CALCULATION OF THE OPERATING FIELD STRENGTH OF A UNIPOLAR CAPACITIVE GENERATOR (UCG)

A. I. Bertinov, B. A. Ponomarev

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

Analytical calculation of the strength of the resultant electric field in the working gap of unipolar capacitive generators, because of the presence of higher harmonics, is a complex problem. In its practical study only the first steps are being taken, and the published theoretical works in this area are few in number and contradictory.

Thus, in [1] the point of view is defended according to which, in a unipolar capacitive generator with a grounded electrode system (UCGG), without commutation, the greatest possible voltage in the working gap, determining the maximum value of the mean field strength, is equal to the excitation voltage. In [2] an expression was obtained for the voltage of a UCGG, indicating its alternating-sign character. This is also confirmed by the empirical relation given in [3], which relates the field strength to the output voltage and the excitation voltage of a unipolar capacitive generator; however, without taking into account the influence of the waveform of the voltage curve and the pole-connection scheme, this formula did not receive experimental substantiation in the work and proved unsuitable for calculating the field strength of a UCGG.

In the present work, the results are set forth of graphical-analytical investigations concerning the estimation of the mean value of the electric-field strength in the working gap of multipole synchronous UCGGs of the disk type with conductor carriers.

The field in the working gap of a capacitive generator at a given instant of rotor motion can be determined if the potentials of the systems of working half-disks are known. The adopted scheme for connecting the poles of the UCGG is such that the rotor \((R)\) remains, throughout the entire operating cycle, under a constant excitation potential, and one of the systems of half-disks \((S_1)\) is grounded. Only the potential of the other system of stator half-disks \((S_2)\), connected to the load, is variable.

Fixing the position of the stator half-disks relative to the moving rotor during operation can be accomplished by simultaneous oscillographic recording of the variable stator potential and of the signal from a rotor-position sensor. As the latter, a device is used whose sensitive element is a photoelement: the light flux falling on the photoresistor from a special light source is periodically interrupted by a disk diaphragm with slits, ensuring a voltage pulse of the signal at the instant when the axes of the rotor poles and the poles of the grounded system of stator half-disks coincide.

The capacitive generator under investigation has \(m = 15\) rotor disks and \(2p = 36\) poles, bounded by diameters \(D_1 = 250\) mm and \(D_2 = 90\) mm, which ensures a nearly sinusoidal character of the variation of its working mutual capacitance.

On the basis of oscillograms taken for several modes of active load, in Fig. 1 the output-voltage curves of the generator have been plotted to a definite scale; as the initial instant of reference there is taken the moment of the surge of the signal from the rotor-position sensor, fed to the input of the electronic oscilloscope through the signal-input switch—a polarized relay.

Fig. 1. Curves of the output voltage of the UCG
for various active-load modes

As can be seen from the graphs presented, the shape of the UCG output-voltage curve changes substantially with a change in load: in a mode close to short circuit (curve 1), it resembles a sinusoid, which becomes increasingly distorted as no-load operation is approached—the negative half-period becomes sharpened, is shortened in duration, and increases in amplitude to a greater extent than the positive half-wave. On the other hand, as the voltage curve moves away from the short-circuit mode, it increasingly lags in phase by a certain angle behind the excitation field, which is rigidly connected with the rotor. This angle is analogous to the angle \(\Theta\) in a synchronous electromagnetic generator.

The maximum field strength in the gap of a capacitive generator should obviously be expected when there is a potential of the system of insulated stator half-disks opposite in sign to the excitation potential. The fact that the negative maximum of the output voltage for the operating modes considered occurs at different instants of rotor motion has a direct influence on the distribution of fields in the gap of a synchronous UCG. Indeed, as analysis of the pole section in the active zone of the generator shows, at the indicated instants of time the distance between the rotor poles and the insulated stator poles does not remain constant. Consequently, the maximum value of the average field strength, which depends not only on the magnitude of the unlike potentials of the poles but also on their mutual position, will likewise occur at different moments for each operating mode.

moments of the operating cycle, which in the general case do not correspond to the moment at which the negative maximum of the output voltage appears.

In the presence of a negative half-wave of voltage, the expression for the average field strength of the resulting field in the UCG gap can be represented in the form

\[ E=-\frac{U_{RS_2}}{\delta}=-\frac{U_{\mathrm{B}}}{\delta}+\frac{u}{\delta}, \tag{1} \]

where

\[ \begin{aligned} U_{RS_2} &\text{ — the potential difference between the poles of the rotor } R \text{ and the stator } S_2;\\ \delta &\text{ — the shortest distance between them;}\\ E_{\mathrm{B}}=-\frac{U_{\mathrm{B}}}{\delta} &\text{ — the component of the resultant field strength due to the excitation field;}\\ U_{\mathrm{B}} &\text{ — the excitation voltage;}\\ E_{\mathrm{H}}=\frac{u}{\delta} &\text{ — the component of the resultant field strength due to the load field;}\\ u &\text{ — the absolute value of the potential of the insulated stator pole at the moment under consideration.} \end{aligned} \]

The quantity \(\delta\) depends on the geometry of the poles and on the magnitude of the working gap of the capacitive generator. The ideal pole profile, satisfying the conditions of the parametric equations for maximum forces over the entire pole surface, is practically difficult to realize in multipole multidisk designs. The pole profile of the generator under investigation is close to a semicircle whose radius is equal to the minimum working gap \(\delta_0\).

Introducing the working-gap coefficient

\[ k_\delta=\delta/\delta_0 \tag{2} \]

and the voltage coefficient

\[ k_u=u/U_{\mathrm{H}}, \tag{3} \]

where \(U_{\mathrm{H}}\) is the rms value of the output voltage, we rewrite expression (1) in the form

\[ E=E_{\mathrm{B}}+E_{\mathrm{H}}=\frac{U_{\mathrm{B}}}{k_\delta\delta_0}+\frac{k_uU_{\mathrm{H}}}{k_\delta\delta_0}. \tag{4} \]

Investigating the dependences \(k_\delta(\omega t)\) and \(k_u(\omega t)\) at those moments of the operating cycle when the generator output voltage passes through a negative maximum, it was established that, as the load increases, the field strength calculated from (4) at first increases, but then, after reaching a certain maximum value, begins to decrease. Moreover, the largest value of the field strength occurs in an operating mode close to the maximum-power mode, to which curve 6 of the generator output voltage corresponds (Fig. 1).

Similar conclusions are obtained from modeling the electric-field pattern in the UCG gap over a section of double pole pitch for several characteristic rotor positions and generator operating modes, carried out on the basis of recorded voltage oscillograms. Since, in the cross section of the specified active zone of the generator, the field can, with a sufficient degree of accuracy, be assumed plane-parallel, the method of modeling on electrically conductive paper was used.

For this purpose, from electrically conductive paper, at a certain scale, a repeating region of the working gap corresponding to the considered instant of rotor motion was cut out. Then certain potentials were supplied to the corresponding sections of the obtained model by means of metal buses, after which, according to the known method [4], graphical constructions of equipotential field lines were carried out with the aid of a dc bridge circuit. Since in disk machines the width of the poles along their length is not constant, the section corresponding to equal pole areas was taken as the basis.

One of the field patterns, recorded for the maximum-power operating mode in the rotor position corresponding to the angle \(-\alpha = \dfrac{\pi}{4}\), is shown in Fig. 2. The modeling results confirmed the obtained regularity of the variation of the field strength in the gap of a synchronous UCG, which makes it possible to generalize it to other designs of multipole machines of this type, for which a sinusoidal variation of the working mutual capacitance is characteristic.

Fig. 2. Electric-field pattern constructed in accordance with the recorded oscillogram of the voltage for the maximum-power operating mode of the UCG

In Fig. 3 the dependences are plotted of the voltage coefficient \(k_{um}\) and the gap coefficient \(k_{\delta m}\), corresponding to the maximum field strength in the gap,

\[ E_m = \frac{U_B + k_{um} U_n}{k_{\delta m}\delta_0}, \tag{5} \]

on the load voltage in relative units

\[ \overset{*}{U} = U_n : U_{xx}, \tag{6} \]

Fig. 3. For calculating the maximum value of the average field strength in the UCG gap

where

\[ U_{\mathrm{xx}} \quad \text{is the rms value of the no-load voltage.} \]

For comparison, the dependence of the amplitude coefficient of the output voltage on the quantity \(U_{\mathrm{n}}^{*}\) is also shown here:

\[ k_{\mathrm{a}} = U_{\mathrm{nm}}/U_{\mathrm{n}}, \tag{7} \]

where

\[ U_{\mathrm{nm}} \quad \text{is the absolute value of the negative amplitude of the output voltage.} \]

We see that, at small loads, the graphs of the functions \(k_{\mathrm{um}}(U_{\mathrm{n}}^{*})\) and \(k_{\mathrm{a}}(U_{\mathrm{n}}^{*})\) coincide (curves 1 and 2, Fig. 3), i.e., the maximum field strength coincides with the moment at which the negative amplitude of the output voltage appears. With a subsequent increase in the load, despite the growth of the negative amplitude of the output voltage (the amplitude coefficient continuously increases), a weakening of the field occurs, since the gap coefficient increases only slightly in this case (curve 3, Fig. 3), while the voltage coefficient drops sharply (curve 2).

Conclusions

1. The voltage at the output of a synchronous UCG is alternating in sign.

2. In a mode close to short circuit, when the influence of higher harmonics is insignificant, the output-voltage curve of the UCG is close in shape to the curve of variation of the operating mutual capacitance and becomes increasingly distorted as the armature reaction increases upon approach to no-load operation.

3. As the load increases, the angle between the beginning of the cycle of variation of the operating mutual capacitance (or, equivalently, the cycle of variation of the short-circuit current) and the beginning of variation of the output voltage is \(\Theta \to 90^\circ\) electrical.

1. The relationship between the maximum value of the average field strength in the working gap of a synchronous UCG and the load is nonlinear.

2. The operating field strength of a multipole synchronous UCG can be calculated using the proposed experimental coefficients.

REFERENCES

1. V. I. Levitov, A. G. Lyapin. Electrostatic Generators with a Rigid Rotor. TsINTIpriborelektroprom, Moscow, 1963.

2. L. S. Polotovsky. High-Voltage Direct-Current Capacitive Machines. Moscow, GEI, 1960.

3. E. A. Lodochnikov, V. P. Luk’yanchuk, V. A. Kufa. Factors determining the specific power of capacitive generators. Elektrotekhnika, 1964, No. 11.

4. G. A. Ryazanov. Experiments and Modeling in the Study of the Electromagnetic Field. Nauka, Moscow, 1966.