BRIDGE INSTRUMENT FOR STUDYING THE DYNAMICS OF COMMUTATORS IN ELECTRIC MACHINES

L. Ya. Zinner, A. I. Skorospelkin

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

At the present time, a number of profilometers using one or another method of measurement have been developed for studying the influence of mechanical factors on the operation of commutator-type electric machines.

The most successful circuits of the profilometers that have been developed [1, 2] operate on the principle of measuring the frequency of a resonant circuit. One of the elements of the circuit, \(L\) or \(C\), is used as the transducer. Other circuits are also possible, though they have no fundamental distinction [3].

Common to all these developments is the fact that, while possessing good sensitivity, they give a significant error when measuring small displacements and a lack of repeatability of results from one experiment to another. This is explained, above all, by the presence in the oscillatory circuit of such nonlinear elements as a vacuum tube and a semiconductor, whose parameters depend on such factors as aging, temperature dependence, spread of characteristics, and instability of the power source. In principle, however, it is possible to reduce these errors to a minimum by complicating the measuring circuits, introducing deep feedback, and increasing the stability of the power sources. On the other hand, making the instrument more complicated entails a reduction in its reliability and requires special knowledge during maintenance.

Consequently, what is needed is an instrument that would be free from the shortcomings inherent in the previously developed ones, would provide sufficient sensitivity to changes in the measured parameter, and would be insensitive to external fields and parasitic couplings.

At first glance, AC bridges best satisfy such requirements, having significant advantages over the others. These are, above all, ease of adjustment, absence of nonlinear elements, simplicity of the measuring circuits, and insensitivity to instability of the frequency of the supply generator and to stability of the power source. However, a substantial drawback of ordinary AC bridges is their sensitivity to external fields and parasitic capacitive couplings, which becomes especially pronounced when operating at high frequencies.

Protecting bridges from parasitic external couplings is a rather difficult task and entails a considerable increase in the dimensions of the instrument, which is not always desirable [4].

To facilitate the conditions of shielding, it is advisable to use—

use as the measuring cell of a bridge with close inductive coupling, which at the same time has higher sensitivity in comparison with ordinary alternating-current bridges.

One of the elements in the calculation of a bridge circuit is the determination of its relative sensitivity. A number of works [5, 6, 7] are devoted to this question; however, all of them consider the general case of measuring complex resistance and the loss-angle tangent of capacitors.

Proceeding from the foregoing, the authors analyzed three possible circuits of the measuring cell, designed and experimentally tested a prototype of a bridge measuring cell with a capacitive sensor of small displacements.

In the present work the basic calculation formulas for bridge measuring cells are given, together with a block diagram and a brief description of the bridge profilometer, as well as some results of its tests.

The possible circuits considered for implementing the measuring cell are shown in Fig. 1.

In Fig. 1 a, b, the measured resistance \(Z_1\) and the standard \(Z_2\) are connected in series with the windings \(L_1\) and \(L_2\).

The measured capacitance \(C_d\) in Fig. 1 c is connected in parallel with the capacitance \(C\) and resistance \(R\). In the circuits of Fig. 1 b, c, \(R\) is a purely active resistance serving for precise balancing of the bridge, for facilitating the conditions of protecting the bridge against the pulsed voltage from the commutator entering the measuring circuit, and, as will be shown below, contributing to an increase in the sensitivity of the bridge.

The supply-voltage generator is connected in the diagonal \(ab\) of the bridge, and the indicator to the winding \(L_3\).

For simplicity of calculation the following assumptions are adopted: parasitic capacitive couplings are absent, the active resistances of the windings are equal to zero, and the coupling coefficient between the windings is \(K=1\).

Fig. 1. Circuit of the measuring cells.

The latter is all the more justified if, in manufacturing the windings, a bifilar winding method is used and the transformer core is made of a ferrite toroid with high magnetic permeability.

To determine \(U_{\text{out}}\), assuming that the input resistance of the indicator \(Z_{\text{out}}=\infty\) and the internal resistance of the generator is zero, we write Kirchhoff’s equations for the loop currents of the circuit:

\[ \begin{aligned} \dot I_1\left(Z_1+j\omega L_1\right) -\dot I_2 j\omega \sqrt{L_1L_2} &=\dot U_g,\\ -\dot I_1 j\omega \sqrt{L_1L_2} +\dot I_2\left(Z_2+j\omega L_2\right) &=\dot U_g. \end{aligned} \tag{1} \]

It is obvious that this system of equations is valid for all three circuits of Fig. 1.

The simultaneous solution of these equations gives the expression for determining the loop currents.

\[ \dot{i}_1=\dot{U}_r\cdot \frac{Z_2+j\omega L_2+j\omega\sqrt{L_1L_2}} {Z_1Z_2+j\omega\left(Z_1L_2+Z_2L_1\right)}, \tag{2} \]

\[ \dot{i}_2=\dot{U}_r\cdot \frac{Z_1+j\omega L_1+j\omega\sqrt{L_1L_2}} {Z_1Z_2+j\omega\left(Z_1L_2+Z_2L_1\right)}. \tag{3} \]

Let us find the output voltage of the bridge circuit for the case when the input resistance of the indicator is infinitely large and it is connected to the indicating winding \(L_3\). In this case

\[ \dot{U}_{\mathrm{out}}=\dot{i}_1j\omega\sqrt{L_1L_3}-\dot{i}_2j\omega\sqrt{L_2L_3}. \tag{4} \]

Substituting the values of \(\dot{i}_1\) and \(\dot{i}_2\) into expression (4), we obtain:

\[ \dot{U}_{\mathrm{out}}=\dot{U}_r\cdot \frac{j\omega\left(Z_2\sqrt{L_1L_3}-Z_1\sqrt{L_2L_3}\right)} {Z_1Z_2+j\omega\left(Z_1L_2+Z_2L_1\right)}. \tag{5} \]

In the case of a symmetrical transformer with close inductive coupling between the arms, the relations \(L_1=L_2=L\). The expression for \(\dot{U}_{\mathrm{out}}\) will take the form:

\[ \dot{U}_{\mathrm{out}}=\dot{U}_r\cdot \frac{\sqrt{L_3}\left(Z_2-Z_1\right)} {\left(Z_1+Z_2+\dfrac{Z_1Z_2}{j\omega L}\right)\cdot\sqrt{2}}. \tag{6} \]

It is evident that, with an unchanged value of the inductances of the primary and secondary windings of the transformer, the ratio \(\dfrac{\sqrt{L_3}}{\sqrt{L}}\) can be replaced by the ratio of the number of turns

\[ \dot{U}_{\mathrm{out}}=\dot{U}_r \frac{n_3\left(1-\dfrac{Z_1}{Z_2}\right)} {n\left(1+\dfrac{Z_1}{Z_2}+\dfrac{Z_1}{j\omega L}\right)}. \tag{7} \]

Formula (7) is valid for all three types of measuring cells. For the circuit of Fig. 1, \(a\), it is assumed that the resistances \(Z_1\) and \(Z_2\) are purely capacitive in character,

\[ Z_1=\frac{1}{j\omega C_d}, \qquad Z_2=\frac{1}{j\omega C}, \quad \text{where} \tag{8} \]

\[ C_d=\frac{S\varepsilon_0}{d} \]
is the capacitance of the transducer,

\(S\) is the area of the transducer plate,

\(d\) is the distance between the transducer plate and the collector surface,

\(\varepsilon_0\) is the dielectric constant of air.

After substituting the values of \(Z_1\) and \(Z_2\) into (7) and carrying out certain transformations, the dependence of \(\dot{U}_{\mathrm{out}}\) on the circuit parameters is expressed by the formula

\[ \dot{U}_{\mathrm{out}}=\dot{U}_r\cdot\frac{n_3}{n}\, \frac{1-\dfrac{C}{C_d}} {1+\dfrac{C}{C_d}-\dfrac{1}{\omega^2LC_d}}. \tag{9} \]

Balancing such a bridge can be carried out in two ways:

1) by adjusting the variable capacitance in the comparison arm;
2) by adjusting the number of turns of the transformer winding in the comparison arm.

Balancing the bridge by the second method, while convenient, has a substantial drawback, namely a reduction in sensitivity. This is easily shown by using the expression for the relative sensitivity of the bridge to a change in capacitance [5].

\[ S_c= \frac{2}{ (p+1)\sqrt{ \left[1-\frac{1}{(p+1)\omega^2LC}\right]^2+ \left[\frac{1}{Q}-\frac{2\omega L(p-1)}{Z_u}+\frac{2}{\omega C Z_u}\right]^2 }} \tag{10} \]

where \(P\) is the arm ratio,

\(Q\) is the quality factor of the system,

\(Z_u\) is the internal resistance of the indicator.

In the case when \(P=1\), the sensitivity of the bridge is maximal. Nevertheless, it should be noted that balancing the bridge by changing the number of turns is expedient in those cases when it becomes necessary to widen the adjustment limits.

Taking into account the assumptions adopted earlier, we obtain an expression for the relative sensitivity of the bridge to a change in capacitance:

\[ S_c=\frac{1}{1-\dfrac{1}{2\omega^2LC}}. \tag{11} \]

It follows from (11) that, for certain values of \(\omega\), \(L\), and \(C\), the denominator may become zero, and \(S_c\) infinite.

Thus, the choice of optimal values of \(\omega\), \(L\), and \(C\) can ensure high sensitivity of the bridge circuit with good linearity of the scale in the specified range of change of the sensor capacitance.

The absolute sensitivity of the bridge to a change in the measured gap, reduced to 1 volt of supply voltage, can be expressed, according to [8], as

\[ S_d^{U_{\text{out}}}=\frac{1}{\dot U_g}\cdot \frac{d(\dot U_{\text{out}})}{d(d)}. \]

Substituting into (9) the value of the sensor capacitance from (8), we have for \(\dot U_{\text{out}}\) and \(S\)

\[ \dot U_{\text{out}}=\dot U_g\,\frac{n_3}{n}\cdot \frac{S-Cd}{S+Cd-\dfrac{d}{\omega^2L}}; \tag{12} \]

\[ S_d^{U_{\text{out}}}= \frac{n_3}{n}\cdot \frac{S-2SC\omega^2L}{ \omega^2L\left(S+Cd-\dfrac{d}{\omega^2L}\right)^2}. \tag{13} \]

For small values of the measured displacements, the value of the capacitance \(C\) is on the order of units of picofarads, and the value of \(L\) is on the order of units of mH; in this case the resonant frequency of the bridge is \(1.5\)—\(2\) MHz. The bridge ensures the best linear dependence of \(S\) on the change in the gap, while at the same time satisfying the requirement of high sensitivity to small changes in the gap, when operating in the subresonant region.

On the basis of the foregoing, one may, without noticeable errors, neglect the terms \(2SC\omega^2LS + Cd\) because of their smallness, and the expression for the absolute sensitivity of the bridge assumes a form convenient for practical calculations:

\[ S_d^{U_{\text{out}}}=\frac{n_3}{n}\cdot \frac{S}{d^2}\cdot \omega^2L. \tag{14} \]

The calculation formulas for the bridge of the type shown in Fig. 1, b are derived in an analogous way.

To simplify the calculation we assume that the shares of active resistance in the compared arms are equal, and that the bridge is unbalanced \((C \ne C_d)\). Then the dependence of the bridge output \(\dot U_{\text{out}}\) on its parameters is expressed by the formula

\[ \dot U_{\text{out}} = \dot U_g \frac{n_3}{n} \frac{ 1 + \dfrac{C}{C_d} }{ 1 + \dfrac{C}{C_d} + \dfrac{(Rj\omega C_d - 2)\cdot(Rj\omega C - 2)}{4\omega^2 L C_d} }. \tag{15} \]

When the bridge is balanced by adjustment of \(R\), the absolute sensitivity to a change in \(R\) is expressed by the formula

\[ S_R^{U_{\text{out}}} = \frac{n_3}{2n}\cdot R\omega^2 \frac{C^2}{C_d}. \tag{16} \]

As is seen from (16), the absolute sensitivity of the bridge of the type shown in Fig. 1, b has a linear dependence on \(R\), whereas when the bridge is balanced by changing the capacitance in the comparison arm, as is easy to show, the absolute sensitivity depends nonlinearly on the value of the balancing capacitance.

On this basis, balancing the measuring cell by means of an active resistance is preferable, since this makes it possible simply to take into account the change in the sensitivity of the circuit when passing from one measuring range to another.

The dependence of the bridge sensitivity on the value of the setting gap is expressed by the formula

\[ S_d^{U_{\text{out}}} = \frac{n_3}{n}\cdot \frac{S}{d^2}\cdot \left( \frac{CS}{d} - R^2\omega^2 C^2 \frac{1}{d} - 4 \right)\omega^2 L. \tag{17} \]

Taking into account that the first two terms in the parentheses have a very insignificant effect on the value of the absolute sensitivity of the bridge because of their smallness, the expression for \(S_d\) is determined as

\[ S_d^{U_{\text{out}}} = \frac{n_3}{n}\cdot \frac{S}{d^2}\cdot \omega^2 L. \tag{18} \]

The dependences \(\dot U_{\text{out}}\), \(S_R^{U_{\text{out}}}\), and \(S_d^{U_{\text{out}}}\) for the bridge of the type shown in Fig. 1, c on its parameters are expressed by the formulas

\[ U_{\text{out}} = U_g \frac{n_3}{n} \frac{ Rj\omega^2 C_1(C_d - C) + 2\omega(C_1 + C - C_d) }{ Rj\omega^2 C_1(C_d - C) + 2\omega(C_1 - C - C_d) + \dfrac{R\omega C_1}{j\omega L} }; \tag{19} \]

\[ S_R^{U_{\text{out}}} = \frac{8n_3}{n}\omega^2 L \left( 2\omega C - 1 + \frac{1}{2}\frac{C_d}{C} \right); \tag{20} \]

\[ S_d^{U_{\text{out}}} = \frac{1}{2}\frac{n_3}{n}\omega^4 L\frac{S}{d^2} \left[ R^2C^2 + 4 - 8C(L+1) \right]. \tag{21} \]

Thus, irrespective of the bridge circuit and the method of balancing, its sensitivity to small displacements depends on the ratio of the number of turns of the primary and secondary windings of the transformer, the capacitance of the capacitor in the comparison arm, the angular frequency, and the initial setting gap.

From expressions (14) and (17) it is evident that bridges of this type (Fig. 1a, b) provide high sensitivity to small displacements; however, these circuits also have a significant drawback. This drawback, which limits the possibility of using them as a measuring cell when inspecting the commutators of electric machines, is the fact that difficulties arise in decoupling the bridge from high-frequency voltage pulsations at the brush–commutator transition. The bridge (Fig. 1c) is to some extent free of this drawback owing to the use of a high-resistance resistor \(R\), the absence of direct electrical contact with the commutator being inspected, and, moreover, it has a higher sensitivity to small displacements. It should also be noted that the bridge in Fig. 1c has one more exceptionally important advantage over the first two, namely that it makes it possible simultaneously to use two identical sensors connected into opposite arms of the bridge, which permits separate investigation of a number of interrelated processes. This may be shown by the following example. In an operating electric machine, the commutator surface is characterized by such components as armature unbalance, commutator eccentricity, commutator ovality, and, finally, elastic and inelastic deformation due to mechanical forces during operation at elevated speeds. With two sensors, it is possible to compensate for the effect of eccentricity, unbalance, and ovality and to obtain a profilogram of the commutator microrelief in pure form.

On the basis of the foregoing, in order to determine the possibility of using a bridge with tight inductive coupling, a capacitive profilometer with a measuring cell according to the circuit of Fig. 1c was built and experimentally tested. The block diagram of the profilometer is shown in Fig. 2. The parameters of the measuring cell are as follows:

\[ n = 40,\quad n_3 = 150,\quad C = C_1 = (2 \div 30)\ \text{pF},\quad R = 680\ \text{k}\Omega. \]

Fig. 2. Block diagram of a bridge profilometer.

A ferrite toroid Ф1000 with a diameter of 55 mm was used as the magnetic core. The secondary winding of the transformer was shielded from the leakage fields of the primary, and the entire transformer was shielded in order to stabilize parasitic capacitive couplings. The measuring cell with the harmonic-oscillation generator 1 and emitter follower 2 (Fig. 2) was mounted as an attachment on a textolite plate fastened to the machine window. On the same plate a capacitive sensor \(C_d\) was fastened in such a way that its plate was located above the longitudinal axis of the commutator and was strictly parallel to it. To set the sensor at a specified initial clearance, it was provided with a micrometer screw, which at the same time made it possible to calibrate the profilometer. The measuring cell was powered from a genera-

...generator 1 with a voltage of 1.5 V and a frequency of 1 MHz. The bridge unbalance signal is taken from the secondary winding of transformer Tr 1 and fed to the input of the measuring device, consisting of two emitter followers 2 and 6, a resonant amplification cascade—3, a detector—4, a video amplifier—5, and an electronic oscilloscope.

For balancing the bridge and monitoring the generator voltage, a millivoltmeter—9—has been introduced into the circuit. The circuit is powered from a stabilized source—8.

According to preliminary tests of the capacitive profilometer, the following results were obtained. With setting gaps of 75, 100, and 150 μm, the circuit provides a constant sensitivity of 1 mV/μm when the gap varies over wide limits, and high resolving power. At smaller setting gaps, the sensitivity of the circuit depends to a considerable extent, nonlinearly, on the change in the gap, which is explained by the nonlinearity of the conversion characteristic of the capacitive probe.

Thus, a capacitive profilometer with a measuring cell in the form of a bridge with close inductive coupling is an instrument that makes possible a comprehensive investigation of the mechanical and dynamic properties of the commutator-and-brush assembly in electric machines. Possessing high sensitivity and resolving power, it makes it possible to reduce the measurement error substantially.

REFERENCES

1. K. K. Namitokov, V. F. Chepura, V. G. Brozinsky. Instrument for investigating dynamic changes in the surface shape of commutators of electric machines. Elektromekhanika, 1, 1966.

2. K. V. Chernyshev. Instrument for measuring small amplitudes. Pribory i tekhnika eksperimenta, 6, 1959.

3. D. A. Kennedi. The measurement of Commutator Profiles. Conf. Commut. Rotat. Mach, London, 1964.

4. K. M. Sobolevsky, Yu. A. Shkoda. Protection of AC bridges. Publishing House of the Academy of Sciences of the Ukrainian SSR, Kiev, 1957.

5. G. V. Gessen. On the sensitivity of certain bridge circuits with inductively coupled arms. Trudy VNIIM, 67 (127), 1962.

6. K. B. Karandeev. Bridge Methods of Measurement. Gostekhizdat USSR, 1953.

7. K. B. Karandeev, F. B. Grinevich, K. M. Sobolevsky. On the properties of one bridge circuit with mutual inductance between two adjacent arms. Proceedings of the Lviv Polytechnic Institute, vol. 1, issue 2, 1955.

8. A. V. Chebotarev, E. Yu. Nebolyubov. On determining the sensitivity of measuring circuits of digital bridges with inductively coupled arms. Sensors and circuits of moisture meters and level meters for the petrochemical industry. Ilim Publishing House, 1965.