EFFECT OF ESG BRUSH DISPLACEMENT ON LOAD CURRENT MAGNITUDE

A. M. Kuptsov

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

Various branches of production, science, and technology require a compact, reliable, inexpensive source of high voltage. One source of high direct voltage is the electrostatic generator (ESG).

Electrostatic generators have proved necessary and have found wide application in the acceleration of charged particles, in X-ray technology, television, and especially in electron-ion technology. Electrostatic ignition generators for automobiles are widely used.

A large number of ESGs, differing from one another both in design and in principle of operation, have been installed and are operating in laboratories and in production. The most widespread are electrostatic generators in which charge transfer is carried out by means of metallic rods—conductors (ESGP)—or by a solid dielectric (ESGD).

The principle of operation and the basic elements of an electrostatic generator with conductor-transporters connected in a cascade circuit are described sufficiently fully in [1].

We have investigated the operation of such an electrostatic generator with a stator made of semiconducting material for various positions of the commutating devices—the brushes.

It was established that when the brushes are displaced relative to the geometric neutral of the generator [2], with the inductors located on the geometric neutral, there is a noticeable decrease in the current and voltage of the generator at the same load resistance.

This decrease in current and voltage is explained by a reduction in the magnitude of the charge difference delivered to the load.

Let us calculate the magnitude of this charge difference when the high-voltage and low-voltage brushes are displaced simultaneously relative to the geometric neutral by the angle

\[ \pm k \cdot \frac{\pi}{m}, \]

where \(k = 1, 2, 3, \ldots, 2m\) is the number of rotor transporters, i.e., by an angle that is a multiple of the angle between two adjacent transporters, under the condition that the brushes do not leave the region of the inductors. Positive angles are counted when the brushes are displaced in the direction of rotor rotation, and negative angles when displaced against the rotation of the rotor.

We shall use the replacement circuit of the ESGP and the method of electrostatic analogy as applied to the calculation of this type of generator, as set forth in [3].

The replacement circuit of a generator with contact commutation, for an induction excitation scheme with a semiconducting stator, is shown in Fig. 1. Here the brushes are displaced relative to the geometric neutral in the direction of rotor rotation by a certain angle characterized by the coefficient \(\kappa\).

Fig. 1. Replacement circuit of an ESG with a semiconducting stator

This circuit, by means of the superposition method [4], is calculated as a direct-current circuit, which makes it possible, by virtue of the formal analogy between the electrostatic field and the direct-current field, to determine the desired charge difference delivered to the load.

Omitting a series of transformations, we have, according to the calculation method [3],

\[ q - q' = \frac{2U_{\mathrm{B}}}{R_c \operatorname{th}\frac{m\gamma}{2}} + \frac{ 4U_{\mathrm{n}} C_2 \operatorname{sh}\frac{m-s+1}{2}\gamma \operatorname{ch} k\gamma }{ (m-s+1)\operatorname{sh}\frac{m\gamma}{2} }, \]

where \(U_c\) is the voltage of the excitation source;

\(R_e\) is the repeated resistance of the four-poles of the chain circuit, equal to

\[ \frac{1}{C_1\sqrt{1+\frac{4C_2}{C_1}}}; \]

\(\gamma\) is the propagation constant of the four-poles, equal to

\[ \ln\left[1+\frac{C_1\left(1+\sqrt{1+4C_2/C_1}\right)}{2C_2}\right]; \]

\(C_1\) and \(C_2\) are the partial capacitances of the transporters;

\(S\) is the number of transporters overlapped by the armature;

\(U_n\) is the load voltage.

In the case where the brushes are located on the geometrical neutral, i.e., for \(\kappa=0\), the magnitude of the charge difference will be:

\[ q_1-q_1'=\frac{2U_b}{R_c\operatorname{th}\frac{m\gamma}{2}} + \frac{4U_n C_2\operatorname{sh}\frac{m-s+1}{2}\gamma} {(m-s+1)\operatorname{sh}\frac{m\gamma}{2}}. \]

The load current of the generator is determined by the magnitude of the calculated charge and is equal to:

\[ I=2mn(q-q'), \]

where \(n\) is the rotational speed of the rotor.

The load voltage of the generator is:

\[ U_n=2mn(q-q')R_n. \]

Let us compare the load currents of the generator when operating with brushes located on the geometrical neutral \((I)\) and when operating with brushes displaced by the angle \(\kappa\cdot\frac{\pi}{m}\) \((I')\), determining their ratio:

\[ \frac{I'}{I}=\frac{1+M}{1+M\operatorname{ch}\kappa\gamma}, \]

where

\[ M= \frac{8mnR_n\operatorname{sh}\frac{m-s+1}{2}\gamma} {(m-s+1)\operatorname{sh}\frac{m\gamma}{2}}. \]

The obtained ratio shows that, when the high-voltage and low-voltage brushes are simultaneously displaced in one direction or the other from the geometrical neutral, the load current decreases.

In an analogous manner one can determine the effect of the displacement of only one of the brushes, either the high-voltage or the low-voltage one, on the operation of this type of generator.

Let the low-voltage brush be displaced by the angle \(\pm\kappa\cdot\frac{\pi}{m}\); then,

calculating the circuit, similar to Fig. 1, we obtain the required charge difference:

\[ q-q'= \frac{U_{\mathrm{в}}\operatorname{sh} m\gamma} {R_c \operatorname{sh}\frac{m+\kappa}{2}\gamma \operatorname{sh}\frac{m-\kappa}{2}\gamma} + \frac{ 4U_{\mathrm{н}}\operatorname{sh}\frac{m-s+1}{2}\gamma\cdot \operatorname{sh}\frac{m\gamma}{2}\operatorname{ch}^{2}\frac{\kappa\gamma}{2} } { (m-s+1)\operatorname{sh}\frac{m+\kappa}{2}\gamma \operatorname{sh}\frac{m-\kappa}{2}\gamma }. \]

Owing to the symmetry of the calculation circuit when the high-voltage brush is displaced, the result will be the same. The expression for the charge difference without brush displacement remains the same, which makes it possible to find the ratio of the currents with brush displacement \((I_1')\) and without brush displacement \((I)\)

\[ \frac{I_1'}{I}= \frac{1+M} { \frac{ \operatorname{sh}\frac{m+\kappa}{2}\gamma\cdot \operatorname{sh}\frac{m-\kappa}{2}\gamma } { \operatorname{sh}^{2}\frac{m\gamma}{2} } + M\operatorname{ch}^{2}\frac{\kappa\gamma}{2} }, \]

or, for \(m \geqslant 10\kappa\), with sufficient accuracy we obtain

\[ \frac{I_1'}{I}= \frac{1+M} {1+M\operatorname{ch}^{2}\frac{\kappa\gamma}{2}}, \]

which indicates a certain decrease in current when the brush is displaced.

The phenomenon of a decrease in the current and load voltage when the brushes are displaced is confirmed experimentally.

The experimental data obtained on an experimental ESGP model, and the calculated data for brush displacement by the angle \(\pm \frac{\pi}{m}\), are given in the table.

Table

Some discrepancy between the calculated data and the experimental data is associated with the difficulty of obtaining a semiconducting stator; nevertheless, the results obtained make it possible to draw a number of conclusions:

1. When the brushes are displaced relative to the geometrical neutral, the magnitude of the charge transferred to the load decreases, and consequently the current and the load voltage of the generator decrease at one and the same load resistance.

1. The relative decrease in load current depends on the magnitude of the load resistance, on the rotor speed, on the number of rotor carriers, and on the partial capacitance between the carriers. The width of the inductor is of considerable importance. The greatest decrease will evidently occur in a mode close to no-load operation.

2. The magnitude of this decrease depends on the nature and magnitude of the brush displacement, and will be greatest when the high-voltage and low-voltage brushes are displaced simultaneously relative to the geometric neutral.

3. From the standpoint of obtaining the maximum load current, the optimal brush position is the geometric neutral.

REFERENCES

1. A. F. Ioffe. Electrostatic generator, ZhTF, vol. IX, 23, 1939.
2. V. I. Levitov, A. G. Lyapin. Electrostatic generators with a rigid rotor, Part I, Tsintipriboelektroprom, Moscow, 1963.
3. V. A. Lukutin, V. V. Patsevich, V. D. Eskov. On the electrical calculation of an ESG, Abstracts of Reports at the VI Interuniversity Conference on Electron Accelerators, Tomsk, TSU Press, 1966.
4. G. V. Zeveke, P. A. Ionkin et al. Fundamentals of Circuit Theory, GEI, 1963.