On a Method for Measuring the Voltage Distribution Across Insulation

V. S. Dmitrevsky

In the investigation of solid dielectrics, and also in the investigation of insulating structures, it is extremely important to know the voltage distribution in the dielectric and over its surface at voltages of various frequencies and under impulses.

All known methods for measuring the voltage distribution may be divided into two groups:

1. Compensation methods.
2. Voltmeter methods.

Compensation methods require complex equipment and a rather large expenditure of time to obtain the voltage-distribution curve. These methods give high accuracy. Such measurements can also be carried out in the presence of corona discharge.

Fig. 1

1—specimen; 2—voltmeter; HV—high-voltage terminal; \(U\)—total voltage on the specimen; \(U'_1\)—voltage on the part of the specimen measured by the instrument; \(Y_1\) and \(Y_2\)—total conductance of the parts of the specimen; \(Y'_1\)—total conductance of the instrument.

“Voltmeter methods” do not require complex equipment, and measurements can be made with less expenditure of time and by less qualified personnel. The accuracy of the measurement depends on the accuracy of the instrument used and on how great is the distortion of the voltage distribution introduced by the conductance of the instrument. The smaller the conductance of the instrument in comparison with the conductance of the specimen along which the voltage distribution is measured, the smaller the measurement error.

In our opinion, the voltmeter method can be made more accurate if measurements are performed twice, connecting the instruments as shown in

Figs. 1a and 2a. The equivalent circuits for these measurements are shown in Figs. 1b and 2b. For these equivalent circuits one may write the following equations:

\[ I_1 = U'_1 (Y'_1 + Y_1) = (U - U'_1)Y_2, \tag{1} \]

\[ I_2 = (U - U'_2)Y_1 = U'_2 (Y'_2 + Y_2), \tag{2} \]

where \(I_1\) and \(I_2\) and the other quantities are defined from Figs. 1 and 2. In the absence of the measuring circuit, when \(Y'_1 = Y'_2 = 0\), the current flowing through the dielectric may be written as

\[ I = Y_1 (U - U_2) = U_2 Y_2. \tag{3} \]

In these equations \(Y'_1\), \(Y'_2\), \(Y_1\), and \(Y_2\) are the admittances of the measuring system and of the sections of the dielectric, defined in operational form:

\[ Y = \frac{1}{R + Lp + \dfrac{1}{Cp}}, \tag{4} \]

where \(R\), \(L\), and \(C\) are the active resistance, inductance, and capacitance of the measuring system or of the section of the specimen for which \(Y\) is calculated.

Fig. 2

1—specimen; 2—voltmeter; H.V.—high-voltage terminal; \(U\)—total voltage across the specimen; \(U'_2\)—voltage on a section of the specimen, measured by the instrument; \(Y_1\) and \(Y_2\)—total admittances of the separate parts of the specimen; \(Y'_2\)—total admittance of the instrument.

In equations 1, 2, and 3 the known quantities are: \(U\), \(U'_1\), \(U'_2\), \(Y'_1\), and \(Y'_2\), while the unknowns are \(Y_1\), \(Y_2\), and \(U_2\). The values of \(Y_1\) and \(Y_2\) found from equations 1 and 2 are substituted into equation 3, and the value of \(U_2\) is found:

\[ U_2 = \frac{ UU'_2Y'_2 - U'_1U'_2 (Y'_2 - Y'_1) }{ U'_1Y'_1 + U'_2Y'_2 }. \tag{5} \]

For most measuring instruments the admittance changes little within the limits of measurement on a single scale, i.e., it may be assumed that \(Y'_1 = Y'_2 = Y'\).

In this case equation (5) takes the simple form:

\[ U_2 = U'_2 \frac{U}{U'_1 + U'_2}. \tag{6} \]

The method described above will be inaccurate in the presence of a corona discharge on the electrodes.

Results of measuring the voltage distribution over a string of insulators

The method for measuring voltage distribution described above was applied to obtain voltage-distribution curves over strings of type P—4.5 insulators.

Figure 3 shows the results of measuring the voltage distribution over a 6-element string of insulators with instruments having different conductivities and producing different degrees of distortion of the electric field. However, when measurement is carried out by the method described above, the measurement error resulting from field distortion is almost completely taken into account in the calculation by formula (6).

Fig. 3. Distribution of the voltage drop over a 6-element string of type P-4.5 insulators at alternating voltage.

× measurement was carried out with spheres of diameter 2 cm.
⊕ measurement was carried out with a voltmeter with an ohmic voltage divider having a resistance of 40 MΩ.

Using formula (6), it is possible fairly quickly and with a high degree of accuracy to obtain the voltage-distribution curve over an insulating structure with the aid of a pointer instrument. Thus, for example, obtaining the voltage-distribution curve over a 6-element string of insulators with the aid of a voltmeter with a voltage divider takes no more than 2 hours, whereas obtaining this curve by the method of constant dis—

moving the spheres, described in the literature [1], about 8 hours were required, with lower measurement accuracy.

Figures 4 and 5 show curves of the voltage distribution over a 13-element string of insulators under alternating voltage and under impulse voltage (wave 1.5/40) of negative polarity. The voltage measurements were made with a sphere gap with a sphere diameter of 2 cm. The curves were taken with rings of various diameters present on the insulator string, and in their absence.

Fig. 4. Distribution of the voltage drop over a 13-element string of type P-4.5 insulators under alternating voltage

⊙ without rings, × with rings of diameter 0.8 m

From Figs. 4 and 5 it is evident that, in the absence of grading rings, the greatest voltage drop falls on the element nearest the conductor and reaches 35–37% of the total voltage. This is considerably greater than the value previously given in the literature [2]. Such a discrepancy with the literature data is explained by the following circumstance.

The voltage at which we measured the voltage distribution over the insulator string remained constant throughout the entire test and, in all tests, was never higher than 50 kV; i.e., the voltage drop on the first element of the string (17 kV) in all measurements was below the corona inception voltage on the element (18–20 kV).

The literature describes measurements carried out by the method of constant separation of the spheres. Prof. A. M. Zalesskii recommends, when measuring by

for this method to take spheres 5 mm in diameter and a distance between the spheres of 0.5 mm. The discharge voltage of such a gap is 2.4 kV. If such a gap is installed on an element of the string located not far from the grounded end, where the voltage drop, as is seen from Fig. 4, is 2–3%, then in order to break down this gap it is necessary to apply to the insulator string a voltage on the order of 80–100 kV; in this case the first element will have on the order of 30 kV, which considerably exceeds the corona-inception voltage on the insulator element and will cause significant measurement errors, which are not taken into account when measurements are made by the method of constant separation of the spheres.

Fig. 5. Distribution of voltage drop over a 13-element string of type P-4.5 insulators under impulse voltage.

⊙ without rings, × with rings 0.8 m in diameter, ⊕ with rings of diameter at the grounded end of the string 0.8 m, at the conductor 0.94 m.

In Figs. 4 and 5 the voltage distribution along an insulator string is shown in the presence of rings that equalize the voltage drop, according to the author’s data. These data show that, in the presence of rings, the maximum voltage drop under impulses occurs on the third and fourth element from the conductor, while at the grounded end the distribution remains similar to the voltage distribution in the absence of rings. Under alternating voltage the rings equalize the voltage distribution along the string, i.e., they achieve the purpose for which they are installed.

Conclusions

A method has been proposed for measuring the distribution of voltage along the length of an insulator.

The method consists in the following: in order to eliminate the influence of distortions introduced by the measuring circuit, the voltage at a given point of the dielectric is measured with respect to two electrodes between which the dielectric is located. By means of a simple mathematical calculation, the true value of the voltage at the given point with respect to either of the electrodes is found.

The method is applicable at any voltage—direct, alternating, and impulse—and permits the use of any measuring instrument. The method gives significant errors when corona is present on the electrodes. The application of this method for measuring the voltage distribution along strings of insulators made it possible to carry out these measurements under impulse voltage and to refine the voltage distribution along an insulator string under alternating voltage.

Our measurements showed that the voltage distribution along an insulator string is more nonuniform than has been described in the literature.

The author expresses his deep gratitude to Doctor of Physical and Mathematical Sciences Professor A. A. Vorob’ev for supervising the present work, and to Doctor of Technical Sciences Docent G. E. Pukhov, who reviewed the manuscript and made a number of valuable comments.

Literature

1. Zalesskii A. M., Sorokin A. V., and Stefagov K. S. Manual for Laboratory Work on High Voltage. Moscow–Leningrad, Kubuch, 1934.
2. High-Voltage Engineering, issue II, edited by Professor L. I. Sirotinskii. Gosenergoizdat, 1939.