Emitter Correction Circuit in a Two-Stage Amplifier Based on Planar Triodes (PT)

I. N. PUSTYNSKII

(Presented by the conference dedicated to the 60th anniversary of the Tomsk Polytechnic Institute)

The equivalent circuit of a two-stage amplifier with emitter correction of high-frequency distortions is shown in Fig. 1. Since the problems of correcting high-frequency distortions by means of an \(RC\) lead network in the emitter circuit in an amplifier stage based on PT with an active signal-source resistance are considered in detail in [1], we shall not dwell here on questions of correction of the first stage with the second disconnected.

Fig. 1.

The signal-source resistance for the second stage differs from an active one. It is equal to the output resistance of the first stage, i.e.

\[ Z_{0}=\frac{Z^{\infty}_{\mathrm{out}.1}\cdot Z_{H1}}{Z^{\infty}_{\mathrm{out}.1}+Z_{H1}}, \tag{1} \]

where \(Z^{\infty}_{\mathrm{out}.1}\) is the output resistance of the first stage at \(Z_{H1}=\infty\); \(Z_{H1}\) is the load resistance of the first stage with the second disconnected \((Z_{H1}=R_{H1})\).

The output resistance of the first stage with the load disconnected, i.e. at \(R_{H1}=\infty\), is equal to

\[ Z_{\mathrm{out}.1}^{\infty} = R_{\mathrm{out}.1}^{\infty}\, \frac{ 1+j\omega \dfrac{1}{a_{0e1}^{0}}\left(\tau'_{\beta 1}+\tau_{01}a_{0e1}^{0}\right) +(j\omega)^2\dfrac{1}{a_{0e1}^{0}}\tau_{01}\tau'_{\beta 1} }{ \left(1+j\omega r_{K1}C_{K1}\right)\cdot \left(1+j\omega \tau_{0c}\dfrac{a_{e1}^{\infty}}{a_{0e1}^{\infty}}\right) }, \tag{2} \]

where

\[ a_{e1}^{0} \simeq 1+\frac{r_{e1}(1+\beta_{01})}{R_{b1}+r_{b1}}, \qquad a_{e1}^{\infty}=1+\frac{r_{e1}}{R_{b1}+r_{b1}} \]

are the coefficients of the internal feedback (due to \(r_{e1}\)) of the first stage at mid-frequencies, respectively for \(R_{H1}=0\) and \(R_{H1}=\infty\);

\[ a_{0e1}^{0}\simeq 1+\frac{(R_{01}+r_{e1})(1+\beta_{01})}{R_{b1}+r_{b1}}, \qquad a_{0e1}^{\infty}= 1+\frac{R_{01}+r_{e1}}{R_{b1}+r_{b1}} \]

are the coefficients of the total (internal and external) feedback of the first stage at mid-frequencies, respectively for \(R_{H1}=0\) and \(R_{H1}=\infty\);

\[ R_{\mathrm{out}.1}^{\infty} = \frac{r_{K1}}{1+\beta_{01}}\cdot \frac{a_{0e1}^{0}}{a_{0e1}^{\infty}} \]

is the output resistance of the 1st stage at mid-frequencies for \(R_{H1}=\infty\);

\(\beta_{01}\) is the current-transfer coefficient of the triode base in the first stage at low frequency,

\[ \tau_{01}=R_{01}C_{01},\qquad \tau'_{\beta 1}=\tau_{\beta 1}+C_{K1}(R_{01}+r_{e1})(1+\beta_{01}), \]

\[ \tau''_{\beta 1}=\tau_{\beta 1}+C_{K1}r_{e1}(1+\beta_{01}), \]

\(\tau_{\beta 1}\) is the current-transfer constant of the triode base in the first stage.

Substituting the value \(Z_{\mathrm{out}.1}^{\infty}\) from (2) into formula (1), we obtain an expression for the resistance of the signal source of the second stage:

\[ Z_{b2}\simeq R_{b2}\, \frac{ 1+j\Omega_{1}\left(\dfrac{q_{1}}{1+q_{1}}+m_{1}\dfrac{a_{b1}^{0}}{a_{0e1}^{0}}\right) +(j\Omega_{1})^{2}\,m_{1}\dfrac{q_{1}}{1+q_{1}} }{ 1+j\Omega_{1}\left(1+m_{1}\dfrac{a_{e1}^{0}}{a_{0e1}^{0}}\right) +(j\Omega_{1})^{2}m_{1} }, \tag{3} \]

where

\[ \Omega_{1}=\omega\,\frac{\tau_{k\beta 1}}{a_{0e1}^{0}} \]

is the normalized angular frequency;

\[ \tau_{k\beta 1}=\tau_{\beta 1}+C_{K1}R_{H1}(1+\beta_{01}) \]

is the equivalent current-transfer constant of the base in the 1st amplifier stage;

\[ m_{1}=\frac{\tau_{01}}{\tau_{k\beta 1}}\cdot a_{0e1}^{0} \]

is the correction parameter of the 1st stage, \(R_{b2}\simeq R_{H1}\);

\[ q_{1}=\frac{\tau_{\beta 1}}{C_{K1}R_{H1}(1+\beta_{01})} \]

is the coefficient of relative inertia of the triode in the 1st amplifier stage.

Expression (3) has been obtained under the assumption that

\[ C_{K1}(R_{01}+r_{e1})(1+\beta_{01}) \ll \tau_{\beta 1}, \quad \text{i.e.} \quad \tau''_{\beta 1}\simeq \tau'_{\beta 1}\simeq \tau_{\beta 1}, \]

\[ R_{H1}\gg R_{\mathrm{out}.1}^{\infty},\qquad a_{e}^{\infty}\simeq 1 \quad \text{and} \quad a_{0e}^{\infty}\simeq 1, \]

since such relations also often occur in practice.

It follows from (3) that for \(q_{1}\gg 1\), i.e., when the distortion is caused mainly by the inertia of the triode (by processes occurring in the base region), we have \(Z_{b2}\simeq R_{b2}\).

Assuming, for simplicity, that the parameters of both stages are identical, except for the correcting capacitances \(C_{01}\) and \(C_{02}\) (i.e., the parameters \(m_1\) and \(m_2\)), we write the frequency-distortion coefficient for the second stage in the form¹

\[ \frac{K_{\omega 2}}{K_{\omega 02}} = \frac{ 1+j\Omega(1-m_2)+(j\Omega)^2(m_1 m_2)+(j\Omega)^3 m_1 m_2 }{ 1+j\Omega 2+(j\Omega)^2(m_1+m_2+d)+(j\Omega)^3 d(m_1+m_2)+(j\Omega)^4 d m_1 m_2 }, \]

where

\[ m_2=\frac{C_{02}R_b}{\tau_k}, \quad a_{0c}\text{ is the correction parameter of the second stage,} \]

\[ d=\frac{\chi+q}{1+q}, \qquad z=\frac{r_b}{R_b+r_b}. \]

Equating in \(\left|\dfrac{K_{\omega 2}}{K_{\omega 02}}\right|\) the coefficients at \(\Omega^2\), we find the correction parameter of the second stage for which its frequency response is close to optimal:

\[ m_{2u}=-1+\sqrt{4-2d}. \]

At the same approximations, for a stage with active signal-source resistance the correction parameter is

\[ m_{uR}\simeq 0.414. \]

Since \(0<d<1\), we have \(1>m_{2u}>0.414\). Since in real circuits usually \(d\simeq 0.3 \div 0.85\), we obtain

\[ m_{2u}\simeq (1.2 \div 2.0)\cdot m_{uR}. \]

For the entire amplifier of Fig. 1 the frequency-distortion coefficient is equal to

\[ \frac{K_u}{K_{u0}} = \frac{ 1+j\Omega(m_1+m_2)+(j\Omega)^2 m_1 m_2 }{ 1+j\Omega 2+(j\Omega)^2(m_1+m_2+d)+(j\Omega)^3 d(m_1+m_2)+(j\Omega)^4 d m_1 m_2 }. \]

Fig. 2.

Equating in \(\left|\dfrac{K_u}{K_{u0}}\right|\) the coefficients at \(\Omega^2\), we obtain the relation between the correction parameters \(m_1\) and \(m_2\) for which the frequency response of the whole amplifier is close to optimal. This relation is determined by the expression

\[ m_2=-1+\sqrt{5-2d-m_1(m_1+2)}. \]

In the case \(m_1=m_{1R}=0.414\), we obtain

\[ m_2=m_{2u}=-1+\sqrt{4-2d}. \]

It is of interest to determine for what ratios of the correction parameters \(m_1\) and \(m_2\) a larger passband of the amplifier is ensured with a “flat” frequency response.

Figure 2 shows calculated dependences of the modulus

\[ \left|\frac{K_u}{K_{u0}}\right| = f(\Omega). \]

The numbers of the curves correspond to the following cases:

1) \(m_1 = 0,\ m_2 = d = 0.828\) and \(m_2 = 0,\ m_1 = d = 0.828\)—when equality of the coefficients in \(\left|K_u/K_{u0}\right|\) is ensured both for \(\Omega^2\) and for \(\Omega^4\).

2) \(m_1 = m_{1u} = 0.414,\ m_2 = m_{2u} = 0.531\)—when optimal frequency characteristics are ensured in both the first and second stages. Here, as in cases “3,” “4,” and “5,” for purposes of comparison we also take \(d = 0.828\).

3) \(m_1 = m_2 = -1 + \sqrt{3 - d} = 0.471\)—when the first stage is overcorrected and the second is undercorrected. Although curves “2” and “3” practically merge, in this case the limiting frequency is nevertheless somewhat higher.

4) \(m_1 = 0.3,\ m_2 = 0.629\)—when the first stage is undercorrected and the second is overcorrected.

5) \(m_1 = 0,\ m_2 = 0\)—when correcting capacitances are absent in both stages.

Thus, the greatest passband with a “flat” frequency response in a two-stage amplifier with emitter correction is obtained for

\[ m_1 = m_2 = -1 + \sqrt{3 - d}, \]

i.e., when the first stage is overcorrected and the second is undercorrected. Experiment confirms this.

References

1. Pustynskii I. N. High-frequency correction in transistor amplifiers by means of an \(RC\) feedback circuit in the emitter circuit. Izvestiya vuzov SSSR, Radiotekhnika, No. 5, 1960.

1. It is taken into account that
   \[ \frac{z_b+r_b+R_n}{z_k}\ll 1,\quad a_{0c}\simeq a_e,\quad \frac{R_n}{r_k}(1+z_b)\ll 1, \]
   i.e.
   \[ a_{0c}^{b}\simeq a_{0c} \quad\text{and}\quad a_c^b\simeq a_e. \] ↩