ON A METHOD FOR DETERMINING THE PARAMETERS OF PULSE OVERVOLTAGES MOST DANGEROUS FOR INSULATION

I. I. Kalyatskii, V. F. Panin

(Presented by the scientific seminar of the Research Institute of High Voltages)

To select appropriate levels of high-voltage insulation in power systems, data are needed on the electric strength of insulation under pulse overvoltages of various waveforms. Of particular interest are those overvoltage waveforms for which the impulse strength of insulation is minimal.

The possibility of establishing the waveforms of the most dangerous pulse overvoltages arose in connection with the authors’ discovery of an anomalous decrease in the impulse strength of insulating media in certain ranges of the rate of rise of the impulse voltage [1—4].

According to the data of [1—4], the most dangerous for insulation should be considered pulses whose steepnesses lie in the region of critical values (for insulating structures in air the critical steepness is about 7 kV/µs; in transformer oil, about 150 kV/µs; for insulation made of solid dielectrics, about 30 kV/µs).

Typical forms of pulse overvoltages are pulsating, oscillatory, and aperiodic pulses. The parameters of pulse overvoltages directly associated with the rate of voltage rise at a given overvoltage ratio are the frequency of pulsations and oscillations, as well as the front duration. Therefore, in order to evaluate the forms of pulse overvoltages that may present the greatest danger (all other conditions being equal) for insulating structures in various insulating media and for different voltage classes, it is necessary to calculate the frequencies of pulsating and oscillatory pulse overvoltages, as well as the front durations of aperiodic pulses.

With the indicated approach to solving the problem, the front of the impulse voltage at which the minimum discharge gradients occur must be located along the lines of critical steepness. Consequently, the front durations and oscillation (pulsation) frequencies of dangerous pulses are determined by the type of insulation and the voltage class (Fig. 1).

In the interest of a systematic approach, it is advisable to assume a division of the insulation used into three types: insulation in gaseous, solid, and liquid insulating media. Such a division entails a certain idealization of the approach to solving the ques-

growth. First, for dielectrics of the same type there are differences in the course of the prebreakdown process, caused by differences in chemical composition, structure, etc. Second, practical insulation is often a combination of several media.

Differences in the formation of breakdown in dielectrics of the same type apparently have a more quantitative than qualitative character. This is indicated by the very fact that critical steepnesses exist [3, 4].

With regard to the second point of the idealization, it should be noted that up to the present time most work on the study of the behavior of combined insulation (primarily oil-barrier insulation) under overvoltages of various shapes has been reduced to the investigation of the simplest homogeneous structures; moreover, attempts to justify such a possibility are known in the literature [5, et al.].

In estimating the shape of dangerous impulses, it is necessary, for any levels of impulse voltage \(U_a\), to obtain expressions for the front duration \(\tau_\phi\) of aperiodic and oblique-front impulses, and for the frequencies of oscillatory \(f_{1п}\) and pulsating \(f_{cos}\) impulses, corresponding to the critical rate of voltage rise \(a_{\text{cr}}\).

Fig. 1. On the method of estimating the shapes of dangerous pulsating \((a)\) and aperiodic \((b)\) overvoltages; \(\tau_{\phi1}\) and \(\tau_{\phi2}\) are the front durations of overvoltages dangerous for insulation at different nominal voltages.

Let us estimate (by the standard method [7]) the front duration of an oblique-front impulse having amplitude \(U_a = U_p\) and discharge time \(t_p\) (Fig. 2).

According to Fig. 2,

\[ 0.6\tau_\phi = t_2 - t_1, \tag{1} \]

where \(t_1\) and \(t_2\) are the instants of time at which the oblique-front wave reaches the values \(0.3U_a\) and \(0.9U_a\).

Since breakdown occurred on the average at the level \(0.7U'_a\) (Fig. 2), then \(0.3U_a = 0.21U'_a\) and \(0.9U_a = 0.63U'_a\), where \(U'_a\) is the full amplitude of the exponential impulse (with time constant \(\tau'\)) on whose front breakdown took place.

For \(t_1\) we have

\[ 1 - e^{-\frac{t_1}{\tau'}} = 0.21, \tag{2} \]

whence

\[ t_1 = -\tau' \ln 0.79, \tag{3} \]

and, respectively,

\[ t_2=-\tau' \ln 0.37 . \tag{4} \]

Let us express \(\tau'\) in terms of \(t_p\). For \(t=t_p\) we have

\[ 1-e^{-\frac{t_p}{\tau'}}=0.7, \tag{5} \]

whence

\[ \tau'=\frac{t_p}{1.205}. \tag{6} \]

Fig. 2. On the derivation of the ratio \(\dfrac{\tau_\phi}{t_p}\)

Then

\[ t_1=-\frac{t_p}{1.205}\ln 0.79, \tag{7} \]

\[ t_2=\frac{t_p}{1.205}\ln 0.37. \tag{8} \]

Substituting (7) and (8) into (1) and solving the resulting equation with respect to \(\tau_\phi\), we obtain

\[ \frac{\tau_\phi}{t_p}=1.06. \tag{9} \]

In a number of cases, breakdown occurs at levels different from \(0.7U'_a\). In this connection, values of \(\dfrac{\tau_\phi}{t_p}\) were determined for the interval of levels \((0.5—0.9)U'_a\), which is usually used in measurements of this kind. In the indicated interval the values of \(\dfrac{\tau_\phi}{t_p}\) amount to not

1.07–1.027, i.e., close to the value \(\dfrac{\tau_{\mathrm{f}}}{t_p}\) for the level \(0.7U'_a\). Therefore, in what follows relation (9) has been used.

It is of interest to compare the instantaneous rates of rise of the oblique-front and complete exponential pulses (shown in Fig. 2) for various \(\dfrac{U_a}{U'_a}\). Table 1 gives the ratios \(\dfrac{a_{\mathrm{ob}}}{a_{\mathrm{exp}}}\) for two instantaneous voltage values \((0.7U_a\) and \(0.9U_a)\) and a number of values of \(\dfrac{U_a}{U'_a}\).

Table 1

Ratio of the instantaneous rates of voltage rise of an oblique-front and a complete exponential wave \(\left(\dfrac{a_{\mathrm{ob}}}{a_{\mathrm{exp}}}\right)\) for various values of \(\dfrac{U_{\mathrm{inst}}}{U_a}\) and \(\dfrac{U_a}{U'_a}\)

In [6] it is reported that the development of a discharge in long air gaps is determined by the average steepness (close to the standard one), and not by the instantaneous rates of voltage rise. In view of this, the indicated ratios (Table 1) should be considered satisfactory.

Let us relate the front duration corresponding to the critical steepness to the frequencies of pulsating and oscillatory pulses. The voltage rise on the front of a pulsating wave with amplitude \(U\) and frequency \(f_{\cos}\), neglecting damping, is characterized quite completely by the expression

\[ u=\frac{U_a}{2}\left(1-\cos 2\pi f_{\cos} t\right). \tag{10} \]

(Fig. 3)

According to Fig. 3, one can write an equation analogous to (1), where

\[ t_1=\frac{\arccos(1-0.6)}{2\pi f_{\cos}}, \tag{11} \]

\[ t_2=\frac{\arccos(1-1.8)}{2\pi f_{\cos}}. \tag{12} \]

Using expressions (11), (12), and (1), we obtain

\[ \tau_{\mathrm{f}}=\frac{0.355}{f_{\cos}}. \tag{13} \]

The front of an oscillatory pulse (Fig. 4), neglecting damping, is well reproduced by the expression

\[ u=U_a\sin 2\pi f_{\sin} t. \tag{14} \]

A solution analogous to the case of the pulsating wave gives the following relation between the front duration and the frequency of the damped sinusoidal oscillations:

\[ \tau_{\mathrm{f}}=\frac{0.225}{f_{\sin}}. \tag{15} \]

Fig. 3. Toward the derivation of relation (13)

In the figure:

\[ U=\frac{U_a}{2}\left(1-\cos 2\pi f t\right) \]

with marked levels \(0.9U_a\), \(0.3U_a\), and intervals \(t_1\), \(t_2\), \(0.6\tau_{\varphi}\), \(\tau_{\varphi}\), \(T/2\).

Fig. 4. Toward the derivation of relation (15)

In the figure:

\[ U=U_a\sin 2\pi f t \]

with marked levels \(0.9U_a\), \(0.3U_a\), and intervals \(t_1\), \(t_2\), \(0.6\tau_{\varphi}\), \(\tau_{\varphi}\), \(T/4\).

In Fig. 5 pulses of different shape are compared, but with identical amplitudes \(U_a\) and front duration \(\tau_\phi\). As follows from Fig. 5, the difference in the instantaneous rates of voltage rise of the pulsating, oscillatory, and oblique-angled pulses is even smaller than in the case of the oblique-angled and exponential waves.

In Fig. 5:

\[ u=U_a\left(1-e^{-t/\tau}\right) \]

\[ u=\frac{U_a}{2}\left(1-\cos 2\pi f t\right) \]

\[ u=U_a\sin 2\pi f t \]

\[ u=\frac{U_a}{0.7}\left(1-e^{-t/(2.6\tau)}\right) \]

Labels shown in the figure: \(0.9U_a\), \(0.3U_a\), \(t_p\), \(\tau_\phi\), \(T_{\sin}/4\), \(T_{\cos}/2\).

Fig. 5. Comparison of an oblique-angled pulse with pulses of another shape

Since the mean value of the steepness in the experiment was determined as

\[ \frac{U_p}{t_p} \]

\([2—4]\), then, taking (9) into account, we obtain the mean value of the critical steepness

\[ a_{\mathrm{cr}}=\frac{U_a\cdot 1.06}{\tau_\phi}, \tag{16} \]

whence

\[ \tau_\phi=\frac{U_a\cdot 1.06}{a_{\mathrm{cr}}}. \tag{17} \]

Substituting (17) into (13) and (15), we obtain

\[ f_{\cos}=\frac{0.334\cdot a_{\mathrm{cr}}}{U_a}, \tag{18} \]

\[ f_{\sin}=\frac{0.212\cdot a_{\mathrm{cr}}}{U_a}. \tag{19} \]

If in (17), (18), and (19), instead of \(U_a\), one substitutes the value of the level of permissible overvoltages for a high-voltage structure of the corresponding voltage class, then it is possible to estimate the parameters \(\tau_\phi\), \(f_{\cos}\), \(f_{\sin}\) of overvoltages that may pose a danger for the given structure.

Since in the range of critical steepnesses there is a decrease in impulse strength, overvoltages lower than those permitted by GOST become dangerous.

The calculation of $\tau_{\phi}$, $f_{\cos}$, and $f_{\sin}$ for dangerous overvoltages therefore requires an estimate of the permissible overvoltage levels, taking into account the anomalous decrease in the strength of insulating media in the region of $a_{\mathrm{cr}}$. This estimate and the calculation of the parameters of dangerous overvoltages were carried out in [8].

REFERENCES

1. I. I. Kalyatskii, V. F. Panin, Izv. vuzov SSSR—Energetika, No. 6, 1966.

2. V. F. Panin, Izv. TPI, vol. 139, 1965.

3. A. A. Vorob'ev, I. I. Kalyatskii, V. F. Panin, Izv. TPI, vol. 152, 1966.

4. V. F. Panin, Izv. TPI, vol. 159, (in press).

5. P. G. Toshev, Izv. vuzov SSSR—Energetika, No. 3, 1963.

6. I. S. Stekol'nikov, A. V. Shkilev, DAN AN SSSR, 151, No. 5, 1963.

7. GOST-1516-60, “Transformers, apparatus, and insulators for high voltage,” Moscow, 1960.

8. I. I. Kalyatskii, V. F. Panin. Approximate estimation of the parameters of impulse overvoltages most dangerous for high-voltage insulation, (present collection).