STUDY OF THE PHENOMENON OF DOUBLE NATURAL TRANSFER IN AN ESG

A. M. Kuptsov, V. V. Patsevich

(Presented by the joint scientific seminar of the Department of TOE and the REST Division of the Scientific Research Institute of Nuclear Physics)

The use of a part of the rotor surface moving from the high-voltage pole to the excitation pole for energy conversion in an electrostatic generator (ESG), in the absence of recharging in the high-voltage-pole zone, is possible with a dielectric rotor if charges of the opposite sign are present on the rotor surface facing the stator.

By introducing the concept of ionic shielding, it is shown in [1] that, under the natural operating conditions of the generator, these charges must accumulate on the rotor surface, ensuring that the mean value of the electric-field strength is equal to zero at any point of its surface during one revolution for one pair of poles and during \(1/p\) of a revolution for \(p\) pairs of poles. A sufficiently high specific electrical resistance of the transporter material is considered to be the only condition for the manifestation of the ionic-shielding effect [1].

The phenomenon caused by the presence of charges on the “nonworking” surface of the rotor under the natural operating conditions of a generator with a dielectric rotor was called the phenomenon of double natural transfer. An important practical consequence of this phenomenon is that the current of a generator operating without forced recharging with a single excitation source will be the same as in generators with forced recharging, if

\[ U_{\mathrm{v}} = |U_{\mathrm{v}}'| + |U_{\mathrm{v}}''|, \tag{1} \]

where \(U_{\mathrm{v}}\) is the excitation voltage without forced recharging,

\(U_{\mathrm{v}}'\) and \(U_{\mathrm{v}}''\) are the excitation voltages in the charging and discharging systems in the scheme with forced recharging.

If it is taken into account that the practical implementation of a rotor ESG with forced recharging, which makes it possible at most to double the load current, encounters considerable technical and economic difficulties, whereas the use of an excitation source with an increased voltage value requires only a slight increase in the costs of its manufacture, then the practical importance of the consequence noted above for determining and realizing their maximum possible specific energy indices becomes clear. The validity of relation (1) is confirmed by the results of experiments [1], which, however, do not make it possible to draw any sufficiently convincing conclusions about the presence of the ionic-shielding effect [2] and do not reveal the ess—

phenomena underlying this relationship. Presented below are the results of investigations carried out on a generator with a cascade connection of transporters–conductors placed in a dielectric, with the aim of clarifying questions associated with the phenomenon of double natural charge transfer.

Calculation of the Current of an ESG in the Presence of Double Transfer

Considering the character of the electric field on the surface of the dielectric of the rotor, in which the transporters are placed, it is not difficult to establish that, in the presence of the effect of ionic shielding, this surface, like the nonworking surface of the dielectric rotor, must be charged with a charge of opposite sign relative to the charge of the transporter. In this case the magnitude of this charge must be such that the average value of the electric-field intensity on the rotor surface over one operating cycle of the generator is equal to zero. Schematically, the pattern of the electric field in the rotor–stator gap will have the form shown in Fig. 1. Here only one half of the rotor is shown. The lower part, relative to the axial line, will be analogous to the upper part and is not shown here.

Let us denote, for the excitation system at the moment when the transporter is disconnected from the brush, the charge of the transporter by \(q_1\), the charge of the element of the rotor surface corresponding to one transporter by \(q_2\), the charge of the excitation inductor by \(q_3\), the capacitance transporter–surface element by \(C_2\), and the capacitance between the surface element and the inductor by \(C_1\). For the high-voltage system: the charge of the transporter being disconnected from the brush by \(q_4\), the charge of the high-voltage inductor by \(q_5\), and the capacitance between the surface element corresponding to one transporter and the inductor by \(C_3\).

Neglecting the mutual influence of the transporters on one another, as a result of simple calculations (taking into account that the capacitances \(C_1\) and \(C_2\) are connected in series, while the capacitances \(C_2\) and \(C_3\) are connected in parallel), we obtain:

\[ - q_1 = C_0 U_{\mathrm{в}} + k_1 q_2, \tag{2} \]

\[ - q_4 = k_2 q_5, \tag{3} \]

where

\[ C_0 = \frac{C_1 C_2}{C_1 + C_2}, \qquad k_1 = \frac{C_2}{C_1 + C_2}, \qquad k_2 = \frac{C_2}{C_2 + C_3}. \]

In equations (2) and (3), the quantities \(U_{\mathrm{в}}\) and \(q_2\) are taken with their corresponding signs.

Neglecting leakage of charges from the transporters and from the rotor surface, for the generator load current from (2) and (3) we obtain

Fig. 1. Schematic representation of the field in an ESG

\[ I_n = A(C_1 - q_4) = -A[C_0U_v + (k_1 - k_2)q_2]. \tag{4} \]

Here \(A\) is a constant proportional to the number of carriers and to the number of rotor revolutions per second. It follows from equation (4) that, for \(k_1 = k_2\), i.e., \(C_1 = C_3\), the magnitude of the generator load current will be the same as for \(q_2 = 0\), and is equal to \(I_n = I_0 = -AC_0U_v\).

If \(k_1 > k_2\), then \(I_n = I_n' > I_0\), and for \(U_v = 0\), \(I_n' > 0\).

If \(k_1 < k_2\), then \(I_n = I_n'' < I_0\), and for \(U_v = 0\), \(I_n'' < 0\).

Further, if we set \(C_3 = 0\), which occurs when there is no high-voltage inductor or when, in the high-voltage system, there is a wide brush advanced in the direction of rotor rotation in such a way that, at the moment the carrier is disconnected from the brush, the capacitance of the surface element opposite this carrier, relative to the inductor, is sufficiently small, then we obtain:

\[ I_{n1} = -A[C_0U_v + (k_1 - 1)q_2]. \tag{5} \]

If the charge \(q_2\) is due to the effect of ionic shielding, then the condition must be satisfied

\[ q_3 = -q_5. \tag{6} \]

Taking into account that

\[ q_3 = C_0U_v + (k_1 - 1)q_2, \]

\[ q_5 = (1 - k_2)q_2, \]

from (6) we obtain

\[ q_2 = \frac{C_0U_v}{2 - k_1 - k_2}. \tag{7} \]

Substituting (7) into (5) and taking into account that for \(C_2 = 0\), \(k_2 = 1\), we obtain

\[ I_{n1} = 0. \tag{8} \]

From (4), with allowance for (7), we obtain

\[ I_n = -2AC_0U_v \frac{1 - k_2}{2 - k_1 - k_2}, \tag{9} \]

whence it is evident that for any

\[ k_1 < 1,\quad k_2 < 1 \quad \text{and} \quad U_v = 0,\quad I_n = 0. \tag{10} \]

The physical meaning of relations (8) and (10) is that, for \(C_3 = 0\), no field inversion occurs under the high-voltage inductor, or, in other words, there is no redistribution of part of the charge \(q_1\), associated with the charge \(q_2\), between the carrier and the inductor. As a result, in order for the average value of the field strength on the dielectric surface to be equal to zero, it is necessary that \(q_1 = q_2\). If this condition is satisfied, charge transfer to the load is impossible. On the other hand, for \(U_v = 0\), the average value of the field strength will be equal to zero only for \(q_2 = 0\), i.e., the current in the load must also be equal to zero.

Experimental Results

To verify relations (1)--(10), the characteristics

\[ I_{\text{load}} = f(U_v) \]

were taken for a single-disk model of an ESG with a cascade connection of carriers.

The bearings of the rotor disk of the experimental generator were mounted in special supports; the stator was not used during generator operation. The inductors were fastened on separate insulating stands; the rotor was driven in rotation by a direct-current motor at a speed of 1000 rpm. The experiments were carried out with rubbing brushes...

Fig. 2. Short-circuit characteristics with wide brushes

in an air atmosphere with a relative humidity of 39–50%.

Figure 2 shows the dependences \(I_{\mathrm{kz}} = f(U_{\mathrm{v}})\) during operation of the generator with brushes simultaneously covering no fewer than five transporters. Curve 1 was obtained in the absence of high-voltage inductors. Curve 2 was obtained during operation of the generator with a brush protruding beyond the edge of the high-voltage inductors in the direction of rotor rotation. Curve 3 corresponds to the operating mode of the generator according to the scheme in Fig. 1. These dependences were determined both when \(U_{\mathrm{v}}\) was increased and when it was decreased (marked by arrows). According to the results of numerous experiments, it was established that, in the region of the rectilinear part of the dependence, the forward and reverse runs of the characteristics coincide. If the voltage reduction is begun at \(U_{\mathrm{v}} \ge U_{\mathrm{k}}\), where \(U_{\mathrm{k}}\) is the voltage corresponding to the bend point of the characteristic obtained in the absence of high-voltage inductors or with the brush advanced, in the direction of rotation, beyond the edge of the inductors, then the reverse curve for characteristics 1 and 2 runs almost parallel to the forward one, but somewhat below it. The reverse run of characteristic 3 practically coincides with the forward run. The difference between the forward and reverse runs of characteristics 1 and 2 proved to be the greater, the larger \(U_{\mathrm{v}} > U_{\mathrm{k}}\).

In all cases, fairly good stability over time of the reverse-run characteristics was observed. For example, the current value \(I(0)\) at \(U_{\mathrm{v}} = 0\) remained practically constant during generator operation for one hour. In repeated experiments, the characteristic began from the value \(I(0) \ne 0\) and, in the linear section, remained parallel to that obtained earlier.

Fig. 3. Short-circuit characteristics with various brushes.

The phenomenon noted above also occurred when narrow brushes were used (touching one transporter). In Fig. 3, curves 1 and 2 were obtained with an excitation brush covering approximately three transporters, and with a load brush installed without inductors and touching one transporter. Curve 2 was obtained at positive excitation voltage, curve 1 at negative excitation voltage. Curve 3 was obtained with identical brushes touching one transporter, at positive excitation voltage.

The dependences \(I(0)\), obtained by placing a shorting bar between the brush and the excitation inductor, are also presented here.

Figure 4 presents the characteristics for the cases \(C_1 > C_3\) (curve 1) and \(C_1 < C_3\) (curve 2) when increasing (forward run) and decreasing (reverse run) the voltage \(U_{\mathrm{v}}\) with wide brushes. The ratio of the capacitances \(C_1\) and \(C_3\) was determined from the sizes of the gaps between the inductors and the surface of the rotor dielectric.

The character of these dependences remains the same if \(I(0) \ne 0\); in this case only the position of the inflection point changes.

Discussion of the Experimental Data

From the obtained dependences (Figs. 2, 3, 4) it is directly evident that, beginning with the voltage \(U_{\mathrm{v}} \approx U_{\mathrm{k}}\), a certain charge appears on the rotor dielectric.

The magnitude of this charge increases rapidly with increasing \(U_{\mathrm{v}}\) when exceeding ...

Fig. 4. Short-circuit characteristics for \(C_1 \ne C_3\).

…increase in \(U_{\mathrm{k}}\), as evidenced by the increase of the current \(I(0)\) (Fig. 3, curves 1, 2).

However, the good agreement of the forward and reverse runs at the beginning of the characteristics up to \(U_{\mathrm{в}} \approx U_{\mathrm{k}}\), and the nonfulfillment of relations (8, 10), according to which \(I_{\mathrm{н}}=0\) in the absence of a high-voltage inductor or with the brush extended beyond the edge of the inductor in the direction of rotation, make it possible to conclude that the effect of ionic screening is not predominant among the causes giving rise to this charge.

A comparison of characteristics 1, 2, 3 in Fig. 2 makes it possible to represent as follows the nature of the processes occurring as the excitation voltage is increased. When the excitation voltage is raised to \(U_{\mathrm{в}} \leq U_{\mathrm{k}}\), the charge on the dielectric surface will be equal to zero.

For \(U_{\mathrm{в}}>U_{\mathrm{k}}\), critical gradients are reached in the region between the surface and the inductor and a corona arises, as a result of which the dielectric surface is charged to the value

\[ q_2=(U_{\mathrm{в}}-U_{\mathrm{k}})C_2=\Delta U C_2 \]

and the electric-field intensity decreases to a value below the critical one. The conditions for the appearance of the corona probably explain the instability of the characteristics, often observed at \(U_{\mathrm{в}}\approx U_{\mathrm{k}}\) (shown in Fig. 2 for the forward run by a dotted line). The charge of the carrier in this case will be equal to

\[ q_1=C_0U_{\mathrm{k}}+\Delta U C_2 . \]

In the discharge system the charge of the carrier, associated with the surface charge, will be redistributed in direct proportion to the capacitances \(C_2\) and \(C_3\) and will be determined by equation (3).

The short-circuit current of the generator will be equal to

\[ -I=A\left[C_0U_{\mathrm{k}}+\Delta U C_2(1-k_2)\right] \]

or

\[ -I=A\left[C_0U_{\mathrm{k}}+\Delta U C'_0\right], \tag{11} \]

where

\[ C'_0=C_2(1-k_2)=\frac{C_2C_3}{C_3+C_2}. \]

If \(C_1=C_3\), i.e., \(C'_0=C_0\), then, as before, we have

\[ -I = A[C_0(U_k+\Delta U)] = AC_0U_v . \tag{12} \]

When critical gradients are reached in the zone of the high-voltage inductors, a corona also arises there, and part of the charge of the conveyor that has passed onto the inductors is compensated by a charge on the surface outside the load circuit. A constant current value is established in the load circuit, equal to

\[ -I_{\max}=2AC_0U_k . \tag{13} \]

From the nature of the processes described above and from equation (11), it follows that the slope of the characteristic \(I_{kz}=f(U_v)\) for different values of \(C_1\) and \(C_3\) must be different for \(U_v<U_k\) and for \(U_v>U_k\).

If \(C_1>C_3\), then for \(0<U_v<U_k\) the characteristic must be steeper than for \(U_v>U_k\); conversely, if \(C_1<C_3\), then for \(0<U_v<U_k\) the characteristic must be flatter than for \(U_v>U_k\). In addition, if \(q_2\ne0\), then at \(U_v=0\) and \(C_1>C_3\), in the excitation system, as a result of redistribution of charges between the conveyor and the inductor, the charge of the conveyor will be smaller than in the load system. That is, the conveyor will discharge in the excitation system and charge in the load system. As a result of this, a current of opposite sign, compared with the current at \(U_v>0\), will flow in the load circuit. If \(C_1<C_3\), then a current of the same sign will flow in the load circuit.

Thus, when \(C_1\ne C_2\), depending on their ratio, the reverse-run characteristics, i.e., upon decreasing \(U_v\) (with \(q_2\ne0\)), must have fundamentally different forms. The correctness of this explanation is confirmed by Fig. 4. In the same way one can explain the phenomenon of displacement of the generator characteristics during operation.

If, during operation of a generator for which \(C_1>C_3\), the excitation voltage is reduced at \(q_2\ne0\), then its operating mode will pass to another characteristic shifted to the right. To obtain the same current value again, an even higher excitation voltage is required. With a systematic decrease and increase of the current, the generator operating mode will pass to a characteristic approximately parallel to the section \(U_v>U_k\), but shifted considerably to the right, even for an insignificant difference between the capacitances \(C_1\) and \(C_3\). If \(C_1<C_3\), then when the current is reduced the operating mode will shift to the left, and the same current can be obtained at a smaller value of \(U_v\). However, this circumstance cannot be used in practice during prolonged operation of the generator, since, owing to the finite value of the conductivity, the magnitude of the charge on the surface of the dielectric will decrease. Thus, in practice it is most expedient to have \(C_1=C_3\) or \(C_1<C_3\).

These results agree well with the results of studying the current curves in various generator circuits by oscillographic recording, and with the results of studying the law of potential distribution over the rotor conveyors. It should also be noted that the characteristics \(I_{kz}=f(U_v)\) with forced recharging, for the case \(C_1\approx C_3\), completely coincide with the characteristics \(I_{kz}=f(U_v)\) without forced recharging with two inductors, independently of the law of variation of each of the voltages separately, if \(U_v=|U_v'|+|U_v''|\).

It is evident from equation (12) that, when \(C_1\ne C_3\), a generator with forced recharging will have the same characteristics as in Fig. 4 if the excitation voltage is first raised to the value \(U_k\), and then the excitation voltage of the forced-recharging source is increased. These data provide convincing confirmation that the phenomenon of double natural transfer makes it possible to obtain the same practical results as forced recharging of conveyors in a high-voltage system.

On the basis of an analysis of the processes that occur when the excitation voltage is increased, it may be assumed that they also take place during operation of a generator with a stator made of a poorly conducting material.

Under ordinary conditions the conductivity of a gas is, as a rule, less than the conductivity of dielectrics; consequently, the process of ion deposition on the dielectric surface will be compensated by leakage. High gas conductivity will occur only when the critical values of the electric-field strength are reached and ionization appears.

As the excitation voltage is increased, critical gradients will arise in places of increased field nonuniformity in the rotor--stator gap. It is here that surface charging will occur, as a result of which the field strength due to the transporter charge will decrease to a value below the critical value.

In the presence of a poorly conducting stator, for example one made of glass, the course of the processes will be more complex, since the inductors are separated from the rotor surface by a material with a sufficiently high specific resistance. Therefore, with the appearance of ionization in the rotor--stator gap, both the appearance of charges on the rotor surface and the appearance of charges of the opposite sign on the glass surface will be inevitable. The presence of these charges may, under certain conditions, lead to a substantial change in the distribution of potentials over the stator and significantly affect the magnitude of the maximum values of the generator current.

Conclusions

1. The phenomenon of double natural transfer occurs in generators with a cascade connection of transporter-conductors placed in a dielectric (shown for the first time) and in generators with a transporter-dielectric.

2. Generators with a cascade connection of transporter-conductors possess the same positive properties (arising from the phenomena of natural recharging and double natural transfer) as generators with a transporter-dielectric. However, the former make it possible to achieve higher ratios of working volumes (gaps) to the total volume of the generator in their disk design and, consequently, provide higher energy performance.

Literature

1. Felici N. I., Cartner E., Rev. gén. electr., 62, No. 2, 71; 62, No. 3, 136 (1953).

2. V. I. Levitov, A. G. Lyapin. Electrostatic Generators with a Rigid Rotor, Part II, VNIIEM, Moscow, 1963.