On the Measurement of the Electrical Conductivity of Ionic Crystals Using Pulses

K. K. Sonchik

(Presented by Professor Doctor A. A. Vorob′ev)

The study of the electrical conductivity of solid dielectrics in high electric fields under direct voltage is inevitably associated with thermal processes, with polarization, and with a number of other undesirable phenomena. It would be of interest to know how electrons behave in ionic crystals in fields close to breakdown fields, during electrical breakdown. From this point of view, the study of electrical conductivity under pulsed voltage, when the influence of polarization, heating, and other secondary factors is excluded, is an important problem; pulse measurements would make it possible to obtain important data on the dependence of conductivity on time.

The measurement of pulsed electrical conductivity ultimately reduces to measuring, by oscillographic or other means, the current flowing through the specimen. Under direct voltage, investigations are usually carried out with the aid of galvanometers and electrometers. When a pulsed voltage is applied to a dielectric, the indicated instruments, owing to their considerable inertia, prove unsuitable for measuring pulsed currents. In this case an oscilloscope is usually used, as an instrument possessing very small inertia and at the same time making it possible to observe visually the change of current with time. By oscillographing the voltage across a known resistance connected in series with the specimen, one can determine the magnitude of the conduction current.

We were faced with the task of choosing a circuit and a method for measuring the pulsed electrical conductivity of ionic crystals in high electric fields, and of elucidating the possibilities of using certain circuits for measuring, in solid dielectrics, pulsed currents of duration \(10^{-8} \div 10^{-7}\) sec.

In measuring pulsed electrical conductivity, investigators encounter capacitive currents, which occur whenever the voltage across the dielectric changes and which complicate the interpretation of the results.

In the work of Turner and Lyons [1], the electrical conductivity of soda-lime glass was measured under the action of a voltage pulse with a front increasing linearly to the amplitude in \(150\ \mu\text{sec}\). The capacitive current in this case was relatively small and could easily be ...

determined by calculation. The difficulties associated with capacitive currents were also pointed out by Keller [2], who limited his measurements of the electrical conductivity of glass to pulses no shorter than \(8\cdot 10^{-5}\) sec for temperatures not below 60°C.

Applying to the specimen, with a resistance connected in series with it (Fig. 1), a voltage pulse, we detect, in addition to the conduction current of the specimen, also a capacitive current caused by the voltage varying at the leading edge of the pulse. This current, for a pulse-front rise time of the order of \(10^{-7}\div 10^{-8}\) sec, proves to be much larger than the conduction current—for example, of a rock-salt specimen, even at an electric-field strength close to breakdown, at room temperature. Thus, two currents will flow through the specimen and the resistance \(R\): the conduction current and the capacitive current. The problem is to divert, in some manner, the capacitive current that interferes with the measurement, or somehow to compensate it.

Naturally, the question arises of the possibility of compensating the capacitance of the specimen and electrode system \(C_0\) by means of some inductance \(L\). In this case the circuit takes the form shown in Fig. 2, and the inductance \(L\) is determined from the condition for compensation of the reactive components of the voltage drop

\[ -\frac{I}{\omega C_0}=I\omega L \quad \text{or} \quad L=\frac{1}{\omega^2 C_0}, \]

where \(\omega\) is the angular frequency.

Fig. 1. Dielectric circuit with measuring resistance

Fig. 2. Circuit for compensating the capacitive current by an inductance

Most often in the laboratory one deals with high-voltage pulses whose leading edge is an exponential. This exponential can be expanded into sinusoidal components in a Fourier series and, consequently, for a given capacitance \(C_0\) and inductance \(L\), one can speak of compensating only one component. All the others will not be compensated and, consequently, the corresponding capacitive currents will appear.

In 1941 Reiss [3] proposed a circuit for measuring prebreakdown currents in dielectrics with compensation of the capacitive current (Fig. 3).

In essence, this was a bridge circuit. Compensation of capacitive currents by means of bridge circuits was used by a number of authors [3, 4, 5, 6]; however, the conditions under which they worked (lower voltages, longer times, low frequencies, appreciable conductivities) were much easier than when working with solid dielectrics at short times.

Fig. 3. Race circuit

Race’s circuit consisted of two symmetrical halves, in each of which there was a pulse-voltage generator (PVG). In one arm of this bridge circuit a chamber with the specimen under test (SU) was connected, and in the other a regulable loss-free air capacitor \(C_{\mathrm{v}}\). Each of the generators produced voltage pulses of identical amplitude and shape, but of opposite polarity. When the spark gaps were ignited, the circuit operated: voltage pulses of opposite polarity were applied to the test specimen and to the air capacitor \(C_{\mathrm{v}}\), and, if the capacitances of the specimen and the capacitor were set equal, the resulting capacitive current was compensated, producing no voltage drop across the resistance \(R\). Since a dielectric specimen has, although very large, nevertheless finite active resistance, through the arm of the circuit in which it is connected there will flow, in addition to the capacitive current, also a conduction current, which across the resistance \(R\) will create an unbalance voltage drop. This voltage was recorded by an oscillograph.

Race used his circuit mainly for measuring currents in liquids with comparatively high conductivity. In this case the conduction current was sufficient for the voltage drop across the resistance \(R\) to cause a noticeable deflection of the beam on the oscillograph screen.

However, even for these experiments Race had to use a resistance \(R\) equal to \(1\ \mathrm{M\Omega}\). The situation is different with solid dielectrics. The currents in them are very small, and the voltage drop even across a resistance \(R = 1\ \mathrm{M\Omega}\) cannot be observed on the oscillograph tube when the deflection plates are connected directly to the indicated resistance.

Since it is impossible to balance the circuit absolutely precisely, part of the capacitive charging current will flow through the resistance \(R\), which makes it necessary to limit the magnitude of the latter. In addition, as the resistance \(R\) is increased, the oscillograph plates will exert an increasingly shunting action. Precisely as a result of this—

for these reasons Rice was forced to use comparatively long (280 microsec) pulses and a resistance \(R = 1\) megohm.

The operation of the circuit depended entirely on the accuracy with which its parts were balanced. Even with equal capacitances of the EO sample and of the air capacitor \(C_v\), an unbalance current could arise because the shapes and amplitudes of the pulses produced by the two pulse generators could nevertheless differ from one another, for example because of stray capacitances and inductances that are difficult to take into account. In work with liquids, which have comparatively high conductivity, this difference could not strongly affect the measurement results, but in the study of solid dielectrics it could be of decisive importance.

On an apparatus assembled according to Rice’s circuit, we observed a conduction current in glycerin, but in solid dielectrics (rock salt) we were unable to detect a conduction current.

Uncertainty as to the identity of the shapes and amplitudes of the pulses of the two halves of Rice’s circuit and, consequently, uncertainty as to the accuracy of its balancing prompted us to try a circuit for testing with only one pulse (Fig. 4). The idea of compensating the capacitive current remained the same as in Rice’s circuit; however, a voltage pulse from only one pulse generator was applied to bridge points 3--4. Because of the conduction current of the sample, the potentials of points 1 and 2 differed from one another, which was to be recorded on an oscillograph connected in the diagonal 1--2 of the bridge.

Labels in Fig. 4: \(C_0\), \(C_v\), \(R_1\), \(R_2\), 1, 2, 3, 4; “to the amplifier.”

Fig. 4. Bridge circuit with symmetrical output for measuring the electrical conductivity of dielectrics.

Although bridge circuits are good in their idea of compensating the capacitive current, their adjustment at small conduction currents and short pulses is very difficult. We attempted to avoid this difficulty by using a simple series circuit without compensation of the capacitive current (Fig. 5).

The pulse-voltage generator (the part of the circuit to the left of the dashed line) produced single voltage pulses with a steep front and a flat top, which were applied to the chain \(C_0R\) (where \(C_0\) is the capacitance of the sample). With sufficiently small \(C_0\) and \(R\), the pulses were not distorted in shape and could be regarded as acting on the sample. When the front of the pulse acted on the sample, a capacitive current flowed through the \(C_0R\) chain, considerably exceeding the conduction current of the sample. We began the measurements from the instant when the capacitive current ceased, that is, when the flat top of the pulse began; therefore we tried in every way to reduce the length of the pulse front. The shortest front duration in our experiments was \((2 \div 3)\cdot 10^{-8}\) sec.

Measurements of electrical conductivity were made while the “flat” top of the pulse acted on the sample. In this case a capacitive discharge current also flowed, although it was considerably smaller in magnitude. The point is that the top of the pulse is in fact not accurately flat.

flat, but represents an exponential, only with a large time constant \(T_1\), determined by the parameters \(C_1\) and \(R_1\) of the circuit. The larger \(C_1\) and \(R_1\), the flatter the top of the pulse will be. It is impossible to increase \(T_1=C_1R_1\) only by increasing \(R_1\), since the spark will be weak and unstable when the thyratron is fired. The capacitance \(C_1\) must also be increased. In practice this is not always possible, since the capacitor \(C_1\) is usually a high-voltage one, and its dimensions become very large.

Fig. 5. Simple series circuit for measuring the electrical conductivity of dielectrics without compensation of the capacitive current

The capacitive discharge current \(i_c\), when the flat part of the pulse acts on the specimen, can be estimated roughly if the capacitance of the specimen \(C_0\) is known, as well as the magnitude of the decrease in the pulse voltage \(\Delta U\) over the investigated time interval \(\Delta t\) (Fig. 6):

Fig. 6. On determining the capacitive current when the “flat” top of a pulse acts on a specimen

\[ i_c=C_0\,\frac{\Delta U}{\Delta t}, \]

where

\[ \Delta U=U\left(1-e^{-\frac{\Delta t}{T_1}}\right), \]

where

\[ \begin{aligned} U&\text{ is the pulse amplitude at the moment }t_1;\\ \Delta t&=t_2-t_1\text{ is the time interval during which the electrical conductivity is measured;}\\ T_1&=C_1R_1\text{ is the time constant of the decay of the “flat” top of the pulse.} \end{aligned} \]

If the current \(i_c\) is much smaller than the conductivity current, which for rock salt we took to be \(\simeq 10^{-8} a\), then measurement of the latter is possible. Otherwise, if it is permissible under the technical conditions, \(C_1\) and \(R_1\) should be increased.

When working with the circuits considered, the voltage drop caused by the conductivity current of the dielectric across the resistance \(R\) (Fig. 1), across the resistances \(R_1\) and \(R_2\) (Fig. 4), and across the resistance \(R\) (Fig. 5) proved insufficient for observing the phenomenon on the oscilloscope tube when the phenomenon plates were connected directly to the indicated resistances. Increasing the latter could not lead to the desired result, since in most cases the charging time constant of the specimen capacitance increased, that is, the time of flow of the capacitive charging current increased. Therefore, in all the circuits we investigated, a pulse amplifier was used as an intermediate link between the measuring resistance and the oscilloscope tube. To protect the amplifier from overvoltages arising from capacitive currents, as well as in the event of breakdown of the specimen, a neon lamp, serving as a discharge tube, was connected in parallel with the input terminals of the amplifier. It should also be noted that, when working according to the circuit of Fig. 4, it is necessary to use an amplifier with an input symmetrical with respect to ground; moreover, when the pulse front acts, the diagonal 1—2 of the bridge and the amplifier input will be at a high potential with respect to ground.

We were unable to find a commercially manufactured oscilloscope with a built-in amplifier. Usually such oscilloscopes are designed for photographing periodic, not single, pulses, and their amplifiers have a comparatively narrow band of transmitted frequencies and a small amplification factor.

In our experiments we used an oscilloscope manufactured in the laboratory of the Department of TVN of the Tomsk Polytechnic Institute on the basis of a high-voltage oscilloscope tube 13LO2S with an accelerating voltage of 30 kV, and an amplifier for single pulses. The amplifier must be broadband, so as to pass pulses with a steep front without noticeable distortion and, in addition, have a large amplification factor, so that the output voltage of the amplifier is sufficient for a noticeable deflection of the beam when the phenomenon plates of the oscilloscope are connected directly to the amplifier output. The combination in one amplifier of a broad pass band \(\Delta f\) and a large amplification factor \(K\) has its limitations, since for a given amplifier the condition \(\Delta f \cdot K = \mathrm{const}\) must always be satisfied. It is comparatively easy to build an amplifier with a large amplification factor but a narrow pass band, and vice versa. Constructing an amplifier with large \(\Delta f\) and \(K\) is complicated (up to several tens of tubes) and expensive. Its adjustment is even more difficult in view of possible self-excitation. Even such measures as shielding the stages and supplying them separately do not always help. In addition, as the amplification factor is increased, noise in both the circuit resistances and in the amplifier itself becomes increasingly significant. These noises often set a limit to increasing the amplification factor, since they become comparable with the signal being investigated, or greater than it.

Noises arise as a result of the random thermal motion of electrons in the circuit resistances even without the application of voltage. These are thermal noises. They are the more noticeable, the smaller the current being measured and the shorter the time interval of measurement, that is, the broader the band of frequencies passed by the amplifier. If we wish

to measure the electrical conductivity with pulses having a front \(t_{\phi}=10^{-8}\) sec, the bandwidth \(\Delta f\) of the amplifier must be:

\[ \Delta f \geq \frac{0.35}{t_{\phi}} = 35\ \text{Mc}. \]

The voltage of thermal noise on some resistance \(R\) can be calculated from the formula

\[ U_{\mathrm{T}}^{2}=4\kappa T R\Delta f, \]

where \(U_{\mathrm{T}}^{2}\) is the mean value of the square of the thermal-noise voltage,
\(T\) is the absolute temperature, \({}^{\circ}\mathrm{K}\),
\(\kappa\) is Boltzmann’s constant,
\(\Delta f\) is the frequency band.

Usually pulse amplifiers have a high-resistance input, and if the signal is taken from a low-resistance element of the circuit \(R\), then the thermal noise will be determined by this latter element.

For example, for \(R=1000\ \Omega\) and \(\Delta f=35\ \text{Mc}\), at room temperature \((T=300^{\circ}\mathrm{K})\), we have:

\[ U_{\mathrm{T}}=\sqrt{4\kappa TR\Delta f} =\sqrt{4\cdot 1.38\cdot 10^{-23}\cdot 300\cdot 1000\cdot 35\cdot 10^{6}} =24\ \mu\text{V}. \]

Taking for ionic crystals [7] a current in the prebreakdown fields of the order \(i=10^{-8}\ \text{A}\), we find the signal voltage \(U_c\) on the resistance \(R=1000\ \Omega\):

\[ U_c=iR=10^{-8}\cdot 1000=10^{-5}\ \text{V}=10\ \mu\text{V}. \]

Thus, \(U_{\mathrm{T}}>U_c\), and the signal will not be visible because of the noise. Since the thermal-noise voltage is proportional to the square root of the resistance \(R\), while the signal voltage is directly proportional to \(R\), at first glance it might seem that these noises could be relatively reduced by increasing \(R\).

However, this is not always possible. Inaccurate balancing of bridge circuits (Figs. 3, 4) leads to the presence of a large capacitive current. In addition, in the circuits of Fig. 4 and Fig. 5, an increase in \(R\) leads to an increase in the charging time constant of the specimen \(T=C_0R\), where \(C_0\) is the capacitance of the specimen with the electrode system. For \(R=10^{4}\ \Omega\) and \(C_0=10^{-11}\ \text{F}\) (Fig. 5), we obtain \(T=10^{-7}\) sec; that is, under no conditions is it possible to obtain a pulse on the specimen with a front shorter than \(10^{-6}\div 10^{-7}\) sec.

In addition to the noise arising on the measuring resistance \(R\), there is also noise in the amplifier itself. This includes shot-effect noise, mainly of the first amplifier tube. Suppose that the first tube is a 6Zh4 pentode. Then the resistance equivalent to the tube noise, recalculated from the anode circuit to the grid circuit, may be taken as \(R_{\lambda}=720\ \Omega\). Taking the parasitic capacitance at the amplifier input as \(2\cdot 10^{-11}\ \text{F}\), we find the noise due to the shot effect:

\[ U_{\lambda}^{2}=4\kappa TR_{\lambda}\Delta f \left[ 1+\frac{1}{3}(2\pi\Delta fCR)^{2} \right] =31.4\cdot 10^{-10}\ \text{V},\qquad U_{\lambda}=56\ \mu\text{V}, \]

where \(T\), \(R\), and \(\Delta f\) have the values adopted above. The total noise voltage at the amplifier input is

\[ U_n=\sqrt{U_T^2+U_\lambda^2}=60\ \mu\text{V}. \]

Let us take the ratio of the signal voltage to the noise voltage to be equal to 5; then the signal voltage must be no less than \(300\ \mu\text{V}\), and the signal current for \(R=1000\ \Omega\) must be no less than \(3\cdot 10^{-7}\ \text{A}\), for the specified pulse-front duration \((t_f=10^{-7}\div 10^{-8}\ \text{s})\). In order to obtain at the amplifier output a voltage of the order of \(100\ \text{V}\), even with such a current in the resistance \(R\), the required gain factor is

\[ K=\frac{100}{300\cdot 10^{-6}}=3.3\cdot 10^5. \]

It is clear that amplification of currents of the order of \(10^{-8}\div 10^{-9}\ \text{A}\) with a pulse-front duration \(t_f=10^{-8}\ \text{s}\) is not yet to be discussed.

We were interested in the question of measuring the electrical conductivity of rock salt at room temperature. The approximate calculation presented has shown that, with pulses of the indicated duration, this is impossible. Pulse currents can be investigated at longer times or with an artificial increase in the electrical conductivity of the dielectric by heating, X-raying, etc.

We shall not dwell on questions of reducing the amplifier’s own noise, increasing its gain, or widening the passband of transmitted frequencies. Questions of the calculation and design of pulse amplifiers are treated in the relevant literature. We note only that some reduction of noise still does not completely solve the problem posed.

REFERENCES

1. Turner C. H., Lewis W. E. Electrical conductivity of amorphous solid insulating materials. Nature, 159, No. 4036, 334–335, 1947.

2. Keller K. J. Breakdown and electrical conductivity of glass. Physica 17, 5, 511, 1951.

3. Race H. Measurement of pre-breakdown currents in dielectrics by means of a cathode-ray oscillograph. GER 44, 8, 445, 1941.

4. Hagegguth J. H., Liao T. W. Method of detecting pulse ionization. Tr. AIEE 71, 3, 161, 1952.

5. Chainitskii V. M. Oscillographing currents in insulation. Elektrichestvo, No. 4, 76, 1951.

6. Francis G. Growth of non-electron discharges in hydrogen. Proc. Phys. Soc. 68, 3, 423B, 137, 1955.

7. Hippel A. Electrical conductivity of insulating crystals in a high electric field. Phys. Rev. 54, 12, 1096, 1938.