Nonlinear Distortions of a Common-Emitter Transistor Amplifier

M. V. Sergeev

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

It is known that the mechanism by which nonlinear distortions arise in transistor amplifiers is considerably more complex than in tube amplifiers. A characteristic feature of a transistor is the substantial dependence of nonlinear distortions on the signal-source resistance, caused by the nonlinearity of the input circuit.

A considerable number of works have been devoted to the analysis of nonlinear distortions taking these circumstances into account, in particular [1—4]. But in the cited works and in other works known to us, the “internal” feedback through the nonlinear resistance of the collector junction is not taken into account; as will be shown, this feedback can substantially change the generally accepted picture of nonlinear distortions.

The most rational method, in our opinion, for determining the harmonic composition of the output voltage with the use of the distributed parameters of the transistor equivalent circuit is given in [5]. But in that work as well the “internal” nonlinear feedback is not taken into account; furthermore, the distortions caused by the input (passive) and active circuit of the transistor are not clearly separated, since the parameter introduced there, \(g_{21}\) (the transconductance of the transistor), is itself a function of the input resistance. This follows from the known relation [6]

\[ g_{21}=\frac{\beta}{r_{11}}, \tag{1} \]

where \(\beta\) is the current transfer coefficient in the common-emitter circuit;
\(r_{11}\) is the input resistance of a common-emitter transistor.

The aim of the present work is to obtain calculation relations for determining the nonlinear distortions of a common-emitter transistor amplifier, taking into account the remarks set forth above.

Analysis

Figure 1a shows the T-shaped equivalent circuit of a transistor [6]. The expression obtained on its basis by the generalized nodal-voltage method [8] for the voltage gain under the obvious assumptions

\[ r_{K}\gg r_{c}, \qquad \beta \gg 1 \]

can be brought to the form

\[ K=\frac{-S R_{\text{н}}\gamma_1\gamma_2}{1+S R_{\text{н}}\gamma_1\gamma_2\gamma_3} =-\frac{K_0}{1+K_0\gamma_3}, \tag{2} \]

where

\[ \gamma_1=\frac{r}{R+r},\qquad \gamma_2=\frac{r_{\text{к}}}{R_{\text{н}}+r_{\text{к}}},\qquad \gamma_3=\frac{R}{r_{\text{к}}},\qquad S=\frac{1}{r_{\text{э}}}, \]

\[ r=r_{\text{э}}\beta,\qquad R=R_i+r_{\text{б}}. \]

Fig. 1a

Fig. 1b

\(r_{\text{э}}, r_{\text{б}}, r_{\text{к}}, \beta\) are the parameters of the T-shaped equivalent circuit;
\(R_i\) is the resistance of the signal source;
\(R_{\text{н}}\) is the load resistance.

On the basis of expression (2), the structural diagram (Fig. 1b) used in the subsequent analysis has been constructed.

In feedback terms, the numerator and denominator of (2) express, respectively, the value of the forward gain and the depth of feedback, which in this case is due to the finite value of \(r_{\text{к}}\). Let us determine the distortions introduced by the forward-gain channel. It follows from the structural diagram that

\[ K_0(U_1)=S(U_0)\gamma_1(U_0)\gamma_2(U_0). \]

According to the method proposed in [5],

\[ U_0=\int \gamma_1(U_0)\,dU_0,\qquad \gamma_1(U_0)=\frac{r(U_0)}{R+r(U_0)}. \tag{3} \]

In the further analysis we regard \(r_{\text{б}}\) as a linear quantity, which, owing to its small degree of nonlinearity, introduces appreciable errors only when \(R_i<r_{\text{б}}\). Representing \(r(U_0)\) by a Taylor series,

\[ r(U_0)=r+r'U_0+\frac{1}{2}r''U_0^2+\cdots, \tag{4} \]

substituting (4) into (3), taking into account \(U_0 \simeq \gamma_1 U_1\), after elementary transformations, given in [5], we obtain

\[ \gamma_1(U_0)=\gamma_1+A_1U_1+\frac{1}{2}B_1U_1^2; \]

\[ U_0=\gamma_1U_1+\frac{1}{2}A_1U_1^2+\frac{1}{6}B_1U_1^3, \]

where

\[ A_1=\frac{r'}{r}\gamma_1^2(1-\gamma_1),\qquad B_1=\frac{r''}{r}\gamma_1^3(1-\gamma_1). \tag{5} \]

The output voltage without taking into account the shunting of the load by the nonlinear resistance \(r_{\mathrm{k}}\) will be equal to

\[ U_2'=\frac{U_2}{\gamma_2}=-R_{\mathrm{H}}\int S(U_0)\,dU_0 . \]

Representing \(S(U_0)\) by an expansion of the form (4), and, after transformations, retaining only the essential combination products and components up to the third harmonic, we obtain the expression

\[ U_2'=-R_{\mathrm{H}}\left[S\gamma_1U_1+\frac{1}{2}\left(A_1S+S'\gamma_1^2\right)U_1^2+\right. \]

\[ \left.+\frac{1}{6}\left(B_1S+3S'\gamma_1A_1+S''\gamma_1^3\right)U_1^3\right] \]

or, taking into account the notation (5) and the obvious relations

\[ \frac{r'}{r}=\frac{r'_{\mathrm{e}}}{r_{\mathrm{e}}}+\frac{\beta'}{\beta},\qquad \frac{r''}{r}=\frac{r''_{\mathrm{e}}}{r_{\mathrm{e}}}+\frac{\beta''}{\beta}+2\frac{\beta'r'_{\mathrm{e}}}{\beta r_{\mathrm{e}}},\qquad r'_{\mathrm{e}}<0,\quad r''_{\mathrm{e}}<0, \]

\[ S'=-S\frac{r'_{\mathrm{e}}}{r_{\mathrm{e}}},\qquad S''=-S\left[\frac{r''_{\mathrm{e}}}{r_{\mathrm{e}}}-2\left(\frac{r'_{\mathrm{e}}}{r_{\mathrm{e}}}\right)^2\right] \]

we finally write

\[ U_2'=-R_{\mathrm{H}}\left(S\gamma_1U_1+\frac{1}{2}S_i'\gamma_1^2U_1^2+\frac{1}{6}S_i''\gamma_1^3U_1^3\right), \]

where \(S_i'\), \(S_i''\) are generalized nonlinearity parameters of the transistor, equal to

\[ S_i'=S\left[\frac{\beta'}{\beta}(1-\gamma_1)+\gamma_1\frac{r'_{\mathrm{e}}}{r_{\mathrm{e}}}\right]; \tag{6} \]

\[ S_i''=S\left[\frac{\beta''}{\beta}(1-\gamma_1)+\gamma_1\frac{r''_{\mathrm{e}}}{r_{\mathrm{e}}}+(1-\gamma_1)\frac{r'_{\mathrm{e}}}{r_{\mathrm{e}}}\left(\frac{\beta'}{\beta}-3\frac{r'_{\mathrm{e}}}{r_{\mathrm{e}}}\right)\right]. \]

In [5] it is erroneously assumed that the transmission coefficient of the output circuit (in our notation \(\gamma_2\)) is a function of \(U_2'\), and the output voltage is found from the relation

\[ U_2=\int \gamma_2(U_2')\,dU_2' . \]

In fact, however, this notation is not valid, since \(r_{\mathrm{k}}\), and consequently also \(\gamma_2\), are functions of \(U_0\). Therefore in [5] the influence of the nonlinearity of \(r_{\mathrm{k}}\) on the nonlinear distortions of the direct-amplification channel is unjustifiably overestimated.

Since \(\gamma_2\) does not depend on the quantity \(U_2'\), one can write for the output voltage:

\[ U_2=U_2'\gamma_2(U_0). \]

By analogy with \(\gamma_1(U_0)\) and taking into account \(r'_{\mathrm{k}}<0\), \(r''_{\mathrm{k}}<0\)

\[ \gamma_2(U_0)=\gamma_2-A_2U_1-\frac{1}{2}B_2U_1^2, \]

where

\[ A_2=\frac{r'_{\mathrm{k}}}{r_{\mathrm{k}}}\gamma_2^2(1-\gamma_2),\qquad B_2=\frac{r''_{\mathrm{k}}}{r_{\mathrm{k}}}\gamma_2^3(1-\gamma_2), \]

then

\[ \begin{aligned} U_2 &= S R_{\mathrm{H}}\gamma_1\gamma_2 U_1 + \frac{1}{2} R_{\mathrm{H}}\left(S_i'\gamma_1^2\gamma_2 - 2A_2S\gamma_1\right)U_1^2 +{}\\ &\quad + \frac{1}{6} R_{\mathrm{H}}\left(S_i''\gamma_1^3\gamma_2 - 3B_2S\gamma_1 - 3S_i'A_2\gamma_1^2\right)U_1^3 . \end{aligned} \tag{7} \]

It follows from expression (7) that shunting the load by the nonlinear collector resistance leads to a certain compensation of distortion in the forward-gain channel, as is also noted in [4]. But this compensation is insignificant, since in practice the inequality \(r_{\mathrm{K}} \gg R_{\mathrm{H}}\) is almost always satisfied, and therefore

\[ \gamma_2 \simeq 1,\quad A_2 = 0,\quad B_2 \simeq 0 . \]

Taking the foregoing remark into account, one may write

\[ \begin{aligned} K_0(U) &= \frac{U_2}{U_1} = S\gamma_1 R_{\mathrm{H}} + \frac{1}{2}S_i'\gamma_1^2 R_{\mathrm{H}}U_1 + \frac{1}{6}S_i''\gamma_1^3 R_{\mathrm{H}}U_1^2 ={}\\ &= S\gamma_1 R_{\mathrm{H}}(1+\delta_2+\delta_3). \end{aligned} \tag{8} \]

Here

\[ \delta_2=\frac{1}{2}\frac{S_i'}{S}\gamma_1U_1,\qquad \delta_3=\frac{1}{6}\frac{S_i''}{S}\gamma_1^2U_1^2 \]

are the partial coefficients of the second and third harmonics, respectively. The dependence of the “internal” feedback coefficient \(\gamma_3\) on the voltage \(U_0\) can be represented in the form

\[ \gamma_3=\frac{R}{r_{\mathrm{K}}(U_0)} =\frac{R}{r_{\mathrm{K}}\left(1-\frac{r_{\mathrm{K}}'}{r_{\mathrm{K}}}\gamma_1U_1-\frac{1}{2}\frac{r_{\mathrm{K}}''}{r_{\mathrm{K}}}\gamma_1^2U_1^2\right)} . \]

Denoting

\[ \delta_2'=\frac{r_{\mathrm{K}}'}{r_{\mathrm{K}}}\gamma_1U_1,\qquad \delta_3'=\frac{1}{2}\frac{r_{\mathrm{K}}''}{r_{\mathrm{K}}}\gamma_1^2U_1^2 \tag{9} \]

and assuming

\[ \delta_2' + \delta_3' \ll 1, \]

on the basis of the approximate relation

\[ \frac{1}{1+x} \simeq 1-x \quad (\text{for } x \ll 1) \]

one may write

\[ \gamma_3(U_0)=\gamma_3(1+\delta_2'+\delta_3'). \]

Then the gain coefficient with feedback will be equal to

\[ K(U)=\frac{K_0(1+\delta_2+\delta_3)} {1+K_0\gamma_3(1+\delta_2+\delta_3+\delta_2'+\delta_3')} . \]

Assuming \(\delta_2+\delta_3+\delta_2'+\delta_3' \ll 1\), after transformations analogous to the preceding ones, and taking into account the notations \(F=1+T\), \(T=K_0\gamma_3\), we finally write

\[ K(U)=\frac{K_0}{F}\left(1+\frac{\delta_2}{F}-\frac{\delta_2'T}{F} +\frac{\delta_3}{F}-\frac{\delta_3'T}{F}\right), \tag{10} \]

whence it follows that “internal” feedback through the nonlinear resistance \(r_{\mathrm{K}}\) leads to compensation of distortion both in the second and in the third harmonics. The indicated compensation is substantially greater than the compensation due to shunting the load by the resistance \(r_{\mathrm{K}}\).

Let us consider in greater detail the coefficient of the second harmonic, which predominates significantly in the total distortions of the transistor. Taking relation (1) into account, one can write

\[ T = S R_{\mathrm{H}}\gamma_1 r_3 = \frac{S R_{\mathrm{H}} r}{R+r}\cdot \frac{R}{r_{\mathrm{k}}} = \frac{\beta R_{\mathrm{H}}}{r_{\mathrm{k}}}(1-\gamma_1). \]

Then, taking into account the notation (6), (8), and (9), the coefficient of the second harmonic will be equal to

\[ K_2= \frac{U_1\gamma_1}{F} \left[ (1-\gamma_1) \left( \frac{1}{2}\frac{\beta'}{\beta} - \frac{r_{\mathrm{k}}'}{r_{\mathrm{k}}}\frac{\beta R_{\mathrm{H}}}{r_{\mathrm{k}}} \right) + \frac{1}{2}\gamma_1\frac{r_\mathrm{e}'}{r_\mathrm{e}} \right]. \tag{11} \]

Since the comparison of nonlinear distortions for different values of \(R\) is carried out at a constant output voltage, then for each new value \(R=R_j\) and \(\gamma_1=\gamma_{1j}\), in order to keep the output voltage constant it is necessary to increase the value of the input voltage to

\[ U_{1j}=\frac{U_1}{\gamma_{1j}}. \]

It follows from this that the multiplier \(\gamma_1\) at \(U_1\) in expression (11) must be omitted.

Analysis of expression (11) shows that “internal” nonlinear feedback can lead to complete compensation of the second harmonic. Complete compensation is possible only when the condition

\[ \frac{r_{\mathrm{k}}'}{r_{\mathrm{k}}}\cdot \frac{\beta R_{\mathrm{H}}}{r_{\mathrm{k}}} > \frac{1}{2}\frac{\beta'}{\beta} \tag{12} \]

is satisfied, and it occurs at a value of the internal resistance of the signal source equal to

\[ R_0= r\, \frac{\dfrac{1}{2}\dfrac{r_\mathrm{e}'}{r_\mathrm{e}}} {\dfrac{r_{\mathrm{k}}'}{r_{\mathrm{k}}}\dfrac{\beta R_{\mathrm{H}}}{r_{\mathrm{k}}} - \dfrac{1}{2}\dfrac{\beta'}{\beta}} . \tag{13} \]

Fulfillment of condition (12) depends both on the values of the derivatives \(r_{\mathrm{k}}'\), \(\beta'\), and on the absolute value of these parameters and the value of the load. It is known [6] that the base-current transfer coefficient \(\beta\), as the emitter current increases, first grows; then, at a certain value of the latter, it reaches a maximum value and remains almost constant up to very large currents. At the same time, experimentally obtained dependences show that the value of \(\beta'\) remains practically constant over a wide range of variation of the emitter current. It follows from this that the ratio \(\beta'/\beta\) decreases as the current increases. For example, for the 2T602B transistor at a current of \(5\) mA, \(\beta'/\beta=16.7\), and at a current of \(20\) mA, \(\beta'/\beta=8.4\). For the 2T308V transistor at a current of \(1.5\) mA, \(\beta'/\beta=26.7\), while at a current of \(10\) mA, \(\beta'/\beta=11.1\).

According to the graphs \(r_{22}=f(E_{\mathrm{e}})\) given in [5] for the P402 transistor, the values of \(r_{22}'/r_{22}\) increase, though only slightly, as the current increases. At a current value of \(\sim 1.5\) mA, \(r_{22}'/r_{22}=30\), and at a current of \(\sim 10\) mA, \(r_{22}'/r_{22}=33\).

Since the relation [6]

\[ r_{22}=\frac{r_{\mathrm{k}}}{1+\beta}, \]

one may expect that the ratio \(r'_K/r_K\) increases with increasing current faster than \(r'_{22}/r_{22}\).

The factor \(\beta R_H/r_K\) at \(r'_K/r_K\) in expression (12) increases very rapidly with increasing current because \(\beta\) and \(r_K\) vary in opposite directions.

On the basis of the data presented, it may be assumed that condition (12) is realistically attainable for medium- and high-power transistors having small \(r_K\) and operating at large currents.

Conclusions

1. The magnitude of the nonlinear distortions of a transistor at low and medium frequencies is uniquely determined by the parameters \(r_e\), \(r_\delta\), \(r_K\), \(\beta\), and by their derivatives with respect to the base–emitter voltage (or with respect to the emitter current).

2. If condition (12) is not satisfied, then the nonlinear distortion in the second harmonic, as \(R_i\) increases, decreases monotonically, tending for \(R_i \gg r\) to the value

\[ K_2=\frac{U_1}{F}\left(\frac{1}{2}\frac{\beta'}{\beta}-\frac{r'_K}{r_K}\frac{\beta R_H}{r_K}\right); \tag{14} \]

moreover, as the emitter current increases, the absolute value of \(K_2\) decreases.

1. When condition (12) is satisfied, the distortion pattern changes significantly.

Fig. 2. Dependence of the modulus of the normalized value of the second harmonic of the output voltage on the signal-source resistance for different values of the emitter current \((5\ \text{mA},\ 10\ \text{mA},\ 20\ \text{mA})\).

When \(R_i\) changes from zero to \(R_i<R_0\), the distortion in the second harmonic decreases sharply; at \(R_i=R_0\) it becomes equal to zero; and with a further increase in \(R_i\), the phase of the second harmonic changes abruptly by \(180^\circ\), and the distortion increases in modulus, tending to the value determined by expression (14).

It is characteristic that the distortion values decrease with increasing current for \(R_i<R_0\) and increase for \(R_i>R_0\).

1. The strengthening of inequality (12) with increasing collector current leads to the fact that, as the latter increases, the quantity \(R_0\) approxi-

approaches the value of the input resistance (at the given current) \(r\) and, in principle, may be less than the latter.

1. Inversion of the phase of the second harmonic is a very significant factor both in the design of linear amplifiers and in the measurement of the distortion magnitude of the latter by the direct method, since the possible phase opposition of the harmonics of the generator and the amplifier may lead to compensation of distortions at the output of the latter and, consequently, to unreliability of the measurement results. Therefore, the compensation method described in [7] may be considered a more acceptable method for measuring small nonlinear distortions of amplifiers. The theoretical propositions of the present work are clearly confirmed by experiment.

Figure 2 gives the normalized values of the modulus of the second harmonic for the high-voltage silicon transistor 2T602B (\(R_n = 1\ \text{k}\Omega\), \(U_{CE} = 10\ \text{V}\)) as a function of the source resistance for different emitter currents. The harmonic magnitude was determined by the procedure described in [7]. Phase inversion was observed from Lissajous figures.

References

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5. V. D. Ivanchenko. “Nonlinear distortions in transistor amplifiers.” In: Semiconductor Devices in Electrical-Communication Engineering. Ed. by I. F. Nikolaevsky. Issue 2, Svyaz Publishing House, Moscow, 1968.

6. I. F. Nikolaevsky. Operational Parameters and Features of the Application of Transistors. Svyazizdat, Moscow, 1963.

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8. V. P. Sigorsky. Analysis of Electronic Circuits. Gostekhizdat of the Ukrainian SSR, Kiev, 1963.