On the Minimum Voltage for the Onset of Ionization in a Gas Inclusion
Abstract Generated abstract
This paper examines how the size of gas inclusions in solid electrical insulation affects the voltage at which ionization begins, a parameter often used to estimate insulation lifetime. Using gas breakdown relations and dielectric field conditions, it derives expressions for ionization inception voltage in both uniform fields and the nonuniform field of a cylindrical capacitor, and determines the inclusion size corresponding to the minimum voltage. The analysis indicates that inception voltage is not a monotonic function of pore size: it decreases to a minimum and then rises, with the minimum depending on insulation thickness, dielectric permittivity, and field geometry. The paper concludes that inception voltage alone cannot uniquely characterize insulation service life and should be supplemented by other diagnostic methods.
Full Text
On the Minimum Voltage for the Onset of Ionization in a Gas Inclusion
V. S. Dmitrevskii
(Presented by the scientific seminar of the Department of Electrical Insulation and Cable Engineering)
Ensuring the long-term operation of electrical insulation is an important scientific and practical problem. A number of authors \[1, 2, 3\] have shown that the destruction of insulation occurs as a result of ionization in gas inclusions. In connection with this, practical formulas have been obtained that relate the lifetime of insulation to the voltage at which ionization begins. One such formula was proposed by Dellektorskii \[4\]:
\[ t=\frac{\kappa}{fU^{2}\left(1-\frac{U_{\mathrm{i}}}{U}\right)} . \tag{1} \]
As is evident from the formula given, the lifetime of the insulation will be greater the higher \(U_{\mathrm{i}}\) is. The voltage at which ionization begins in a gas inclusion is an indirect measure for estimating the pore size in electrical insulation. It is naturally assumed here that the larger the pore, the lower the ionization inception voltage and the shorter the lifetime of the insulation. The purpose of the present work is to establish the relationship between pore size and ionization inception voltage.
It has been shown in the literature \[1\] that the breakdown voltage in a gas inclusion is approximately equal to the breakdown voltage of the gas between metallic electrodes when the distance between the electrodes is equal to the thickness of the gas inclusions.
Then, assuming that the field in the gas inclusion is uniform, the breakdown voltage of the gas is found from \[5\]:
\[ U_{\mathrm{br.g}}= \frac{Bpx}{ \ln \frac{Apx}{\ln\left(\frac{1}{\gamma}+1\right)} }, \tag{2} \]
where \(x\) is the thickness of the gas pore in the direction of the electric field.
The voltage applied to the insulation up to breakdown of the gas inclusion is found as
\[ U=U_{\mathrm{g}}+U_{\mathrm{s}} . \tag{3} \]
In this case \(U_{\mathrm{g}}\) and \(U_{\mathrm{s}}\) are the voltage drops attributable to the gas-
gas inclusion and on the solid dielectric, respectively. Breakdown of the gas inclusion will occur when \(U_{\mathrm{g}}=U_{\mathrm{br.g}}\). Let us consider how the ionization inception voltage will vary with the size of the gas inclusion.
a) Uniform field.
At the boundary between two dielectric media, the field strength changes discontinuously:
\[ E_{\mathrm{g}}=\xi E_{\mathrm{t}}. \tag{4} \]
Obviously, for the conditions of a uniform field,
\[ \begin{aligned} U_{\mathrm{g}}&=E_{\mathrm{g}}x,\\ U_{\mathrm{t}}&=E_{\mathrm{t}}(l-x), \end{aligned} \tag{5} \]
where \(l\) is the distance between the electrodes;
\(x\) is the thickness of the air inclusion.
Using expressions (2), (3), (4), (5), we find the discharge inception voltage in the gas inclusion
\[ U_{\mathrm{i}}= \frac{Bp}{\displaystyle \frac{Apx}{\ln\frac{1}{\ln\left(\frac{1}{\gamma}+1\right)}}} \left(x-\frac{x}{\xi}+\frac{l}{\xi}\right). \tag{6} \]
The function in equation (6) has a minimum, which can be found by taking the first derivative with respect to \(x\) and setting it equal to zero,
\[ \frac{dU_{\mathrm{i}}}{dx}=0. \tag{7} \]
Solving equation (6) gives
\[ \frac{l}{\xi-1} = x_\mu \left[ \ln \frac{Apx_\mu}{\ln\left(\frac{1}{\gamma}+1\right)} -1 \right], \tag{8} \]
where \(x_\mu\) is the thickness of the inclusion corresponding to the minimum ionization inception voltage.
Figure 1 shows the dependence of \(x_\mu\) on the thickness for different dielectric constants of the dielectric under the condition \(\gamma=4\), \(A=17.5\ \text{1/cm·mm Hg}\), \(p=760\ \text{mm Hg}\) [4]. As can be seen from Fig. 1 and formula (8), the pore size corresponding to the minimum value of the ionization inception voltage increases as \(\xi\) decreases and as the insulation thickness increases.
Figure 2 shows the dependences of the ionization inception voltage on the size of gas inclusions at an insulation thickness \(l=1\ \text{cm}\), \(\xi=3\), and \(B=87.2\ \text{V/cm·mm Hg}\).
It is evident from Fig. 2 that the ionization inception voltage can have the same value for two values of the gas-inclusion size. At larger gas-inclusion sizes, insulation wear will be more intense, since after breakdown of the gas inclusion a greater field strength is produced in the solid dielectric. Thus, at \(U_{\mathrm{i}}=1.5\ \text{kV}\), for a gas-inclusion size of \(0.009\ \text{cm}\), \(E_{\mathrm{t}}=1.5\ \text{kV/cm}\), whereas at \(x=0.3\ \text{cm}\), \(E_{\mathrm{t}}=2.14\ \text{kV/cm}\).
b) Field of a cylindrical capacitor.
A cylindrical capacitor represents a field quite frequently encountered in electrical-insulation structures. Suppose that the gas inclusion is located at the electrode of smaller
Fig. 1. Dependence of the thickness of the gas inclusion, corresponding to the minimum ionization inception voltage, on the thickness of the dielectric under uniform-field conditions for different dielectric permittivities of the dielectrics. The numbers beside the curves indicate the dielectric permittivity of the solid dielectric.
Fig. 2. Dependence of the ionization inception voltage on the thickness of the gas inclusion for uniform (1) and nonuniform (2) fields.
radius. Assuming that the field in the gas inclusion is uniform, one may write
\[ U_{\mathrm{pr.r.}}= \frac{Bp\,(r_2-r_1)} {\ln\frac{Ap\,(r_2-r_1)}{\ln\left(\frac{1}{\gamma}+1\right)}} \tag{2a} \]
The voltage drop across the layers of solid dielectric and gas is found as
\[ \begin{aligned} U_{\mathrm{g}}&=E_{\mathrm{g\,max}}r_1\ln\frac{r_2}{r_1}\\ U_{\mathrm{t}}&=E_{\mathrm{t\,max}}r_2\ln\frac{r_3}{r_2} \end{aligned} \tag{9} \]
Solving equations (3), (4), (9), and (2a) simultaneously, and assuming that ionization in the gas inclusion begins under the condition \(E_{\mathrm{g\,max}}=E_{\mathrm{pr.r.}}\), we find the ionization inception voltage
\[ U_{\mathrm{n}}= \frac{Bp r_1} {\ln\frac{Ap\,(r_2-r_1)}{\ln\left(\frac{1}{\gamma}+1\right)}} \left(\ln\frac{r_2}{r_1}+\frac{1}{\xi}\ln\frac{r_3}{r_2}\right) \tag{10} \]
The conditions for the minimum value of \(U_{\mathrm{n}}\) are found by solving equation (7). As a result of the solution we obtain
\[ \frac{r_2}{r_2-r_1} \left(\xi\ln\frac{r_2}{r_1}+\ln\frac{r_3}{r_2}\right) = (\xi-1)\ln \frac{Ap\,(r_2-r_1)} {\ln\left(\frac{1}{\gamma}+1\right)} \tag{11} \]
or, putting \(r_2-r_1=x\),
\[ \frac{r_1+x_\mu}{x_\mu} \left(\xi\ln\frac{r_1+x_\mu}{r_1}+\ln\frac{r_3}{r_1+x_\mu}\right) = (\xi-1)\ln \frac{Ap x_\mu} {\ln\left(\frac{1}{\gamma}+1\right)} . \tag{11a} \]
Figure 3 shows the dependence of the size of the gas inclusion corresponding to the minimum ionization voltage on the insulation thickness \(\Delta=r_3-r_1\) for the case \(r_1=0.5\ \mathrm{cm}\), at different \(\xi\).
Comparing Figs. 1 and 3 with one another, we see that the dependence \(x=f(\Delta)\) for a nonuniform field is considerably more gradual than in the case of a uniform field.
Figure 2 shows the dependence of the ionization inception voltage on the size of the gas inclusions for cylindrical specimens at \(\xi=3\).
The analysis carried out shows that the dependence of the ionization inception voltage on the size of gas inclusions has a minimum, dependent on the dimensions of the insulation and its dielectric permittivity. In this case, as the pore size increases, the ionization inception voltage decreases (the left branch of the curves in Fig. 2), and then increases. Taking into account that insulation wear increases with increasing size of the gas inclusion, one should expect accelerated insulation wear at small gas inclusions when \(U_{\mathrm{n}}\) decreases, and at large gas inclusions accelerated insulation wear when \(U_{\mathrm{n}}\) increases. Obviously, formula (1) will be valid only for small sizes of a gas inclusion in insulation.
It has been shown that the inception voltage of ionization cannot unambiguously determine the service life of insulation, and that it must necessarily be supplemented by other methods.
Fig. 3. Dependence of the thickness of the gas inclusion corresponding to the minimum ionization inception voltage on the thickness of the dielectric under conditions of a nonuniform field, for various dielectric constants of the solid dielectric. The numbers at the curves indicate the dielectric constant of the solid dielectric.
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