Volt-Ampere Characteristics of a Sliding Contact in Static and Dynamic Operating Modes

B. I. Kostylev, R. F. Bekishev, A. I. Skorospelkin

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

A detailed study of the physical properties of the sliding brush contact of commutator electric machines is one of the most important conditions for solving complex problems of commutation. In recent years, a large number of works have been published on the study of the properties of the sliding contact and, in particular, of the conditions for the formation of the surface film of the commutators of electric machines.

For the theoretical study of commutation, both static and dynamic volt-ampere characteristics are used. Study of the structure of the surface film makes it possible to explain the physical nature of the volt-ampere characteristics.

Electron-microscopic investigation of the surface film of commutators [1], as well as data from microchemical analysis [2], show that the structure of the commutator film consists mainly of cuprous oxide \(\mathrm{Cu_2O}\) with inclusions of graphite (\(\mathrm{Cu_2O}\)—65.8 percent, \(\mathrm{C}\)—22.1 percent). According to [2], the thickness of the commutator film varies within the range \(50\text{--}400\,\text{\AA}\).

The presence in the film of a large amount of cuprous oxide, which has semiconductor properties, makes it possible in the investigation to draw a parallel with a cuprous-oxide rectifier. Indeed, the cuprous-oxide film on the commutator of an electric machine forms with the copper of the commutator a blocking layer having the properties of a \(P\)-type semiconductor.

According to the investigations [3, 4], the authors of which analyze the polar properties of electric brushes from static volt-ampere characteristics taken on a short-circuited commutator, the equivalent circuit of the sliding brush contact may be represented by the parallel connection of the rectifier resistance (\(R_{\mathrm{direct}}\) and \(R_{\mathrm{reverse}}\)) and the resistance of the point regions of the film, \(r\), conducting in both directions (Fig. 1a). In this case, for the cathode brush the \(\mathrm{Cu_2O}\) film constitutes an insulating surface. Naturally, on the commutator of an electric machine the blocking layer between the film and the copper, although formed over the entire surface, is very thin and has individual conducting regions caused by the passage of current (\(\alpha\)-spots), inclusions of graphite, and friction. Under the anode brush the barrier layer is conducting (\(R_{\mathrm{direct}} \gg R_{\mathrm{reverse}}\)).

Considering the equivalent circuit of the sliding contact proposed—

Fig. 1. Equivalent circuits of a sliding brush contact

... in [3], it should be noted that representing the cuprous-oxide rectifier only by ohmic resistances \(R_{\mathrm{pr}}\) and \(R_{\mathrm{obr}}\) corresponds to the static operating mode; therefore only static characteristics, taken on a short-circuited collector with direct current flowing, will be equivalent to the characteristic of the indicated equivalent circuit.

However, describing the properties of the brush contact by a static characteristic does not correspond to reality, since under real operating conditions of commutator DC electric machines the lamellar current is alternating, changing at a frequency of \(1\)—\(3\) kHz and higher. In this case, according to semiconductor theory [5, 6, 8], the blocking layer must be represented by an equivalent circuit that includes a parallel connection of the capacitance of the blocking layer \(C\) and the variable ohmic resistance of the layer \(R_{\mathrm{a.c.}}\). The capacitance of the blocking layer depends on the magnitude and polarity of the applied voltage and consists of two components: barrier capacitance and diffusion capacitance. The latter has a considerable influence at forward voltages. According to [8], for cuprous-oxide rectifiers the magnitude of the diffusion capacitance is of the same order as the barrier capacitance even at zero voltage. At reverse voltages the influence of the diffusion capacitance disappears.

The thickness of the copper-oxide film on the collector of an electric machine (the collector polish) is much smaller than the \(\mathrm{Cu_2O}\) layer in cuprous-oxide rectifiers. Accordingly, the magnitude of the blocking layer is also smaller. Therefore the specific capacitance may reach several hundred \(\mu\mathrm{F}/\mathrm{cm}^2\) and higher. Such a capacitance value is in agreement with the data of [7, 9],

obtained in the study of commutation, according to which the capacitance can reach values of \(10^{-4}\)–\(10^{-1}\) F/cm\(^2\) and even 0.5 F/cm\(^2\). Thus, the fundamental difference between static and dynamic volt-ampere characteristics lies in the fact that, in the dynamic operating mode of a brush contact, a significant effect is exerted by the capacitance of the blocking layer, which depends on the magnitude and polarity of the applied voltage. The ohmic resistance of the barrier layer also depends on the magnitude and direction of the applied voltage.

If one proceeds from the above premises, the brush contact can be represented by an equivalent circuit (Fig. 1, b), the characteristics of which will be equivalent both to the static characteristics taken on a short-circuited commutator and to the dynamic volt-ampere characteristics of a real electrical machine.

Let us explain the parameters of the equivalent circuit.

The resistance between the brush and the commutator film, which depends on the type of brushes and on the mechanics of the commutator-brush assembly, is determined by the series-connected resistance \(r_{\mathrm{shch}}\). The resistance of the regions of active conductivity of the disturbed film, as well as the total point resistance of \(\alpha\)-spots formed by fritting, is taken into account by the leakage resistance \(R_{\mathrm{ut}}\).

The variable ohmic resistance of the blocking semiconducting layer, which depends on the magnitude and polarity of the applied voltage, is determined by the resistance \(R_{\mathrm{ac}}\).

\(C\) is the total capacitance, including the barrier and diffusion capacitances connected in parallel.

The resistance of the body of the semiconducting commutator film \(\mathrm{Cu}_2\mathrm{O}\) is determined by the resistance \(r_{\mathrm{p}}\) connected in series with the \(R\)–\(C\) circuit. Since the thickness of the commutator film is very small (50–400 Å), the value of the resistance \(r_{\mathrm{p}}\) is insignificant, and in calculating the equivalent circuit it may be neglected. Then the equivalent circuit is considerably simplified (Fig. 1, c).

The parameters of the equivalent circuit may be determined from static volt-ampere characteristics taken on a short-circuited commutator. In this case the resistance between the brush and the commutator film \(r_{\mathrm{shch}}\) may be determined either from the cathodic or from the anodic volt-ampere characteristic at a high current density, when the leakage resistances and the resistance of the semiconducting layer may be neglected with sufficient accuracy for practical calculations. It should be noted that the resistance \(r_{\mathrm{shch}}\) takes into account the average mechanical condition of the commutator on which the characteristics were taken, and depends on the contact area of the lamella with the brush. Therefore, on another commutator its value may be different. If the parameters of the commutator-brush assembly of the machine are known, then the mechanics can be taken into account by a corresponding change of \(r_{\mathrm{shch}}\) by lamellae.

The volt-ampere characteristic of the nonlinear leakage resistance is determined from the found value of \(r_{\mathrm{shch}}\) and the volt-ampere characteristic for the cathodic brush, the resistance of the semiconducting layer under which may be taken as infinitely large, as under a reverse applied voltage.

Finally, the volt-ampere characteristic of the semiconducting layer under a forward applied voltage is determined from the volt-ampere characteristic for the anodic brush and from the known \(r_{\mathrm{shch}}\) and \(R_{\mathrm{ut}}\). This resistance does not depend on the contact area of the lamella with the brush, since it is determined by the area of the barrier layer. Thus, all active resistances of the equivalent circuit are determined on the basis of static volt-ampere characteristics.

The determination of the contact capacitance is considerably more complicated, since up to the present time semiconductor theory has had no acceptable method for calculating the capacitance, and its value is determined experimentally from the phase shift during operation of the rectifier on alternating current. Therefore, in order to determine the capacitance value at a given voltage in the contact (preferably at a voltage across the capacitance equal to zero), it is necessary to have the dynamic volt-ampere characteristic, or the corresponding part of it, for one of the lamellae.

The character of the variation of capacitance with voltage for a given type of semiconductor is the same and does not depend on the dimensions and construction of the rectifier [5]. For example, for cuprous-oxide rectifiers the specific capacitance may be written as

\[ C = \frac{A}{2 - U_{\mathrm{sl}}}, \tag{1} \]

where

\[ A = 2C_0, \]

\(C_0\) is the capacitance of the semiconductor at zero voltage,

\(U_{\mathrm{sl}}\) is the voltage of the barrier layer, positive under forward voltage.

The dependence of the capacitance on the magnitude and polarity of the applied voltage is given in Fig. 2.

According to the equivalent circuit (Fig. 1,c), the lamella current is determined by

\[ i = i_c + i_{\mathrm{ut}} + i_{\mathrm{ac}}, \tag{2} \]

where the capacitive current is

\[ i_c = \frac{dq}{dt} = C \frac{dU_{\mathrm{sl}}}{dt} + U_{\mathrm{sl}} \frac{dC}{dt}, \]

the leakage current is

\[ i_{\mathrm{ut}} = f(SU_{\mathrm{sl}}), \]

\(S\) is the area of contact between the lamella and the brush, \(i_{\mathrm{ac}} = \varphi(U_{\mathrm{sl}})\) is the current of the semiconducting layer; \(C\) is the capacitance of the blocking layer in farads. The current \(i_{\mathrm{ac}}\) for a cathode brush is equal to zero, since a reverse voltage is applied.

Then expression (2), taking (1) into account, may be written as

\[ i = \frac{A}{2 - U_{\mathrm{sl}}}\cdot \frac{dU_{\mathrm{sl}}}{dt} + \frac{AU_{\mathrm{sl}}}{(2 - U_{\mathrm{sl}})^2} + f(SU_{\mathrm{sl}}) + \varphi(U_{\mathrm{sl}}) \tag{3} \]

or, after transformations,

\[ i = \frac{2A}{(2 - U_{\mathrm{sl}})^2}\cdot \frac{dU_{\mathrm{sl}}}{dt} + f(SU_{\mathrm{sl}}) + \varphi(U_{\mathrm{sl}}). \tag{4} \]

This expression is valid for brushes of both polarities, with the sole exception that, in the case of a cathode brush, the last term is absent from expression (4), since the current of the semiconducting layer is equal to zero. Since the voltage of the brush contact is created by the current through the lamella, which depends on the nature of the commutation, although it, in turn, affects the magnitude and nature of the change of this current, expression (4) must be solved with respect to \(U_{\mathrm{sl}}\) for known current through the lamella and area of contact of the lamella with the brush. Since expression (4) is a nonlinear differential equation, the solution must be carried out by a numerical method on an electronic digital computer. The total voltage of the brush contact is determined, for the corresponding polarity, as

\[ U_K = U_{\mathrm{sl.K}} + U_{\mathrm{sh}} = U_{\mathrm{sl.K}} + \frac{r_{\mathrm{sh}}}{S}\, i \]

Determining the capacitance of the contact is considerably more complicated, since up to the present time semiconductor theory has lacked an acceptable method for calculating capacitance, and its magnitude is determined experimentally from the phase shift during operation of a rectifier on alternating current. Therefore, to determine the magnitude of the capacitance at a given voltage in the contact (preferably at a voltage on the capacitance equal to zero), it is necessary to have the dynamic volt-ampere characteristic, or the corresponding part of it, for one of the lamellae.

The nature of the variation of capacitance with voltage for a given type of semiconductor is the same and does not depend on the dimensions and design of the rectifier [5]. For example, for copper-oxide rectifiers the specific capacitance may be written as

\[ C = \frac{A}{2 - U_{\text{sl}}}, \tag{1} \]

where

\[ A = 2C_0, \]

\(C_0\) is the capacitance of the semiconductor at zero voltage,

\(U_{\text{sl}}\) is the voltage of the barrier layer, positive for forward voltage.

The dependence of the capacitance on the magnitude and polarity of the applied voltage is shown in Fig. 2.

According to the equivalent circuit (Fig. 1c), the lamella current is determined as

\[ i = i_c + i_{\text{leak}} + i_{ac}, \tag{2} \]

where the capacitive current is

\[ i_c = \frac{dq}{dt} = C\frac{dU_{\text{sl}}}{dt} + U_{\text{sl}}\frac{dC}{dt}, \]

the leakage current is

\[ i_{\text{leak}} = f(SU_{\text{sl}}), \]

\(S\) is the area of contact of the lamella with the brush; the current of the semiconducting layer is \(i_{ac} = \varphi(U_{\text{sl}})\); \(C\) is the capacitance of the blocking layer in farads. The current \(i_{ac}\) for the cathode brush is equal to zero, since a reverse voltage is applied.

Then expression (2), taking (1) into account, may be written as

\[ i = \frac{A}{2 - U_{\text{sl}}}\cdot \frac{dU_{\text{sl}}}{dt} + \frac{AU_{\text{sl}}}{(2 - U_{\text{sl}})^2} + f(SU_{\text{sl}}) + \varphi(U_{\text{sl}}) \tag{3} \]

or, after transformations,

\[ i = \frac{2A}{(2 - U_{\text{sl}})^2}\cdot \frac{dU_{\text{sl}}}{dt} + f(SU_{\text{sl}}) + \varphi(U_{\text{sl}}). \tag{4} \]

This expression is valid for brushes of both polarities, with the sole exception that in the case of the cathode brush the last term is absent from expression (4), since the current of the semiconducting layer is equal to zero. Since the voltage of the brush contact is produced by the current through the lamella, which depends on the nature of commutation, although it, in turn, affects the magnitude and nature of the variation of this current, expression (4) must be solved with respect to \(U_{\text{sl}}\) for a known current through the lamella and area of contact of the lamella with the brush. Since expression (4) is a nonlinear differential equation, the solution must be carried out by a numerical method on a digital computer. The total voltage of the brush contact is determined for the corresponding polarity as

\[ U_k = U_{\text{sl.k}} + U_{\text{sh}} = U_{\text{sl.k}} + \frac{r_{\text{sh}}}{S} i \]

Fig. 2

\[ U_A = U_{\mathrm{sl}.A} + U_{\mathrm{sh}} - U_{\mathrm{sl}.A} + \frac{r_{\mathrm{sh}}}{S} i . \tag{5} \]

The volt-ampere characteristics obtained by this method will be nothing other than the dynamic volt-ampere characteristics of the brush contact, depending on the magnitude and nature of the change in current in the contact.

An obvious advantage of this method is that the calculation uses static volt-ampere characteristics, which are sufficiently stable and relatively easy to approximate. The dynamic volt-ampere characteristics are obtained as derivatives of the static ones, and are not specified for the calculation from experiment.

For verification, calculations were carried out of dynamic volt-ampere characteristics for different modes of commutation in an EMU-12A for EG-8 brushes, whose static volt-ampere characteristics are shown in Fig. 3. The number of brushes per brush bolt is 2; the brush dimensions are \(0.5 \times 1.5\ \mathrm{cm}^2\); the brush overlap is \(\beta = 1.5\).

The parameters of the equivalent circuit, determined from the static volt-ampere characteristics, are as follows:

\[ r_{\mathrm{sh}} = 0.07\ \Omega, \]

\[ i_{\mathrm{ut}} = 2S U_{\mathrm{sl}}^3, \]

\[ i_{\mathrm{ac}} = 3U_{\mathrm{sl}}^3 . \]

The specific capacitance at zero voltage, according to experimental data, is equal to \(C_0 = 250\ \mu\mathrm{F}/\mathrm{cm}^2\); then \(A = 2C_0 = 500\ \mu\mathrm{F}/\mathrm{cm}^2\). Since the area of the semiconducting layer on the lamella is equal to \(1\ \mathrm{cm}^2\), the capacitance is determined as:

Fig. 3

\[ C=\frac{500}{2-U_{\text{sl}}}\,[\mu\text{F}]. \]

Figures 4 and 5 show experimental curves of the current and voltage drop of the brush contact for EMU-12A under accelerated and retarded commutation for brushes of both polarities. Also shown there are calculated curves of the voltage drop obtained from expression (4). Since the purpose of the present work is to verify the proposed physical model of the sliding brush contact, the lamella current obtained experimentally was taken as the initial data in solving equation (4). In the case of a complete calculation of the commutation process, the values of the lamella current are determined directly in the course of the calculation when solving the system of differential equations of commutation. In addition, Figs. 4 and 5 show dynamic volt-ampere characteristics, representing the dependence of the instantaneous values of the contact voltage drop on the instantaneous values of the current density. As can be seen from the data presented, the calculated and experimental curves of the voltage drop in the contact agree sufficiently accurately both qualitatively and quantitatively. Consequently, the proposed equivalent circuit of the brush contact, taking into account its capacitance, correctly reflects the properties of the contact and can be used in calculations of the commutation process. Moreover, by the proposed method it is also possible to verify the influence on commutation of various factors affecting the properties of the brush contact, for example, mechanical vibrations of the brush through the resistance \(r_{\text{sh}}\) and temperature through the resistance of the semiconducting layer \(R_{\text{a.s}}\), as is done in semiconductors.

Fig. 4. Commutation during acceleration. Solid lines—calculated; dashed lines—experimental.

Conclusions

1. A physical model of the sliding brush contact is proposed, explaining the properties of the static and dynamic volt-ampere characteristics.

2. A method is proposed for calculating dynamic volt-ampere characteristics with allowance for the real properties of the sliding contact, on the basis of which a complete calculation of the commutation process can be carried out.

References

1. E. B. Granovskii, V. S. Titov. Electron-microscopic study of the surface film of collectors of electrical machines. Elektrotekhnika, No. 1, 1968.

Fig. 5. Delayed commutation. Solid lines—calculated; dashed lines—experimental.

1. Van Brunt C., Savage R. Collector films, parts I, II. General Electric Review, 47, 8, 1945; 46, 7, 1944.

2. V. I. Nællin. On the polar properties of electric brushes. Collection of scientific works of OMIIT, vol. XXIV, 1957.

3. F. Schroeter. Transient resistance of a sliding contact. ETZ-A, No. 14, 1955.

4. G. Henisch. Semiconductor Rectifiers. IIL, 1951.

5. L. S. Berman. Nonlinear Semiconductor Capacitance. Fizmatgiz, 1963.

6. Parker D. A. The Surface Properties of Copper—Graphite Sliding Contacts. Conf. Commut. Rotat. Mach., London, 1964.

7. J. N. Shive. Physical Properties and Design of Semiconductor Devices. GEI, 1963.

8. Kuczogi E., Schultze A. Discussion concerning the article “A Contribution to the Investigation of the Dynamic Characteristics of a Brush Contact,” ETZ-A, H-25, 1964.