Direct Current Amplifiers

G. I. Berleev

Introduction

Direct current amplifiers are electronic devices which, when connected to a galvanometer or another electrical measuring instrument, make it possible to measure many of the fundamental electrical quantities with a high degree of accuracy. For this reason their development is of great importance for the further progress of a number of branches of science.

Modern amplifiers make it possible to measure direct currents down to \(10^{-18}\) a, i.e., smaller than the limiting currents accessible to measurement by classical methods; they make it possible to measure potential differences, with an arbitrarily large internal resistance of the source, down to \(10^{-6}\) v, as well as large resistances, capacitances, dielectric constants, etc.

Classical methods of measuring the fundamental electrical quantities with the aid of galvanometers or electrometers are limited either by insufficient sensitivity or by extreme inconvenience in operation.

According to present-day data, the greatest current sensitivity has been achieved in magnetoelectric-system galvanometers and amounts to \(10^{5}\) mm/µµa, which, for a large internal resistance of the source, permits currents down to \(10^{-11}\) a to be measured when the galvanometer scale is one meter from the mirror. Along with high sensitivity, such instruments have a very large period of oscillation (\(\sim 40\) sec.). If one takes into account that, in order to obtain a steady deflection, a time of not less than \(1.36T_{0}\) is required even with the most advantageous damping coefficient, then all the inconveniences of working with them become obvious (\(T_{0}\) is the period of oscillation of the galvanometer). Thus the advantages in sensitivity are achieved at the cost of increasing the inertia of the moving system.

The difficulties in constructing highly sensitive galvanometers arise both for technical reasons and because of the limitations of the method.

It is known that the angle of steady deflection of the frame in magnetoelectric galvanometers is equal to

\[ \varphi=\frac{HS}{D}I, \]

where \(\varphi\) is the angle of rotation in radians, \(H\) is the density of the magnetic flux through the area of the contour \(S\), \(HS\) is the total flux linkage in maxwells, \(I\) is the current in the galvanometer circuit in absolute electromagnetic units, and \(D\) is the specific torsional moment of the suspension in dyne·cm/radian.

In this formula,

\[ \frac{HS}{D}=\alpha \]

is the current sensitivity of the galvanometer.

It follows from this expression that the sensitivity is proportional to the magnetic flux through the contour of the frame and inversely proportional to the specific torsional moment. It would seem that, by indefinitely increasing the magnetic flux and decreasing the specific torsional moment of the suspension, one could raise the sensitivity as much as desired; but this method is applicable only within rather narrow limits. First, the use of strong fields is hindered by constructional difficulties and, second, the stronger the field, the more slowly the coil reaches its new steady position. With a considerable increase in the field, owing to electromagnetic damping, the time of deflection of the frame becomes so large that it becomes impossible to work with such an instrument \(^{1,2}\).

Some extension of the measurement limits of the fundamental electrical quantities is achieved by the use of electrometers. Among the many systems of electrometers, the vacuum quadrant electrometer of Hoffmann \(^{3}\) has the greatest sensitivity. The limited use of such electrometers is due to the difficulty of adjustment, the lack of sharpness of the light spot, and their extremely high cost.

All the other types of electrometers have lower sensitivity and therefore cannot satisfy the ever-increasing need for measuring ultrasmall currents.

The difficulty is overcome by using direct-current amplifiers. In their properties they can fully replace both the most sensitive electrometers and highly sensitive galvanometers. The advisability of introducing direct-current amplifiers into the practice of scientific-research institutions and into industry is due not only to the fact that they make it possible to measure small currents with a high degree of accuracy, but also because, in mass production, this device is very inexpensive—considerably cheaper than electrometers.

I. PRINCIPLE OF OPERATION OF THE AMPLIFIER AND ELECTROMETER TUBES

1. Types of amplifier circuits

Various circuits of direct-current amplifiers have been described in a number of works4, 5, 6, 7, 8, 9, etc. These circuits are shown in Fig. 1. In addition to the circuits presented here, there are also others, which represent certain minor modifications of them. The circuits presented here are designed to operate with a four-electrode electrometer tube. The exceptions are the circuits shown in Fig. 1,a and 1,b, in which the cathode-grid circuit may be omitted, and then they permit the use of three-electrode electrometer tubes. In this case one obtains the circuit shown in Fig. 1,3.

The amplifier circuits presented here are not the earliest; they arose after a considerable series of works devoted to the study of the properties of amplifiers and the conditions for their stabilization.

At first, compensating two-tube direct-current amplifiers were developed. An example is the circuit shown in Fig. 2 and described in works 7, 8. This type of amplifier suffers from very major shortcomings. According to the original idea, instability arising simultaneously in both arms of the bridge is automatically extinguished. In reality the matter is much more complicated. In order for the instability that has arisen to be extinguished, it is necessary that the tubes operating in a pair possess properties identical or very close to one another. However, this necessary condition is still not sufficient. It is necessary that changes in the parameters of these tubes, chiefly the cathode emission, be identical or very close. Otherwise the circuit is difficult to balance and very quickly loses stability. To overcome these very great difficulties, paired tubes have recently begun to be manufactured9, 10.

A characteristic feature of these tubes is a common directly heated cathode. This circumstance is noteworthy in that a change in the emission current, associated with aging of the tube or with a decrease in the filament current, causes an identical change in the compensating arms, and therefore the state of stability of the circuit does not change. Such circuits possess a number of advantages.

2. Principle of operation of the amplifier

The circuits of direct-current amplifiers shown in Fig. 1 are varieties of the Wheatstone bridge. As an example, let us consider the circuit shown in Fig. 1,b. In it the arms of the bridge are:

1. The resistance between the cathode grid and the anode of the electrometer tube.

Fig. 1.

1. The total resistance, consisting of the filament \(R_{\mathrm{н}}\), \(R_2\), and \(R_3\).

2. The resistance \(R_4\).

3. The resistance \(R_a\).

If the circuit is balanced, then there is no current in the galvanometer circuit. However, if some voltage is applied to the input of the tube, the magnitude of the anode current will change and the entire system will depart from equilibrium. A current proportional to the grid bias will appear in the diagonal of the bridge.

Let the grid potential \(U_c\) change by \(\Delta U_c\); then the change in the anode current may be written as follows:

\[ \Delta I_a=\Delta U_c\,\frac{\partial I_a}{\partial U_c}=\Delta U_c S. \tag{1,1} \]

Here

\[ \left(\frac{\partial I_a}{\partial U_c}\right)_{U_a}=S \]

is the slope of the tube characteristic.

Fig. 2.

The change in grid potential can be expressed through the change in grid current,

\[ \Delta U_c=R_c\Delta I_c, \tag{1,2} \]

where \(R_c\) is the resistance in the tube grid circuit and \(\Delta I_c\) is the change in grid current.

After substituting (1,2) into (1,1), the change in anode current may be written as follows:

\[ \Delta I_a=R_c\Delta I_c S. \tag{1,3} \]

\(\Delta I_c\) in expression (1,3) is the measured current, whose magnitude, in the case of operation with the circuit shown in Fig. 1, \(b\), may range from \(10^{-12}\) to \(10^{-16}\) A. For example, if the slope of the tube characteristic reaches \(130\,\mu\text{A}/\text{V}\), then in order that the above-indicated values of \(\Delta I_c\) could be measured with a mirror galvanometer with a constant of \(10^{-9}\,\text{A}/\text{mm}\), the grid resistance \(R_c\) must have a value on the order of \(10^{10}\div 10^{11}\) ohms.

When measuring such small currents, a necessary condition is the stability of the circuit; to achieve this, a definite selection of the circuit elements and of the amplifier supply conditions is required. Investigations on this question are set forth in works \(^{11,12,13,14}\).

Starting from expression (1,3), we obtain the current amplification factor, equal to

\[ K=\frac{\Delta I_a}{\Delta I_c}=R_cS. \tag{1,4} \]

From expression (1.4) it follows that the current amplification coefficient depends on the value of the grid resistance and on the steepness of the tube characteristic. Thus, in order to obtain reliable measurements it is necessary to ensure the stability of the grid resistance and of the steepness of the tube characteristic. The stability of \(R_c\) depends on the value of the temperature coefficient of resistance \(\alpha\) and on the constancy of the temperature during the measurement:

\[ dR_c=\alpha R_{c0}\,dT . \tag{1.5} \]

Here \(R_{c0}\) is the value of the resistance \(R_c\) at \(0^\circ\mathrm{C}\), and \(dT\) is the change in temperature.

But a more substantial cause of amplifier instability is the change in the steepness of the tube characteristic \(S\) due to changes in the supply voltages.

Indeed, the steepness of the characteristic is a function of the filament, anode, and grid voltages:

\[ S=\varphi(U_{\mathrm{n}},U_a,U_c). \tag{1.6} \]

Hence the total differential of the function \(S\) is

\[ dS=\frac{\partial S}{\partial U_{\mathrm{n}}}\,dU_{\mathrm{n}}+ \frac{\partial S}{\partial U_a}\,dU_a+ \frac{\partial S}{\partial U_c}\,dU_c . \tag{1.7} \]

Carrying out joint transformations of (1.6) and (1.7) and substituting numerical values of the quantities found experimentally, we find1 that

\[ \frac{dS}{S} = 10\,\frac{dU_{\mathrm{n}}}{U_{\mathrm{n}}} - \frac{D\,dU_a}{DU_a+U_c} - \frac{dU_c}{DU_a+U_c}. \tag{1.8} \]

Here \(D\) is the permeability of the tube. For electrometer tubes \(D\) is close to unity. Analyzing (1.8), it is not difficult to see that the main influence on the change in the steepness of the characteristic is exerted by the change in the filament voltage; therefore special attention must be paid to the quality of the power supplies. This fact is noted in a number of works2,3.

3. Types of electrometer tubes

Electrometer tubes differ from other electron tubes in that their grid current is very small and fluctuates around \(10^{-15}\) a. Most of the electrometer tubes produced are four-electrode tubes with two grids. The first grid is called the cathode grid or space-charge grid and is located near the cathode. The second grid is the control grid. A sufficiently large positive potential is applied to the cathode grid, which promotes the dispersion of the electron cloud near the cathode,

Table I

Characteristics of electrometric tubes

Type	Filament voltage, V	Filament current, mA	Anode voltage, V	Cathode-grid voltage \(U_{\mathrm{e}}\), V	Control-grid voltage \(U_c\), V	Anode current, μA	Characteristic slope, μA/V	Grid current, A	Amplification factor \(\mu\)	Grid capacitance, μμF	Literature references
EM-2	2	80	10	6	−4	300	55	\(10^{-13}\)	1.65	2.5	10
EM-3	3	122	6	4	−3	55	50	\(\sim 10^{-15}\)	1.15	5.2
Hauser tube	3	—	8	8	−4	300	300	\(10^{-12}\)	—	—	19
T-113	3	100	10	10	−3	—	180	\(10^{-13}\)	2.5	—	20
T-114	2	90	6	4	−4	—	55	\(10^{-14}\)	1.0	—	17, 20
T-115	2.8	500	12	12	−3	—	200	\(10^{-11}\)	2.5	—	17, 20
D96475	1.0	270	4	4	−3	—	40	\(10^{-15}\)	—	—	21
FP-54	2.5	90	6	4	−4	40	25	\(5 \cdot 10^{-15}\)	1	6	22
UX-54	2.5	90	6	4	−4	40	25	\(5 \cdot 10^{-15}\)	1	6	22
VX-41	1.25	10	4.5	4.5	−3	250	20	\(5 \cdot 10^{-15}\)	1	—	23
4060	0.56	1100	4	—	−4.5	50	30	\(2 \cdot 10^{-15}\)	0.5	—	17
RH505	2.0	250	6	—	−3	300	75	\(10^{-15}\)	1.0	3	24
RH506	2.5	250	6	—	−3	400	90	\(10^{-15}\)	0.8	4	24
RH507	2.0	60	6	—	−3	200	60	\(10^{-12}\)	0.8	4	24
A151A	1—1.25	170—195	4.75	—	−6	350	50—60	\(2 \cdot 10^{-14}\)	—	—	23
CK570AX	0.625	20	12	—	−3	220	125	\(5 \cdot 10^{-13}\)	1.5	—	23

which allows the electrons emitted by the cathode to reach the anode at a very low anode voltage. This circumstance, in turn, permits operation at potentials much lower than the gas ionization potential, as a result of which the measured current is freed from the ionic component of the grid current.

Of the three-electrode electrometer tubes, the most widespread are the “platrons,” in which the anode and the control electrode are made in the form of plates located on both sides of the cathode.

The cathodes of electrometer tubes operate at comparatively low temperatures. As a rule, electrometer tubes employ cathodes made of thoriated or oxide-coated filaments. Tubes with a thoriated cathode operate in the electrometer mode at a temperature of \(1500\text{—}1700^\circ\mathrm{K}\), and those with an oxide-coated cathode at \(750\text{—}900^\circ\mathrm{K}\).

It should be noted that electrometer tubes with oxide-coated cathodes exhibit smaller fluctuations of the emission current than tubes with a thoriated cathode and, at identical filament temperatures, give a larger emission current.

The insulation resistance of electrometer tubes reaches \(10^{14}\text{—}10^{15}\) ohms. In order to further reduce the leakage of current through the insulation resistance, a guard ring is placed on the neck of the tube\(^{10,17}\). For this purpose, after treating the surface of the tube with acetone or benzine and alcohol, a graphite ring is applied (preferably with the aid of aquadag). A metal ring is placed over the graphite ring; to this ring a voltage is applied equal in magnitude and opposite in sign to the constant bias of the control electrode. To reduce photoemission of the grid, and also to reduce conduction through the insulation resistance of the tube due to condensation of moisture on it, the tube is placed in a special impermeable metal enclosure, in which either a vacuum is produced or a moisture absorber is placed.

The characteristics of electrometer tubes are collected in Table I (see p. 99).

II. SELECTION OF AMPLIFIER ELEMENTS AND OF THE STABLE OPERATING REGIME

A stable operating regime of a direct-current amplifier should be taken to mean one in which no, or almost no, slipping of the galvanometer zero (drift) and fluctuations are observed. These conditions can be fulfilled only when the elements of the amplifier (its resistances) and the supply regime have been selected in accordance with definite conditions.

Let us consider, in sequence, the principles underlying the solution of the problem posed.

DC AMPLIFIERS

1. First method of selecting the elements of the amplifier and the stable operating mode

The present variant of selecting the elements of the amplifier was considered in work²³. The basis of the consideration was the Wheatstone bridge (see Figs. 1,b and 3). As a result of simple reasoning, two relations were obtained expressing the equilibrium of the amplifier, from which approximate values of the sought elements are determined.

These conditions are as follows:

In order that no current flow in the diagonal of the bridge, the equalities must be satisfied:

\[ \frac{R_7}{R_6}=\frac{R_T}{R_8} \tag{2,1} \]

and

\[ E_B=\frac{R_8}{R_T+R_8}\,E_T \tag{2,2} \]

or

\[ R_8=\frac{E_B}{E_T-E_B}\,R_T. \tag{2,2a} \]

Fig. 3.

Here \(R_T\) is the equivalent resistance cathode grid—anode of the tube, \(R_8=R_a\), \(R_6=R_4\), \(R_7=R_2+R_3+\) the resistance of the filament \(R_n\), \(E_T\) is a fictitious source of emf. The magnitude \(E_T\) is equal to the voltage drop across the resistance \(R_T\), and \(E_B\) is a compensating battery with a voltage of approximately 1.5 V.

The value of the plate resistance \(R_8\) (in the equivalent circuit) is found from formula (2,2a). However, the path proposed by the authors for finding this quantity is not strictly theoretical, since it assumes knowledge of certain empirical relations which, for the given tube, are found experimentally.

First, the dependence of the plate current (in µA) on the filament current is taken, and from the linear part of the curve, in its middle, the best value of the filament current is established.

Fig. 4.

Vertical axis: Plate current in µA.
Horizontal axis: Voltage of the supply batteries in V.

Then the dependence of the plate current at the found filament current on the variation of \(R_1\), i.e. on the grid bias, is taken. As a result, the dependence shown in Fig. 4 is obtained. Here on the abscissa axis are plotted the values of the voltage of the supply battery at changes of the value of \(R_1\) from zero and higher, at which it is maintained-

one and the same filament current (\(\sim 100\) mA) is maintained. The curve obtained is analogous to the preceding one. The middle of the linear part of this curve indicates the best value of the supply-battery voltage, while the intersection of the tangent to the curve at this point with the abscissa axis shows the initial operating point corresponding to the voltage drop across \(R_T\) in the equivalent circuit. For the tube FP54, \(E_T\) is equal to 8.6 V. Having found the values of the resistances \(R_2\), \(R_3\), and \(R_4\) on the basis of the tube parameters and substituting them into the system of equations (2.1) and (2.2), we find that \(R_T\) is equal to \(3.38 \cdot 10^4\) ohms, whence \(R_8\) is also found. To solve this system, the value of the voltage of the compensating element \(E_B\) must still be added to the values found. The method set forth here for selecting the amplifier parameters is neither easy nor exact, since after the calculations have been performed the true values have to be found by practical adjustment.

2. Second method of selecting the amplifier elements and the stable operating mode

What is set forth in the present paragraph pertains to the circuits shown in Figs. 1,б and 1,з4. Circuit 1,з is obtained by removing the cathode-grid circuit in circuit 1,б as applied to a three-electrode tube. In this case the resistance \(R_2\) of circuit 1,з is equal to the sum of the resistances \(R_2\) and \(R_3\) in circuit 1,б.

The principal parameter with respect to which the circuit elements are sought is the filament current. Conditions are found under which small changes in the filament current do not cause a displacement of the galvanometer zero. Mathematically these conditions may be written as follows:

\[ I_a R_a - I_{\mathrm{н}} R_4 = E_B, \tag{2.3} \]

\[ (I_a+\Delta I_a)R_a-(I_{\mathrm{н}}+\Delta I_{\mathrm{н}})R_4=E_B. \tag{2.4} \]

Here \(I_{\mathrm{н}}\) and \(I_a\) are, respectively, the filament current and the anode current; \(\Delta I_{\mathrm{н}}\) and \(\Delta I_a\) are changes in the filament and anode currents; \(E_B\) is the voltage of the compensating battery in the bridge diagonal.

It should be noted that, with unchanged anode and grid voltages, the cause of the change in anode current is the change in filament current. Solving the system of equations (2.3) and (2.4) with respect to \(R_a\) and \(R_4\), we find

\[ R_a= \frac{ E_B \dfrac{\Delta I_{\mathrm{н}}}{\Delta I_a} }{ I_a \dfrac{\Delta I_{\mathrm{н}}}{\Delta I_a}-I_{\mathrm{н}} }, \tag{2.5} \]

\[ R_4= \frac{ E_B }{ I_a \dfrac{\Delta I_{\mathrm{н}}}{\Delta I_a}-I_{\mathrm{н}} }. \tag{2.6} \]

All the remaining parameters of the amplifier are found on the basis of the parameters of the tube used.

To calculate the resistance values, according to formulas (2.5) and (2.6), it is necessary to know the value of the derivative \(\dfrac{\Delta I_a}{\Delta I_{\mathrm{h}}}\), as well as the heater current and the anode current for the point for which this derivative has been found. These data are obtained by taking the curve of the dependence of the anode current on the heater current (Fig. 5). The taking of this curve is carried out according to the circuit shown in Fig. 6.

Fig. 5.

Fig. 6.

The resistance values that can be found in this way are approximate and require additional experimental verification.

3. Third method for selecting amplifier components and the stable operating mode

The present method for selecting the stable operating mode of a direct-current amplifier is described in paper \(^{2)}\). In this work attention was directed above all to the role of constancy of the ambient temperature. Temperature changes cause not only a drift of the galvanometer zero, but also a change in the filament current \(I_{\mathrm{f}}\), on which the stability of the circuit depends. Figure 7 shows the displacement of the stability curve when the temperature changes by two degrees. We see that, for the mode for which the dependence of stability on the temperature of the external medium was obtained, the minimum of the curve not only shifted by 600 millimeters along the scale, but also came to correspond to another filament current. It follows from this that the constancy of the zero is affected not only by internal factors, but also by external ones, among which temperature is foremost. The amount of drift of the galvanometer zero, which depends on changes in the external temperature, can be made arbitrarily small by means of thermal insulation, but it is practically impossible to eliminate it completely.

Fig. 7.

In the event of drift of the galvanometer zero, it is returned to its former position by changing the anode resistance.

The initial investigations were carried out on the circuit shown in Fig. 1, б. As a result of the investigations a new circuit arose, possessing certain advantages (see Fig. 1, г). In this circuit the cathode grid has a positive potential, and a current flows in its circuit 4–5 times greater than the anode current. If, under the action of an external cause, the filament current changes, this causes such changes in the currents in the circuits of the anode and the cathode grid that their ratio remains constant, owing to which the position of the galvanometer zero is preserved. The potential difference at the galvanometer terminals is equal to

\[ U = R_{\mathrm{B}} I_{\mathrm{c}} - R_0 I_{\mathrm{a}} . \tag{2,7} \]

The current flowing through the galvanometer under the action of this potential difference is small in comparison with the currents \(I_{\mathrm{c}}\) and \(I_{\mathrm{a}}\). \(R_0'\) is also small and is connected in series with \(R_0\) for fine adjustment. The current through the galvanometer will be zero \((U=0)\) under the condition that

\[ \frac{I_{\mathrm{a}}}{I_{\mathrm{c}}} = \frac{R_{\mathrm{B}}}{R_0}. \tag{2,8} \]

If, during fluctuations of the filament current, the ratio \(\dfrac{I_a}{I_c}\) remains constant, then the second condition for the independence of \(U\) from changes in the battery voltage and, consequently, of the filament current \(I_{\mathrm{f}}\), is

\[ \frac{dU}{dI_{\mathrm{f}}}=0. \tag{2,9} \]

Taking the derivative of (2,7) and taking (2,9) into account, we find that

\[ \frac{dI_a}{dI_{\mathrm{f}}} = \frac{R_{\mathrm{B}}}{R_0}\, \frac{dI_c}{dI_{\mathrm{f}}}. \tag{2,10} \]

The characteristic curves for the FP-54 tube in the circuit of Fig. 1,2 are shown in Fig. 8.

Fig. 8.
Vertical axis: Anode current in \(\mu\text{A}\). Horizontal axis: Filament current in mA. Annotation: Neutral points.

For this case, the following stabilization method is proposed:

1. By means of the ballast resistance in the filament circuit \(R_3\), set the filament current \(I_{\mathrm{f}}\) according to the tube parameters, and, by means of the anode resistance \(R_0+R'_0\), bring the galvanometer to zero. The galvanometer should be connected so that a decrease of \(R_0\) is accompanied by a decrease of the deflection.

Fig. 9.
Vertical axis: Galvanometer deflection in scale divisions. Horizontal axis: Filament current in mA. Curve labels: \(R_2=34\,\Omega\), \(R_2=44\,\Omega\).

1. Connect shunts to the galvanometer which make its sensitivity equal to 0.1; 0.01, etc., of its normal sensitivity. Passing gradually from lower sensitivities to higher ones, use \(R_3\) to adjust \(I_{\mathrm{f}}\) so that the galvanometer deflections decrease for small changes of \(I_{\mathrm{f}}\) by \(\Delta I_{\mathrm{f}}\). The galvanometer readings must pass through a minimum. If the galvanometer deflections lie outside the scale, the zero of the galvanometer must be shifted by changing \(R_0\).

2. If the values of \(I_{\mathrm{f}}\) at the minimum differ from the normal value by more than 3–4%, then \(R_2\) must be changed and the whole procedure repeated from the beginning. The main parameters can be found after 3–4 trials. The typical displacement of the minimum of the stabilization curve when the resistance \(R_2\) is changed is shown in Fig. 9.

1. With the galvanometer at full sensitivity, find its minimum by means of \(R_3\), and by means of \(R_0\) bring the light spot onto the required point of the scale. After this the circuit will continue to drift for another 15–20 minutes. Subsequently, the final balancing must be carried out by small changes of \(R_0\).

4. Further investigations of stabilization conditions

A further investigation of a somewhat modified circuit, shown in Fig. 1,2, is described in paper \({}^{9}\).

It was found that if the dependence of the galvanometer deflection on the filament current is recorded, then two equilibrium points \(X\) and \(Y\), shown in Fig. 10, can be found, where

\[ \frac{d\alpha}{dI_{\mathrm{н}}}=0. \]

In the equilibrium position, the current passing through the galvanometer must depend only on the potential of the control electrode, provided that the change in current in the anode circuit of the tube is proportional to the change in current in the cathode-grid circuit.

Fig. 10.

Fig. 11.

To increase the resolving power, i.e. to increase the amplifier constant, the voltage obtained in the bridge diagonal is sometimes applied to the input of subsequent amplification stages. Such is the work \({}^{38}\), in which the use of such an amplifier for comparing ion currents in mass spectrography by the null method is described, and also \({}^{14,29}\), in which the stabilization conditions of such an amplifier are given. The basic circuit is shown in Fig. 11. In this circuit an electrometer tube is placed at the input, while in the subsequent amplification stages ordinary electron tubes are used. In order for the amplifier to retain stability, the resistances are selected so that the anode potential does not depend on the voltage of the supply batteries and on the emission of the tube. The values of the resistances \(R_1\), \(R_2\), \(R_3\), \(R_{g_2}\), and \(R_a\) are found from the tube parameters and the following conditions.

Let \(U_{\mathrm{н}}\), \(U_{g_1}\), \(U_{g_2}\), and \(U_a\) be the potentials of the positive end of the filament, of the control electrode, of the cathode grid, and of the anode of the elec-

galvanometric lamp, measured relative to the negative end of the filament. Let \(U_0\) denote the output voltage, i.e., the potential difference between the anode and the negative end of the resistor \(R_1\). In the steady state, under the condition that the currents \(I_{g2}\) and \(I_a\) are small in comparison with the filament current \(I_{\mathrm{н}}\), the following system of equations is satisfied:

\[ U_{g1}=-I_{\mathrm{н}}R_1, \tag{2,11} \]

\[ U_{g2}=U_{\mathrm{н}}+I_{\mathrm{н}}R_2-I_{g2}R_{g2}, \tag{2,12} \]

\[ U_a=U_{\mathrm{н}}+I_{\mathrm{н}}R_3-I_aR_a, \tag{2,13} \]

\[ E=U_{\mathrm{н}}+I_{\mathrm{н}}(R_1+R_3), \tag{2,14} \]

\[ U_0=I_{\mathrm{н}}R_1+U_a. \tag{2,15} \]

Here \(I_{\mathrm{н}}\) is the filament current, \(I_a\) the anode current, \(I_{g2}\) the cathode-grid current, \(U_{g2}\) the cathode-grid voltage, \(U_{\mathrm{н}}\) the filament voltage, \(U_{g1}\) the constant bias of the control electrode, \(U_a\) the anode voltage, and \(U_0\) the output voltage.

The anode current and the cathode-grid current are functions of the potentials of the anode, the cathode grid, and the control electrode, as well as of the filament current. Expanding them in a series and retaining first-order terms, we obtain

\[ I_a=I_{a0}+\frac{\partial I_a}{\partial U_{g1}}\Delta U_{g1} +\frac{\partial I_a}{\partial I_{\mathrm{н}}}\Delta I_{\mathrm{н}} +\frac{\partial I_a}{\partial U_{g2}}\Delta U_{g2} +\frac{\partial I_a}{\partial U_a}\Delta U_a, \]

or

\[ \Delta I_a=P_{g1}\Delta U_{g1}+P_{\mathrm{н}}\Delta I_{\mathrm{н}} +P_{g2}\Delta U_{g2}+P_a\Delta U_a; \tag{2,16} \]

here

\[ P_{g1}=\frac{\partial I_a}{\partial U_{g1}};\qquad P_{\mathrm{н}}=\frac{\partial I_a}{\partial I_{\mathrm{н}}};\qquad P_{g2}=\frac{\partial I_a}{\partial U_{g2}};\qquad P_a=\frac{\partial I_a}{\partial U_a}; \]

and similarly for the increment of the cathode-grid current

\[ \Delta I_{g2}=S_{g1}\Delta U_{g1}+S_{\mathrm{н}}\Delta I_{\mathrm{н}} +S_{g2}\Delta U_{g2}+S_a\Delta U_a; \tag{2,17} \]

here

\[ S_{g1}=\frac{\partial I_{g2}}{\partial U_{g1}};\qquad S_{\mathrm{н}}=\frac{\partial I_{g2}}{\partial I_{\mathrm{н}}};\qquad S_{g2}=\frac{\partial I_{g2}}{\partial U_{g2}};\qquad S_a=\frac{\partial I_{g2}}{\partial U_a}. \]

From equations (2,11), (2,12), and (2,13) it follows that

\[ \Delta U_{g1}=-R_1\Delta I_{\mathrm{н}}, \tag{2,18} \]

\[ \Delta U_{g2}=(r_{\mathrm{н}}+R_2)\Delta I_{\mathrm{н}}-R_{g2}\Delta I_{g2}, \tag{2,19} \]

\[ \Delta U_a=(r_{\mathrm{н}}+R_3)\Delta I_{\mathrm{н}}-R_a\Delta I_a. \tag{2,20} \]

The resistance of the filament can be expressed as \(r_{\mathrm{n}}=\dfrac{\partial U_{\mathrm{n}}}{\partial I_{\mathrm{n}}}\). Substituting into (2.17) the value of \(\Delta U_{g_1}\) from (2.18) and \(\Delta I_{g_2}\) from (2.19), one can find

\[ \Delta U_{g_2}= \frac{r_{\mathrm{n}}+R_2+R_{g_2}(S_{g_1}R_1-S_{\mathrm{n}})} {1+S_{g_2}R_{g_2}}\Delta I_{\mathrm{n}} - \frac{S_aR_{g_2}}{1+S_{g_2}R_{g_2}}\Delta U_a . \tag{2.21} \]

Similarly,

\[ \Delta U_a= \frac{r_{\mathrm{n}}+R_3+R_a(P_{g_1}R_1-P_{\mathrm{n}})} {1+P_aR_a}\Delta I_{\mathrm{n}} - \frac{P_{g_2}R_a}{1+P_aR_a}\Delta U_{g_2}. \tag{2.22} \]

Then, from equations (2.21) and (2.22), eliminating \(\Delta U_{g_2}\), one can express \(\Delta U_a\)

\[ \Delta U_a= \left\{ \frac{(1+S_{g_2}R_{g_2})[r_{\mathrm{n}}+R_3+R_a(P_{g_1}R_1-P_{\mathrm{n}})]} {(1+R_aP_a)(1+R_{g_2}S_{g_2})-P_{g_2}R_aS_aR_{g_2}} \right. \]

\[ \left. -\frac{P_{g_2}R_a[r_{\mathrm{n}}+R_2+R_{g_2}(S_{g_1}R_1-S_{\mathrm{n}})]} {(1+R_aP_a)(1+R_{g_2}S_{g_2})-P_{g_2}R_aS_aR_{g_2}} \right\}\Delta I_{\mathrm{n}} . \tag{2.23} \]

The condition for the independence of the output voltage \(U_0\) from the battery voltage \(E\) can be written, according to equation (2.15), as

\[ \Delta U_0=R_1\Delta I_{\mathrm{n}}+\Delta U_a=0. \tag{2.15a} \]

Substituting into (2.15a) the value of \(\Delta U_a\) from (2.23), we obtain the following equation:

\[ (1+S_{g_2}R_{g_2})[r_{\mathrm{n}}+R_3+R_a(P_{g_1}R_1-P_{\mathrm{n}})]- \]

\[ -P_{g_2}R_a[r_{\mathrm{n}}+R_2+R_{g_2}(S_{g_1}R_1-S_{\mathrm{n}})]+ \]

\[ +R_1[(1+P_aR_a)(1+S_{g_2}R_{g_2})-P_{g_2}S_aR_aR_{g_2}]=0. \tag{2.24} \]

Equations (2.11), (2.12), (2.13), and (2.24) constitute a system of four equations with five unknowns \(R_1\), \(R_2\), \(R_3\), \(R_a\), and \(R_{g_2}\). This system can be reduced to a single equation with two unknowns. The currents and potentials enter this equation as parameters chosen on the basis of the data of the electrometer tube. If, for example, \(R_1\), \(R_a\), and \(R_{g_2}\) are eliminated from this system, an equation is obtained containing the unknowns \(R_2\) and \(R_3\), which can be expressed one through the other. Knowing the normal filament current and voltage, one can determine from equations (2.14) the necessary voltages of the power supply. If the calculation gives \(R_3\) smaller than \(R_2\), then, with the found values of the remaining resistances, the ends of the resistances \(R_a\) and \(R_{g_2}\) connected to the resistances \(R_2\) and \(R_3\) should be interchanged (see Fig. 11).

The parameters entering into the system of equations presented are determined experimentally. Measured for one of the GR-54 tubes, they are

led to the following values:

\[ \begin{aligned} P_{g_1}&=15.6\cdot 10^{-6}\ \mathrm{a/v},& P_{\mathrm{n}}&=0.0026\ \mathrm{a/v},\\ P_{g_2}&=13.5\cdot 10^{-6}\ \mathrm{a/v},& P_a&=16.2\cdot 10^{-6}\ \mathrm{a/v},\\ S_{g_1}&=-8.75\cdot 10^{-6}\ \mathrm{a/v},& S_{\mathrm{n}}&=0.0185\ \mathrm{a/v},\\ S_{g_2}&=65\cdot 10^{-6}\ \mathrm{a/v},& S_a&=11\cdot 10^{-6}\ \mathrm{a/v},\\ I_a&=34.9\cdot 10^{-6}\ \mathrm{a},& U_{\mathrm{n}}&=2.47\ \mathrm{v},\\ I_{g_2}&=219\cdot 10^{-6}\ \mathrm{a},& r_{\mathrm{n}}&=25\ \Omega . \end{aligned} \]

The operating voltages of this tube in the electrometric regime are as follows:

\[ U_a=6\ \mathrm{v};\qquad U_{g_2}=4\ \mathrm{v};\qquad U_{g_1}=-4\ \mathrm{v};\qquad I_{\mathrm{n}}=0.09\ \mathrm{v}. \]

If these data are substituted into equations (2.11), (2.12), and (2.13), it is found that

\[ R_1=44.4\ \Omega,\quad R_{g_2}=411R_2-6990\ \Omega,\quad \text{and}\quad R_a=2580R_3-101000\ \Omega. \]

After substituting these values into (2.24), \(R_2\) can be expressed through \(R_3\):

\[ R_2=\frac{7.62R_3-371}{0.19R_3-3.62}\ \Omega . \tag{2.25} \]

The minimum value of \(R_2\) is found from the assumption that \(R_{g_2}\) is equal to zero. In this case \(R_2\) should be about \(17\ \Omega\), while \(R_3\) turns out to be greater than \(70\ \Omega\). From these conditions and on the basis of equations (2.14), \(E\) must be greater than \(14.8\ \mathrm{v}\).

Fig. 12.

Fig. 13.

By varying the value of \(R_3\) so that \(R_{g_2}\) remains positive, one can calculate the values of the remaining resistances of the circuit from the formulas given above. These data are subsequently checked experimentally by means of the circuit shown in Fig. 12. By varying the battery voltage \(E\) and taking the dependence of the current flowing through the galvanometer on the filament current, one can judge the compensating properties of the circuit (Fig. 13).

The curves in Fig. 13 were obtained under the following conditions:

Curve \(A\): \(R_3 = 70\ \Omega,\)  \(R_2 = 17\ \Omega,\)

\[ R_{g2}=0,\qquad R_a = 92.5\cdot 10^3\ \Omega, \]

Curve \(B\): \(R_3 = 75\ \Omega,\)  \(R_2 = 20.5\ \Omega,\)

\[ R_{g2}=1430\ \Omega,\qquad R_a = 92.5\cdot 10^3\ \Omega, \]

Curve \(V\): \(R_3 = 100\ \Omega,\)  \(R_2 = 27.5\ \Omega,\)

\[ R_{g2}=4310\ \Omega,\qquad R_a = 157\cdot 10^3\ \Omega, \]

Curve \(G\): \(R_3 = 400\ \Omega,\)  \(R_2 = 39.6\ \Omega,\)

\[ R_{g2}=9350\ \Omega,\qquad R_a = 931\cdot 10^3\ \Omega. \]

From the curves presented it is evident that the stability condition is best satisfied in mode \(A\). This may be judged by the magnitude of the plateau, along which changes in the filament current, and consequently in the emission current, do not affect the magnitude of the output voltage.

Fig. 14.

The method described here for compensating instability has a number of advantages over other methods, but it is not without shortcomings. The chief drawback is due to the large difference between the currents flowing in the anode circuit and in the cathode-grid circuit of the electrometer tube. From this it is not difficult to see that relatively small changes in the supply circuit of an amplifier in which the biases of the circuit elements are set potentiometrically may create conditions for a large change in the emission current and, consequently, a disturbance of the regime of stable operation. This occurs because, when the emission current changes, the ratio of the anode and screen-grid currents is not maintained. For these reasons the search for ways of better stabilizing the emission current is continuing. In particular, the search is directed toward the creation of double electrometer tubes for compensation circuits. In the case of single-tube amplifiers, methods are also being developed for compensating instability by introducing negative feedback. Such circuits are distinguished by a smaller amplification factor; however, for measuring currents that are not very small they may be recommended \(^{28, 29, 30, 31, 32, 33, 34, 48}\). As an example, let us consider the circuit shown in Fig. 14.

The parameters of this circuit can be calculated from the following system of equations:

\[ U_0+\mu U_g=I_a(R_i+R_{\mathrm{н}})+I_3R_3, \tag{2,26} \]

\[ U_0=I_2R_2+I_1R_1, \tag{2,27} \]

\[ Ir=I_3R_3-I_2R_2, \tag{2,28} \]

\[ I_2=I_1+I, \tag{2,29} \]

\[ I_a=I+I_3. \tag{2,30} \]

Here \(\mu\) is the static amplification factor of the tube, \(U_0\) is the voltage drop across the resistances \(R_1\) and \(R_2\), \(I\) is the current in the galvanometer circuit, and \(U_g\) is the potential difference between the grid and the cathode of the tube.

Substituting \(U_g=U_c-I_uR_c\) into equation (2,26) and taking into account that \(U_c=I_uR_c\), where \(I_u\) is the measured current, as a result of solving the above system with respect to \(I\) we obtain

\[ I=\frac{-R_3(R_1+R_2)\mu U_c+U_0\{-R_1R_3+R_2[R_i+R_{\mathrm{н}}(1+\mu)]\}} {R_3^2(R_1+R_2)-[R_i+R_3+R_{\mathrm{н}}(1+\mu)]\{(R_2+R_3+r)(R_1+R_2)-R_2^2\}}. \tag{2,31} \]

In the case where the tube operates on the linear part of its characteristic, i.e. the internal resistance \(R_i\) is independent, within sufficiently wide limits, of the supply voltage, from the expression for the current \(I\) one can find the condition under which it does not depend on the magnitude of the supply voltage \(U_0\), and consequently the coefficient at \(U_0\) in the numerator of (2,31) must become zero:

\[ -R_1R_3+R_2[R_i+R_{\mathrm{н}}(1+\mu)]=0. \tag{2,32} \]

For \(R_2=R_3\), condition (2,32) can be rewritten as

\[ R_1-[R_i+R_{\mathrm{н}}(1+\mu)]=0. \tag{2,33} \]

Since for non-electrometer tubes \(\mu\gg 1\), (2,33) can be written in its final form as

\[ R_1=[R_i+R_{\mathrm{н}}(1+\mu)]=R_i+\mu R_{\mathrm{н}}. \tag{2,34} \]

If the internal resistance of the galvanometer \(r\ll R_i\), and also \(r\ll R_2\) and \(r\ll R_3\), then after substituting into (2,31) the expressions (2,32), (2,33), and (2,34), and taking into account the conditions given here, the expression for the current \(I\) will take the following form:

\[ I=\frac{\mu U_c}{2(R_i+\mu R_{\mathrm{н}})}. \tag{2,35} \]

The current amplification factor for this amplifier is expressed as

\[ k=\frac{I}{I_u}=\frac{\mu R_c}{2(R_i+\mu R_{\mathrm{н}})}. \tag{2,36} \]

For an ordinary direct-current amplifier,

\[ k_0 = SR_c, \tag{2.37} \]

where \(S\) is the slope of the tube characteristic.

The ratio \(\dfrac{k_0}{k}\) makes it possible to compare the amplifying capabilities of these circuits:

\[ \frac{k_0}{k}=\frac{2S\left(R_i+\mu R_n\right)}{\mu}. \tag{2.38} \]

But \(S=\dfrac{\mu}{R_i}\), and therefore (2.38) is finally rewritten in the form

\[ \frac{k_0}{k}=2\left(1+SR_n\right). \tag{2.39} \]

From expression (2.39) it is clear that circuits with compensation of instability are less sensitive than circuits without compensation. Even in the absence of negative feedback (\(R_n=0\)), circuits with compensation of instability have a current gain coefficient that is half the corresponding gain coefficient for circuits without compensation. In the case \(R_n>0\), the gain coefficient \(k\) becomes still smaller. However, the value of amplifiers of this type is not reduced when they are used for measuring not very small currents, such as, for example, those arising in the tube of an ionization manometer.

Constancy of the current in the galvanometer circuit when the voltage of the power supply fluctuates is possible only with strict constancy of the internal resistance of the tube \(R_i\) and independence of the static amplification coefficient \(\mu\) from the battery voltage. In practice this condition is not fulfilled for electron tubes (especially shielded ones). Compensation of the instability of the power-supply voltage is effected only partially, since when the tube parameters change the bridge compensation is disturbed owing to the fact that the resistance of one arm changes, while the resistance of the other arms remains unchanged. This condition was the reason for the development of two-tube circuits.

5. Present state of the question of the principles of selecting amplifier elements and the conditions of stabilization

The principles set forth above for selecting the elements of a direct-current amplifier and the stabilization conditions suffer from an internal limitation, due to the fact that they were developed without taking into account the nature of the instability. Meanwhile, knowledge of the causes that give rise to instability makes it possible to make the correct choice of parameters that may be regarded as independent, as well as of the stabilization rules.

It is necessary to distinguish two kinds of instability: slow drift and rapid fluctuations of the galvanometer zero. The causes that give rise to

instability can be divided into two groups: external (not depending on the state of the amplifier) and internal (connected with the amplifier itself). Each of these groups of causes can produce both slow drift and rapid oscillations 9, 10, 38, 39, 40.

Among the most essential external causes one must include such things as variable electric and magnetic fields, variable ambient temperature, vibrations of the electrometer tube caused by external oscillations, etc.

The influence of variable external electric and magnetic fields, as well as the influence of variable ambient temperature, is eliminated by creating appropriate shielding. The influence of vibrations is eliminated with the aid of shock absorbers.

A much more substantial role is played by internal causes, which produce displacement of the zero of the galvanometer (drift) and small oscillations. Drift of the galvanometer zero is due to a gradual increase or decrease of the anode current without any change in all the other parameters of the circuit. It may also be due to an increased rate of self-discharge of the supply battery.

The first cause of this phenomenon is as follows.

As is known, all electrometer tubes are directly heated tubes and, consequently, there is a certain voltage drop between the ends of the cathode. The anode and grid voltages are specified relative to the negative end of the filament. Owing to this circumstance, the emission of electrons from the negative end is greater than from its positive end.

The total anode current depends on the temperature of the cathode, and the latter depends on the filament current and the emission current. Since the emission from the negative end of the filament is greater than from the positive end, the negative end is heated more strongly than the positive. This, in turn, causes an increase in the emission of electrons from the cathode, and so on. A process of self-heating of the cathode takes place, as a result of which the anode current increases, while the anode and grid biases remain unchanged. This process of self-heating continues until a state of thermal equilibrium is reached between the cathode and the surrounding medium. The duration of such a process reaches several tens of hours. During all this time drift is observed.

Small oscillations are due to the following causes:

1) the shot effect of emission; 2) insufficient activation of the cathode; 3) the current of positive ions emitted by the cathode; 4) instability of the grid input resistance 9, 39, 40.

The influence of small oscillations can be considerably reduced by connecting at the galvanometer output a $C$ of relatively large period.

Thus, what is the limitation of the stabilization methods described above?

In all the preceding cases, when finding the stable operating mode of an amplifier, the filament current was chosen as the independent variable parameter. In all these cases, circuits with potentiometric feeding of the amplifier elements were meant. When the filament current is changed, the emission of the cathode changes, but along with it the grid and anode voltages also change. Since small changes in the bias of the control electrode correspond to large changes in the anode current, there exists such a relation between the filament current and the biases of the control electrode and the anode for which the curve of the change in the current in the diagonal of the bridge has a maximum (Fig. 15). With a further increase of the filament current, the current in the diagonal of the bridge not only does not increase, but begins to decrease. The stable mode in this case is determined by the width of the maximum. Therefore, if in the process of self-heating of the lamp cathode the anode current increases by an amount smaller than that by which it should have increased for a change in the filament current equal to half the width of the maximum, then the amplifier will remain in a stable state; but if, in the process of self-heating of the cathode, the emission current exceeds this limit, then after some time drift will again begin to be observed, and therefore it is necessary to repeat the entire stabilization procedure again from the beginning. Thus, the proposed measure does not guarantee a mode of stable operation. At the same time, it excludes the possibility of varying the sensitivity of the amplifier at the experimenter’s discretion.

Fig. 15.

Indeed, as was established above, the current sensitivity of an amplifier depends on the values of the output grid resistance and the slope of the tube characteristic. But the latter depends to a considerable extent on the filament current. It increases, and rather rapidly, with increasing filament current. Thus, by changing the filament current, we also change with it the slope of the characteristic and, consequently, the sensitivity of the amplifier. In addition, at the moment when balancing is completed the system remains in a state in which there is no thermal equilibrium between the cathode and the surrounding medium.

Summarizing all that has been set forth above, the following conclusions must be drawn:

1. Stabilization of an amplifier cannot be carried out by means of the filament current, since any change in the filament current takes the tube operating mode out of the state of thermal equilibrium.

1. By means of the filament current there must be set the sensitivity that is necessary to the experimenter for the given measurements, and subsequently this operating condition must be strictly maintained.

2. The amplifier must remain switched on throughout the entire time of the measurements. Any change in the supply conditions must be followed by a subsequent settling time.

3. After an equilibrium condition has been established, stabilization of the amplifier must consist in bringing the galvanometer zero to the scale by means of the anode resistance and the compensation resistance. In the case of the circuit shown in Figs. 1 and 3, the compensation resistance is the arm \(R_4\).

III. COMPENSATION CIRCUITS OF DIRECT-CURRENT AMPLIFIERS WITH TWO OR DOUBLE TUBES

1. Compensation circuits with two electrometer tubes

In the early stage of the development of direct-current amplifiers and tubes for them, as a measure for combating instability, the idea was put forward of compensating rapid fluctuations of emission. The essence of this idea is that if the disturbances arising simultaneously in both tubes are directed toward each other, they will cancel, and at the output these disturbances will not be observed \(^{35,36}\) (Fig. 16). As is evident from this figure, the amplifier circuit is a bridge, two arms of which are resistances of specified value and the other two are tubes. Before the measured quantity is applied, the bridge is balanced. This is achieved by selecting the filament current.

Indeed, the grid biases relative to the cathodes of the tubes are set by a common battery, and then the imbalance in the arms can be due only to different emission currents, which, in turn, depend on the filament current.

Fig. 16.

At the input of the first tube, a large resistance (\(\sim 10^{10}\) ohms) is placed in the grid (control-electrode) circuit. The small current that must be measured is passed through this resistance. Owing to the large value of the resistance, the small current produces a large voltage drop across it, which strongly changes the anode current, and the circuit goes out of equilibrium. The magnitude of the measured current can be read from the deflection of the galvanometer, if these deflections have been calibrated against some constant source. Such

amplifier permits measuring currents down to \(10^{-11}\) a. However, the accuracy is low because of instability, the cause of which is that the supplied current flows not only through the grid resistance, but also through the glass of the tube, whose resistance does not remain constant. In addition, fluctuations of emission in the tubes cannot be strictly identical.

A further improvement consisted in making the influence of the leakage resistance constant and known by means of a grid bias from a battery connected through the grid resistance. With such a circuit the harmful role of large grid currents of the tube was first revealed, and then the idea arose of creating special electrometric tubes with a small grid current.

Let us consider in somewhat greater detail the influence of grid currents on the measured value of the current.

As was already noted above, the main source of fluctuations of the current in the bridge diagonal is noise in the tube and in the input grid resistance. Among the causes producing noise in electrometric tubes, and especially in tubes with an oxidized cathode operating at a very low temperature, is the shot effect of emission.

The absolute magnitude of the mean value of the cathode-emission fluctuation of the tube can be calculated from the formula

\[ \overline{U}_{\mathrm{фэ}}=R_c\sqrt{2 i_{\mathrm{эм}} e \Delta f}. \tag{3,1} \]

Here \(R_c\) is the input resistance of the grid of the electrometric tube; \(i_{\mathrm{эм}}\) is the cathode emission current; \(\Delta f\) is the width of the frequency band passed by the amplifier, and \(e\) is the charge of the electron. \(\Delta f\) is related to the input time constant of the amplifier \(\theta=R_c C\), where \(C\) is the input capacitance, by the following relation:

\[ \Delta f=\frac{1}{2\theta}. \tag{3,2} \]

After substituting (3,2) into (3,1) we obtain:

\[ \overline{U}_{\mathrm{фэ}}=R_c\sqrt{\frac{e\cdot i_{\mathrm{эм}}}{\theta}}. \tag{3,3} \]

The voltage \(U_c\), developed by the measured current \(I_c\) across the resistance \(R_c\), is equal to

\[ U_c=I_cR_c. \tag{3,4} \]

Thus, the ratio of the magnitude of the voltage fluctuation to the useful signal is

\[ \frac{U_{\mathrm{фэ}}}{U_c}=\sqrt{\frac{e\cdot i_{\mathrm{эм}}}{I_c^2\theta}}. \tag{3,5} \]

The influence of fluctuations of the emission current may be neglected if one sets

\[ \frac{\overline{U}_{\phi э}}{U_c}=\frac{1}{10}. \]

In this case

\[ \frac{I_c^2}{i_{эм}}=\frac{16{,}0\cdot 10^{-18}}{\theta}, \]

or

\[ I_c=4\cdot 10^{-9}\sqrt{\frac{i_{эм}}{\theta}}. \tag{3,6} \]

It follows from expression (3,6) that the limiting value of the current accessible to measurement by the given amplifier decreases as the time constant of the amplifier input increases. It would seem that, proceeding in this way, one could obtain the possibility of measuring arbitrarily small current values. In reality, however, this path is limited. As \(\theta\) increases, the inertia of the system also increases. The time required for the steady deflection becomes so large that making measurements becomes impossible.

Using Nyquist’s formula, one may write the value of the effective voltage of thermal fluctuations \(\overline{U}_T\) as

\[ \overline{U}_T=2\sqrt{kTR_c\Delta f}. \tag{3,7} \]

Here \(k\) is Boltzmann’s constant, equal to \(1{,}37\cdot 10^{-23}\) watt/degree; \(T\) is the absolute temperature. For room temperatures,

\[ \overline{U}_T=8{,}8\cdot 10^{-11}\sqrt{\frac{R_c}{\theta}}. \tag{3,8} \]

The ratio of (3,8) to (3,4) gives

\[ \frac{\overline{U}_T}{U_c}=8{,}8\cdot 10^{-11}\frac{1}{I_cR_c\sqrt{C}}. \tag{3,9} \]

Assuming, as before, that

\[ \frac{\overline{U}_T}{U_c}=\frac{1}{10}, \]

we obtain

\[ I_c=8{,}8\cdot 10^{-10}\frac{1}{R_c\sqrt{C}} =8{,}8\cdot 10^{-10}\sqrt{\frac{1}{R_c\theta}}. \tag{3,10} \]

From expression (3,10) it is seen that, in order to increase the current sensitivity of the amplifier, it is necessary to increase \(R_c\) and decrease \(C\).

Taking into account both of the above factors, the minimum current accessible to measurement can be expressed as follows. The total fluctuation voltage at the output is

\[ \sqrt{\overline{U}_{ш}^{\,2}} = \sqrt{\overline{U}_T^{\,2}+\overline{U}_{\phi э}^{\,2}} = \sqrt{\frac{kT}{C}+\frac{eI_c}{2}\cdot\frac{I_c}{C}}, \tag{3,11} \]

Taking into account (3.4), we find the relation

\[ \frac{U_c}{\sqrt{\overline{U_{\mathrm{ш}}^2}}} = \frac{I_c R_c}{\sqrt{\dfrac{kT}{C}+\dfrac{eI_c}{2}\cdot\dfrac{R_c}{C}}}, \tag{3.12} \]

or

\[ \frac{U_c}{\sqrt{\overline{U_{\mathrm{ш}}^2}}} = \frac{I_c}{\sqrt{\dfrac{kT}{R_c^2 C}+\dfrac{eI_c}{2}\cdot\dfrac{1}{R_c C}}}. \tag{3.12'} \]

On the basis of the accepted ratio between the useful signal and the noise,

\[ I_c = 100 \sqrt{\frac{kT}{R_c^2 C}+\frac{eI_c}{2}\cdot\frac{1}{R_c C}} . \tag{3.13} \]

If it is taken into account that \(R_c\) may be equal to the lamp’s own leakage resistance and \(C\) to its input capacitance, then in the present case \(I_c\) is the lamp’s own grid current. As is seen from (3.13), it is the smaller, the larger the time constant of the amplifier input. Thus, the grid current and the time constant of the amplifier input are competing quantities that limit the possibility of measuring small current values. For this purpose special lamps with small grid current (\(\sim 10^{-15}\) a) and a small input time constant, called electrometric lamps, have been constructed.

Fig. 17.

In connection with the appearance of special electrometric lamps, several bridge-type two-lamp circuits were proposed\(^{37,38}\). One such circuit, shown in Fig. 17, was described in work\(^{39}\). With a grid resistance equal to \(10^{10}\) ohms, it makes it possible to measure currents down to \(5 \cdot 10^{-18}\) a, or 30 electrons per second. The accuracy of the measurements is 15%. The upper limit, as the author states, reaches \(5 \cdot 10^{-19}\) a, or 3 electrons per second. By using a short-period galvanometer, it is possible to measure the presence of a single electron.

For the purposes of practical measurement of small currents and large resistances, a two-lamp compensation circuit was proposed\(^{8}\), shown in Fig. 2. In principle this circuit does not differ in any way from the circuit shown in Fig. 17, but thanks to spe-

tial selection of parameters, it is very stable, and a galvanometer with a sensitivity of \(10^{-10}\) A/mm can be used at its output when the grid resistance is \(10^{11}\) ohms. Under these conditions the current sensitivity can be brought to a value equal to \(8 \cdot 10^{-17}\) A/mm. The amplification factor can be made equal to \(1.2 \cdot 10^{6}\). Such an amplifier makes it possible to measure potentials down to \(10^{-4}\) V and resistances up to \(10^{13}\) ohms.

Along with the development of compensating two-tube circuits using special electrometer tubes, compensating circuits using tubes of types 954 and 959 (acorn) were also developed\(^{11,46,47,48,49}\). Amplifiers assembled with these tubes showed sufficient sensitivity and stability. They make it possible to measure currents from \(10^{-7}\) to \(10^{-13}\) A.

As has already been noted, complete stability of compensating amplifiers assembled with two tubes cannot be achieved because of unequal changes in the emission currents of the two tubes during self-heating, i.e., the drift of a two-tube amplifier is caused by the difference in the cathode properties of the different tubes.

Fig. 18.

In parallel with the described measures for stabilizing two-tube direct-current amplifiers, the following one is also used. The influence of the inequality of fluctuations of the emission current in the tubes can be suppressed if the cathode grid of one tube is used to control the anode current\(^{50,51}\). Such a circuit is shown in Fig. 18. The principle of anode-current control is implemented by introducing a series resistance into the cathode-grid circuit. The voltage actually applied to it will be less than the operating voltage by the amount of the voltage drop across this resistance. As the filament heating of the tube increases, the emission current grows, and at the same time the current of the cathode grid increases. The voltage drop across the resistance increases, and correspondingly the positive potential of the cathode grid decreases. Thus the cathode grid controls the emission current, since a drop in its potential causes a decrease in the current. By an appropriate choice of the resistance in the cathode-grid circuit of one tube, one can arrange that a decrease in the emission current causes an identical change (weakening) of the anode current of both tubes. In this way it was possible to obtain compensation for different filament heating of the tubes.

As can be seen from Fig. 18, the grid circuits contain resistances \(R_3\) and \(R_4\), of which \(R_4\) is variable. The full anode voltage \(U_a\) is used as the supply voltage for the cathode grids. It has been

this is because the voltage drop of a separate battery feeding the cathode grids would affect the anode current to different degrees and would disturb the equilibrium. When the anode battery is used to feed the cathode grids, however, no disturbance of the bridge equilibrium occurs, since the different effect of the voltage drop of this battery on the two tubes has already been compensated in advance by the selection of the resistances \(R_1\) and \(R_2\).

The initial grid biases \(U_{g_1}\) and \(U_{g_2}\) are supplied from one and the same battery. To change the sensitivity, the galvanometer is shunted.

According to the data of work \(^{49}\), with proper adjustment a good constancy of the galvanometer zero position is achieved. Thus, over five hours of operation, as the authors write, the galvanometer zero shifted by an amount corresponding to a bias of 0.5 millivolt applied to the grid. The authors do not state what the voltage sensitivity was, but if one assumes that it was of the order of \(10^{-5}\ \mathrm{V/mm}\), this corresponds to a creep of the galvanometer zero by 50 mm, or a drift of \(10\ \mathrm{mm/hour}\). In the absence of fluctuation, such a drift may be considered quite acceptable.

2. Compensation circuits using dual tubes

To eliminate the difficulty that arose in stabilizing two-tube amplifiers, dual tubes with a common cathode were constructed. These are dual triodes and dual tetrodes \(^{9, 10, 50, 52}\). These tubes are made in such a way that two tubes are mounted in one envelope and on a common cathode, i.e., there are two anodes and two control grids. In a bridge compensation circuit assembled with such a dual tube, sufficiently complete compensation of emission fluctuations is achieved.

Dual four-electrode electrometric tubes make it possible to realize bridge circuits, as shown in Fig. 19. Fluctuation of the emission current causes a coordinated change of the anode current in both anode circuits. Thus, if the bridge is balanced, its stability does not depend on the emission current. The equilibrium conditions of the circuit can be written as follows:

\[ I_1 = I_2, \tag{3,14} \]

\[ \frac{\partial I_1}{\partial E} = \frac{\partial I_2}{\partial E}, \tag{3,15} \]

\[ R_1 - r = R_2. \tag{3,16} \]

Expression (3,14) is the best condition for stability in the absence of drift and fluctuations of the emission current.

Expression (3,15) satisfies the condition of balance independently of small fluctuations of the voltage of the supply battery.

Assuming that equation (3,14) is satisfied, the requirement of balance in the bridge is given by expression (3,16). According to Fig. 20 the current

the filament is regulated by the resistance \(R_{15}\). The resistances \(R_8\) and \(R_9\) make it possible to select the anode voltage and the cathode-grid voltage. The grid bias is regulated by means of \(R_{11}\) and \(R_{12}\). The balancing method consists in satisfying conditions (3,14)—(3,16). In the normal regime the grid bias must be set in accordance with the characteristic of the tube. The filament current \(I_3\) is changed by varying \(R_{15}\); at the same time a change in \(I_1\) and \(I_2\) is observed. If \(I_1\) is greater than \(I_2\), then \(R_{14}\) must be increased; if \(I_1\) is less than \(I_2\), then \(R_{14}\) must be decreased.

Fig. 19.

Fig. 20.

By changing \(I_3\) in one direction and the other, one achieves that equality (3,14) is satisfied \((I_1 = I_2)\). This operation is continued until the current \(I_1\) remains constant when \(I_3\) is changed by approximately \(15\%\). In order that condition (3,16) be satisfied, in the left part of the circuit the resistance \(R_{10}\) is supplemented by the resistances \(R_3\) and \(R_4\). The final balancing is carried out by observing the behavior of the galvanometer.

The ultimate task is the complete elimination of drift, which disappears when \(I_1\) becomes equal to \(I_2\). The final elimination of drift is carried out with the aid of \(R_{13}\) and \(R_{14}\), with \(R_{13}\) serving as the fine adjustment of \(R_{14}\).

CONCLUSION

Direct-current amplifiers operating with electrometer tubes are very sensitive and stable measuring instruments, permitting measurements of many electrical quantities with a high degree of accuracy.

The sensitivity of direct-current amplifiers is higher than that of the most sensitive electrometers themselves.

Taking into account the simplicity of handling them, their universality and relative cheapness, one may hope that in the very near future direct-current amplifiers will become an indispensable measuring instrument in scientific-research institutes, laboratories, and in production.

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On pp. 122 and 123, in the list of cited literature, owing to a typographical error, the numbers of all bibliographic references beginning with 18 have mistakenly been decreased by one (one should read 19 instead of 18, etc.).

Order No. 1461.

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