Analysis of Generator Circuits in Tolerance Inspection of the Dimensions and Electrical Conductivity of Nonmagnetic Products by the Eddy Current Method

D. V. Milyaev, V. K. Zhukov

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

The problem of developing measuring apparatus becomes especially important in measurements of nonelectrical quantities, when, in addition to measuring transducers, a whole complex of radio-electronic devices is involved. The progress of radio electronics has created the conditions for the further development of such areas in measurement technology as nondestructive testing of products by the eddy-current method.

Without dwelling on the essence of the eddy-current method, we note that the development of measuring apparatus for monitoring the dimensions and electrical conductivity of nonmagnetic products remains a pressing problem to this day.

As is known from the theory of eddy currents, a change in the dimensions and properties of a product, its structure, or a disturbance of it leads to a change in the parameters of the electromagnetic field in which the product under test is placed. The sensing element that converts changes in the product parameters into an electrical quantity is an inductive sensor. When the measured quantity acts on the measuring transducer, its parameters change—active and reactive resistance and the associated inductance, quality factor, etc. The task of the measuring circuit is to register these changes as accurately as possible.

Among the known and widely used measuring circuits—bridge, resonant, and the like—the most widespread are generator circuits, in which the inductive sensor is an element of an oscillatory circuit, and a change in its parameter (inductance) is converted into a change in the frequency or amplitude of the self-oscillator. The subsequent measuring circuit can be constructed on the basis of measuring either amplitude or frequency. From the standpoint of sensitivity and measurement accuracy, the frequency method is more promising. In this article an analysis is proposed of generator circuits for monitoring the dimensions and electrical conductivity of products by frequency.

The simplest way to implement frequency measurement is according to the block diagram shown in Fig. 1. The inductive sensor D1, included in the circuit of generator G, senses the action of the measured nonelectrical quantity, converting it into a change in inductance; this leads to a change in the frequency of the self-oscillator, which is recorded by the frequency meter FM. Such a circuit cannot provide the required measurement accuracy because of its low sensitivity; therefore, it is often used—

... a differential or two-generator circuit is used (Fig. 2), in which one of the generators (\(\Gamma_1\)) is the measuring one with sensor \(D_1\), and the other is an auxiliary one (\(\Gamma_2\)) with reference sensor \(D_2\). The difference frequency obtained at the output of the mixer is monitored by the frequency meter IF. The difference frequency is equal to 0 when the effect on both the reference and the measuring sensors is the same. When the controlled quantity acts on the measuring sensor, the frequency meter registers a frequency proportional to this measured quantity. The measurement accuracy is determined mainly by the stability of the frequencies of both generators. However, changes in the temperature of the supply voltages, aging of the generator elements, etc., lead to different frequency drift, which requires introducing additional adjustments and calibrations, and, in the case of rapid changes in the factors named, makes the measurement process altogether impossible.

Fig. 1.

Fig. 2.

Fig. 3.

The influence of temperature on the measuring circuit must be noted especially. If active measures can be used to stabilize the supply voltages, then the use of thermostating is not always convenient or possible. When generator circuits operate at high frequencies (tens of megahertz), which, as a rule, is required in monitoring geometrical dimensions, the frequency drift caused by a change in temperature may prove greater than the frequency drift caused by the parameter being monitored. Thus, to increase measurement accuracy it is necessary to construct self-oscillators with a small temperature coefficient of frequency (TCF). The TCFs of self-oscillators operating in the range of units to tens of megahertz usually do not exceed \(10^{-5}\), even with the use of phase correction. Such stability can be achieved only when the quality factor of the circuits is sufficiently high. Therefore, the requirement of monitoring small changes in nonelectrical quantities and the requirement for accuracy in their measurement lead to the necessity of constructing new circuits that are only slightly subject to the influence of destabilizing factors.

One variant in the development of differential generator circuits is the circuit in Fig. 3. The main feature of this circuit is that, in order to eliminate the influence of changes in the parameters of the active element of the self-oscillator (tube, semiconductor triode), the measuring and compensation sensors are periodically connected to the measuring generator as the inductance of its oscillatory ...

circuit. At the generator input, voltage bursts are obtained whose frequencies are proportional to the switched inductances. After conversion by the mixer (Mix), the difference between the burst frequencies is detected (FD) and measured by the indicating instrument (I).

Periodic connection of the inductances is carried out by means of a commutator that includes the actual switches (mechanical or contactless) and the control circuit for these switches. The principal requirement imposed on the switches is their relatively low direct resistance and identity. Nonidentity of the direct resistances of the switches and their instability over time determine the main inspection error.

Another source of error is the frequency detector, with its inherent shortcomings: instability of the transition frequency (zero drift), nonlinearity of the transfer characteristic, influence of input-voltage instability, etc. To reduce the inspection error caused by the frequency meter, automatic frequency tuning can be used, acting in the intervals between measurements.

Another variant of the measuring circuit operating by the method of periodic comparison is shown in Fig. 4. In this circuit it is not the inductances (sensors) that are periodically switched, but the oscillatory circuits. Moreover, each circuit can first be thermally compensated separately, which is especially important when inspecting nonelectrical quantities, when the inductive sensors (measuring and compensating) may be separated by considerable distances and be under different temperature conditions. The influence of nonidentity of the commutator switches and changes in the direct resistances will have practically no effect on the amplitude and frequency of the generator, because the resistance of the switch is connected in series with the large resonant resistance of the circuit.

The error of the circuit caused by the meter of the difference between the burst frequencies is substantially reduced through the use of a capacitor frequency meter with commutation.

Figure 5 presents the schematic diagram of a prototype instrument for inspecting the geometric dimensions or electrical conductivity of nonmagnetic products.

The measuring sensor \(L_x\) and the compensating sensor \(L_0\), together with the capacitances \(C_{10}\) and \(C_{11}\), respectively form two oscillatory circuits, which are periodically connected to the measuring generator (\(T_1\)) by means of a commutator including the actual switches (diodes \(D_3 \div D_6\)) and a control circuit—a multivibrator (\(T_7\) and \(T_8\)). The high-frequency voltage bursts are fed to the mixer (\(T_2\)), to which voltage is continuously supplied from the auxiliary generator (\(T_3\)). Both generators are assembled according to the same type of circuit, using the inertial properties of a semiconductor triode. The inclusion of a high-frequency choke in the collector circuit of the measuring generator makes it possible, when switching the oscillatory circuits, not to disturb the transistor’s dc operating mode. The inductance of the choke is selected so that the choke impedance at the generation frequency is considerably greater than the resonant resistance of the circuits. Consequently, the choke has little influence on the sensitivity of the sensors. After conversion, the resonant-frequency bursts arrive at the capacitor frequency meter with commutation. The first two stages (\(T_4, T_5\)) are limiting amplifiers that convert the sinusoidal voltage into a rectangular voltage, necessary for operation of the frequency meter output switch (\(T_6\)).

[Figure: block diagram with labeled blocks and connections.]

Fig. 4.

[Figure: electrical circuit diagram with component designations.]

Fig. 5.

Unlike an ordinary capacitor frequency meter, a frequency meter with commutation has two pairs of charge–discharge circuits. In this case, the discharge current of one circuit is proportional to the frequency of one packet, and the discharge current of the other circuit is proportional to the frequency of the other packet. The current flowing through the output instrument is determined by the difference of the discharge currents; consequently, the readings of the instrument are proportional to the difference of the packet frequencies.

The readings of the instrument can be related to the measured frequency through the conversion coefficients of the corresponding elements by the expression

\[ \alpha = S_i \cdot S_f \cdot f_x, \tag{1} \]

in which \(S_f\) is the function of converting frequency into direct current (voltage), \(S_i\) is the sensitivity of the measuring instrument to current, and

\[ f_x = f_1 - f_2 \tag{2} \]

is the difference of the packet frequencies, proportional to the measured quantity. The smallest value of the measured quantity that can be measured by the instrument is determined by the maximum error. Neglecting the error of the inductive transducer, the relative error of the circuit can be determined as the ratio of the absolute error \(\Delta L\) to the total inductance of the transducer \(L_x\)

\[ \gamma = \frac{\Delta L}{L_x}. \tag{3} \]

The principal sources of error are the error caused by incomplete thermal compensation of the circuits,

\[ \gamma_1 = \pm \frac{1}{2}\left[(\alpha_{L_x} - \kappa_1 \alpha_{C_{11}}) + (\alpha_{L_0} - \kappa_2 \alpha_{C_{10}})\right]\Delta t \tag{4} \]

and the error of the frequency-meter circuit. In expression (4), \(\alpha_{L_x}\), \(\alpha_{C_{11}}\), \(\alpha_{L_0}\), and \(\alpha_{C_{10}}\) are the temperature coefficients of \(L_x\), \(L_0\), \(C_{11}\), and \(C_{10}\), respectively; \(\kappa_1\) and \(\kappa_2\) are coefficients that take into account the incomplete compensation of the circuits; \(\Delta t\) is the deviation of the ambient temperature from the temperature at which the circuits were thermally compensated.

It follows from expression (1) that, for \(f_x = \mathrm{const}\),

\[ \frac{\Delta \alpha}{\alpha} = \frac{\Delta S_i}{S_i} + \frac{\Delta S_f}{S_f} \tag{5} \]

and, for \(\alpha = \mathrm{const}\),

\[ \frac{\Delta f_x}{f_x} = \frac{\Delta S_i}{S_i} + \frac{\Delta S_f}{S_f}. \tag{6} \]

The ratio \(\delta_i = \dfrac{\Delta S_i}{S_i}\) is the relative error of the measuring instrument. The error \(\dfrac{\Delta S_f}{S_f}\) expresses the change in the frequency sensitivity of the circuit, which ultimately leads to the same change in the current through the instrument, i.e.,

\[ \frac{\Delta S_f}{S_f} = \frac{\Delta I_{\mathrm{pr}}}{I_{\mathrm{pr}}} \tag{7} \]

Thus,

\[ \frac{\Delta f_x}{f_x} = \delta_i + \frac{\Delta I_{\mathrm{pr}}}{I_{\mathrm{pr}}} \tag{8} \]

is the relative error of the frequency meter, or, referred to the frequency of the measuring generator,

\[ \frac{\Delta f_x}{f_r}=\left(\delta_i+\frac{\Delta I_{\mathrm{pr}}}{I_{\mathrm{pr}}}\right)\frac{f_x}{f_r}. \tag{9} \]

To determine the value \(\Delta I_{\mathrm{pr}}/I_{\mathrm{pr}}\), we use the expression

\[ I_{\mathrm{pr}}=C_1 f_1 E_1 Q_1 \kappa_3 \kappa_4 - C_2 f_2 E_2 Q_2 \kappa_5 \kappa_6, \tag{10} \]

which relates the output current of the instrument to the parameters of the frequency meter and to the frequency difference being monitored.

In expression (10)

\[ E_1=(U\ \text{charge}_1-U\ \text{discharge}_1),\quad E_2=(U\ \text{charge}_2-U\ \text{discharge}_2) \tag{11} \]

\[ Q_1=\frac{1-e^{-\tau_1 f_1}}{1+e^{-\tau_1 f_1}};\quad Q_2=\frac{1-e^{-\tau_2 f_2}}{1+e^{-\tau_2 f_2}}. \tag{12} \]

\[ \kappa_3=\frac{R_1}{R_1+R_2+R_{\mathrm{pr}}};\quad \kappa_6=\frac{R_2}{R_1+R_2+R_{\mathrm{pr}}}. \tag{13} \]

\(\tau_1\) and \(\tau_2\) are the discharge time constants of the frequency-meter circuits; \(\kappa_4\) and \(\kappa_5\) are coefficients that take into account the inequality of the switching times of the frequency-meter circuits, caused by the difference in the half-periods of the multivibrator. A change in the temperature of the supply voltage, aging of elements, etc., leads to a change in the current through the instrument

\[ \Delta I_{\mathrm{pr}}= f_1 E_1 Q_1 \kappa_3 \kappa_4 \Delta C_1 + f_1 C_1 Q_1 \kappa_3 \kappa_4 \Delta E_1+ \]

\[ + \ldots + C_1 f_1 E_1 \kappa_3 \Delta \kappa_4 - f_2 E_2 Q_2 \kappa_5 \kappa_6 \Delta C_2 - \]

\[ - \ldots - C_2 f_2 E_2 Q_2 \kappa_5 \Delta \kappa_6. \tag{14} \]

Dividing (14) by (10) and transforming, we obtain the relative change in the current through the instrument caused by a change in the parameters.

\[ \frac{\Delta I_{\mathrm{pr}}}{I_{\mathrm{pr}}} = \frac{\Delta C_1 f_1}{C_1(f_1-\eta_1 f_2)} + \frac{\Delta E_1}{E_1}\frac{f_1}{(f_1-\eta_1 f_2)} +\ldots \]

\[ + \frac{\Delta \kappa_4}{\kappa_4}\frac{f_1}{(f_1-\eta_1 f_2)} - \frac{\Delta C_2}{C_2}\frac{f_2}{(\eta_2 f_1-f_2)} -\ldots \]

\[ - \frac{\Delta \kappa_6}{\kappa_6}\frac{f_2}{(\eta_2 f_1-f_2)}, \tag{15} \]

where

\[ \eta_1=\frac{C_2 E_2 Q_2 \kappa_5 \kappa_6}{C_1 E_1 Q_1 \kappa_3 \kappa_4} \tag{16} \]

\[ \eta_2=\frac{1}{\eta_1}. \tag{17} \]

In the first approximation \(\eta_1=\eta_2\), then

\[ \frac{\Delta I_{\mathrm{pr}}}{I_{\mathrm{pr}}} = \frac{f_1}{f_1-f_2} \left[ \left(\frac{\Delta C_1}{C_1} - \frac{\Delta C_2}{C_2}\frac{f_2}{f_1}\right) + \right. \]

\[ \left. + \left(\frac{\Delta E_1}{E_1} - \frac{\Delta E_2}{E_2}\frac{f_2}{f_1}\right) +\ldots+ \left(\frac{\Delta \kappa_4}{\kappa_4} - \frac{\Delta \kappa_6}{\kappa_6}\frac{f_2}{f_1}\right) \right]. \tag{18} \]

In the general case, the relative changes \(\dfrac{\Delta C_1}{C_1}\) and \(\dfrac{\Delta C_2}{C_2}\), \(\dfrac{\Delta E_1}{E_1}\) and \(\dfrac{\Delta E_2}{E_2}\), etc., may have both different magnitudes and different signs. In tolerance inspection, the frequencies \(f_1\) and \(f_2\) may also differ insignificantly. Taking this into account, expression (18) can be rewritten as

\[ \frac{\Delta I_{\mathrm{pr}}}{I_{\mathrm{pr}}} \cong \frac{\Delta C}{C} + \frac{\Delta E}{E} + \frac{\Delta Q}{Q} + \frac{\Delta \kappa'_3}{\kappa'_3} + \frac{\Delta \kappa'_6}{\kappa'_6}. \tag{19} \]

Then

\[ \frac{\Delta f_x}{f_r} = \left( \delta_{\mathrm{i}} + \frac{\Delta C}{C} + \frac{\Delta E}{E} + \frac{\Delta Q}{Q} + \frac{\Delta \kappa'_3}{\kappa'_3} + \frac{\Delta \kappa'_6}{\kappa'_6} \right) \frac{f_x}{f_r}. \tag{20} \]

As was already stated above, the error of the circuit is expressed in terms of the relative change in the inductance of the sensor. The total error of the circuit is

\[ \gamma = \pm \left[ (\alpha_{L_x}-\kappa_1 \alpha_{\mathrm{cu}} - {} +(\alpha_{L_0}-\kappa_2 \alpha_{\mathrm{cu}})) \right]\Delta t + \]

\[ + 2 \left( \delta_{\mathrm{i}} + \frac{\Delta C}{C} + \frac{\Delta E}{E} + \frac{\Delta Q}{Q} + \frac{\Delta \kappa'_3}{\kappa'_3} + \frac{\Delta \kappa'_6}{\kappa'_6} \right) \frac{f_x}{f_r}. \tag{21} \]

It follows from expression (21) that the circuit error is determined mainly by the stability of the elements, and first of all by the elements of the oscillatory circuits. The error introduced by the frequency-measuring circuit depends on the values of the frequencies \(f_x\) and \(f_r\): the higher the generator frequency, the smaller the total error. In practice, the first component of the error can be brought to a value on the order of \(10^{-6}/^\circ\mathrm{C}\).

To verify the capabilities of the circuit considered, experimental studies were carried out. The temperature coefficients of the frequency of each circuit at \(20^\circ\mathrm{C}\) were of the same sign and equal in magnitude to \((10^{-4}/^\circ\mathrm{C})\). When the temperature changed to \(60^\circ\mathrm{C}\), the drift of the difference frequency (at a measuring-generator frequency of \(30\ \mathrm{MHz}\)) did not exceed \(12\ \mathrm{kHz}\), which is equivalent to a relative change in the inductance of the sensor of no more than \(10^{-5}/^\circ\mathrm{C}\).

The discrepancy between the calculated and experimentally found error is explained by the fact that, with increasing temperature, the temperature coefficients of the frequency of both oscillatory circuits turned out to be of the same sign.