NONLINEAR THEORY OF A CATHODE-COUPLED OSCILLATOR

I. A. Suslov

(Presented by the scientific seminar of the Radio Engineering Faculty)

Introduction

The cathode-coupled oscillator [13, Fig. 1] is finding ever wider application in measuring, testing, and other radio-engineering apparatus [1, 2, 3, 4, 5]. It is distinguished by good stability, a low harmonic content, and has no feedback elements that are critical in magnitude. When several bands are present, the transition from one band to another is not associated here with the need to switch feedback elements, which makes it possible to use a simpler switch. In comparison with a transitron oscillator, which is distinguished by the same feature, the circuit under consideration has greater constancy of amplitude within the band and does not require a special selection of the operating mode. The negative resistance developed by a cathode-coupled oscillator, depending on the tube used and the parameters of the circuit, lies within the range \(300\ \Omega\)—\(3\ \mathrm{k}\Omega\). This makes possible the use of circuits with low resonant resistance and ensures good operation of the oscillator up to frequencies of the order of \(200\)—\(300\ \mathrm{MHz}\).

Questions of the nonlinear theory of the cathode-coupled oscillator have not been sufficiently developed in the available literature. Two works devoted to this subject are known. One of them belongs to P. G. Sul’zer [6], the other to G. A. Khavkin [7].

The basis of P. G. Sul’zer’s work is the approximation used by him for the volt-ampere characteristic (Fig. 1) of a two-terminal network with cathode coupling [13, Fig. 2] by means of a third-degree polynomial

\[ i = -\alpha u + \beta u^3 \tag{1} \]

and the differential equation of the oscillator corresponding to this approximation, in the form

\[ \frac{d^2 y}{d x^2} -\varepsilon \left(1-y^2\right)\frac{d y}{d x} + y = 0. \tag{2} \]

Since approximation (1) and equation (2) underlie the classical nonlinear theory of tube oscillators [8, 9], the formulas and conclusions obtained by P. G. Sul'tser are simply a repetition of what has long been known from that theory. These formulas and conclusions pertain to oscillators in general, do not take into account the special features of the circuit under consideration, and cannot provide recommendations for choosing its parameters.

Fig. 1

G. A. Khavkin’s work [7] likewise represents an attempt to apply an existing theory to the new circuit—in this case, the theory of oscillators with inertial nonlinearity. The author even tries to adapt the oscillator under consideration to the named theory. To this end he recommends using small cathode resistances \(R_k\) (of the order of \(20\text{--}30\ \Omega\)). Such a recommendation is motivated by the desire to increase the frequency stability of the oscillator through the expected reduction of the nonlinear correction to the period. Thus the influence of changes in the nonlinear correction is exaggerated; as is known [10, 11], these cause much smaller frequency drifts than drifts due to other causes (changes in temperature, space-charge capacitance, supply conditions, grid currents, etc.). A sharp reduction of \(R_k\) relative to the usually employed values will intensify the action of all these causes (as well as that of the nonlinear correction), and will almost completely eliminate the negative feedbacks to which the circuit owes its high stability.

The choice of a small \(R_k\) forces G. A. Khavkin to make yet another error—to connect the thermistor in the anode of the first tube. Such a connection weakens the stabilizing action of the thermistor.

In his effort to adapt the oscillator under consideration to the theory of oscillators with inertial nonlinearity, G. A. Khavkin completely ignores the literary material [3] available to him, pays no attention to the fact that in real circuits fairly large \(R_k\) values are used, and that when a thermistor is used, the latter is connected as a cathode resistance.

Experimental verification shows that, with the circuit parameters chosen according to G. A. Khavkin, the cathode-coupled oscillator operates in an extremely unstable manner.

Thus, of the two works devoted to the nonlinear theory of the cathode-coupled oscillator, one [6] in fact has no direct bearing on this oscillator, since it does not take its special features into account, while the other [7] is erroneous. In the present article an attempt is made to construct a nonlinear theory of the cathode-coupled oscillator that would make it possible to take into account the characteristic features of the circuit under consideration and could serve as a basis for choosing its parameters and for calculating the oscillator.

Current–voltage characteristic of a cathode-coupled two-terminal network

The basic nonlinear dependence for the oscillator under consideration is the current–voltage characteristic of the cathode-coupled two-terminal network¹. This characteristic has a “falling” middle section and two rectilinear branches with positive slope (Fig. 1). The slopes of the individual sections of the current–voltage characteristic are determined by the values, corresponding to these sections, of the conductance

\[ \frac{di}{du} \]

of the cathode-coupled two-terminal network. This conductance can be expressed in the following way through the gain factor \(K\) of the cathode-coupled amplifier (with capacitor \(C_1\) disconnected) and its output resistance \(Z_{\text{out}}\). Denote by \(\Delta u\) the increment of the voltage \(u\) applied to the amplifier input. Then the increment of the voltage at the amplifier output will be \(K\Delta u\), and the open-circuit voltage between the amplifier input and output will change by the amount

\[ \Delta u_{\text{o.c.}} = \Delta u - K\Delta u . \]

If now the output is connected to the input by means of a capacitor \(C_1\) (Fig. 2) of sufficiently large capacitance, then, owing to \(\Delta u\), a current will flow through this capacitor

\[ \Delta i = \frac{\Delta u_{\text{o.c.}}}{Z_{\text{out}}} = \frac{\Delta u(1 - K)}{Z_{\text{out}}}. \tag{3} \]

From (3) one can find the conductance of the cathode-coupled two-terminal network

\[ \frac{\Delta i}{\Delta u} = \sigma = \frac{1 - K}{Z_{\text{out}}}. \]

At \(u=0\), \(K\) is maximal (or close to the maximum), and \(\sigma\) takes its maximum negative value in absolute magnitude,² corresponding to the middle falling section of the current–voltage characteristic. At values of \(u\) (positive and negative) that make \(K\) equal to unity, \(\sigma=0\). These correspond to the horizontal sections in the region of the bends of the current–voltage characteristic. At positive and negative \(u\) that cut off the second or the first tube—

therefore, \(K=0\) and \(z=\dfrac{1}{Z_{\text{out}}}\). Thus straight-line branches of the characteristic with positive slope are obtained.

Fig. 2

The experimental determination of the volt–ampere characteristic can be carried out by means of a circuit operating on direct current. For this purpose, instead of the capacitor \(C_1\), it is necessary to connect a constant separating voltage replacing the potential difference that exists on the plates of the capacitor. The magnitude of the required voltage can be selected by ensuring that, at an input voltage \(u=0\), the current \(i\) of the resulting circuit is equal to zero. A drawback of this method is its sensitivity to changes in the supply voltages and in the voltage of the separating source.

It is very convenient to obtain the volt–ampere characteristics of a cathode-coupled two-terminal network by oscillographic means. For this purpose, the voltage from the oscillator circuit must be applied to the horizontal-deflection amplifier of the oscilloscope, and to the vertical-deflection amplifier—from a small resistance connected in series with the circuit.

The circuit should be taken with a large resonant resistance, so that the oscillator gives a large amplitude of oscillations, making it possible to record the characteristic over sufficiently wide limits of variation of \(u\).

Approximation of the Volt–Ampere Characteristic

A characteristic feature of the nonlinear dependence fundamental for the oscillator under consideration—the volt–ampere characteristic of a cathode-coupled two-terminal network—is the presence in it of straight-line

branches with positive slope. The function \(i=\varphi(u)\), approximating the indicated dependence, must therefore reproduce these branches well. In addition, it must be sufficiently simple so that, in deriving formulas for the amplitude of oscillations, the nonlinear correction to the period, and the harmonics, the necessary quadratures can be carried out. The approximating function must be such that the approximation coefficients are in a simple dependence on the parameters of the circuit and of the operating mode and thus would make it possible to use the formulas of nonlinear theory for the rational selection of the oscillator elements.

As we shall see below, the differential equation of the oscillator contains not only the function \(i=\varphi(u)\), but also its derivative, and the latter plays the principal role. We shall therefore approximate the dependence

\[ \frac{di}{du}=\varphi'(u). \]

The expression for the current-voltage characteristic itself will then be found by integration.

Fig. 3

Figure 3 shows the curve

\[ \frac{di}{du}=\varphi'(u) \]

for a cathode-coupled two-terminal network using a 6N8 tube, for which \(R_k=2\,\text{k}\Omega\), \(R_a=10\,\text{k}\Omega\).

The form of the curves in Figs. 1 and 3 suggests approximation by broken lines. However, such piecewise approximations prove inconvenient. The point is that the “cutoff angles” determining the positions of the breaks in the approximating line will depend on the amplitude of the oscillations, which is still to be found. Eliminating these angles leads to transcendental equations for the amplitude, which in the general case can be solved only graphically.

The common approximation by means of power polynomials is not the best in our case. Such an approximation reproduces well only the falling portion of the current-voltage characteristic and poorly reproduces its rectilinear branches. As the number of...

terms, the difficulties in determining the coefficients of the power polynomial rapidly increase. At the same time, the formulas of the nonlinear theory become more complicated, making them inconvenient for calculations and opaque for analysis. The approximation coefficients then turn out not to be directly connected (except for the coefficient of the first degree in \(u\)) with the circuit parameters. This does not make it possible, when analyzing the resulting formulas, to trace the influence of the circuit parameters on the amplitude of the oscillations, the harmonics, etc.

We shall use an approximation of the slope of the volt-ampere characteristic by means of the expression

\[ \frac{di}{du}=\varphi'(u)=\alpha-\frac{\beta}{(1+\gamma u^2)^2}. \tag{4} \]

In this case the volt-ampere characteristic itself will be determined by the formula

\[ i=\varphi(u)=\alpha u-\frac{\beta u}{2(1+\gamma u^2)}-\frac{\beta}{2\sqrt{\gamma}}\arctg \sqrt{\gamma}\,u . \tag{5} \]

To use the adopted approximation, it is necessary to determine the coefficients \(\alpha\), \(\beta\), and \(\gamma\).

For this purpose, let \(|u|=\infty\) in (4); we obtain

\[ \left.\frac{di}{du}\right|_{|u|\to\infty}=\alpha . \]

Consequently, \(\alpha\) is determined by the slope of the branches. On the other hand, from (3) we obtain

\[ \left.\frac{di}{du}\right|_{|u|\to\infty}=\frac{1}{Z_{\mathrm{out}}}. \]

Thus,

\[ \alpha=\frac{1}{Z_{\mathrm{out}}}. \]

The slopes of the branches of the real characteristic

\[ \frac{di}{du}=\frac{1}{Z_{\mathrm{out}}} \]

are not identical, since for the left branch, corresponding to the blocked left tube \(L_1\),

\[ \frac{1}{Z_{\mathrm{out}}}=\frac{1}{R_a}+\frac{1}{R_i+R_k(1+\mu)}, \]

whereas for the right branch, obtained with tube \(L_2\) blocked,

\[ \frac{1}{Z_{\mathrm{out}}}=\frac{1}{R_a}. \]

The chosen approximation does not allow one to take into account the difference in the slope of the branches. For sufficiently large \(R_i\), \(R_k\), and \(\mu\), and not very large \(R_a\), when it may be assumed that \(R_a \ll R_i+R_k(1+\mu)\), this difference may be small. We shall therefore neglect it and shall assume

\[ \alpha=\frac{1}{R_a}. \tag{6} \]

For \(u=0\), formula (4) gives

\[ \left(\frac{di}{du}\right)_{u=0}=\alpha-\beta . \]

On the other hand, in this case

\[ \left(\frac{di}{du}\right)_{u=0}=-\frac{1}{\rho}, \]

where \(\rho\) is the absolute value of the negative resistance for the middle “falling” section. Thus

Thus,

\[ \alpha-\beta=-\frac{1}{\rho}\quad \text{or}\quad \beta=\alpha+\frac{1}{\rho}. \tag{6'} \]

The quantity \(\rho\) can always be calculated using the formulas for the conductance of a two-terminal network with cathode coupling [13].

The coefficient \(\gamma\) determines the curvature of the volt-ampere characteristic. The position of its bends and, consequently, the position of its branches depend on it. We shall choose the value of the coefficient \(\gamma\) in such a way as to obtain coincidence of the asymptotes of the approximating function (5) with the rectilinear branches of the volt-ampere characteristic. To do this, let us find the equation of the right-hand branch of the volt-ampere characteristic corresponding to the locked second triode \((i_{a2}=0)\). From the circuit of Fig. 2, for \(i_{a2}=0\), we directly obtain

\[ i'_{a2}=-i \quad \text{and}\quad u+u_c=iR_a+E_a . \tag{7} \]

If the capacitance \(C_1\) is sufficiently large and represents a small resistance for currents of the generated frequency, then it may be assumed that, in the case of a symmetrical volt-ampere characteristic, the voltage \(u_c\) remains constant and equal to the voltage at the output of the amplifier with cathode coupling\(^1\) for \(u=0\).

\[ u_c=E_a-i_{a20}R_a, \tag{8} \]

where \(i_{a20}\) is the dc component of the anode current of the right-hand triode in the amplifier with cathode coupling.

Substituting (8) into (7), we find the equation of the right-hand branch

\[ i=-\frac{u}{R_a}-i_{a20}. \tag{9} \]

The equation of the corresponding asymptote of function (5) will be

\[ i=-\frac{u}{R_a}-\frac{\pi\beta}{4\sqrt{\gamma}}. \tag{10} \]

Equating the constant terms in (7) and (8), we obtain

\[ \gamma=\left(\frac{\pi\beta}{4i_{a20}}\right)^2 . \tag{11} \]

The value \(i_{a20}\), for chosen \(R_k\) and \(R_a\), can be found graphically [14]. Figure 4 shows an example of the approximation described.

Differential Equation of the Oscillator

In setting up the differential equation of the oscillator, we shall proceed from its equivalent circuit shown in Fig. 5. Here the two-terminal network with cathode coupling is replaced by a nonlinear active resistance \(\rho\), having exactly the same volt-ampere characteristic as the two-terminal network. We shall not take into account the change in the shape of this characteristic due to the action of parasitic capacitances on

\(^1\) That is, when \(C_1\) is disconnected.

Fig. 4. Approximation of the volt-ampere characteristic of a cathode-coupled oscillator. 1—the volt-ampere characteristic; 2—plot of the approximating function.

high frequencies. We shall assume its form to be the same for all frequencies. On the basis of Kirchhoff’s second law, for the oscillatory circuit of the circuit in Fig. 5 one may write:

Fig. 5

\[ \frac{1}{C}\int i_k\,dt+L\frac{di_k}{dt}+Ri_k = L\frac{di}{dt}+Ri . \tag{12} \]

Since

\[ \frac{1}{C}\int i_k\,dt=u, \]

and \(i=\varphi(u)\), and consequently,

\[ i_k=C\frac{du}{dt},\qquad \frac{di}{dt}=\varphi'(u), \]

equation (12) may also be rewritten as

\[ \frac{d^2u}{dt^2} + \left[ \frac{R}{L}+\frac{1}{C}\varphi'(u) \right]\frac{du}{dt} + \frac{R}{LC}\varphi(u) + \frac{1}{LC}u =0 . \tag{12′} \]

Denoting \(\dfrac{1}{\sqrt{LC}}=\omega_0\) and \(\omega_0t=x\), one may reduce the last equation to the form

\[ \frac{d^2u}{dx^2} + d\left[1+Z_p\varphi'(u)\right]\frac{du}{dx} + d^2Z_p\varphi(u) + u =0, \tag{12″} \]

where \(d=\dfrac{R}{\omega_0L}\) is the damping of the circuit, and \(Z_p\) is its resonant resistance. The penultimate term in (12″) is small in comparison with the adjacent terms \(\left|d^2Z_p\varphi(u)=d^2Z_p i\simeq d^2u\ll u\right|\). Neglecting it, one can obtain the oscillator equation in a simpler form

\[ \frac{d^2u}{dx^2} + d\left[1+Z_p\varphi'(u)\right]\frac{du}{dx} + u =0 . \tag{13} \]

The legitimacy of the indicated neglect, however, raises doubts, since the other nonlinear term of (12″),

\[ d\left[1+Z_p\varphi'(u)\right]\frac{du}{dx}, \]

for oscillations close to sinusoidal, is also small in comparison with \(u\), and it cannot be said in advance that discarding \(d^2Z_p\varphi(u)\) will not lead to appreciable errors in the formulas of the nonlinear theory. Therefore the calculations of the amplitude \(U_m\), the nonlinear correction to the period \(\tau\), and the harmonic coefficients \(k_{fn}\) were carried out on the basis of the oscillator equation in the form (12″). It turned out that retaining the term \(d^2Z_p\varphi(u)\) is not reflected in the results of calculating the amplitude of the zeroth approximation. In calculating the nonlinear correction and the harmonics, however, neglecting

\(d^2 Z_p\varphi(u)\) is inadmissible, since it would mean discarding terms of the same order as those retained. Taking the foregoing into account, in computing the amplitude in the present paper we shall proceed from the simplified equation (13), while in computing \(\tau\) and \(k_{fn}\) we shall use the complete equation \((12'')\).¹

Amplitude of Oscillations

In computing the amplitude of oscillations and the nonlinear correction to the period, we shall use the method of the small parameter in the form in which it is set forth in the well-known work of A. A. Andronov and C. E. Chaikin, Theory of Oscillations [12]. To use the indicated method, let us write the oscillator equation (13) in the form

\[ \frac{d^2u}{dx^2}+u=-d\left[1+Z_p\varphi'(u)\right]\frac{du}{dx} =df\left(u,\frac{du}{dx}\right). \tag{13'} \]

As the small parameter we shall take the damping of the circuit, \(d\). Owing to the smallness of this quantity, the solution of equation \((13')\) will be close to the so-called generating solution (obtained for \(d=0\)). The latter may be represented in the form

\[ u=U_m\cos x=\psi_0(x), \tag{14} \]

since the initial phase, owing to the autonomy of the system, is arbitrary. The solution of the complete equation \((13')\), when using the small-parameter method, is sought in the form of the series

\[ u=\psi_0(x)+A\beta_1+B\beta_2+Cd+D\beta_1d+E\beta_2d+Fd^2+\ldots, \tag{15} \]

where \(A, B, C, D\ldots\) are periodic functions of \(x\), and the small parameters \(\beta_1\) and \(\beta_2\) are determined by the initial conditions for \(d\ne0\):

\[ u(0)=\psi_0(0)+\beta_1 \quad\text{and}\quad u'(0)=\psi'_0(0)+\beta_2. \tag{16} \]

The functions \(A\) and \(B\) turn out to be purely harmonic: \(A=\cos x\), \(B=\sin x\). The remaining coefficients of the series are expressed, as is known, by integrals of the form

\[ y=\int_{0}^{x}\Phi(v)\sin(x-v)\,dv. \tag{17} \]

The amplitude of oscillations of the fundamental frequency—the so-called “amplitude of the first approximation” when using the small-parameter method—is determined from the equation \(C(2\pi)=0\), which follows from the requirement of periodicity of the solution (15).

Since for what follows it is necessary to know \(C(x)\), we shall begin with the calculation of this function, and the required \(C(2\pi)\) will then be found as a particular value. As is known [12], \(C(x)\) is determined by the integral

\[ C(x)=\int_{0}^{x} f\left(u,\frac{du}{dx}\right)\sin(x-v)\,dv . \]

In our case

\[ f\left(u,\frac{du}{dx}\right)=-\left[1+Z_p\varphi'(u)\right]\cdot\frac{du}{dx}, \tag{18} \]

where \(\varphi'(u)\) is determined by formula (4). Taking into account (11), (4), and (15), we find

\[ C(x)=\int_{0}^{x}\left[1+\alpha Z_p-\frac{\beta Z_p}{\left(1+\gamma U_m^2\cos^2 v\right)^2}\right]U_m\sin v\sin(x-v)\,dv . \]

To compute this integral, let us split it into four simpler ones:

\[ C(x)=C_1(x)+C_2(x)+C_3(x)+C_4(x). \]

\[ C_1(x)=(1+\alpha Z_p)U_m\sin x\int_{0}^{x}\sin v\cos v\,dv =(1+\alpha Z_p)\frac{U_m}{2}\sin^3 x . \tag{19} \]

\[ C_2(x)=-(1+\alpha Z_p)U_m\cos x\int_{0}^{x}\sin^2 v\,dv= \]

\[ =(1+\alpha Z_p)\frac{U_m}{2}\left(\sin x\cos^2 x-x\cos x\right). \tag{20} \]

\[ C_3(x)=-\beta Z_p U_m\sin x\int_{0}^{x}\frac{\sin v\cos v\,dv}{\left(1+\gamma U_m^2\cos^2 v\right)^2}= \]

\[ =\frac{\beta Z_p\sin x}{2\gamma U_m(1+\gamma U_m^2)} -\frac{\beta Z_p\sin x}{2\gamma U_m\left(1+\gamma U_m^2\cos^2 x\right)} . \tag{21} \]

\[ C_4(x)=\beta Z_p U_m\cos x\int_{0}^{x} \frac{\sin^2 v\,dv}{\left(1+\gamma U_m^2\cos^2 v\right)^2}= \]

\[ =-\frac{\beta Z_p U_m\sin x\cos^2 x}{2\left(1+\gamma U_m^2\cos^2 x\right)} +\frac{\beta Z_p U_m\cos x}{2\sqrt{1+\gamma U_m^2}} \operatorname{arctg}\frac{\operatorname{tg}x}{\sqrt{1+\gamma U_m^2}} . \tag{22} \]

Adding (19), (20), (21), and (22), we obtain

\[ C(x)=(1+\alpha Z_p)\frac{U_m}{2}(\sin x-x\cos x) -\frac{\beta Z_p U_m\sin x}{2(1+\gamma U_m^2)}+ \]

\[ +\frac{\beta Z_p U_m \cos x}{2\sqrt{1+\gamma U_m^2}}\, \arctan \frac{\tan x}{\sqrt{1+\gamma U_m^2}} . \tag{23} \]

Substituting \(x=2\pi\) into (23) and equating the resulting expression to zero, we find the equation for the amplitudes

\[ U_m\left(1+\alpha Z_p-\frac{\beta Z_p}{\sqrt{1+\gamma U_m^2}}\right)=0. \tag{24} \]

This equation has the roots

\[ U_{m1}=0,\qquad U_{m2}=\sqrt{\frac{\left(\dfrac{\beta Z_p}{1+\alpha Z_p}\right)^2-1}{\gamma}} \quad \text{and} \]

\[ U_{m3}=-\sqrt{\frac{\left(\dfrac{\beta Z_p}{1+\alpha Z_p}\right)^2-1}{\gamma}} . \tag{25} \]

The third root (25) has no physical meaning, since the amplitude of oscillations cannot be negative.

Stability of the Solution Obtained

In order to solve the problem of the oscillations produced by the generator, it is necessary not only to determine the periodic solution of the differential equation of the generator, but also to investigate the stability of the solution obtained. The oscillation regime will be stable if the oscillatory system is capable of restoring the regime when it is disturbed as a result of the action of external perturbations. Otherwise the regime under investigation will be unstable and cannot continue for a long time.

The condition for stability of the solution (15) has the form [12]:

\[ \int_0^{2\pi} f'_{u'}(\psi,\psi')\,dx<0. \tag{26} \]

In our case

\[ f'_{u'}(\psi,\psi') = -\left[1+Z_p\varphi'(\psi)\right] = -\left[ 1+\alpha Z_p - \frac{\beta Z_p}{\left(1+\gamma U_m^2\cos^2 x\right)^2} \right], \]

and condition (26) is written as follows:

\[ -\left(1+\alpha Z_p\right)\cdot 2\pi+\beta Z_p \int_0^{2\pi} \frac{dx}{\left(1+\gamma U_m^2\cos^2 x\right)^2}<0. \tag{26'} \]

Let us evaluate the integral entering the last inequality, first taking the upper limit to be \(x\).

\[ \int_0^x \frac{dx}{\left(1+\gamma U_m^2\cos^2 x\right)^2} = \frac{1+\dfrac{\gamma U_m^2}{2}}{\left(1+\gamma U_m^2\right)^{3/2}} \arctg \frac{\tg x}{\sqrt{1+\gamma U_m^2}} - \]

\[ -\frac{\gamma U_m^2}{1+\gamma U_m^2}\cdot \frac{\cos x\sin x}{2\left(1+\gamma U_m^2\cos^2 x\right)}. \]

Putting here \(x=2\pi\) and substituting into \((26')\), we obtain the stability condition in the form:

\[ \frac{\beta Z_p\left(1+\dfrac{1}{2}\gamma U_m^2\right)} {\left(1+\gamma U_m^2\right)^{3/2}} <1+\alpha Z_p. \]

For the first root of equation (13), \(U_{m1}=0\), we shall have:

\[ \beta Z_p<1+\alpha Z_p. \]

For the second root \(U_{m2}\), taking expression (25) into account, we obtain

\[ \beta Z_p>1+\alpha Z_p. \tag{27} \]

This inequality is the condition for self-excitation, since when it is not satisfied the amplitude of the oscillations is equal to zero and differs from zero when condition (27) is satisfied.

Dividing both sides of inequality (27) by \(Z_p\) and using relation (6), we obtain the self-excitation condition in the form

\[ \frac{1}{\rho}>\frac{1}{Z_p}. \]

Thus, for oscillations to exist, we have the evident condition: the conductance of the cathode-coupled two-terminal network must be greater than the conductance of the parallel circuit connected to it.

Nonlinear Correction to the Period

We shall now consider the question of the influence of the nonlinearity of the volt-ampere characteristic of the cathode-coupled two-terminal network on the generator frequency. The small-parameter method used here makes it possible to determine the nonlinear correction from the periodicity condition for solution (15) and its derivative.

The first-approximation expression for the nonlinear correction to the period has, as is known [12], the following form:

\[ \tau = d \cdot \frac{C'(2\pi)}{U_m}, \tag{28} \]

where \(d\) is the damping of the circuit, \(U_m\) is the oscillation amplitude, and

\[ C'(2\pi)=\int\limits_0^{2\pi} f[\psi_0(v),\psi'_0(v)]\cos v\,dv . \tag{29} \]

The quantity \(f(\psi_0,\psi'_0)\) entering (29), according to equation \((12'')\), is equal to

\[ f(\psi_0,\psi'_0)=-[1+Z_p\varphi'(\psi_0)]\psi'_0-dZ_p\varphi(\psi_0). \tag{30} \]

The term \(d^2 Z_p\varphi(\psi_0)\) can no longer be neglected when finding the small nonlinear correction. Substituting (30), (4), (5), and (14) into (29), we obtain

\[ \begin{aligned} C'(2\pi) &=\int\limits_0^{2\pi} \Biggl\{ \Biggl[ 1+aZ_p-\frac{\beta Z_p}{(1+\gamma U_m^2\cos^2 v)^2} \Biggr]U_m\sin v \\ &\qquad -dZ_p\Biggl[ aU_m\cos v -\frac{\beta U_m\cos v}{2(1+\gamma U_m^2\cos^2 v)} -\frac{\beta}{2\sqrt{\gamma}}\operatorname{arctg}(\sqrt{\gamma}U_m\cos v) \Biggr] \Biggr\}\cos v\,dv \\ &=C'_1(2\pi)+C'_2(2\pi)+C'_3(2\pi)+C'_4(2\pi). \end{aligned} \]

Here

\[ C'_1(2\pi)= \int\limits_0^{2\pi} \left\{ 1+Z_p\left[ a-\frac{\beta}{(1+\gamma U_m^2\cos^2 v)^2} \right] \right\} U_m\sin v\cos v\,dv=0 \]

\[ C'_2(2\pi)= -dZ_p aU_m\int\limits_0^{2\pi}\cos^2 v\,dv =-dZ_p aU_m\pi, \tag{31} \]

\[ C'_3(2\pi)= \frac{dZ_p\beta U_m}{2} \int\limits_0^{2\pi} \frac{\cos^2 v\,dv}{1+\gamma U_m^2\cos^2 v} = \]

\[ = \frac{dZ_p\beta\pi}{\gamma U_m} \left( 1-\frac{1}{\sqrt{1+\gamma U_m^2}} \right), \tag{32} \]

\[ C'_{\overline{4}}(2\pi)=\frac{d Z_p \beta}{2\sqrt{\gamma}} \int_{0}^{2\pi}\cos v\, \operatorname{arctg}(\sqrt{\gamma}\,U_m\cos v)\,dv =-\frac{d Z_p \beta\pi}{\gamma U_m} \cdot\left(1-\sqrt{1+\gamma U_m^2}\right). \tag{33} \]

Adding (31), (32), and (33), and taking (24) into account, we find

\[ C'(2\pi)=\pi d U_m . \tag{34} \]

The coefficient \(C'(2\pi)\) thus turns out to be a quantity of the first order of smallness. This means that formula (28) cannot be used in the present case. Indeed, it was obtained by discarding terms of the second and higher orders of smallness in the conditions

\[ -U_m\frac{\tau^2}{2}+C(2\pi)d+C'(2\pi)\tau d+D(2\pi)\beta_1 d+F(2\pi)d^2+\cdots \tag{35} \]

and

\[ -U_m\tau+C'(2\pi)d+C''(2\pi)\tau d+D'(2\pi)\beta_1 d+F'(2\pi)d^2+\cdots \tag{36} \]

for the periodicity of the solution (15).

In our case it is necessary also to discard the term \(C'(2\pi)d=\pi U_m d^2\). As a result, instead of (28) one obtains the equation \(U_m\tau=0\). The latter indicates that the nonlinear correction is of the second order of smallness. Discarding in (35) and (36) terms of the third and higher orders of smallness and substituting \(C(2\pi)=0\), we obtain

\[ \beta_1=-\frac{F(2\pi)}{D(2\pi)}\,d, \tag{37} \]

\[ \tau=d\cdot \frac{ C'(2\pi)-\dfrac{D'(2\pi)}{D(2\pi)}F(2\pi)d+F'(2\pi)d }{ U_m }. \tag{38} \]

The coefficient \(D(2\pi)\) for \(C(2\pi)=0\) is expressed [12] by the formula

\[ D(2\pi)=\int_{0}^{2\pi}\frac{\partial f}{\partial \psi_0}\,dv. \]

In our case it is equal to:

\[ D(2\pi)=-\int_{0}^{2\pi} \left[ 1+\alpha Z_p-\frac{\beta Z_p}{\left(1+\gamma U_m^2\cos^2 v\right)^2} \right]\,dv== \]

\[ = -\frac{\pi \beta \gamma Z_p U_m^2}{(1+\gamma U_m^2)^{3/2}} . \tag{39} \]

We shall now find \(D'(2\pi)\). This coefficient can be determined [12] by the formula:

\[ D'(2\pi)=\int_{0}^{2\pi}\frac{\partial f}{\partial \psi_0}\,dv-\frac{C'(2\pi)}{U_m}. \]

For our case:

\[ D'(2\pi)=2\beta Z_p\gamma U_m^2 \int_{0}^{2\pi} \frac{\cos v \sin v\,dv}{(1+\gamma U_m^2\cos^2 v)^3} - \]

\[ -dZ_p\int_{0}^{2\pi} \left[ \alpha-\frac{\beta}{(1+\gamma U_m^2\cos^2 v)^2} \right]\,dv - \]

\[ -\pi d = \pi d \left[ 1-\frac{\beta\gamma Z_p U_m^2}{(1+\gamma U_m^2)^{3/2}} \right]. \tag{40} \]

The coefficient \(F(2\pi)\) is determined [12] by the integral:

\[ F(2\pi)= -\int_{0}^{2\pi} \left[ \frac{\partial f}{\partial \psi_0}C(v) + \frac{\partial f}{\partial \psi'_0}C'(v) \right]\sin v\,dv. \tag{41} \]

The function \(f(\psi_0,\psi'_0)\) entering here is expressed by formula (30), while \(C(v)\) must be computed taking into account the term \(dZ_p\varphi(\psi_0)\) in the function \(f(\psi_0,\psi'_0)\).

Let us denote \(C(v)=C_1(v)+dC_2(v)\).

Here \(C_1(v)\) is the approximate value of \(C(v)\) obtained when the term \(dZ_p\varphi(\psi_0)\) in (30) is neglected, while \(dC_2(v)\) is added when this term is taken into account. \(C_1(v)\) was computed earlier [formula (23)].

Let us further denote

\[ g(v)=g_1[\psi_0(v)]=-\{1+Z_p\varphi'[\psi_0(v)]\} \]

and

\[ g_2[\psi_0(v)]=-Z_p\varphi[\psi_0(v)]. \tag{42} \]

Then

\[ f[\psi_0(v),\psi'_0(v)]=g_1(\psi_0)\psi'_0(v)+d\,g_2(\psi_0). \]

Formula (41) can now be written in the form

\[ F(2\pi)= -\int_{0}^{2\pi} \frac{d}{dv}\,[g(v)C_1(v)]\sin v\,dv - d\int_{0}^{2\pi} \frac{d}{dv}\,[g(v)C_2(v)]. \]

\[ \cdot \sin \vartheta\, d\vartheta - d \int_{0}^{2\pi} g_2'[\psi_0(\vartheta)] C_1(\vartheta)\sin\vartheta\,d\vartheta - d^2 \int_{0}^{2\pi} g_2[\psi_0(\vartheta)]\cdot \]

\[ \cdot C_2(\vartheta)\sin\vartheta\,d\vartheta = F_1(2\pi)+dF_2(2\pi)+dF_3(2\pi)+d^2F_4(2\pi). \]

Since \(\dfrac{D'(2\pi)}{D(2\pi)}\), according to (39) and (40), is of the first order of smallness with respect to \(d\), in formula (38) \(F_2(2\pi)\), \(F_3(2\pi)\), and \(F_4(2\pi)\) form terms of the third order of smallness, which we have neglected from the very beginning. Thus,

\[ F(2\pi) \simeq F_1(2\pi) = -\int_{0}^{2\pi}\frac{d}{d\vartheta}\,[g(\vartheta)C_1(\vartheta)]\sin\vartheta\,d\vartheta. \tag{43} \]

The form obtained for the expression for \(F(2\pi)\) makes it possible to apply integration by parts and thereby avoid differentiating \(C_1(\vartheta)\), which would lead to a considerable complication of the integrand in (41).

\[ F(2\pi)=\int_{0}^{2\pi} g(\vartheta)C_1(\vartheta)\cos\vartheta\,d\vartheta. \tag{43′} \]

Substituting here the expanded expressions for \(C_1(\vartheta)\) and \(g(\vartheta)\) from (23), (41), (4), and (14), we obtain:

\[ F(2\pi)= -\int_{0}^{2\pi} \left[ 1+\alpha Z_p -\frac{\beta Z_p}{\left(1+\gamma U_m^2\cos^2\vartheta\right)^2} \right] \]

\[ \times \left[ (1+\alpha Z_p)\frac{U_m}{2}(\sin\vartheta-\vartheta\cos\vartheta) -\frac{\beta Z_p U_m}{2(1+\gamma U_m^2)}\sin\vartheta +\right. \]

\[ \left. +\frac{\beta Z_p U_m}{2\sqrt{1+\gamma U_m^2}} \cos\vartheta\,\arctg \frac{\tg\vartheta}{\sqrt{1+\gamma U_m^2}} \right] \cos\vartheta\,d\vartheta. \tag{44} \]

Integral (44) can be split into 5 simpler ones,

\[ F(2\pi)\simeq F_1(2\pi) =F_{11}(2\pi)+F_{12}(2\pi)+F_{13}(2\pi)+F_{14}(2\pi)+F_{15}(2\pi). \]

Here

\[ F_{11}(2\pi)=-\frac{U_m}{2}\int\limits_0^{2\pi} \left[(1+\alpha Z_p)-\frac{\beta Z_p}{(1+\gamma U_m^2)}\right] \left[1+\alpha Z_p-\frac{\beta Z_p}{(1+\gamma U_m^2\cos^2 v)^2}\right] \sin v\cos v\,dv=0, \]

\[ F_{12}(2\pi)=(1+\alpha Z_p)^2\frac{U_m}{2} \int\limits_0^{2\pi} v\cos^2 v\,dv =\frac{\pi^2 U_m}{2}(1+\alpha Z_p)^2, \tag{45} \]

\[ F_{13}(2\pi)=-(1+\alpha Z_p)\frac{\beta Z_p U_m}{2} \int\limits_0^{2\pi} \frac{v\cos^2 v\,dv}{(1+\gamma U_m^2\cos^2 v)^2} = \]

\[ =-\frac{\pi^2\beta Z_p(1+\alpha Z_p)U_m} {2(1+\gamma U_m^2)^{3/2}}, \tag{46} \]

\[ F_{14}(2\pi)=- \frac{(1+\alpha Z_p)\beta Z_pU_m} {2\sqrt{1+\gamma U_m^2}} \int\limits_0^{2\pi}\cos^2 v\,\operatorname{arc\,tg} \frac{\operatorname{tg} v}{\sqrt{1+\gamma U_m^2}}\,dv = \]

\[ =-\frac{\pi^2(1+\alpha Z_p)\beta Z_pU_m} {2\sqrt{1+\gamma U_m^2}}, \tag{47} \]

\[ F_{15}(2\pi)= \frac{\beta^2 Z_p^2 U_m}{2\sqrt{1+\gamma U_m^2}} \int\limits_0^{2\pi} \operatorname{arc\,tg}\frac{\operatorname{tg} v}{\sqrt{1+\gamma U_m^2}} \cdot \frac{\cos^2 v\,dv}{(1+\gamma U_m^2\cos^2 v)^2} = \]

\[ =\frac{\pi^2\beta^2 Z_p^2 U_m}{2(1+\gamma U_m^2)^2}. \tag{48} \]

Adding (45), (46), (47), and (48), and taking (24) into account, we obtain

\[ F(2\pi)=0. \]

The nonlinear correction to the period will now be determined by the formula

\[ \tau=\frac{C'(2\pi)d+F'(2\pi)d^2}{U_m}. \tag{49} \]

It is sufficient to compute the coefficient \(F'(2\pi)\) with accuracy up to terms of order \(d\). This coefficient, as is known [12], is determined by the formula

\[ F'(2\pi)=\int\limits_{0}^{2\pi} \left[ \frac{\partial f}{\partial \psi_0} C(v) + \frac{\partial f}{\partial \psi'_0} C'(v) \right]\cos v\,dv . \tag{50} \]

The part of the integrand enclosed in square brackets here is exactly the same as in (41). We can therefore, analogously to the preceding, write our integral in the form

\[ F'(2\pi)=\int\limits_{0}^{2\pi} \frac{d}{dv}\,[g(v)C_1(v)]\cos v\,dv = \int\limits_{0}^{2\pi} g(v)C_1(v)\sin v\,dv . \]

Substituting here the expanded expressions for the functions \(g(v)\) and \(C_1(v)\), we shall have

\[ F'(2\pi)=- \int\limits_{0}^{2\pi} \left[ 1+\alpha Z_p - \frac{\beta Z_p}{(1+\gamma U_m^2\cos^2 v)^2} \right] \left[ (1+\alpha Z_p)\frac{U_m}{2}(\sin v-v\cos v) \right. \]

\[ \left. -\frac{\beta Z_p U_m}{2(1+\gamma U_m^2)}\sin v -\frac{\beta Z_p U_m}{2\sqrt{1+\gamma U_m^2}} \cos v\,\operatorname{arctg} \frac{\operatorname{tg} v}{\sqrt{1+\gamma U_m^2}} \right]\sin v\,dv . \]

This integral can be represented as the sum of six simpler integrals

\[ F'(2\pi)=F'_{11}(2\pi)+F'_{12}(2\pi)+F'_{13}(2\pi)+F'_{14}(2\pi)+F'_{15}(2\pi)+F'_{16}(2\pi). \]

Here

\[ F'_{11}(2\pi)= -(1+\alpha Z_p) \left( 1+\alpha Z_p-\frac{\beta Z_p}{1+\gamma U_m^2} \right) \frac{U_m}{2} \int\limits_{0}^{2\pi}\sin^2 v\,dv = \]

\[ = -(1+\alpha Z_p) \left( 1+\alpha Z_p-\frac{\beta Z_p}{1+\gamma U_m^2} \right) \frac{U_m\pi}{2}, \tag{51} \]

\[ F'_{12}(2\pi)= \beta Z_p\frac{U_m}{2} \left( 1+\alpha Z_p-\frac{\beta Z_p}{1+\gamma U_m^2} \right) \int\limits_{0}^{2\pi} \frac{\sin^2 v\,dv}{(1+\gamma U_m^2\cos^2 v)^2} = \]

\[ = \frac{\beta Z_p U_m}{2}\left(1+\alpha Z_p-\frac{\beta Z_p}{1+\gamma U_m^2}\right)\frac{\pi}{\sqrt{1+\gamma U_m^2}} . \tag{52} \]

\[ F'_{13}(2\pi)=(1+\alpha Z_p)^2\frac{U_m}{2}\int_0^{2\pi} v\cos v\sin v\,dv= \]

\[ =-(1+\alpha Z_p)^2\frac{\pi U_m}{4}, \tag{53} \]

\[ F'_{14}(2\pi)=-(1+\alpha Z_p)\frac{\beta Z_p U_m}{2} \int_0^{2\pi}\frac{v\cos v\sin v\,dv}{(1+\gamma U_m^2\cos^2 v)^2}= \]

\[ =-\frac{(1+\alpha Z_p)\beta Z_p}{2(1+\gamma U_m^2)\gamma U_m} \left(1-\sqrt{1+\gamma U_m^2}\right), \tag{54} \]

\[ F'_{15}(2\pi)=- \frac{(1+\alpha Z_p)\beta Z_p U_m}{2\sqrt{1+\gamma U_m^2}} \int_0^\pi \sin v\cos v\arctan \frac{\tan v}{\sqrt{1+\gamma U_m^2}}\,dv= \]

\[ =-(1+\alpha Z_p)\frac{\beta Z_p\pi}{2\gamma U_m} \left(\sqrt{1+\gamma U_m^2}-1\right), \tag{55} \]

\[ F'_{16}(2\pi)= \frac{\beta^2 Z_p^2 U_m}{2\sqrt{1+\gamma U_m^2}} \int_0^{2\pi}\arctan\frac{\tan v}{\sqrt{1+\gamma U_m^2}}\cdot \frac{\sin v\cos v\,dv}{(1+\gamma U_m^2\cos^2 v)}= \]

\[ =-\frac{\pi\beta^2 Z_p^2}{4(1+\gamma U_m^2)^{3/2}} . \tag{56} \]

Adding (51), (52), (53), (54), (55), and (56), we obtain

\[ F'(2\pi)=(1+\alpha Z_p)^2\frac{\pi U_m}{4} \left( 1-\frac{1}{\sqrt{1+\gamma U_m^2}} +\frac{4}{\gamma U_m^2} -\frac{2}{\gamma U_m^2\sqrt{1+\gamma U_m^2}} -\frac{2}{\gamma U_m^2}\sqrt{1+\gamma U_m^2} \right). \tag{57} \]

Let us now denote

\[ \frac{1}{\sqrt{1+\gamma U_m^2}}=\frac{(1+\alpha Z_p)}{\beta Z_p}=\varepsilon . \tag{58} \]

Then

\[ F'(2\pi)=(1+\alpha Z_p)^2\,\frac{\pi U_m}{4}\cdot\frac{(1-\varepsilon)^2}{(1+\varepsilon)} . \tag{59} \]

Substituting (34) and (59) into (49), we obtain

\[ \tau=\pi d^2\left[1+\frac{(1+\alpha Z_p)^2(1-\varepsilon)^2}{4(1+\varepsilon)}\right]. \tag{60} \]

Hence the relative change in frequency is equal to:

\[ \frac{\Delta f}{f}=-\frac{\Delta T}{T}=-\frac{\tau}{2\pi} =-\frac{d^2}{2}\left[1+\frac{(1+\alpha Z_p)^2(1-\varepsilon)^2}{4(1+\varepsilon)}\right]. \tag{61} \]

Here the first term, \(-d^2/2\), not connected with the approximation coefficients, represents the “linear correction” due to the presence of active resistance in the inductive branch of the circuit. Indeed, the exact frequency of the circuit shown in Fig. 5 is determined by the formula

\[ \omega=\frac{1}{\sqrt{LC}}\sqrt{1-\left(\frac{R}{\omega_0 L}\right)^2} \simeq \omega_0\left(1-\frac{d^2}{2}\right). \]

Discarding \(-d^2/2\), for the nonlinear correction proper we find

\[ \frac{\Delta f}{f} =-\frac{d^2}{8}\,\frac{(1+\alpha Z_p)^2(1-\varepsilon)^2}{(1+\varepsilon)} = \]

\[ -\frac{w_0^2\left(\frac{1}{R_a}+\frac{1}{Z_p}\right)^2(1-\varepsilon)^2}{8(1+\varepsilon)}, \tag{62} \]

where \(w_0\) is the characteristic impedance of the circuit.

It is seen from formula (62) that the nonlinear correction becomes smaller the smaller the damping of the circuit and its characteristic impedance. It tends to zero as the circuit approaches the threshold of self-excitation and decreases with increasing \(R_a\).

Oscillator Harmonics

We shall calculate the nonlinearity coefficients for the individual harmonics using the zeroth approximation for the amplitude of the fundamental oscillation and the first approximation for the amplitudes of the harmonics. The solution of the first approximation is obtained by discarding terms

of order \(d^2\) and terms of higher orders of smallness. In accordance with (15) and (37), it is equal to

\[ u=U_m\cos v+dC(v)\simeq U_m\cos v+dC_1(v)=U_m\cos v+ \]

\[ +d\left[(1+\alpha Z_p)\frac{U_m}{2}(\sin v-v\cos v) -\frac{\beta Z_pU_m}{2(1+\gamma U_m^2)}\sin v+ \right. \]

\[ \left. +\frac{\beta Z_pU_m}{2\sqrt{1+\gamma U_m^2}}\cos v\operatorname{arc\,tg} \frac{\operatorname{tg}v}{\sqrt{1+\gamma U_m^2}} \right]. \]

In order to use this solution for determining the nonlinearity coefficients for the individual harmonics, it is necessary to expand the periodic function \(C_1(v)\) in a Fourier series:

\[ C_1(v)=\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos nv+b_n\sin nv). \]

The constant term of this expansion is evidently equal to zero. By direct integration one can verify that the coefficient

\[ a_n=\frac{1}{\pi}\int_{0}^{2\pi} C_1(v)\cos nv\,dv \]

is also equal to zero for \(n\ne1\). We shall not compute the coefficients \(a_1\) and \(b_1\), since the small correction

\[ \Delta U_m=d\sqrt{a_1^2+b_1^2} \]

to the amplitude of the fundamental oscillation, given by the first approximation, has practically no effect on the magnitude of the nonlinearity coefficients.

We now proceed to determine \(b_n\).

\[ b_n=\frac{1}{\pi}\int_{0}^{2\pi} C_1(v)\sin nv\,dv=b_{n1}+b_{n2}+b_{n3}. \]

Here

\[ b_{n1}=\frac{1}{\pi}\left[(1+\alpha Z_p)-\frac{\beta Z_p}{1+\gamma U_m^2}\right] \frac{U_m}{2}\int_{0}^{2\pi}\sin v\sin nv\,dv=0, \]

\[ b_{n2}=-(1+\alpha Z_p)\frac{U_m}{2\pi} \int_{0}^{2\pi}v\cos v\sin nv\,dv =(1+\alpha Z_p)U_m\frac{n}{n^2-1}, \tag{63} \]

\[ b_{n3}=\frac{\beta Z_pU_m}{2\pi\sqrt{1+\gamma U_m^2}} \int_{0}^{2\pi}\cos v\sin nv\,\operatorname{arc\,tg} \frac{\operatorname{tg}v}{\sqrt{1+\gamma U_m^2}}\,dv= \]

\[ = - \frac{\beta Z_0 U_m}{\sqrt{1+\gamma U_m^2}}\cdot \frac{n}{n^2-1} + \frac{\beta Z_0 U_m}{4\pi} \cdot \int_0^{2\pi} \left[ \frac{\cos (n+1)v}{(n+1)} + \frac{\cos (n-1)v}{(n-1)} \right] \cdot \frac{dv}{1+\gamma U_{2m}\cos^2 v}. \tag{64} \]

Adding (63) and (64) and using equality (24), we obtain

\[ b_n = -\frac{\beta Z_0 U_m}{4\pi} \left[ \frac{1}{n+1} \int_0^{2\pi} \frac{\cos (n+1)v}{1+\gamma U_m^2\cos^2 v}\,dv + \frac{1}{n-1} \int_0^{2\pi} \frac{\cos (n-1)v}{1+\gamma U_m^2\cos^2 v}\,dv \right]. \tag{65} \]

In the case of even \(n\), the integrals (65) are equal to zero. For odd \(n\) the coefficients \((n+1)\) and \((n-1)\) turn out to be even, and the integrals (65) take the form

\[ h_k = \frac{1}{2k} \int_0^{2\pi} \frac{\cos 2kv\,dv}{1+\gamma U_m^2\cos^2 v} = \frac{1}{2k} \int_0^{2\pi} \frac{\cos 2kv\,dv}{1+\dfrac{1-\varepsilon^2}{\varepsilon^2}\cdot \cos^2 v}. \tag{66} \]

For such integrals it is not difficult to obtain the recurrence formula

\[ (k+1)h_{k+1} = -2k\,\frac{1+\varepsilon^2}{1-\varepsilon^2}\,h_k - (k-1)h_{k-1}. \tag{67} \]

Using this formula, we find

\[ h_k = (-1)^k \frac{\pi\varepsilon(1-\varepsilon)^k}{k(1+\varepsilon)^k}. \tag{68} \]

On the basis of (65) and (68), for odd \(n\) we have

\[ b_n = \frac{\beta Z_0 U_m \varepsilon}{2} \left[ \frac{(-1)^{\frac{n+1}{2}}}{n+1} \left( \frac{1-\varepsilon}{1+\varepsilon} \right)^{\frac{n+1}{2}} + \frac{(-1)^{\frac{n-1}{2}}}{n-1} \left( \frac{1-\varepsilon}{1+\varepsilon} \right)^{\frac{n-1}{2}} \right]. \]

Whence the coefficient of nonlinearity for the \(n\)-th harmonic is equal to

\[ k_{fn}=\frac{d b_n}{U_m} =-\frac{d\beta Z_p \varepsilon}{2} \left[ \frac{1}{n+1}\left(-\frac{1-\varepsilon}{1+\varepsilon}\right)^{\frac{n+1}{2}} + \frac{1}{n-1}\left(-\frac{1-\varepsilon}{1+\varepsilon}\right)^{\frac{n-1}{2}} \right]. \tag{69} \]

Denoting

\[ \frac{1-\varepsilon}{1+\varepsilon} = \frac{\frac{1}{\rho}-\frac{1}{Z_p}} {\frac{1}{\rho}+\frac{1}{Z_p}+\frac{2}{R_a}} =\xi, \]

it can be given an even simpler form

\[ k_{fn}= \frac{w_0}{2} \left( \frac{1}{Z_p}+\frac{1}{R_a} \right) \left[ \frac{(-\xi)^{\frac{n+1}{2}}}{n+1} + \frac{(-\xi)^{\frac{n-1}{2}}}{n-1} \right]. \tag{70} \]

For individual harmonics we obtain:

\[ k_{f3}= \frac{w_0}{2} \left( \frac{1}{Z_p}+\frac{1}{R_a} \right) \left( \frac{\xi^2}{2}-\xi \right), \]

\[ k_{f5}= \frac{w_0}{2} \left( \frac{1}{Z_p}+\frac{1}{R_a} \right) \left( \frac{\xi^2}{2}-\frac{\xi^3}{3} \right) \quad \text{and so on.} \]

The formulas for the nonlinearity coefficients for individual harmonics show that the nonlinear distortions decrease as the wave resistance of the circuit decreases and as the threshold of oscillation is approached, while, in operation far beyond the threshold, they decrease as the resonant resistance and the quality factor of the circuit increase. Furthermore, it is evident from these formulas that the relative share of a given harmonic is the smaller the higher its order. For higher orders this share, when approaching the threshold of oscillation, decreases more rapidly than for lower ones.

Analysis and experimental verification of the obtained formulas

Let us analyze the formula obtained in § 5 for the amplitude of oscillations. To this end we substitute into it the values (6), (61), and (14) for the approximation coefficients. We obtain:

\[ U_m=\frac{4}{\pi} i_{a20} \sqrt{ \frac{1}{\left(\frac{1}{Z_p}+\frac{1}{R_a}\right)^2} - \frac{1}{\left(\frac{1}{\rho}+\frac{1}{R_a}\right)^2} }. \tag{71} \]

It follows from formula (71) that oscillations arise (\(U_m\) is finite) when \(Z_p>\rho\). A gradual increase of \(Z_p\) leads to a smooth—

to the increase of the amplitude. As long as \(Z_p\) remains less than or comparable with \(R_a\), the amplitude grows rapidly. This growth gradually slows down, and for \(Z_p \gg R_a\) a further increase in the resonant resistance of the circuit hardly changes the amplitude. What has been said is confirmed by the curves in Fig. 6. The quantity \(R_a\) has an analogous effect on the amplitude of oscillations. Usually \(R_a\) is several times, or even many times, greater than \(\rho\). Therefore the second term under the radical in (71) changes hardly at all as \(R_a\) increases. The first term, however, increases and leads to a noticeable growth of the amplitude only until \(R_a\) becomes several times greater than the resonant resistance of the circuit \(Z_p\).

The current \(i_{a20}\) entering into (71) is determined, in the graphical calculation of the cathode-coupled amplifier, by the ordinate of the point of the characteristic \(i_{a2} = f(-u_k)\) lying on the same vertical line as the point of intersection of the curve \(i_k = f_1(-u_k)\) and the straight line \(i_k = -\dfrac{u_g}{R_k}\) (Fig. 7). This current and the amplitude \(U_m\), as is seen from the figure, decrease as \(R_k\) increases. An increase in \(R_a\) leads to a certain decrease in \(i_{a20}\). In the case \(R_a \gg Z_p\), this is one of the reasons for the decrease in amplitude with increasing \(R_a\) observed experimentally (Fig. 8).

The tube in formula (71) is represented through \(i_{a20}\) and \(\rho\). The quantity \(i_{a20}\) is the larger, the greater the slopes of the dynamic characteristics \(i_k = f_1(-u_k)\) and \(i_{a2} = f(-u_k)\), and the greater their separation. The resistance \(\rho\) is approximately equal to \(\rho \simeq \dfrac{2}{S}\). Thus, the amplitude will be greater when a tube with small \(\mu\) and high transconductance is used.

When the anode supply voltage \(E_a\) changes, \(i_{a20}\) changes proportionally, since the characteristics \(i_k = f_1(-u_k)\) and \(i_{a2} = f(-u_k)\)

Fig. 6

Dependence of the oscillation amplitude on the resonant resistance of the circuit.
1. \(R_a = 10\ \text{k}\Omega,\ R_k = 1\ \text{k}\Omega;\) 2. \(R_a = 10\ \text{k}\Omega,\ R_k = 3\ \text{k}\Omega;\) 3. \(R_a = 100\ \text{k}\Omega,\ R_k = 3\ \text{k}\Omega\)

are thereby shifted, remaining approximately parallel to their previous positions. This means that the change in \(E_a\) must lead to proportional changes in the amplitude.

Fig. 7

Fig. 9, in which the experimental curves \(U_m(E_a)\) are presented, shows that such

Fig. 8. Dependence of the oscillation amplitude on the resistances \(R_a\) and \(R_k\).
1. \(R_k = 0.5\ \text{k}\Omega\), 2. \(R_k = 1\ \text{k}\Omega\), 3. \(R_k = 2\ \text{k}\Omega\), 4. \(R_k = 3\ \text{k}\Omega\), 5. \(R_k = 5\ \text{k}\Omega\), 6. \(R_k = 10\ \text{k}\Omega\), 7. \(E_a = 240\ \text{V}\), \(E_f = 6.3\ \text{V}\), \(Z_p = 14.7\ \text{k}\Omega\), tube 6N8.

proportionality does indeed hold. Only with a considerable decrease of \(E_a\), when \(\rho\) increases so much that the second term under the radical in (71) becomes comparable with the first, is a more rapid decrease of the amplitude observed than according to the law of direct proportionality.

Fig. 9

When the filament voltage \(E_f\) is changed, the amplitude will also change owing to changes in \(i_{a20}\) and \(\rho\). The curves \(U_m(E_f)\) are shown in Fig. 10. The curves \(i_{a20}(E_f)\) have approximately the same form. The resistance \(\rho\), when \(E_f\) is varied within the limits 4–8 volts, changes little. Only a lower—

Fig. 10

increase of \(E_f\) to 3–4 volts leads to a noticeable increase in \(\rho\), which accelerates the drop in amplitude. The numerals beside the curves in Figs. 9 and 10 denote: the upper one—the value of \(R_a\), the lower one—the value of \(R_k\) (both in kilohms). The character of the curves \(U_m(E_a)\) and \(U_m(E_f)\) at lower frequencies, and also for other values of the resonant resistance of the circuit \((Z_p > \rho)\), remains the same as in Figs. 9 and 10.

The data presented indicate that formula (71) correctly reflects the influence of the tube, the circuit parameters, and the supply voltages on the magnitude of the oscillation amplitude.

Table 1 gives data for the calculation of \(U_m\) by formula (71) and data from an experimental check for an oscillator using a 6N8 tube.

It is seen from the table that, when the conditions of the theory set forth here are fulfilled (a symmetrical volt-ampere characteristic), formula (71) gives good quantitative agreement with the experimental results. The discrepancy with experiment increases for larger \(R_a\), when the operating point is shifted relative to the middle of the “falling” section of the volt-ampere characteristic.

In the experimental verification of the formulas for the nonlinear correction, the frequency drifts of the oscillator resulting from changes in the resistances \(R_a\) and \(Z_p\) were measured. Figure 11 presents the results of an investigation of the influence of the value of the resistance \(R_a\) on the correction

Table 1

Data from an experimental check of formula (71) for the oscillation amplitude

\(L\), μH	\(C\), pF	\(Z_p\), kΩ	\(R_a\), kΩ	Oscillation amplitude, calculated	Oscillation amplitude, experimental
1.93	140	5.37	10	19.9	20.3
1.93	140	5.37	20	17.62	22.4
1.93	140	5.37	40	12.5	21.5
1.93	140	5.37	50	11.22	20.5
6.8	140	10.6	10	31.0	22.6
6.8	140	10.6	20	30.0	26.4
6.8	140	10.6	40	23.3	26.1
6.8	140	10.6	50	21.4	24.7
1.93	520	4.16	10	15.6	13.5
1.93	830	3.3	10	12.0	10.6
1.93	1160	2.78	10	9.45	8.05

\[ R_k = 1\ \text{k}\Omega,\qquad E_a = 250\ \text{V}\qquad E_f = 6.3\ \text{V} \]

\[ \frac{\Delta f}{f}. \]

As the initial value of \(R_a\), \(R_a =\) was taken

\(=100\ \text{k}\Omega\). The corresponding correction \(\dfrac{\Delta f}{f}\) was calculated by formula (60). For other values of \(R_a\), the frequency drifts observed experimentally were added to this correction. Thus the solid curves in Fig. 11 were obtained. The dashed curves are plotted from points calculated with the aid of formula (60).

To investigate the dependence of the nonlinear correction on the resonant resistance of the circuit \(Z_p\), active resistances were introduced into the circuit, and the corresponding values of \(Z_p\) and changes in frequency \(\dfrac{\Delta f}{f}\) were measured. The results of such an experiment and of the calculation performed in parallel with it are presented in Fig. 12. Here, for the solid curves corresponding to the experimental data, the initial values of the correction, just as in the preceding case, were calculated, while the remaining ones were obtained by adding the experimentally observed frequency deviations. In the experiment described, both the nonlinear and the linear correction changed. Therefore the use of the complete formula (60) is absolutely necessary. A comparison of the calculated curves with the experimental ones (for which one point was in fact also obtained by calculation) shows that formula (60) and the formulas following directly from it give a good approximation to reality.

The change in the nonlinear correction when the supply voltages change is one of the causes of frequency instability of the oscillator. Since in formula (61) only \(\rho\) depends on the supply voltages, the change in the nonlinear correction will be equal to

Fig. 11

\[ \Delta \frac{1}{f} = \frac{d}{d\varepsilon}\left(\frac{\Delta f}{f}\right) \frac{d\varepsilon}{d\rho}\,\Delta \rho . \]

Fig. 12.

Carrying out the differentiation and omitting the second-order term \(\Delta\) in \(\Delta f/f\), we obtain the following expression for the frequency drift:

\[ \frac{\Delta f}{f} = \frac{\omega_0^2}{8} \left( \frac{1}{R_a}+\frac{1}{Z_p} \right) \cdot \frac{\varepsilon^2(1-\varepsilon)(3+\varepsilon)}{(1+\varepsilon)^2} \cdot \frac{\Delta p}{p^2}. \]

Calculations by this formula show that the frequency drift produced by the change in the nonlinear correction is 1–2 orders of magnitude smaller than the drifts observed experimentally and associated with the action of other causes that produce instability of the oscillator frequency.

Conclusions

1. The proposed approximation of the voltage-current characteristic of the cathode-coupled two-terminal network reproduces well the characteristic features of this nonlinear dependence, which is fundamental for the oscillator under consideration, and at the same time makes it possible to carry out the quadratures needed for solving nonlinear problems.

2. The formulas obtained have a simple form, represent well the influence of the tube, the circuit parameters, and also the supply voltages, and make it possible to calculate the amplitude of oscillations, harmonic content, and nonlinear correction to the period with accuracy sufficient for practical purposes.

LITERATURE

1. Kulikovskii A. A. New Developments in Radio Reception Engineering, Gosenergoizdat, 1951.

2. Ratbon R. Pulse generator with variable repetition frequency, Problems of Radiolocation Engineering, No. 3 (9), Foreign Literature Publishing House, 1952.

1. Pullen K. A. The Cathode-Coupled Amplifier, Proc. Inst. Radio Eng., v. 34, No. 6, 1946.

2. Crosby M. G. The-Terminal Oscillator, Electronics, No. 5, 1946.

3. Watts H. M., Television Front-End Design, Electronics, No. 4—5, 1949.

4. Sulzer P. G., Frequency and Amplitude Stability of the Cathode-Coupled Oscillator, Proc. Inst. Radio Eng., v. 38, No. 5, 1950.

5. Khavkin G. A. Theory of oscillators using an amplifier with cathode coupling, ZhTF, vol. 18, issue 11, 1948.

6. Krylov N. M. and Bogolyubov N. N. New methods of nonlinear mechanics in their application to the study of the operation of electronic oscillators, GTTI, 1934.

7. Van der Pol B. Nonlinear theory of electric oscillations. Svyazizdat, 1935.

8. Anzhelovich E. S. Problems of stability of oscillator circuits, IEST No. 6, 1935.

9. Arshinov S. S. Temperature stability of the frequency of vacuum-tube oscillators, Gosenergoizdat, 1952.

10. Andronov A. A. and Khaikin S. E., Theory of Oscillations, ONTI, 1937.

11. Suslov I. A. Conductance of an active two-terminal network in the circuit of an oscillator with cathode coupling, Izvestiya TPI, vol. 86, 1958.

12. Suslov I. A. Graphical calculation of amplifiers with cathode coupling, Izv. TPI, vol. 73, 1952.

1. P. G. Sul’zer everywhere uses the simplified equation (13), which leads to an error in computing \(\tau\) and \(k_{fn}\). ↩↩

2. Usually, at \(u=0\), \(K \gg 1\). ↩