Response to a Letter by V. D. Ivanchenko

In article [1] it was noted that in work [5] the “internal” nonlinear feedback, caused by the presence of the nonlinear resistance of the transistor collector junction \(r_k\), was not taken into account, and the components of the distortion introduced by the input (passive) and active transistor circuits were not clearly separated, since the parameter used in [5],

\(g_{21}=\beta/r_{11}\), is a “through” parameter. Our comments were based on the following.

In [5] the initial formula for the transfer coefficient of an amplifier with emitter feedback has the following form:

\[ K=\frac{r_{11}}{R_c+r_{11}}-\frac{R}{R_{\mathrm{H}}+r_{22}}\cdot g_{21}\cdot r_{22}. \tag{1} \]

According to the method for determining the dependence \(r_{22}(U_b)\), given in [5] on p. 35, \(r_{22}\) is understood to mean the output resistance of a transistor with emitter feedback, i.e.,

\[ r_{22}=\frac{r_k}{1+\beta}. \]

But for this value of \(r_{22}\) expression (1) is incorrect. As has been shown,* formula (1) is valid if \(r_{22}\) is understood as the following:

\[ r'_{22}=\frac{r_k}{1+\beta}\cdot \frac{r_{\mathrm{e}}(1+\beta)+r_b+R}{r_{\mathrm{e}}+r_b+r_c}. \]

It is clear from this that \(r'_{22}\to r_{22}\) only when \(R_c \gg r_{\mathrm{e}}(1+\beta)+r_b\), whereas the author of [5] extends the results obtained to arbitrary values of \(R_c\). The use of (1) as the initial relation led to the fact that in [5] the “internal” feedback through the nonlinear resistance \(r_k\) is not taken into account at all, which substantially distorts the real picture of the occurrence of nonlinear distortions.

The choice of a system of parameters and of the form in which the result is presented is a matter for each author, provided that well-founded conclusions are drawn.

* M. S. Roitman, V. M. Sergeev. Some questions in the analysis and rational design of transistor amplifiers with voltage emitter feedback. Avtometriya, 1971, No. 6.

What conclusions, then, did the author of the letter arrive at in [5]? He obtained a relation for the nonlinearity parameter with respect to the second harmonic in the form

\[ H_2 \simeq \frac{r'_{11}}{r_{11}}(1-K_1)-\frac{g'_{21}}{g_{21}} . \tag{2} \]

The author of [5] asserts without any reservations or proofs that the signs of \(r'_{11}\) and \(g'_{21}\) are opposite, which indicates the possibility of complete or partial compensation of second-harmonic distortions. It is easy to show that the opposition of the signs of \(r'_{11}\) and \(g'_{21}\) occurs only under quite definite conditions.

The author of the letter knows the relation \(g_{21}=\beta/r_{11}\), from which it follows that \(g'_{21}/g_{21}=\beta'/\beta-r'_{11}/r_{11}\), or, taking into account that \(r'_{11}<0\), \(g'_{21}/g_{21}=\pm \beta'/\beta+r'_{11}/r_{11}\). It is obvious that the sign of \(g'_{21}/g_{21}\) depends on the sign of \(\beta'\), which may be either positive or negative. Thus compensation is not always possible, something the author of [5] does not specify. While nevertheless noting the possibility of compensation, the author of [5] does not recommend that the condition of complete compensation be carried out in practice, “... since it is fulfilled in the region of small collector currents, where the gain factor does not exceed a few units.”

Let us show the absolute incorrectness of this conclusion, which, incidentally, was made without any proof. Substituting the value of \(g'_{21}/g_{21}\) into (2), taking into account \(r'_{11}<0\), we obtain

\[ H_2 \simeq \left|\frac{r'_{11}}{r_{11}}\cdot K_1\right|+\frac{\beta'}{\beta}. \tag{3} \]

It is clearly seen that compensation is possible only for \(\beta'<0\), which occurs in the region of large currents, where a decrease in \(\beta\) is observed as the collector current increases. The indicated compensation effect in the region of large currents is an effective means of linearizing powerful transistor amplifiers, and it is very unfortunate that the erroneous conclusions of the author of [5] may mislead designers. At the same time, in modern high-frequency transistors there is a weak dependence \(\beta(i_k)\), i.e., \(\beta'\simeq 0\), while \(\beta'<0\) in the region of currents exceeding the nominal ones. In this case compensation, according to [5], is altogether impossible. At the same time, the presence of internal nonlinear feedback through \(r_k\) leads, as shown in [1], to the possibility of compensation also for \(\beta'\geq 0\).

In our paper [1] the final result has the form

\[ K_2=\frac{U_1}{F}\left[(1-\gamma_1)\left(\frac{1}{2}\frac{\beta'}{\beta} -\left|\frac{r'_k}{r_k}\right|\frac{\beta R_{\mathrm{n}}}{r_k} +\frac{1}{2}\gamma_1\left|\frac{r'_{\mathrm{e}}}{r_{\mathrm{e}}}\right|\right)\right]. \tag{4} \]

Here the distortion components introduced by the input \(r'_{\mathrm{e}}/r_{\mathrm{e}}\) and output circuit \((\beta', r'_k)\) are clearly separated, and the possibility of compensation both for \(\beta'<0\) and for \(\beta'\geq 0\) is also readily evident.

With regard to the notation \(U_2=\int \gamma_2(U'_2)\,dU'_2\) and to the conclusions accompanying this notation, made in [1] with reference to [5], in fact there is no such relation in [5], and it was cited by us incorrectly, which was facilitated by the ambiguity of the notation adopted in [5]; namely, the following notation is present:

\[ K_1=\frac{r_{11}}{r_{11}+R_{\mathrm{c}}} \quad \text{and} \quad K(t)=\frac{r_{11}(t)}{r_{11}(t+R_{\mathrm{c}})} , \]

at the same time

\[ K_2=\frac{R_{\mathrm{n}}}{r_{22}+R_{\mathrm{n}}},\quad \text{whereas}\quad K_2(t)=\frac{R_{\mathrm{n}}}{r_{22}(t)+R_{\mathrm{n}}}\cdot g_{21}(t)\cdot r_{22}(t)? \]

Nevertheless, this does not relieve us of the need to offer our apologies in this connection.

The author of the letter calls into question the principal relation obtained by us in [1], since he was unable to reduce it to the form

\[ K_{2\mathrm{g}}=\frac{1}{4}\,\frac{g_{21}^{1}}{g_{21}}\,U_1 \quad (\text{for } R_{\mathrm{n}}=0,\ R_{\mathrm{c}}=0). \]

From the logic of the analysis in [1] it follows that the coefficient \(K_2\), represented by expression (4), is the “partial” coefficient of the second harmonic and expresses the error of the transfer coefficient due to nonlinearity. Thus, the instantaneous transfer coefficient of the amplifier has the form (without taking into account the partial coefficients of the third and higher harmonics)

\[ K(U_1)=K_0[1+K_2(U_1)+\ldots]. \]

Hence \(U_2=U_1\cdot K(U_1)=K_0U_1+K_2(U_1)U_1\). Substituting here the value of \(K_2(U_1)\) from (4) and \(U_1=U_{1m}\cos\omega t\), it is easy to obtain that the true coefficient of the second harmonic will be equal to

\[ K_{2\mathrm{g}}=\frac{1}{2}K_2. \tag{5} \]

For \(R_{\mathrm{c}}=R_{\mathrm{n}}=0\), taking (5) into account, we obtain, according to the notation adopted in [1]:

\[ K_{2\mathrm{g}}=\frac{1}{4}\left|\frac{r_{\mathrm{b}}}{r_{\mathrm{b}}+r_{\mathrm{e}}\beta}\cdot\frac{\beta'}{\beta}+ \frac{r_{\mathrm{e}}\beta}{r_{\mathrm{b}}+r_{\mathrm{e}}\beta}\left|\frac{r_{\mathrm{e}}'}{r_{\mathrm{e}}}\right|\right|U'_{1m}. \tag{6} \]

Since \(g_{21}\simeq \beta/[r_{\mathrm{e}}\beta+r_{\mathrm{b}}]\), and in [1] \(r_{\mathrm{b}}\) was considered linear, then, taking \(r_{\mathrm{e}}<0\) into account,

\[ \frac{g_{21}^{1}}{g_{21}}= \frac{r_{\mathrm{b}}}{r_{\mathrm{e}}\beta+r_{\mathrm{b}}}\cdot\frac{\beta'}{\beta}+ \frac{r_{\mathrm{e}}\beta}{r_{\mathrm{e}}\beta+r_{\mathrm{b}}}\left|\frac{r_{\mathrm{e}}'}{r_{\mathrm{e}}}\right|, \tag{7} \]

substituting (7) into (6), we obtain

\[ K_{2\mathrm{g}}=\frac{1}{4}U_{1m}\frac{g_{21}^{1}}{g_{21}}, \]

which is what was required to be proved. Since the qualitative picture does not depend on the numerical coefficient, and the transformations presented are obvious, we did not dwell specifically on these points in [1].

Thus, all the claims of the author of the letter, with the exception of item 3, are, in our opinion, unfounded, and it remains only for us to express regret that Comrade V. D. Ivanchenko did not wish to confine himself to private correspondence and demanded that his letter be placed on the journal’s scarce pages.

Respectfully, V. M. Sergeev