DEVICE FOR MEASURING SMALL DISPLACEMENTS

L. Ya. Zinner, A. I. Skorospeshkin

(Recommended by the seminar of the Departments of Electrical Machines and General Electrical Engineering)

To study the influence of mechanical factors on the commutation of collector machines, a number of instruments using one method or another have now been developed. Earlier we proposed a profilometer using the amplitude-modulation method [1, 2]. As elements sensitive to small displacements, eddy-current and capacitive sensors with bridge and transformer connection into the measuring circuit were used.

In the case of transformer connection, the output quantity of the sensor is the emf of the measuring winding; in this case the measuring circuit responds to one of the parameters of the emf, which may be the amplitude, phase, or the real and imaginary components. An eddy-current sensor provides the greatest sensitivity to small irregularities of the commutator surface when tuned to resonance at the generator frequency. In this case the measuring circuit responds both to a change in the emf of the signal winding and to a change in its impedance and quality factor.

A profilometer with transformer connection of an eddy-current sensor, while providing high sensitivity, has a number of substantial shortcomings. These are, first of all, the need to ensure high stability of the current of the driving generator and to linearize the conversion characteristic of the sensor, which leads to an increase in circuit error.

A profilometer with bridge connection of a capacitive sensor [2] is in many respects free from the shortcomings inherent in the one described above.

The latter is characterized by the fact that a profilometer with a capacitive sensor provides greater reliability, higher resolution, and smaller circuit error. Tests carried out at the Elektromashina plant showed that a profilometer with a capacitive sensor satisfies all the requirements imposed on meters of small displacements. However, despite its high operating qualities, this instrument is also not free from shortcomings inherent in the amplitude-measurement method itself, since amplitude monitoring does not provide the required accuracy in continuous measurements because of the action of destabilizing factors; moreover, amplitude measurement is difficult to carry out with sufficient accuracy.

In practice, measurements are usually made by frequency, since frequency stability, even with parametric stabilization, is significantly higher than amplitude stability, and, consequently, frequency measurements provide greater accuracy.

Taking the foregoing into account, we have developed a profilometer based on measuring the oscillator frequency when one of the parameters of the grid circuit, \(L\) or \(C\), is changed. A single-winding eddy-current and capacitive sensor with a guard ring is used as the transducer.

It should be noted that at present there already exists an instrument for measuring surface irregularities of a commutator that uses the method of measuring frequency deviation [3]. However, the instrument described suffers from one important drawback, namely the presence of only one measuring oscillator, which leads to frequency instability over time. Frequency drift is determined mainly by the temperature instability of the elements of the oscillator circuits and by instability of the power supply; this is especially important in the present case, when the sensor capacitance is only a few units, while the measured capacitance deviations are hundredths and thousandths of a picofarad.

In order to reduce the error caused by drift of the measuring-oscillator frequency, we used a beat-frequency circuit (Fig. 1), in which the comparison principle is implemented not in the circuit containing the sensor, but in the measuring circuit itself.

When applied to capacitive and inductive sensors, the method is based on comparing the frequencies of the measuring and reference oscillators.

The measuring circuit proper consists of: a reference oscillator \(L_{2}\) (6S4P), operating at frequency \(f_{1}\); a measuring oscillator \(L_{1}\) (6S4P) with frequency \(f_{2}\), varying as a function of the measured displacement; two cathode followers \(L_{3}, L_{4}\) (6Zh1P); a mixer \(L_{5}\) (6I1P), in whose anode circuit the intermediate frequency is selected; an intermediate-frequency limiter-amplifier \(L_{6}\) (6Zh1P); a limiter \(L_{7}\) (6Zh1P); a discriminator \(L_{8}\) (6Kh2P), and two stages of video amplifier \(L_{9,10}\) (6N1P).

To set the operating point of the discriminator and the initial sensor gap, a tube millivoltmeter \(L_{14}\) (6N1P), equipped with an M-24 microammeter, is included in the profilometer circuit.

At the same time, the millivoltmeter serves to measure the total runout of the commutator under static conditions.

The profilometer is powered from an electronic voltage stabilizer assembled with tubes \(L_{11}\) (6N3), \(L_{12}\) (6Zh1P), \(L_{13}\) (SG-4S).

In order to stabilize the tube filaments and the cathode-to-filament capacitance, 5-\(\mu\text{H}\) chokes and 1500-pF capacitors are introduced into the filament circuit.

The measures listed above made it possible to reduce the circuit error to a minimum.

The relative error in measuring the capacitance and inductance of the sensors is equal to twice the error in determining the frequency:

\[ \frac{dC}{C} = -2 \frac{df}{f}, \]

\[ \frac{dL}{L} = -2 \frac{df}{f}. \tag{1} \]

When operating at a frequency of 105 MHz, as is the case in this instrument, \(\frac{df}{f}\) is very small, the sensitivity to a change in gap is high, and the relative error can be reduced to thousandths of a percent.

In practice, quartz stabilization of the reference oscillator is usually employed, which does not eliminate the influence of all destabilizing factors on the frequency of the measuring oscillator.

Fig. 1. Schematic circuit diagram of the small-displacement meter.

In-figure label: Mains.

In our view, the use of two generators—measuring and reference—identical in circuit design, assembly, and parts used, is more effective, since this compensates for the frequency change caused by destabilizing factors. As an indicator of the frequency change when small displacements are varied, the circuit uses a frequency discriminator with a pair of inductively coupled circuits \(Л_8(6Х2П)\), producing at the output a voltage proportional to the departure of the frequency from resonance.

\[ U_{\text{out}} = k(f - f_{\text{res}}). \tag{2} \]

The discriminator circuit is tuned to the intermediate frequency of 500 kHz and is inductively coupled to the limiter circuit \(Л_7(6Ж1П)\).

\[ U_{\text{out}} = U_0 \cos \Theta \psi_{(a)}, \tag{3} \]

where \(U_0\) is the voltage on the limiter circuit,

\(\Theta\) is the cutoff angle of the diode detector,

\(\psi_{(a)}\) is a function depending on the parameters of the coupled circuits.

It follows from (3) that it is necessary to introduce a limiter into the circuit, since the output voltage of the discriminator depends not only on frequency drift, but also on the voltage on the circuit.

The sensitivity of the circuit to changes in the capacitance or inductance of the sensor is determined by the slope of the discriminator characteristic, which increases with increasing resonant frequency.

However, when selecting the resonant frequency, it is necessary to take into account the fact that, as it increases, the range of measured displacements decreases.

The dependence of the output voltage for the given circuit on changes in the capacitance and inductance of the sensors when the gap changes is determined by substituting into (2) the expression for \(f\) in terms of the circuit parameters:

\[ U_{\text{out}} = \frac{k}{\sqrt{L_{\text{res}}}} \left( \frac{1}{\sqrt{C_{\text{res}}}} - \frac{1}{\sqrt{C_{\text{res}}+\Delta C}} \right); \tag{4} \]

\[ U_{\text{out}} = \frac{k}{\sqrt{C_{\text{res}}}} \left( \frac{1}{\sqrt{L_{\text{res}}}} - \frac{1}{\sqrt{L_{\text{res}}+\Delta L}} \right), \tag{5} \]

where \(k\) is the proportionality coefficient,

\(L_{\text{res}}, C_{\text{res}}\) are the resonant values of the circuit parameters in \(\mu\text{H}\) and pF,

\(\Delta L, \Delta C\) are the increments of the capacitance and inductance of the sensors when the gap changes.

For (4), expressing \(\Delta C\) through the change in the gap between the collectors and the sensor plate, we have

\[ U_{\text{out}} = \frac{k}{\sqrt{L_{\text{res}} C_{\text{res}}}} \left( 1 - \frac{1}{\sqrt{1-\dfrac{h}{d-\delta}}} \right), \tag{6} \]

where \(d\) is the initial gap in m,

\(h\) is the change in the gap in m.

In the case where an eddy-current sensor is used, obtaining an expression describing the dependence of \(U_{\text{out}}\) as a function of the measured gap is associated with considerable difficulties, especially since a sensor with a ferrite core is used. The works presently available [4, 5, 6, 7] give a solution for the field of a surface-mounted sensor in general form and do not make it possible to select the optimum sensor dimensions and supply parameters in order to ensure maximum sensitivity to the measured parameter.

In the case of using a sensor with a magnetic screen and a ferrite-

with a concentrator, the solution for the introduced impedance is associated with insurmountable mathematical difficulties, which makes it necessary in each particular case to select experimentally the parameters of the transducer and of the supply in order to attain maximum sensitivity.

For a qualitative representation of the dependence of \(U_{\text{out}}\) on the frequency deviation, let us take the simplest case, when the transducer is a low-turn coil of finite dimensions.

For this case the inductance of the transducer in air can be determined as

\[ L_0 = \pi \mu_0 n^2 \frac{D}{l} k_A \ldots, \tag{7} \]

where \(\mu_0\) is the magnetic permeability of vacuum,
\(n\) is the number of turns of the transducer winding,
\(D\) is the transducer diameter, \(m\),
\(l\) is the winding height, \(m\),
\(k_A\) is a coefficient found from Table 6–2 [8].

The introduced impedance when the transducer is placed above the surface being monitored is determined from the expression for the introduced e.m.f. [4]:

\[ Z_{\text{in}} = -\frac{E_{\text{in}}}{I_1} = j \omega \mu_0 n^2 D e^{-6\frac{h}{D}} F_1(\beta_0), \tag{8} \]

where \(h\) is the gap between the transducer and the surface being monitored, \(m\),

\[ jF_1(\beta_0) = \frac{1}{2\beta_0^2 + j + 2\beta_0\sqrt{\beta_0^2 + j}} \tag{9} \]

is a function carrying information about the properties of the material of the monitored surface;

\[ \beta_0 = \frac{3}{D\sqrt{\omega\mu_0\delta}} \]

is a generalized parameter;

\(\delta\) is the electrical conductivity of the surface material.

As shown by Skarlet [4], the function \(jF_1(\beta_0)\) can, with an error of 5%, be approximated by an exponential of the form

\[ jF_1(\beta_0) = 1.55\beta_0 e^{-2\beta_0} - j e^{-2\beta_0}. \tag{10} \]

After substituting (10) into (8) and carrying out certain transformations, we obtain the expression for the introduced inductance:

\[ L_{\text{in}} = \mu_0 n^2 D e^{-\left(6\frac{h}{D} + 2\beta_0\right)}. \tag{11} \]

Assuming that the discriminator output voltage is zero when the transducer is removed from the conducting surface, the resonant frequency of the generator is determined from the expression

\[ f_2 = \frac{1}{ 1.77\, n D \sqrt{\frac{\mu_0}{l} k_A C_{\text{res}}} }. \tag{12} \]

When the transducer is placed above a conducting surface, the new value of the transducer inductance will be

\[ L = \mu_0 n^2 D \left[ \frac{k_A}{l}D - e^{-\left(6\frac{h}{D} + 2\beta_0\right)} \right]. \tag{13} \]

and the new value of the generator frequency

\[ f_2'= \frac{1}{ \sqrt{ C_{\mathrm{res}} \left[ \mu_0\frac{\pi k_A}{l} -\frac{1}{D}\mu_0 e^{-\left(6\frac{h}{D}+2\beta_0\right)} \right] } }. \tag{14} \]

Starting from (12) and (14), we find an expression describing the dependence of \(U_{\mathrm{out}}\) on changes in the parameters of the sensor:

\[ U_{\mathrm{out}}= \frac{k}{ 1.77D \sqrt{\mu_0\frac{k_A}{l}C_{\mathrm{res}}} } \left[ 1- \frac{1}{ \sqrt{ 1-\frac{1}{\pi D k_A} e^{-\left(6\frac{h}{D}+2\beta_0\right)} } } \right]. \tag{15} \]

The sensitivity of the circuit to a change in the distance from the conducting surface to the capacitive and inductive sensors is defined as

\[ S_C= \frac{d}{d-h}\cdot \frac{k}{ \sqrt{ L_{\mathrm{res}}C_{\mathrm{res}} \frac{(d-2h)^3}{d-h} } }, \tag{16} \]

\[ S_L= \frac{31k\,e^{-\left(6\frac{h}{D}+2\beta_0\right)}}{ k_1 \left[ 1-\frac{1}{\pi D k_A} e^{-\left(6\frac{h}{D}+2\beta_0\right)} \right]^{3/2} }, \tag{17} \]

where

\[ k_1=1.77\,k_A D^3 n\, \sqrt{\mu_0\frac{k_A}{l}C_{\mathrm{res}}}. \tag{18} \]

Thus, from the expressions for the sensitivity and \(U_{\mathrm{out}}\) of the eddy-current sensor, it is evident that its value is affected both by a change in the gap \(h\) and by a change in the electrical conductivity of the material. However, considering the inhomogeneity of the collector materials to be insignificant, and especially since at the operating frequency of 10 MHz the sensitivity of the circuit to changes in electrical conductivity is negligibly small, one may confine oneself to considering the effect on \(U_{\mathrm{out}}\) and \(S_L\) of the diameter of the sensor coil, the number of turns, and the gap.

Increasing the sensitivity by increasing the diameter of the sensor coil does not appear possible, since this conflicts with the need to ensure high resolving power over the width of a collector lamella. Changing the number of turns of the sensor is also limited by the choice of operating frequency and by detuning from the influence of electrical conductivity.

Thus, in the displacement meter developed, the magnitude of \(U_{\mathrm{out}}\) is affected only by the change in the distance from the collector to the sensor, which in turn determines the sensitivity of the circuit. It should also be noted that, although the inductance of the sensor changes exponentially with a change in the gap, it is always possible to choose a section of the sensor characteristic where the dependence of \(U_{\mathrm{out}}\) on the gap will be practically linear.

The resolving power required to distinguish individual collector plates is achieved by shielding the eddy-current sensor and by using a ferrite concentrator to localize the electromagnetic field. If a capacitive sensor is used, the latter is made with a guard ring, which makes it possible to eliminate the edge effect and thereby ensure the necessary resolving power.

Figure 2 shows the dependence of \(U_{\mathrm{out}}\) on the change in the gap between the collector and the capacitive sensor for an initial gap of \(310\,\mu\).

At a gap of 370 μm, \(U_{\text{out}}\) is equal to zero, and a further increase in the gap is associated with a change in the polarity of the output signal. As can be seen from Fig. 2, the device developed provides good linearity of the conversion characteristic and a constant high sensitivity of 10 mV/μm over the range of expected changes in the collector surface.

Fig. 3 shows a sweep of the profile of the collector of a P-42 machine, taken with the profilometer described above. It is evident from Fig. 3 that the profilometer provides good resolution and makes it possible to distinguish the relative protrusion both of a group of plates and of two adjacent plates. If it is necessary to increase the resolving power of the instrument, the lining of the capacitive sensor can be reduced to a point probe.

Thus, summarizing the above, it may be said that the profilometer developed by the authors is largely free of the shortcomings of previously proposed profilometers and has high sensitivity and resolving power.

Fig. 2. Dependence of \(U_{\text{out}}\) on the gap.

Fig. 3. Profilogram of the collector of a P-42 machine.

REFERENCES

1. A. I. Skorospeshkin, L. Ya. Zinner, A. I. Proshin. A device for investigating mechanical factors in commutator electrical machines. Izvestiya TPI, vol. 160, 1967.

2. L. Ya. Zinner, A. I. Skorospeshkin. A bridge device for investigating the dynamics of commutators of electrical machines. Izvestiya TPI, vol. 160, 1966.

3. K. K. Namitokov, V. F. Chepura, V. G. Brezinsky. A device for investigating dynamic changes in the surface shape of commutators of electrical machines. Elektromekhanika, No. 1, 1966.

4. Yu. M. Shkarlet. Nondestructive Methods for Testing Materials and Products. Optipribor, 1964.

5. V. V. Klyuev. Some problems in the calculation of high-frequency surface-mounted eddy-current sensors. Defektoskopiya, No. 4, 1966.

6. I. I. Kifer, V. S. Fastritsky. On the choice of operating modes for a surface-mounted sensor located above a ferromagnetic half-space. Defektoskopiya, No. 4, 1966.

7. V. S. Sobolev. On the choice of sensor parameters for contactless measurements of the electrical conductivity of materials by the eddy-current method. Izmeritel’naya tekhnika, No. 3, 1964.

8. P. L. Kalantarov, L. A. Tseitlin. Calculation of Inductances. Gosenergoizdat, 1955.