PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE named after S. M. KIROV

Vol. 159
1967

INFLUENCE OF THE DIMENSIONS OF CONVEYORS AND INDUCTORS ON THE ELECTRICAL CHARACTERISTICS OF DISK ESGs

A. P. KONONOV

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

In studying the operation of experimental models of disk-type ESGs with a cascade connection of conductor-conveyors, it was observed that the electrical characteristics of the ESG are very strongly affected by the geometrical dimensions of certain elements. These include: the dimensions of the rotor disk, the dimensions of the inductors, and those of the conductor-conveyors.

A number of works [1, 2] give calculations for selecting the best electrode profile in order to create a uniform electric field in an ESG.

However, in the works cited there is no analysis of the influence of the geometrical dimensions of the electrodes on the electrical characteristics of the generator. This is probably explained by the difficulty of analytical calculation, since simultaneous allowance for many factors (the size and shape of the inductor, the size and shape of the conveyor, the distance between conveyors, the air gap, etc.) is impossible.

Therefore, in order to determine the best geometrical ratios of certain elements, an experimental approach was chosen. An experimental single-disk generator with a cascade connection of conductor-conveyors was designed and built. The design of this generator makes it possible simply and sufficiently quickly to carry out a number of operations, such as: changing the rotor, changing the number of pole pairs, replacing the inductors, changing the position and shape of the brushes, and changing the gap between the rotor and the stator.

For this generator, inductors of three sizes and seven rotors with different numbers and cross-sectional shapes of conveyors embedded in epoxy resin were made. The outside diameter of all rotors is 140 mm, and the length of the conveyors is 25 mm. All investigations of the operation of the experimental generator were carried out in an atmosphere of carbon dioxide at a pressure of 4 atm.

In analyzing the dependences \(I_{\text{kz}} = f(U_{\text{v}})\) (Fig. 1), it was noted that the magnitude of the short-circuit current increases with an increase in the number of conveyors and a decrease in their radius.

To determine the best conveyor radius at which the short-circuit current will have a maximum value, an analysis of the following equation [4] was carried out:

\[ I_{\text{kz}} = K_i \cdot n \cdot 2 \cdot m C_0 U_{\text{v}}, \tag{1} \]

where \(K_i\) is the recharging coefficient,
\(n\) is the number of revolutions per minute,
\(m\) is the number of conductor-conveyors,

\(C_0\) — partial capacitance of one conveyor relative to the stator,

\(U_{\mathrm{v}}\) — excitation voltage.

The conveyors were taken to have a circular cross section because this electrode shape is the most effective when charges are moved in a region of high gradient [5].

Fig. 1. Graph of the dependence \(I_{\mathrm{kz}}=f(U_{\mathrm{v}})\) of rotors with different numbers of conveyors: 1 — for a rotor with \(m=27\), 2 — for a rotor with \(m=45\), 3 — for a rotor with \(m=54\), 4 — for a rotor with \(m=74\).

The value of the recharging coefficient can be determined from the formula [6]

\[ K_i=\sqrt{1+4\frac{C_1}{C_0}}, \tag{2} \]

where \(C_1\) is the partial capacitance between adjacent conveyors.

The dependences of the partial capacitances \(C_0\) and \(C_1\) were determined by modeling in an electrolytic bath. As a result of processing the experimental data, empirical dependences were obtained for the indicated quantities for a single-row arrangement of conveyor conductors:

\[ C_0=\frac{4.3\,\mathrm{E}\,la}{D_{\mathrm{pr}}}, \tag{3} \]

\[ \frac{C_1}{C_0}= \frac{1}{3} \left( \frac{D_{\mathrm{pr}}}{2r}-1 \right) \left( \mathrm{sc}\,\frac{\pi r}{a}-1 \right)^{0.8}, \tag{4} \]

where \(\mathrm{E}\) is the dielectric constant of the resin,

\(l\) — length of the conveyor,

\(a\) — distance between the axes of the conveyors,

\(r\) — radius of the conveyor.

\[ D_{\mathrm{pr}}=d_{\mathrm{sm}}+2r+\mathrm{E}d_{\mathrm{voz}}, \]

\(d_{\mathrm{sm}}\) — thickness of the dielectric,

\(d_{\mathrm{voz}}\) — air gap.

Taking into account dependences (3) and (4), equation (1) can be written as:

\[ I_{\mathrm{kz}}= U_{\mathrm{v}}\cdot n\cdot 2\cdot m\, \frac{4.3\,\mathrm{E}\,la}{D_{\mathrm{pr}}} \sqrt{ 1+4\left[ \frac{1}{3} \left( \frac{D_{\mathrm{pr}}}{2r}-1 \right) \left( \mathrm{sc}\,\frac{\pi r}{a}-1 \right)^{0.8} \right] } \]

or

\[ I_{\mathrm{kz}}= \frac{8.6\,U_{\mathrm{v}}nm\mathrm{E}la} {d_{\mathrm{sm}}+2r+\mathrm{E}d_{\mathrm{voz}}} \sqrt{ 1+\frac{4}{3} \left( \frac{d_{\mathrm{sm}}+2r+\mathrm{E}d_{\mathrm{voz}}}{2r}-1 \right) \left( \mathrm{sc}\,\frac{\pi r}{a}-1 \right)^{0.8} }. \tag{5} \]

When the conveyor radius \(r\) is varied, while leaving \(U_{\mathrm{v}}, n, a, l, d_{\mathrm{sm}}, d_{\mathrm{voz}}=\mathrm{const}\), it is seen that \(I_{\mathrm{kz}}\) will increase as the radius \(r\) decreases and as the number of conveyors \(m\) increases. Therefore, in order to obtain the maximum short-circuit current in a generator with an unchanged rotor disk radius, it is necessary to increase the number of conveyor conductors and decrease their radius. However, reducing the conveyor radius is associated with manufacturing difficulties.

At TPI two methods have been developed for manufacturing ESG rotors with conveyors of small radius.

1. Manufacturing an ESG rotor by means of printed circuits.
2. Manufacturing a rotor by winding a thin wire onto the rotor disk.

These two methods can be used in mass production of generators.

In addition, the dimensions of the conveyors and their number affect the ripple of the output voltage. In the experiment, the voltage curve was oscillographed. The oscillograms show that the greatest voltage ripple in the generator is observed with a small number of conveyors. As the number of conveyors increases and their radius decreases, the voltage ripple decreases.

To determine the optimal size of the inductor, the dependences \(I_{\text{sc}} = f(U_{\text{g}})\) (Fig. 2) were taken for one rotor, but with different inductors.

Experience has shown that the inductor has a certain optimal dimension in width. The length of the inductor must not be less than the length of the conveyor.

On the basis of experimental data for various geometrical ratios of inductors and conveyors, a graph was constructed for the dependence of the short-circuit current on the ratio of the width of the inductor to the width (or diameter) of the conveyor (Fig. 3). It is evident from the graph that, for the experimental generator, the short-circuit current has its maximum value in the case when this ratio is equal to \(3.5 \div 4\).

To determine the radius of the rotor disk, let us use the condition of the electric strength of the dielectric (resin). Proceeding from this condition, one can determine the size of the gap that can withstand the specified voltage at the dielectric strength \(E_{\text{br}}\) of the dielectric.

\[ d = \frac{U}{E_{\text{br}}}, \tag{7} \]

where \(d\) is the distance between the inductors along the arc of the inner radius \((R_{\text{in}})\) of the disk \((d=\pi R_{\text{in}})\) through the clean dielectric,

\(U\) is the generator voltage,

\(E_{\text{br}}\) is the electric strength of the dielectric.

When placing conveyors in the rotor disk, it is necessary to have the total length \(d\) of all resin gaps along the inner radius between the inductors such that this length will withstand the specified voltage at the dielectric strength \(E_{\text{br}}\) of the dielectric for a nonuniform field.

Fig. 2. Characteristics \(I_{\text{sc}} = f(U_{\text{g}})\) for different inductors: 1—characteristic with a wide inductor, 2—characteristic with a medium inductor, 3—characteristic with a narrow inductor. —— rotor with number of conveyors \(m=45\), —·—·— rotor with number of conveyors \(m=54\).

The length of the circumference of radius \(R_{\mathrm{in}}\) is equal to \(2\pi R_{\mathrm{in}}\); the number of conveyors \(m\) is equal to

\[ m=\frac{2\pi R_{\mathrm{in}}}{a}, \]

then the quantity \(d\) will be equal to \(d=\frac{1}{2}m\cdot b\), where \(b\) is the distance between adjacent conveyors through the dielectric (resin), and \(m\) is the number of conveyors in the disk.

\[ d=\frac{1}{2}\frac{2\pi\cdot R_{\mathrm{in}}}{a}\cdot b =\frac{\pi R_{\mathrm{in}}}{a}\cdot b. \tag{8} \]

Comparing equations (7) and (8), we obtain

\[ R_{\mathrm{in}}=\frac{U\cdot a}{\pi E_{\mathrm{br}}}\cdot b. \tag{9} \]

The value of the outer diameter is determined from the condition of maximum power [3], since

\[ P_{\mathrm{n}}=U_{\mathrm{n}}\cdot I_{\mathrm{n}}, \]

and

\[ U_{\mathrm{n}}=E_{\tau}\pi R_{\mathrm{in}} \qquad \text{and} \qquad I_{\mathrm{n}}=n\sigma\pi\left(R_{\mathrm{out}}^{2}-R_{\mathrm{in}}^{2}\right) \]

\[ P=E_{\tau}\pi R_{\mathrm{in}}n\sigma\pi\left(R_{\mathrm{out}}^{2}-R_{\mathrm{in}}^{2}\right), \]

\[ \frac{\partial P}{\partial R_{\mathrm{in}}} =R_{\mathrm{out}}^{2}-2R_{\mathrm{in}}^{2}=0 \]

\[ R_{\mathrm{out}}=\sqrt{3}\,R_{\mathrm{in}}. \tag{10} \]

After determining \(R_{\mathrm{out}}\), it is necessary to check its value against the condition of the critical linear speed. The critical linear speed for a disk made of epoxy resin is \(V_{\mathrm{k}}=100\ \mathrm{m/sec}\) [5, 7].

Fig. 3. Dependence of the short-circuit current on the ratio of the inductor width to the conveyor width for various rotors. — rotor with number of conveyors \(m=45\); —·—·— rotor with number of conveyors \(m=54\).

\[ V_{\mathrm{k}}=\frac{\pi n\cdot R_{\mathrm{out}}}{30}, \]

where \(n\) is the number of revolutions per minute (we take \(n = 3000\) rpm, since most AC motors have this number of revolutions).

Then

\[ R_{\mathrm{n}}=\frac{V_{\mathrm{k}}\cdot 30}{n\cdot \pi}=320\ \mathrm{mm}. \]

The quantity \(a\) can be expressed by the following relation

\[ a=2r+b \qquad (\text{since } a>2r). \]

The above-mentioned methods of manufacturing ESG rotors make it possible to manufacture a rotor with a transporter radius equal to \(0.15\ \mathrm{mm}\) \((r=0.15\ \mathrm{mm})\).

The distance between transporters in the resin \((b)\) is determined by the possibility of embedding the electrodes in epoxy resin.

As practice has shown, the smallest distance between transporters can be obtained equal to \(0.1\ \mathrm{mm}\) \((b=0.1\ \mathrm{mm})\).

Using equation (9), we determine the value of \(R_{\mathrm{v}}\) at \(E_{\mathrm{br}}=50\ \mathrm{kV/cm}\) [8, 9] for the following series of voltages.

Table 1.

When comparing \(R_{\mathrm{n}}\) (Table 1) with the outer radius calculated from the condition of critical speed at \(n=3000\) rpm, we see that all radii for the indicated series are significantly smaller than the calculated one.

In addition, in order to reach critical dimensions these radii can be increased several more times.

For the voltage series \(U=100\div 300\ \mathrm{kV}\), the value of the generator current can be increased by increasing the disk radius, that is, by increasing the number of transporters.

Since the current value is

\[ I=\pi\left(R_{\mathrm{n}}^{2}-R_{\mathrm{v}}^{2}\right), \]

then, when the radii are doubled, we obtain a fourfold increase in current.

For the voltage series \(U=100\div 200\ \mathrm{kV}\), when the radii are increased threefold, a ninefold increase in current is obtained.

With such an increase in the disk radii, with an unchanged number of pole pairs, a further increase in voltage occurs by the same factor by which the disk radius increases. If the voltage is kept unchanged, then when the disk is doubled in size two pole pairs can be installed; when it is increased threefold, correspondingly, three pole pairs.

Table 2

As a result, the current value in the generator can increase in the first case by 8 times, and in the second by 27 times relative to the initial value.

The path of increasing the dimensions of the rotor disk is tempting, but disk growth is associated with a number of difficulties. One of the main ones may be considered manufactur-

...the pressure cylinder of large radius. There is another way—to increase not the size of the disks, but their number. But here difficulties of their own arise, which must also be solved.

Thus, on the basis of the foregoing, the following conclusions may be drawn:

1. In the rotor disk it is necessary to have as many conveyors as possible at a very small radius; in order, when reducing the radius of the conveyors, not to make the disk too thin, the disk must be manufactured with a two-row arrangement of conveyors. The methods indicated above provide for this.

2. The inductor must have an optimal size and must overlap (in width) no fewer than three conveyors; i.e., the ratio of the inductor width to the conveyor width must be equal to 4.

3. The radius of the rotor disk, for a given rotational speed, should be chosen as large as possible.

REFERENCES

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

2. N. Felici. Les Surfaces a champ electrique constant, Rev. Gen. Electricite 59, No. 11, 1950.

3. L. S. Polotovskii. Capacitive High-Voltage Direct-Current Machines. GEI, 1960.

4. V. V. Patsevich. Certain Problems in the Theory and Calculation of ESGs with Conveyor Conductors. Dissertation, TPI, 1964.

5. V. I. Levitov, A. G. Lyapin. The Application of Electric-Field Forces in Industry and Agriculture. VNIIEM, 1964.

6. V. A. Lukutin, V. V. Patsevich, V. D. Es’kov. Electrical Calculation of ESGs. Abstracts of Reports at the VI Inter-University Scientific Conference on Accelerators.

7. V. I. Levitov, A. G. Lyapin. Electrostatic Generators with a Rigid Rotor, Part 1, TsINTIPRIBORELEKTROPROM, 1963.

8. D. V. Razevig et al. High-Voltage Engineering. Energiya, 1964.

9. D. Meek, J. Craggs. Electrical Breakdown of Gases. IL, 1960.