Cathode Tubes, Their Theory and Principal Applications

S. N. Rzhevkin.

The investigations that led to the discovery of cathode tubes in their modern form are closely connected with the general development of the question of the ionization of gases and the passage of electric current through them. As early as 1853, Becquerel discovered that air at a temperature of 1,500° becomes a conductor of electricity. Elster and Geitel in 1880 found that an insulated plate placed near a filament heated by an electric current loses its charge. In 1883 T. Edison found that if an insulated electrode, connected through a galvanometer with the incandescent filament, is introduced into the bulb of an incandescent lamp, then in this circuit an electric current is detected, flowing from the filament through the galvanometer to the plate and then, as it were, closing through the vacuum of the lamp. This phenomenon, called the Edison effect, is the prototype of the modern cathode tube. Fleming, investigating this phenomenon, showed that the insulated plate inside the incandescent lamp becomes charged negatively up to a certain potential; a considerable current in the plate circuit flows when there is a \(+\) potential on it, whereas the current ceases completely when there is a \(-\) potential. Only in 1904 did the same Fleming have the fortunate idea of applying this one-sided conductivity inside the incandescent lamp for the purpose of detecting (rectifying) the high-frequency oscillations used in radiotelegraphy, and of replacing the inconvenient and not very sensitive coherer with it.

The crystal detector was discovered in 1906 and, thanks to its simplicity and sensitivity, became widely used, somewhat weakening interest in Fleming’s “valve.” Almost simultaneously with Fleming, De Forest in America patented a detector in which the one-sided conductivity of the flame of a gas burner is used, and in 1905 a detector with an incandescent filament in a gas atmosphere; he also made the next step in this field by introducing a third electrode—the grid—thanks to which it became possible considerably to increase the detector action and, moreover, to obtain amplification. The name “audion,” given by De Forest to his detector, is still applied to the basic detector circuit. The idea of using a cathode tube as

the generator of undamped oscillations belongs to the German scientists Lieben and Reisz (1906); the Lieben–Reisz tube was furnished with a Wehnelt cathode coated with alkaline-earth metal oxides.

In view of the importance of the use of cathode tubes in military affairs, the development of their design became a military secret of each individual state, and very few works in this interesting field appeared in the scientific literature. The European war showed how much scientists of all countries had succeeded in doing in this field, and now a very large number of papers in journals on electrical engineering, radiotelegraphy, and physics are devoted to questions of developing the theory of cathode tubes, their improvement, and their applications. With all the variety of applications of cathode tubes, usually only three of their basic properties are used: detector, amplifying, and generator action. The exposition of these basic properties is the subject of the present article.

I. THE TWO-ELECTRODE CATHODE TUBE—DIODE

A two-electrode cathode tube consists of an incandescent filament (cathode), placed in a vessel with rarefied gas, and a second electrode (anode), of one form or another, insulated from the filament and connected to the positive pole of a high-voltage battery; the vessel is usually made of glass, but it is also possible to make it metallic. As the investigations of Richardson have shown, in a vacuum the clean surface of a metal at high temperature is capable of emitting electrons and giving a thermoelectronic current in the anode circuit. This purely thermal property of metals does not depend on the surrounding medium (since it does not affect the structure of the surface layer) and is explained by the fact that electrons moving inside the metal with the average kinetic energy of its particles can, if they possess a certain velocity, overcome the surface potential difference and pass out through the surface; the work which the electron must expend for this is equal to \(e\psi\), where \(e\) is the charge of the electron, equal to \(4.77 \cdot 10^{-10}\ CgSE\), and \(\psi\) is the surface potential difference, different in different metals; the quantity \(e\psi\) may be conveniently converted into heat units and called the heat of evaporation of an electron from the metal. Evidently only those electrons can pass through the surface which have a velocity \(v\) sufficient to overcome the potential difference \(e\psi\), i.e. for which \(\frac{mv^{2}}{2} > e\psi\); the kinetic energy can be measured by that potential difference, overcoming which the electron reduces its velocity to zero.

\[ v = \sqrt{2\frac{e}{m}V} = 6.0 \cdot 10^{7}\sqrt{V}\ \frac{cm}{sec} \]

where \(V\) is the potential difference in volts.

When the temperature \(T\) is raised, the mean velocity of the electrons increases, a larger number of them can penetrate through the surface, and the thermoelectronic current increases. The mean kinetic energy of an electron is, at absolute temperatures:

\[ \begin{array}{rcl} 1{,}000^\circ & \ldots & 0.13\ \text{volt} \\ 10{,}000^\circ & \ldots & 1.3\ \text{volt} \\ 100{,}000^\circ & \ldots & 13\ \text{volt} \end{array} \]

The measurement of the surface difference of potentials is attended by great difficulties, and different observers give widely divergent results:

TABLE I.

Metal.	\(\psi\) Volt.	\(b=\dfrac{e\psi}{k}\) deg.	Observer.
Platinum	6.25	\(7.25 \cdot 10^4\)	Wilson.
”	5.85	\(6.78 \cdot 10^4\)	Richardson.
”	6.9	\(8.0 \cdot 10^4\)	Langmuir.
”	5.3	\(6.15 \cdot 10^4\)	Compton.
Tungsten	4.53	\(5.25 \cdot 10^4\)	Langmuir.
”	4.4	\(5.10 \cdot 10^4\)	Compton.
Tantalum	4.3	\(5.0 \cdot 10^4\)	Langmuir.
CaO	3.48	\(4.03 \cdot 10^4\)	Compton.
Na	2.05	\(2.38 \cdot 10^4\)	Compton.

The greatest weight, apparently, should be assigned to Langmuir’s data.

At \(3{,}000^\circ K.\) (abs.) the velocity of the electrons will on the average be \(0.4\) volt; for \(W\), a velocity sufficient to overcome the difference of potentials of \(4.5\) volts, i.e. more than 10 times greater than the average, will be possessed by a very small percentage of the electrons; according to Maxwell’s law this will be \(0.00014\%\); but for the surface of \(Na\) (\(\psi = 2.05\) volt) a sufficient velocity will already be possessed by \(0.19\%\) of all electrons, whence it follows that electron emission, at the same \(T\), from the surface of \(Na\) will be 1300 times stronger. Tungsten has an enormous advantage over other metals in view of its high melting temperature (\(3{,}270^\circ K.\)) with a comparatively small \(\psi\). The emission of electrons is greatly increased when coating

Recent theoretical and experimental investigations by Dushman1 lead to the conclusion that thermoelectronic emission is more accurately expressed by the following formula:

\[ J = AT^2 e^{-\frac{b}{T}} \quad (1'), \qquad \text{where } A=\frac{2\pi k^2me}{h^3}, \]

where \(k\) is the gas constant per molecule: \(\frac{R}{N}\); \(m\) and \(e\) are the mass and charge of the electron; \(h\) is Planck’s constant, \(h=6.55\cdot 10^{-27}\ \frac{erg}{sec}\). The constant \(A\) is a universal constant and turns out to be the same for all metals:

\[ A=60.2\ \frac{amp}{cm^2\,grad^2}; \]

it characterizes the general thermal properties of all metals. The equation in the form \((1')\) had already been proposed by Richardson, but he did not succeed in finding the correct value of the constant \(A\), which became possible only with the aid of quantum theory. In this formula Dushman assumes for W \(b_o=52{,}600\); it is interesting to note that this gives considerably smaller saturation-current values than in Table II, which should evidently be explained by the great care of Dushman’s experiments, which excluded any influence of gas residues thanks to an extremely good vacuum, as well as by the rigorous verification of the temperature scale for W carried out in Langmuir’s laboratory. To explain the discrepancies it is sufficient to assume a difference of temperatures of several tens of degrees between the old and new data, which is quite probable given the low accuracy of measuring such high \(T\).

The stream of electrons moving from the cathode to the anode carries a certain quantity of negative electricity and therefore forms a certain spatially distributed charge, which produces a field directed oppositely to the anode field, retards the motion of the electrons, and can force them to move back toward the filament—the cathode. This electron cloud, or atmosphere, increases its density near the surface of the filament to a large value, as a result of which the field near the very surface has the opposite direction—from cathode to anode—and only faster electrons can break through this barrier; at appreciable potentials on the anode the region of the reverse field extends to a distance of less than \(0.1\ mm\) from the surface of the cathode. Farther on the field passes through zero and assumes the direction from anode to cathode. For simplicity of calculation let us assume that the field is equal to zero at the very surface of the cathode (which corresponds to the emission

electrons without initial velocity) and find the relation between the potential difference \(V_a\) at the anode and cathode and the strength of the thermoelectron current \(I\). Let us consider the case of a plane cathode and an anode of infinite surface, arranged parallel to one another at a distance \(a\), and let us seek the current \(J\) per \(1\ \mathrm{cm}\) of surface. Since all the lines of force issuing from the anode and terminating on the electrons of the space charge (at the surface of the cathode the field is equal to zero and therefore the lines of force do not reach it) will go perpendicular to the surface of the anode and cathode, we may place the \(x\)-axis of the coordinate system in this direction and assume that the potential \(V_x\) at a given point at distance \(x\) from the cathode will depend exclusively on \(x\), and not on the two other coordinates \(y\) and \(z\). By Poisson’s theorem we have:

\[ \frac{d^2 V_x}{d x^2}=4\pi \rho_x=4\pi n_x e, \]

where \(n_x\) is the number of electrons in one \(\mathrm{cm}^3\) and is a function of \(x\); the current strength \(J=n_x e v_x\) (through \(1\ \mathrm{cm}^2\)), where \(v_x\) is the velocity of the electrons at distance \(x\) from the cathode; \(J\), of course, cannot depend on \(x\). On the other hand, the kinetic energy of an electron is

\[ \frac{m v_x^2}{2}=e V_x, \]

whence:

\[ v_x=\sqrt{2}\sqrt{\frac{e}{m}}\sqrt{V_x}. \]

Substituting this quantity into the expression \(n_x=\dfrac{J}{e v_x}\) and introducing \(n_x\) into Poisson’s formula, we find:

\[ \frac{d^2 V_x}{d x^2} = \frac{4\pi J}{\sqrt{2}\sqrt{\dfrac{e}{m}}\sqrt{V_x}} \]

This differential equation is satisfied by the solution:

\[ V_x= \left( \frac{9\pi J}{\sqrt{2}\sqrt{\dfrac{e}{m}}} \right)^{2/3} x^{4/3} \]

introducing the notation:

\[ B=\frac{\sqrt{2}\sqrt{\dfrac{e}{m}}}{9\pi} \]

we obtain the desired relation between \(J\) and \(V_a\) in the form

\[ J=\frac{I_{\text{tot}}}{S_{\text{area}}}=B\frac{V_a^{3/2}}{a^2} \tag{2} \]

where \(S\) is the area of the anode and cathode; in practical units:

\[ J=2.33\cdot 10^{-6}\frac{V_a^{3/2}}{a^2}\ \frac{\text{amp.}}{\text{cm}^2}, \]

if \(V_a\) is expressed in volts and \(a\) in centimeters. This formula was derived in 1911 by Child; its complete interpretation was given only by Langmuir2, on account of which it is for the most part called the “Langmuir three-halves law.”

For a given anode potential the density of the electron atmosphere at each point will be, as is readily seen,

\[ n_x=\frac{1}{9\pi}\left(\frac{J}{B}\right)x^{-3/2} \tag{2} \]

If the thermoelectronic emission is increased by raising the temperature, the current strength will not increase from this, since all the extra electrons (contrary to formula (2)) will not be allowed through by the retarding layer at the surface and will return back into the metal; thus formula (2) gives the maximum current value \(\max J\) for the given potential difference.

In an analogous way the solution of the problem is easily obtained for the total current strength in a diode with a cylindrical anode of radius \(r\) and a filament along its axis:

\[ J_{\text{tot}}= 2\pi r\frac{\sqrt{2}\sqrt[4]{\frac{e}{m}}}{9\pi} \frac{V^{3/2}}{r^2}l = 2\pi B\frac{V^{3/2}}{r}l = 14.65\,l\frac{V^{3/2}}{r}\ \text{amp.} \tag{3} \]

Fig. 1 (taken from Langmuir’s work) illustrates formulas 2 and 3. At low temperatures the current strength is determined by Richardson’s formula, and when the maximum current given by Langmuir’s law is reached, any further increase of the current strength with increasing \(T\) ceases; when the anode voltage is increased, the part of the curve in accordance with Richardson’s formula will extend to higher \(T\); the maximum values of all the curves obey the three-halves law.

In Fig. 3, relating to a three-electrode tube, the curve \((J_a + J_g)\) gives a fairly accurate representation of the “characteristic” of a diode, i.e., of the dependence of the thermoelectronic current on the potential difference between the anode and the cathode2. For small values of \(V_a\), the current varies in accordance with the three-halves law, until all the thermoelectrons emitted by the cathode have been utilized; after that an increase in voltage will not cause an increase in current. This limiting current \(J_s\), corresponding to a given temperature, is called the saturation current. With an increase in temperature the saturation current increases according to Richardson’s formula.

In view of the fact that along the filament there is a fall of potential \(V_k\)—usually several volts—at anode potentials smaller than the potential of the \((+)\) pole of the filament, the current begins to fall very sharply, since electrons will reach the anode no longer from the whole filament, but only from that part of it whose potential is lower than the anode potential.

Fig. 1. Dependence of the thermoelectronic current on the temperature of the cathode at various anode potentials.

Since thermoelectrons have certain initial velocities, distributed according to Maxwell’s law, some of the fastest of them can reach the anode even with a small negative voltage on it; thus, for negative \(V_a\), a weak current will be obtained, falling very rapidly with an increase in the negative potential of the filament. Taking the mean velocity of the thermoelectrons as 0.5 volt, we may assume that at \(V_a = -1\) volt only an insignificant fraction of the electrons, having a velocity greater than twice the mean, will be able to reach the anode; practically, it may be considered that at \(-1\) volt the anode current ceases. The law of variation of the current at small positive—

at positive and negative anode potentials will be approximately the following:

\[ J=J_{0}^{\prime} e^{-\frac{V_a}{V_0}} \quad \text{where } V_0=8.6\cdot 10^{-5}T^{1}) \]

At \(T=2300^\circ\), \(V_0=0.2\) volt, i.e. the current will decrease by a factor of 2.7 when \(V_a\) is decreased by 0.2 volt.

The presence of gas introduces very considerable changes into the picture of thermionic emission. Owing to collisions of electrons with gas molecules, positive ions may be formed, which neutralize the space charge and eliminate its retarding action, as a result of which the current in the diode increases strongly. Thus, for example,\(^{2}\) mercury vapor at a pressure of only \(10^{-5}\) mm made it possible to obtain in a diode at 25 volts a current of 0.1 amp., attainable in a full vacuum only at 200 volts. Langmuir gives the extremely interesting observation that with a tungsten cathode, oxygen and oxygen-containing gases do not increase the current, but decrease it.

Deviations from the \(3/2\) law, when the voltage at the anode is raised above several tens of volts, make it possible to estimate the degree of perfection of the vacuum.

As the investigations of Franck and Hertz, and also a whole series of other investigators in recent years, have shown, when an electron strikes a molecule it may excite it to luminescence; for this the electron must have an energy \(eV\) not less than \(h\nu_0\), while it itself loses all its velocity; here \(\nu_0\) is the frequency of radiation of the first line of the absorption series of the given gas, \(h\) is Planck’s constant. This resonance potential has a definite value for each gas. \((Hg—4.68\ \mathrm{v.};\ N_2—17—18\ \mathrm{v.};\ O_2—15.5\ \mathrm{v.};\ H_2—16\ \mathrm{v.},\ He—25\ \mathrm{v.};\ Ar—15—18\ \mathrm{v.};\ Ne—21—23\ \mathrm{v.}).\) It is clear that, when a potential equal to the resonance potential is applied to the anode, near the anode there will immediately occur a stoppage of a large number of electrons, which will introduce changes into the distribution of potential and will be reflected in the magnitude of the current. Indeed, in gas-filled tubes sharp jumps and breaks in the characteristic are observed, from which the values of the resonance and ionization potentials can be found.\(^{3}\) The possibility of applying these properties of the gas-filled tube will be discussed in Chapter III of the present article.

The diode is especially convenient to use as a rectifier of high-voltage alternating currents. The first diodes intended for this purpose were constructed by Langmuir under the name “Kenotron”; one of the types gave, for example, at 200 volts—1 amp. and at 180,000 volts—0.25 amp.

\(^{1}\) Barkhausen. Jahrb. d. Drahtl. Tel. B. XIV, S. 27. 1919.
\(^{2}\) Langmuir. Proc. Inst. Radioeng. 3, p. 3. 1915.
\(^{3}\) Raund. Phys. Rev. 15, p. 132. 1920.
Stead a. Glossing. Phil. Mag. 40. 1920.

(for X-ray apparatus). At present, the General Electric Co. in America, where Langmuir works, is building rectifiers of almost unlimited power, in which the electron current in vacuum reaches several tens of amperes; such rectifiers are made in a metal vessel—the anode—which has the shape of a cylinder and is placed in a cooling reservoir with oil; glass is soldered into the bases of the anode, serving as insulation for the current lead to the tungsten wire—the cathode; in the largest models this latter has a thickness of more than 1 cm and requires a current of hundreds of amperes for incandescence. In view of their exceptional stability and the possibility of parallel connection, kenotrons successfully compete with mercury rectifiers.

The circuit for connecting a kenotron into an alternating-current circuit is given in Fig. 2. The choke \(L\) and the condenser \(C\), with a capacitance of \(0.001\) mfd., serve to delay the alternating component of the rectified current; fluctuations of the current strength in this circuit are less than \(1\%\).

Fig. 2. Hull circuit for obtaining direct current by means of kenotrons.

It is interesting to note that, for the rectification of alternating currents under laboratory conditions, a 3-electrode tube (triode) may also serve, if its grid is connected directly to the anode. The author succeeded in obtaining from a 120-volt and 50-cycle-per-second mains supply, through two French receiving tubes according to the circuit of Fig. 2 (Hull circuit),2 a direct current of up to 20–30 milliamperes, with the filament also supplied by alternating current through a special transformer—reducer (5 volt); such rectified current could even be used to supply amplifiers. With such tubes, with a good vacuum, one may rectify voltages up to 1,000 volt and even higher, if the insulation of the leads sealed into the glass withstands breakdown.

II. THE THREE-ELECTRODE CATHODE TUBE—TRIODE

GENERAL PROPERTIES.

The introduction into the cathode tube of a third electrode completely changes the picture of the distribution of potential inside it; this third electrode is intended, by means of its potential, to regulate

the current in the anode circuit and, in foreign literature, is often called the control electrode; it is most often made (though not always) in the form of a more or less dense mesh lying between the anode and the cathode. It is natural to suppose that the strength of the thermionic current will be directly dependent on the magnitude of the field \(E\) at the surface of the cathode, i.e. on the charge \(q_k\) induced per unit surface, since \(E=4\pi q_k\). The total charge on the surface \(s\) of the cathode will be equal to:

\[ Q_k=s q_k=C_{ak}V_a+C_{gk}V_g, \]

where \(C_{ak}\) and \(C_{gk}\) are the mutual capacitances between the anode and cathode and between the control electrode and cathode; \(V_a\) and \(V_g\) are the potentials of the anode and of the control electrode, the cathode potential being taken as equal to 0 (the symbol \(g\) is adopted in English and German literature as the initial letter of the word: Gitter, Grid). In view of the fact that the control electrode is for the most part a grid, we shall always use this term, making a reservation only in those cases when it has another form.

Writing the above expression in the form

\[ Q_k=C_{ak}\,[V_a+kV_g], \quad \text{where } k=\frac{C_{gk}}{C_{ak}}, \]

we may suppose that the action of the anode and the grid together will be the same as that of a single anode if it had a certain equivalent potential \((V_a+kV_g)\). Langmuir was the first to propose, for calculating the current in a triode, a formula analogous to (2) and (3):

\[ J=A\,(V_a+kV_g)^{\frac{3}{2}} \tag{4} \]

where \(A\) is a certain constant determined experimentally, and \(k\) is determined by the geometrical forms of the triode and by the density of the grid; it is clear that this dependence can be valid only up to the onset of saturation current. Formula (4) calculates the entire current produced by the thermoelectrons leaving the cathode and distributed between the grid and the anode depending on their potentials and shape; thus, denoting by \(J_a\) and \(J_g\) the current in the anode and grid circuits, we have:

\[ J=J_a+J_g. \]

When the grid potentials are considerably smaller than the anode potential, the current \(J_g\) in the grid is considerably smaller than in the anode \((<10\%)\), and therefore one may take \(J=J_a\). If the grid potential becomes larger, a considerable current \(J_g\) appears in it, which may exceed \(J_a\) when \(V_g>V_a\)\(^1\); in this case Langmuir’s law will be valid only

\(^1\) The considerable increase of the current in the grid circuit and the decrease of it in the anode circuit is also explained by the appearance of secondary electrons when primary electrons strike the anode surface (Barkhausen); when \(V_g>V_a\), all the secondary electrons will fall on the grid and will produce a considerable increase of the current \(J_g\) and a decrease of \(J_a\) as \(V_g\) increases.

with respect to the sum \((J_s + J_a)\), but not to any one anode current. In addition, it is clear that Langmuir’s law is valid only for currents smaller than the saturation current.

The curve expressing the dependence of the current in the anode circuit \(J_a\) on the grid potential \(V_g\), at constant \(V_a\), is usually called the “characteristic” of the triode; Fig. 3 gives an example of such a characteristic*) for \(V_a = 80\) volt; on the same graph are plotted the current in the grid \(J_g\) and the total current \(J\) as functions of \(V_g\). In taking the characteristic, all potentials are reckoned relative to the negative end of the filament.

Fig. 3. Characteristics of a French amplifying tube.

With regard to the current in the grid it should be noted that it appears at \(V_g\) greater than \(-1\) volt (see above), increases rather rapidly, then remains almost unchanged (simultaneously with the anode current), and for \(V_g > V_a\) again increases rapidly. The total current \(J_a + J_g\) initially increases according to Langmuir’s law and, upon reaching the saturation current \(J_s\), remains constant.

The anode current \(J_a\) at \(V_g = 0\) is not equal to 0, since it is a function of \((V_a + k V_g)^{\frac{3}{2}}\); \(J_a\) will be equal to 0 when \(V_g = -\dfrac{V_a}{k}\). Thus it is clear that the characteristic is shifted, when the anode potential \(V_a\) is increased, by the amount \(\dfrac{\Delta V_a}{k}\) toward negative potentials (to the left); in Fig. 3 the dashed lines show the characteristics for \(V_a = 100\) V and 60 V; the first is shifted

*) For a French amplifying tube.

to the left by 3 volts, the second by the same amount to the right; the magnitude of the saturation current to the anode remains almost unchanged for all three potentials. From the magnitude of the shift one can calculate the quantity

\[ k=\frac{\Delta V_a}{\Delta V_g}=\frac{20}{3}=6.7. \]

The quantity \(k\) characterizes how many times more strongly the grid potential \(V_g\) acts (on the anode current) than the anode potential; it is called the “voltage amplification factor” in the triode; obviously, a certain increment of the grid potential is equivalent to a \(k\)-fold increment of the anode potential. In accepted designs the quantity \(k\) varies within wide limits. In the Bina lamp with a control electrode in the form of a plate, \(k\) is close to 1; \(C_{ak}=C_{gk}\). In amplifier tubes of cylindrical construction \(k\) has a value of 5–10; in generator tubes with a dense grid \(k\) has a value of 20–100 and more; the calculation of the quantity \(k\) will be discussed below.

A more rigorously grounded derivation of Langmuir’s formula (4) was given by Eccles\(^{1}\) for a triode of plane form and for a cylindrical triode with a grid wound along a helical line around the filament at a distance \(r_g\); for the cylindrical triode the following expression is obtained for the current strength per unit length:

\[ J=-\frac{2\pi B}{(1+g)^{\frac{3}{2}}r_g}(V_a+gV_g)^{\frac{3}{2}} =-\frac{14.65}{(1+g)^{\frac{3}{2}}r_g}(V_a+gV_g)^{\frac{3}{2}}\frac{amp}{cm} \tag{5} \]

Here the letter \(g\) denotes the voltage amplification factor (\(k\)), which can be calculated if \(r_g\) and the number of grid turns per 1 cm are known. It should be noted that Eccles’s expression for the coefficient \(A\) in formula (4) is determined by the radius of the grid, not of the anode.

German authors treat the question in a somewhat different plane. Barkhausen assumes that the field in the space between the grid and the filament, which determines the emission current of the electrons, is determined chiefly by the field of the grid and only to a small extent by \(DV_a\) of the anode field (where \(D\) is a proper fraction) penetrating through the grid; thus the grid is, as it were, a semitransparent screen with “permeability” \(D\) (Durchgriff). The entire emission current \((J_a+J_g)\) will be determined in the same way as in a diode, with the anode in the place of the grid and with the potential on it equal to \(V_g+DV_a\):

\[ J=2\pi B\frac{(V_g+DV_a)^{\frac{3}{2}}}{r_g} =\frac{14.65}{r_g}(V_g+DV_a)^{\frac{3}{2}} \tag{5′} \]

\(^{1}\) Radio Rev. Oct., Nov., Dec. 1919.
\(^{2}\) Jahrb. Drahtl. Tel. B, XIV, S. 27 1919.

but with the difference that this current will go chiefly to the anode, and only a small part of it will reach the grid. If in formula (5) we take \(D\) outside the brackets and denote \(\frac{1}{D}=g\), then the expression obtained is:

\[ J=\frac{2\pi B}{g^{\frac{3}{2}}}-\frac{(V_a+gV_g)^{\frac{3}{2}}}{r_g} \]

for large values of \(g\), almost not differing in practice from Eckles’s formula; thus, for small permeabilities \(D<10\%\), one may always take \(D=\frac{1}{g}\). As rigorous analysis shows2, both formulae (5) and (5′) cannot be regarded as exact, since the magnitude of the mutual capacitances \(C_{ak}\) and \(C_{gk}\) \(\left(D=\frac{C_{ak}}{C_{gk}}\right)\) varies greatly (by a factor of 2–4) depending on the space charge, i.e. on the current strength, and therefore also on the potentials \(V_g\) and \(V_a\); consequently, we are not entitled to regard the quantity \(D\) as a constant when \(V_g\) and \(V_a\) vary.

The structural design of the triode is most often encountered with a cylindrical anode and grid and with the filament along the axis of the cylinder; a plane form is also often encountered (Langmuir’s pliotron, the amplifier tubes of the Nizhny Novgorod Radio Laboratory); above there was already mentioned Wien’s tube with a control electrode in the form of a plate placed on the other side of the filament (with respect to the anode), intended for operation with low voltage on the anode; it is also interesting to mention Vigand’s tube with a control electrode in the form of a cylinder placed over the tube from the outside (complete elimination of the current \(J_g\)).

The calculation of the magnitude of the “permeability” \(D\) or the “voltage amplification coefficient” \(g=\frac{1}{D}\) has served as a subject for the work of many prominent theorists3. For the cylindrical triode we give J. J. Thomson’s formula as the simplest:

\[ g=\frac{\pi d N \lg \frac{D}{d}}{\lg\left(\frac{1}{\pi \delta N}\right)} \tag{6} \]

where \(D\) is the diameter of the anode, \(l\) the grid, \(N\) the number of turns of the grid per \(1\ \mathrm{cm}\); \(\delta\) is the diameter of the grid wire; introducing the magnitude of the distance between the turns of the grid

\[ b=\frac{1}{N}, \]

we obtain:

\[ g=\frac{\pi \dfrac{d}{b}\,l\,\dfrac{D}{d}}{\lg \dfrac{1}{\pi}\dfrac{b}{\delta}} \tag{6'} \]

This formula shows that \(g\) is the greater (\(D\) the smaller) 1) the larger the diameter of the anode in comparison with the diameter of the grid, 2) the larger the diameter of the grid in comparison with the spacing of its turns, i.e. the denser the grid, and 3) the smaller the spacing of the turns in comparison with their diameter \(\left(\dfrac{b}{\delta}\right)\), i.e. the smaller the free opening between the turns of the grid.

For most applications of the triode it is very important that the steepest, working part of the characteristic should lie on both sides of the initial potential of the grid, whereby the greatest amplifying effect, the least distortion of the current during amplification, and easy self-excitation of the generator are attained. If the characteristic lies too far to the right, then the intended aim can be achieved by giving the grid a certain positive potential, but this is very disadvantageous, since a considerable parasitic (useless) current will appear in the grid; therefore it is preferable to make the initial potential of the grid \(0\) or \(-1\) volt and to shift the characteristic to the left, which, as we have seen, can be achieved by increasing the product \(D V_a\), i.e. by increasing either \(D\) or \(V_a\); thus, for example, a shift of \(10\) volt at \(D=5\%\) can be produced by increasing \(V_a\) by \(200\) volt, or at \(D=10\%\), by increasing \(V_a\) by \(100\) volt.

Let us determine at what grid potential \((V_g)_o\) the steepest part of the characteristic lies, in the absence of potential on the anode, i.e. with a short circuit between anode and cathode (strictly speaking, Langmuir’s law in this case is no longer applicable to a single \(J_a\), but only to the total current \((J_a+J_g)\)); it is obvious that at \(V_a=0\) the characteristic will begin at \(V_g>0\) and will reach the saturation current \(J_s\) at some positive potential on the grid; let us assume that \((V_g)_o\) will correspond to one half of the saturation current \(\dfrac{J_s}{2}\). On the basis of Tables II and III, the saturation currents for a filament of diameter \(0.125\ \mathrm{mm}\) will be as follows:

\[ \text{weak incandescence } 2300^\circ \;—\; J_s=0.006\ \frac{\mathrm{Amp}}{\mathrm{cm}} \]

\[ \text{strong incandescence } 2500^\circ \;—\; J_s=0.200\ \frac{\mathrm{Amp}}{\mathrm{cm}} \]

Applying formula (5), we obtain that:

\[ (V_a)_0=\left(\frac{J r_0}{214{,}65\cdot 10^{-6}}\right)^{2/3} \]

Assigning various values to \(r_0\), we obtain the following table:

Table IV.*

\(r_0\), cm.	\((V_a)_0\) Volt. 2,300°	\((V_a)_0\) Volt. 2,500°
0.1 cm.	7.4	21.5
0.2	11.7	34.0
0.3	15.5	45.0
0.5	22	63.0
1.0	35	100.0

A French amplifying tube, having \(r_0=0.2\) cm and \(D=0.15\), at weak incandescence requires a shift of the characteristics by 11.7 volt, which is obtained at

\[ V_a=\frac{11.7}{D}=\frac{11.7}{15}\,100=78\ \text{volt}; \]

as is known, this voltage is usually used. Obtaining potentials of such magnitude is accompanied by considerable difficulties (a large battery weight, the difficulty of charging them), and therefore many attempts were made to construct triodes giving a considerable amplifying action at a smaller anode potential; for this purpose M. Wien constructed a tube, already mentioned, having \(D=1\); it is clear that for this tube the anode voltage which will produce a shift of 11.7 volt will be equal to

\[ \frac{11.7}{1}=11.7\ \text{volt}; \]

indeed, these tubes, with 10–12 volt on the anode, give good amplification.

For a large generator tube with \(r_0=1\) cm we obtain the necessary shift of 100 volt (strong incandescence) at

\[ V_a=\frac{100}{D}; \]

* Jahrb. Drahtl. Tel. 15, S. 27, 1920.

if

\[ \begin{aligned} D&=10\% \;-\; V_a = 1{.}000\ \text{volt}.\\ D&= 4\% \;-\; V_a = 2{,}500\ \text{volt}.\\ D&= 2\% \;-\; V_a = 5{.}000\ \text{volt}.\\ D&= 1\% \;-\; V_a = 10{.}000\ \text{volt}. \end{aligned} \]

Thus it is clear that, at high anode voltages, one has to use a small \(D\), i.e. a dense grid, which is what we always see in generator tubes designed for high power.

The steep, rising part of the characteristic has an almost strictly rectilinear form; as Eccles (loc. cit.) has shown, this form is quite compatible with Langmuir’s law if one takes into account the nonuniform temperature of the filament owing to the cooling of its ends: when the grid potential is increased, only the middle part of the filament follows the \(3/2\) law, while the ends give a saturation current.

The rectilinearity of the characteristic allows it to be expressed in a simple mathematical form, convenient for approximate calculations (for changes of potential small enough not to pass beyond the limits of the rectilinear part):

\[ J_a = aV_a + bV_g + c^{1}), \]

where \(c\) is a small quantity giving the current at \(V_a = V_g = 0\), depending on the fact that the electrons have a certain initial velocity different from zero.

Let us determine the meaning of the coefficients \(a\) and \(b\). In the most general form the equation of the characteristic may be represented as

\[ J_a = F(V_a, V_g). \]

The increment of the current \(dJ_a\) is related to the increments of the anode and grid potentials by the following relation:

\[ dJ_a=\left(\frac{\partial J_a}{\partial V_a}\right)_{v_g} dV_a +\left(\frac{\partial J_a}{\partial V_g}\right)_{v_a} dV_g \]

where \(\left(\dfrac{\partial J_a}{\partial V_a}\right)_{v_g}\) gives the change of anode current when \(V_a\) is changed by one volt at constant \(V_g\), and \(\left(\dfrac{\partial J_a}{\partial V_g}\right)_{v_a}\) gives the change of anode current when \(V_g\) is changed by one volt at constant \(V_a\); it is clear that both these partial derivatives have the dimension of conductance, or \((\text{resistance})^{-1}\); Barkhausen (loc. cit.) introduces for the second of them the notation \(S\).

\(^{1}\) Vallauri. Jahrb. Drahtl. Tel. B. 12, S. 349, 1917.

(steepness), it characterizes the steepness of the characteristic at the given point, and \(\frac{1}{R_i}\) for the first, where \(R_i\) characterizes the internal resistance of the tube, for at \(V_g=\text{const}\) and with the anode circuit short-circuited,

\[ dJ_a=\frac{dV_a}{R_i}. \]

The quantity \(S\) is found from the characteristics geometrically as the \(\tg\) of the angle of inclination of the curve at the given point to the axis of abscissae; in amplifier tubes of the French type \(S\) has, in the steepest part, a value of the order of \(4\cdot 10^{-4}\,\frac{\mathrm{amp}}{\mathrm{volt}}\); in German ones—of the order of \(1\cdot 10^{-4}\,\frac{\mathrm{amp}}{\mathrm{volt}}\); \(R_i\) can likewise be found for each \(V_g\) geometrically, if there are two characteristics taken at different anode potentials, as is shown, for example, in Fig. 3 by the dashed curves; in amplifier tubes of the French type \(R_i \simeq 30{,}000\,\Omega\), in German tubes with a flat grid and anode \(R_i \simeq 100{,}000\,\Omega\). Experimentally \(R_i\) can be measured by means of a Wheatstone bridge, connecting the tube as one arm of the bridge.

The partial derivative

\[ -\left(\frac{\partial V_a}{\partial V_g}\right)_{J_a} \]

will show by how many volts \(V_a\) must be decreased when \(V_g\) is increased by one volt, in order that the anode-current strength remain unchanged, i.e., in other words, how many times greater a change of the anode potential is equivalent to the given change of the grid potential; this quantity is customarily called, as was already said above, the “voltage amplification factor” and denoted by the letter \(g\); the reciprocal quantity \(\frac{1}{g}\), for large \(g\), is sufficiently accurately equal to the “permeability” of the grid:

\[ g=-\left(\frac{\partial V_a}{\partial V_g}\right)_{J_a} \quad \text{and} \quad D=-\left(\frac{\partial V_g}{\partial V_a}\right)_{J_a}; \]

Taking into account the known relation between partial derivatives:

\[ \left(\frac{\partial J_a}{\partial V_a}\right)_{v_g} = -\left(\frac{\partial J_a}{\partial V_g}\right)_{v_a} \left(\frac{\partial V_g}{\partial V_a}\right)_{v_a}, \]

we find the relation between the quantities \(S\), \(R_i\), and \(D\) or \(g\):

\[ \begin{aligned} 1/R_i &= SD \quad \text{or} \quad SDR_i=1,\\ 1/R_i &= \frac{S}{g} \quad \text{or} \quad g=SR_i, \end{aligned} \tag{6} \]

The rectilinear part of the characteristic, in Barkhausen’s notation, can be expressed by the equation:

\[ J_a = SV_g+\frac{1}{R_i}V_a+c, \tag{7} \]

where \(S\) and \(R_i\) have a constant value throughout the entire rectilinear part (it is precisely these values that were given above).

For small oscillations of the variable voltages on the grid and anode we obtain, denoting by lowercase letters \(i_a\), \(v_g\), and \(v_a\) the amplitudes of the corresponding variable quantities, the following relation:

\[ i_a = S v_g + \frac{1}{R_i} v_a \tag{7'} \]

Langmuir’s equation (4) applies only to a purely electronic process, for which a very perfect vacuum is an essential condition. In the presence of traces of gas, electrons having sufficient velocity can ionize gas molecules, as a result of which, on the one hand, the number of carriers of electric current increases, and, on the other, the negative space charge is neutralized, which in general leads to a considerable increase in the current in the anode circuit; when ionization appears, the \(3/2\) law, of course, ceases to hold. Since the ionization potentials of gases do not exceed 25 volt, in gas-filled tubes there is always observed a sharp increase of the anode-current strength at potentials \(V_g\) exceeding several tens of volts and, accordingly, a very steep characteristic (large \(S\)). Although a steep characteristic makes it possible to obtain large amplification, gas-filled tubes, despite this advantage, prove to be very unstable in operation, since their regime, first, depends on temperature and, second, is subject to gradual changes over time (hardening or softening); thus, for example, tubes that work well in a warm room refuse to operate in frost or when raised in an airplane.

Sudden breaks in the course of the characteristic are explained, as was stated earlier, by the stopping of electrons that have attained sufficient velocity to excite gas molecules to luminescence (resonance potential); these breaks in the characteristic are successfully used for detector action (see below).

The presence of gas in a tube can be detected by the following method: if a considerable negative potential is applied to the grid and a positive potential to the anode, then in an evacuated tube we shall obtain no current in the grid circuit; in the presence of gas, a small number of electrons that have passed by the negatively charged grid immediately enter the field of the anode, accelerate their motion, and can produce ionization; positive ions are directed toward the grid and give a current \(J_g\); thus, if at negative grid potentials a current of the reverse direction flows in it, this serves as an indication of the presence of gas. Since in ionization one \(+\) ion and one electron are formed, which will move toward the anode, the anode current increases by a magni-

grid \(J_g\); the current due to the primary electrons will be equal to \(J_a - J_g\). The ratio

\[ \frac{J_g}{J_a - J_g} \simeq \frac{J_g}{J_a} \]

will be proportional to the number of ions formed per one primary electron per unit time; if the length of the path of the primary electrons along which they can produce ionization is denoted by \(d\) \((d = r_a - r_g)\), then the quantity \(\dfrac{J_g}{d J_a}\) will be proportional to the number of ions formed in 1 sec. over 1 unit length of the path of the primary electron; this quantity will be proportional to the gas pressure, or, conversely: \(p = k \cdot \dfrac{J_g}{d J_a}\), where \(k\) is an empirical constant depending on the nature of the gas; as the pressure increases, \(k\) decreases slightly.

Kaufman and Serov3 give a method for the absolute determination of the quantity \(k\) from the geometrical dimensions, the potentials of the grid and anode, making use of a certain function which gives the number of ions per 1 cm of the electron path as a function of the potential difference through which it has passed; this function has a different form for each gas and was determined for various gases by Mayer. When a sensitive galvanometer is used in the grid circuit, it is possible to measure discharges down to \(10^{-6}\) mm. In Telefunken generator tubes the authors found pressures of the order of \(10^{-6}\) mm.

So long as comparatively thin filaments and small heating currents were used in cathode tubes, in calculations it was possible to neglect the magnitude of the magnetic field of the heating current. As Hull1 has shown, with thick wires—cathodes (up to 1 cm in diameter), used in powerful rectifiers and generators, the magnetic field of the heating current produces a strong disturbing action on the flight of the electrons and may force them, after describing some curve, to return back to the cathode without reaching the anode; for each heating current \(J_k\) there exists a definite critical potential at the anode, below which the anode current is 0. At \(T = 2500^\circ\) and a cathode-wire diameter of \(0.25\) mm the critical potential is small:

\[ \begin{aligned} &\text{for a diameter of } 5\ \text{mm} \;—\; \text{it is } 10{,}000\ \text{volt},\\ &\text{〃}\quad\text{〃}\quad 10\ \text{mm} \;—\; \text{it is } 100{,}000\ \text{volt}. \end{aligned} \]

On this principle Langmuir constructed a new system for generating high-frequency currents, in which the “control” current is the heating current.

III. DETECTOR ACTION OF THE CATHODE TUBE.

Spark radio stations give radiation in the form of damped oscillations; one damped train follows another with a certain audio frequency. High-frequency oscillations are not perceived directly by the ear, and in order to perceive them aurally they are rectified by means of various devices called detectors, as a result of which from each train of oscillations there is obtained a series of one-sided impulses, following one another at a high radio frequency and merging in the telephone into one common impulse that bends the telephone diaphragm; since the trains follow one another at regular intervals of time, the rhythmic impulses in the telephone together give a musical note, with a frequency equal to the number of damped trains emitted by the transmitter in one second.

The simplest among the large number of detectors is undoubtedly the crystal detector, which is a contact of a metal with a crystal (copper wire—galena, carborundum—steel plate) or of two different crystals (zincite–chalcopyrite, etc.); the weak side of crystal detectors is their instability—mechanical and electrical shocks (for example, atmospheric discharges), even weak ones, can completely disrupt the operation of the detector and require its readjustment. Fleming proposed using for purposes of detection the one-sided conductivity of a two-electrode tube with rarefied gas (Fleming valve); he discovered that certain points of the characteristic curve, in which there are sharp bends, are especially sensitive for detection.

A triode can also serve as a detector in any part of its characteristic except the rectilinear one; for example, in the initial part of the characteristic (Fig. 3), when a sinusoidal voltage of small amplitude is applied to the grid, in the anode circuit there is obtained a greater increase of current in the case of the positive half-wave than the decrease in the case of the negative half-wave; the mean current in the anode circuit, when a train of oscillations is applied, will increase.

As is easy to see, expanding the equation of the characteristic \(J_a = F(V_g)\) in a Taylor series and taking the mean value, the increment of current in the anode circuit when oscillations with amplitude \(v_g\) are applied to the grid will be equal to:

\[ \Delta J_a = \frac{\partial^2 J_a}{\partial V_g^2}\,\frac{v_g^2}{4} \tag{8} \]

The condition for detector action, therefore, is \(\dfrac{\partial^2 J_a}{\partial V_g^2} \gtrless 0\); the first condition will be satisfied on the lower, initial part of the characte-

characteristic, the second—on its upper bend as it passes to the saturation current; the second case of detection is of little advantage, since it is obtained with a considerable additional potential on the grid, which causes a parasitic current in it.

A much stronger action is produced by the detector circuit first used by de Forest in his “Audion”; it may be called, by the breadth of its application, the basic detector circuit. The chief role in this circuit (Fig. 4) is played by a small capacitor \(C_g\) of \(50\text{–}100\ \mathrm{cm}\), connected in the grid circuit and shunted by a large ohmic resistance \(R_g\) of several megohms; the grid has an initial potential of \(0\). The incoming oscillations excite the \(LC\) circuit and cause an alternating voltage to appear on the grid; in view of the fact that, as the potential increases, the current in the grid rises sharply (according to the law \(e^{\frac{v_g}{v_0}}\), according to Barkhausen), during the \((+)\) half-wave on the grid a considerable stream of electrons reaches it, charging it negatively; during the \((-)\) half-wave the current in the grid is small and almost no change of potential occurs.

Fig. 4. Audion circuit.

A series of oscillations gradually charges the grid to the maximum negative potential at which all current in it ceases—that is, approximately to \(-1\) volt (Fig. 5); the resistance \(R_g\) serves so that the charge of the grid can gradually leak off from it, and by the beginning of the next series the grid is again ready to receive oscillations. The telephone connected in the anode circuit registers only the average variations of the strength of the anode current, which have audio frequency. The detector action according to the audion circuit is considerably greater than according to the circuit without a capacitor; the inclusion of a special resistance can often be avoided, since sufficient leakage is provided by the lamp base and by the insulation of the capacitor. The resistance \(R_g\) is usually made in the form of a strip of paper coated with India ink or graphite, the ends of which are pressed by metal plates.

The triode detector circuit, when acted upon by undamped oscillations, gives a certain constant decrease in the current strength in the circuit:

the anode, which will remain completely imperceptible when listening by telephone. In order to make the undamped oscillations audible, it is necessary to break them up in some way into parts; this is often done by means of a purely mechanical interruption, but a far more convenient method is to superpose upon the receiving circuit local undamped oscillations of small amplitude with a frequency different from that of the received signals. The incoming oscillations with frequency \(N\) give, with the local frequency \(N'\), beats with a frequency equal to the difference \(N - N'\), which can easily be chosen so that it lies within the limits of the ear’s greatest sensitivity. A local generator (Ueberlagerer, heterodyne) is also easily implemented with the aid of a cathode lamp (see below). An even more convenient construction can be made by causing the detector tube simultaneously to generate oscillations in the receiving circuit, tuned to a frequency slightly different from the received frequency; then the incoming waves excite forced oscillations in the receiving circuit, which, as has already been explained above, give beats with the local oscillations and, in this way, can be heard: this arrangement is known as the “autodyne” or “oscillating audion.”

Fig. 5. Sensitivity of a detector with mercury vapor.

Very advantageous conditions for detection are provided by gas-containing tubes, because their characteristic has points with sharp rises and bends. Fig. 52 gives the characteristic of a tube containing mercury vapor. The detecting action of a gas tube will depend on the initial potential of the grid, which determines the position on the characteristic, and on the magnitude of the amplitude; in Fig. 5 the magnitude of the detecting action is shown at various points of the characteristic. At the present time many firms manufacture special detector tubes containing, for the most part, mercury vapor.

VI. The Cathode Tube as an Amplifier

Before the invention of the cathode tube, the problem of amplifying weak alternating currents was almost insoluble, and here the situation was much worse than with signals supplied by direct current, where, by means of a relay, it was possible to achieve the detection and manifestation, on an enlarged scale, of the action of extremely weak currents. The cathode tube made it possible to make a relay for alternating current in the sense that the amplification of oscillations is obtained at the expense of some source of energy, the expenditure of which is under the control of the oscillations being amplified.

The idea of using the cathode tube as an amplifier is explained in Fig. 6. Weak alternating voltages (for example, from a telephone line) are applied between the cathode and the grid through a transformer (or directly) and produce considerable oscillations of current strength in the anode circuit, which can be detected by inserting a telephone into it (directly or through a transformer); in order that there should be no parasitic current in the grid circuit, a small additional voltage \((V_g)\) may be included.

Fig. 6. Triode amplifier.

For small amplitudes, the alternating current \(i_a\) arising in the anode circuit (in what follows we shall denote the amplitudes of alternating quantities by lower-case letters) will be determined by equation (7).

\[ i_a = S v_g + \frac{1}{R_i} v_a . \]

A change in the anode voltage \(v_a\) will appear as a result of a change in the current strength in the anode circuit if some resistance \(r_a\) is included in it [by \(r_a\) we shall henceforth understand the total resistance to alternating current—impedance, which in the particular case may be ohmic—\(R_a\)] and will be equal to:

\[ v_a = - i_a r_a . \]

Substituting this expression into formula (7) and determining \(i_a\) from it, we obtain:

\[ i_a = \frac{v_g S R_i}{R_i + r_a} = \frac{v_g}{D}\,\frac{1}{R_i + r_a} = \frac{g v_g}{R_i + r_a} \tag{9} \]

This formula, according to Barkhausen, was first derived by Schottky (earlier, in 1919); in the English literature it was derived by Belti (1919) and is called “Belti’s theorem.” This formula, which is fundamental for the theory of amplifying action, states that the triode acts in the anode circuit as an alternating-current generator with electromotive force \(gv_g=\dfrac{v_g}{D}\), having an internal resistance \(R_i\); with such an interpretation of the question the term “voltage amplification coefficient” (denoted by the letter \(g\)) becomes quite obvious2.

For a purely ohmic resistance \(r_a=R_a\) we obtain the formula:

\[ i_a=\frac{v_g}{D}\frac{1}{R_i+R_a}=\frac{g v_g}{R_i+R_a} \tag{9'} \]

Remaining on the basis of the geometrical picture, we can represent Eq. (9′) graphically in the form of a certain characteristic for alternating current, which in the case of small amplitudes will obviously be a straight line passing through the origin and having the slope:

\[ S'=\left(\frac{di_a'}{dv_g}\right)_{v_a} =\frac{1}{D(R_i+R_a)} =\frac{1}{DR_i\left(1+\frac{R_a}{R_i}\right)} =\frac{S}{1+DSR_a} \tag{10}, \]

i.e. the slope of the operating characteristic is reduced by a factor of \(1+DSR_a\) in comparison with the original one.

In the general case \(r_a\) may consist of a combination of an ohmic resistance \(R\), a capacitance \(C\), and a self-inductance \(L\); in the case of an oscillatory circuit (see Fig. 8), inserted, for example, in the anode circuit of the generator, we shall have:

\[ r_a= \frac{(i\omega L+R)\dfrac{1}{i\omega c}} {i\omega L+R+\dfrac{1}{i\omega c}} \tag{11} \]

where \(i=\sqrt{-1}\); \(\omega=\dfrac{2\pi}{T}\) is the (angular) frequency of the alternating current excited in the anode circuit. As is not difficult to prove, in this case the operating characteristic will have the form of an ellipse about the origin, described counterclockwise if the current in the anode circuit leads the voltage on the grid (capacitance predominates), and clockwise if the current lags (self-inductance predominates). In the case of resonance \(\omega=\dfrac{1}{\sqrt{LC}}\)

CATHODE TUBES AND THEIR APPLICATIONS

and for a small resistance \(R\) one obtains approximately:

\[ R_a \simeq \frac{L}{CR}, \tag{12} \]

i.e. the oscillatory circuit is equivalent to a certain ohmic resistance, and there is no phase shift between \(v_g\) and \(i_a\).

On the basis of elementary calculations, from formula (9′) we shall find that the greatest power will be released in the anode circuit under the condition \(R_a=R_i\); telephones are made with a resistance rarely greater than 4,000 ohms; in amplifier tubes of the German type \(R_i=100{,}000\) ohms, and connecting a telephone directly into the anode circuit proves to be of little advantage (only \(1/7\) of the maximum energy is utilized for \(R_a=4{,}000\,\Omega\), and \(1/25\) for \(R_a=1000\,\Omega\)); in the French amplifying tube \(R_i=30{,}000\,\Omega\), and for \(R_a=4{,}000\,\Omega\) approximately \(3/8\) of the maximum energy is utilized. In view of this it is more advantageous to take off the current by means of a transformer lowering the voltage several times; then a telephone with resistance \(R_a\) in the secondary winding will give in the primary an apparent resistance \(u^2 R_a\) (where \(u\) is the transformation coefficient), which can be made equal to \(R_i\) and the maximum energy obtained from the tube. It must be taken into account that every transformer gives losses, and therefore its use when \(R_a\) differs from \(R_i\) by less than 5–6 times has no sense; in French tubes very often no output transformer is used. The situation is otherwise if an instrument with a small resistance must be connected (e.g. a string galvanometer, an oscillograph); then the use of a transformer is advantageous; in this latter case it is preferable to use tubes with small internal resistance (Wien’s tube).

In the case of amplification of high-frequency oscillations the capacitance \(C_{ak}\) between anode and cathode creates a certain shunt resistance to the receiving apparatus; taking \(C_{ak}=10\) cm, we find that at a wavelength

\[ \lambda = 300\ \text{m}, \]

i.e. frequency \(10^6\), this shunt will be equal to

\[ \frac{1}{C\omega}=15{,}000\ \text{ohms}; \]

for \(R_a=R_i\) (the most advantageous case), for the French tube \(R_i=30{,}000\,\Omega\), only \(1/3\) of the entire current enters the indicator; in this case also, tubes with small internal resistance will give a considerable gain.

Let us now examine the question of the most advantageous method of applying current to the grid. We shall regard as the best circuit that which gives the highest voltage on the grid \(v_g\); when the grid potential is below 1 volt, the current in it ceases completely, and the resistance of the grid–cathode gap may be considered equal to infinity; of course, there is always some insulation resistance \(R_g\), which, in parallel with the capacitance \(C_{gk}\), gives a certain “effective resistance” of the grid \(r_g\). The alternating voltage \(v_g\) on the grid is the greater, the greater \(r_g\) is (since the voltage—

voltage is equal to \(i_g r_g\); but increasing \(r_g\) too much proves disadvantageous, since in that case the amplifier becomes excessively sensitive to various extraneous influences (induction from wires, lightning discharges), and for the most part the grid is deliberately shunted by a large resistance of the order of several megohms. The greatest voltage on the grid will be obtained when the external resistance is equal to the resistance of the grid; if the external resistance is small (a telephone line), it is advantageous to install an input transformer that raises the voltage \(u\) times; then the “effective resistance” in the secondary winding will be \(u^2\) times greater and, by choosing \(u\), it can be made comparable with the “effective resistance” of the grid \(r_g\); it should not be forgotten that the transformer also gives losses here, so that its use makes sense only when the external and internal resistances differ greatly. In the case of tuning to resonance with the incoming oscillations, the transformer gives a considerable increase in voltage; for sharpness of resonance \(R_g\) must be large.

The amplifying capacity of a cathode tube is naturally characterized as the ratio of the maximum power delivered in the indicator \((N_v)_{max}\) to the supplied unamplified power \(N_u\). Barkhausen calls the quantity

\[ W=\sqrt{\frac{(N_v)_{max}}{N_u}} \]

the linear amplification.

Let us find the magnitude of the amplification as a function of the quality of the tube and of the circuit.

The power delivered in the anode circuit is equal to:

\[ N_a=\frac{i_a^2 r_a}{2}, \]

from formula (9) we shall find the greatest power

\[ (N_a)_{max}=\frac{v_g^2}{S D^2 R_i}=\frac{1}{4}\,\frac{S}{D}(v_g)^2_{\mathrm{eff}}, \quad \text{putting } r_a=R_i . \]

Barkhausen calls the quantity \(\dfrac{S}{D}\) the “goodness” of the tube (Güte); it is equal to four times the greatest power delivered in the anode circuit when the alternating effective voltage on the grid is equal to 1 volt. Denoting below the efficiency of the output transformer by \(\eta_a\), we shall find the greatest power delivered in the indicator:

\[ (Nv)_{max}=\eta_a\,\frac{1}{4}\,\frac{S}{D}(v_g)^2_{\mathrm{eff}} . \]

CATHODE LAMPS AND THEIR APPLICATIONS

The power expended in the grid circuit will be equal to:

\[ N_g=(v_g)_c(i_g)_c=\frac{(v_g)^2_{eff}}{r_g}; \]

if the efficiency of the input transformer is \(\eta_g\), then the supplied power is

\[ N_u=\frac{N_g}{\eta_g}=\frac{(v_g)^2_{eff}}{\eta_g r_g}; \]

substituting the expressions obtained, we obtain the following formula for the linear amplification:

\[ W=\sqrt{\frac{(N_g)_{max}}{N_u}} =\frac{1}{2}\sqrt{\eta_g\eta_a}\cdot\sqrt{\frac{S}{D}}\cdot\sqrt{r_g} \tag{12} \]

For given efficiencies of the input and output transformers, the amplification is the greater, the greater the quality factor of the lamp and the more perfect the insulation in the grid circuit. Taking

\[ \eta_g\eta_a=40\% \quad \text{and} \quad \frac{S}{D}=10^{-3}\ \frac{watt}{volt^2} \]

for the German lamp; (for the French amplifying lamp

\[ \frac{S}{D}=3\cdot10^{-3}\ \frac{watt}{volt^2}); \]

below we shall give the corresponding data for the French lamp in parentheses, and obtain for the magnitude of the amplification:

\[ W=\sqrt{\frac{r_g}{10.000}}\left(\sqrt{\frac{r_g}{3.300}}\right). \]

If \(r_g<10.000\,\Omega\) \((3.300\,\Omega)\), then \(W<1\), i.e., no amplification will be obtained; at \(r_g=100.000\) \((33.000)\,\Omega\) an amplification of 3.2 times will be obtained; at \(10^6(3.3\cdot10^5)\,\Omega\), an amplification of 10 times, etc.

In amplifying radio frequencies, the leakage to the grid is produced not only by the ohmic resistance \(R_g\), but also by the capacitive one—equal to \(\frac{1}{\omega C_{gk}}\); if \(C_{gk}=10\ \text{cm}\) is assumed, then this capacitive leakage becomes equal to \(10.000\,\Omega\) at \(\lambda=600\ mt\) (for the German lamp), and thus at waves shorter than 600 \(mt\) it is not possible to obtain amplification; working with a transformer tuned to resonance, it is possible to obtain amplification also for shorter waves.

Determining the amplification (linear) is conveniently done according to the scheme proposed by Pirani2: the alternating current of some generator flows successively through certain resistances \(r_1\) and \(r_2\), with \(r_2\) made considerably smaller than \(r_1\); from \(r_1\), through a switch, the current can be fed to the telephone, while from \(r_2\) the current is fed to the amplifier and, after amplification, through a switch to the same telephone. By alternately switching the telephone to the amplified and unamplified current and selecting the resistance values so that the sound intensity does not change, one can easily calculate the voltage amplification for the given frequency. When transformers are used, the amplification at different frequencies is usually far from uniform (they possess sharply resonant properties); moreover, the amplification of small amplitudes is much greater than that of large ones.

By feeding the amplified oscillations back into the grid circuit by means of inductive or capacitive coupling, it is possible to obtain a considerable increase in the amplifying action at the expense of the battery energy in the anode circuit; the limit of amplification is set by the circumstance that, with strong feedback, the tube begins to operate as a generator (see Ch. IV) with a certain natural period, and weak external oscillations are drowned out by strong local ones. This ingenious method was first applied by Armstrong3 and is now widely used in the construction of so-called regenerative receivers for radiotelegraphy; the regenerative method enabled Armstrong to obtain from one tube an amplification of 100 times. Strong feedback, giving large amplification, has the disadvantage that a strong electrical impulse may bring about the excitation of generation, disrupting the operation of the tube as an amplifier; moreover, the oscillations have the property of persisting after the signals have ended.

Mathematically, feedback may be represented as a certain negative resistance introduced into the receiving circuit and reducing its damping to an arbitrarily small value; when the resistance becomes less than zero, self-excitation occurs. The regenerative method is remarkable in that it makes it possible to compensate for all losses in the grid and receiving circuits (due to heating, leakage, hysteresis, etc.) and allows the coils of radio receivers to be made from fine wire, antennas with large resistance and poor insulation to be used, and even coils with iron cores to be employed in receivers.1

The use of feedback at low frequencies causes sounds to drag out (after-sound) and a sharp accentuation of resonant

properties, so that in amplifying telephone currents one has to abandon the regenerative principle.

Various whistles in amplifiers are explained, for the most part, by self-excitation of oscillations in some circuit with a low frequency, caused by the influence of feedback; in order to combat this evil, one has to be satisfied with sensitivity and reduce the resistance in the grid circuit, and, moreover, carefully shield (by metallic, earthed envelopes and by the appropriate arrangement) the parts of the circuit carrying unamplified currents from the amplified currents.

Fig. 7. Three-stage amplifier.

Amplified oscillations from the anode circuit can be transferred to the grid of the second tube, in whose anode circuit further amplification will be obtained (Fig. 7). The transfer may take place either by means of resistances \(R_a\) (from the 1st tube to the 2nd) of large value (of the order of the internal resistance of the tube) through the capacitor \(C''\), or by means of a transformer (from the 2nd tube to the 3rd). The first method is usually used for amplifying radio frequencies and, for low frequencies, in those cases when it is desirable to reduce distortion; this method is disadvantageous in that it requires a large voltage of the anode battery \(V_B\), so that, after subtracting the potential drop across the resistance \(R_a\), a sufficiently high potential remains on the anode. The first and second tubes (Fig. 7) simultaneously play the role of amplifier and detector; in the second tube, to charge the grid with negative potential, a \(-1\) volt cell is inserted in series with a large resistance \(R_i\). It is possible to use feedback from the anode circuit of the second tube to the grid of the first (the possible feedback by means of a capacitor \(C'\) of small ... is shown by a dotted line).

capacitance, which can be varied). In transmission by means of resistances the amplification will be equal to the ratio of the voltage in the anode circuit to the voltage in the grid circuit; but the voltage in the grid circuit of the following tube is equal to the voltage drop across the resistance of the preceding tube, i.e. \((V_a)_1 = (V_g)_2\); substituting the value \(V_a = i_a R_a\) from formula (9′), we obtain the expression for the amplification in passing from the first tube to the second:

\[ W=\frac{(V_g)_{i2}}{(V_a)_1} =\frac{1}{D\left(1+\frac{R_a}{R_i}\right)} =\frac{1}{D+\frac{1}{S R_a}} \tag{13} \]

If \(R_a > R_i\), then \(W=\frac{1}{D}=g\), i.e. the amplification will be equal to the reciprocal of the permeability; for a German amplifying tube \(D=10\%\), i.e. \(W=10\), for a French one \(D=15\%\), i.e. \(W=6\). Double-grid tubes, having small \(D\), give greater amplification in this circuit.

Transmission by means of a transformer is used at low frequencies. Sharp resonance properties can be avoided by tuning the various transformers of the amplifier to different frequencies. The dotted line shows the metallic shielding of the transformers, the input grid circuit, and the output circuit, done in order to avoid self-excitation and the occurrence of whistling.

Besides the distortions introduced by resonance properties, when the amplitude is increased distortions arise caused by the nonlinearity of the characteristic; to eliminate this one has to lengthen the rectilinear part by increasing the filament current and the anode voltage. The telephone diaphragm, having its own period, of course also contributes its share of distortion. All these circumstances make the task of large amplifications, for a loud-speaking telephone, very difficult.

The possibility of amplification with an increase in the number of tubes is theoretically almost unlimited. Schottky2 believes that the limit of amplification is due to the fact that the flight of individual electrons becomes audible. Amplifications of several million times are quite possible. In practice, amplifiers with more than 8–9 tubes are not built, since together with the signals harmful noises are also amplified, making reception extremely unintelligible. Moreover, multistage amplifiers are so sensitive to the action of feedback that the slightest movements near it or touches to parts of the amplifier cause a complete detuning of its operation. Barkhausen gives the following example: self-excitation is easy to obtain with capacitive feedback \(C_{ag}=50\ \mathrm{cm}\); if an amplification of 1000 times is obtained, then a capacitance of \(0.5\ \mathrm{mm}\), i.e. a sphere with a diameter of \(1\ \mathrm{mm}\), will be sufficient to excite generation; from this—

it is clear that displacement of the observer’s body can also cause sufficiently large changes in the mutual capacitances.

Amplifiers with 3–4 tubes are less capricious and more stable in operation, which is why they have become widely used.

Tubes with a double grid2.

As we have seen, the amplification is proportional to the quantity \(\sqrt{\dfrac{S}{D}}\); double-grid tubes precisely make it possible to reduce \(D\) and increase \(S\), which gives greater amplification. For a double-grid tube one may write an equation analogous to (5); if the potential of the 1st grid, nearest to the cathode, is denoted by \(V_{g1}\) and its permeability by \(D_g\), the potential of the second grid by \(V_{g2}\) and its permeability by \(D_a\), then:

\[ J_a = A'\left(V_{g1}+D_g V_{g2}+D_g D_a V_a\right)^{3/2} \tag{14} \]

The incoming oscillations are applied to the first grid; the second grid serves only so that, by its charge, it gives the necessary shift of the characteristics to the left; therefore the permeability \(D_g\) of the first grid is made large (a sparse grid), so that a small positive voltage \(V_{g2}\) on the 2nd grid gives a sufficient shift \(D_g V_{g2}\). The reverse action of the anode voltage \(D_g D_a V_a\) can be made very small if \(D_a\) is made small (a dense grid); in this way one can greatly increase the “merit” of the tube and, at the same time, the amplification it gives; in order that there should be no strong current in the 2nd grid charged with a positive potential, the voltage on the anode should be made several volts higher than on the 2nd grid. Let the shift of the characteristic be 6 volts; choosing \(D_g = 0.3\), we find \(V_{g2}=\dfrac{6}{0.3}=20\) volts; for \(W_a\) a voltage of 30 volts is sufficient; if \(D_a=0.033\) is made, then the permeability of the system of two grids is \(D=D_gD_a=0.3\cdot0.033=0.01\); the internal resistance will be very large. A single-grid tube with \(D=0.01\) would require, for the same shift \(V_a\), a voltage of 600 volts, i.e. 20 times greater. With the same voltage on the anode one can increase the merit, using a double-grid tube, by 20 times, i.e. the amplification by \(\sqrt{20}=4.5\) times; true, in view of the large value of the internal resistance, it is necessary to carry out the corresponding transformation in order to use the greatest power in the anode circuit.

It is possible to use a double-grid tube in another way as well. The first grid is charged with a positive potential of 10 volts, so that the thermionic current reaches almost the saturation value. Taking—

the oscillations being amplified are imposed on the 2nd grid. Between the 1st and 2nd grids a considerable space charge is produced; when the potential of the second grid is negative, the electrons are repelled from it and fall mainly onto the first grid; when the negative potential of the 2nd grid is decreased, the electrons can easily pass to the anode, since, in being distributed between the 1st and 2nd grids, they are subjected to the action of the anode field much more strongly than when grouped near the cathode. Thanks to this, a very large steepness of the characteristic \(S\) is obtained, and only a small displacement of it is required. With a high permeability of the second grid, no large value of anode voltage is needed, so that the first grid and the anode may be charged to the same potential. The internal resistance of this tube is somewhat smaller than in a single-grid tube, while the steepness is almost 10 times greater, which makes it possible to obtain an amplification \(\sqrt{10}=3.2\) times greater, or, in other words, at 10 volt on the anode to obtain the same amplification as at 100 volt with a single-grid tube.

The advantages of tubes of both types can be combined in a three-grid tube.

V. The Cathode Tube as a Generator of Undamped Oscillations.

As was already mentioned in Ch. IV, in the presence of strong feedback between the anode and grid circuits, the cathode tube becomes a generator of undamped oscillations with a frequency corresponding to the natural frequency of the circuits included in the circuit. Fig. 8 shows the simplest circuit of a generator with an oscillatory circuit \(LC\) in the anode circuit, with inductive feedback (coefficient of mutual inductance \(M\)) with the grid circuit. The process of the occurrence of oscillations in this case is explained in general terms as follows: suppose some electrical impulse, say the closing of the battery \(V_B\) or \(V_K\), causes the occurrence of oscillations in the circuit \(LC\); their frequency will be

Fig. 8. Triode generator.

\[ N=\frac{1}{2\pi\sqrt{LC}}, \]

in the absence of the tube they would very quickly cease, owing to the damping of the circuit. If the coupling coil with the grid circuit \(L'\) is sufficiently

is strongly coupled to the anode circuit, and if it is connected in such a way that, when the current in the circuit flows clockwise, a positive potential is induced on the grid, while in the opposite direction a negative potential is induced (for this the coil \(L'\) must, as it were, serve as a continuation of \(L\) in the winding), then under these conditions, owing to the valve action of the cathode tube, the battery gives an impulse agreeing with the direction of the current in the circuit; whereas in the opposite direction, owing to the negative potential of the grid, the battery cannot supply current. Thus, once in every period the battery will reinforce the oscillations that have arisen in the circuit \(LC\), and replenish their energy, in exactly the same way as the spring in a clock does with respect to the oscillations of the pendulum. If the coupling is sufficiently strong, the energy drawn from the battery compensates the natural damping of the circuit, and the oscillations become undamped. The strength of the battery impulses will depend primarily on the battery voltage \(V_B\), on the filament current \(J_k\), and, of course, on the coefficient of coupling between the anode and grid circuits

\[ k=\frac{M^{2}}{LL'} . \]

If the coupling is insufficient for generation, the oscillations, once they have arisen in the circuit, will indeed decay, but with a smaller decrement, as though the tube were introducing into the circuit a certain negative resistance; in this case, when weak external oscillations are applied, they produce a considerable current in the circuit, since the circuit can be made (by means of the tube) almost devoid of resistance—something which is used, as was indicated above, in regenerative receivers.

With strong coupling, the oscillations, once they have arisen, begin to grow (the damping of the circuit becomes, as it were, \(<0\)); the amplification of the oscillations causes an increase in the amplitude of the oscillations on the grid, and this in turn strengthens the impulses supplied by the battery, and hence also the strength of the oscillations in the circuit; the limit of amplification of the oscillations is set by the saturation current, since in this case an increase in the amplitude of the voltage oscillations on the grid \(V_g\) will not cause an increase in the battery impulses. In addition to the basic circuit of Fig. 8, one may also imagine an innumerable multitude of circuits with oscillatory circuits in the grid and anode circuits, with capacitive, inductive, or mixed feedback, which in particular cases have advantages over the simplest circuit.

A complete theory of the cathode generator presents considerable mathematical difficulties, since the characteristic equation is difficult to represent in full by any single mathematical law. Under the simplifying assumption of a straight-line characteristic, Balauri2 gave a theory of the generator leading to the solution

systems of linear differential equations with constant coefficients; the solution of this system leads to an expression for the current strength and voltage in the oscillatory circuit in the form of a damped oscillation with period \(T=2\pi\sqrt{LC}\) and damping decrement:

\[ \delta= \frac{\dfrac{L}{R_i}+RC-MS}{2LC} \]

where \(R\) is the ohmic resistance, \(L\) the self-inductance, \(R_i\) and \(S\) the internal resistance and steepness of the tube; the remaining notation is clear from Fig. 8.

In order that the oscillations be undamped, the necessary condition is: \(\delta \leq 0\), or

\[ \frac{L}{R_i}+RC\leq MS \tag{15} \]

This condition is nothing other than the condition for self-excitation of the generator. Condition (15) can also easily be obtained from other considerations. Suppose that, during generation, a weak alternating current \(i_a\) circulates in the anode circuit (in addition to some direct current); then an alternating voltage \(v_g\) will be induced on the grid, determined from the relation \(\dfrac{v_g}{v_l}=\dfrac{M}{L}\), where \(v_l\) is the potential difference at the ends of the coil \(L\), equal to \(v_l=i_a r_a=i_a\dfrac{L}{CR}\) (see formula 12); thus

\[ v_g=i_a r_a\frac{M}{L}=i_a\frac{M}{CR}. \]

This alternating voltage on the grid will, in turn, produce in the anode circuit an alternating current of strength \(i_a'=S'v_g=S'i_a\dfrac{M}{RC}\), where \(S'\) is the steepness of the oscillatory characteristic (see formula 10), equal to:

\[ S'=\frac{S}{DR_i\left(1+\dfrac{r_a}{R_i}\right)}. \]

It is quite clear that, in order that the weak oscillations which have arisen should not die out, it is necessary that

\[ i_a'\geq i_a \quad \text{or} \quad S'\geq \frac{RC}{M}, \]

whence, substituting the value of \(S'\) and \(r_a\) (formula 12),

\[ \frac{L}{R_i}+RC\leq MS; \]

i.e. Barkhausen’s condition.

If we draw through the origin of coordinates a straight line at the angle

\[ \alpha=\operatorname{arctg}\frac{CR}{M}, \]

characterizing the feedback, then the condition for self-excitation will be that this straight line lie below the characteristic, i.e., that \(\tg\alpha<S'\); if \(\tg\alpha=S'\), i.e. the feedback line is inclined to the abscissa axis at the same angle as the characteristic, the oscillations will remain of unchanged strength; if \(\tg\alpha>S'\), the oscillations will die out, i.e. self-excitation will not occur.

For stronger oscillations the use of a rectilinear characteristic is not legitimate; increasing the amplitude \(v_g\) beyond the value corresponding to the onset of the saturation current does not increase the current strength, and the oscillatory characteristic becomes parallel to the abscissa axis (Fig. 9); at still larger potentials \(v_g\), when \(v_g>V_a\), a considerable current appears in the grid circuit, and, moreover, the phenomenon of secondary cathode rays becomes sharply manifest, which in general leads to a strong decrease in the current strength in the anode circuit and to a decrease in the amplitude of the alternating component of the anode current \(i_a\)—the oscillatory characteristic bends toward the abscissa axis2. A generator operating at grid voltages smaller than the onset of the fall of the anode current is called “undervolted” (unterspannt), and at larger voltages—“overvolted” (überspannt). If the characteristic is shifted somewhat to the left (small anode voltage, dense grid), or if an additional negative voltage is applied to the grid, then the initial part of the oscillatory characteristic (for small amplitudes) rises less steeply than the further part (for large amplitudes); precisely this case is shown in Fig. 9. With a sufficient shift, the oscillatory characteristic has the greatest steepness immediately from zero.

Fig. 9. Oscillatory characteristic

The oscillatory circuit in the anode circuit has the resistance:

\[ R_a=\frac{L}{RC}, \]

the greater this resistance, the lower the maximum reached by the oscillatory characteristic.

If one draws on the graph (Fig. 9) the feedback line:

\[ a=\operatorname{arctg}\frac{CR}{M}, \]

then, for grid-voltage amplitudes smaller than that corresponding to point \(A\), the oscillations that arise will die out \((i_a<i'_a)\), i.e. a generator with such a characteristic as in Fig. 9 will not self-excite; it is precisely this kind of characteristic that it is desirable to have for a regenerative receiver. At amplitudes \(v_g\) corresponding exactly to point \(A\), the oscillations will be undamped but unstable; an accidental decrease of the amplitude \(v_g\) will lead to damping, an increase—to growth of the oscillations. A stable regime will be established only at amplitudes \(v_g\) corresponding to point \(B\). The ordinate of point \(B\) will give us the maximum amplitude of the alternating current in the anode circuit for the given feedback.

A generator with a characteristic similar to Fig. 9 does not excite itself, but it can be excited by a strong impulse; such generators were often encountered in the earliest designs (the ROBT and T generator, the receiver of the Tver radio station), and special methods were recommended for their excitation, for example, opening and closing the battery \(V_B\) or increasing the filament current.

As we see, the greatest amplitude of oscillations will be obtained if the feedback is chosen so that the line

\[ a=\operatorname{arctg}\frac{RC}{M} \]

intersects the oscillation characteristic at the highest point; in this case the oscillation of the current in the anode circuit \(J_a\) will take place from the value of the saturation current \(I_s\) to \(0\), i.e. the amplitude of the alternating component will be \(\frac{I_s}{2}\); the oscillations of the voltage on the anode will then take place from the value \(2V_B\) to \(0\), i.e. the amplitude of the alternating voltage will be \(V_B\); these considerations will serve us further for calculating the power of the generator.

The current in the oscillatory circuit \(i_L\) can be easily calculated if the current \(i_a\) in the anode circuit is known (its alternating component); the potential drop on the coil \(L\) due to the current in the circuit will be equal to \(i_L L\omega\), where \(i_L\) is the amplitude of the current in the circuit; it must be equal to the potential drop occurring when the current \(i_a\) passes through the circuit:

\[ i_L L\omega=i_a\frac{L}{RC}; \]

whence:

\[ i_L=i_a\frac{L\omega}{R}=i_a\frac{1}{RC\omega}=\frac{\pi}{d}\,i_a, \tag{16} \]

where \(d\) is the logarithmic decrement of damping of the circuit \(\dfrac{R}{2LT}\). For constructing a generator it is advantageous to take circuits with small damping; with \(d=0.01\), we obtain

\[ i_L=i_a\times 100. \]

We see that in this typical case of current resonance, in the circuit one can obtain a considerable current with a very weak feeding alternating current \(i_a\); when the circuit operates into some load, this will, of course, be equivalent to introducing an ohmic resistance into the circuit, and the amplitude \(i_L\) will decrease.

Having the oscillatory characteristic, it is easy to construct the curve of the distribution of current in the anode circuit \(I_a\) over the time of a period. At small voltages on the grid, the oscillatory characteristic is close to a straight line2, and the anode current will be expressed almost exactly by a half-sinusoid during the time \(+v_g\) and will be equal to 0 during the time \(-v_g\). When the saturation current is reached, the distribution of \(I_a\) will be expressed by a half-sinusoid with its top cut off (approximately a trapezoid); with overvoltage, a dip forms at the top of the trapezoid, caused by the transfer of current from the anode circuit into the grid circuit.

Knowing the magnitude of the current \(I_a\) as a function of time, one can calculate the power expended over the period \(T\) by the battery:

\[ N_B=V_a\frac{1}{T}\int_0^T I_a\,dt. \tag{17} \]

In order to find the power in the oscillatory circuit

\[ N_a=\frac{i_a v_a}{2}=\frac{i_a^2 r_a}{2}=i_a^2\frac{L}{2CR}, \tag{18} \]

we must know the amplitude of the alternating component of the anode current \(i_a\); disregarding the action of overtones (calculation shows that they are very weak), we shall regard as \(i_a\) the coefficient of the first variable term in the expansion of the current \(I_a\) in a Fourier series; the coefficients of the following terms will give the amplitudes of the overtones.

Having found \(N_a\) and \(N_B\), one can calculate the efficiency of the generator

\[ \eta=\frac{N_a}{N_B}. \tag{19} \]

Such a calculation was first made by Meissner2, the inventor of the self-excited generator; for a 20-watt lamp the efficiency proved, according to his calculations, to be equal to 56%.

Using the method of oscillatory characteristics, one can exactly solve the question of the greatest useful power \(N_a\) that can be obtained at a given direct-current voltage \(V_B\). For this purpose one must find the maximum of the function \(N_a=\dfrac{i_a v_a}{2}\) as a function of two variables \(r_a\) and \(v_a\); geometrically, \(N_a\) is represented by a certain surface, the highest point of which will give us the value \((N_a)_{\max}\); the values \(r_a=\dfrac{L}{RC}\) and \(v_a\) for this point will indicate to us which circuit should be included and what feedback should be taken in order to obtain this greatest power.

It may be assumed (approximately)3 that the greatest power is obtained when \(i_a=\dfrac{I_s}{2}\) and \(v_a=V_B\)—the voltage at the anode will vary from 0 to \(2V_B\), whence one can calculate the circuit resistance at which the lamp gives the greatest power in the form of oscillatory energy:

\[ (r_a)_{opt}=\frac{2V_B}{I_s} \tag{20} \]

\[ (N_a)_{\max}=\frac{V_B^2}{2(r_a)_{opt}}=\frac{V_B I_s}{4}. \tag{21} \]

Thus, for an amplifying lamp operating at 200 volt and with increased incandescence \((I_s=20\cdot 10^{-3}\ \mathrm{amp})\), the greatest power proves to be equal to 1 watt.

If \(r_a<(r_a)_{opt}\), then the useful power is smaller. With several lamps \((n)\) operating in parallel, \((r_a)_{opt}\) decreases \(n\) times (since the saturation current is \(n\) times smaller); if for one lamp \((r_a)_{opt}\) is greater than \(r_a\) in the anode circuit, then by increasing the number of lamps we shall achieve an increase in useful power; conversely, if \((r_a)_{opt}<r_a\), then increasing the number of lamps will give nothing.

By increasing the incandescence of the cathode and the anode voltage, one can, theoretically speaking, increase the power delivered by the lamp without limit. In practice the limit is set by the difficulty of removing from the anode the large quantity of heat released as a result of its bombardment by electrons; thus, at a power of 10 kw with an efficiency of 50%, about 1 large calorie per second will have to be removed from the anode, which will require vigorous

water cooling. It is possible to build tubes without water cooling only up to a power of no more than \(1\)–\(2\ \mathrm{kw}\). (anode voltage up to 10,000 volts). It is also necessary to guarantee good insulation of the tube electrodes, since the alternating voltages may reach a large magnitude.

A calculation shows that the efficiency of a tube generator will be the higher, the shorter the time during which a positive potential predominates on the grid (short current pulses); with a small value of the useful power, the efficiency can be brought almost to 100%. The indicated condition will be achieved by applying an additional negative potential to the grid.

A capacitor inserted in the grid circuit, shunted by a large resistance, is charged to a negative potential, and, consequently, it will also increase the efficiency of the generating system. Moreover, such an insertion will prevent the possibility of the formation of strong currents in the grid circuit (in an overvoltage generator), will considerably increase the efficiency, and will reduce the heating of the anode, which is why it is used in powerful generators.

It is clear that the resistance \(R_g\), shunting the capacitor \(C_g\), should preferably be chosen sufficiently large; too great an increase of \(R_g\), however, is impossible, since in that case such a high average negative potential will be obtained on the grid that the oscillations that arise become damped (see Fig. 9) and quickly cease; when the charge leaks off the capacitor \(C_g\), oscillations may arise again. The oscillatory process will thus alternate with pauses at regular intervals of time; the duration of the pauses is proportional to the capacitance \(C_g\). If the periods of generation are short in comparison with the pauses, then it may be considered that the interval of time between two successive moments of generation is proportional to the product \(R_g C_g\). Each excitation of generation can be detected by ear, by means of a telephone connected with the generator. Counting the number of excitations of generation during a known interval of time, it is possible, as the author has shown together with B. A. Vvedenskii2, to measure, over wide ranges, the magnitudes of capacitances (from \(1\ \mathrm{mf}\) and less) and large resistances (from \(0.5 \cdot 10^8\) ohms to \(10^{11}\) ohms).

The technique of applying cathode tubes has in recent years advanced so far that the present article cannot give a complete picture in this field. Therefore, without claiming completeness, we shall confine ourselves to the above foundations of the theory of cathode tubes, which are necessary in analyzing all their applications.

ADDENDUM. ELECTRON EMISSION FROM THE SURFACE OF THORIATED TUNGSTEN.

Langmuir, as early as 1914, observed an increase in the emission of electrons from the surface of \(W\) containing a small percentage of \(Th\). This phenomenon was investigated in detail in his laboratory over a number of years; a summary of these most interesting works was given by Langmuir in Zschr. Phys., v. XXII, p. 357, 1923. The process of thoriating \(W\) is as follows. If a \(W\) filament contains a certain amount of thorium oxide, then by raising its temperature to \(2800^\circ\) for several seconds one can partly convert the oxide of \(Th\) into metallic \(Th\); this latter diffuses to the surface, but at high \(T\) it is not retained on the surface and immediately evaporates. The diffusion process also proceeds at lower temperatures, though more slowly. Experiment has shown that at \(2000^\circ\)—\(2100^\circ\) the diffusion of \(Th\) to the surface still occurs quite vigorously, while evaporation is extremely reduced; by keeping the filament for a long time at this \(T\), one can cover its surface to 80—90% with metallic \(Th\), which is arranged in a layer one atom thick. This layer of \(Th\) causes an extraordinary increase in electron emission (see table), by several hundred thousand and even more than a million times, so that at temperatures of about \(1500^\circ\) one obtains the same emission as at \(2400^\circ\) from pure \(W\); it is clear that such a lowering of the operating \(T\) increases the life of the incandescent filament practically almost without limit.

The reason for such an action of the layer of \(Th\) atoms lies in the lowering of the surface potential difference from 4.5 volt (\(b = 52.600\)) for pure \(W\) to 2.94 volt (\(b = 34.1000\)) for the thoriated surface; the layer of \(Th\) atoms lying on the filament evidently has a positive potential and facilitates the escape of electrons from the surface. With an increase of \(T\) of the thoriated filament its emission strongly increases up to \(2100^\circ\), but then, when, owing to intensified evaporation, the thorium flies off from the surface and does not have time to be replenished by fresh thorium from the inner parts of the filament, the emission falls considerably and thereafter gradually reaches the value characteristic of pure \(W\); such an overheated hair can be reactivated at \(2000^\circ\). The layer of \(Th\) can, besides overheating, be destroyed by the action of positive bombardment and by chemical action (residual gases). In the following table are given the quantities characteristic of a thoriated filament \(d = 0.039\) mm with a \(Th\) content of 1% at various \(T\); the quantity \(\theta\) denotes the part of the surface occupied by \(Th\) atoms; \(i_T\) is the emission current from the thoriated surface, \(i_W\) from pure tungsten; \(T\) is the time in hours during which the total supply of \(Th\) in the filament, as a result of evaporation, decreases by \(\dfrac{1}{e}\) part (lifetime).

Cathode Lamps and Their Applications

\(T^\circ K\)	\(\theta\)	\(i_T\), \(\dfrac{\mathrm{amp}}{\mathrm{cm}^2}\)	\(i_x\), \(\dfrac{\mathrm{amp}}{\mathrm{cm}^2}\)	\(i_T / i_x\)	\(\tau\), hours
1300	0,99997	\(4,14 \cdot 10^{-4}\)	\(2,71 \cdot 10^{-10}\)	\(15,3 \cdot 10^5\)	—
1400	0,99975	\(3,12 \cdot 10^{-3}\)	\(5,68 \cdot 10^{-9}\)	\(5,49 \cdot 10^5\)	—
1500	0,99878	0,0179	\(7,97 \cdot 10^{-8}\)	\(2,25 \cdot 10^5\)	—
1600	0,9528	0,0812	\(8,14 \cdot 10^{-7}\)	\(0,997 \cdot 10^5\)	—
1700	0,9848	0,287	\(6,27 \cdot 10^{-6}\)	\(0,459 \cdot 10^5\)	—
1800	0,9605	0,772	\(4,13 \cdot 10^{-5}\)	\(0,186 \cdot 10^5\)	720.000
1900	0,9191	1,59	\(2,06 \cdot 10^{-4}\)	7.710	91.000
2000	0,8713	2,89	\(0,91 \cdot 10^{-3}\)	3.150	15.100
2100	0,781	3,43	\(3,5 \cdot 10^{-3}\)	980	2.897
2200	0,551	1,24	0,012	103	643
2300	0,139	0,114	0,037	3,09	164
2400	0,0601	0,168	0,104	1,61	47
2500	0,0355	0,357	0,264	1,35	14.6
2600	0,0207	0,773	0,665	1,16	5.01
2800	0,0088	3,48	3,270	1,064	0,74
3000	0,0011	13,5	13,15	1,026	0,14

A reserve of \(Th\) under emission of \(772\) mil. amp. per sq. cm at \(T — 1800^\circ\), which corresponds to the emission of pure \(W\) at \(2600^\circ\), suffices for 720,000 hours, i.e. theoretically for an infinitely long time. In reality the thorium layer is destroyed chiefly through impacts of positive ions much more rapidly. This process occurs especially rapidly at high anode voltages, that is, in generator tubes, which is why it has not yet been possible to construct elegant tubes with thoriated filaments.

Langmuir also succeeded*) in coating the surface of \(W\) with another “activating” substance, namely cesium \(Cs\). Cesium is still more volatile than \(Th\), and on the surface of \(W\) it is retained only up to \(900^\circ\). Owing to the small value of the surface potential difference, only \(1,34\ V\) \((b = 15.500)\), a cesiated filament gives an electron emission of \(0,3\ \dfrac{\mathrm{amp}}{\mathrm{cm}^2}\) at \(900^\circ\), which corresponds to the emission of \(W\) at \(2500^\circ\). At higher \(T\) the cesium evaporates and another extremely interesting phenomenon arises: the atoms of \(Cs\), detaching from the surface of \(W\), pro-

*) Phys. Rev. v. 21. p. 380, 1923.

become ionized; since the vapor \(Cs\) is continuously deposited on the filament, and at the same time evaporation takes place, the cesiated filament becomes a source of positive thermionic emission. The detachment of an electron by a \(Cs\) atom is explained by the fact that its ionization potential, \(3.9\) Volt, is less than the surface potential difference \(W = 4.5\) Volt. An electrode emitting \(+\) ions in this way was called by Langmuir a “genode.”

1. P. Kuksenko recently constructed a very compact regenerative receiver with iron. Report at RORI in February 1923. ↩↩↩

2. S. Rzhevkin and B. Vvedenskii. Telegraphy and Telephony without Wires, No. 11, p. 67, 1921; also Phys. Zschr. XXIII, p. 150, 1921. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩

3. As Barkhausen does. ↩↩↩↩