GRAPHICAL CALCULATION OF THE DYNAMIC CHARACTERISTICS OF TRIODE CATHODE-COUPLED AMPLIFIERS

I. A. Suslov

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

Previously, [1] described a method for the graphical calculation of the dynamic characteristics of a cathode-coupled amplifier, giving an accuracy comparable with that obtained in the experimental determination of these characteristics. However, calculations by this method are complicated by the use of two auxiliary curves, drawn separately in addition to the constructions performed on the family of anode characteristics.

In [2] another method is given, proposed by G. N. Alyab’eva. Here the operations of the graphical calculation are limited to constructions on the anode characteristics, but they are accompanied by computations of a number of quantities—the current of the first tube $i_{a1}$, the amplified voltage $u_1$, and the output voltages $u_{\text{out}1}$ and $u_{\text{out}2}$—which increases the labor involved in the calculations.

The present article is devoted to presenting a simplified variant of the method [1], making it possible to reduce computations to a minimum and to dispense with the construction of auxiliary curves.

Calculation of the dynamic characteristic of a single-ended stage

The operation of a single-ended cathode-coupled amplifier (Fig. 1a) is described by the following system of equations, evident from the figure:

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

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

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

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

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

\[ u_{\text{out}} = E_a - i_{a1} R_a, \tag{6} \]

\[ u_{a2} = u_{\text{out}} - u_k. \tag{7} \]

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

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

Relations (8) and (9) are the equations of the anode characteristics, which we shall assume to be identical for both triodes. The resistances \(R_k\) and \(R_a\), the bias voltages \(u_{10}\) and \(u_{20}\), and the voltage of the anode-supply source \(E_a\) will be assumed known.

Fig. 1.

The dynamic characteristic \(i_{a2}=f(u_1)\) [or \(u_{\mathrm{out}}=F(u_1)\)] can be found from the system of equations (1–9) by successive elimination of all variables that do not enter this dependence. We shall carry out the elimination graphically. At first we shall assume the bias voltages \(u_{10}\) and \(u_{20}\) to be identical, \(u_{10}=u_{20}=u_0\). On the family of anode characteristics of the triode, through the point \(O'(E_a,0)\), draw the straight lines 1, 2, and 3—the load lines for load resistances \(R_a\), \(R_a+R_k\), and \(R_k\), respectively (Fig. 2). On the characteristic \(u_g=0\) mark the point \(A\), lying to the left of line 1 at a distance \(u_0\) from it, measured parallel to the horizontal axis. To the left of the point \(O'(E_a,0)\) lay off the segment \(O'Q\), equal to \(|u_{g2}|+u_0\), where \(u_{g2}\) is the voltage \(u_g\) corresponding to the static characteristic passing through the point \(Q\). Join \(A\) and \(Q\) by the straight line \(I\). It differs little [1] from the curve \(i_{a2}=\Phi_2(u_{a2})\). On the line \(I\) mark the points \(A,B,C\), etc., of intersection with the anode characteristics.

Mark the point \(M\), the intersection of \(I\) with 2, and the point \(M'\), lying at the same level as it, on line 3. Join \(M'\) with \(Q\) by the straight line \(I'\). This straight line differs little [1] from the curve \(i_{a2}=\Psi_2(-u_k)\), if the point \(O'(E_a,0)\) is taken as the origin of coordinates. On \(I'\) mark the points \(A_1,B_1,C_1\), etc., lying at the same levels as \(A,B,C\), etc.

Denote by \(N\) the point of line 3 lying on the same vertical as \(Q\), and by \(m\) the projection of \(M'\) on the horizontal axis. The straight line \(II\), joining \(N\) and \(m\), is [1] an approximate representation of the dependence \(i_{a1}=\Phi_1(u_{a1})\). On it mark the points \(A_2,B_2,C_2\), etc., having the same abscissas as \(A_1,B_1,C_1\), etc.

The ordinates \(A,B,C\), etc., and \(A_2,B_2,C_2\ldots\) give the values of the currents \(i_{a2}\) and \(i_{a1}\). The corresponding voltages \(u_1\), on the basis of equations (3) and (4), are determined by the difference

Fig. 2.

\[ u_1=u_{g1}-u_{g2}. \tag{10} \]

Here \(u_{g2}\) are the grid voltages corresponding to the points \(A,B,C\), etc., of straight line 1. They can be read directly on the characteristics passing through these points. The points \(A_2,B_2,C_2\), etc., as a rule, do not lie on the characteristics. The corresponding voltages \(u_{g1}\) are found by interpolation. The segment of a horizontal straight line drawn, for example, through point \(B_2\) and bounded by two adjacent characteristics of the family is divided by this point into parts proportional to the differences between the voltages corresponding to the adjacent characteristics and the voltage for the given point \((B_2)\).

The found values of \(u_{g2}\) and \(u_{g1}\), and the values of the corresponding currents \(i_{a2}\) and \(i_{a1}\), are conveniently recorded in tabular form. From it one can easily construct the dynamic characteristic \(i_{a2}=f(u_1)\) and the dependence \(i_{a1}=f_1(u_1)\).

To determine the voltage \(u_{\text{out}}\) at the amplifier output, we mark, at the levels \(A, B, C\), etc., points \(A', B', C'\), etc., on straight line 1, and points \(A'', B'', C''\), etc., on perpendicular 4 to the voltage axis drawn through point \(O'\). The segments \(A'A''\), \(B'B''\), \(C'C''\), etc., determine the values of the output voltage \(u_{\text{out}}\), which it is also expedient to enter in the table. Taking from the latter \(u_{\text{out}}\) and the corresponding values of \(u_1\), one can plot the characteristic \(u_{\text{out}} = F(u_1)\).

Calculation of the dynamic characteristics of a push-pull stage

The basic system of equations for currents and voltages in a push-pull stage with cathode coupling (Fig. 1b) differs from the corresponding system of equations for a single-ended stage only in that equation (5) is replaced by the equation

Fig. 3.

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

Since the output voltage in the push-pull circuit is taken from the interval between the anodes, the characteristic

\[ u_{p.\,\text{out}} = u_{\text{out}1} - u_{\text{out}2} = u_{a1} - u_{a2} = g(u_1) \]

is of greatest value.

In the graphical calculation of the dynamic characteristics, the straight lines \(1\), \(1'\), and \(II\) are constructed, exactly as in the case of a single-ended circuit. The straight line \(II\), however, will no longer represent the dependence of the current \(i_{a1}\) on \(u_{a1}\), since \(u_{a1}\) is now determined by equation \((5')\), and not by (5). To find the graphical representation of the dependence \(i_{a1}=\Phi_1(u_{a1})\), we draw from the point \(O'\) the straight line \(2'\), corresponding to the load \(R_{a1}+R_k\) (Fig. 3). On it we mark the point \(N'\), lying at the same level as the point \(N\) of the straight line \(II\). Connecting \(N'\) with \(m\), we obtain the straight line \(II'\), which is an approximate expression of the dependence \(i_{a1}=\Phi_1(u_{a1})\). On the straight line \(II'\) we mark the points \(A_3,B_3,C_3\), etc., lying at the same levels as \(A_2,B_2,C_2\), etc. of the straight line \(II\). By interpolation one can find the grid voltages \(u_{g1}\) for the characteristics passing through these points, while the ordinates of the points give the corresponding values of \(i_{a1}\).

On the load lines \(1\) and \(1'\) (for the plate resistances \(R_{a2}\) and \(R_{a1}\)) we mark the points \(A',B',C'\ldots\) and \(A'',B'',C'',\ldots\), lying at the same levels as \(A,B,C,\ldots\) and \(A_3,B_3,C_3,\ldots\). The distances from these points to the vertical straight line 4 determine the output voltages \(u_{\text{out}2}\) and \(u_{\text{out}1}\), respectively.

From the points \(A,B,C\ldots\), \(A_3,B_3,C_3\ldots\) one can compile a table of mutually related values of \(u_{g2}\), \(i_{a2}\), \(u_{g1}\), and \(i_{a1}\). It can be supplemented with the calculated values \(u_1=u_{g1}-u_2\) and with the values \(u_{\text{out}1}\) and \(u_{\text{out}2}\), found from the points \(A',B',C'\ldots\) and \(A'',B'',C''\ldots\). The data of the table make it possible to draw the curve \(u_{\text{out}1}-u_{\text{out}2}=g(u_1)\)—the dynamic characteristic.

Calculation of the Operating Mode of a Stage with Cathode Coupling

In amplifiers with cathode coupling, comparatively large \(R_k\) are usually used, making it possible to obtain sufficiently strong coupling between stages. However, with large \(R_k\), the operating points of the tubes shift into the region of the lower portions of the plate characteristics, the transconductances of the tubes and the amplification factor decrease, and the range of the dynamic characteristic is reduced. This makes it necessary to apply additional positive biases \(u_{10}\) and \(u_{20}\), which make it possible to ensure the most advantageous operating mode for the amplifier tubes. The magnitudes of these biases are limited by the appearance of grid currents when the amplified voltage varies within the working portion of the dynamic characteristic, i.e., the portion over which neither of the tubes is cut off.

Let us denote by \(u_{0\max}\) the value of \(u_0\) at which grid current appears at one of the boundaries of the working portion of the dynamic characteristic. As shown in [1], in a single-ended circuit \(u_{0\max}\) will correspond to the appearance of grid current in the second tube at the moment when the first is cut off, and in a push-pull circuit—to grid current in the first tube when the second is cut off.

The graphical method makes it easy to determine \(u_{0\max}\). In the case of a single-ended circuit, for this it is necessary to mark the point \(P\) of intersection of straight line 2 with the plate characteristic \(u_g=\text{const}=0\) (Fig. 4). The distance from this point to straight line 1, measured in the horizontal direction, gives the bias \(u_0\) at which \(u_{g2}\) does not exceed zero at the moment when the first tube is cut off. Similarly, one can find the bias \(u_0\) at which \(u_{g2}\) does not exceed, for example, \(1\ \text{V}\). For this purpose it is necessary to measure the distance from the point \(P_1\), the intersection of straight line 2 with the characteristic \(u_g=\text{const}=-1\ \text{V}\), to the point \(P'_1\), having the same ordinate and lying on straight line 1 (Fig. 4).

When determining the maximum permissible bias \(u_{0\max}\) in the case of a push-pull circuit, point \(N'\) (Fig. 3) must be placed on the plate characteristic \(u_g=\mathrm{const}=0\). For this purpose we draw straight line \(2'\), mark the point \(N'\) of intersection of line \(2'\) with the characteristic \(u_g=\mathrm{const}=0\), and the point \(N\), lying at the same level, on line 3. Projecting point \(N\) onto the voltage axis, we obtain point \(Q\). By interpolation we determine the voltage \(u_g=u_{g\mathrm{otc}}\) corresponding to this point, and lay off to the right from \(Q\) the segment \(Qm\), equal to the voltage \(u_{g\mathrm{otc}}\) found. Then \(mO'\) gives the maximum permissible bias \(u_{0\max}\), which must be applied to both grids.

Fig. 4.

On the dynamic characteristic obtained by the described method of calculation, it is necessary to choose the position of the operating point. The latter, depending on the nature of the signals being amplified (symmetrical signals, positive or negative pulses), must lie in the middle of the operating portion, or in its upper or lower region. Suppose that the required position of the operating point on the characteristic, calculated under the assumption that \(u_{10}=u_{20}=u_0\), corresponds to the abscissa \(u_1=u_{1p}\). Then, in order to ensure the required operating condition, the bias on the grid of the second tube must be taken equal to \(u_{20}=u_{0\max}\), and on the first, \(u_{0\max}+u_{1p}\).

Fig. 5.

In the case of amplification of alternating voltages, instead of applying additional biases from separate sources or from dividers, the circuits of Fig. 5 a and b are often used. 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. In this case the calculation described must be supplemented by determining the resistance values \(R_{k1}, R_{k2}, R_{k3}\) (Figs. 5a and b), which ensure the selected operating mode. After the biases \(u_{20}=u_{0\max}\) and \(u_{10}=u_{0\max}+u_{1p}\) have been determined, from the characteristics \(i_{a1}=f_1(u_1)\) and \(i_{a2}=f(u_1)\) (or from the data of the corresponding table) we find \(i_{a1p}\) and \(i_{a2p}\), corresponding to the selected operating point \(u_{1p}\), and calculate \(i_{kp}=i_{a1p}+i_{a2p}\). Let us denote \(R_{k1}+R_{k2}+R_{k3}=R_k\). Only the bias produced by the current \(i_{kp}\) across the resistance \(R_{k1}\) is applied to the grid of the second tube. The bias \(i_{kp}(R_{k2}+R_{k3})\) across the resistances \(R_{k2}\) and \(R_{k3}\) is blocked by the capacitor \(C_{g2}\). In the circuits of Fig. 1, for the same loads \(R_k\) and \(R_a\) and the same mode, the action of the corresponding part of the cathode bias is compensated by the positive bias \(u_{0\max}\). We may therefore write

\[ u_{0\max}=i_{kp}(R_{k2}+R_{k3})=i_{kp}(R_k-R_{k1}). \tag{11} \]

Similarly, for the first tube we have

\[ u_{0\max}+u_{1p}=i_{kp}R_{k3} \tag{12} \]

or

\[ u_{1p}=-i_{kp}R_{k2}.^{1)} \tag{12'} \]

From equations (11) and (12′) one can find \(R_{k1}\) and \(R_{k2}\), and then

\[ R_{k3}=R_k-R_{k1}-R_{k2}. \]

Conclusion

The proposed method makes it possible to dispense with the auxiliary curves III and IV used in [1]. Compared with the method of G. N. Alyab’eva [2], it has the advantage that the computational part occupies much less space and is limited only to determining the input voltage by formula (10) and the output voltage (in the case of a push-pull circuit) by the formula

\[ u_{\mathrm{out}}=u_{\mathrm{out}1}-u_{\mathrm{out}2}. \]

The method can be used in calculating amplifiers and limiters with cathode coupling.

REFERENCES

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

2. A. A. Bulgakov, Electronic Devices of Automatic Control, 2nd ed., Gosenergoizdat, 1958.

\(^{1)}\) It is assumed that \(u_{1p}<0\). For \(u_{1p}>0\), the positions of the taps from the resistance \(R_k\) to the grids of the tubes change (Figs. 5a and b), and equations (11) and (12′) are replaced by the equations

\[ u_{0\max}=i_{kp}R_{k3}, \tag{11'} \]

\[ u_{1p}=i_{kp}R_{k2}. \tag{12''} \]