \frac{\Delta X}{|X_{\mathrm{idle}}|} = f\left( \frac{\Delta R}{|X_{\mathrm{idle}}|} \right) \]

for a single-coil transducer and

\[ \frac{\mathcal{E}_{\mathrm{mn}}}{|\mathcal{E}_{\mathrm{idle}}|} = f\left( \frac{\mathcal{E}_{\mathrm{subst}}}{|\mathcal{E}_{\mathrm{idle}}|} \right) \]

for a two-coil transducer.

The form of these complex planes for ferromagnetic cylinders is shown in Fig. 1 (the complex plane

\[ \frac{\Delta X}{|X_{\mathrm{idle}}|} = f\left( \frac{\Delta R}{|X_{\mathrm{idle}}|} \right) \]

differs from the plane

\[ \frac{\mathcal{E}_{\mathrm{mn}}}{|\mathcal{E}_{\mathrm{idle}}|} = f\left( \frac{\mathcal{E}_{\mathrm{subst}}}{|\mathcal{E}_{\mathrm{idle}}|} \right) \]

only in that the abscissa axis is shifted upward by one unit).

Such normalization, carried out by division by the modulus of the reactance of the idle transducer for a single-coil design and by the modulus of the emf of the idle transducer for a two-coil design, makes it possible to eliminate completely the influence of the winding data of the transducer and also, by using the laws of similarity, to extend the graphs obtained to transducers and specimens of any geometrical dimensions; that is, the requirement of universality is fulfilled here in the best possible way. However, in this case the circumstance is taken into account that, for different meth-

Fig. 1

with the methods of connecting the transducer, the directly measurable quantities may be the most diverse electrical parameters of the transducer itself.

Therefore, along with the complex planes considered, it is reasonable to introduce additional planes characterizing the principal methods of connecting the transducer.

In the case of a single-winding transducer, such methods include the following:

1. Connecting the transducer as an inductance in a resonant circuit (resonant and self-oscillating measuring circuits).
   The directly measurable parameters of the transducer here are its reactance and quality factor. Taking this into account, the most convenient complex plane here is of the form

\[ \frac{X}{|X_{\text{idle}}|}=f\left(\frac{R}{|X|}\right), \]

since it gives a visual representation of the change in the reactance of the transducer (ordinate axis) and of the reciprocal of the quality factor (abscissa axis), and at the same time satisfies to some extent the requirement of universality. The form of this plane is shown in Fig. 2.

1. Connecting the transducer in a balanced bridge circuit. The output quantities in this case are determined by the relative increments of the active and reactive resistances of the transducer and are most fully characterized by the complex plane

\[ \frac{\Delta X}{|X_{\text{idle}}|}=f\left(\frac{\Delta R}{R_{\text{idle}}}\right), \]

shown in Fig. 3.

1. Connecting the transducer in an unbalanced bridge circuit. The output quantity here is the bridge unbalance voltage, which is completely determined by the complex quantity

\[ \frac{\Delta Z}{Z_{\text{idle}}}. \]

Therefore, here it is most convenient to use the complex plane

\[ f\left(\frac{\Delta Z}{Z_{\text{idle}}}\right)_{J_m} = f\left(\frac{\Delta Z}{Z_{\text{idle}}}\right)_{R_e}. \]

It is shown in Fig. 4.

For a two-winding transducer, two principal methods of connection can be distinguished: simple and mixed. With simple connection, the output signal is determined only by the magnitude and phase of the emf arising in the measuring winding of the transducer, which will be completely characterized by the normalized complex plane

\[ \frac{\mathcal{E}}{|\mathcal{E}_{\text{idle}}|}, \]

shown in Fig. 1. With mixed connection (the measuring winding of the transducer is connected into a resonant circuit), the output signal will depend both on the emf arising in the measuring winding and on the total impedance of this winding. This method of connection can be characterized by the complex plane

\[ \frac{U_{k\cdot \text{mn}}}{|U_{\text{res}}|} = f\left(\frac{U_{k\cdot \text{real}}}{|U_{\text{res}}|}\right), \]

but in this case it is necessary to specify at what values of the product parameters (or, possibly, in the absence of the product altogether) the circuit is tuned to resonance.

The construction of all these complex planes makes it possible to present the relationships between the directly measured parameters of the transducer for various methods of connecting it and the parameters of the inspected product in a form that is visual and convenient for analysis.

Fig. 2

Fig. 3

Fig. 4

Axes:

\[ \left(\frac{\Delta Z}{Z_{\mathrm{hol}}}\right)_{\mathrm{Im}} \]

\[ \left(\frac{\Delta Z}{Z_{\mathrm{hol}}}\right)_{\mathrm{Re}} \]

Visible labels and annotations in the figure include:

\[ d_0: 10\ \mathrm{mm} \]

\[ \sigma: 10\ \Omega\cdot \mathrm{mm}^2/\mathrm{m} \]

\[ \mu = 40,\qquad \mu = 200 \]

\[ d_0: 8 \]

\[ \mu_0: 89\ (2\ \mathrm{kHz}) \]

\[ (500\ \mathrm{Hz}),\quad 4.15,\quad 3.54,\quad 4.46,\quad 7.1,\quad 8.9,\quad 12.6,\quad 10,\quad 16,\quad 20,\quad 23,\quad 32,\quad 40,\quad 56,\quad 63,\quad 89,\quad 126,\quad 150\ \mathrm{Hz},\quad 202,\quad 203 \]

\[ \delta: 20^\circ \]

Following from this, we are faced with a question that constitutes the second basic aspect of multiparameter separate inspection—the question of obtaining information of sufficient dimensionality.

If one approaches this question from the mathematical side, the following can be established.

1. To obtain the solution of a system of equations of the form (1), it is necessary to have \(n\) independent equations.

2. The solution can be carried out on the basis of approximating the given functions by one or another algebraic function.

3. The number of possible solutions depends on the form of the approximating functions.

The first condition indicates that, when the number of unknown parameters of the product is \(n\), we must have the same number of measuring channels, measuring \(n\) independent electrical parameters of the transducer. However, when harmonic unmodulated fields are used, there can be only two independent parameters of the transducer, while all the others will be merely certain functional combinations of these two, since the source of information about the inspected product is ultimately the modulus and phase of the total magnetic flux embraced by the transducer circuit.

If the number of unknown parameters is greater than two, it is necessary in some way to increase the dimensionality of the information obtained. This can be done by changing the excitation conditions. In the simplest case, this leads to the necessity of using several operating frequencies. Indeed, the values of the directly measured parameters of the transducer \((X_i)\) depend not only on the values of the product parameters \((Y_i)\), but also on the frequency of the exciting field. In this case, the frequency may be regarded not only as an additional independent variable in equations (1), but as a parameter influencing the very form of the functions

\[ X_i = F_i(Y_1, Y_2, \ldots, Y_n). \]

It remains only to find such frequencies at which these functions will be independent; i.e., the additional measuring channels we need can be obtained by measuring the same parameters of the transducer, but at different frequencies. The conditions for independence of the functional relationships obtained at different frequencies can be expressed analytically and will be determined by the form of the approximating functions. When approximating by algebraic polynomials, this condition is expressed by the nonvanishing of the Jacobian of the systems of equations obtained. In particular, when approximating by first-degree polynomials, this condition reduces to the nonvanishing of the determinant of the system (this case occurs in tolerance inspection).

When the conditions of independence of the approximating equations comprising the given system are fulfilled, solutions of this system with respect to the unknown product parameters can obviously be found. Finding these solutions constitutes the essence of the third basic aspect of multiparameter inspection—the decoding of the information obtained. In the case of approximation by first-degree polynomials, this solution is unique. But when approximation is made by polynomials of higher degrees, which occurs in the presence of substantial nonlinearities in the initial relationships, we shall already have \(p^n\) possible solutions of the system (where \(p\) is the order of the polynomials, and \(n\) is the number of equations). To obtain an unambiguous solution, it is obviously necessary to impose some additional conditions on this system.

Thus, one of the ways of decoding the information in multiparameter separate inspection is the mathematical processing of the results obtained when measuring the individual electrical parameters of the transducer at different frequencies.

However, the use of this method leads to the need to develop complex multichannel circuits including an automatic computing device. In this case, regardless of the number of parameters of the inspected product that are of interest to us, the number of measuring channels must be equal to the number of unknown parameters; therefore, in our view, it is more expedient to use “hybrid” circuits that combine mathematical processing and correction of the results with the various methods now widely used for detuning from the influence of interfering parameters, such as, for example, the use of circuits with parametric or phase selectivity, etc. The synthesis of such circuits can be carried out only on the basis of a comprehensive and thorough analysis of the complex planes presented above, as well as of the sensor sensitivity diagrams.

This methodology makes it possible to carry out not only tolerance inspection, but also the separate measurement of the absolute values of various parameters of ferromagnetic products.

REFERENCES

1. V. E. Dreizin, V. K. Zhukov. On the use of the complex magnetic permeability of a substance in calculating electromagnetic fields in ferromagnetic bodies, Defektoskopiya, No. 2, 1966.

2. V. E. Dreizin. Theoretical foundations of inspection of ferromagnetic materials by eddy currents, Izvestiya TPI, vol. 141, 1966.

3. V. K. Arkad’ev. Electromagnetic processes in metals, ONTI, Part II, 1936.