UDC 621.319.3.001.037

ELECTRICAL MODELING OF THE ESGS FIELD

I. P. Guk, V. A. Lukutin

(Presented by the scientific seminar of the Department of TOE)

One of the basic conditions for increasing the specific power of electrostatic generators with carrier-conveyors (ESGS) is obtaining the maximum possible value of the average electric-field intensity; this can be achieved by increasing the electric strength of the medium and by equalizing the field in the working volume of the machine [1].

Achieving maximum uniformity of the electric field, in turn, requires a careful study of it, taking into account the physical properties of the materials from which the main elements of the generator are made, the geometrical dimensions of the electrodes, and their mutual arrangement. The best solution to this problem would undoubtedly be a complete mathematical description of the field; however, in such a multi-electrode system as an ESG, with a large number of carrier-conveyors, analytical calculations lead to very complicated expressions that are difficult to use for analysis. Direct experimentation on operating specimens in order to determine the differential characteristics of the field is practically impossible because of the small thickness of the air gap (fractions of a millimeter). At the same time, the use of electrical modeling methods makes it possible to carry out not only qualitative but also quantitative analysis with an accuracy sufficient for engineering purposes. And since, up to the present time, the field of the ESG has practically not been studied [2, 3], the physical clarity of the solution is of great importance; this is achieved by constructing field patterns with the aid of an electrolytic bath—the simplest and most economical method of electrical modeling, based on the analogy between the mathematical equations of the current field in a conducting medium and the electrostatic field.

In studying an ESG, the process of converting mechanical energy into electrical energy is of primary interest; therefore, at the first stage it is quite sufficient to confine ourselves to constructing a model for studying the electric field of a generator operating in a steady-state regime, for the instant of time corresponding to a definite arrangement of the stator and rotor elements.

The electric field of the generator under consideration satisfies the conditions of quasistationarity; consequently, for any instant of time it can be described by Poisson’s equation, and in modeling it the similarity of the phenomena in the actual object and in the model is obtained automati-

... only when geometric similarity is observed and the boundary conditions are specified correctly [4].

Below, a method is presented for modeling, in an electrolytic bath, the field of a cylindrical ESG with carrier-conductors, the operating principle and structural elements of which are described in [1, 2, 3] and in a number of other works.

Figure 1 shows the schematic diagram, arrangement, and geometric dimensions of the ESG elements. Since the length of the generator is large in comparison with the carrier–stator distance, then in modeling

Fig. 1

the edge effect may be neglected and the field may be considered plane-parallel. As is seen from the figure, the rotor consists of \(2m\) carrier-conductors arranged at equal spacing and insulated from one another. If one assumes the absence of leakage through the insulation, which must hold in a well-designed generator, and contact commutation, then in the steady-state regime, at the instant when the 1st and \((m+1)\)-st carriers separate from the corresponding brushes, carriers \(1, 2, 3 \ldots m\) have charge \(q\), while \((m+1), (m+2), \ldots, 2m\) have charge \(q'\). The difference of these charges is delivered to the load, whose resistance is \(R_{\mathrm{n}}\), and thereby both the current \((I_{\mathrm{n}})\) and the voltage \((U_{\mathrm{n}})\) of the load are determined. For a more uniform distribution of voltage over the stator, the latter is usually made of a semiconducting material (glass). Thus, the realization of the boundary conditions in modeling the ESG field will consist, as it were, in solving two problems: first, in specifying the voltage distribution over the stator (if the charges of the carriers are taken to be zero, this will be the Dirichlet problem for Laplace’s equation), and second, in specifying current sources that model the right-hand side of Poisson’s equation.

In solving the first problem, the boundary conditions are usually specified discretely by means of equipotential electrodes [7, 8]. Such a discrete approximation of a continuous potential function, although universal, has a number of shortcomings, the chief of which are the “sagging” of the potential between electrodes and the need to shift the actual boundaries of the model [10], which leads

to significant errors in the case of modeling the field in narrow regions, for example, in the air gap of electromagnetic and electrostatic machines. Attempts to eliminate these shortcomings by increasing the number of electrodes that define the boundary function lead, correspondingly, to an increase in the number of variable resistances and measuring instruments. And if one takes into account that in an ESG model it is also necessary to implement a large number of current sources, it becomes obvious that modeling in an electrolytic bath loses such advantages as simplicity and convenience in setting up the model and in operating it.

To eliminate the errors that arise in the discrete approximation of the boundary function by the minimally permissible number of equipotential electrodes, one can use a thin strip of constant cross section made of a poorly conducting material (for example, rubber with increased conductivity). For this purpose it is necessary to place the strip along the contour of the region under study and to connect to it, ensuring good contact, the boundary electrodes, the number of which may be taken to be half as large as in the known method [8]. In some cases it is even possible to dispense with boundary electrodes altogether, as is done in modeling with conducting paper [9]. Thus, for example, when specifying in the model a linear distribution of voltages, it is sufficient to cut out a strip of constant width and connect a voltage source to its ends.

Fig. 2. Modeling circuit in load mode: A — ammeter, P — switch, LI — low-voltage inductor, HI — high-voltage inductor

The specific volume conductivity of the poorly conducting material is selected experimentally, on the basis of the following condition: it must be considerably less than the conductivity of the electrodes, but greater by approximately two orders of magnitude than the conductivity of the electrolyte.

In an electrostatic generator it is desirable to have the voltage distribution over the stator be linear [1, 2, 3, etc.]; in this case, modeling of the boundary function by the method described will be simple and at the same time free of the shortcomings inherent in the discrete specification of boundary conditions.

Implementation of charged transporters presents no difficulty: in the model it is sufficient to represent them by conductors connected through large variable resistances to a voltage source. Considerable inconvenience arises, however, when setting up the model, since it is necessary to have a large number of milliammeters (equal to the number of transporters) for measuring the currents. Therefore, switches were introduced into the modeling circuits shown in Figs. 2 and 3; these make it possible to use one measuring instrument, or two if the charges of the transporters have the same sign. If, when the switch is rotated

one of the electrodes is connected through a resistor and a milliammeter to the terminal of the voltage source, while all the others are connected to this terminal, bypassing the measuring instrument. The current values are adjusted by variable resistors.

For a prescribed distribution of voltages over the stator and a known excitation voltage \((U_{\mathrm{в}})\), the model must be adjusted so that

\[ I_1 = I_2 = \ldots = I_m \quad \text{and} \quad I_{m+1}=I_{m+2}=\ldots=I_{2m}, \]

and the ratio of the magnitudes of these currents is such that the potential difference between the high-voltage inductor and the transporter located beneath it \((m+1)\) is equal to zero. This is explained by the fact that, in the original, during commutation they are connected by means of a brush.

Fig. 3. Modeling circuit in short-circuit mode

In the event of a change in the operating mode of the ESG, field modeling must be carried out by the method described, with the corresponding specification of the boundary and initial conditions.

Experiments to obtain patterns of the electric field of the ESG were carried out in an electrolytic bath measuring \(1200 \times 1200 \times 1200\) mm. The bath was welded from sheets of vinyl plastic 10 mm thick and reinforced with stiffening ribs, while the measuring circuit was assembled on a separate plate of organic glass. As the electrolyte, a mixture of distilled and purified tap water was used, with a specific volume conductivity on the order of

\[ 10^{-4}\,\frac{1}{\Omega\,\text{cm}}. \]

To reduce the galvanic effect, all electrodes were made of brass.

The accuracy of the modeling was determined on models of capacitors of different configurations (plane—plane, cylinder—cylinder, etc.), the field patterns of which can be obtained by calculation. At the same time, the influence of the frequency of the alternating current used to feed the model on the measurement results was checked. The scatter of the latter when the frequency was varied from 50 to 2000 Hz was no more than 3% for internal problems of field theory and no more than 15% for external ones; and since the ESG field belongs to the former, subsequently the supply was carried out with current of industrial frequency.

The theoretical justification of the electrolytic-bath method, descriptions of the devices on which this method is implemented, sources of errors and ways of eliminating them are set out in detail in [4, 5, 6] and elsewhere; therefore we have dwelt only on the points directly relating to modeling the ESG field.

Before proceeding to a detailed study of the electric field of the ESG, field patterns were obtained throughout the rotor–stator gap under the assumption of homogeneity of the medium (the dielectric permittivity of the air gap and of the rotor material were taken to be the same—).

coil). The experiments were carried out for different operating modes of the generator while varying the number of conveyors—conductors of circular cross section in the case of the conduction excitation circuit. As an example, the paper presents field patterns of an ESG with 32 conveyors for the short-circuit, load, and no-load modes.

In all the patterns, the excitation voltage is taken as unity, i.e., the potential of the conveyor located under the low-voltage inductor, while the potential of the latter is equal to zero. The potential difference between any two adjacent equipotentials in Figs. 4 and 5 is \(0.4 U_{\mathrm{v}}\), and in Fig. 6 it is \(0.32 U_{\mathrm{v}}\).

Fig. 4

From the field pattern for the short-circuit mode (Fig. 4) it is evident that the potentials in one half of the gap are above zero, and in the other below zero. This can occur only when the charges of the forward- and reverse-motion conveyors have different signs. This phenomenon is explained by the coupling between the commutating conveyor and the neighboring ones and is called natural recharging. With positive charges on all conveyors in the model, it is impossible to establish a potential difference equal to zero between the high-voltage inductor and the \((m+1)\)-th conveyor.

To determine the influence of the number of conveyors on the field, the model was made with 8, 16, 24, and 32 conveyors of the same radius \((r_{\mathrm{t}})\). It was found that, as the distance between conveyors decreases, for the same charge difference \(q\) and \(q'\), the field becomes more uniform: the normal component of the electric-field intensity decreases, while the tangential component increases. This means that the number of conveyors, at a constant rotor diameter, should be increased until the maximum short-circuit current begins to decrease owing to discharges along the collector. Thus, it is necessary to find such a rational generator design as to obtain maximum field uniformity with a minimum tangential component of the electric-field intensity.

Figure 5 presents the field pattern for the no-load mode of an ideal generator. In most of the working volume of the machine the field is fairly uniform, and its distortion results from the large diameter of the conveyors. The greatest potential gradients are reached in the zone of the low-voltage inductors, and the smallest in the zone of the high-voltage ones. The latter explains the armature reaction, which, by analogy with electromagnetic machines, is understood as a decrease in current when the load voltage is increased.

If the superposition method is used, then in the load mode—and, consequently, also in the no-load mode (the latter may be considered as the limiting load mode)—the electric field

in the working volume of the ESG can be decomposed into two components: the field \(\overline{E}_c\), created by the distribution of voltages over the internal surfaces of the semiconducting stator, and the rotor field \(\overline{E}_p\)—the field from the linear charges of the carriers, i.e.,

\[ \overline{E}=\overline{E}_c+\overline{E}_p . \]

The best generator is considered to be one in which the intensity \(\overline{E}_c\) at any point of the air gap would be perpendicular to \(\overline{E}_p\). Unfortunately, in an ESG with conducting carriers this cannot be achieved; under load conditions there appears a normal component of the stator field, the magnitude of which depends on the geometrical and physical parameters of the ESG. Thus, by modeling it has been established that \(E_{cn}\) increases with an increase in the thickness of the rotor and of the air gap, in the number of carriers and their radii, and also that \(E_{cn}\) is the larger, the more significantly the dielectric constants of the air gap and of the rotor body material differ. It was also found that, on a line passing through the centers of the inductors, in the region of carrier discharge the intensities \(\overline{E}_{cn}\) and \(\overline{E}_{pn}\) are subtracted, while in the charge zone they are added arithmetically, which leads to undesirable effects.

Fig. 5

A more detailed investigation of the ESG field showed that with an increase in \(\overline{E}_{cn}\) the theoretical voltage multiplication factor decreases,

\[ \kappa_U=U'_{xx}/U_{\mathrm{в}} . \]

Modeling of the ESG showed: under load conditions the fields can be considered as intermediate between the short-circuit field and the no-load field; consequently, all regularities inherent in the latter will be valid for the generator field in the load regime.

The field pattern (Fig. 6) was obtained for the case in which the charge was taken as equal to zero, since in this load regime it is easier to detect the general regularities. In the working part of the rotor (forward stroke), both the normal and tangential components of the field are large, especially in the zone adjacent to the excitation inductors. From the field pattern it is seen that three carriers approaching the discharge brush have a potential higher than the potential of the high-voltage inductor. It was also noted that the number of such carriers and their potentials increase as the distance between the carriers decreases at a constant rotor diameter. The converging part of the rotor (return stroke) is under more favorable conditions: here \(E_n\) is very small.

Summing up the consideration of the generator field patterns, it should be noted that changing the same parameters leads to both positive and negative phenomena: natural recharging and armature reaction, equalization of the field in the rotor–stator gap and a decrease in the coefficient \(k_v\), etc. Therefore, in order to obtain

Fig. 6

maximum specific power, it is necessary to carry out a more detailed study of the electric field of the ESG in combination with theoretical and experimental investigations. In this connection, first of all it is necessary to clarify the influence of the material of the stator and rotor, the medium in the gap, the geometry of the stator electrodes and the conveyors on the magnitude of the maximum field strength in the gas gap of the generator.

REFERENCES

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