On the Calculation of the External Characteristic of an Electrostatic Generator with a Cascade Connection of Conductor Conveyors

A. P. Kononov, V. A. Lukutin, V. V. Patsevich

(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering)

The creation of an industrial-type electrostatic generator with conductor conveyors is associated with solving a whole range of problems, one of which is the calculation of the external characteristic of the generator. In the present work an approximate method for this calculation is given, and an attempt is made to analyze the phenomenon of natural recharging [3], which to some extent affects the external characteristic of the generator.

The magnitude of the short-circuit current for a generator of this type, without taking natural recharging into account, is determined from the equation [1–2]:

\[ I_0 = 2m \cdot n q_0 = 2mnC_0U_{\mathrm{v}}, \tag{1} \]

where \(m\) is the number of conductor conveyors,

\(n\) is the number of rotor revolutions per second,

\(q_0\) is the charge of one conductor conveyor,

\(C_0\) is the charging capacitance of a conveyor,

\(U_{\mathrm{v}}\) is the excitation voltage,

Figure 1. Dependence \(I_{\mathrm{kz}} = f(U_{\mathrm{v}})\).
1. Experimental dependence when one conveyor is bridged by a brush.
2. Calculated dependence without taking the recharging current into account.
3. Calculated dependence taking the recharging current into account.
4. Experimental dependence when three conveyors are bridged by a brush.

The calculated values of the short-circuit current, computed by formula (1), and the experimental data in the form of the dependence \(I_{\mathrm{kz}} = f(U_{\mathrm{v}})\), are presented in Fig. 1. It is seen from the graph that the value of the short-circuit current at the same excitation voltage, calculated from equation (1), is substantially smaller than that determined experimentally. The increase in the short-circuit current in the electrostatic generator in comparison with the calculated value occurs owing to natural recharging [3].

This increase in current in a real generator can be explained as follows.

In the short-circuit mode, when conveyor \(m_1\) (Fig. 2) approaches the high-voltage grounded brush, its charge \(+q_0\) goes to ground. But next to conveyor \(m_1\) there is a neighboring conductor \(m\), having a charge \(+q_0\) at a high positive potential. This conveyor \(m\) will induce on the grounded conveyor \(m_1\) a certain charge of opposite sign \((-q_1)\). Thus, a charge \(Q = q_0 + q_1\) will flow in the grounding conductor, which leads to an increase in the short-circuit current of the generator.

Fig. 2.

For the investigation of the recharging phenomenon, an experimental disk-type generator with a cascade connection of conveyor-conductors was constructed, in which provision was made for changing the position and width of the brushes, as well as for measuring the current of the departing conveyors.

We recorded the dependence of the recharging current for the experimental generator, which is presented in Fig. 3 (dependence 1).

In analyzing the natural recharging occurring in the electrostatic generator, we came to the following conclusion.

The magnitude of the charge \(-q_1\) will be determined by the capacitance between the neighboring conveyors \(m\) and \(m_1\) (Fig. 2) and by the potential difference between them. Therefore the charge \(q_1\), induced on conveyor \(m_1\), can be determined from the equation:

\[ q_1 = C_1 \cdot \Delta U, \tag{2} \]

where \(C_1\) is the capacitance between neighboring conveyors, \(\Delta U\) is the potential difference between conveyors \(m\) and \(m_1\).

This charge \(q_1\), in the absence of leakage, remains unchanged on the departing conveyors \(m_2\), \(m_3\), etc. (Fig. 2) and determines the recharging current in the generator, the value of which was measured experimentally.

The magnitude of the recharging current in the generator can be calculated from an expression analogous to equation (1),

Fig. 3. Dependence \(I_{\text{per}} = f(U_{\mathrm{v}})\).
1. Experimental dependence with one conveyor covered by the brush.
2. Calculated dependence.
3. Experimental dependence with three conveyors covered by the brush.

\[ I_{\text{per}} = 2mnq_1. \tag{3} \]

The calculated dependence of the recharging current is presented in Fig. 3 (dependence 2).

From the graph in Fig. 3 it is evident that the experimental dependence has values much greater than those calculated from equation (3). This can be explained by the fact that, in determining the charge \(q_1\), only the capacitance between neighboring conveyors was taken into account, while the influence of other conductors was not taken into account in the calculation.

Taking the recharge current into account, the short-circuit current of the generator can be determined from the equation

\[ I_{\mathrm{sc}}=I_0+I_{\mathrm{re}}. \tag{4} \]

The calculated dependence of the short-circuit current, found from formula (4), is shown in Fig. 1 (dependence 3), which almost coincides with the experimental one (dependence 1).

The definition of natural recharging given above applies to the case when the brush covers only one conveyor and, in this case, the recharge current has its maximum value. If, however, the brush covers two, three, or more conveyors, then the recharge current decreases.

This can be explained as follows.

When conveyor \(m\), with charge \(q_0\), approaches a wide brush (Fig. 4), a phenomenon occurs analogous to that described above. But in this case, with the further motion of the conveyor, it continues to remain connected to the grounded brush.

Fig. 4.

When the conductor occupies position \(m_2\), it has zero potential and is already separated from the positively charged approaching conveyor \(m\) by one intermediate conveyor \(m_1\) with potential equal to zero.

Naturally, the effect on it of the positive charge \(q_0\) will be considerably weakened. It will be weakened still more when the conveyor occupies position \(m_3\).

Since the conveyors covered by the brush screen one another, the electrostatic influence of the conductor approaching the brush on the departing conveyor is very small. Therefore the recharging phenomenon is not observed, and the conveyors leaving the high-voltage brush will have no charge.

The experimental dependences \(I_{\mathrm{re}}=f(U_{\mathrm{v}})\) and \(I_{\mathrm{sc}}=f(U_{\mathrm{v}})\), when the brush covers three conveyors, are presented respectively in Fig. 3 (dependence 3) and in Fig. 1 (dependence 4).

Fig. 5.

As is seen from Fig. 3, the recharge current is almost absent, while Fig. 1 shows good agreement between the experimental dependence and the calculated dependence computed from equation (1) without taking recharging into account.

The recharging phenomenon is also observed in the excitation system [3], which is not considered in the present work and therefore was eliminated in the experimental generator by installing a wide brush covering three conveyors.

When the generator operates into a load (load mode), the value of the current flowing through the load will decrease, and at no load it will be equal to zero. In other words, in load mode the charge \(q_0\) approaching the high-voltage brush will not be completely removed from the conveyor; part of it will remain on the departing conveyors. Let us call this charge residual and denote it by \(q_{\mathrm{res}}\).

To determine the residual charge \(q_{\mathrm{res}}\), an experiment was carried out on an actual generator with a brush covering three conveyors, since in this case there is no recharge current.

This made it possible to obtain the change in the residual charge \(q_{\text{res}}\), which in turn gives the residual current \(I_{\text{res}}\).

In studying the operation of the generator it was established that the magnitude of the residual current (in the absence of recharging) varies linearly from zero under short-circuit conditions to a value equal to \(I_0\) under no-load conditions (Fig. 6, dependence 4).

Fig. 6. Dependence \(U_{\text{l}} = f(I_{\text{l}})\).
1. Calculated dependence, determined from equation (9). 2. Experimental dependence. 3. Value of the current \(I_0\), calculated from equation (1). 4. Value of the residual current \(I_{\text{res}}\). 5. Value of the recharging current \(I_{\text{rech}}\). 6. Value of the outgoing current \(I_{\text{out}}\).

On the basis of the experiment, a dependence was obtained that makes it possible to calculate the magnitude of the residual current in the form of the equality

\[ I_{\text{res}} = I_0 \frac{U_{\text{l}}}{U_{\text{xx}}}, \tag{5} \]

where \(I_0\) is the value of the current determined from equation (1);

\(U_{\text{l}}\) is the load voltage;

\(U_{\text{xx}}\) is the no-load voltage.

In the short-circuit regime \(U_{\text{l}} = 0\), and therefore \(I_{\text{res}} = 0\). In the no-load regime \(U_{\text{l}} = U_{\text{xx}}\), and therefore \(I_{\text{res}} = I_0\). This is confirmed by experiment (dependence 4, Fig. 6).

When analyzing the load regime, in the case where the brush covers only one conveyor, it turned out that there is a charge on the outgoing conductors, which we shall call the outgoing charge \((q_{\text{out}})\), and it determines the outgoing current \(I_{\text{out}}\).

Then, when considering the load regime (Fig. 5), we shall have three currents—the current \(I_0\) approaching the brush, \(I_{\text{l}}\), the load current, and \(I_{\text{out}}\), the outgoing current.

The value of the load current can then be determined from the equation:

\[ I_{\text{l}} = I_0 - I_{\text{out}}. \tag{7} \]

The experimental dependence of the outgoing current is presented in Fig. 6 (dependence 6).

The magnitude of this current can be determined using the superposition principle, according to the equality

\[ I_{\text{out}} = I_{\text{res}} - I_{\text{rech}}. \tag{8} \]

The value of the recharging current is taken with a minus sign because this current is due to a charge having the opposite sign.

Combining equations (7) and (8), we obtain the dependence for determining the generator load current

\[ I_{\text{l}} = I_0 - I_{\text{out}} = I_0 - (I_{\text{res}} - I_{\text{rech}}) = I_0 + I_{\text{rech}} - I_{\text{res}}. \tag{9} \]

The calculated dependence \(U_{\text{l}} = f(I_{\text{l}})\), computed from equation (9), and the experimental dependence, shown in Fig. 6, agree well.

Conclusion

1. The phenomenon of recharge is observed not only in the short-circuit mode, but also occurs under load conditions.

2. Natural recharge is noted as a positive phenomenon, useful in generator operation at high currents, but at low voltage.

3. The magnitude of the recharge current in the high-voltage system, to a first approximation, can be calculated from equation (3).

References

1. A. A. Vorob’ev et al. High-Voltage Testing Equipment and Measurements, GEI, 1960.

2. A. F. Ioffe. ZhTF 9, No. 23, 2071, 1939.

3. V. V. Patsevich. Some Problems in the Theory and Calculation of ESG with Conductor Conveyors, Dissertation, TPI, 1964.