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AMPLIFIER BASED ON ABSORPTION
V. M. Lopukhin and A. A. Vedenov
1. INTRODUCTION
In recent years the problem of amplification of decimeter and centimeter radio waves has been widely discussed in the Soviet as well as the foreign press[^1].
Of great importance are devices of the traveling-wave-tube and electron-wave-tube type, in which an electron stream having mean velocity \(v_0\) interacts with a slowed electromagnetic wave whose mean velocity, in the system without the electron stream, \(u_0\), is close to \(v_0\).
The requirement of synchronism between the electrons and the wave (\(v_0 \simeq u_0\)) is essential for the operation of traveling-wave-tube devices.
Recently, reports have appeared in the literature[^2][^3] concerning a new type of amplifier of centimeter radio waves—the amplifier based on absorption (hereafter, for brevity, AA).
This device has an amplification coefficient approximately the same as that of a traveling-wave tube or an electron-wave tube (\(\sim 30\ \mathrm{db}\) in power); however, it has a considerably broader pass band, which, depending on the type of tube, may be of the order of 70–120% relative to the carrier frequency. The amplifier is also characterized by the practically complete absence of internal feedback, which makes harmful parasitic generation impossible.
The amplification of an AA also depends only very weakly on the operating regime of the device: the current and the beam potential.
The absorption amplifier described in the literature consists of a round cylindrical waveguide with absorbing walls. Along the axis of the waveguide an electron stream is passed, in which a charge-density wave is excited. This wave interacts with the resultant field of the waveguide, so that, on the average over a period, the stream gives energy to the wave.
The absorption amplifier uses a new physical principle, which consists in using the phase shift between the electron current and the alternating components of the field that arise—
… due to the presence of absorbing walls*) (the material of the walls is characterized by a complex permeability).
The slowed wave, as well as the synchronism between the electrons and the wave in a system without an electron flow, is not obligatory in an absorption amplifier.
II. DESCRIPTION OF THE OPERATING ABSORPTION-AMPLIFIER TUBES
The absorption amplifier described in the literature (Fig. 1) was a thin-walled (wall thickness \(0.12\ \mathrm{mm}\)) cylindrical glass tube \(250\ \mathrm{mm}\) long**), on the inner side of which an absorbing layer of tin oxide of thickness \(2\cdot 10^{-4}\ \mathrm{mm}\) had been deposited.
Fig. 1. General view of the absorption amplifier.
Inside the tube, along its axis, an electron beam moved, the voltage of which could be varied in the interval from 300 to 1000 V. The total current in the beam varied from 0 to 20 mA. Interaction with the beam at the entrance to the system (modulation of the flow) and at the exit was accomplished, for one model, by means of conducting spirals each \(7.5\ \mathrm{cm}\) long, and for another tube by means of cavity resonators.
The measurements were carried out in the range \(10\ \mathrm{cm}<\lambda<30\ \mathrm{cm}\).
*) Naturally, the presence of absorbing walls also leads to thermal losses of part of the energy. However, these losses prove to be considerably smaller than the energy entering the field from the electron flow.
**) The exact diameter of the tube is not indicated; however, from the figure given one may conclude that the tube has an internal diameter of approximately \(3\text{–}4\ \mathrm{mm}\).
The total amplification of the experimental tube containing conducting helices was composed of the amplification associated with the presence of the absorbing layer and the amplification of the traveling-wave tube, which is formed by the sections of the helix penetrated by the electron beam.
When the absorbing section of the amplifier was replaced by a well-conducting waveguide, the amplification dropped sharply, although it still remained large.
Fig. 2. Dependence of the amplification on the current of the electron beam: a — for the amplifier with a modulating helix, b — the absorbing section of the amplifier is replaced by a waveguide.
Fig. 3. Same as Fig. 2, for the amplifier with modulating resonators.
units. In this case it was due to the effect of the traveling-wave tube.
The amplification associated with the presence of the absorbing layer is found as the difference between the total amplification and the amplification of the traveling-wave tube. The following experimental curves were taken:
1) Amplification as a function of beam current with beam modulation by means of a conducting helix (Fig. 2) and by means of resonators (Fig. 3).
Curve a corresponds to the presence of the absorbing section; curve b to the case in which the latter is replaced by a well-conducting waveguide.
2) Dependence of the amplification on the potential of the conducting layer, again for two types of modulation of the electron flow (Figs. 4 and 5). Curves a and b in Fig. 5 correspond to the cases of strong and weak current in the presence of the absorbing section; curve c correspond—
Fig. 4. Dependence of gain on the accelerating potential for a TWT with modulating helices.
Fig. 5. Same as Fig. 4, for a TWT with modulating resonators: \(a\)—strong current, \(b\)—weak current, \(c\)—weak current; the absorbing section is replaced by a waveguide.
Fig. 6. Frequency characteristic of a TWT: \(a\)—for each frequency the change in potentials was chosen to give the maximum possible gain; \(b\)—the potentials remained constant for all frequencies.
Fig. 7. Dependence of output power on input power: \(a\)—beam current \(25\ \mathrm{mA}\), \(b\)—\(20\ \mathrm{mA}\), \(c\)—\(15\ \mathrm{mA}\).
will be weak; in addition, the absorbing section has been replaced by a waveguide.
3) Gain as a function of frequency (Fig. 6). Curve a—all potentials were selected so that the gain assumed its maximum value for the given frequency; curve b corresponds to the case where the potentials were kept constant.
4) Output power of the tube as a function of the input-signal power (Fig. 7) for beam-current values of 15, 20, and 25 mA, frequency \(\nu=3000\) MHz, and stream potential \(V=650\) V.
The graphs clearly show the presence of gain, the role of the absorbing section, and also the wide band characterizing the device.
III. ELEMENTARY THEORY OF THE ABSORPTION AMPLIFIER
Let an infinite conducting medium, characterized by conductivity \(\sigma\) and dielectric permittivity \(\varepsilon\), be penetrated by an electron stream of mean velocity \(v_0\) with current density \(j_0=\rho_0 v_0\), where \(\rho_0\) is the charge density. We direct the \(z\)-axis along the electron velocity.
We shall set ourselves the task of finding the proper electromagnetic waves of the system. We shall seek the solution in the form
\[ E_z=A\cdot e^{i(\omega t-\beta z)}, \tag{1} \]
where \(A\) is the amplitude, and \(\omega\) and \(\beta\) are the angular frequency and the propagation constant.
We consider the frequency \(\omega\) as given. The unknown is the propagation constant \(\beta\), which in the general case is complex, \(\beta=\operatorname{Re}\beta+i\operatorname{Im}\beta\). A wave for which \(\operatorname{Im}\beta>0\) increases with the coordinate \(z\) according to the law \(e^{\operatorname{Im}\beta\cdot z}\).
Everywhere below we shall assume that the mean charge of the electrons is compensated by the ionic background.
Let us consider the first field equation
\[ \operatorname{rot}\mathbf H=\mathbf J+\varepsilon_k i\omega \mathbf E, \tag{2} \]
where \(\mathbf E\) and \(\mathbf H\) are the intensities of the electric and magnetic fields,
\[ \varepsilon_k=\varepsilon\left(1+\frac{\sigma}{i\omega\varepsilon}\right), \]
\(\sigma\) being the conductivity of the medium.
Calculating the operation \(\operatorname{div}\) of both sides of (2), we have:
\[ \operatorname{div}(\mathbf J+\varepsilon_k i\omega\mathbf E)=0. \tag{3} \]
For the one-dimensional problem \((J=J_z,\ E=E_z)\), taking into account the dependence on \(z\) in the form \(e^{-i\beta z}\), we obtain:
\[ J+\varepsilon_k i\omega E=0, \tag{4} \]
i.e., in the case under consideration the convection current is completely compensated by the displacement current. Substituting instead of \(\varepsilon_k\) its expression \(\varepsilon\left(1+\dfrac{\sigma}{i\omega\varepsilon}\right)\), we have:
\[ J=-i\omega\varepsilon E-\sigma E. \tag{5} \]
Expression (5) indicates that the current \(J\) contains a component \((-\sigma E)\) which is in antiphase with the field.
Let us find an expression for the current \(J\), using the equations of motion of the electrons. We shall assume that \(J\), \(\rho\), and \(v\) have the form \(J+\tilde J_0\), \(\rho+\tilde\rho_0\), \(v+\tilde v_0\), where the variable components \(\tilde J\), \(\tilde\rho\), and \(\tilde v\) are much smaller than the constant components \(J_0\), \(\rho_0\), \(v_0\), so that \(\tilde J \ll J_0\), \(\tilde\rho \ll \rho_0\), and \(\tilde v \ll v_0\) (i.e., the theory of small amplitudes is valid). The equations of motion and continuity and the expression for the current take the form
\[ \frac{\partial \tilde v}{\partial t}+v_0\frac{\partial \tilde v}{\partial z} =\frac{e}{m}E, \tag{6} \]
\[ \operatorname{div}\tilde J+\frac{\partial\tilde\rho}{\partial t}=0, \tag{7} \]
\[ \tilde J=\rho_0\tilde v+v_0\tilde\rho. \tag{8} \]
In equations (6) and (8), terms of second order of smallness have been omitted.
Assuming that all quantities are proportional to the factor \(e^{i(\omega t-\beta z)}\), we have:
\[ i(\omega-\beta v_0)\tilde v=\frac{e}{m}E, \tag{9} \]
\[ -i\beta\tilde J+i\omega\tilde\rho=0, \tag{10} \]
\[ \tilde J=\rho_0\tilde v+v_0\tilde\rho. \tag{11} \]
Solving this system for \(\tilde J\), we obtain:
\[ \tilde J= \frac{\omega\rho_0\,\dfrac{e}{m}\,E} {i(\omega-\beta v_0)^2}. \tag{12} \]
From (4) and (12) follows the dispersion equation of the one-dimensional problem under consideration
\[ 1= \frac{\omega_0^2} {\varepsilon'\left(1+\dfrac{\sigma}{i\omega\varepsilon}\right)(\omega-\beta v_0)^2}, \tag{13} \]
where \(\omega_0^2=\dfrac{e\rho_0}{m\varepsilon_0}\), \(\varepsilon'=\dfrac{\varepsilon}{\varepsilon_0}\), \(\varepsilon_0\) is the dielectric permittivity of vacuum.
The solution of equation (13) has the form
\[ \beta=\beta_e \pm \frac{\beta_\rho}{\sqrt{\dfrac{\varepsilon_k}{\varepsilon_0}}}, \tag{14} \]
where
\[ \beta_e=\frac{\omega}{v_0}; \quad \beta_\rho=\frac{\omega_0}{v_0}; \quad \frac{\varepsilon_k}{\varepsilon_0} =\varepsilon'\left(1+\frac{\sigma}{i\omega\varepsilon'\varepsilon_0}\right). \]
It is convenient to write expression (14) in the form
\[ \beta=\beta_e \pm \beta_\rho \frac{p+iq}{\sqrt{\varepsilon'}}, \tag{15} \]
where the dimensionless coefficients \(p\) and \(q\), as functions of the parameter \(\dfrac{\omega\varepsilon'\varepsilon_0}{\sigma}\), are given in Fig. 8. A wave for which \(\operatorname{Im}\beta>0\) grows
Fig. 8. Dependence of \(p\) and \(q\) for the crude model of the TWT on \(\dfrac{\omega\varepsilon'\varepsilon_0}{\sigma}\).
exponentially with the coordinate \(z\). The gain \(G\) of the wave is equal to
\[ G=8.69\beta_\rho \frac{q}{\sqrt{\varepsilon'}} \ \text{db/m}. \tag{16} \]
The maximum value of \(G\) occurs at \(\dfrac{\omega\varepsilon_0\varepsilon'}{\sigma}=\dfrac{1}{\sqrt{3}}\) (see Fig. 8):
\[ G_{\max}\simeq \frac{3}{\sqrt{\varepsilon'}}\beta_\rho \ \text{db/m}. \]
Assuming, for example, \(J_0=0.3\ \text{a}/\text{cm}^2\), \(V_0=400\ \text{v}\), we obtain \(G_{\max}=5.8\ \text{db}/\text{cm}\). According to formula (16), the gain \(G\) depends only weakly on the electronics, which enters the expression for \(G\) only through \(\beta_\rho=\dfrac{\omega_0}{v_0}\). This fact, as noted above, was confirmed experimentally (see Fig. 4).
The elementary theory presented also gives a qualitatively correct dependence of the gain coefficient \(G\), proportional to \(q\), on the frequency \(\omega\) (see Fig. 8 and Fig. 6).
The physical reason for amplification in the one-dimensional problem is that, in accordance with (5), the current \(\tilde J\) contains a component that is in antiphase with the field \(E\). As a consequence, the average power \(\{P\}\) of interaction between the field and the current, of the form \(\frac{1}{2}\operatorname{Re} J^{*}E\), proves to be negative:
\[ \{P\}=\frac{1}{2}\operatorname{Re} J^{*}E =\frac{1}{2}\operatorname{Re}(i\omega\varepsilon-\sigma)EE^{*} =-\frac{1}{2}\sigma EE^{*}<0. \]
Thus, on the average, the electrons are slowed down; their energy is transferred to the electromagnetic field.
IV. A MORE COMPLETE THEORY OF THE ABSORPTION AMPLIFIER
A. Formulation of the Problem
Let an electron beam of radius \(r=a\), mean velocity \(v_0\), and charge density \(\rho_0\) pass through a circular cylindrical waveguide of radius \(r=a\).
Let the waveguide possess a known value of the conductivity
\[ Y=\left.\frac{H_{\varphi}}{E_z}\right|_{r=a}=G+iB, \]
where \(H_{\varphi}\) and \(E_z\) are components of the magnetic and electric fields in a cylindrical coordinate system whose axis coincides with the axis of the waveguide, and \(G\) and \(B\) are the real and imaginary components of the input conductivity.
It is required to find the possible solutions in such a system which have the form of electric-type waves, for which \(E_z\ne 0\), \(H_z=0\). We shall seek the solutions in the form \(E_z=Ae^{i(\omega t-\beta z)}\), where \(A\) is the amplitude of the electric-field intensity, while \(\omega\) and \(\beta\) are the angular frequency and the propagation constant of the wave.
The problem thus formulated includes, as special cases, both the theory of the absorption amplifier and the theory of the traveling-wave tube.
To pass to these special problems, one must choose \(G\) and \(B\) as the required functions of the frequency and of the parameters of the system.
B. Input Conductivity of the Electron Beam \(Y_1\)
To calculate
\[ Y_1=Y_1(\omega,\beta)=\left.\frac{H_{\varphi 1}}{E_{z1}}\right|_{r=a} \]
it is necessary to solve simultaneously the equations of the electromagnetic field and the equations of motion of the electrons.
The solutions of the field equations for \(E\)-waves will be written with the aid of the electric polarization potential \(\Pi\):
\[ \boldsymbol{E}=-i\beta \operatorname{grad}\Pi+k^2\Pi, \tag{17} \]
\[ \boldsymbol{H}=i\omega\varepsilon_0 \operatorname{rot}\Pi, \tag{18} \]
\[ \frac{\partial^2 \Pi}{\partial r^2}+\frac{1}{r}\frac{\partial \Pi}{\partial r} +\frac{1}{r^2}\frac{\partial^2 \Pi}{\partial \varphi^2} +(k^2-\beta^2)\Pi =-\frac{J}{i\omega\varepsilon_0}, \tag{19} \]
where \(\boldsymbol{E}\) and \(\boldsymbol{H}\) are the electric and magnetic field intensities, \(\Pi=\Pi_z\) is the component of the polarization potential directed along the \(z\)-axis, \(J\) is the electron current density, \(k=\omega\sqrt{\varepsilon_0\mu_0}\), and \(\varepsilon_0\) and \(\mu_0\) are the dielectric and magnetic permeabilities of vacuum.
For the fundamental wave, characterized by axial symmetry, \(\dfrac{\partial}{\partial\varphi}=0\), we have:
\[ E_z=(k^2-\beta^2)\Pi, \tag{20} \]
\[ H_\varphi=-i\omega\varepsilon_0\frac{\partial \Pi}{\partial r}, \tag{21} \]
\[ \frac{\partial^2 \Pi}{\partial r^2}+\frac{1}{r}\frac{\partial \Pi}{\partial r} +(k^2-\beta^2)\Pi =-\frac{J}{i\omega\varepsilon_0}. \tag{22} \]
The expression for the conductivity \(Y_1\) of the space inside the waveguide has the form
\[ Y_1=\frac{H_\varphi}{E_z} =-\frac{i\omega\varepsilon_0\dfrac{\partial \Pi}{\partial r}} {(k^2-\beta^2)\Pi}, \tag{23} \]
where \(\Pi\) is the solution of equation (22), whose right-hand side contains the expression for the density of the convection current of the electrons.
Assuming the small-amplitude theory to be valid and using (12) and (20), we write (22) in the form
\[ \frac{\partial^2 \Pi}{\partial r^2}+\frac{1}{r}\frac{\partial \Pi}{\partial r} +\Gamma^2\Pi=0, \tag{24} \]
where
\[ \Gamma^2=(k^2-\beta^2)\left(1-\frac{\beta_\rho^2}{(\beta-\beta_e)^2}\right), \tag{25} \]
\[ \beta_\rho=\frac{\omega_0}{v_0},\qquad \beta_e=\frac{\omega}{v_0},\qquad \beta=\frac{\omega}{v},\qquad \omega_0^2=\frac{e\rho_0}{m\varepsilon_0}, \]
\(v_0\) is the mean velocity of the electrons, and \(\omega_0\) is the resonance frequency of plasma oscillations.
The solution of equation (24), finite at zero, has the form
\[ \Pi = B J_0(\Gamma r), \tag{26} \]
where \(B\) is an arbitrary constant; \(J_0(\Gamma r)\) is a Bessel function of zeroth order.
Taking into account the identity
\[ \frac{dJ_0(\Gamma r)}{dr}=-\Gamma J_1(\Gamma r), \]
on the basis of (23) and (26) we obtain:
\[ Y_1\big|_{r=a}=\frac{i\omega\varepsilon_0\Gamma J_1(\Gamma a)} {(k^2-\beta^2)J_0(\Gamma a)}. \tag{27} \]
B. Dispersion equation
Equating (27) to the input conductance of the waveguide \(Y_2\big|_{r=a}=G+iB\), we obtain the dispersion equation determining \(\beta=\beta(\omega)\),
\[ \frac{i\omega\varepsilon_0 a\cdot \Gamma a J_1(\Gamma a)} {a^2(k^2-\beta^2)J_0(\Gamma a)}=G+iB, \tag{28} \]
where \(\Gamma\) is given by expression (25), while \(G\) and \(B\) are regarded as known. Introduce the following notation:
\[ f(\Gamma a)=\frac{\Gamma a J_1(\Gamma a)}{J_0(\Gamma a)} =re^{i\theta};\qquad \Gamma a=\beta_e a(m+in). \tag{29} \]
In Fig. 9 are plotted the values of \(r\) and \(\theta\), obtained with the aid of an electrolytic bath, as functions of \(\operatorname{Re}\Gamma a\) and \(\operatorname{Im}\Gamma a\), i.e. \(\beta_e a m\)
Fig. 9. Function
\[ f(z)=\frac{zJ_1(z)}{J_0(z)}. \]
The grid gives the modulus and argument of the function
\[ f(z)=re^{i\theta}, \]
and \(\beta_e a n\). If \(G\) and \(B\) are known, then with the aid of the graph in Fig. 9 one can find \(\operatorname{Re}\Gamma a\) and \(\operatorname{Im}\Gamma a\) and, further, with the aid of (25), also \(\operatorname{Re}\beta\) and \(\operatorname{Im}\beta\).
This method of calculation encounters a number of difficulties when one attempts to use it, for example, in the theory of a traveling-wave tube, containing
... as a slow-wave structure a conducting helix. In this case the conductivity of the helix is a function strongly dependent on \(\beta\), which makes it difficult to solve equation (28).
For the problems that will be considered below, \(Y=G+iB\) is a slowly varying function of \(\beta\), and moreover \(\beta \simeq \beta_e=\dfrac{\omega}{v_0}\). The latter means that we restrict ourselves to considering such waves in the system whose phase velocity is close to \(v_0\).
Putting \(\beta=\beta_e\) in the right-hand side of (28) and neglecting \(k^2 \ll \beta_e^2\), we obtain:
\[ f(\Gamma a)=-\frac{(\beta_e a)^2}{i\omega\varepsilon_0 a}\,Y(\beta_e). \tag{30} \]
Under these same assumptions (25) gives:
\[ \beta=\beta_e \pm \beta_p\left(1+\frac{\Gamma^2}{\beta_e^2}\right)^{-1/2}. \tag{31} \]
This equation can be written in the form
\[ \beta=\beta_e \pm \beta_p(p+iq), \tag{32} \]
where \(p\) and \(q\) are connected with the previously introduced coefficients \(m\) and \(n\) by the relation
\[ p+iq=\left[1+(m+in)^2\right]^{-1/2}. \tag{33} \]
In Fig. 10 the values of \(p\) and \(q\) are given as functions of \(m\) and \(n\). The parameter \(q\) is related to the value of the gain by the former relation
\[ G=8.693\,\beta_p q\ \mathrm{db}/\mathrm{m}. \tag{34} \]
The parameter \(p\) determines the change of the phase velocity in the system
\[ v=\frac{\omega}{\beta}=\frac{\omega}{\beta_e+\beta_p p} =\frac{v_0}{1+p\,\dfrac{v_0}{\omega}}. \tag{35} \]
Let us consider various special cases.
a) \(Y\to\infty\), which physically corresponds to ideally conducting waveguide walls.
It follows from equation (30) that \(J_0(\Gamma a)\to 0\). This means that \(\Gamma a\) takes a discrete series of values that are roots of the equation \(J_0(\Gamma a)=0\), i.e. \(\Gamma a=2.4;\ 5.52;\ldots\). Using the expression \(\Gamma a=\beta_e a(m+in)\), we note that \(n=0\). Consequently, in the graph of Fig. 10 the corresponding points lie on the horizontal axis. This corresponds to \(q=0\). Thus, in the problem under consideration there is no gain.
b) \(Y\to 0\), which means the complete absence of conductivity of the walls. Here, as before, \(\Gamma a\) takes a discrete series of values \(\Gamma a=3.83;\ 7.01;\ldots\), which are roots of the equation \(J_1(\Gamma a)=0\).
In complete analogy with the preceding case we have \(n=0\) and, consequently, \(q=0\), i.e. in the problem under consideration the amplification is zero.
в) \(Y=iB_C,\ B_C>0\), which corresponds to a capacitive loading of the waveguide walls.
From equation (30) it is clear that \(f(\Gamma a)=\dfrac{\Gamma a J_1(\Gamma a)}{J_0(\Gamma a)}\) is purely real and positive; consequently, in the expression \(f(\Gamma a)=r e^{i\theta}\), \(\theta=0\).
From the graph in Fig. 9 it is clear that in this case \(n=0\); consequently,
Fig. 10. The function \(p+iq=[1+(m+in)^2]^{-1/2}\); the net of curves \(p=\mathrm{const}\), \(q=\mathrm{const}\) makes it possible to compute \(p\) and \(q\), if \(m\) and \(n\) are known.
in accordance with Fig. 10 \(q=0\); here too the amplification is absent.
г) \(Y=-iB_L\ (B_L>0)\), inductive loading of the waveguide walls.
For inductive walls the dispersion equation (28) takes the form
\[ \Gamma a\,\frac{J_1(\Gamma a)}{J_0(\Gamma a)} = -\frac{(\beta_e a)^2}{\omega\varepsilon_0 a}\,B_L, \tag{36} \]
so that the left-hand side of (36) must be a real negative number. Consequently, \(\theta=180^\circ\), which corresponds to the points of the vertical axis of Fig. 9. Thus \(\Gamma a\) is obtained as purely imaginary \((m=0,\ n<0)\). From Fig. 10 it is clear that it is necessary to distinguish two cases: a) \(0>n>-1\), here \(q=0,\ p\ne0\); b) \(-1>n>-\infty\), here \(q\ne0,\ p=0\).
Change of \(q\) and \(p\) as a function of \(n\) is shown in Fig. 11. From this figure it is clear that, for \(n=-1\), \(p \to \infty\) and \(q \to \infty\). This means that the solution is an exponentially growing wave with an infinite value of the growth coefficient.
Let us compute the corresponding value of the inductance \(B_L\). For \(m=0\) and \(n=-1\), \(\Gamma a=-i\beta_e a\), and (36) gives
\[ B_L=\frac{\omega \varepsilon_0 a I_1(\beta_e a)}{\beta_e a I_0(\beta_e a)} . \tag{37} \]
For \(\beta_e a=1\), for example, we obtain \(B_L=0.45\,\omega\varepsilon_0 a\). In this case the theoretical coefficient of wave growth is equal to infinity. Under real conditions, owing to thermal losses, the amplification coefficient has a finite value. As was indicated earlier, the conducting helix used in the traveling-wave tube represents, for the electron stream (provided that synchronism between the electron stream and the slowed wave in the helix is satisfied), an inductive load\(^*\). When the synchronism conditions are violated, the load of the electron stream changes sign, becomes capacitive, and the amplification coefficient of the traveling-wave tube falls to zero.
Fig. 11. Dependence of \(q\) and \(p\) on \(n\) for an inductive load of the electron stream.
d) \(Y=G\), the load is of a purely active character. The dispersion equation (30) takes the form
\[ f(\Gamma a)=\frac{(\beta_e a)^2}{i\omega\varepsilon_0 a}\,G = r e^{i\theta}. \tag{38} \]
From (38) it is clear that
\[ \theta=-90^\circ,\qquad r=\frac{(\beta_e a)^2}{\omega\varepsilon_0 a}\,G . \]
Next, using the graphs of Figs. 9 and 10, one can compute \(q\) for various \(\beta_e a\) as a function of \(\dfrac{\omega\varepsilon_0 a}{G}\). The results of these computations are given in Fig. 12. The maximum value
\[ q_{\max}=\frac{1}{2\sqrt{2}}, \]
which occurs at
\[ \frac{\omega\varepsilon_0 a}{G}=\frac{2}{\sqrt{3}}, \]
is sufficiently large to use this type of amplifier for practical purposes.
\(^*\) In a real helix the load is inductive-active. For details, see, for example, \(^1\).
V. M. LOPUKHIN AND A. A. VEDENOV
The largest value of the amplification occurs at \(\beta_e a \ll 1\). Physically this corresponds to an electron beam of small radius. A real model satisfying the condition \(\beta_e a \ll 1\) must include a number of parallel electron beams flying through a set of parallel channels, each having a small cross section.
Fig. 12. Dependence of \(q\) on the parameter \(\dfrac{\omega\varepsilon_0 a}{G}\) under an active load of the electron beam.
The amplification coefficient \(q\) of an amplifier with active walls depends only weakly on the velocity of the electrons. Indeed, when \(\beta_e a\) is changed from the value 1 to the value 2, which corresponds to changing the electron velocity by a factor of two, the amplification coefficient \(q\) also changes by approximately a factor of two.
Let us also note that for
\[ \frac{\omega\varepsilon_0 a}{G} \simeq 1 \]
(in this case the amplification is close to maximal) the tube has an attenuation of approximately \(40\, d\delta/\lambda\), where \(\lambda\) is the wavelength in free space. This indicates that the output and input of the tube are practically completely decoupled; self-excitation and parasitic generation in the tube are impossible.
e) A real model of a traveling-wave amplifier in which a thin absorbing layer of thickness \(d\) and conductivity \(\sigma\) is applied from the inside to a glass tube of thickness \(g\) has, at the boundary of the electron beam, a conductance \(Y(a)\), expressed with sufficient accuracy by the formula \(Y=\sigma d+Y(b)\), where \(Y(b)\) is the conductance calculated on the inner surface of the glass tube.
Fig. 13. Dependence of the effective dielectric permittivity of the wall of the glass tube \(\varepsilon'_{\mathrm{eff}}\) on the value \(\beta_e g\).
The solution of the field equations shows\(^2\) that \(Y(b)\) has a purely capacitive character and can be represented in the form
\[ Y(b)=\frac{i\omega\varepsilon_0}{\beta_e}\,\varepsilon'_{\mathrm{eff}}, \tag{39} \]
where \(\varepsilon'_{\mathrm{eff}}\) is determined from Fig. 13. As the wall thickness of the tube increases, \(\varepsilon'_{\mathrm{eff}}\) increases.
The presence of capacitive conductance in the section \(r=b\) reduces the amplification coefficient (recall that with purely capacitive conductance there is no amplification).
ABSORPTION AMPLIFIER
In order to obtain an amplification \(q \ne 0\), it is necessary to take the thickness of the layer \(g\) sufficiently small. For this reason, in the experimental lamp \(g\) was of the order of \(0.012\ \mathrm{cm}\).
Figure 14 gives the dependence of \(q\) on \(\beta_e a\) for the capacitively active model. The absorbing layer is assumed to be very thin; it is supported by an insulator tube with permittivity \(\varepsilon=\varepsilon_c\). The product \(\dfrac{1}{\sigma_2 d} v_0 \varepsilon_0\) is chosen equal to 0.44, which leads to the maximum value of \(q\).
Fig. 14. Amplification coefficient \(q\) as a function of \(\beta_e a\) for the capacitively active model.
g) Finally, let us consider one more model of an absorption amplifier.
An electron flow of mean current density \(J_0\) and velocity \(v_0\) flies through a cylindrical channel of radius \(r=a\), drilled in an absorbing medium of conductivity \(\sigma\) and permittivity \(\varepsilon_0\). We shall regard the absorbing medium as unbounded.
Let us calculate the conductance
\[ Y_2(a)=\left.\frac{H_\varphi}{E_z}\right|_{r=a} \]
in the absorbing medium.
The fields \(E_z\) and \(H_\varphi\) have the form (we omit the factor \(e^{i(\omega t-\beta z)}\)):
\[ E_z=(k^2-\beta^2)\Pi(r), \tag{40} \]
\[ H_\varphi=-i\omega\varepsilon_k \frac{\partial \Pi}{\partial r}, \tag{41} \]
where \(\varepsilon_k\) is the complex dielectric permittivity
\[ \varepsilon_k=\varepsilon_0\left(1+\frac{\sigma}{i\omega\varepsilon_0}\right),\qquad k^2=\omega^2\varepsilon_k\mu, \]
and \(\Pi(r)\) is the \(z\)-component of the polarization potential, satisfying the equation
\[ \frac{\partial^2\Pi}{\partial r^2} +\frac{1}{r}\frac{\partial\Pi}{\partial r} +\frac{1}{r^2}\frac{\partial^2\Pi}{\partial \varphi^2} +(k^2-\beta^2)\Pi=0. \tag{42} \]
The axially symmetric solution of equation (42), corresponding to a wave going away from the electron flow as \(r\to\infty\), has the form
\[ \Pi(r)=C H_0^{(2)}\left(\sqrt{k^2-\beta^2}\,r\right), \]
where \(C\) is a constant, and \(H_0^{(2)}\) is the Hankel function of zero order of the second kind. Using the identity
\[ \frac{dH_0^{(2)}(x)}{dx}=-H_1^{(2)}(x), \]
we have:
\[ Y(a)=- \frac{i\omega\varepsilon_k}{\sqrt{k^2-\beta^2}}\, \frac{H_1^{(2)}\left(\sqrt{k^2-\beta^2}\,a\right)} {H_0^{(2)}\left(\sqrt{k^2-\beta^2}\,a\right)}. \tag{43} \]
Fig. 15. Dependence of the amplification factor \(q\) on
\[ \xi=\frac{\sigma}{\omega \varepsilon_0} \]
for various values of \(\beta_e a\).
Fig. 16. Dependence of \(q\) on \(\beta_e a\) at the optimal \(\xi=2.0\).
Fig. 17. Theoretical frequency response of the AP (dependence of the amplification factor \(q\) on \(\omega\)).
Using (43), we write the dispersion equation (38) in the form
\[ \Gamma a \frac{J_1(\Gamma a)}{J_0(\Gamma a)} = - i \frac{\varepsilon_k}{\varepsilon_0}\,\beta_e a\, \frac{H_1^{(2)}(i\beta_e a)}{H_0^{(2)}(i\beta_e a)} . \tag{44} \]
In the right-hand side of (44) we have set \(\beta \simeq \beta_e\) and have discarded terms \(\sim |k^2|/\beta_e^2\). Taking into account the expression for \(\varepsilon_k=\varepsilon_0\left(1+\dfrac{\sigma}{i\omega\varepsilon_0}\right)\), we write (44) in the form
\[ \Gamma a \frac{J_1(\Gamma a)}{J_0(\Gamma a)} = (1-i\zeta)\Psi(\beta_e a), \tag{45} \]
where \(\zeta=\dfrac{\sigma}{\omega\varepsilon_0}\); \(\Psi(\beta_e a)=-i\beta_e a\,\dfrac{H_1^{(2)}(i\beta_e a)}{H_0^{(2)}(i\beta_e a)}\) is a real, easily tabulated function of \(\beta_e a\).
Solving equation (45) for \(\beta\) in accordance with the procedure considered in detail earlier, we arrive at the graphs shown in Figs. 15, 16, 17. The meaning of these curves is clear from the captions to the figures. The theoretical curves of Figs. 16 and 17 qualitatively coincide with the experimental curves of Figs. 6 and 4.
Quantitative agreement between experiment and the theory developed in the present section cannot be required, since the experimental absorption amplifier corresponds more closely to another theoretical problem, set forth in item e.
V. COMPARISON OF THE ABSORPTION AMPLIFIER WITH A TRAVELING-WAVE TUBE AND AN ELECTRON-WAVE TUBE
The absorption amplifier possesses certain advantages in comparison with a traveling-wave tube and an electron-wave tube.
Thus, in the absorption amplifier the input and output of the tube are completely decoupled; internal feedback is absent. This is explained by the fact that the backward wave of the amplifier is attenuating (only the forward wave, close to the electron stream, increases with the coordinate \(z\)). This attenuation, caused by absorption of the electromagnetic wave in a poorly conducting layer, has a magnitude on the order of several tens of decibels.
This advantage is especially noticeable in comparison with a traveling-wave tube, in which, in order to reduce the internal feedback caused by the wave in the helix reflected from the “output end” of the tube, the helix must be made of a poorly conducting material. For an electron-wave tube, containing two electron streams, feedback through the electron streams at low current densities in the beams is absent, since in this case all four waves in the system are forward waves.
Another important advantage of the AA is the weak dependence of the gain on the mean velocity of the electron stream. Let us recall—
… that a considerably sharper dependence of the gain on the electron-current potential in a traveling-wave tube is explained by the requirement of synchronism between the electron stream and the slow wave in the system. For a two-beam tube operating in the amplification regime, the condition of synchronism must likewise be fulfilled between the electron stream and one of the waves propagating in the other electron stream.
The weak dependence of the gain in an absorption tube on the electron-current potential is explained by the fact that the operation of this device is based not on the principle of synchronism of the electron stream with the slow wave in the system, but on the use of the phase shift between the convection current and the electric-field intensity that arises owing to the conductivity of the absorbing section.
Among the disadvantages of the absorption amplifier we note the following.
The manufacture of this amplifier is associated with considerable technical difficulties (let us recall that in the experimental tube the absorbing layer was deposited on a glass tube with walls of very small thickness, \(g \simeq 0.12\) mm).
For effective operation of the amplifier, the modulating and receiving spirals must occupy a considerable part of the total length of the tube (in the experimental tube this ratio was approximately 0.5). With further lengthening of the spirals, the absorption amplifier will be close to an ordinary traveling-wave tube.
The first models of the absorption amplifier had a rather large value of the noise factor—of the order of 20 db. This value is close to that by which the first traveling-wave tubes were characterized in terms of noise. Apparently, with further improvement of the absorption amplifier, the noise factor can be reduced to a value of the order of 10 db, which characterizes modern traveling-wave tubes.
In summary, it should be said that the absorption amplifier is a new and interesting physical device which, in a number of special problems, will find application with the further development of radiophysics.
CITED LITERATURE
- V. M. Lopukhin, Excitation of Electromagnetic Oscillations and Waves by Electron Streams, Gostekhizdat, 1953.
- C. K. Birdsall and J. R. Whinnery, Journ. Appl. Phys. 24, 314 (1953).
- C. K. Birdsall, G. R. Brewer and A. V. Haeff, PIRE 41, 865 (1953).