ANALYSIS OF CIRCUITS WITH FEEDBACK VIA THE SCREEN GRID

L. M. ANAN’EV, L. E. BARABOSHKIN

(Presented by the scientific seminar of the Research Institute of Nuclear Physics, Electronics, and Automation at TPI)

In the technique of amplifying electrical signals, additional electrodes of electron tubes are often used both for taking the feedback voltage from the output and for applying it to the input of a circuit. In this case the feedback circuit consumes little signal energy, and the nonlinear dependence of the anode current on the voltages at the individual electrodes can be used for effective compensation of nonlinear distortions in individual harmonics. In the methodological and technical literature insufficient attention is paid to the analysis and calculation of such feedbacks. In particular, negative feedback via the screen grid is widely used in the output stages of modern sound-reproducing equipment. This is the so-called “ultralinear stage” (Fig. 1), whose tube parameters have an intermediate value, while its volt-ampere characteristics have an intermediate configuration between pentode operation and triode operation [1, 4, 2].

At the same time, nonlinear distortions may be smaller than in triode operation.

Fig. 1. Single-ended circuit of an ultralinear stage

Another example of circuits with feedback via the screen grid, as a result of which the tube changes its properties, is provided by cathode followers, in which the alternating potential of the screen grid has an intermediate value between the cathode potential and the anode (ground) potential.

Control of the anode current by means of the screen grid is also used in generators employing the Schembehl circuit, in phantastron and transitron generators, and in other circuits.

The well-developed theory of feedback for linear circuits cannot in some cases give a satisfactory explanation of the phenomena or recommend a calculation method for the circuits mentioned.

In this article, using the ultralinear stage as an example, an analysis is given of tube circuits in which, because of large signals, the feedback via the screen grid should be regarded as nonlinear. On the basis of such an analysis it will not be difficult to calculate similar circuits.

When designing the output stage of a low-frequency power amplifier, it is generally necessary to:

1) choose the initial operating mode of the tube,
2) calculate the oscillatory power delivered by the tube,
3) determine the nonlinear-distortion coefficient,
4) calculate the frequency response,
5) calculate the electrical parameters of the output transformer.

To determine the first three quantities, one must have a family of tube characteristics. For the ultralinear mode, such characteristics are not available in the literature.

The last two requirements can be fulfilled by analyzing the equivalent circuit at all audio frequencies and the equivalent internal resistance of the tube, which will also be intermediate between the triode and pentode values.

For single-tube circuits covered by external feedback, equivalent anode characteristics are sometimes constructed and the calculation is carried out from them [3]. In this case it is necessary to have the anode characteristics of the tube and the feedback coefficient \( \beta \). As applied to the circuits of Fig. 1, the voltage feedback coefficient, referred to the first grid, is

\[ \beta = \frac{U_{\mathrm{fb}}}{U_{\mathrm{out}}} = \frac{U_{s m}}{\mu_{\mathrm{tr}}\left(U_{s m}+U'_{a m}\right)} = \frac{1}{\mu_{\mathrm{tr}}}\cdot \frac{(J_a+J_s)R_s}{(J_a+J_s)R_s+J_aR_a} = \frac{1}{\mu_{\mathrm{tr}}}\cdot \]

\[ \cdot \frac{n\sqrt{p}\,(R_s+R_a)} {n\sqrt{p}\,(R_s+R_a)+(1-\sqrt{p})(R_a+R_s)} = \frac{1}{\mu_{\mathrm{tr}}}\cdot \frac{1} {1+\frac{1}{n}\left(\frac{1}{\sqrt{p}}-1\right)} = \]

\[ = \frac{1}{\mu_{\mathrm{tr}}}\cdot \frac{1} {1+\left(1-\frac{\mu_{\mathrm{tr}}}{A}\right)\cdot\left(\frac{1}{\sqrt{p}}-1\right)}, \tag{1} \]

where

\[ n=\frac{J_a+J_s}{J_a} \]
is a coefficient allowing for the current distribution in the tube;

\[ A=\frac{1}{\mu_{\mathrm{tr}}}\frac{J_a+J_s}{J_s} \]
is an auxiliary coefficient [2];

\[ P=\left(\frac{W_s}{W_s+W'_a}\right)^2 \]
is the screen-grid connection coefficient [1];

\(w_s;\; w'_a\) are the numbers of turns between the middle and end terminals of the primary winding of the output transformer (Fig. 1);

\(U_{s m};\; U_{a m}\) are the voltage amplitudes, respectively, on turns \(w_s\) and \(w'_a\) (Fig. 1);

\[ U_{\mathrm{fb}}=\frac{U_{s m}}{\mu_{\mathrm{tr}}} \]
is the feedback voltage, referred from the screen grid to the control grid;

\[ U_{\mathrm{out}}=U_{s m}+U'_{a m} \]
is the output voltage in the anode circuit.

\[ \sqrt{p}=\frac{w_s}{w_s+w'_a}=\frac{R_s}{R_a+R_s}. \tag{2} \]

Here \(R_e\) is the equivalent resistance at midrange frequencies, consisting of the recalculated resistances of the secondary and the entire primary winding in the triode part of the circuit (Fig. 4), i.e., it is the load for the triode connection of the tube; \(R_a\) is the resistance recalculated to winding \(w'_a\) from the remaining parts of the transformer.

The mutual-induction resistance is taken into account in this case as discussed at the end of the article.

The values of \(\mu_{tr}\) and \(A\) for 20 common domestic tubes are given in [2] on p. 300. The experimentally determined optimal value of \(p\), from the standpoint of minimum nonlinear distortion, is \(P=0.185\) for the 6P14P and 6P3S tubes; for the 6P1P and 6P6S tubes, \(P=0.05\) [4, 1, 2].

When constructing each point of the equivalent characteristic, it is desirable each time to find \(\mu_{tr}\) graphically from the triode characteristics of the tube, while pentode characteristics should be used as the initial characteristics. In this way the nonlinearity of the negative feedback is taken into account. A serious shortcoming of the method is that it does not take into account the part of the output power that appears because the signal passes through the feedback circuit.

Figure labels: \(I_a+I_e\), \(U_e\), \(I_{am}^{tr}\), \(U'_{em}\), \(E'_a\), \(U''_{em}\); curve labels \(0\), \(-4\), \(-8\), \(-12\), \(-16\), \(-20\); points \(1\), \(2\).

Fig. 2. Anode characteristics of a tube in triode connection.

Another method of transformer-analytical calculation, in our opinion simpler and more accurate, is as follows. Dynamic lines of alternating current are plotted simultaneously on the triode and pentode characteristics, as shown in Figs. 2 and 3. The initial operating points are chosen depending on the specified anode-supply source and on the type of operating mode—\(A\) or \(AB\) for a push-pull circuit. The slope of the load lines on the triode and pentode characteristics is determined by the magnitude of the total load and by the connection coefficient \(P\), which is expressed by formula (2).

Thus, the resultant action of the circuit in Fig. 1 is proposed to be considered as the joint operation of two tubes connected in series: the triode in Fig. 4 and the pentode, covered by feedback, in Fig. 5. Moreover, the amplitude of the equivalent alternating voltage applied to the control grid in the pentode is found as

\[ U'_{\mathrm{in},c}=U_{cm}+\frac{U'_{em}}{\mu_{tr}}, \tag{3} \]

where \(U_{cm}\) is the amplitude of the signal applied to the control grid.

\(U'_{em}\) is the amplitude of the voltage on the screen grid, i.e., on the load in Fig. 4.

Of course, \(U_{cm}\) and \(U_{em}\) are in opposite phase. The change in the anode potential relative to the screen-grid potential (the reaction of the anode to the screen-grid circuit) in the circuit of Fig. 4, for the working portion of the pentode characteristics, may be neglected and it may be assumed that the anode and screen grid are short-circuited.

This assumption is admissible, since it is known from the course on vacuum-tube devices that the influence of the anode potential in screen-grid tubes on the magnitude of the anode current is negligibly small in comparison with the action of the potential on the screen grid. The validity of such an approach was specially verified experimentally.

Fig. 3. Anode characteristics of the tube in pentode connection

Let us describe the oscillatory regime.

For the positive amplitude of the signal, the screen voltage decreases by \(U'_{\text{em}}\) (Fig. 2). This will cause a vertical “compression” of the pentode characteristics. The operating point in Fig. 3 will move upward along the line to the static characteristic \(U_{c_1}\) at the reduced screen voltage, which is equivalent, at an unchanged voltage \(U_{\text{e}} = E'\), to an equivalent characteristic with a voltage on the first grid

\[ U_{c_1} - \frac{1}{\mu_{\text{tr}_1}} U'_{\text{em}} . \]

For the negative half-wave of the signal at the input, the pentode characteristics “expand,” and the operating point will reach the equivalent curve

\[ U_{c_2} + \frac{1}{\mu_{\text{tr}_2}} U'_{\text{em}} . \]

In these expressions, \(\mu_{\text{tr}_1}\) and \(\mu_{\text{tr}_2}\) are determined graphically, respectively, at point 1 and at point 2 in Fig. 2. In the case where the family of static pentode cha-

Fig. 4. Triode part of the circuit of Fig. 1

Fig. 5. Pentode part of the circuit of Fig. 1

If the set of characteristics in the handbook does not correspond to the selected value of the power-supply source, each of the characteristics should be shifted vertically by the amount \(\dfrac{\Delta U_{\mathrm{e}}'}{\mu_{\mathrm{tr}}}\), where \(\Delta U_{\mathrm{e}}'\) is the difference between the selected \(E_{\mathrm{a}}\) and the screen voltage \(U_{\mathrm{e}}\) indicated for the family; \(\mu_{\mathrm{tr}}\) for each point being retuned is determined graphically from the family of triode characteristics.

The resulting oscillatory power of the entire stage and the nonlinear distortions for the individual harmonics are found by algebraically summing these quantities, determined separately from the two graphs.

\[ P_{\sim}=P_{\sim}^{\mathrm{tr}}+P_{\sim}^{\mathrm{p}}, \tag{3a} \]

\[ \gamma_{3}=\gamma_{3}^{\mathrm{tr}}+\gamma_{3}^{\mathrm{p}}, \tag{3в} \]

\[ \gamma_{2}=\gamma_{2}^{\mathrm{tr}}+\gamma_{2}^{\mathrm{p}}, \tag{3б} \]

\[ \gamma=\sqrt{\gamma_{2}^{2}+\gamma_{3}^{2}}. \tag{3г} \]

Here the index “tr” corresponds to the triode graph of Fig. 2; the index “p” to the pentode graph of Fig. 3; \(\gamma_2\), \(\gamma_3\), and \(\gamma\) are the coefficients of nonlinear distortion for the second and third harmonics and the total coefficient, determined by the five-coordinate method; \(P_{\sim}\) is the oscillatory power.

In the pentode characteristics of Fig. 3, the weakening of the influence of the nonlinear dependence of the anode current on the grid voltage is clearly visible, owing to the crowding of the curves—at the bottom and in the region of the line of the critical regime—at the top. In pentodes not encompassed by feedback, these nonlinearities cause an increase in the third harmonic as the anode load is increased. Unlike the action of linear negative feedback, in the present case, in addition to the useful distortion of the voltage waveform \(V_{\mathrm{c}}\), the feedback coefficient \(\beta\), which depends on \(\mu_{\mathrm{tr}}\) and “n,” also changes simultaneously. Indeed, from formula (1),

\[ \beta=\frac{1}{\mu_{\mathrm{tr}}}\, \frac{1}{1+\dfrac{1-\sqrt{p}}{\sqrt{p}}\cdot\dfrac{1}{\mathrm{n}}}, \tag{1a} \]

it follows that in the lower region of the pentode characteristics, where the nonlinearity is large, \(\mu_{\mathrm{tr}}\) decreases (the triode characteristics in Fig. 2 crowd together at the bottom because of the island effect), while the coefficient \(\beta\) and the depth of feedback increase. In the upper part, when nonlinearity appears because of current distribution in the pentode, the depth of feedback also increases, since in formula (1a) the denominator of the second factor decreases faster (because of the increase of \(\mathrm{n}=\dfrac{J_{\mathrm{a}}+J_{\mathrm{e}}}{J_{\mathrm{a}}}\)) than the denominator of the first factor increases, and the feedback coefficient \(\beta\) increases.

Thus the nonlinear distortions for the odd harmonics are reduced. The distortions for the second harmonic can be reduced to a lesser extent in this circuit, since the triode part of the circuit not covered by feedback, at small anode load, is a rather intense generator of even harmonics.

Compensation of the latter is accomplished by using push-pull circuits and by correctly selecting the anode-load resistance for the pentode part of the circuit, \(R_{\mathrm{a}}\).

As is known from the course on amplifiers, as the pentode load is increased, the phase of the second harmonic changes to the opposite one (when

the load line lies below the upper “knee” of the static characteristic \(U_c=0\ \text{V}\). Therefore, for an appropriate value of \(R_a\), in formula (36) for calculating the nonlinearity in the second harmonic it is possible to obtain terms \(\gamma_2^{\mathrm{tr}}\) and \(\gamma_2^{\mathrm{p}}\) of different signs. Having calculated from the graphs several variants, for different loads \(R_a\) and \(R_\mathrm{э}\), of the nonlinear distortions and oscillatory power, one can select the so-called “ultralinear” operating mode of the given tube.

The equivalent circuit of an ultralinear stage is shown in Fig. 6.

Designations: \(Z_\mathrm{э}\) is the equivalent resistance in the plate circuit of the triode part of the circuit of Fig. 4 for all frequencies in the passband,

\(Z_a\) is the equivalent resistance in the plate circuit of the pentode part of the circuit of Fig. 5.

The two-terminal networks \(Z_a\) and \(Z_\mathrm{э}\) contain the active load and reactive elements, referred respectively to the windings \(w_a\) and \(w_\mathrm{э}\).

Fig. 6. Equivalent circuit of an ultralinear stage

\(R_{in}\) and \(R_{i\mathrm{tr}}\) are the internal resistances of the tube in pentode and triode modes,

\(\mu_\mathrm{p}\) is the static amplification factor of the pentode,

\[ U_{\mathrm{in}_{c\,v}}=U_{\mathrm{in}}-\frac{(J_\mathrm{э}+J_a)Z_\mathrm{э}}{\mu_\mathrm{tr}}— \tag{7} \]

the equivalent input voltage in the pentode part of the circuit. In addition to calculating the frequency response and the parameters of the output transformer, it is often necessary to know the expression for the gain factor (for example, when determining the input dynamic capacitance of the final stage), and the output resistance. These formulas are not difficult to derive from the equivalent circuit.

By the mesh-current method we determine:

\[ \mu_\mathrm{p} U_{\mathrm{in}_{c\,v}}=J_a(R_{in}+Z_a), \]

\[ \mu_\mathrm{tr}U_{\mathrm{in}}=(J_a+J_\mathrm{э})(R_{i\mathrm{tr}}+Z_\mathrm{э}). \tag{8} \]

From relations (7) and (8), the plate current is

\[ J_a=\mu_\mathrm{p}U_{\mathrm{in}}\cdot \frac{1}{R_{in}+Z_a}\cdot \frac{R_{i\mathrm{tr}}}{R_{i\mathrm{tr}}+Z_\mathrm{э}}. \tag{9} \]

The plate and screen currents in sum:

\[ J_a+J_\mathrm{э}=\mu_\mathrm{tr}U_{\mathrm{in}} \frac{1}{R_{i\mathrm{tr}}+Z_\mathrm{э}}. \tag{10} \]

The output voltage after transformations is

\[ U_{\mathrm{out}}=J_aZ_a+(J_a+J_\mathrm{э})Z_\mathrm{э}= \]

\[ =\mu_\mathrm{p}U_{\mathrm{in}}\cdot \frac{Z_a}{Z_a+R_{in}}\cdot \frac{R_{i\mathrm{tr}}}{R_{i\mathrm{tr}}+Z_\mathrm{э}} +\mu_\mathrm{tr}U_{\mathrm{in}}\cdot \frac{Z_\mathrm{э}}{R_{i\mathrm{tr}}+Z_\mathrm{э}}. \tag{11} \]

And finally, transforming expression (11), we ultimately obtain the gain of the stage:

\[ K=\frac{U_{\text{out}}}{U_{\text{in}}} = S_{\mathrm{tr}} Z_{i\mathrm{e}} + S_{\mathrm{p}} Z_{ia}\cdot \frac{1}{1+\dfrac{Z_{\mathrm{e}}}{R_{i\mathrm{tr}}}}, \tag{12} \]

where

\[ Z_{i\mathrm{e}}=\frac{Z_{\mathrm{e}} R_{i\mathrm{tr}}}{R_{i\mathrm{tr}}+Z_{\mathrm{e}}}; \qquad Z_{ia}=\frac{Z_a R_{i\mathrm{p}}}{R_{i\mathrm{p}}+Z_a}; \]

\(S_{\mathrm{tr}}\) is the transconductance of the tube in triode mode,
\(S_{\mathrm{p}}\) is the transconductance of the tube in pentode connection.

The quantity \(S_{\mathrm{tr}}\cdot Z_{i\mathrm{e}}\) is the coefficient of direct signal transmission through the feedback circuit from the amplifier input to the output, i.e., in the present case the feedback circuit is a four-terminal network active in the forward direction and passive in the reverse direction. The quantity \(S_{\mathrm{p}}\cdot Z_{ia}\) is the transmission coefficient of the pentode part of the circuit with load \(Z_a\) and constant screen voltage. The depth of negative feedback in this case is equal to

\[ A=1+\frac{Z_{\mathrm{e}}}{R_{i\mathrm{tr}}}. \tag{13} \]

The feedback coefficient is

\[ \beta=\frac{A-1}{S_{\mathrm{p}}Z_{ia}} =\frac{Z_{\mathrm{e}}}{R_{i\mathrm{tr}}\cdot S_{\mathrm{p}}\cdot Z_{ia}} =\frac{1}{\mu_{\mathrm{tr}}}\cdot \frac{1}{\dfrac{S_{\mathrm{p}}}{S_{\mathrm{tr}}}\cdot \dfrac{Z_{ia}}{Z_{\mathrm{e}}}} \simeq \]

\[ \simeq \frac{1}{\mu_{\mathrm{tr}}}\cdot \frac{1}{\dfrac{1}{n}\left(\dfrac{Z_a+Z_{\mathrm{e}}}{Z_{\mathrm{e}}}-1\right)} = \frac{1}{\mu_{\mathrm{tr}}}\cdot \frac{1}{\dfrac{1}{n}\left(\dfrac{1}{\sqrt{p}}-1\right)}, \]

which corresponds to formula (1), if in it one takes

\[ \beta=\frac{U_{\mathrm{em}}}{U'_{\mathrm{am}}}\cdot \frac{1}{\mu_{\mathrm{tr}}}. \]

The analysis given did not take into account, in each of the two circuits, the EMF of mutual induction. If this circumstance is taken into consideration, then in each part of the equivalent circuit of Fig. 6 it is necessary to include, in series with the load, an EMF generator: for the triode part \(J'_a\cdot Z_{\mathrm{sv}}\), and for the pentode part \((J_a+J_{\mathrm{e}})'\cdot Z_{\mathrm{sv}}\).

The current values in such a more complicated circuit have the form

\[ J'_a=\mu_{\mathrm{p}} U_{\text{in}}\, \frac{1}{ (R_{i\mathrm{p}}+Z_a)(R_{i\mathrm{tr}}+Z_{\mathrm{e}}) - \dfrac{1}{ R_{i\mathrm{tr}}-Z_{\mathrm{sv}}\dfrac{\mu_{\mathrm{tr}}}{\mu_{\mathrm{p}}} -Z_{\mathrm{sv}}\dfrac{\mu_{\mathrm{p}}}{\mu_{\mathrm{tr}}} } }, \tag{14} \]

\[ (J_a+J_{\mathrm{e}})'= \mu_{\mathrm{tr}} U_{\text{in}}\cdot \frac{1}{ (R_{i\mathrm{tr}}+Z_{\mathrm{e}}) + \dfrac{Z_{\mathrm{sv}}}{R_{i\mathrm{p}}+Z_a}\cdot \left(\dfrac{\mu_{\mathrm{p}}}{\mu_{\mathrm{tr}}}\cdot R_{i\mathrm{tr}}-Z_{\mathrm{sv}}\right) }. \tag{15} \]

Here \(Z_{\mathrm{sv}}\) is the mutual-induction resistance.

Formulas (14) and (15) turn into (9) and (10) if one sets \(Z_{\mathrm{sv}}=0\) in them. The introduction of dependent generators into the circuits, owing to the action of magnetic coupling, is equivalent to an increase in the active resist-

...of the voltage. We compute the equivalent loads \(Z'_a\) and \(Z'_{\mathrm e}\), which must be placed respectively instead of \(Z_a\) and \(Z_{\mathrm e}\) in Fig. 6. The output voltages are

\[ U'_{\mathrm e}=(J_a+J_{\mathrm e})'\cdot Z_{\mathrm e}+J_a\cdot Z_{\mathrm{fb}} =(J_a+J_{\mathrm e})'\cdot\left(Z_{\mathrm e}+\frac{1}{n'}\cdot Z_{\mathrm{fb}}\right), \tag{16} \]

\[ U'_a=J'_a\left[Z_a+\frac{(J_a+J_{\mathrm e})'}{J'_a}\cdot Z_{\mathrm{fb}}\right] =J'_a\left(Z_a+n'Z_{\mathrm{fb}}\right), \tag{17} \]

where

\[ n'=\frac{(J_a+J_{\mathrm e})'}{J'_a}. \]

Thus, the coefficients of the currents in formulas (16) and (17) are the equivalent resistances

\[ \begin{aligned} Z'_{\mathrm e}&=Z_{\mathrm e}+\frac{1}{n'}Z_{\mathrm{fb}}\\ Z'_a&=Z_a+n'Z_{\mathrm{fb}}. \end{aligned} \tag{18} \]

At medium frequencies the resistances \(Z'_{\mathrm e}\) and \(Z'_a\) in the circuit are active in character and, in a graphical calculation, are determined from the slope of the load line.

To calculate the transformation coefficients it is necessary to know the recalculated resistances \(Z_{\mathrm e}\) and \(Z_a\), which are also taken as active.

From equations (18), taking into account that \(Z_{\mathrm{fb}}=\sqrt{Z_a\cdot Z_{\mathrm e}}\), we find the relation

\[ \frac{Z_{\mathrm{fb}}}{Z_a}=\frac{w_{\mathrm e}}{w_1} =\frac{Z'_{\mathrm e}}{Z_{\mathrm e}}\cdot n'. \tag{19} \]

It is then not difficult to determine \(Z_a\) and \(Z_{\mathrm e}\) separately. The value of the coefficient \(n'\) is found from expressions (14), (15), or (9) and (10) without magnetic coupling,

\[ n'=\frac{R_{i\mathrm{l}}+Z_a}{\dfrac{\mu_{\mathrm p}}{\mu_{\mathrm{tr}}}\cdot R_{i\mathrm{tr}}-Z_{\mathrm{fb}}}; \tag{20} \]

for \(Z_{\mathrm{fb}}=0\),

\[ n=\frac{\mu_{\mathrm{tr}}}{\mu_{\mathrm p}}\cdot \frac{R_{i\mathrm{l}}+Z_a}{R_{i\mathrm{tr}}} =\frac{S_{\mathrm{tr}}}{S_{\mathrm{pd}}}, \tag{20a} \]

where

\[ S_{\mathrm{pd}}=\frac{S_{\mathrm p}}{1+\dfrac{Z_a}{R_{i\mathrm{l}}}} \]

is the dynamic transconductance in pentode operation.

Calculation of the principal quantities \((P_{\sim},\gamma)\) by the method considered, and experimental verification for a typical single-ended ultralinear circuit using a 6P14P tube, showed discrepancies of 16%.

REFERENCES

1. V. K. Labutin. New Developments in the Technique of High-Quality Amplification. Gosenergoizdat, 1957.
2. G. S. Tsykin. Electronic Amplifiers. Svyazizdat, 1960.
3. G. V. Voishvillo. Low-Frequency Amplifiers on Electron Tubes. Svyazizdat, 1963.
4. V. K. Labutin. “Ultralinear Amplifiers.” Radio, No. 11, 1958.