Graphical Calculation of Cathode-Coupled Amplifiers

I. A. Suslov

1. Introduction

In recent years, cathode-coupled amplifiers have found wide application in radio equipment. They are used as resonant high-frequency amplifiers, direct-current and low-frequency amplifiers, broadband amplifiers, phase inverters, limiters, differential amplifiers, etc.

Figure 1 shows a single-ended cathode-coupled amplifier. This is a two-stage amplifier in which the first stage is a cathode follower and the second is an amplifier with a grounded grid. The coupling element between the stages is the common cathode resistance \(R_k\).

In the existing literature [1; 2; 3], linear methods of analysis and calculation of cathode-coupled circuits predominate, based on representing tube characteristics by straight lines. If the anode and cathode load resistances to direct current are small, i.e., the latter are formed by resonant circuits, chokes, transformers, etc., linear methods give sufficiently accurate results. In this case the operating mode of the tubes will depend little on the load resistances, and it can be chosen so that operation takes place in the linear region, where the tube parameters are close to the typical values given in handbooks. However, most cathode-coupled amplifier circuits contain active \(R_k\) and \(R_a\), and the values of the latter are chosen sufficiently large. In this case the operating mode of the tubes is established automatically, their operating points are located somewhere on the curvilinear portions of the tube characteristics, and the values of the tube parameters may differ very greatly from the typical ones.

The linear method does not make it possible to determine the actual values of the tube parameters at the operating points; therefore one has to use tabulated values. This can lead to considerable errors, especially with large \(R_k\) and in the absence of additional positive grid biases. With positive biases that do not yet produce grid currents, the operating points of tubes \(L_1\) and \(L_2\) move into the region of more linear portions of the characteristics, and the data of the linear calculation prove to be closer to the experimental results. Experience shows, however, that the accuracy of the formulas of the linear method for large \(R_k\) and \(R_a\), used in such cases, often is still inferior to the accuracy of an approximate estimate. The linear method cannot give a satisfactory answer to the question of the maximum amplitudes amplified without distortion, nor of the degree of distortion.

The most complete idea of the properties of a cathode-coupled amplifier can be obtained from its dynamic characteristic, i.e., the dependence of the output voltage or current in the load on the input voltage. It makes it possible to find the amplification factor, limiting thresholds,

distortions, etc. The calculation of the amplifier can be reduced to a justified choice of the resistances \(R_k\) and \(R_a\) and to finding this characteristic, or its separate elements, from the family of plate characteristics and the selected \(R_k\) and \(R_a\).

The only computational method that makes it possible to find the dynamic characteristic is the graphical method. It operates with the actual characteristics of tubes and potentially can have an accuracy of the same order as the accuracy of the experiment.

In application to cathode-coupled amplifiers, graphical methods have been least developed. This is explained by the difficulty of graphical analysis of the circuits under consideration. To form an idea of them, one may compare them with a cathode follower or with an amplifier with feedback through a cathode resistor, where analogous problems stand before the graphical calculation. A considerable number of works have been devoted to the graphical calculation of the indicated circuits. All of them, in one way or another, make use of the fact that through the tube, the plate and cathode resistances in these circuits carry one and the same current—the plate current of the tube. In a cathode-coupled amplifier there are two plate circuits with a common section—the cathode resistance \(R_k\). This resistance produces 100-percent negative feedback in the first stage and negative current feedback in the second, provides direct coupling of the first stage with the second, and positive feedback of the second stage with the first. Owing to such an interweaving of couplings, it is difficult to split the circuit, for purposes of analysis, into two stages, and the methods used in calculating amplifiers with feedback through a cathode resistor (including cathode followers) are inapplicable here. Evidently, the difficulties of graphical calculation of cathode-coupled amplifiers explain both the absence in the literature of works devoted to this question and the indications found in the literature of the necessity of experimentally determining tube parameters in order to use them in the formulas of the linear method [4]. The latter makes the application of calculation completely unjustified, since the experiment serving to aid the calculation proves in this case to be more complicated than the experiment instead of the calculation.

We have proposed several graphical methods for calculating cathode-coupled amplifiers. The present article is devoted to the exposition of one of them.

2. Dynamic Characteristic of a Cathode-Coupled Amplifier

The complexity of the circuit under consideration for analysis and the impossibility¹ of dividing it into two stages compel us to regard the circuit as a single whole and to use resultant characteristics. Such a characteristic can be the dynamic characteristic of the amplifier already mentioned. We shall define it as the dependence of the plate current \(i_{a2}\), or of the voltage \(u\) at the output of the circuit, on the magnitude of the voltage \(u_1\) applied to the input (see Fig. 1). Figure 2 gives examples of such characteristics for an amplifier using a 6N8 tube.

At large negative voltages \(u_1\), the first triode \(L_1\) is cut off. The current \(i_{a2}\) then has a maximum value, which is determined by the conditions of equilibrium of this current with the bias created by it on the grid of the second triode \(L_2\). It remains constant for all values of \(u_1\) that cut off the first triode. This corresponds to the left-hand section of the dynamic characteristic, parallel to the voltage axis \(u_1\). At larger values of \(u_1\), the first triode opens; its current \(i_{a1}\) creates an additional negative bias on the grid of the second triode, as a result of which

Fig. 1

Fig. 2

\[ \begin{aligned} a)\quad & i_{a2}=f(u_1)\\ b)\quad & u=F(u_1) \end{aligned} \qquad \left\{ \begin{aligned} R_k&=5\ \text{k}\Omega,\\ R_a&=25\ \text{k}\Omega,\\ u_0&=29\ \text{V} \end{aligned} \right. \]

\[ \begin{aligned} a_1)\quad & i_{a2}=f(u_1)\\ b_1)\quad & u=F(u_1) \end{aligned} \qquad \left\{ \begin{aligned} R_k&=5\ \text{k}\Omega,\\ R_a&=100\ \text{k}\Omega,\\ u_0&=12\ \text{V} \end{aligned} \right. \]

the current \(i_{a2}\) decreases. Thus one obtains the descending, middle segment of the dynamic characteristic \(i_{a2}=f(u_1)\) and the rising segment of the characteristic \(u=F(u_1)\). At a certain positive voltage \(u_1\), the current \(i_{a1}\) creates across \(R_k\) a voltage drop that completely cuts off the right-hand triode. In this case, and for still larger \(u_1\), the current of the second triode is zero, and the output voltage is equal to the constant voltage of the supply source. This corresponds to the right-hand segment of the dynamic characteristic, parallel to the \(u_1\) axis (the characteristic \(u=F(u_1)\)) or coincident with it (the characteristic \(i_{a2}=f(u_1)\)).

The slope of the characteristic \(i_{a2}=f(u_1)\) determines the transconductance of the amplifier

\[ \frac{di_{a2}}{du_1}, \]

and the slope of the characteristic \(u=F(u_1)\) determines the amplification factor.

3. Initial System of Equations

In beginning the analysis of the amplifier, it is first necessary to establish the basic relations connecting the currents and voltages in the individual elements of the circuit. For this purpose we introduce the designations indicated in Fig. 1. We readily obtain the system of equations evident from the figure:

\[ i_k=i_{a1}+i_{a2}, \tag{1} \]

\[ u_k=i_k R_k, \tag{2} \]

\[ u_1+u_{10}=u_{g1}+u_k, \tag{3} \]

\[ u_{20}=u_{g2}+u_k, \tag{4} \]

\[ u_{a1}=E-u_k, \tag{5} \]

\[ u=E-i_{a2}R_{a1}, \tag{6} \]

\[ u_{a2}=u-u_k, \tag{7} \]

\[ i_{a1}=\Theta(u_{a1},u_{g1}), \tag{8} \]

\[ i_{a2}=\Theta(u_{a2},u_{g2}). \tag{9} \]

The last two relations are the equations of the anode characteristics, which we shall regard as identical for both triodes.\(^1\) The values of the resistances \(R_k\) and \(R_a\), the biases \(u_{10}\) and \(u_{20}\), and the voltage of the anode-supply source \(E\) will be assumed known. The action of parasitic capacitances, lead inductances, and electron transit time is not taken into account.

We have in all 9 equations with 10 unknowns. Such a system makes it possible to find the relation between two of the required quantities. For example, one can obtain the dependencies \(i_{a2}=f(u_1)\) or \(u=F(u_1)\), i.e., the dynamic characteristics of the amplifier.

4. Graphical Determination of the Dynamic Characteristic

The dynamic characteristic \(i_{a2}=f(u_1)\) can be obtained from the initial system of equations (1)……(9) by successive elimination of all variables not entering into this dependence. The elimination operation can be divided into two stages. At first we shall assume the biases \(u_{10}\) and \(u_{20}\) to be identical, \(u_{10}=u_{20}=u_0\). In particular, they may be equal to zero.

\(^1\) This restriction is not obligatory.

First stage. On the family of plate characteristics of the second triode, through the point \(O'(E,0)\) we draw the straight line 1 (Fig. 3) at an angle \(\beta\), for which

Fig. 3

\[ \operatorname{ctg}\beta=-R_a. \]

It will represent equation (6) on the drawing. On each characteristic of the family we mark the point lying to the left of straight line 1 at a distance \(|u_{g2}|+u_0\) from it, measured parallel to the horizontal axis and equal to the voltage on grid \(g_2\) at which the corresponding plate characteristic was taken. The points obtained in this way determine curve I:

\[ i_{a2}=\Phi_2(u_{a2}). \]

They make it possible also to construct the curve

\[ i_{a2}=\Psi_2(-u_k)=\Psi_2(u_{g2}-u_0). \]

For this purpose, taking \(O'(E,0)\) as the origin of coordinates, we plot on the drawing points with the same ordinates as those of the obtained points of the curve \(i_{a2}=\Phi_2(u_{a2})\), and with abscissas equal to the corresponding val—

—to the voltage values \(u_{g2}=-u_0\) (or \(-u_k\)). Connecting them by a smooth line, we obtain the curve \(i_{a2}=\Psi_2(-u_k)\) (curve I′).

From the point \(O'(E,0)\) we then draw the straight line 2 at an angle \(\beta_1\), for which \(\operatorname{ctg}\beta_1=-R_k\) (the load line for a load resistance equal to \(R_k\)). It is the graphical representation of equation (2) in the coordinate system \(-u_k,\ i_k\), whose origin is placed at the point \(O'\). This straight line intersects curve I′ at the point \(M\) (Fig. 3). At the latter, the current \(i_{a2}\) alone, without the current \(i_{a1}\), satisfies equation (2), \(u_k=i_kR_k\). This point therefore determines the current \(i_{a2}\) and the voltage \(u_k\) when the first tube is cut off. At the \(u_k\) corresponding to the point \(Q\) where the current \(i_{a2}\) is cut off, the second tube is cut off, and in equation (2) the cathode current \(i_k\) is equal to the current \(i_{a1}\), whose magnitude is determined in this case by the length of the segment \(NQ\). With the second tube not cut off, the current \(i_{a1}\) as a function of the voltage \(u_{g2}=-u_k\),

\[ i_{a1}=\Phi_{10}(-u_k) \]

(curve II), is obtained by subtracting the ordinates of curve I′ from the corresponding ordinates of straight line 2. This same curve, in the coordinate system \(u_a,\ i_{a1}\), in accordance with equation (5), gives the dependence

\[ i_{a1}=\Phi_{10}(E-u_k)=\Phi_1(u_{a1})^{1}. \]

The points of intersection of curve II with the anode characteristics make it possible to construct the function \(i_{a1}=\Omega_1(u_{g1})\), i.e., from the current \(i_{a1}\) to find the voltage \(u_{g1}\) between grid and cathode of the first tube. If too few such points are obtained, their number can be increased by marking on curve II points corresponding to intermediate values of \(u_{g1}\). To do this, the horizontally placed ruler is moved into such a position that curve II divides the distance between two adjacent characteristics, measured along the edge of the ruler, in half. The corresponding point of curve II will then belong to an intermediate characteristic (the mean one relative to the indicated adjacent characteristics).

Fig. 4

Second stage. On the same drawing with the family of anode characteristics, or, better, separately on millimeter paper, we construct the dependences

\[ i_{a2}=\Omega_2(u_{g2}) \]

and

\[ i_{a1}=\Omega_1(u_{g1}) \]

(curves III and IV in Fig. 4). For this purpose we lay off along the voltage axis the values of \(u_g\) corresponding to the anode characteristics intersected by curves I and II, and along the current axis—the ordinates of the points of intersection (respectively \(i_{a2}\) and \(i_{a1}\)). Through the point \(O_k(u_0,0)^{2}\) we draw the load line corresponding to the load resistance equal to \(R_k\) (straight line 2).

Take some point of curve III, for example, \(A\) (Fig. 4). Its ordinate \(A_a\) will represent the current \(i_{a2}\), and the abscissa \(aO\) the voltage \(u_{g2}\). The segment \(AA'\), in accordance with equation (1), will represent the anode current \(i_{a1}\) of the first triode. The current in the cathode resistance will be represented by the segment \(aA'\), and the voltage drop across it, \(u_k\), which according to equation (4) can be expressed by the formula \(u_k=u_0-u_{g2}\), will be equal to the length of the segment—

\(^{1}\) The functions \(\Phi_{10}(E-u_k)\) and \(\Phi_1(u_{a1})\) coincide in the case of the single-ended circuit considered.

\(^{2}\) For large \(u_0\)—through the point

\[ P_k\left(0,\frac{u_0}{R_k}\right). \]

segment \(aO_k\). The coordinates of point \(M\), the intersection of straight line 2 with curve III, determine the maximum voltage \(u^{*}_{g2}\) between the grid and cathode and the maximum plate current \(i^{*}_{a2}\) of the second triode.

The grid voltage and plate current of the first triode \((u^{*}_{g1}\) and \(i^{*}_{a1})\) at the instant when the second tube is cut off are determined as follows. At point \(Q\) of the intercept of the current \(i_{a2}\) we erect a perpendicular to the horizontal axis, which intersects straight line 2 at point \(q\). Drawing through the latter a straight line parallel to the horizontal axis until it intersects curve IV, we find point \(N\), whose coordinates are \(u^{*}_{g1}\) and \(i^{*}_{a1}\). The same can be done with the aid of dividers. For this it is necessary to set the opening of the dividers equal to \(Qq\) and, moving the legs, find the point of curve IV with ordinate equal to the opening.

To obtain the dynamic characteristic of the amplifier, on curve III between point \(M\) and point \(Q\) of the intercept of current \(i_{a2}\) we mark several arbitrarily located points \(A, B, C, D\), etc., such that they describe the course of the curve in sufficient detail. For each of these points we determine the ordinate \(i_{a2}\) and the vertical distance to straight line 2. For example, for point \(B\) these will be the segments \(Bb\) and \(BB'\). On curve IV we find point \(B_1\) with ordinate \(B_1b_1\) equal to \(BB'\). The distance between the projections \(b\) and \(b_1\) of points \(B\) and \(B_1\) gives the corresponding value of the amplified voltage \(u_1\). Indeed, subtracting equation (4) term by term from equation (3) for \(u_{10}=u_{20}=u_0\), we are convinced that \(u_1=u_{g1}-u_{g2}=Ob_1-Ob=bb_1\). Thus the value of the argument of the sought function \(f(u_1)\) and simultaneously of the function \(i_{a1}=f_1(u_1)\) is found. The values of the functions themselves are determined by the ordinates of the points marked on curves III and IV \((B\) and \(B_1)\). In an analogous way we obtain a number of other points of the characteristics \(i_{a2}=f(u_1)\) and \(i_{a1}=f_1(u_1)\), corresponding to other selected points of curve III. The operations of the second stage of the calculation can be performed with the aid of dividers, similarly to how this was done when determining point \(N\) \((u^{*}_{g1}, i^{*}_{a1})\). The negative limiting voltage (the lower limiting threshold) is determined by the segment \(mQ_1\) (Fig. 4), and the positive voltage (the upper threshold) by the segment \(Qn\).

5. Dynamic characteristic with respect to the voltage on the grid of the second tube

Sometimes in a cathode-coupled amplifier different external voltages are applied to the grids of both triodes; for example, \(u_1\) to the first grid and \(u_2\) to the second. In this form the circuit is used, in particular, when it is required to amplify the difference of two voltages \(u_1\) and \(u_2\), independently of their common level (differential amplifiers).

We shall now consider the simpler case, when the external voltage acts only on the grid of the second tube, while the first grid is grounded; this case, together with the one considered earlier, makes it possible to carry out a graphical analysis of a circuit acted upon by two voltages \((u_1\) and \(u_2)\), and, moreover, may also have independent significance. It will differ from the case considered earlier in that equations (3) and (4) of the original system are replaced by the equations:

\[ u_{10}=u_{g1}+u_k, \tag{3'} \]

\[ u_2+u_{20}=u_{g2}+u_k. \tag{4'} \]

The dynamic characteristic \(i_{a2}=\varphi(u_2)\) is obtained from the original system by eliminating all variables except \(i_{a2}\) and \(u_2\). The graphical operations of such elimination, analogously to the preceding case, can be divided into two stages.

First stage. On the curves of the family of anode characteristics of the first tube we mark points located to the left of the straight line \(u_{a1}=E\), at distances \(u_k=u_0-u_{g1}=u_0+|u_{g1}|\). Here \(u_0\) is the bias on the grids, the same for both tubes, and \(u_{g1}\) are the grid voltages at which the corresponding characteristics were taken.

Fig. 5

corresponding characteristics. Joining these points by a smooth line, we find the curve \(i_{a1}=\Phi_{10}(E-u_k)=\Phi_1(u_{a1})\) (Fig. 5, curve 11). It represents a cathode characteristic for an input voltage equal to \(u_0\).¹

Through the point \(O'(E,0)\) we draw straight line 2 at an angle whose cotangent is equal to \(R_k\) (equation (2), in the coordinate system \(-u_k,\, i_k\), where the point \(O'(E,0)\) is the origin). Points \(N\) and \(M\) of this straight line correspond: the first to cutoff of the second tube, and the second to cutoff of the first. Between them

the points belonging to the working portion of the dynamic characteristic are located. Subtracting from the ordinates of straight line 2 the ordinates of curve II at equal values of \(u_k\) (within the indicated portion), we obtain the curve \(i_{a2}=\Psi_2(E-u_k)\) (curve I′). From the point \(O'(E,0)\) we draw straight line 1 at an angle \(\beta\) such that \(\operatorname{ctg}\beta=-R_a\). For several arbitrary values of the current \(i_{a2}\), from curve I′ we find the corresponding values of \(u_k\) and lay them off to the left from the points of straight line 1 corresponding to the same values of \(i_{a2}\). The smooth curve passing through the points obtained in this way will determine the dependence \(i_{a2}=\Phi_2(u_{a2})\) (curve I in Fig. 5).

Second stage. From the points of intersection of curves II and I with the anode characteristics we construct the curves \(i_{a1}=\Omega_1(u_{g1})\) and \(i_{a2}=\Omega_2(u_{g2})\), i.e., curves IV and III. The subsequent operations will be analogous to the preceding case, only curves III and IV change roles.

The point \(M_1\) of intersection of straight line 2 with curve IV will determine \(u_{g1}^{*}\) and \(i_{a1}^{*}\)—the grid voltage and anode current of the first triode with the second cut off. In order to find the maximum value of the grid voltage and anode current of the second triode within the working portion of the dynamic characteristic, i.e., \(u_{g2}^{*}\) and \(i_{a2}^{*}\), we mark the point \(q_1\) of straight line 2, for which the point \(Q_1\) of the intercept of the current \(i_{a1}\) is a projection on the voltage axis. The point \(N_1\) of curve III, lying at the same level as \(q_1\), has coordinates \(u_{g2}^{*}, i_{a2}^{*}\). For a more detailed determination of the dynamic characteristic, we divide, by means of a number of points \(A, B, C, D\), etc., now curve IV (previously curve III was divided), and find the points \(A_1, B_1, C_1, D_1\), etc., on curve III (Fig. 6). The abscissas of the first points will give the values \(u_{g1}\), and the abscissas of the second—the corresponding values \(u_{g2}\).

Fig. 6

The voltage \(u_2\) at \(u_{10}=u_{20}=u_0\) will be found as the difference \(u_2=u_{g2}-u_{g1}\). The ordinates of the points \(A, B, C, D \ldots\) and \(A_1, B_1, C_1, D_1 \ldots\) determine, respectively, the ordinates \(i_{a1}\) and \(i_{a2}\) of the characteristics being calculated

\[ i_{a1}=\varphi_1(u) \quad \text{and} \quad i_{a2}=\varphi(u_2). \]

The negative limiting voltage in this case will be equal to the length of the segment \(m_1Q\), and the positive one to \(Q_1n_1\) (Fig. 6).

6. Influence of constant biases on the dynamic characteristics

The circuit of Fig. 1 can amplify only when the input voltage \(u_1\) varies within the inclined portion of the dynamic characteristic. Outside it, limiting occurs: the output voltage does not change when the input changes. At \(u_{10}=u_{20}=0\), the extent of the inclined portion, the slope, and the dimensions of its linear part are small, since both grids are under the negative potential \(-u_{k0}\) relative to the cathode, shifting the operating points of the tubes into the region of the lower, curvilinear and flatter portions of the anode-grid characteristics. The application of additional positive biases from separate sources

…or by means of dividers—from the plate-supply source makes it possible to weaken the action of negative automatic grid bias, to extend the sloping portion, and to increase its steepness and linearity.

Let us consider various ways of applying additional biases.

1. The amplified voltage and the constant bias are applied to one and the same grid. It is obvious that the form of the dynamic characteristic will not change in this case; only the position of the operating point will change, being shifted by the amount of the bias. If, for example, \(u_{10} \ne 0\), then the characteristic \(i_{a2}=f(u_1)\) will be the same as for \(u_{10}=u_{20}=0\), only the initial operating point will be shifted by the amount \(u_{10}\). The bias \(u_{20}\) has a similar effect on the characteristic \(i_{a2}=\varphi(u_2)\). The calculation of the dynamic characteristics is performed in this case as in the absence of biases, except that the change in the position of the operating point is taken into account.

2. Equal biases are applied to the grids of both tubes. This case was the basic one in the exposition of the proposed method.

3. Unequal biases \(u_{10}\ne u_{20}\) are applied to the grids \(g_1\) and \(g_2\) of the amplifier. In this case the characteristic \(i_{a2}=f(u_1)\) will be the same as for the bias \(u_{20}\) applied to both grids, only the operating point will be shifted by \(u_{10}-u_{20}\). Indeed, the result will not change if we apply equal voltages \(u_{20}\) to both grids and then bring the bias on grid \(g_1\) to the value \(u_{10}\), applying to it an additional constant voltage \(u_{10}-u_{20}\). The first step corresponds to the preceding case: it changes the dimensions and the steepness of the sloping portion of the dynamic characteristic, while the second only causes the operating point to move by a distance equal to \(u_{10}-u_{20}\). Consequently, in the graphical determination of the dynamic characteristic \(i_{a2}=f(u_1)\) in this case one should proceed as in the presence of the identical bias \(u_{20}\) on both grids, but the operating point on the already calculated characteristic should be shifted by a distance equal to \(u_{10}-u_{20}\) along the voltage axis.

Analogously, the characteristic \(i_{a2}=\varphi(u_2)\) for unequal biases will be the same as for the bias \(u_{10}\) applied to both grids; only the operating point will be shifted on it along the voltage axis by a distance equal to \(u_{20}-u_{10}\).

1. The bias is applied to one grid and the amplified voltage to the other. This case, like 1 and 2, is a special case of the more general case 3. The dynamic characteristic with respect to the amplified voltage will be the same as for a bias applied to both grids, only the operating point will be shifted along the voltage axis by the amount of the bias taken with the opposite sign. For example, if \(u_{10}=0\) and \(u_{20}\ne 0\), then the characteristic \(i_{a2}=f(u_1)\) will be the same as when the bias \(u_{20}\) is applied to both grids, but the operating point will be shifted by \(-u_{20}\).

7. Determination of the Maximum Permissible Positive Biases

Let us determine the maximum permissible positive biases on the grids \(g_1\) and \(g_2\), at which grid currents do not appear within the sloping portion of the dynamic characteristic. These currents are undesirable because, in the presence of large resistances in the grid circuits—for example, in the circuits of Fig. 7 described below—they lead to distortion; moreover, the grid current of the first tube always causes a sharp decrease in the input resistance of the amplifier.

The sloping portion of the dynamic characteristic is enclosed between the limiting thresholds. The positive limiting voltage corresponds to the maximum value, for this portion, of the voltage between…

grid \(g_1\) and the cathode \((u_{g1}^{*})\), and to the negative half-cycle—the maximum current \(i_{a2}\) and the maximum voltage between grid \(g_2\) and the cathode \((u_{g2}^{*})\). It is necessary that these maximum voltages not be positive or not exceed, say, \(-1\) volt. The voltages \(u_{g1}^{*}\) and \(u_{g2}^{*}\) are usually not equal.

This means that if one of the tubes, within the limits of the sloping portion, is used completely (the voltage on its grid reaches the maximum permissible value and does not exceed it), then the other is either used incompletely or is overloaded. Since the second case is impermissible, one must be limited to full utilization of only one of the tubes.

The bias \(u_0\) at which the voltage \(u_{g2}^{*}\) is equal to zero is easily determined. To do this, from the point \(O'(E,0)\) we draw straight lines 1 and 3 at angles \(\beta\) and \(\beta_2\), for which \(\operatorname{ctg}\beta=-R_a\) and \(\operatorname{ctg}\beta_2=-(R_a+R_k)\) (Fig. 3). Mark the point \(P\) of intersection of straight line 3 with the anode characteristic \(i_{a2}=\Theta(u_{a2},0)\).

The distance from this point to straight line 1, measured in a direction parallel to the axis of abscissas, gives the positive bias \(u_0\) at which the voltage \(u_{g2}\) does not exceed zero.

In analogous fashion one can determine the bias \(u_0\) at which \(u_{g2}\) does not exceed, for example, \(-1\). For this purpose it is necessary to measure the distance from the point \(P_1\), the intersection of straight line 3 with the characteristic \(i_{a2}=\Theta(u_{a2},-1)\), to the point \(P'_1\), having the same ordinate and lying on straight line 1. The voltage \(u^{*}_{g1}\) at the moment when the second tube is cut off, for the value of \(u_0\) found in this way, can be determined from the position of the point \(N\) of straight line 2, lying on the same vertical as the point \(Q\) of cutoff of the current \(i_{a2}\). To do this, by interpolation one must find the grid voltage corresponding to the anode characteristic passing through this point. For \(R_k\) of the order of a thousand ohms or more, the grid current, as \(u_0\) is increased in a push-pull circuit (Fig. 1), first appears in the second tube¹. Therefore it is not necessary to seek \(u^{*}_{g1}\). If it is required to find the bias \(u_0\) at which \(u^{*}_{g1}=0\), then we draw straight line 2 and mark on it the point \(N_1\) of intersection with the anode characteristic \(i_{a1}=\Theta(u_{a1},0)\) (Fig. 3). Let the projection of \(N_1\) on the axis of abscissas be called \(Q_2\). To the right of \(Q_2\) we lay off the segment \(Q_2S\), equal to the voltage \(u_g\) for this point \((Q_2)\). Then \(SO'\) gives the required value of the bias \(u_0\). The maximum voltage on the grid of the second tube \((u^{*}_{g2})\) may in this case turn out to be positive or negative, depending on the magnitudes of the resistances \(R_k\) and \(R_a\).

8. Features of the calculation with reduced automatic bias

The application to the amplifier grids of additional positive biases is not the only method that makes it possible to weaken the action of the negative automatic bias \(-u_{k0}\), which reduces the slope of the dynamic characteristic and its linearity. In the case of amplification of alternating voltages, circuits are often used in which only a part of the constant component \(-u_{k0}\) of the voltage formed across the cathode resistance is applied to both grids; at the same time the alternating component \(u_{k\sim}\) of this voltage is transmitted to both grids in full. Examples of such circuits are given in Figs. 7a and b. Here the capacitances \(C_{g1}\), \(C_{g2}\), \(C_1\), and \(C_2\), and the resistances \(R_{g1}\), \(R_{g2}\), \(R_1\), and \(R_2\), are chosen sufficiently large so that the indicated resistances are much greater than the resistance \(R_k\), while the capacitors present small resistances for alternating currents.

If the point \(a\) in Fig. 7a divides the resistance \(R_k\) into \(R'_k\) and \(R''_k\), with \(R'_k=\alpha R_k\), then its potential relative to the cathode is equal to \(-\alpha(u_{k0}+u_{k\sim})\). The common lead (point 3) will have, relative to \(a\), a potential \(-(1-\alpha)(u_{k0}+u_{k\sim})\). The potential difference \(-(1-\alpha)(u_{k0}+u_{k\sim})\) is applied to the circuit containing \(C_{g1}\) and \(R_{g1}\), which for direct current has an infinitely large resistance (the resistance of the capacitor \(C_{g1}\)), while for alternating current it presents a resistance practically equal to \(R_{g1}\) (if the internal resistance of the source \(G\) of the oscillations being amplified is much smaller than \(R_{g1}\)). Therefore the constant part \(-(1-\alpha)u_{k0}\) of this potential difference will be applied to the plates of the capacitor \(C_{g1}\), and the alternating part \(-(1-\alpha)u_{k\sim}\) to the resistance \(R_{g1}\). Consequently, between the grid and cathode of the first triode there will act a feedback voltage equal to the sum of the voltages across \(R'_k\) and \(R_{g1}\), i.e. equal to \(-(\alpha u_{k0}+u_{k\sim})\). The same voltage will be applied between the grid and cathode of the second triode if the circuit \(C_{g2}R_{g2}\) is connected to the same point \(a\).

The circuit of Fig. 7b is analogous to the one considered. It is more convenient if the source of the amplified voltage has a high output resistance. In this circuit, across the resistances \(R_1\) and \(R_2\) there is formed the same potential difference, \(-(1-\alpha)u_{k\sim}\), as across the resistance \(R_k''\). Across the other two resistances in the grid circuits (\(R_{g1}\) and \(R_{g2}\)), the voltage \(u_k'\) creates no potential differences. The first of them is the load for the source of the amplified voltage.

From the described operating features of the circuits of Figs. 7a and b it is not difficult to see that the latter are similar to the previously considered circuit in which positive biases are applied to both grids. In the present case these biases must be regarded as equal to \((1-\alpha)u_{k0}\). Therefore the graphical calculation of such circuits is performed in the same way as in the case of applying constant biases from external sources, and differs only in the need to determine the connection points of the resistances \(R_{g1}\) and \(R_{g2}\), or \(R_1\) and \(R_2\), which make it possible to realize the constant biases found by calculation (for example, from the condition of absence of grid currents) or the specified constant biases. For this it is necessary to know the dc component \(i_{k0}\) of the cathode current \(i_k\), or the voltage drop \(u_{k0}\) which it produces. The value \(i_{k0}\) is found from the dynamic characteristic of the amplifier after the graphical calculation of the latter has been carried out. Knowing \(i_{k0}\) and the bias voltage \(u_0\), it is not difficult to find

\[ \alpha=\frac{R_k'}{R_k} \]

from the relation \(u_0=(1-\alpha)R_k i_{k0}\), and thereby to determine the position of point \(a\).

9. Push-pull circuit

Push-pull amplifiers with cathode coupling (Fig. 8) have found wide application as sweep amplifiers in oscillographs, as phase inverters, differential amplifiers, etc. They differ from single-ended amplifiers in that in push-pull amplifiers the first stage, just like the second, contains an anode load. This makes it possible to obtain a balanced output if the external load is connected between the anodes.

The method set forth above for the graphical calculation of amplifiers with cathode coupling is also applicable to push-pull circuits. Let us dwell on some features. The basic system of equations differs in the present case only in that equation (5) is replaced by the equation:

\[ u_{a1}=E-i_{a1}R_{a1}-u_k. \tag{5′} \]

Fig. 8

Since the output voltage is taken from the interval between the anodes, the dynamic characteristic \(u_{a1}-u_{a2}=\) has the greatest value.

\(=g(u_1)\). It is easily obtained, knowing the dependences \(i_{a2}=f(u_1)\) and \(i_{a1}=f_1(u_1)\), from the relation \(g(u_1)=u_{a1}-u_{a2}=i_{a2}R_{a2}-i_{a1}R_{a1}\). The curve \(g(u_1)\) in the region of negative values of \(u_1\) lies above the \(u_1\) axis. Here it has a horizontal section corresponding to the voltages that lock the first tube. For positive values of \(u_1\) the characteristic \(g(u_1)\) passes below the voltage axis and has no horizontal section, since the current \(i_{a1}\) invariably increases with increasing voltage \(u_1\) and limiting does not occur.

In the graphical calculation of the dynamic characteristics \(i_{a1}=f_1(u_1)\) and \(i_{a2}=f(u_1)\), the curves I, I′, and II are constructed in exactly the same way as in the case of a single-ended circuit. The difference will be that curve II, \(i_{a1}=\Phi_{10}(E-u_k)\), will no longer represent the dependence of the current \(i_{a1}\) on \(u_{a1}\), since \(u_{a1}=E-i_{a1}R_{a1}-u_k\), and not \(E-u_k\), as was the case in the single-ended circuit. To find this dependence, on curve II we choose several arbitrary points so that they reproduce the course of the curve sufficiently well. In the coordinate axes \(u_a, i_a\) we mark points with the same ordinates, lying to the left of the straight line

\[ i_a=\frac{E-u_a}{R_{a1}} \]

(straight line 1′, Fig. 9) at distances \(u_k\), measured parallel to the horizontal axis. Here \(u_k\) represents the abscissas of the selected points of curve II. Drawing a smooth line through the points thus found, we obtain curve II′, \(i_{a1}=\Phi_1(u_{a1})\). The method of constructing curve II′ shows that it represents curve II, redrawn from the rectangular coordinates \(u_a, i_a\) into oblique coordinates, for which the \(u_a\) axis is the axis of abscissas, and the straight line 1′ is the axis of ordinates. It resembles the construction of curve I from I′ in calculating the characteristic \(i_{a2}=\varphi(u_2)\) (Fig. 5).

Fig. 9

The second stage of the calculation differs only in that curve IV is constructed from the points of intersection of curve II′, and not curve II, with the anode characteristics. In conclusion, from the obtained points of the dependences \(i_{a1}=f_1(u_1)\) and \(i_{a2}=f(u_1)\), the dynamic characteristic \(g(u_1)=i_{a2}R_{a2}-i_{a1}R_{a1}\) is constructed.

The calculation of the dynamic characteristic with respect to the grid voltage of the second tube in the case of a push-pull circuit, owing to its symmetry, does not differ in any way from that described. Only the tubes \(L_1\) and \(L_2\) exchange roles.

The determination of the maximum values \(u_{g1}^{*}\) and \(u_{g2}^{*}\), assumed by the grid voltages \(u_{g1}\) and \(u_{g2}\) within the working portion of the dynamic characteristic, and the associated determination of the largest biases at which grid currents are absent, has the following difference from the case of a single-ended circuit. The voltage \(u_{g1}^{*}\) is determined not by the position of point \(N\) of straight line 2, but by point \(N′\) of straight line 3′ (the load line for a resistance equal to \(R_{a1}+R_k\)), having the same ordinate. To find this point, it is necessary from \(Q\) (Fig. 9) to draw a straight line parallel to I′ until it intersects straight line 3′. Thus, the position of point \(N′\) in the oblique coordinates \(u_a, 1′\) is determined in the same way as the position of point \(N\) in the rectangular coordinates \(u_a, i_a\). \(u_{g2}^{*}\) is found as the grid voltage-

the voltage corresponding to the anode characteristic passing through the point \(P\) of intersection of curve I with straight line 3.

The resistance \(R_{a1}\) is more often taken to be of the same order as, or equal to, \(R_{a2}\). In this case \(u_g^{*}{}_1\) will be greater than \(u_g^{*}{}_2\), and grid current within the working portion of the dynamic characteristic appears, as the positive bias \(u_{10}=u_{20}=u_0\) is increased, first in the first tube. Indeed, the second tube is locked only by means of the bias \(-i_a^{*}{}_1 R_k\), whereas the first is locked as a result of the combined action of the bias \(-i_a^{*}{}_2 R_k\) and the negative external voltage \(u_1\)

\[ u_{\text{cut}2}=-i_a^{*}{}_1 R_k,\qquad u_{\text{cut}1}=-i_a^{*}{}_2 R_k-u_1 . \tag{10} \]

Here \(i_a^{*}{}_1\) and \(i_a^{*}{}_2\) are the currents corresponding to the voltages \(u_g^{*}{}_1\) and \(u_g^{*}{}_2\), while \(u_{\text{cut}1}\) and \(u_{\text{cut}2}\) are the cutoff voltages of the anode current of the first and second tubes. From the graphs of Fig. 9 it is not difficult to see that \(u_{\text{cut}1}\approx u_{\text{cut}2}\). From equations (10), on the basis of this approximate equality, we obtain:

\[ i_a^{*}{}_1 R_k \approx i_a^{*}{}_2 R_k+u_1 . \]

Discarding the positive quantity \(u_1\), we arrive at the inequality

\[ i_a^{*}{}_1>i_a^{*}{}_2 . \]

The reasoning given is valid for both the single-ended and the push-pull circuit. Therefore the current \(i_a^{*}{}_1\) for both circuits will be greater than the current \(i_a^{*}{}_2\). The points \(N'(u_{g1}=u_g^{*}{}_1,\ i_{a1}=i_a^{*}{}_1)\) and \(P(u_{g2}=u_g^{*}{}_2,\ i_a^{*}{}_2)\) lie on one straight line (when \(R_{a1}=R_{a2}\)) or on close ones (when \(R_{a1}<R_{a2}\),¹) and since for the corresponding currents we have \(i_a^{*}{}_1>i_a^{*}{}_2\), it follows that \(u_g^{*}{}_1\) must be greater than \(u_g^{*}{}_2\). In the single-ended circuit \(i_a^{*}{}_1\) is determined by the point \(N\), which lies on straight line 2, having a considerably greater slope to the abscissa axis than straight line \(3'\), on which the point \(N'\) lies. In this case usually \(u_g^{*}{}_1<u_g^{*}{}_2\), i.e., as the bias \(u_0\) is increased the second tube is overloaded first. Since for push-pull circuits \(u_g^{*}{}_1>u_g^{*}{}_2\), it is sufficient to ensure the absence of grid current for the first tube.

To find the maximum bias \(u_0\) at which grid currents are absent, the point \(N'\) must be placed on the anode characteristic \(u_g=\text{const}=0\). For this purpose we draw the straight lines \(1'\) and \(3'\) (Fig. 9). From the point of intersection of the straight line \(3'\) with the anode characteristic \(u_g=\text{const}=0\), we draw a straight line parallel to \(1'\). Let it intersect the abscissa axis at the point \(Q\). By the method of interpolation we determine the voltage \(u_g\) corresponding to this point, and lay off to the right from it a segment \(Qm\) equal to the found \(u_g=u_{g\,\text{cut}1}\). Then \(mO'\) gives the maximum permissible bias \(u_0\).

Push-pull amplifiers usually have larger values of the resistances \(R_{a1}\), \(R_{a2}\), and \(R_k\). This makes it possible to obtain a more linear dynamic characteristic and to achieve greater push-pull symmetry of the output voltage.

10. Simplification of the Method

To find the dynamic characteristic, we had to draw three straight lines (1, 2, 3) and construct the curves I, I′, II, III, and IV. The graphical

¹ The anode resistances \(R_{a1}\) and \(R_{a2}\) of push-pull circuits are usually close in magnitude.

the operations turn out, as we see, to be rather complicated. Particular inconvenience is caused by the construction of the auxiliary curves. However, the method described admits considerable simplification. It turns out that curves I and II (Fig. 3) differ little from straight lines and may be replaced by them. In this case curve I′ will also be replaced by a straight line, and only curves III and IV will remain. We shall retain, for the straight lines replacing curves I and II, the designations adopted by us for the latter.

The graphical calculation of the dynamic characteristic of a single-ended amplifier will now be carried out in the following order. On the family of anode characteristics we draw straight lines 1 and 3. If a constant bias \(u_0\) has been chosen in advance, we find the point \(P_0\) of the anode characteristic \(u_g=\mathrm{const}=0\), lying to the left of straight line 1 at the distance \(u_0\), measured in the horizontal direction. On the lower anode characteristics we mark 2–3 points, as in the previously described construction of curve I. Drawing through them a smooth line up to its intersection with the \(u_a\) axis, we find the point \(Q\). The points \(P_0\) and \(Q\) determine straight line I. On it we mark the point \(M'\) of intersection with straight line 3.

If the given quantity is the maximum voltage \(u_{g2}^{*}\) on the grid of the second triode, we immediately find the point \(M'\) as the intersection of straight line 3 with the anode characteristic \(u_g=\mathrm{const}=u_{g2}^{*}\). Straight line I is now determined by the points \(M'\) and \(Q\). The latter is found in the same way as in the preceding case. Next we draw straight line 2 and mark on it the point \(N\) on the same vertical as \(Q\) and the point \(M\)—at the same level as \(M'\). The projection of \(M\) onto the \(u_a\) axis determines the point \(m\). Connecting \(N\) and \(m\), we obtain straight line II. From the points of intersection of straight lines I and II with the anode characteristics we construct curves III and IV in the same way as was done earlier with the aid of curves I and II. The subsequent operations of finding the dynamic characteristic with the aid of curves III and IV differ in no way from those described earlier.

Let us show that the error arising from the “straightening” of curves I and II will be very small. In passing from points of curves I and II to the points of the corresponding straight lines (I and II) lying at the same levels, the voltages \(u_{g1}\) and \(u_{g2}\) receive increments \(\Delta u_{g1}\) and \(\Delta u_{g2}\), \(\mu\) times smaller than the increments of the anode voltages \(u_{a1}\) and \(u_{a2}\). Since the currents \(i_{a1}\) and \(i_{a2}\) remain unchanged in this process, the ordinate \(i_{a2}\) of the approximate dynamic characteristic remains the same as for the exact one. To find the change of the abscissa, we subtract equation (4) term by term from equation (3). This gives \(u_1=u_{g1}-u_{g2}\), and the change of \(u_1\) may be represented in the form \(\Delta u_1=\Delta u_{g1}-\Delta u_{g2}\). The increments \(\Delta u_{g1}\) and \(\Delta u_{g2}\), with the indicated method of drawing straight lines I and II, are positive. The maximum (within the sloping portion of the dynamic characteristic) values of \(\Delta u_{a1}\) and \(\Delta u_{g1}\) correspond to comparatively large currents \(i_{a1}\); they correspond to small values of \(i_{a2}\) and, as is seen from Fig. 3, to the maximum increments \(\Delta u_{a2}\) and \(\Delta u_{g2}\). As a result, the difference of the increments \(\Delta u_{g1}\) and \(\Delta u_{g2}\), i.e. \(\Delta u_1\), and the maximum displacement along the \(u_1\) axis of the approximate curve relative to the exact one may be several times smaller than \(\Delta u_{g\max}=\dfrac{\Delta u_{a\max}}{\mu}\). Since the dynamic characteristic runs obliquely to the \(u_1\) axis, the shortest distance between the two curves at the place where they differ most will be still smaller.

In one of the calculation examples carried out for comparison by the “exact” and by the “approximate” methods (6N8 tube, \(R_k=5\ \text{k}\Omega\), \(R_a=25\ \text{k}\Omega\)), \(\Delta u_{a\max}\) was equal to \(2\ \text{V}\). This corresponds to

\[ \Delta u_{g\max}=\frac{\Delta u_{a\max}}{\mu}=0.1\ \text{V} \quad \text{and} \quad \Delta u_1<0.1\ \text{V}. \]

With a scale along the \(u_1\) axis equal to \(1\ \mathrm{V}\) per \(1\ \mathrm{cm}\), it was impossible to distinguish these two curves, since the error in drawing the curves through the points proves to be greater than the error caused by straightening the auxiliary curves.

When calculating the characteristic with respect to the voltage on the second grid, straight line II is constructed similarly to I in the preceding case, with the only difference that the role of straight line 1 is played by \(u_a = E\), and that of straight line 3 by straight line 2. To construct straight line \(I'\), mark the point \(M\) of straight line 2, lying on the same vertical with the point \(Q_1\) of the current intercept \(i_{a1}\), and the point \(N\) of intersection of straight lines II and 2 (Fig. 5). Denote the projection of \(N\) on the \(u_a\) axis by \(n\). The points \(M\) and \(n\) determine straight line \(I'\), while \(n\) and the point \(M'\) of straight line 3, lying at the same level as \(M\), determine straight line I. From the points of intersection of straight lines II and I with the anode characteristics we construct curves III and IV. The subsequent calculation will be analogous to that described in paragraph 5.

In the case of a push-pull amplifier, straight line I is constructed in the same way as in the case of a single-ended one. To obtain straight line \(II'\), from the point \(Q\), where I meets the voltage axis, we draw a straight line parallel to \(1'\) until it intersects \(3'\) at the point \(N'\). Mark the point \(M'\), the intersection of I and 3, and the point \(M\) of straight line 2 lying at the same level as it. By connecting its projection \(m\) on the abscissa axis with the point \(N'\), we obtain straight line \(II'\). From the points of intersection of straight lines I and \(II'\) with the anode characteristics we construct curves III and IV. The subsequent calculation is the same as in the case of the single-ended circuit. The calculation of the characteristic with respect to the voltage on the grid of the second triode, owing to the symmetry of the circuit, will be analogous to the one just considered.

11. Experimental Verification

Verification of the proposed graphical calculation method was carried out as follows. The 6N8 and 6N9 tubes were placed on “aging” for several days in order to eliminate the influence of “drift,” which is especially noticeable in the first hours of operation of a new tube. Then families of anode characteristics were taken separately for each triode of each tube. The grid voltages were varied in steps of \(1\ \mathrm{V}\) (for the 6N9 tube, in steps of \(0.5\ \mathrm{V}\)), and the anode voltages in steps of \(10\ \mathrm{V}\). The voltmeter measuring the voltage of the anode-supply source had a scale division of \(2\ \mathrm{V}\), the milliammeter in the anode circuit \(0.1\ \mathrm{mA}\), and the voltmeter in the grid circuit \(0.1\ \mathrm{V}\). The filament voltage was \(6.3\ \mathrm{V}\). The anode characteristics were plotted on graph paper with a scale along the \(u_a\) axis equal to \(10\ \mathrm{V}/\mathrm{cm}\), and along the \(i_a\) axis equal to \(1\ \mathrm{mA}\) per \(1\ \mathrm{cm}\). The dynamic characteristics of single-ended and push-pull amplifiers were calculated for various values of the resistances entering into them and various fixed biases. Curves II and \(II'\) (or the corresponding straight lines) were constructed on the family of anode characteristics of the first triode of the tube, and curve I (or straight line I) on the family of the second; curves III and IV were drawn on a separate plot. This was done in order to eliminate a possible error that could arise if the existing (although small) difference in the characteristics of the two triodes of the tube were not taken into account. The experimental determination of the dynamic characteristics was performed with the aid of the same instruments as were used for taking the anode characteristics.

It turned out that, when the above conditions were observed (“aged” tubes, use in the calculation of the characteristics of both triodes, and use in the experimental verification of the same instruments as in taking the anode characteristics), the experimentally found dynamic characteristics coincided exactly with the calculated ones. The error of the graphical calculation proves to be of the same order as the error of reading the instruments during the experimental verification.

12. Conclusion

Experimental verification has shown that the dynamic characteristics of a cathode-coupled amplifier can be calculated with high accuracy by means of the method described. This method makes it possible to solve problems in which knowledge of the entire dynamic characteristic is required, and not merely of its individual elements (for example, the transconductance of the amplifier or the gain), and it permits an exact calculation to be carried out where analytical methods may give an error of 100–200% or more (determination of limiting thresholds, determination of the gain for large \(R_k\) and in the absence of positive bias). It supplements the existing analytical methods and substantially broadens the possibilities for calculating cathode-coupled amplifiers.

Owing to the presence of deep feedbacks, the influence of the spread in tube parameters on the dynamic characteristics of the amplifier is considerably reduced. This justifies the use of exact calculation methods. The method set forth can also be successfully applied in such demanding cases as the use of the circuits considered in computing-and-solving devices, where high accuracy is required and where specially selected tube specimens with a small spread of parameters are used.

The simplifications described in Section 10 practically do not reduce the accuracy of the method; therefore they should always be used. In this form, the proposed method proves to be only slightly more complicated than the graphical methods used in calculating cathode followers, grounded-grid amplifiers, etc., although the circuit itself is considerably more complex.

References

1. Rizkin A. A. Fundamentals of the Theory of Amplifier Circuits, Sovetskoye Radio Publishing House, Moscow, 1951.
2. Tube Amplifiers, Vol. II, Sovetskoye Radio Publishing House, Moscow, 1951.
3. Packl O. S. Sweep Generators, Gosenergoizdat, 1948.
4. Sulzer P. G. A cathode-coupled circuit possessing negative resistance. PIRE, No. 8, August, 1948.

1. For small \(R_k\), on the contrary, first in the first. ↩↩↩