On the Possibility of Frequency Self-Correction in a Two-Stage Direct-Coupled Transistor Amplifier
Abstract Generated abstract
This paper examines low-frequency distortion correction in a two-stage direct-coupled transistor amplifier intended for measuring-amplifier applications. It argues that the emitter circuits of the two stages need not be treated independently: the gain reduction caused by the first-stage emitter network can be compensated by the frequency-dependent increase of the second-stage input resistance. Using an equivalent-circuit analysis, the authors derive the first-stage gain, formulate conditions for optimal frequency self-correction, and obtain expressions for selecting the first-stage emitter resistance and the emitter bypass capacitance of the second stage. The result is a calculation method that can reduce the required capacitance values compared with conventional single-stage design approaches while assigning the remaining permissible frequency distortion primarily to the output-stage emitter circuit.
Full Text
ON THE POSSIBILITY OF FREQUENCY SELF-CORRECTION IN A TWO-STAGE TRANSISTOR AMPLIFIER WITH DIRECT COUPLING
M. S. ROITMAN, V. M. SERGEEV
(Presented by the scientific-technical seminar of the Department of Radio Engineering)
In [1] a two-stage transistor amplifier, shown in Fig. 1, is recommended as a functional unit for the construction of measuring amplifiers (MAs) with a large gain factor.
One of the most important requirements imposed on MAs is small frequency distortion over a wide frequency range. The known methods for calculating the value of the capacitors in the emitter circuit of a rheostat stage [2, 3, 4] lead, under the stated requirements, to considerable values of the latter (on the order of several hundred microfarads), even when deep negative feedback (NFB) is taken into account. In addition, the calculation formulas given in [2, 3, 4] are derived for a single stage and do not take into account the mutual influence of the indicated circuits in a multistage amplifier.
At the same time, a qualitative analysis of the action of the circuits \(R_{\mathrm{e}1} — C_{\mathrm{e}1}\) and \(R_{\mathrm{e}2} — C_{\mathrm{e}2}\) of the amplifier under consideration shows that, while because of the influence of \(R_{\mathrm{e}1} — C_{\mathrm{e}1}\) the gain factor of the first stage tends to decrease, the increase in the input resistance of the second stage due to the action of the circuit \(R_{\mathrm{e}2} — C_{\mathrm{e}2}\) produces the opposite effect in the first stage. Hence it follows that, with a certain selection of the time constants of both circuits, one can obtain optimal correction of the frequency characteristic of the first stage. Then the frequency distortions permissible for such a functional unit will be entirely attributable to one capacitor \(C_{\mathrm{e}2}\), which will determine its smaller value. We shall show this by analysis.
Fig. 1.
The gain factor of the first stage at low frequencies is determined as [5]
\[ K_{\mathrm{I}}=\frac{Z_{\mathrm{n}}\Delta_{12}}{Z_{\mathrm{n}}\Delta_{22}+\Delta}, \]
where
\[ Z_n = [R_{\mathrm{in}2} + (1+\beta_2) Z_{e2}] \parallel R_{k1}; \]
\(R_{\mathrm{in}2}\) is the input resistance of the second stage at midrange frequencies;
\(\beta_2\) is the current gain of transistor \(T\);
\(\Delta, \Delta_{12}\), and \(\Delta_{22}\) are the determinant and the algebraic cofactors of the resistance matrix for the circuit of Fig. 2.
In analyzing the gain, local feedback in the first stage through the resistance \(R_0\) is taken into account.
The resistance matrix \(\Delta\) for the circuit of Fig. 2 has the form
Fig. 2
\[ \frac{Z_{e1}+R_0+r_e+r_b}{-Z_{e1}-R_0-r_e+\alpha r_k} \;\middle|\; \frac{-Z_{e1}-R_0-r_e}{Z_{e1}+R_0+r_e+r_k(1-\alpha)} , \]
where \(\alpha, r_e, r_b, r_k\) are the parameters of the physical equivalent circuit of the triode for midrange frequencies,
\[ \Delta = Z(r_k+r_b)+r_k r_b(1-\alpha), \qquad Z = Z_{e1}+r_e+R_0. \]
\[ \Delta_{12}=-(Z+\alpha r_k); \qquad \Delta_{22}=Z+r_b. \]
Taking into account that \(r_k \gg r_b\) and \(\alpha r_k \gg Z\), one may write
\[ K_n = - \frac{\alpha r_k Z_n} {Z_n(Z+r_b)+r_k[Z+r_b(1-\alpha)]}, \]
practically \(r_k(1-\alpha)>Z_n\); then, neglecting the first term in the denominator, we obtain
\[ K_n = -\frac{\alpha Z_n}{Z+r_b(1-\alpha)} . \tag{1} \]
Writing for expression (1)
\[ Z_n = \frac{ R_{k1}\left[R_{\mathrm{in}2}+Z_{e2}(1+\beta_2)\right] }{ R_{k1}+R_{\mathrm{in}2}+Z_{e2}(1+\beta_2) }, \]
where
\[ Z_{e2}=\frac{R_{e2}}{1+j\omega \tau_2}, \qquad \tau_2=R_{e2}C_{e2}, \]
and, taking \(R_{\mathrm{k}_1}=(\gamma-1)R_{\mathrm{in}_2}\), we finally obtain
\[ K_{\mathrm{v}}= -\frac{\alpha R_{\mathrm{k}_1}\left(R_{\Sigma}+j\omega\tau_2 R_{\mathrm{in}_2}\right)(1+j\omega\tau_1)} {\left(R_{\Sigma}^{1}+j\omega\tau_2\gamma R_{\mathrm{in}_2}\right)\left(r_{\Sigma}^{1}+j\omega\tau_1 R_{\Sigma}\right)}, \tag{2} \]
where
\[ R_{\Sigma}=R_{\mathrm{in}_2}+R_{\mathrm{e}_2}(1+\beta_2);\qquad R_{\Sigma}^{1}=r_{\mathrm{k}_1}+R_{\Sigma}; \]
\[ r_{\Sigma}=r_{\mathrm{e}}+R_0+r_{\mathrm{b}}(1-\alpha);\qquad r_{\Sigma}^{1}=R_{\mathrm{e}_1}+r_{\Sigma};\qquad \tau_1=R_{\mathrm{e}_1}C_{\mathrm{e}_1}. \]
The mid-frequency gain \(K_0\) can be obtained from (1) by substituting, instead,
\[ Z_{\mathrm{n}}\to \frac{R_{\mathrm{k}_1}}{\gamma} \quad \text{and} \quad Z\to r_{\mathrm{e}}+R_0. \]
Taking the foregoing into account, we obtain
\[ K_0=-\frac{\alpha R_{\mathrm{n}}}{r_{\Sigma}}. \tag{3} \]
The coefficient of frequency distortion at the lower frequency of the operating range \(\omega_{\mathrm{n}}\) is determined as
\[ M_{\mathrm{n}}=\frac{K_0}{K_{\mathrm{v}}} = \frac{R_{\Sigma}^{1}r_{\Sigma}^{1}-\omega_{\mathrm{n}}^2\tau_1\tau_2\gamma R_{\mathrm{in}_2}r_{\Sigma} +j\omega_{\mathrm{n}}\left(\tau_2\gamma R_{\mathrm{in}_2}r_{\Sigma}^{1}+\tau_1 r_{\Sigma}R_{\Sigma}^{1}\right)} {\gamma R_{\Sigma}r_{\Sigma}-\omega_{\mathrm{n}}^2\tau_1\tau_2\gamma R_{\mathrm{in}_2}r_{\Sigma} +j\omega_{\mathrm{n}}\left(\tau_2\gamma R_{\mathrm{in}_2}r_{\Sigma}+\tau_1\gamma r_{\Sigma}R_{\Sigma}\right)}. \tag{4} \]
If the condition
\[ R_{\Sigma}^{1}r_{\Sigma}^{1}=\gamma R_{\Sigma}r_{\Sigma}, \tag{5} \]
is satisfied, then expression (4) can be represented in the form
\[ M_{\mathrm{n}}=\frac{1+j\omega A}{1+j\omega B}, \tag{6} \]
where \(A\), \(B\) are the ratios of the imaginary part to the real part of the numerator and denominator, respectively.
The frequency response of the form (6) can be optimized according to Braude under the condition
\[ A=B. \tag{7} \]
Thus, for optimal correction it is necessary that conditions (5) and (7) be fulfilled.
Let us expand these conditions.
In [1] an optimal calculation, from the standpoint of obtaining the maximum depth of negative feedback, is given for the two-stage amplifier under consideration. As a result of the calculation presented there, all elements of the amplifier are uniquely determined, except for the value of the resistance \(R_{\mathrm{e}_1}\). We shall use equality (5) to determine its value. It is easy to show that requirement (5) is satisfied when
\[ R_{\mathrm{e}_1} = r_{\Sigma} \frac{(\gamma-1)(1+\beta_2)R_{\mathrm{e}_2}} {\gamma R_{\mathrm{in}_2}+R_{\mathrm{e}_2}(1+\beta_2)}. \tag{8} \]
Equality (7) can be represented in the form
\[ \tau_2\gamma R_{\mathrm{in}_2}R_{\mathrm{e}_1} = \tau_1 r_{\Sigma}\left(\gamma R_{\Sigma}-R_{\Sigma}^{1}\right). \tag{9} \]
When the indicated correction is carried out, frequency distortions will be introduced only by the emitter circuit of the output stage. The modulus of the frequency-distortion coefficient due to this circuit is equal to the modulus of the reciprocal difference of the negative feedback due to \(Z_{\mathrm{e}_2}\).
Using the simplified method for determining the feedback difference \(F\) given in [1], one may write
\[ M_{\mathrm{e}_2}=F=1+\frac{Z_{\mathrm{e}_2}(1+\beta_2)}{\gamma R_{\mathrm{in}_2}} . \tag{10} \]
It should be taken into account that the coefficient \(M_{\mathrm{e}_2}\) and the frequency-distortion coefficient specified for the entire amplifier, \(M_0\), are related by the known relation
\[ M_{\mathrm{e}_2}=1+(M_0-1)\cdot F_0, \]
where \(F_0\) is the feedback ratio for the entire amplifier due to the resistance \(R_0\). The quantity \(F_0\) can be calculated by the formula given in [1]. After obvious transformations, from (10) one can obtain the required value of the time constant of the \(R_{\mathrm{e}_2}\)—\(C_{\mathrm{e}_2}\) circuit:
\[ \tau_2=\frac{\sqrt{\delta^2-M_{\mathrm{e}_2}^2}}{\omega_{\mathrm{n}}\sqrt{M_{\mathrm{e}_2}^2-1}}, \tag{11} \]
where
\[ \delta=1+\frac{R_{\mathrm{e}_2}(1+\beta_2)}{\gamma R_{\mathrm{in}_2}} ; \]
substituting (11) into (9), we obtain the formula for determining the required capacitance of the capacitor \(C_{\mathrm{e}_2}\):
\[ C_{\mathrm{e}_2}= \frac{\gamma R^{\mathrm{in}}_1\sqrt{\delta^2-M_{\mathrm{e}_2}^2}} {\omega_{\mathrm{n}} r_y(\gamma R_y-R_1^{2})\sqrt{M_{\mathrm{e}_2}^2-1}} . \tag{12} \]
REFERENCES
- M. S. Roitman, V. M. Sergeev. A decade measuring amplifier. Izvestiya TPI, vol. 171, 1968.
- G. S. Tsykin. Electronic Amplifiers. Svyazizdat, 1965.
- A. V. Tsykina. Design of Transistor Amplifiers. “Svyaz,” 1965.
- G. R. Shi. Calculation of Transistor Circuits. “Svyaz,” 1963.
- I. P. Sigorskii. Calculation of Electronic Circuits. Svyazizdat, 1963.
NOTICED MISPRINTS
| Page | Line | Printed | Should read |
|---|---|---|---|
| 3* | 18 from top | $10$ MHz and up to $5 \cdot 10^{-4}$— —$5 \cdot 10^{-3}\%$ |
$10$ MHz up to $5 \cdot 10^{-4}$— —$5 \cdot 10^{-3}$ |
| 3 | 7 from bottom | 2 | 3 |
| 3 | 7 from bottom | 3 | 2 |
| 16 | 4 from bottom | Collection of papers of TIRiETa (in press) |
Izvestiya TPI, vol. 171, 1968 |
| 30 | 5 from top | $T$ | $\mathrm{PT}_2$ |
| 31 | 5 from top | $r_\Sigma +$ | $r_\Sigma =$ |
| 34–35 | in all cases | $\Delta_f$ | $\Delta_\varphi$ |