EFFECT OF BRUSH SHIFT ON THE EXTERNAL CHARACTERISTIC OF AN ESGS

V. M. FEDYAKIN, V. D. ESKOV

(Presented at the scientific seminar of the Department of TOE)

In theoretical studies of rod electrostatic generators (ESGS), designs are usually considered in which the inductors are arranged symmetrically along the circumference of the stator, and it is assumed that commutation of the carriers occurs opposite the middle of the inductors.

The purpose of the present article is to estimate the effect of brush shift (and, correspondingly, of the commutation point relative to the middle of the inductors) on the output parameters of the machine when it operates on the linear portion of the external characteristic.

To solve this problem one may use a technique based on applying the method of electrostatic analogy to equations with partial capacitances, as set forth in [1]. That work also gives expressions for the potentials of the carriers of an ESGS with one pair of poles, in the absence of brush shift, which cover one carrier each at the moment of interruption of commutation simultaneously in the excitation and load systems.

These expressions take into account the values of the principal partial capacitances of a carrier (with respect to the opposite stator element \(C_1\), a neighboring carrier \(C_2\), the stator element opposite the neighboring carrier \(C_3\), grounded structural elements \(C_0\)), the total number of carriers \(2m\) and the number of carriers covered by each inductor, \(s\), the excitation circuit, the magnitudes of the excitation voltage \(U_{\mathrm{в}}\) and load voltage \(U_{\mathrm{н}}\), and the position of the carrier (or, more precisely, its number, the count being taken in the direction of rotor rotation from the carrier that has just been commutated in the excitation system).

If, in accordance with the superposition principle, the potential of the \(n\)-th carrier is represented as the sum of two components:

\[ U'_n = U'_{nc} + U'_{np}. \tag{1} \]

where \(U'_{nc}\) is determined only by the stator field in the absence of charges on the carriers,

and \(U'_{np}\) only by the charges of the carriers with the stator grounded, then, preserving the notation adopted in [1], one may write:

\[ U'_{np} = -\frac{1}{2C_{\mathrm{э}}} \left[ q_z + \Delta q \frac{2f_1 - 1}{k_i} \right], \tag{2} \]

\[ U'_{nc}= \begin{cases} U_n F_c, & \text{for the conduction excitation circuit,}\\[2mm] (U_n+U_v)F_c-\left(1-\dfrac{C_0}{C_e}\right), & \text{for the induction circuit,} \end{cases} \tag{3} \]

where

\[ \kappa_i=\left(\operatorname{th}\frac{m\gamma}{2}\operatorname{th}\frac{\gamma}{2}\right)^{-1} \]
is the coefficient of natural recharging,

\[ \gamma=f\!\left(\frac{C_2}{C_e}\right),\qquad C_e=C_0+C_1+2C_3, \]

\(q_\Sigma=q+q'\), \(\Delta q=q-q'\); \(q\) and \(q'\) are the charges of the transporters, respectively, on the path from the excitation system to the load system and back.

\[ F_c=f_2\left(1-\frac{C_0}{C_e}\right)+\frac{f_1}{2}\left(\frac{1}{\kappa_u}-\frac{C_0}{C_e}\right), \]

\[ \kappa_u= \left[ \frac{C_0}{C_e} + \frac{ 2(C_1+2C_3\operatorname{ch}\gamma)\operatorname{sh}\dfrac{m-s+1}{2}\gamma }{ C_e(m-s+1)\operatorname{sh}\gamma\,\operatorname{ch}\dfrac{m\gamma}{2} } \right]^{-1} \]
is the voltage-increase coefficient without brush displacement,

\(f_1, f_2, f_4\) are coefficients depending on \(n\), which are not expanded here because of the cumbersomeness of the expressions, and in [1] are reduced to a table.

We shall consider a displacement of the brushes by an integer number of transporters \(h\) simultaneously in the excitation and load systems and in one and the same direction. In doing so, the counting of \(n\) will still be begun from the transporter just commutated in the excitation system. In this case, obviously, to determine \(U_{np}\), expression (2) may be used without change, while to find \(U_{nc}\) in formula (3) it is necessary to replace \(n\) by \(n'=n\pm h\), respectively for displacement of the brushes in the direction of rotor rotation and in the opposite direction.

To determine the charges of the transporters, the following boundary conditions should be used:

\[ U_1= \begin{cases} U_v, & \text{for the conduction excitation circuit,}\\ 0, & \text{for the induction circuit.} \end{cases} \tag{4} \]

\[ U'_{m-1}=U_n \]

The simultaneous solution of equations (1–4), taking into account the above replacement of \(n\) and \(0\le h\le \dfrac{s-1}{2}\), gives: for the conduction excitation circuit

\[ q_\Sigma=U_v C_e+U_n C_0 \]

\[ \Delta q = U_v\kappa_i C_e \left(1-\frac{U_n\operatorname{ch}\gamma h}{\kappa_u U_v}\right) + U_n\kappa_i C_0(\operatorname{ch}\gamma h-1) \tag{5} \]

and for the induction circuit

\[ q_\Sigma=U_n(C_e-C_0)+U'_v C_0 \]

\[ \Delta q = U_v\kappa_i C_e \left[ 1-\frac{(U_n+U'_v)\operatorname{ch}\gamma h}{\kappa_u U_v} \right] + (U_n+U'_v)\kappa_i C_0(\operatorname{ch}\gamma h-1), \tag{6} \]

It follows from this that the sum of the charges of the transporters \(q\) and \(q'\) does not change when the brushes are displaced, whereas the charge delivered to the load, \(\Delta q\), and correspondingly the ESGS current \(I_n=2mN\Delta q\) (\(N\) is the number of rotor rev-

(revolutions per minute) decrease with increasing \(h\), regardless of the direction of displacement. Only in the short-circuit mode of the generator with a conduction excitation circuit does \(\Delta q\) not depend on \(h\). As for the no-load mode (\(\Delta q = 0\)), the voltage-increase coefficient decreases with increasing \(h\). Thus, with a conduction excitation circuit,

\[ \kappa_{uh}^{-1} = \frac{U_{\mathrm{в}}}{U_{xx}} = \operatorname{ch}^{2} h \left( \frac{1}{\kappa_u} - \frac{C_0}{C_{\mathrm{э}}} \right) + \frac{C_0}{C_{\mathrm{э}}}, \]

whereas with induction excitation, as in [1],

\[ \kappa'_{uh} = \kappa_{uh} - 1 . \]

Thus, when the point of commutation of the transporters is shifted relative to the midpoint of the inductors, the output parameters of the ESGS deteriorate the more, the greater this displacement.

REFERENCES

1. V. A. Lukutin, V. D. Eskov. On the calculation of electrostatic generators with a cascade connection of transporter-conductors. Izvestiya AN SSSR, “Energetika i transport”, No. 2, 1967.