CONDUCTANCE OF AN ACTIVE TWO-TERMINAL NETWORK IN THE CIRCUIT OF A CATHODE-COUPLED OSCILLATOR

I. A. Suslov

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

Introduction

A sinusoidal-oscillation generator constructed on the basis of an amplifier with cathode coupling (Fig. 1) is conveniently regarded as an oscillatory circuit connected to a nonlinear active two-terminal network possessing negative resistance. Let us denote the conductance of such a two-terminal network (we shall hereafter call it a two-terminal network with cathode coupling) by

\[ Y_{ab}=\sigma_{ab}+j\omega C_{ab}. \]

Fig. 1

The active component \(\sigma_{ab}\) enters into the self-excitation condition of the generator and affects the amplitude of its oscillations.

The reactive component \(j\omega C_{ab}\), associated with the stray capacitances present in the two-terminal network circuit, has a direct effect on the oscillator frequency and its stability.

Figure 2a shows the stray capacitances of a two-terminal network with cathode coupling. Some of these capacitances can be combined, since they are connected in parallel with one another. If the feedback capacitance \(C_1\) is sufficiently large and its resistance at the generated frequencies can be neglected, the circuit of Fig. 2a can be replaced by the simpler circuit of Fig. 2b, where \(C_k = C_{k1} + C_{k2} + C_{gk2} + C_{ak1}\), \(C_{gk}=C_{gk1}+C_{ak2}\), and \(C_a=C_{ga1}+C_{ga2}\).

P. G. Sul’tser [1] investigated the conductance of a two-terminal network with cathode coupling. However, the formulas he obtained are cumbersome and inconvenient for analysis and calculation. Moreover, they apply only to cases in which the capacitances \(C_{gk}\) and \(C_k\) (Fig. 2b) may either be completely neglected or only the action of one of them is taken into account. These capacitances are of the same order of magnitude, and there is no reason to neglect the effect of one of them in comparison with the other.

In the present article the general case of the simultaneous action of all stray capacitances of the circuit is considered.

Total conductance of a two-terminal network with cathode coupling

Let us determine, for the circuit of Fig. 2b, the conductance between terminals \(a\) and \(b\)—the conductance of the active two-terminal network with cathode coupling. To do this, we shall construct an equivalent circuit in which the tubes are replaced by equivalent current generators. We obtain the circuit of Fig. 3a. The conductance between points \(a\) and \(b\), \(Y_{ab}\), will be equal to

\[ Y_{ab}=j\omega C_a+\frac{1}{R_a}+Y_{cd}. \]

\(Y_{cd}\) can be found from the simpler equivalent circuit of Fig. 3b. For this circuit we write the node-voltage equations. The use of the node-voltage method in combination with replacing the tubes by equivalent current generators makes it possible to get by with a smaller number of equations and to express the desired quantities through a determinant of lower order than when using equivalent emf generators and the loop-current method [2, 3]. The circuit of Fig. 3b is described by two equations:

\[ U_1\left(j\omega C_{gk}+\frac{1}{R_{i2}}\right) - U_k\left(j\omega C_{gk}+\frac{1}{R_{i2}}\right) = i_{\mathrm{in}}+S_2U_k, \]

\[ -U_1\left(j\omega C_{gk}+\frac{1}{R_{i2}}\right) + U_k\left[ j\omega(C_{gk}+C_k) + \frac{1}{R_{i1}} + \frac{1}{R_{i2}} + \frac{1}{R_k} \right] = S_1U_{g1}-S_2U_k. \]

Substituting \(U_{g1}=U_1-U_k\), we obtain:

\[ U_1\left(j\omega C_{gk}+\frac{1}{R_{i2}}\right) - U_k\left(j\omega C_{gk}+\frac{1}{R_{i2}}+S_2\right) = i_{\mathrm{in}}, \]

Fig. 2

\[ - U_1\left(j\omega C_{gk}+\frac{1}{R_{i2}}+S_1\right)+U_k\left[j\omega(C_{gk}+C_k)+\right. \]

\[ \left.+\frac{1}{R_{i1}}+\frac{1}{R_{i2}}+\frac{1}{R_k}+S_1+S_2\right]=0 . \]

From this system one can find the conductance \(Y_{cd}\)

\[ Y_{cd} = -\frac{LP+\omega^2(MP-NQ)}{P^2+\omega^2Q^2} +j\omega\frac{NP+LQ+\omega^2MQ}{P^2+\omega^2Q^2}, \]

where

\[ L=S_1S_2-\frac{1}{R_{i2}}\left(\frac{1}{R_k}+\frac{1}{R_{i1}}\right), \qquad M=C_{gk}C_k, \tag{1} \]

\[ N=C_{gk}\left(\frac{1}{R_k}+\frac{1}{R_{i1}}\right)+\frac{C_k}{R_{i2}}, \]

\[ P=S_1+S_2+\frac{1}{R_{i1}}+\frac{1}{R_{i2}}+\frac{1}{R_k} \]

and

\[ Q=C_{gk}+C_k . \]

Fig. 3

Let us denote

\[ -\frac{LP+\omega^2(MP-NQ)}{P^2+\omega^2Q^2}=g_{cd} \tag{2} \]

\[ -\frac{NP+LQ+\omega^{2}MQ}{P^{2}+\omega^{2}Q^{2}}=C_{cd}. \tag{3} \]

Then \(Y_{cd}=\sigma_{cd}+j\omega C_{cd}\).

Let us first transform the expression for the active component \(\sigma_{cd}\) of the conductance \(Y_{cd}\).

Active Component of the Conductance of a Cathode-Coupled Two-Terminal Network

The expressions (1), which enter into the formulas for the negative conductance \(\sigma_{cd}\) and the capacitance \(C_{cd}\), are simplified if the parameters of both tubes are taken to be identical. The inequality of the parameters in tubes of the same type is due to the fact that in the anode circuit of the second tube there is a resistance \(R_a\), which is absent from the anode circuit of the first. However, the difference in parameters caused by this resistance is small if the resistance \(R_a\) itself is chosen to be relatively small (of the order of \(10\ \text{k}\Omega\)). Considering \(R_a\) not very large \((R_a<2R)\), let us put in expressions (1) \(S_1=S_2=S,\ R_{i1}=R_{i2}=R_i\).

Then

\[ L=S^{2}-\frac{1}{R_i}\left(\frac{1}{R_k}+\frac{1}{R_i}\right), \]

\[ M=C_{gk}C_k, \]

\[ N=C_{gk}\left(\frac{1}{R_k}+\frac{1}{R_i}\right)+\frac{C_k}{R_i}, \]

\[ P=2S+\frac{2}{R_i}+\frac{1}{R_k}, \]

\[ Q=C_{gk}+C_k. \tag{1} \]

Using (1′), let us express the ratio of the terms of the numerator and denominator of (2) that contain \(\omega^2\), and the ratio of the terms of (2) that do not depend on frequency, i.e.,

\[ \frac{MP-NQ}{Q^{2}} \quad \text{and} \quad \frac{L}{P}, \]

in terms of the tube and circuit parameters.

\[ \frac{L}{P}= \frac{ S^{2}-\dfrac{1}{R_i}\left(\dfrac{1}{R_k}+\dfrac{1}{R_i}\right) }{ 2S+\dfrac{1}{R_k}+\dfrac{1}{R_i} }, \tag{4} \]

\[ \frac{MP-NQ}{Q^{2}}= \frac{ 2\nu S-\nu^{2}\left(\dfrac{1}{R_k}+\dfrac{1}{R_i}\right)-\dfrac{1}{R_i} }{ (1+\nu)^{2} }, \tag{5} \]

where

\[ \nu=\frac{C_{gk}}{C_k}. \]

Carrying out the division of the polynomials in (4), we find:

\[ \frac{L}{P} = -\frac{S}{2} -\frac{1}{4R_k} -\frac{1}{2R_i} + \frac{\dfrac{1}{4R_k^2}} {2S+\dfrac{1}{R_k}+\dfrac{2}{R_i}} . \tag{4′} \]

The capacitances \(C_{gk}\) and \(C_k\) are of the same order. For equal capacitances \((C_{gk}=C_k)\), \(\gamma=1\), and

\[ \frac{MP-NQ}{Q^2} = -\frac{S}{2} -\frac{1}{4R_k} -\frac{1}{2R_i}. \tag{5′} \]

Comparison of (5′) with (4′) shows that the two expressions differ by a small quantity

\[ \Delta\vartheta = \frac{\dfrac{1}{4R_k^2}} {2S+\dfrac{1}{R_k}+\dfrac{2}{R_i}} . \tag{5″} \]

Let us denote

\[ \frac{L}{P}=\vartheta . \tag{4″} \]

Then

\[ \frac{MP-NQ}{Q^2}=\vartheta-\Delta\vartheta . \tag{5‴} \]

The relative error when replacing (5′) by (4′) will be

\[ \frac{\Delta\vartheta}{\vartheta} = \frac{\dfrac{1}{4R_k^2}} {S^2-\dfrac{1}{R_iR_k}-\dfrac{1}{R_i^2}} = \]

\[ = \frac{1} {(2SR_k)^2-\dfrac{4SR_k}{\mu}-\dfrac{(2SR_k)^2}{\mu^2}}, \tag{6} \]

where \(\mu\) is the internal amplification factor of the tube.

As \(R_k\) increases, the product \(SR_k\) increases. As formula (6) shows, the relative error \(\dfrac{\Delta\vartheta}{\vartheta}\) will then decrease. For a two-terminal network using a 6N8 tube, with \(R_k=1\ \mathrm{k\Omega}\) and \(R_a=10\ \mathrm{k\Omega}\), \(SR_k=1.7\), \(\dfrac{\Delta\vartheta}{\vartheta}=0.09\); with \(R_k=10\ \mathrm{k\Omega}\) and \(R_a=10\ \mathrm{k\Omega}\), \(SR_k=4.3\), \(\dfrac{\Delta\vartheta}{\vartheta}=0.014\). In the case of a two-terminal network using a 6N9 tube, with \(R_k=1\ \mathrm{k\Omega}\) and \(R_a=10\ \mathrm{k\Omega}\), \(SR_k=1.4\), \(\dfrac{\Delta\vartheta}{\vartheta}=0.13\), while with \(R_k=10\ \mathrm{k\Omega}\) and \(R_a=10\ \mathrm{k\Omega}\), \(\dfrac{\Delta\vartheta}{\vartheta}=0.006\);

The examples given show that the relative difference between expressions (4′) and (5′) decreases as \(R_k\) increases. At \(R_k = 10\) kΩ it is of the order of \(1\%\).

The small difference between expressions (4′) and (5′) for \(\nu = 1\), and the presence of a “spread” in the values of \(\nu\) for real circuits, suggest the following way of simplifying formula (2). Suppose we have the proportion \(\dfrac{a}{b} = \dfrac{c}{d}\); then an expression of the form \(\dfrac{a+\omega^2 c}{b+\omega^2 d}\) will be exactly equal to each of the two ratios written. We have seen that, for \(\nu = 1\), the expressions \(\dfrac{L}{P}\) and \(\dfrac{MP-NQ}{Q^2}\) prove to be close. We shall now find that \(\nu=\nu_0\) for which their difference will be minimal. Since \(\dfrac{L}{P}\) does not depend on \(\nu\), this will occur at the maximum of the expression \(\dfrac{MP-NQ}{Q^2}\). The latter is easily determined by setting the partial derivative equal to zero:

\[ \frac{\partial}{\partial \nu}\frac{MP-NQ}{Q^2} = \frac{\partial}{\partial \nu} \frac{ 2\nu S-\nu^2\left(\frac{1}{R_k}+\frac{1}{R_i}\right)-\frac{1}{R_i} }{ (1+\nu)^2 } =0. \]

Carrying out the simple mathematical operations, we obtain

\[ \nu_0= \frac{ S+\frac{1}{R_i} }{ S+\frac{1}{R_k}+\frac{1}{R_i} } = \]

\[ = \frac{ SR_k+\frac{SR_k}{\mu} }{ SR_k+\frac{SR_k}{\mu}+1 }. \tag{7} \]

As the resistance \(R_k\) is increased, the product \(SR_k\) increases. In this case \(\nu_0\) approaches unity. However, for any \(R_k\) it remains less than unity.

We shall now show that for \(\nu=\nu_0\), \(\dfrac{MP-NQ}{Q^2}\) is equal to \(\dfrac{L}{P}\) and their difference \(\Delta \vartheta\) is equal to zero. Substitute into (5), in place of \(\nu\), the expression for \(\nu_0\) from (7). We obtain:

\[ \left(\frac{MP-NQ}{Q^2}\right)_{\max} = \frac{ S^2-\frac{1}{R_k R_i}-\frac{1}{R_i^2} }{ 2S+\frac{2}{R_i}+\frac{1}{R_k} } = \frac{L}{P}. \tag{8} \]

Equality (8) makes it possible, on the basis of the known properties of derived proportions, to simplify substantially expression (2) for the conductance \(g_{cd}\). The latter will now have the following form:

\[ \sigma_{cd}=-\frac{L}{P}= -\frac{ S^2-\dfrac{1}{R_i}\left(\dfrac{1}{R_k}+\dfrac{1}{R_i}\right) }{ 2S+\dfrac{2}{R_i}+\dfrac{1}{R_k} }. \tag{9} \]

For the active component \(\sigma_{ab}\) of the admittance \(Y_{ab}\), the following expression is then obtained:

\[ \sigma_{ab}= \frac{1}{R_a} - \frac{ S^2-\dfrac{1}{R_i}\left(\dfrac{1}{R_k}+\dfrac{1}{R_i}\right) }{ 2S+\dfrac{2}{R_i}+\dfrac{1}{R_k} }. \tag{10} \]

Thus, at \(\nu=\nu_0\), the negative conductance of the cathode-coupled two-terminal network does not depend on frequency.

Formula (10) admits further simplifications. In the numerator one may neglect

\[ \frac{1}{R_i^2}=\frac{S^2}{\mu^2} \]

in comparison with \(S^2\). Then

\[ \sigma_{ab}= \frac{1}{R_a} - \frac{ S^2-\dfrac{1}{R_iR_k} }{ 2S+\dfrac{2}{R_i}+\dfrac{1}{R_k} }. \]

If great accuracy is not required, one may also neglect in the numerator the term

\[ \frac{1}{R_iR_k}=\frac{S^2}{\mu S R_k} \]

in comparison with \(S^2\), and in the denominator

\[ \frac{2}{R_i}=\frac{2S}{\mu} \]

in comparison with \(2S\) (this gives an additional error of a few percent); then the simpler formula is obtained:

\[ \sigma_{ab}= \frac{1}{R_a} - \frac{S^2}{2S+\dfrac{1}{R_k}}. \]

For rough calculations, when \(R_k>1\ \text{k}\Omega\) and \(R_a>10\ \text{k}\Omega\), one may use the formula

\[ \sigma_{ab}\simeq -\frac{S}{2}. \]

Let us see how the conductance of the cathode-coupled two-terminal network changes with frequency if \(\nu\ne\nu_0\). From expression (2) for \(\sigma_{cd}\) it follows that for small \(\omega\) this conductance is equal to

\[ -\frac{L}{P}, \]

and for large \(\omega\),

\[ -\frac{MP-NQ}{Q^2}. \]

If \(\nu\ne\nu_0\), then \(-L/P\) and

\[ -\frac{MP-NQ}{Q^2} \]

are not equal, and as \(\omega\) increases from \(0\) to \(\infty\), \(\sigma_{cd}\) changes from \(-L/P\) to

\[ -\frac{MP-NQ}{Q^2}. \]

This change will be monotonic, since the derivative

\[ \frac{d\sigma_{cd}}{d\omega} = -\frac{(MP^3-NP^2Q-LPQ^2)\,2\omega}{(P^2+\omega^2Q^2)^2} \]

will not vanish in the intervals between the indicated limits. Since, according to the preceding, for \(\nu \ne \nu_0\)

\[ \left|-\frac{L}{P}\right|\geq \left|-\frac{MP-NQ}{Q^2}\right|, \]

the conductance \(\sigma_{cd}\) (and \(\sigma_{ab}\)) decreases with increasing frequency, and the conditions for self-excitation become worse.

Let us find the values of \(\nu\) for which

\[ \frac{MP-NQ}{Q^2}=0, \]

i.e., for which the numerator in formula (5) becomes zero:

\[ 2\nu S-\nu^2\left(\frac{1}{R_k}+\frac{1}{R_i}\right)-\frac{1}{R_i}=0. \tag{11} \]

Solving the quadratic equation (11), we obtain

\[ \nu_{1,2}= \frac{\mu SR_k}{\mu+SR_k} \left( 1\pm \sqrt{1-\frac{1}{\mu SR_k}-\frac{1}{\mu^2}} \right). \]

Expanding

\[ \sqrt{1-\frac{1}{\mu SR_k}-\frac{1}{\mu^2}} \]

by Newton’s binomial formula and neglecting terms of second and higher orders of smallness, we find

\[ \nu_1= \frac{1}{2(\mu+SR_k)} + \frac{1}{2\mu\left(1+\frac{\mu}{SR_k}\right)}, \]

\[ \nu_2= \frac{2\mu SR_k}{\mu+SR_k} -\nu_1 \approx \frac{2\mu SR_k}{\mu+SR_k}. \]

For a circuit using a 6N8 tube, at \(R_k=1\ \text{k}\Omega\) the following values of the coefficients \(\nu_1\) and \(\nu_2\) are obtained: \(\nu_1=0.025,\ \nu_2=3.11\), and at \(R_k=10\ \text{k}\Omega\), \(\nu_1=0.026,\ \nu_2=7.32\).

If \(\nu<\nu_1\) or \(\nu>\nu_2\), then at sufficiently high frequencies, when

\[ \left|\omega^2(MP-NQ)\right|\gg LP, \]

the cathode-coupled two-terminal network will no longer present a negative resistance, and the circuit will not oscillate under any conditions. In fact, oscillation proves impossible at frequencies much lower than

\[ \omega= \sqrt{\frac{LP}{|MP-NQ|}}, \]

since, according to the conditions for self-excitation, the negative conductance \(\sigma_{cd}\) must be greater than the total conductance of the resistance \(R_a\) and the conductance of the parallel circuit \(LC\). The latter, at high frequencies, as is known, increases (the resonant resistance decreases).

Figure 4 shows a family of curves \(\sigma_{cd}(\nu,\omega C_k)\). The curves were obtained for a circuit using a 6N8 tube at \(R_k=1\ \text{k}\Omega\) and \(R_a=10\ \text{k}\Omega\). They indicate that for \(\nu\ne\nu_0\) the negative conductance \(\sigma_{cd}\) with increasing—

with increasing frequency decreases. This decrease occurs the faster, the more \(\nu\) differs from \(\nu_0\). For \(\nu=\nu_0\) the circuit has the maximum frequency limit, and the conductance \(\sigma_{cd}\), up to very high frequencies at which the influence of the electron transit time and the effect of lead inductances must be taken into account, remains the same as at the very lowest frequencies.

Fig. 4

In the expression for the capacitance \(C_k\)

\[ C_k = C_{gk} + C_{ak1} + C_{k1} + C_{k2} \]

the first two terms are approximately equal to \(C_{gk}\)

\[ C_{gk} = C_{gk1} + C_{ak2}. \]

The total capacitance of the cathodes with respect to ground (the heater), \(C_{k1}+C_{k2}\), is no less than \(C_{gk}\). Thus, \(\nu\) must be no greater than \(1/\tau\). The mounting capacitances somewhat level out the capacitances \(C_{gk}\) and \(C_k\), and in an actual circuit \(\nu\) may be close to \(\nu_0\).

Let us estimate the error when using formula (9) for the conductance \(\sigma_{cd}\). Denote

\[ P=x,\quad \omega Q=y,\quad \frac{y^2}{x^2}=\varkappa . \tag{12} \]

Then, taking into account \((4'')\) and \((5''')\), we obtain

\[ \sigma_{cd} = -\frac{LP+\omega^2(MP-NQ)}{P^2+\omega^2Q^2} = -\vartheta+\frac{\Delta\vartheta\, y^2}{x^2+y^2} = \]

\[ =\sigma_{cd0}+\Delta\sigma_{cd}. \]

The relative error in replacing $\sigma_{cd}$ by $-\vartheta=\sigma_{cd0}$, i.e., in discarding the terms with $\omega^2$ in expression (2), will be equal to

\[ \left|\frac{\Delta\sigma_{cd}}{\sigma_{cd}}\right| = \frac{\Delta\vartheta}{\vartheta}\cdot\frac{\varkappa}{1+\varkappa}. \tag{13} \]

It will be small at not very high frequencies, when $\varkappa \ll 1$, and at $\nu$ close to $\nu_0$, when

\[ \frac{\Delta\vartheta}{\vartheta}\ll 1. \]

Figure 5 shows plots of the variation of $\dfrac{\Delta\vartheta}{\vartheta}$ as a function of $\delta=\dfrac{\nu}{\nu_0}$.

The curves of this figure show that at $\nu=\nu_0$ the quantity $\dfrac{\Delta\vartheta}{\vartheta}$ has a rather flat minimum and that the condition $\nu=\nu_0$, under which the conductance of the cathode-coupled two-terminal network is maximal, is not critical.

Fig. 5

Dynamic capacitance of a two-terminal network with cathode coupling

The dynamic capacitance of a two-terminal network with cathode coupling consists of the capacitance $C_a$ and the capacitance $C_{cd}$ between terminals $c$ and $d$ (Figs. 3a and 6).

\[ C_{ab}=C_a+C_{cd}. \]

The capacitance $C_{cd}$ is determined by formula (3). Substituting into this formula the values of the quantities entering into it from (1′), we obtain:

\[ C_{cd}=(C_{gk}+C_k)\cdot \frac{ \left(S+\frac{1}{R_k}\right)^2 + \frac{\nu}{\nu+1} \left(2S+\frac{2}{R_i}+\frac{1}{R_k}\right)\frac{1}{R_k} + \omega^2\nu C_k^2 }{ \left(2S+\frac{2}{R_i}+\frac{1}{R_k}\right)^2 + \omega^2(1+\nu)^2 C_k^2 }, \tag{14} \]

where

\[ \nu=\frac{C_{gk}}{C_k}. \]

Let us denote

\[ \frac{NP+LQ}{P^2}=\eta,\qquad \frac{M}{Q}=\eta_1 \quad \text{and} \quad \eta-\eta_1=\Delta\eta . \tag{15} \]

Let us find the value of \(\nu\) for which the ratio \(\eta\) of the terms of expression (9) that do not depend on frequency will be closest to the ratio \(\eta_1\) of the terms containing \(\omega^2\), i.e., the difference

\[ \frac{NP+LQ}{P^2}-\frac{M}{Q}=\eta-\eta_1= \]

\[ = \frac{ \left(S+\frac{1}{R_i}\right)^2+ \frac{\nu}{1+\nu} \left(2S+\frac{2}{R_i}+\frac{1}{R_k}\right)\frac{1}{R_k} }{ \left(2S+\frac{2}{R_i}+\frac{1}{R_k}\right)^2 } - \frac{\nu}{(1+\nu)^2} \tag{16} \]

will be minimal.

For this purpose we set the partial derivative equal to zero,

\[ \frac{\partial}{\partial \nu} \left( \frac{NP+LQ}{P^2}-\frac{M}{Q} \right)=0 . \]

Solving the resulting equation

\[ \frac{\frac{1}{R_k}}{2S+\frac{2}{R_i}+\frac{1}{R_k}} = \frac{1-\nu}{1+\nu} \]

with respect to \(\nu\), we find:

\[ \nu= \frac{S+\frac{1}{R_i}}{S+\frac{1}{R_k}+\frac{1}{R_i}} . \]

This expression coincides with (7).

Substituting \(\nu=\nu_0\) into expression (16), we determine the minimum value of the difference \(\Delta\eta=\eta-\eta_1\). It proves to be equal to zero,

\[ (\Delta\eta)_{\nu=\nu_0}=0, \]

and the ratios \(\eta\) and \(\eta_1\) are equal\(^1\)

\[ \eta=\eta_1= \frac{ \left(S+\frac{1}{R_i}\right) \left(S+\frac{1}{R_i}+\frac{1}{R_k}\right) }{ \left(2S+\frac{2}{R_i}+\frac{1}{R_k}\right)^2 }. \]

The coincidence of \(\eta\) and \(\eta_1\) signifies the independence of expression (3) for the capacitance \(C_{cd}\) from frequency.

\[ {}^1\ \text{See (15).} \]

For the value of the capacitance \(C_{cd}\) at \(\nu=\nu_0\), we easily obtain the following expression:

\[ C_{cd}=(C_{gk}+C_k)\eta=(C_{gk}+C_k)\eta_1=(C_{gk}+C_k)\frac{\nu_0}{(1+\nu_0)^2} \]

or, finally,

\[ C_{cd}=\frac{C_{gk}C_k}{C_{gk}+C_k}. \tag{17} \]

Thus, at \(\nu=\nu_0\), the dynamic capacitance of a cathode-coupled two-terminal network does not depend on frequency and is equal to its static capacitance.

If \(\nu\ne\nu_0\), then \(\eta\) and \(\eta_1\) are not equal and \(\Delta\eta\ne0\). In this case \(C_{cd}\) is determined by the general formula (14). At not very high frequencies, when \(x\ll1\), the terms containing \(\omega^2\) may be discarded. Then, for the capacitance \(C_{cd}\), we shall have the simpler expression

\[ C_{cd}=(C_{gk}+C_k) \frac{ \left(S+\frac{1}{R_i}\right)^2+ \frac{\nu}{1+\nu}\left(2S+\frac{2}{R_i}+\frac{1}{R_k}\right)\frac{1}{R_k} }{ \left(2S+\frac{2}{R_i}+\frac{1}{R_k}\right)^2 }, \tag{18} \]

and the error due to the discarded terms can be found as follows.

According to (12) and (15), the expression for the capacitance \(C_{cd}\) may be represented in the form:

\[ C_{cd}=(C_{gk}+C_k)\left(\eta-\Delta\eta\frac{x}{1+x}\right)=C_{cd0}-\Delta C_{cd}. \]

The relative error of the indicated neglect will be equal to

\[ \frac{\Delta C_{cd}}{C_{cd}}\simeq \frac{\Delta C_{cd}}{C_{d0}}= \frac{\Delta\eta}{\eta}\frac{x}{1+x}. \tag{19} \]

The last expression shows that when \(\nu\) is close to \(\nu_0\), when \(\frac{\Delta\eta}{\eta}\) is small, the terms containing \(\omega^2\) may be neglected even if they are comparable with the other terms. Conversely, when \(\nu\) differs greatly from \(\nu_0\), formula (18) is valid only when \(x\ll1\), i.e., at not very high frequencies.

At frequencies for which the terms containing \(\omega^2\) in (14) exceed the terms not containing \(\omega^2\), formula (17) proves to be more accurate than (18). It should be preferred (independently of frequency) in those cases when \(\nu=\nu_0\).

Let us find the error from using the simplified formula (17). To do this, using the notation (12) and (15), we represent expression (3) for the capacitance \(C_{cd}\) in the form

\[ C_{cd}=\left(\eta_1+\frac{\Delta\eta}{1+x}\right)(C_{gk}+C_k). \]

Let us denote the difference between \(C_{cd}\) and its approximate value \(\eta_1\) by \(\Delta C_{cd}\):

\[ \Delta C_{cd}=C_{cd}-(C_{gk}+C_k)\eta_1= \frac{\Delta \eta}{1+x}(C_{gk}+C_k). \]

The relative error when using formula (17) will be equal to

\[ \frac{\Delta C_{cd}}{C_{cd}} = \frac{\Delta C_{cd}}{\eta-\Delta\eta+\Delta C_{cd}} \approx \frac{\Delta\eta}{\eta}\cdot \frac{1}{1+x}. \tag{20} \]

Thus it will be small for \(\nu\) close to \(\nu_0\) (\(\Delta\eta\) is small), and decreases with increasing frequency (when \(x\) grows).

We have seen that the error when using the simplified formulas (17) and (18) depends to a considerable degree on the magnitude \(\dfrac{\Delta\eta}{\eta}\).

Let us see how \(\dfrac{\Delta\eta}{\eta}\) changes with a change in \(\nu\). Put \(\nu=\delta\nu_0\). The relative difference \(\dfrac{\Delta\eta}{\eta}\) may be represented in the form

\[ \frac{\Delta\eta}{\eta}=1-\frac{\eta_1}{\eta}. \]

Substituting here the values of \(\eta\) and \(\eta_1\), we obtain

\[ \frac{\Delta\eta}{\eta} = 1- \frac{\delta\nu_0}{(1+\delta\nu_0)^2} \cdot \frac{\left(2S+\dfrac{2}{R_i}+\dfrac{1}{R_k}\right)^2} { \left(S+\dfrac{1}{R_i}\right)^2 + \dfrac{\delta\nu_0}{1+\delta\nu_0} \left(2S+\dfrac{2}{R_i}+\dfrac{1}{R_k}\right) \dfrac{1}{R_k} }. \]

From (7) it is not difficult to see that

\[ 2S+\frac{2}{R_i}+\frac{1}{R_k} = \frac{\nu_0+1}{\nu_0} \left(S+\frac{1}{R_i}\right), \]

and

\[ \frac{\dfrac{1}{R_k}}{S+\dfrac{1}{R_i}} = \frac{1-\nu_0}{\nu_0}. \tag{21} \]

Using first the first and then the second of these relations, we find that

\[ \frac{\Delta\eta}{\eta} = 1- \frac{(1+\nu_0)^2} {\left(\nu_0+\delta\right)\left(\nu_0+\dfrac{1}{\delta}\right)}. \]

Figure 6 shows the plot of the dependence of \(\dfrac{\Delta \eta}{\eta}\) on \(\delta\), obtained for a two-terminal network using a 6N8 tube¹. In constructing this plot, \(S\) and \(R_i\) were determined experimentally, while \(\nu_0\) was calculated from formula (7). The curve in the figure shows that \(\dfrac{\Delta \eta}{\eta}\) remains small within the known limits of variation of \(\nu\), and, consequently, the value \(\nu=\nu_0\) is not critical.

Fig. 6

Case of unequal tube parameters

Let us see how allowing for the inequality of the tube parameters, which is necessary for relatively large \(R_a\), may affect the conclusions obtained earlier.

For the active conductance \(\xi_{cd}\) we had the general expression (2); the quantities entering into it have the values (1). As before, let us compare the ratio \(\dfrac{L}{P}\) of the terms of the numerator and denominator that do not depend on frequency with the ratio \(\dfrac{MP-NQ}{Q^2}\) of the terms containing \(\omega^2\). To do this, we take the difference of the indicated ratios. It is equal to

\[ \frac{L}{P} - \frac{MP - NQ}{Q^2} = \frac{ S_1 S_2 - \dfrac{1}{R_{i2}}\left(\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}\right) }{ S_1+S_2+\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}+\dfrac{1}{R_{i2}} } - \frac{ \nu\left(S_1+S_2+\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}+\dfrac{1}{R_{i2}}\right) }{ (1+\nu)^2 } - \frac{ (1+\nu)\left[\nu\left(\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}\right)+\dfrac{1}{R_{i2}}\right] }{ (1+\nu)^2 }, \tag{22} \]

where \(\nu=\dfrac{C_{gk}}{C_k}\).

Let us find the value \(\nu=\nu_0\) for which the difference (22) has a minimum. Since \(\dfrac{L}{P}\) does not depend on \(\nu\), this will occur at the maximum of \(\dfrac{MP-NQ}{Q^2}\).

To find the latter, we set the partial derivative equal to zero

\[ \frac{\partial}{\partial \nu} \left( \frac{MP-NQ}{Q^2} \right) =0. \]

As a result of elementary mathematical operations we obtain:

\[ \nu_0= \frac{ S_1+S_2+\dfrac{2}{R_{i2}} }{ S_1+S_2+2\left(\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}\right) }. \tag{7′} \]

Let us now find the magnitude of the difference (22) for \(\nu=\nu_0\). For this purpose we first rewrite \(\dfrac{MP-NQ}{Q^2}\) as follows:

\[ \frac{MP-NQ}{Q^2} = \frac{1}{1+\nu_0} \left[ \frac{\nu_0}{1+\nu_0} \left( S_1+S_2+\frac{1}{R_k}+\frac{1}{R_{i1}}+\frac{1}{R_{i2}} \right) - \nu_0\left(\frac{1}{R_k}+\frac{1}{R_{i1}}\right) - \frac{1}{R_{i2}} \right]. \tag{23} \]

On the basis of (7′) we can write

\[ \frac{ S_1+S_2+\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}+\dfrac{1}{R_{i2}} }{ 1+\nu_0 } = \frac{ S_1+S_2+2\left(\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}\right) }{2}. \]

Substituting this expression into the first term of (23), we obtain

\[ \frac{MP-QN}{Q^2} = \frac{1}{1+\nu_0} \left[ \frac{\nu_0(S_1+S_2)}{2} - \frac{1}{R_{i2}} \right] \]

or, after replacing \(\nu_0\) by formula (7′),

\[ \frac{MP-NQ}{Q^2} = \frac{ \left(\dfrac{S_1+S_2}{2}\right)^2 - \dfrac{1}{R_{i2}} \left(\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}\right) }{ S_1+S_2+\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}+\dfrac{1}{R_{i2}} }. \tag{24} \]

Comparing (24) with

\[ \frac{L}{P} = \frac{S_1 S_2 - \dfrac{1}{R_{i2}}\left(\dfrac{1}{R_k} + \dfrac{1}{R_{i1}}\right)} {S_1 + S_2 + \dfrac{1}{R_k} + \dfrac{1}{R_{i1}} + \dfrac{1}{R_{i2}}} \]

we note that \(\dfrac{MP - NQ}{Q^2}\), which has a maximum at \(\nu = \nu_0\), proves, at the indicated value of \(\nu\), to be larger than \(\dfrac{L}{P}\), since the arithmetic mean of two unequal quantities \(S_1\) and \(S_2\) is greater than their geometric mean.

The relative change in the difference between \(\dfrac{L}{P} = \vartheta\) and \(\dfrac{MP - NQ}{Q^2}\)

\[ = \vartheta - \Delta\vartheta, \]

i.e., \(\dfrac{\Delta\vartheta}{\vartheta}\), as a function of the change \(\delta = \dfrac{\nu}{\nu_0}\), is presented for the case of unequal tube parameters in Fig. 7. In comparison with the case considered earlier, here we obtain a more asymmetric curve, which, moreover, falls below the abscissa axis. As the oscillation frequency increases, the role of the terms (2) containing \(\omega\) increases. In this case, the conductance of the cathode-coupled two-terminal network approaches

Fig. 7.

\[ \frac{1}{R_a} - \frac{MP - NQ}{Q^2}. \]

If \(\nu = \nu_0\), it increases in absolute value, and the self-excitation conditions are thereby improved. A deterioration of the self-excitation conditions with increasing frequency occurs only in the case of \(\nu\) values that differ noticeably from \(\nu_0\). The foregoing il-

is illustrated by the curves of Fig. 8, which presents the dependence of the conductance \(\sigma_{ab}\) on \(\omega C_k\) for various values of \(\nu\). These curves were obtained with the aid of the formula

\[ \sigma_{ab} = \frac{1}{R_a} - \frac{ \left[ S_1 S_2 - \frac{1}{R_{i2}} \left( \frac{1}{R_k} + \frac{1}{R_{i1}} \right) \right] \left( S_1+S_2+\frac{1}{R_k}+\frac{1}{R_{i1}}+\frac{1}{R_{i2}} \right) }{ \left( S_1+S_2+\frac{1}{R_k}+\frac{1}{R_{i1}}+\frac{1}{R_{i2}} \right)^2 + \omega^2 C_k^2(1-\nu)^2 } + \]

\[ + \frac{ \omega^2 C_k^2 \left\{ \nu \left( S_1+S_2+\frac{1}{R_k}+\frac{1}{R_{i1}}+\frac{1}{R_{i2}} \right) \right. }{ \left( S_1+S_2+\frac{1}{R_k}+\frac{1}{R_{i1}}+\frac{1}{R_{i2}} \right)^2 + \omega^2 C_k^2(1+\nu)^2 } - \]

\[ - \frac{ (1-\nu) \left[ \nu \left( \frac{1}{R_k} + \frac{1}{R_{i1}} \right) - \frac{1}{R_{i2}} \right] \right\} }{ \left( S_1+S_2+\frac{1}{R_k}+\frac{1}{R_{i1}}+\frac{1}{R_{i2}} \right)^2 + \omega^2 C_k^2(1+\nu)^2 } \]

for a two-terminal network using a 6N8 tube, for which \(R_k=2\ \text{k}\Omega\) and \(R_a=60\ \text{k}\Omega\).

Fig. 8

Let us now consider how the inequality of the tube parameters is reflected in the dynamic capacitance of a two-terminal network with cathode coupling. We shall examine expression (3) for this capacitance by the same methods as before. Let us write the difference between the ratios of the terms of (3) that do not depend on the frequency \(\omega\), and the terms containing \(\omega^2\)

\[ \frac{NP+LQ}{P^2}-\frac{M}{Q}=(C_{gk}+C_k)\left\{ \frac{ \left(S_1+S_2+\frac{1}{R_k}+\frac{1}{R_{i1}}+\frac{1}{R_{i2}}\right)\ldots }{ \left(S_1+S_2+ \right. } \right. \]

\[ \left. \frac{ \ldots\left[ \frac{\nu}{1+\nu}\left(\frac{1}{R_k}+\frac{1}{R_{i1}}\right) +\frac{1}{1+\nu}\cdot\frac{1}{R_{i2}} \right] +\left[ S_1S_2-\frac{1}{R_{i2}}\left(\frac{1}{R_k}+\frac{1}{R_{i1}}\right) \right] }{ \left. +\frac{1}{R_k}+\frac{1}{R_{i1}}+\frac{1}{R_{i2}} \right)^2 } -\frac{\nu}{(1+\nu)^2} \right\}. \tag{25} \]

Let us find that value of \(\nu\) for which difference (25) is minimal. To do this, we set equal to zero the partial derivative of (25):

\[ \frac{\partial}{\partial \nu}\left(\frac{NP+LQ}{P^2}-\frac{M}{Q}\right)=0 \]

or

\[ (C_{gk}+C_k)\left[ \frac{ \frac{1}{R_k}+\frac{1}{R_{i1}}+\frac{1}{R_{i2}} }{ (1+\nu)^2\left(S_1+S_2+\frac{1}{R_k}+\frac{1}{R_{i1}}+\frac{1}{R_{i2}}\right) } -\frac{(1-\nu)}{(1+\nu)^3} \right]=0. \]

Hence \(\nu\) is obtained as

\[ \nu=\nu_0= \frac{ S_1+S_2+\frac{2}{R_{i2}} }{ S_1+S_2+2\left(\frac{1}{R_k}+\frac{1}{R_{i1}}\right) }. \tag{7'} \]

It coincides with that value of \(\nu\) which proves most advantageous from the point of view of the self-excitation conditions.

Let us now determine the sign of the second derivative of difference (25). This derivative is equal to

\[ \frac{\partial^2}{\partial \nu^2}\left(\frac{NP+LQ}{P^2}-\frac{M}{Q}\right) =(C_{gk}+C_k)\cdot \frac{2}{(1+\nu_0)^4}>0. \]

Thus it has been shown that the extremum found is a minimum.

Let us now see what difference (25) is equal to for \(\nu=\nu_0\). Substituting for this purpose \(\nu=\nu_0\) from (7′) into (25), after simple transformations we find

\[ \frac{NP+LQ}{P^2}-\frac{M}{Q} =(C_{gk}+C_k)\left\{ \frac{ \left[ S_1S_2+\left(\frac{S_1+S_2}{2}\right) \left(\frac{1}{R_k}+\frac{1}{R_{i1}}+\frac{1}{R_{i2}}\right)+\ldots \right. }{ \ldots } \right. \]

\[ \ldots \frac{ +\dfrac{1}{R_{i2}}\left(\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}\right) }{ \left(S_1+S_2+\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}+\dfrac{1}{R_{i2}}\right)^2 } - \left[ \left(\dfrac{S_1+S_2}{2}\right)^2 + \left(\dfrac{S_1+S_2}{2}\right) \left(\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}+\dfrac{1}{R_{i2}}\right) + \ldots + \dfrac{1}{R_{i2}}\left(\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}\right) \right] \right\}. \tag{26} \]

The expressions in the square brackets in (26) differ only in the first terms in the numerators. In the minuend this term is equal to \(S_1S_2\), while the corresponding term in the subtrahend is \(\left(\dfrac{S_1+S_2}{2}\right)^2\).

Since, for unequal \(S_1\) and \(S_2\),

\[ S_1S_2<\left(\dfrac{S_1+S_2}{2}\right)^2, \]

then for \(\nu=\nu_0\)

\[ \eta=\frac{NP+LQ}{P^2}<\frac{M}{Q}=\eta_1 . \]

In expression (25) the subtrahend

\[ \frac{\nu}{(1+\nu)^2} \]

tends to zero both as \(\nu\to 0\) and as \(\nu\to\infty\). Between these limiting values of \(\nu\) and the value \(\nu=\nu_0\), there are points at which the subtrahend is equal to the minuend and equality holds between the two ratios we are comparing,

\[ \left(\frac{NP+LQ}{P^2}\ \text{and}\ \frac{M}{Q}\right). \]

Fig. 9

The curve of the dependence of their relative difference

\[ \frac{\Delta\eta}{\eta}=\frac{\eta-\eta_1}{\eta} \]

on

\[ \delta=\frac{\nu}{\nu_0} \]

is shown in Fig. 9. It descends below the \(\delta\)-axis, cross-

intersecting the latter at two points

\[ \delta_1=-\frac{\nu_1}{\nu_0}\quad \text{and}\quad \delta_2=-\frac{\nu_2}{\nu_0}. \]

The error introduced by discarding, in the expressions for the conductance and the dynamic capacitance of a cathode-coupled two-terminal network, terms that do not depend on frequency, or terms containing \(\omega^2\), is determined by the same formulas (13), (19), and (20), which were obtained for the case of identical parameters. The only difference is that the quantities entering them, \(\dfrac{\Delta \vartheta}{\vartheta}\) and \(\dfrac{\Delta \eta}{\eta}\), depend differently on \(\delta=\dfrac{\nu}{\nu_0}\).

Frequency instability of a cathode-coupled oscillator under changes in supply voltages

The dependence of the dynamic capacitance \(C_{ab}=C_a+C_{cd}\) on the tube parameters leads to the fact that, in an oscillator of the \(LC\) type built on the basis of the two-terminal network under consideration, the oscillation frequency becomes nonconstant when the supply mode is changed. The “static” parasitic capacitances \(C_{gk}\), \(C_k\), and \(C_a\) (Fig. 26) also depend on the operating mode. It affects them through thermal effects and through the change in the density of space charges inside the tube. It is very difficult to take these indicated “mode” changes of the capacitances \(C_{gk}\), \(C_k\), and \(C_a\) into account analytically. We shall therefore treat these capacitances as constant and shall study only that influence of the mode on the dynamic capacitance \(C_{ab}\) which is due to the presence, in the analytical formulas for \(C_{ab}\) or \(C_{cd}\), of terms connected with the tube parameters.

The character of the change in frequency when the supply voltages change proves to be different under different conditions. Among such conditions one must include the magnitude of the oscillation frequency. From formula (14) for the capacitance \(C_{cd}\) it may be seen that, as the frequency \(\omega\) increases, the role of the terms of this formula containing \(\omega^2\) and not depending directly on the tube parameters increases. If the influence of the mode through thermal effects and changes of the capacitances of the space charges turns out to be weaker than that caused by changes in the tube parameters, then the stability of the oscillator should improve as the frequency is increased.

As we have already seen, independence of the dynamic capacitance \(C_{cd}\) from the tube parameters, and therefore its stability, can be achieved by choosing the ratio \(\nu=\dfrac{C_{gk}}{C_k}\) equal to \(\nu_0\). If the causes not taken into account by the proposed linear theory preserve, in this case, some frequency instability, it can be reduced by a corresponding underfulfillment or overfulfillment of the compensation conditions.

To verify this, let us turn to formula (14). Bearing in mind not very high frequencies, we discard the terms with \(\omega^2\), which lead only to smoothing of the action of the “mode” terms depending on the tube parameters. Thus we arrive at the simplified formula (18). With the aid of relations (21), this formula is easily transformed into the form:

\[ C_{cd}=(C_{gk}+C_k)\frac{\nu_0^2+\nu}{(\nu_0+1)^2(1+\nu)}. \tag{27} \]

Here the only “mode” parameter is \(\nu_0\). Therefore expression (27) turns out to be very convenient for investigating the influence of changes in the

...changes in the supply voltages. When the anode voltage is decreased, just as when the filament voltage is decreased, the transconductance decreases. The parameter \(\gamma_0\) also decreases in this case. In comparison with \(S\), the change in \(\gamma_0\) occurs, as is evident from (7), considerably more slowly, the more slowly the larger \(R_k\) is. Hence, in accordance with formula (27), the conclusion is obtained: as \(R_k\) is increased, the stability of the dynamic capacitance and of the oscillator frequency increases.

Let us examine expression (27) for extrema, regarding the “operating” parameter \(\gamma_0\) as variable. For this purpose we set the partial derivative equal to zero:

\[ \frac{\partial C_{cd}}{\partial \gamma_0} = (C_{gk}+C_k)\frac{2(\gamma_0-\gamma)}{(1+\gamma_0)^3} =0. \]

The only extremal value is obtained for \(\gamma_0=\gamma\). Taking the second partial derivative and substituting \(\gamma_0=\gamma\) into it, we find

\[ \left(\frac{\partial^2 C_{cd}}{\partial \gamma_0^2}\right)_{\gamma_0=\gamma} = \frac{2(C_{gk}+C_k)}{(1+\gamma)^3} >0. \]

Thus, for \(\gamma_0=\gamma\) we have the minimum capacitance \(C_{cd}\). Substitution of \(\gamma_0=\gamma\) into expression (27) transforms it into (17) and makes it independent of the operating point. Owing to the flat minimum, small deviations of \(\gamma_0\) from \(\gamma\) do not cause any noticeable changes in the capacitance \(C_{cd}\) or in the frequency. Thus, by a somewhat different route, we arrive at the previously obtained condition \(\gamma=\gamma_0\) for frequency stability of the circuit.

If \(\gamma\) is chosen larger than \(\gamma_0(E_a,E_f)\), calculated for normal values of the supply voltages, then when the supply voltages are lowered \(\gamma_0\) will decrease and move still farther away from the optimum value, equal to \(\gamma\). In this case the capacitance \(C_{cd}\) will increase (since \(\gamma_0=\gamma\) corresponds to the minimum capacitance, and we are moving away from it), and the frequency will decrease.

If \(\gamma\) is chosen smaller than \(\gamma_0(E_a,E_f)\) for normal values of the supply voltages, then as the latter are decreased \(\gamma_0\) will, decreasing, approach the optimum value \(\gamma_0=\gamma\), and the capacitance \(C_{cd}\) will approach the minimum. In this case the oscillation frequency will increase.

Thus, by controlling the ratio of the capacitances \(\gamma=\dfrac{C_{gk}}{C_k}\) (for example, by artificially increasing one of them), one can change the course of the frequency-instability curve.

Figure 10 shows the relative change in the dynamic capacitance \(C_{cd}\) when the anode voltage is varied, for a two-terminal network using a 6N8 tube, having \(R_k=1\ \text{k}\Omega\) and \(R_a=10\ \text{k}\Omega\). The curves in the figure were obtained for different \(\gamma\) by means of the formula

\[ C_{cd}=(C_{gk}+C_k) \left[ \frac{ \dfrac{\gamma}{1+\gamma}\left(\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}\right) + \dfrac{1}{1+\gamma}\cdot\dfrac{1}{R_{i2}} }{ S_1+S_2+\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}+\dfrac{1}{R_{i2}} } + \frac{ S_1S_2-\dfrac{1}{R_{i2}}\left(\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}\right) }{ \left(S_1+S_2+\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}+\dfrac{1}{R_{i2}}\right)^2 } \right]. \tag{28} \]

This formula takes into account the nonuniformity of the parameters of the tubes of the two-terminal network. The first term in the square brackets increases as the plate voltage is decreased, while the second decreases. The choice of $\nu$ makes it possible to regulate the ratio between these terms and thereby to change the shape of the curve

$$ \frac{\Delta C_{cd}}{C_{cd}} = F(E_a). $$

In considering the question of the instability of the dynamic capacitance, we discarded the terms with $\omega^2$ that do not depend directly on the operating condition. In the case $\nu = \nu_0$, they cannot change the magnitude of this capacitance.

Fig. 10

If $\nu \ne \nu_0$, the terms with $\omega^2$ act as leveling terms: owing to them the capacitance changes less than it would in their absence. The influence of the terms with $\omega^2$ can be estimated by the error that is obtained when they are discarded. For this purpose one may use formula (19). The factor $\dfrac{x}{1+x}$ in this formula is a measure of the indicated leveling action.

Under real conditions, adjustment of the ratio between the capacitances $C_{gk}$ and $C_k$ may not lead to obtaining the desired high stability of the oscillator, or it may require stronger adjustment (another value of $\nu$) than would be expected on the basis of the linear theory described here. To some extent the results of this theory are distorted as a consequence of the dependence, noted above, of the parasitic capacitances themselves $C_{gk}$, $C_k$, and $C_a$ on the operating condition. However, the principal reasons that can not only distort the effect of adjustment by means of $\nu$, but also make this adjustment ineffective, are the following.

1. The change, averaged over a period, in the dynamic capacitance of the two-terminal network, associated with the inequality of the dynamic capacitances on the branches and on the falling portion of its current–voltage characteristic.

2. The action of grid currents, not taken into account by the proposed theory. These currents may appear for comparatively small \(R_k\) (of the order of \(1\ \text{k}\Omega\) and less) and large amplitudes of oscillation, which, in turn, may occur for large \(Z_p\)² and \(R_a\). Increasing \(R_k\) or decreasing \(Z_p\) and \(R_a\) makes it possible easily to avoid grid currents and thus to eliminate this cause interfering with the realization of compensation.

Let us consider in somewhat greater detail the question of the mean dynamic capacitance. The theory set forth above applies to conditions at the operating points. Outside them the tube parameters already have other values, and so do the dynamic capacitances. During the negative half-cycle of the voltage on the circuit, when the magnitude of this voltage is sufficiently large, the first tube \(L_1\) (Fig. 1) is cut off. In this case \(S_1=0\), \(R_{i1}=\infty\), and the active conductance and dynamic capacitance of the two-terminal network with cathode coupling, in accordance with (2), (3), and (1), are expressed by the formulas:

\[ \sigma_{ab-} = \frac{1}{R_a} + \frac{ \left(S_2+\frac{1}{R_k}+\frac{1}{R_{i2}}\right)\frac{1}{R_k R_{i2}} - \omega^2\left(C_{gk}C_kS_2-\frac{C_{gk}^2}{R_k}-\frac{C_k^2}{R_{i2}}\right) }{ \left(S_2+\frac{1}{R_k}+\frac{1}{R_{i2}}\right)^2 + \omega^2(C_{gk}+C_k)^2 }, \]

\[ C_{ab-} = C_a + \frac{ \left(\frac{C_{gk}}{R_k}+\frac{C_k}{R_{i2}}\right)S_2 + \frac{C_{gk}}{R_k^2} + \frac{C_k}{R_{i2}^2} + \omega^2 C_{gk}C_k(C_{gk}+C_k) }{ \left(S_2+\frac{1}{R_k}+\frac{1}{R_{i2}}\right)^2 + \omega^2(C_{gk}+C_k)^2 }. \tag{29} \]

At the peaks of the positive half-cycle of the voltage on the circuit, the second tube \(L_2\) is cut off, and \(S_2=0\), while \(R_{i2}=\infty\). For the active component of the input conductance and for the dynamic capacitance of the two-terminal network in this case we have:

\[ \sigma_{ab+} = \frac{1}{R_a} - \omega^2 \frac{ C_{gk}C_kS_1-\frac{C_{gk}^2}{R_k}-\frac{C_k^2}{R_{i1}} }{ \left(S_1+\frac{1}{R_k}+\frac{1}{R_{i1}}\right)^2 + \omega^2(C_{gk}+C_k)^2 }, \]

\[ C_{ab+} = C_a + \frac{ C_{gk}\left(\frac{1}{R_k}+\frac{1}{R_{i1}}\right) \left(S_1+\frac{1}{R_k}+\frac{1}{R_{i1}}\right) }{ \left(S_1+\frac{1}{R_k}+\frac{1}{R_{i1}}\right)^2 + \omega^2(C_{gk}+C_k)^2 } + \]

\[ + \frac{ \omega^2 C_{gk}C_k(C_{gk}+C_k) }{ \left(S_1+\frac{1}{R_k}+\frac{1}{R_{i1}}\right)^2 + \omega^2(C_{gk}+C_k)^2 }. \tag{30} \]

At not very high frequencies, when terms containing \(\omega^2\) may be neglected, the formulas given take the simpler form

\[ g_{ab-}=-\frac{1}{R_a}+\frac{1}{R_{i2}+R_k(1+\mu_2)} , \]

\[ g_{ab+}=\frac{1}{R_a} , \]

\[ C_{ab-}=C_a+ \frac{ \dfrac{C_{gk}}{R_k}\left(S_2+\dfrac{1}{R_k}\right) + \dfrac{C_k}{R_{i2}}\left(S_2+\dfrac{1}{R_{i2}}\right) }{ \left(S_2+\dfrac{1}{R_k}+\dfrac{1}{R_{i2}}\right)^2 }, \tag{29'} \]

\[ C_{ab+}=C_a+ \frac{ C_{gk}\left(\dfrac{1}{R_k}+\dfrac{1}{R_{i1}}\right) }{ S_1+\dfrac{1}{R_k}+\dfrac{1}{R_{i1}} }. \tag{30'} \]

The quantities \(S_1\), \(S_2\), \(\dfrac{1}{R_{i1}}\), and \(\dfrac{1}{R_{i2}}\) entering expressions \((29')\) and \((30')\) always decrease as the supply voltages are reduced. This leads to an increase in the capacitances \(C_{ab+}\) and \(C_{ab-}\), and to a decrease in frequency.

Thus, the dynamic capacitances corresponding to the branches of the volt-ampere characteristic always increase as the supply voltages are reduced. If \(\nu=\nu'\) is such that a theory taking into account only the conditions at the operating points predicts a decrease in frequency with decreasing supply voltages, then the reduction in \(\nu\) necessary to obtain a stable frequency in fact proves insufficient. An even greater reduction in \(\nu\) is required to compensate for the lowering of the frequency due to the increase in the dynamic capacitance on the branches of the volt-ampere characteristic corresponding to the cutoff states of the first and second tubes.

Results of the Experimental Verification

An experimental verification of the principal conclusions of the present article was carried out both on oscillator circuits and with the aid of low-frequency models of a two-terminal network with cathode coupling. Some data from this verification are given below. The capacitances \(C_{gk}\) and \(C_k\) in the model of the two-terminal network were deliberately assigned large values (thousands of picofarads). The two-terminal network was constructed using a 6N8 tube; the resistances \(R_k\) and \(R_a\) were equal to \(R_k=2\,\mathrm{k\Omega}\), \(R_a=10\,\mathrm{k\Omega}\). The measurements were performed by means of a bridge circuit borrowed from the work of Kh. I. Cherne [4], devoted to the determination of total resistances with a negative real part. This circuit is shown in Fig. 11.

Audio-frequency oscillations were applied to terminals \(ab\) of the six-terminal network in Fig. 11. At the output, an oscillograph was connected to terminals \(cd\) as an indicator. The measured resistance \(z_x=r_x+jx_x\) was connected between terminals \(a\) and \(c\). For this circuit, according to Kh. I. Cherne, the resistances \(r_x\) and \(x_x\) can be calculated, through the parameters of the six-terminal network corresponding to the bridge balance, by means of the following expressions:

\[ r_x = - (r_1 + r_3), \]

\[ x_x = - \frac{1}{\omega C_x}, \tag{31} \]

where

\[ C_x = \frac{1}{\omega^2 r_1 r_3 C_2}. \]

Fig. 11

Since, in deriving the formulas of the present article, the input resistance of the cathode-coupled two-terminal network was represented as a parallel connection of a negative resistance and a certain capacitance, it is necessary further to pass from the series combination of active and reactive resistances, which underlies formulas (31), to the equivalent parallel one. This transition leads to the expressions:

\[ \rho = - (r_1 + r_3)\left[1 + \left(\frac{\omega C_2 r_1 r_3}{r_1 + r_3}\right)^2\right], \]

\[ C_{cd} = \frac{r_1 r_3 C_2}{(r_1 + r_3)^2 + (\omega r_1 r_3 C_2)^2}. \tag{32} \]

When using formulas (2) and (14), the values of the tube parameters at the operating points were first determined. Table 1 gives the results of measurements of \(\rho\) and \(C_{cd}\) for a model of a cathode-coupled two-terminal network, performed by means of the bridge of Fig. 11. The two-terminal network had \(R_k = 2\ \text{k}\Omega\), \(R_a = 10\ \text{k}\Omega\); the anode supply voltage was \(E_a = 200\ \text{V}\), and the filament voltage \(E_f = 6.3\ \text{V}\). In the last columns of the table are given the values of \(\rho\) and \(C_{cd}\) calculated for the same conditions by means of formulas (2) and (14) of the present article. The figures in the table show that the theory presented correctly reflects the influence of the parasitic capacitances \(C_{gk}\) and \(C_k\) on the active conductance and dynamic capacitance of a cathode-coupled two-terminal network.

To check the influence of

\[ \nu = \frac{C_{gk}}{C_k} \]

on the shape of the instability curves, a large number of curves were taken showing the dependence of the relative frequency deviation of the cathode-coupled oscillator on changes in the supply voltages. The measurements were made at audio and radio frequencies for different values of the parameters of the two-terminal-network circuit, different \(\nu\), and different magnitudes of the resonant resistance of the oscillatory circuit.

It proved possible to compensate the instability under changes of the supply voltages not only by selecting the appropriate

Table 1

Results of measurements of \(\rho\) and \(C_{cd}\) on a model two-terminal network with cathode coupling

Parasitic capacitances \(C_{gk}\), pF	Parasitic capacitances \(C_k\), pF	\(\rho\)	\(\dfrac{C_{gk}C_k}{C_{gk}+C_k}\), pF	Bridge-circuit parameters \(r_1\), ohm	Bridge-circuit parameters \(r\), ohm	Bridge-circuit parameters \(C_n\), pF	Frequency \(f\), Hz	Input resistance and input capacitance, measured by bridge \(\rho\), ohm	Input resistance and input capacitance, measured by bridge \(C_{cd}\), pF	Input resistance and input capacitance, calculated from formulas \(\rho\), ohm	Input resistance and input capacitance, calculated from formulas \(C_{cd}\), pF
3700	7000	0,53	2420	1140	1210	12200	10000	2830	2430	2820	2440
7100	7000	1,01	3530	1000	1000	20700	10000	2840	3650	2770	3600
11500	7000	1,65	4360	1100	510	42500	10000	3000	4940	2940	5130
16500	7000	2,36	4920	970	280	88000	10000	3140	6100	3190	6430
21500	7000	3,07	5300	680	220	161800	10000	3460	7820	3550	7700
7100	11500	0,616	4380	1000	500	37700	12000	2780	4450	2740	4300
7100	17000	0,417	5030	800	600	47000	10000	2842	5660	2870	5300
7100	22500	0,315	5400	790	520	58000	10000	3030	6020	3000	6070
7100	27000	0,263	5630	860	320	90000	10000	3240	6500	3180	6580

values of $\nu$, but also by selecting $R_k$, $R_a$, or the feedback capacitance $C_1$. The values of $\nu$ that ensured compensation of the instability turned out to differ sharply from those that might have been expected on the basis of the theory set forth above; moreover, contrary to the latter, there proved to be two such values: one much less than unity, the other considerably greater. Such compensation was observed only in cases where $Z_p \gg |\rho|$. For $Z_p$ comparable with $|\rho|$ (at short wavelengths), compensation of the instability by selecting $\nu$ could not be obtained. Both the connection of additional capacitors in parallel with $C_k$ and the artificial increase of $C_{gk}$ invariably led to an increase in the instability.

Fig. 12

Legend in the figure: tube 6Н8; $R_k=3\,\text{k}\Omega$, $R_a=100\,\text{k}\Omega$;
1 — $C'_k=4\,\text{pF}$; 2 — $C'_k=15\,\text{pF}$; 3 — $C'_k=27\,\text{pF}$;
4 — $C'_{gk}=9\,\text{pF}$; 5 — $C'_{gk}=15\,\text{pF}$; 6 — $C'_{gk}=27\,\text{pF}$;
$f=6.3$ [[unclear: unit]]; with $C'_{gk}=C'_k=0$, $f=10$ [[unclear: unit]], $U_a=38\,\text{V}$.

In Fig. 12 are shown the curves

\[ \frac{\Delta f}{f}=F(E_a), \]

taken for a generator using a 6Н8 tube at various values of capacitances connected in parallel with $C_{gk}$ and $C_k$. These capacitances added to the circuit will be denoted respectively by $C'_{gk}$ and $C'_k$. From the curves in Fig. 12 it is evident that both the addition of $C'_{gk}$ and the connection of $C'_k$ lead to an increase in the instability of the circuit: the curves bend downward still more. The action of the capacitance $C'_{gk}$ is in this case stronger than that of $C'_k$. In this one can see only some tribute to the theory set forth earlier, which, in order to raise the instability curve, required a decrease in the ratio

\[ \nu=\frac{C_{gk}}{C_k} \]

or, in other words, an increase in $C_k$. The course of the curves in Fig. 12 cannot be explained by the action of such causes as a change in the temperature regime or in the space-charge capacitance, a change in the nonlinear correction to the period, or instability of the tube parameters at the operating points. Indeed, with increasing temperature the capacitances of capacitors usually increase. In our case, with increasing temperature in the tube, the total capacitance in the circuit also increases. This is indicated by the curves of the frequency drift produced by heating of the tube after switching on (Fig. 13). Therefore, due to chan-

of a change in the temperature regime the frequency should increase as the supply voltages are lowered. In reality, however, the opposite picture is observed (curves 1 in Figs. 12, 14, and 15).

The space-charge capacitance, when \(E_a\) is lowered and \(E_f=\mathrm{const}\), increases, since the number of electrons drawn off by the anode decreases. This could provide a qualitative explanation of the behavior of the curves \(\Delta f/f=F(E_a)\). However, if this cause were decisive, one would expect that, with a simultaneous proportional change in \(E_a\) and \(E_f\), the curves \(\Delta f/f=F_1(E_a,E_f)\) would run above the curves \(\Delta f/f=F(E_a)\), since lowering \(E_f\) leads to a decrease in the space-charge density and in the capacitance due to it. In fact, the simultaneous decrease of \(E_a\) and \(E_f\) is accompanied by an increase in the instability. The curves bend downward even more strongly.

Fig. 13

The connection of additional capacitances \(C'_{gk}\) and \(C'_k\) should not have any direct effect either on the temperature regime or on the space-charge capacitance. Therefore there is no basis for associating with them the observed lowering of the instability curves and the failure of compensation by selecting \(\gamma\).

Owing to the change in the nonlinear correction to the period, the frequency should increase as the supply voltages are lowered, since there is an approach to the threshold of oscillation. The latter, as is known, is accompanied by a decrease in the harmonic content and by an increase in frequency.

Calculations, moreover, show that the frequency drift due to the nonlinear correction is two orders of magnitude smaller than that observed experimentally. Thus, this cause also does not explain the observed curves.

It was noted earlier that the effect of the cathode-coupled two-terminal network on the oscillator frequency must be estimated by means of the two-terminal network’s dynamic capacitance averaged over a period. This capacitance is determined by the average action of different portions of the current-voltage characteristic that are swept out in the course of the oscillations and that have unequal dynamic capacitances. Substitution of the concrete values of the quantities entering formulas (31), (32), and (33) shows that the dynamic capacitance on the falling section is greater than on the branches. As the supply voltages are changed, the amplitude of the oscillations changes, as do the capacitances of the individual sections and their share in the average dynamic capacitance; consequently, the magnitude of the latter also changes.

When \(E_a\) is decreased, the oscillation amplitude falls somewhat faster than \(E_a\) [5]; the share of the middle section therefore increases, and the “integral” dynamic capacitance increases. As a result, as \(E_a\) is lowered the frequency decreases (curve 1, Fig. 12). If, simultaneously with \(E_a\), \(E_f\) is decreased proportionally, the changes in the tube parameters will be deeper and the fall in amplitude more rapid. This will lead to an even greater sag in the instability curve. The connection of \(C'_{gk}\) or \(C'_k\) will somewhat worsen the conditions for self-excitation and will lead to a more rapid decrease of amplitude as \(E_a\) and \(E_f\) are lowered. This strengthens the effect of the average dynamic capacitance and leads to an increase in the instability.

Thus, at least qualitatively, the course of the observed instability curves and the failure of compensation of frequency drift by proper selection of \(\gamma\) are explained by changes in the average dynamic capacitance.

In taking the curves of Fig. 12 we varied \(\gamma\), connecting to the oscillator only \(C'_{gk}\) or only \(C'_k\), so as not to lower the circuit frequency too much. If both additional capacitances \(C'_{gk}\) and \(C'_k\) are introduced into the circuit simultaneously, the compensation of instability predicted by the theory can readily be obtained. The success of the compensation in this case is connected with a reduction of the average-capacitance effect. In expressions (29) and (30) for the dynamic capacitances on the branches, the terms with \(\omega^2\), when \(C_{gk}\) and \(C_k\) are increased by means of the additional capacitances, acquire the predominant value, and the capacitances \(C_{cd-}\) and \(C_{cd+}\) will be approximately equal to

\[ \frac{C_k C_k}{C_{gk}+C_k}. \]

The capacitance on the falling section will be the same. The dynamic capacitance averaged over a period will then depend almost not at all on the amplitude. Its influence on the frequency stability is reduced.

In Fig. 14, curve 1 represents the dependence of the instability on \(E_a\) for an oscillator without additional capacitances. When \(C'_{gk}=37\ \mathrm{pF}\) and \(C'_k=80\ \mathrm{pF}\) are connected into the circuit, the instability is determined already by curve 2. The frequency is then lowered from 10.85 to 8.25 MHz owing to the increase of the total capacitance in the circuit. If the same lowering of frequency is obtained by increasing the capacitance of the main capacitor of the circuit, the instability curve proves to be only slightly better than with the minimum capacitance in the circuit (curve 3, Fig. 14).

The required ratio of capacitances was obtained as follows. The capacitor \(C'_k\) was taken equal to 80 pF, and the capacitance \(C'_{gk}\) was adjusted so that, at the anode voltage \(E_a=160\ \mathrm{V}\), a maxi-

\(^{1)}\) Here both the parasitic and the additional capacitances are included in \(C_{gk}\) and \(C_k\).

small amplitude. The latter was measured with a tube voltmeter connected to the tank circuit. Without any additional adjustment, curve 2 in Fig. 14 was obtained. This is a confirmation of the theory set forth above, which predicted the coincidence of the most favorable conditions for self-excitation (and, consequently, the conditions for obtaining the maximum oscillation amplitude) with the conditions for the best frequency stability.

Fig. 14

Curves analogous to those shown in Fig. 14, but obtained with simultaneous proportional variation of both supply voltages \(E_a\) and \(E_f\), are shown in Fig. 15. In this case as well, compensation of the instability is easily achieved by increasing the capacitances \(C_{э к}\) and \(C_k\) and by selecting the proper ratio between them.

Fig. 15

Thus, the parametric stabilization considered here makes it possible to ensure constancy of frequency under variations of the supply voltages over wide limits. It, however, has no relation to frequency drifts associated with changes in the parameters of the tank circuit itself as a result of temperature, mechanical, and other effects. With respect to these effects, a cathode-coupled oscillator differs in no way from other circuits. Therefore their influence should be weakened by the usual methods of increasing the constancy of the oscillatory circuit standard (by using temperature compensation, improving the design of the circuit, etc.).

Conclusions

1. By an appropriate choice of the ratio

\[ \gamma=\frac{C_{gk}}{C_k} \]

between the capacitances \(C_{gk}\) and \(C_k\) of a cathode-coupled two-terminal network, one can obtain the most advantageous conditions for self-excitation of the oscillator formed by the two-terminal network and the oscillatory circuit connected to it.

1. These conditions, for small plate resistances \(R_a^{1)}\) (of the order of \(10\ \text{k}\Omega\)), do not depend on frequency and remain unchanged up to very high frequencies, at which the lead inductances and the finite electron transit time play a large role. In the case of larger \(R_a^{2)}\) (of the order of \(100\ \text{k}\Omega\)), with the indicated optimum value \(\gamma=\gamma_0\), there is even an improvement of the self-excitation conditions as the frequency is increased.

2. For values of \(\gamma\) differing appreciably from \(\gamma=\gamma_0\), the self-excitation conditions deteriorate as the frequency is increased.

3. The dynamic input capacitance of a cathode-coupled two-terminal network at \(\gamma=\gamma_0\) does not depend on the parameters of the tubes or on the oscillation frequency and has a minimum value equal to the static capacitance of the two-terminal network.

4. The value \(\gamma=\gamma_0\) corresponds to the best frequency stability of the cathode-coupled oscillator.

5. At short wavelengths, when the resonant resistance of the circuit \(Z_p\) is comparable with the negative resistance of the two-terminal network \(r\), compensation by changing only one capacitance, \(C_{gk}\) or \(C_k\), cannot be obtained. However, a simultaneous increase of both capacitances and the choice of an appropriate ratio between them make it possible to compensate the instability easily. In this case the oscillation amplitude, in agreement with the theory, proves to be maximal, and the numerical values of \(\gamma_0\) are close to the theoretical ones.

6. The compensation of instability discussed in the preceding point is not critical. It ensures constancy of frequency under variations of the supply voltages over wide limits.

7. The experimentally observed deviations from the theory are explained by the action of causes not taken into account by it. Of these, the greatest significance is the change in the dynamic capacitance averaged over the oscillation period.

1) Or for identical tube parameters \((S_1=S_2)\).
2) Or for different tube parameters \((S_1\ne S_2)\).

References

1. Sulzer, P. G. Cathode-Coupled Negative-Resistance Circuit, PJRE, v. 36, No. 8, 1948.

2. Kolosov, A. A. Resonant Systems and Resonant Amplifiers, Svyazizdat, Moscow, 1949.

3. Bode. Network Theory and the Design of Feedback Amplifiers, translated from English under the editorship of A. A. Kolosov and L. A. Meerovich, Foreign Literature Publishing House, Moscow, 1948.

4. Cherne, Kh. I. Measurement circuits for total resistances of the shunt type with a negative real part, Radiotekhnika, vol. 4, No. 5, 1949.

5. Sulzer, I. A. Nonlinear theory of a cathode-coupled oscillator, Proceedings of TPI, vol. 86, 1958.

1. Two curves of the dependence of \(\dfrac{\Delta \eta}{\eta}\) on \(\delta\) were calculated (for \(R_k=1\,\text{k}\Omega\) and \(R_k=4\,\text{k}\Omega\)). They differ so little, however, that at the scale used they merge. ↩

2. \(Z_p\) is the resonant resistance of the circuit. ↩