VOLTAGE DISTRIBUTION ALONG THE SURFACE OF A DIELECTRIC UNDER PULSES

V. S. Dmitrevskii

(Presented by Professor, Doctor A. A. Vorob'ev)

The problem of voltage distribution along the surface of a dielectric under pulses has not been sufficiently studied \[8, 7\]. Meanwhile, knowledge of the voltage distribution along the surface of a dielectric is necessary in designing high-voltage insulation, in developing impulse-voltage dividers, and in a number of other cases.

The insufficient number of measurements of voltage distribution along the surface of a dielectric is apparently explained by considerable experimental difficulties. The method we developed for measuring voltage distribution \[1, 2, 3\] made it possible to investigate the voltage distribution along the surface of a dielectric under pulses with smaller error.

Measurements of voltage distribution along the surface of a dielectric were carried out on cylindrical specimens.

The source of impulse voltage was a GIN rated at 1400 kV, installed in the laboratory of the Department of High Voltage Engineering of the Tomsk Polytechnic Institute.

Analytical calculation of the voltage distribution along the surface of a dielectric

To calculate the voltage distribution along the surface of a dielectric under pulses, we proposed the equivalent circuit shown in Fig. 1. The equivalent circuits developed by A. M. Zalesskii \[4\] and M. I. Mantrov \[5\] are special cases of the circuit shown in Fig. 1.

For the equivalent circuit presented in Fig. 1, the following differential equation can be written:

\[ \frac{\partial^2 U_2}{\partial x^2} = U_2 \frac{y_{s1}}{y_0} - U \frac{y_{s2}}{y_0} + U_2 \frac{y_{s2}}{y_0}. \tag{1} \]

In this equation:
\(U_2\) — the voltage at a distance \(x\) from the grounded electrode;
\(U\) — the applied voltage;

\(y_{s1}\)—the total conductance per unit length of the surface of the dielectric located at a distance \(x\) from the grounded electrode, with respect to the grounded electrode;

\(y_{s2}\)—the total conductance per unit length of the surface of the dielectric located at a distance \(x\) from the grounded electrode, with respect to the high-voltage electrode;

\(y_v\)—the total volume conductance per unit length of the dielectric.

Solving equation 1, we obtain:

\[ U_2 = \frac{U}{(y_{s1}+y_{s2})sh\gamma l} \left[ y_{s1}sh\gamma x + y_{s2}sh\gamma l - y_{s2}sh\gamma(l-x) \right]. \tag{2} \]

In this equation:

\(l\)—the distance between the electrodes,

\[ \gamma = \sqrt{\frac{y_{s1}+y_{s2}}{y_v}}. \tag{3} \]

Analysis of equation (2) shows that the voltage distribution along the surface of the dielectric will be the more uniform, the smaller \(\gamma\) is, i.e., the smaller \(y_{s1}\) and \(y_{s2}\) are. When \(y_{s1}=y_{s2}=0\), the voltage distribution along the surface of the dielectric will be rectilinear, and equation (2) takes the form:

\[ U_2 = U\frac{x}{l}. \tag{4} \]

Fig. 1. Equivalent circuit of a solid dielectric and the medium surrounding it

Influence of the surrounding medium and the dielectric material on the voltage distribution along its surface

The influence of the properties of the solid dielectric and of the medium surrounding it on the voltage distribution along the surface of the dielectric can be clarified by analyzing the coefficient \(\gamma\). From formula (3) it is evident that the coefficient \(\gamma\) depends only on the magnitudes of the total conductances, whose general expression has the following form:

\[ y = \frac{1}{R} + jpC. \]

Since, under pulses, the active component of the conductance of the solid dielectric and of the medium surrounding it is much smaller than the capacitive component of the conductance, in what follows we shall take into account only the latter in our calculations.

With the assumption we have made, the values of the conductances entering equation (3) can be written in the following form:

\[ y_{s1}=jpC_{s1}=jK_1p\varepsilon_s d, \]

where \(K_1\) is a coefficient taking into account the shape of the grounded electrode,

\(\varepsilon_s\) — dielectric permittivity of the surrounding medium,
\(d\) — diameter of the solid dielectric,

\[ p \text{ — operator, equal to } p=\frac{\dfrac{\partial U}{\partial t}}{U}, \]

\[ y_{s2}=j C_{s2}p=jK_2p\varepsilon_s d, \]

where \(K_2\) is a coefficient taking into account the shape of the high-voltage electrode

\[ y_v=jC_vp=jKp\varepsilon_v d^2. \]

In this formula:
\(K\) — coefficient taking into account the shape of the solid dielectric,
\(\varepsilon_v\) — dielectric permittivity of the solid dielectric.

After substituting the values of the total conductance into equations (2) and (3), we obtain:

\[ \gamma=\sqrt{\frac{\varepsilon_s(K_1+K_2)}{K\varepsilon_v d}}, \tag{3a} \]

\[ U_x=\frac{U}{(K_1+K_2)\operatorname{sh}\gamma l} \left[ K_1\operatorname{sh}\gamma x+K_2\operatorname{sh}\gamma l-K_2\operatorname{sh}\gamma(l-x) \right]. \tag{2a} \]

Fig. 2. Voltage distribution along the surface of a textolite rod under pulses of positive polarity in air (1) and transformer oil (2).

Fig. 3. Voltage distribution along the surface of a glass tube under pulses of positive polarity in air:
\(o\) — tube filled with TiO\(_2\),
\(x\) — tube filled with air,
\(\oplus\) — tube filled with water.

From formula (2a) it is evident that \(\varepsilon_s\) and \(\varepsilon_v\) enter only into the coefficient \(\gamma\), i.e., the physical properties of the materials are taken into account only by the coeffici-

In this case. From formula (3a) it is seen that the larger \(\varepsilon_s\) is and the smaller \(\varepsilon_v\) is, the larger \(\gamma\) will be, i.e., the more nonuniform the voltage distribution along the surface of the dielectric will be.

It is not possible to calculate the voltage distribution along the surface of the dielectric from formula (2a), owing to the unknown coefficients \(K_2\) and \(K_1\). Therefore, the voltage distribution along the surface of a dielectric under pulses was studied by us experimentally.

Figure 2 presents the voltage distribution along the surface of a textolite rod in air \((\varepsilon_s = 1)\) and in transformer oil \((\varepsilon_s = 2.4)\). This experiment and all subsequent ones were carried out on cylindrical specimens with ring electrodes. It is seen from Fig. 2 that, as the dielectric permittivity of the surrounding medium increases, the voltage distribution along the surface of the dielectric becomes increasingly nonuniform.

Figure 3 presents the results of measuring the voltage distribution along the surface of a glass tube 750 mm long, with an internal diameter of 19 mm and an external diameter of 23 mm.

The surrounding medium in these experiments was air. The dielectric permittivity of the solid dielectric was varied by placing inside the tube water \((\varepsilon_v = 80)\), titanium dioxide (the dielectric permittivity in the polycrystalline state is equal to 114), and air \((\varepsilon_v = 1)\).

It is seen from Fig. 3 that, as \(\varepsilon_v\) increases, the voltage distribution along the surface of the dielectric becomes more uniform. The deviation from the indicated regularity in measurements when titanium dioxide is present in the tube is apparent. In our experiments, titanium dioxide was used in the form of a finely crystalline powder poured into the tube, i.e., \(\mathrm{TiO}_2\) was a porous, inhomogeneous material.

For such an inhomogeneous material, the dielectric permittivity under pulses will be 2–2.5 times smaller than for a monolithic structure [6], i.e., the dielectric permittivity of titanium dioxide will be of the order of 50–60.

Figure 4 gives the voltage distribution along the surfaces of a textolite rod \((\varepsilon_v = 5 \div 7)\), a plexiglass rod \((\varepsilon_v = 2.7 \div 3.2)\), and a glass tube filled with air (dielectric permittivity of glass, 6.5–7; of air, 1). The surrounding medium was air. It is seen from Fig. 4 that, as \(\varepsilon_v\) increases, the voltage distribution along the surface of the dielectric under pulses becomes more uniform.

Fig. 4. Voltage distribution in air along the surface of a glass tube (○), a plexiglass rod (×), and a textolite rod (⊂) under pulses of positive polarity.

The experiments performed show that, with an increase in $\varepsilon_v$ and a decrease in $\varepsilon_s$, the voltage distribution along the surface of the dielectric under pulses becomes more uniform. The experimental material is in good qualitative agreement with the analytical conclusions given above.

Dependence of the voltage distribution along the surface of a dielectric on the interelectrode distance

It follows from formula (2) that the voltage drop at the high-voltage electrode depends only slightly on the interelectrode distance. This can be shown by differentiating formula (2) with respect to $x$ for $x \to l$.

We carried out experiments to check the stated assumption. Fig. 5 shows the voltage distribution along the surface of a glass tube under pulses. The surrounding medium was air. From Fig. 5 it is seen that the near-electrode voltage drop remains constant when the interelectrode distance is varied from 300 to 1000 mm. Similar results were obtained for textolite and plexiglass rods in air and in transformer oil.

Fig. 5. Voltage distribution along the surface of a glass tube in air.

The experiments performed confirm the assumption stated by us.

Influence of the magnitude of the applied voltage on the voltage distribution along the surface of a dielectric

It follows from equation (2) that the voltage distribution along the surface of a dielectric, expressed as fractions of the applied voltage, should not depend on its magnitude (this is easily shown if the left- and right-hand sides of equation (2) are divided by $U$).

We measured the voltage distribution along the surface of a dielectric for different magnitudes of the applied voltage.

Figures 6 and 7 show the voltage distribution along the surface of a glass tube in air under pulses of positive (Fig. 6) and negative (Fig. 7) polarity.

It is seen from Figs. 6 and 7 that the voltage distribution along the surface of the glass tube, in fractions of the applied voltage, becomes more uniform as the latter increases.

Figure 8 gives the dependence of the field strength at the high-voltage and grounded electrodes on the magnitude of the applied voltage.

Fig. 6. Voltage distribution along the surface of a glass tube in air under pulses of positive polarity
\(o\) — \(U = 40\) kV, \(x\) — \(U = 68\) kV,
\(\oplus\) — \(U = 153.8\) kV.

Fig. 7. Voltage distribution along the surface of a glass tube in air under pulses of negative polarity
\(o\) — \(U = 32\) kV, \(x\) — \(U = 63.4\) kV,
\(\oplus\) — \(U = 164\) kV.

voltage. It is seen from Fig. 8 that the nature of the function \(E = f(U)\) depends on the polarity of the applied pulse. Under pulses of negative polarity, as the voltage is increased, there is first a rapid increase in the field strength at the high-voltage electrode and a slow increase at the grounded electrode; then the increase in field strength at the high-voltage electrode decreases, while at the grounded electrode it increases. Under pulses of positive polarity, as the applied voltage is increased, a decrease in the field strength at the high-voltage electrode and an increase at the grounded electrode are observed.

The obtained form of the dependences \(E = f(U)\) permits the following supposition.

In a certain voltage range, \(E \equiv U\), up to the voltage at which corona and creeping discharges appear. Since our measurements were carried out at voltage magnitudes exceeding \(U_{\mathrm{cor}}\), the function \(E = f(U)\) is explained by the different character of discharge development from the positive and negative electrode.

Fig. 8. Dependence of the field intensity at the electrodes on the magnitude of the applied voltage (glass tube in air)

1 — negative impulse, 2 — positive impulse.

\(x\) — \(E = f(U)\) — at the grounded electrode,
\(o\) — \(E = f(U)\) — at the high-voltage electrode.

Fig. 10. Dependence of the field intensity at the electrodes on the magnitude of the applied voltage for impulses of positive polarity (bakelite-paper tube in transformer oil)

1 — \(E = f(U)\) — at the high-voltage electrode,
2 — \(E = f(U)\) — at the grounded electrode.

Fig. 9. Voltage distribution along the surface of a bakelite-paper tube under impulses of positive polarity in transformer oil

1 — \(U = 68.8\ \mathrm{kV}\), 2 — \(U = 138\ \mathrm{kV}\),
3 — \(U = 217\ \mathrm{kV}\), 4 — \(U = 306\ \mathrm{kV}\).

Figure 9 shows the voltage distribution along the surface of a bakelite-paper tube in transformer oil under pulses of positive polarity.

It is seen from Fig. 9 that, as the voltage is increased, its distribution along the surface of the dielectric becomes more uniform.

In Fig. 10 the dependence of the field strength at the high-voltage and grounded electrodes on the magnitude of the applied voltage is plotted.

It is seen from Fig. 10 that, as the voltage is raised, the field strength at the high-voltage electrode increases rectilinearly up to \(U = 138\) kV, and with a further increase in voltage it decreases. Inspection of the bakelite-paper cylinder showed that, at a voltage of 217 kV, clear traces of a sliding discharge were observed.

Measurements of the voltage distribution along the surface of a textolite and a plexiglass rod in air and in transformer oil gave analogous results.

On the basis of the experiments carried out, the following conclusion may be drawn: equation (2) well reflects the dependence of the voltage distribution along the surface of dielectrics, in fractions of the applied voltage, on the magnitude of the latter, up to the voltage at which sliding discharges appear.

Conclusions

The experiments carried out make it possible to draw the following conclusions.

The voltage distribution along the surface of a dielectric under pulses has the same character as under alternating voltage. The voltage distribution along the surface of a dielectric under pulses becomes more nonuniform with an increase in the dielectric permittivity of the surrounding medium \(\varepsilon_s\) and with a decrease in the dielectric permittivity of the solid dielectric \(\varepsilon_v\).

The voltage drop near the electrode remains unchanged when the dielectric length is increased from 300 to 1000 mm.

When the applied voltage is increased, the voltage distribution along the length of the dielectric becomes more uniform. This is apparently explained by a developing corona discharge at the electrodes.

The obtained formula (2) is qualitatively well confirmed by the experiments carried out, up to the voltage at which corona discharges appear.

References

1. Dmitrevskii V. S. “Method for measuring voltage distribution.” Elektrichestvo, No. 10, 73, 1954.

2. Dmitrevskii V. S. “On a method for measuring the voltage distribution over insulation.” Izv. Tomskogo politekh. in-ta, vol. 76, 142, 1954.

3. Dmitrevskii V. S. “Analysis of errors in measuring voltage distribution by the two-measurement method” (present collection).

4. Zalesskii A. M., Sorokin A. V., Stefanov K. S. Guide to Laboratory Work in High-Voltage Engineering. Izd. KUBUCH, 1934.

5. Mantrov M. I. “Electric discharges on the surface of dielectrics.” Elektrichestvo, No. 9, 54, 1940.

6. Vorob’ev A. A. High-Voltage Engineering. Gosenergoizdat, 1945.

7. Levshunov R. T. “Investigation of insulators coated with a semiconducting glaze.” Elektricheskie stantsii, No. 4, 36, 1954.

8. Fedchenko I. K. “Voltage distribution along an insulator string under pulses.” Elektricheskie stantsii, No. 3, 1936.