On the Influence of Parasitic Capacitances on the Operation of Circuits for Measuring Pulsed Currents in Solid Dielectrics

A. V. Baranov

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

In the study of the pulsed electrical conductivity of dielectrics, circuits are used that make it possible to compensate the capacitive component of the dielectric current [1–3]. In the cited works, the measurements were carried out on pulses with a front of several hundred microseconds, when the capacitive currents are not very large. However, in this case the secondary effects that occur (high-voltage polarization, volume charges, thermal effects) may substantially complicate the interpretation of the measurement results. It is therefore expedient to use pulses with a short front (of the order of fractions of a microsecond). In this case it is necessary to take into account the role of the parasitic capacitances of the circuit [4] and to find the conditions that determine the minimum duration of the voltage-pulse front for given parameters of the measuring circuit and a specified magnitude of the error.

Of practical interest are the circuits shown in Figs. 1 and 2. In the first circuit the measured signal is taken from the resistance \(R_{\text{out}}\).

Fig. 1. Circuit using pulses of opposite polarity.
(\(C_d\) — dielectric capacitance, \(R_d\) — dielectric resistance, \(C_b\) — compensating capacitance, \(R_{\text{out}}\) — output resistance, \(C_p\) — parasitic capacitance of the measuring circuit).

Fig. 2. Bridge circuit.
(\(C_d\) — dielectric capacitance, \(R_d\) — dielectric resistance, \(C_b\) — compensating capacitance).

In this case two pulses of opposite polarity are applied to the circuit. To the second circuit one voltage pulse is applied, and the measured signal is taken between points \(a\) and \(b\).

Let pulses of opposite polarity of the form

\[ U(t)=\pm U_0(1-e^{-\beta t}) \tag{1} \]

be applied to the circuit of Fig. 1. The signs \(+\) and \(-\) correspond, respectively, to the left and right arms of the circuit. Then at the output we shall have

\[ I_1(t)=U_0\left[ \frac{1}{R_d+R_{\rm out}} + \frac{R_d C_d-R_{\rm out}(C_p+C_b)} {R_{\rm out}(R_d+R_{\rm out})(C_b+C_d+C_p)} \cdot e^{-p_1 t} - \frac{C_b e^{-p_2 t}} {R_{\rm out}(C_b+C_d+C_p)} \right](1-e^{-\beta t}), \tag{2} \]

where

\[ p_1=\frac{R_d+R_{\rm out}} {R_{\rm out}R_d(C_d+C_b+C_p)} \quad \text{and} \quad p_2=\frac{1}{R_{\rm out}(C_d+C_b+C_p)} . \]

An impulse of the form (1) is applied to the circuit of Fig. 2; at the output we have

\[ I_2(t)=U_0\left( \frac{1}{R_d+R_1} + \frac{R_d C_d-R_1 C_{p1}} {R_1(R_d+R_1)(C_d+C_{p1})}\cdot e^{-p_1 t} - \frac{C_b}{R_1(C_b+C_{p2})}\cdot e^{-p_2 t} \right)(1-e^{-\beta t}), \tag{3} \]

where

\[ p_1=\frac{R_d+R_1}{R_dR_1(C_d+C_{p1})} \quad \text{and} \quad p_2=\frac{1}{R_2(C_b+C_{p2})}. \]

If in (3) we set \(C_{p1}=C_p+C_b\), \(C_{p2}=C_p+C_b\), and \(R_1=R_{\rm out}\), then we see that the signal at the output of these circuits is given by one and the same expression, i.e., the circuits can be reduced to a common equivalent circuit. We shall assume that the pulses of opposite polarity applied to the first circuit are strictly identical in shape and magnitude. Then the conditions for compensation of the capacitive current for this circuit will have the form

\[ \begin{aligned} &1)\quad C_b=C_d=C,\\ &2)\quad R_d\gg R_{\rm out}, \end{aligned} \tag{4} \]

for the second circuit

\[ \begin{aligned} &1)\quad C_dR_1=C_bR_2,\\ &2)\quad C_{p1}=C_{p2}=C_p,\\ &3)\quad R_d\gg R_1. \end{aligned} \tag{5} \]

When conditions (4) and (5) are satisfied, the signal at the output of the equivalent circuit is

\[ I(t)=\frac{U_0}{R_d}(1-e^{-\beta t}) - \frac{U_0}{R_d}ae^{-pt}(1-e^{-\beta t}) = i_{\rm act}(t)-i_{\rm cap}(t). \]

Here \(p=p_1=p_2\) for both circuits,

\[ a=\frac{C_d+C_p}{C_p+2C_d} \quad \text{for the first circuit and} \quad a=\frac{C_p}{C_p+C_d} \quad \text{for the second.} \]

From (5) we see that, when a voltage pulse with a long front is used, the measured current is

\[ I(t)=\frac{U_0}{R_d}(1-e^{-\beta t}), \]

i.e., it is equal to the active—

component. For \(\beta \gg p\), i.e., when the front is sufficiently short, there will be no complete compensation of the capacitive current. The distortion is introduced by the term

\[ \delta(t)=ae^{-pt}, \tag{7} \]

which, for sufficiently large \(t=t_p\), can become arbitrarily small. The time \(t_p\) can be determined if one specifies the permissible value of the relative error

\[ t_p=\frac{1}{p}\ln\frac{a}{\delta(t_p)}. \tag{8} \]

Figure 3 shows the components of the measured current: \(i_{\text{act}}^{(t)}\) (solid curves), \(i_{\text{cap}}^{(t)}\) (dashed curves), and \(I(t)\)—dash-dotted curves. It is clear from this figure that in the interval \(0<t<t_p\) the error may be considerable; at \(t=t_p\) it is equal to the permissible value, and for \(t>t_p\) it is less than the permissible value. The correction for the time \(0<t<t_p\) can be determined by calculation from formula (5).

Thus, reliable measurements with an error smaller than the specified one correspond to times \(t \ge t_p\). This same condition may be taken as the minimum duration of the front of the voltage pulse.

Fig. 3. Dependence on time of \(i_{\text{act}}\), \(i_{\text{cap}}(t)\), and \(I(t)=i_{\text{act}}(t)+i_{\text{cap}}(t)\).

As an example, let us determine the smallest duration of the front of the voltage pulse in the bridge circuit [4]. We have: \(R_1=1\ \text{k}\Omega\), \(C_p=C_{ck}+C_{ce}+C_m\). For the 6Zh5P tube used in the circuit, \(C_{ck}=8.5\ \text{pF}\) (capacitance between the cathode and the control grid), \(C_{ce}=2.2\ \text{pF}\) (capacitance between the control and guard grids), \(C_m\simeq 3\ \text{pF}\)—mounting capacitance. Suppose that \(\delta(t_p)\simeq 0.1\%\); then \(t_p=\tau_{\text{fr.min.}}=1.3\cdot 10^{-7}\ \text{s}\). Consequently, correct reading of the current values from the oscillogram corresponds to the time \(t\ge 1.3\cdot 10^{-7}\ \text{s}\). The use of a shorter pulse front is determined by practical necessity, and in that case the calculated correction must be introduced.

REFERENCES

1. N. H. Rase. G. E. R. 44, 8, 1941.
2. E. A. Konorova, D. G. Lebedev. PTE. 4, 68, 1958.
3. V. M. Fainitskii. Elektrichestvo. 4, 76, 1952.
4. A. V. Baranov. Collection of Proceedings of the Interuniversity Conference on Breakdown of Dielectrics and Semiconductors. GEI, 1964.