Issues of Linearization and Stabilization of Vacuum-Tube Stages

L. M. ANAN'EV, L. E. BARABOSHKIN

(Presented by the scientific seminar of the Department of Industrial Electronics)

The purpose of the present work is to draw attention to the expediency of employing the principles of the successfully developing theory of sensitivity [1] and the theory of invariant nonlinear automatic-control systems [2] for constructing single-tube devices with improved characteristics. We shall illustrate this with the example of linearization of a vacuum-tube stage using control by the screen grid, Fig. 1.

We shall understand linearization of a circuit with respect to a given index as the approximation of the transfer function describing the process of obtaining this index to a constant value over as large as possible a range of variations of the input actions, disturbances, and the output index itself.

Nonlinearity is reduced in two ways: the differential parameters are stabilized (negative feedback is used), or the static characteristic of the transfer function is compensated by an opposite nonlinearity (including nonlinear feedback [5, 6]). In the latter case it is often economically more advantageous to use the different character of parameter variations in the multielectrode tube itself and to achieve parametric invariance and, consequently, compensation of disturbances from the power-supply sources.

Fig. 1. Circuit of a triode-connected stage in which part of the load is included in the common supply circuit of the anode and the screen grid

Let us estimate the possibility of linearizing the output voltage in the circuit of Fig. 1, which should be regarded as a nonlinear regulating system. The condition for invariance of \(U_a\) with respect to \(\Delta E_a\) for a given

structure of the system is the presence of two channels and the implementation of a transfer function linking the coordinate under consideration and the external disturbance [2, p. 101]. As follows from the diagram in Fig. 1,

\[ \Delta V_a=\Delta V_{\mathrm{e}}-\Delta V_{R_a}. \tag{1} \]

For qualitative reasoning we shall use a linear model of the tube and apply the superposition principle. The results of calculating the transfer coefficients and the quantities \(\Delta V_{\mathrm{e}},\ -\Delta V_{R_a}\) are presented in the form of the block diagram in Fig. 2, where, taking account of the reaction of the anode \(R_{in}\ne\infty\), the formulas are rewritten as \((R_k=0;\ R_{\mathrm{e}}=0)\):

Fig. 2. Block diagram of the vacuum-tube stage according to Fig. 1

\[ \Delta V_{\mathrm{e}} = \Delta V_{\mathrm{e}}^{\prime\prime} - \Delta V_{\mathrm{e}}^{\prime} = \frac{ \Delta E_a\left(1+\dfrac{R_a}{nR_{in}}\right) - S_{\mathrm{tr}}^{\prime}R_{\mathrm{e}}\Delta V_c }{ 1+\dfrac{R_a}{nR_{in}} +\dfrac{R_{\mathrm{e}}}{R_{i\mathrm{tr}}^{\prime}} +\dfrac{R_{\mathrm{e}}}{R_{in}^{\prime}} }; \]

\[ \Delta V_{R_a} = -\Delta V_{R_a}^{\prime\prime} - \Delta V_{R_a}^{\prime} = -R_a \frac{ \Delta E_a\left(S_{a\mathrm{e}}+\dfrac{1}{R_{in}}\right) + S_{ac}\Delta V_c }{ 1+S_{a\mathrm{e}}\cdot n\cdot R_{\mathrm{e}} +\dfrac{R_n}{R_{in}} }; \ldots \tag{2} \]

In these expressions:

\[ n=\frac{\Delta I_{\Sigma}}{\Delta I_a} = \frac{\Delta I_a+\Delta I_{\mathrm{e}}}{\Delta I_a} \]

is the current-distribution coefficient;

\[ \frac{1}{R_{i\mathrm{tr}}};\quad \frac{\partial I_{\Sigma}}{\partial V_a};\quad S_{\mathrm{tr}}=\frac{\partial I_{\Sigma}}{\partial V_c} \]

are the conductance and transconductance of the tube in triode connection (i.e., \(V_a=V_{\mathrm{e}}\));

\[ \frac{1}{R_{i\mathrm{tr}}^{\prime}},\quad S_{\mathrm{tr}}^{\prime} \]

are the conductance and transconductance of the total current in the presence of \(R_a\) \((V_a\ne V_{\mathrm{e}})\);

\[ R_{in}=\frac{\partial V_a}{\partial I_a} \]

is the internal resistance of the tube with respect to the anode current, taking into account current distribution;

\[ R_{in}^{\prime}=\frac{\partial V_a}{\partial I_{\mathrm{e}}} \]

is the internal resistance of the tube with respect to the anode current without

account of current distribution; \(R_a+nR_E=R_n\) is the equivalent total resistance for the anode current; \(S_{ac}=\dfrac{\partial I_a}{\partial U_c}\) is the transconductance of the anode current with respect to the first grid; \(S_{a\mathrm{э}}=\dfrac{\partial I_a}{\partial V_{\mathrm{э}}}\) is the transconductance of the anode current with respect to the screen grid.

With accuracy sufficient for practice, it may be assumed that

\[ S'_{tr}\simeq S_{tr}=n\cdot S_{ac};\quad R'_{itr}\simeq R_{itr};\quad S_{a\mathrm{э}}=\frac{1}{n\cdot R'_{itr}};\quad nR'_{in}=R_{in}; \]

\[ \frac{R'_{itr}}{R'_{itr}+R_E}=\frac{1}{1+nS_{a\mathrm{э}}R_E}. \]

As follows from the circuit of Fig. 2, the condition of invariance of \(V_{\text{out}}\) with respect to \(\Delta E_a\) is written as:

\[ \Delta V''_a=\Delta V''_{\mathrm{э}}-\Delta V''_{R_a}=0,\quad \text{i.e. } 1-S_{a\mathrm{э}}R_a=0 \tag{3} \]

or

\[ 1-\frac{R_a}{n\cdot R'_{itr}}=0. \tag{3a} \]

Expressions (3) and (3a) are simultaneously the condition of ideal filtering, i.e., equality of the filtration coefficient (4) to infinity,

\[ K_\phi=\left.\frac{\Delta E_a}{\Delta V''_a}\right|_{V_c=\mathrm{const}} = \frac{1}{\dfrac{\Delta V''_{\mathrm{э}}}{\Delta E_a}-\dfrac{\Delta V''_{R_a}}{\Delta E_a}} = \frac{1+S_{a\mathrm{э}}\cdot n\cdot R_E+\dfrac{R_n}{R_{in}}} {1+\dfrac{R_a}{R_{in}}-\left(S_{a\mathrm{э}}+\dfrac{1}{R_{in}}\right)\cdot R_a} \]

or

\[ K_\phi\simeq \frac{1+S_{a\mathrm{э}}\cdot n\cdot R_{\mathrm{э}}}{1-S_{a\mathrm{э}}\cdot R_a}, \tag{4} \]

and the condition that the output differential resistance be equal to zero,

\[ \rho_d= \left.\frac{\Delta V''_a}{\Delta I''_a}\right|_{V_c=\mathrm{const}} = \frac{\Delta V''_{\mathrm{э}}-\Delta V''_{R_a}}{\Delta I''_a} = \frac{1-S_{a\mathrm{э}}R_a}{S_{a\mathrm{э}}+\dfrac{1}{R_{in}}}\ldots \tag{5} \]

The analytical expressions for \(I_a=f(V_a)|_{V_c=\mathrm{const}}\) and \(I_{\mathrm{э}}=f(V_a)|_{V_c=\mathrm{const}}\) in the circuit of Fig. 1, which are easily obtained from (1) and (2) if \(\Delta V_c=0\) is set, describe nonlinear curves. The construction of the indicated curves is conveniently carried out graphically, using a family of anode characteristics in pentode and triode modes, as shown in Fig. 3. Experimentally measured characteristics \(I_a=f(V_a)\) for the 6P3S tube, shown in Fig. 4, practically coincide with those constructed according to Fig. 3.

As follows from expression (3), invariance can be realized by choosing \(R_a\), or, for a given \(R_a\), by selecting the operating mode where \(S_{a\mathrm{э}}=\dfrac{1}{R_a}\).

In the triode mode

\[ R_a=0\quad \text{and}\quad \rho_d=\frac{1}{S_{a\mathrm{э}}+\dfrac{1}{R_{in}}}=R_{itr}>0. \]

In pentodes usually, if \(R_{\mathrm{э}}=0\), \(R_a\) is chosen large, so that

\(1 - S_{aE}R_a < 0\) and \(\rho_d < 0\). However, in the latter case there are nevertheless two portions of the characteristic where the conditions \(1 - \dfrac{R_a}{nR_{itr}} > 0\) are satisfied. These portions are located in the cutoff region of the tube (\(R_{itr}\) is large) and in the region of high currents, where \(n\) increases sharply, i.e., the volt-ampere characteristics have an S-shaped form. By choosing the values of \(R_a\) and \(R_E\)

Fig. 3. Graphical construction of volt-ampere characteristics. \(\mu_{tr1}\) and \(\mu_{tr2}\) are the static amplification factors of the tube, found graphically at points 1 and 2.

one can obtain an operating mode in which the S-shaped curves are approximated in the best way by straight lines perpendicular to the \(V_a\) axis. The form and degree of linearization of the output characteristics can be judged from the dependence of the filtering coefficient \(K_f\) (4), which characterizes

Fig. 4. Experimentally measured volt-ampere characteristics of the 6P3S tube according to the circuit of Fig. 1.

the transfer function \(\Delta V_a = f(\Delta E_a)\) at \(V_c = \text{const}\), as a function of the voltage \(V_c\). It follows from (4) that, in order to expand the range of variation of \(V_c\), where the condi-

for compensation to occur constructively, it is necessary, first, to replace the resistance \(R_a\) of the same type of tube in triode connection; then

\[ 1-S_{a\varepsilon}\cdot R_a = 1-\frac{R_{i\mathrm{tr}}}{nR'_{i\mathrm{tr}}}\simeq 0,\ldots \tag{6} \]

since \(n\) is close to 1 (especially for power tubes 6P3S, 6P14P, etc.). If \(n\) is large (pentodes with a dense screen grid, for example, 6Zh10B), then the screen current and its conductance can be reduced by including a limiting resistance \(R_\varepsilon\) in the screen-current circuit (Fig. 1), since for \(V_c=\mathrm{const}\) and \(S_{a\varepsilon}=1/R_a\) the coefficient \(n=R_a/R_{i\varepsilon}+1\), where \(R_{i\varepsilon}=\partial V_\varepsilon/\partial I_\varepsilon\) is the internal resistance with respect to \(I_\varepsilon\).

In this case the resistance \(R_E\) need not be included, i.e., only the pentode mode may be used.

Second, the parameter \(S_{a\varepsilon}\) can be stabilized by feedback by including resistors, for example \(R_E\) and \(R_k\), in the total-current circuit \(I_e=I_a+I_\varepsilon\). Then expressions (4) and (5), taking into account the presence of the resistance \(R_\varepsilon\), can be written in the form

\[ K_\phi = \frac{1+\dfrac{R_n}{R_{in\,\mathrm{fb}}}} {1-S_{a\varepsilon\mathrm{fb}}\cdot R_n} \quad\text{and}\quad \rho_d = \frac{1-S_{a\varepsilon\mathrm{fb}}\cdot R_n} {S_{a\varepsilon\mathrm{fb}}+\dfrac{1}{R_{in\,\mathrm{fb}}}}, \tag{7} \]

where

\[ R_{in\,\mathrm{fb}}= R_{in}\{1+n\cdot S\cdot R_k+S_{a\varepsilon}[\,nR_E+R_\varepsilon(n-1)\,]\}, \]

\[ S_{a\varepsilon\mathrm{fb}}= \frac{S_{a\varepsilon}} {1+nS_{a\varepsilon}R_E+S_{a\varepsilon}R_\varepsilon(n-1)+n\cdot S\cdot R_k}; \quad S=\left.\frac{\partial I_a}{\partial U_c}\right|_{V^a=\mathrm{const}}^{V^\varepsilon=\mathrm{const}} . \]

It follows from expressions (7) that as the feedback depth increases (i.e., as \(R_E, R_k\) increase), it is necessary to increase \(R_a\). The limiting factors are the maximum value of the quantity \(\dfrac{R'_{i\mathrm{tr}}+R_\varepsilon}{R'_{i\mathrm{tr}}}\) [4], the allowable reduction of the utilization factor of the source \(E_a\), and the fulfillment of other specified conditions [3]. Figures 5 and 6 show the ex-

Fig. 5. Dependence of the filtration coefficient on the potential of the first grid in the circuit of Fig. 1 using a 6P14P tube.

experimental data confirming the analysis set forth above. The bias of the load tube 6P14P in triode connection (see the curve \(R_a = R_{\text{tr}}, E_a = 300\text{ V}, R_{\text{e}} = 0\)) was chosen as \(-0.45\text{ V}\). The ripple coefficient for all cases is

\[ \frac{\Delta E_a}{E_a} = 0.015. \]

The initial operating mode of the three cases in Fig. 6 is typical (\(E_{\text{cm}} = -3\text{ V}, V_{\text{e}} = 120\text{ V}, V_{a0} = 120\text{ V}\)); the gain coefficients are identical (\(K \simeq 22\)); the ripple attenuation in the triode and pentode circuits due to the anode filter is equal to 40.

Conclusions

1. To increase the stability and linearity of vacuum-tube amplifier stages, it is convenient to use the principles of constructing invariant automatic-control systems that are insensitive to changes in parameters.

2. Using the control properties of the screen grid and taking measures to linearize the parameters, vacuum-tube stages, in addition to their main purpose (amplification, generation), can simultaneously be used as electronic filters and anode-voltage stabilizers.

Fig. 6. Dependence of the dc component of the anode voltage on changes in the EMF of the power-supply source in the circuit of Fig. 1 (stabilized stage), with fulfillment of the stabilization condition, and in ordinary triode and pentode circuits

The authors express their gratitude to assistant R. A. Baraboshkina for preparing the graphic material based on the experimental results.

REFERENCES

1. A. A. Voronov. Fundamentals of Automatic Control Theory, Part II, Moscow, Chap. IX, 1966.

2. B. M. Menskii. The Principle of Invariance in Automatic Control (textbook), Vol. (VZEI), 1956.

3. G. A. Shevtsov. On the optimal value of the elements of a rheostatic amplifier circuit. Scientific Notes of the Lvov Polytechnic Institute, issue 27, ser. radio engineering, 1955.

4. E. S. Antseliovich. The influence of nonlinearity of the internal resistance of a tube in the calculation of resistance amplifiers. Radiotekhnika, No. 3, 1955.

5. L. M. Anan’ev, L. E. Baraboshkin. Analysis of circuits with feedback through the screen grid. Proceedings of TPI (present collection).

6. L. M. Anan’ev, L. E. Baraboshkin, Ya. S. Pekker. A rheostatic stage insensitive to changes in the power-supply source. [[unclear: publication details not visible]]

7. L. E. Baraboshkin. Ultrlinear operating mode of screened tubes. Proceedings of the Scientific and Technical Conference Dedicated to Radio Day (brief contents). Tomsk, 1966.