On the Comparison of Experimental and Calculated Short-Circuit Characteristics of a Disk-Type Rod Electrostatic Generator
Abstract Generated abstract
This paper compares calculated and experimental short-circuit characteristics of a single-disk rod electrostatic generator operating with a weakly conducting glass stator, extending earlier validation performed under metallic stator conditions. The authors measured short-circuit current characteristics over rotor to stator air gaps from 0.2 to 4 mm, using narrow and wide brushes to evaluate the influence of natural recharging, and compared the slopes of the linear characteristic regions with values obtained from an existing calculation method. The experimental and calculated dependences showed similar trends, with discrepancies generally within about 20 percent, although calculated values diverged at very small gaps and tended to underestimate results over much of the remaining range. The study concludes that the method is suitable for approximate calculation of short-circuit characteristics in electrostatic generators with weakly conducting stators.
Full Text
Proceedings of the Tomsk Polytechnic Institute, Awarded the Order of the October Revolution and the Order of the Red Banner of Labor
Volume 206 1973
On the Comparison of Experimental and Calculated Short-Circuit Characteristics of a Disk-and-Rod Electrostatic Generator
A. F. Kalganov, E. A. Podpletnyeva, Yu. N. Sivkov
(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering and the sector of rotor electrostatic generators of the Research Institute of Nuclear Physics, Electronics, and Automation)
The new method proposed in [1] for calculating the linear portions of the characteristics of rod electrostatic generators has been well confirmed for the short-circuit mode of a disk electrostatic generator with an equipotential (metallic) stator [2], i.e., under specific, artificial conditions.
We have carried out a comparison of the short-circuit characteristics of a disk electrostatic generator calculated according to [1] and obtained experimentally in the presence of a stator made of a poorly conducting material, satisfying operating conditions. The experiments were conducted on an experimental single-disk electrostatic generator with a double stator made of glass with a specific volume resistivity \(\rho_v = 3 \cdot 10^{12}\ \Omega/\text{cm}\). The generator had the following geometrical parameters:
outer working diameter of the rotor
\[
D_{\mathrm{n}} = 140\ \text{mm};
\]
inner working diameter of the rotor
\[
D_{\mathrm{vn}} = 78\ \text{mm};
\]
radius of the carriers
\[
r = 0.2\ \text{mm};
\]
arrangement of the carriers — two-row;
number of carriers in one row
\[
2m = 149;
\]
distance between carriers along the mean diameter of the rotor
\[
a = 2.3\ \text{mm};
\]
depth of embedment of the carriers in the dielectric body of the rotor
\[
d_2 = 1\ \text{mm};
\]
thickness of the glass stator disks
\[
d_4 = 3\ \text{mm}.
\]
On the outer sides of the stator disks there were inductors in the form of duralumin sectors 4 mm thick. The generator was driven in rotation by a DT-75 asynchronous electric motor \((n = 2870\ \text{rpm} = 47.6\ \text{rps})\).
The generator was investigated in air at atmospheric pressure, with contact commutation, under conditions in which the relative humidity of the air did not exceed \(50 \div 60\%\).
The short-circuit characteristics were taken both with narrow brushes, covering only one carrier (when the phenomenon of natural recharging appeared), and with wide brushes, covering several (3 or more) carriers, when, according to theory and experiment [1, 2], natural recharging does not appear or appears very weakly. The narrow brushes were made of steel mesh, and the wide ones of thin copper wire. Adjustment of the brush-collector system before experiments with narrow brushes was carried out until the greatest steepness of the short-circuit characteristic was obtained, i.e., according to the current maximum, then
as with wide brushes—by the current minimum. In the latter case it was assumed that natural recharging is absent.
In studying the short-circuit characteristics, the value of the rotor–stator gas gap \(d_1\) was chosen as the variable parameter.
Figure 1 shows experimentally obtained short-circuit characteristics for variation of \(d_1\) from 0.2 to 4 mm. A sufficiently good reproducibility of the results was obtained; it was evaluated by taking each of the characteristics at least twice.
Fig. 1. Short-circuit characteristics obtained for the following rotor–stator gap values:
1 — 0.2 mm; 2 — 0.4 mm; 3 — 0.6 mm; 4 — 0.8 mm; 5 — 1.5 mm; 6 — 3.0 mm; 7 — 4.0 mm.
Fig. 2. Dependence of the slope of the short-circuit characteristic on the size of the air gap: 1 — experimental; 2 — calculated by formula [1].
The quality of the characteristics is assessed by their slope, determined by the ratio of the increment of the short-circuit current \(I_{\text{s.c.}}\) to the increment of the excitation voltage \(U_{\mathrm{v}}\). For linear (initial) portions of the short-circuit characteristics, their slope is determined by the ratio of the absolute values of \(I_{\text{s.c.}}\) and \(U_{\mathrm{v}}\).
The calculated short-circuit characteristic was determined according to [1] as
\[ I_{\text{s.c.}}=\kappa_i 2mn C_{\mathrm{e}} U'_{\mathrm{v}}=A U'_{\mathrm{v}}, \tag{1} \]
where \(A=\kappa_i 2mn C_{\mathrm{e}}\) is the slope of the characteristic.
Here the coefficient of natural recharging was calculated from the expression
\[ \kappa_i=\sqrt{\,1+4\frac{d_{\mathrm{pr}}}{\kappa r} \left(Sc\frac{\pi r}{a}-1\right)^{0.8}}, \tag{2} \]
and the total capacitance of one conveyor relative to all elements of the rotor–stator system from the expression
\[ C_{\mathrm{e}}= \frac{4\pi l\varepsilon_0 \varepsilon_r} {\ln\!\left[2\left(\frac{d_{\mathrm{pr}}}{r}+1\right)\right]} \left[ 1-e^{-\frac{a}{2r}\left(\frac{0.9r}{d_{\mathrm{pr}}+r}+0.02\right)} \right], \tag{3} \]
where
\[ l=\frac{D_{\mathrm{n}}-D_{\mathrm{in}}}{2} \]
is the length of the conveyors,
\(\varepsilon_r\) is the relative dielectric permittivity of the rotor material,
\(d_{\mathrm{pr}}=d_2+\varepsilon_r d_1\) is the so-called reduced rotor–stator gap,
\(\kappa\) is a coefficient equal to \(4/3\) for a two-row arrangement of conveyors 2, 3.
Figure 2 gives the dependences of the slope \(A\) of the linear portions of the short-circuit characteristics on the value of \(d_1\), both from experimental data (curve 1) and from calculated data (curve 2).
From the data presented in Figs. 1 and 2 it follows that the course of the experimental dependence \(A=f(d_1)\) is analogous to the course of the calculated dependence, and their discrepancy over almost the entire range of variation of the gap \(d_1\) is comparatively small. At the same time, the calculated values for gaps \(d_1>0.3\ \text{mm}\) prove to be smaller than the experimental ones, whereas for gaps \(d_1<0.3\ \text{mm}\) they increase sharply and become larger than the experimental ones.
It follows from the expression for \(A\) that, for constant values of \(m\) and \(n\), the quantity \(A\) is determined by the product \(k_i\cdot C_e\). In accordance with (2) and (3), as \(d_1\) changes the quantities \(k_i\) and \(C_e\) change in different directions: with a decrease, for example, of \(k_i\), the quantity \(C_e\) increases, and vice versa.
For understanding the reasons for the discrepancy of the curves (Fig. 2), it would have been necessary to determine the nature of the discrepancy between the experimental and calculated dependences \(k_i=f(d_1)\) and \(C_e=f(d_1)\).
In Fig. 3 are presented the experimental (curve 1) and calculated (curve 2) dependences \(k_i=f(d_1)\). Each experimental value of \(k_i\) was determined as the ratio of the slope of the linear part of the short-circuit characteristic taken with narrow brushes to the slope of the linear part of the short-circuit characteristic obtained with wide brushes. As can be seen, here too an intersection of the calculated and experimental curves is observed. However, this intersection occurs in the middle part of the range of variation of \(d_1\) (in the region of about \(1.6\ \text{mm}\)), with only a slight change in the slope of the curves over the entire range of variation of \(d_1\). It follows from this that the character of the variation of \(A=f(d_1)\) depends to a greater degree on the character of the variation of \(C_e=f(d_1)\). For the theoretical dependence \(A=f(d_1)\), this is confirmed by the course of the calculated dependence \(C_e=f(d_1)\). Unfortunately, experimental determination of the dependence \(C_e=f(d_1)\) on the operating generator is not possible. From a qualitative estimate of such a dependence it follows that the actual values of \(C_e\) should be somewhat smaller than the calculated ones, since the stator material is not a conductor. Taking this into account, the experimental curve 1 (Fig. 2) should pass below the calculated curve 2. In practice, for the greater part of the range of variation of \(d_1\), the opposite picture is observed.
Fig. 3. Dependence of the coefficient of natural recharging on the size of the air gap: 1 — experimental; 2 — calculated.
However, the comparatively small discrepancy of the curves (Figs. 2, 3), which may be attributed both to insufficient accuracy of the empirical formulas (2) and (3) and to errors in determining the experimental values of the slope \(A\), and which varies from 0 to \(\pm 20\%\) for different values of \(d_1\), should be assessed as satisfactory agreement between the calculated and experimental data. This result makes it possible to conclude that the method of [1–3] can be used for approximate calculations of the short-circuit characteristics of an electrostatic generator with a weakly conducting stator.
REFERENCES
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