ON THE NATURE OF GENERATOR RESISTANCE IN THE INTERMEDIATE STAGE OF A MULTISTAGE VIDEO AMPLIFIER WITH A PARALLEL CORRECTION CIRCUIT

I. N. PUSTYNSKY

(Presented at the scientific and technical conference dedicated to the 90th anniversary of the birth of V. I. Lenin)

In the intermediate \((i\)-th) stage of a transistor video amplifier, the generator resistance is equal to

\[ Z_{g_i}=\frac{Z_{H_{i-1}}\cdot Z_{\mathrm{out}_{i-1}}} {Z_{H_{i-1}}+Z_{\mathrm{out}_{i-1}}} , \tag{1} \]

where \(Z_{H_{i-1}}\), \(Z_{\mathrm{out}_{i-1}}\) are, respectively, the load resistance and the output resistance of the preceding \((i-1)\)-th stage.

As is evident from expression (1), the generator resistance of the \(i\)-th stage depends to a considerable extent on the output resistance of the \(i-1\)-th stage, which in turn depends on the output resistance of the \(i-2\)-th stage, and so on.

As a result, decoupling of the stages in semiconductor amplifiers is very difficult.

Investigation of the nature of the resistance of the \(i\)-th stage makes it possible to draw a number of useful practical conclusions.

With a parallel correction circuit,

\[ Z_{H_i} \approx R_{H_{i-1}} + j\omega L_{i-1}. \tag{2} \]

The output resistance of an amplifying stage in a common-emitter circuit is equal to

\[ Z_{\mathrm{out}_{i-1}}=\frac{Z_{k_{i-1}}}{1+\beta_{i-1}}\cdot \frac{F_{0_{i-1}}}{F_{\infty_{i-1}}}, \tag{3} \]

where

\[ Z_{k_{i-1}}=\frac{r_{k_{i-1}}}{1+j\omega r_{k_{i-1}} C_{K_{i-1}}} \quad \text{is the collector impedance,} \]

\(F_{0_{i-1}}, F_{\infty_{i-1}}\) are the feedback coefficient (more precisely, the return difference) of the preceding stage, respectively for \(Z_H=0\) and \(Z_H=\infty\), with

\[ F_{0\,i-1}=1+\frac{r_{e\,i-1}(1+\beta_{i-1})\left[1+\dfrac{Z_{e\,i-1}+r_{\delta\,i-1}}{Z_{\kappa\,i-1}}\right]}{Z_{2\,i-1}+r_{\delta\,i-1}} \]

or

\[ F_{\infty\,i-1}=1+\frac{r_{e\,i-1}}{Z_{2\,i-1}+r_{\delta\,i-1}} . \]

Assuming, for simplicity, that the generator resistance of the preceding stage is close to the active resistance, i.e., \(Z_{2\,i-1}\approx R_{2\,i-1}\), and taking into account that usually \(\tau_i \gg C_\kappa r_e(1+\beta_0)\), we obtain

\[ F_{0\,i-1}\approx a_{0\,i-1}\, \frac{1+j\omega\dfrac{\tau_{i-1}}{a_{0\,i-1}}}{1+j\omega \tau_{i-1}} \quad \text{and} \quad F_{\infty\,i-1}\approx a_{\infty\,i-1}, \]

where

\[ a_{0\,i-1}=1+\frac{r_{e\,i-1}(1+\beta_{0\,i-1})}{r_{\delta\,i-1}+R_{2\,i-1}} \]

is the value of the coefficient \(F_{0\,i-1}\) at the mid-frequency,

\[ a_{\infty\,i-1}=1+\frac{r_{e\,i-1}}{R_{2\,i-1}+r_{\delta\,i-1}} . \]

Thus,

\[ Z_{\mathrm{out}\,x\,i-1}=R_{\mathrm{out}\,x\,i-1}\, \frac{1+j\omega\dfrac{\tau_{i-1}}{a_{0\,i-1}}} {1+j\omega R_{\mathrm{out}\,x\,i-1}\cdot C_{\mathrm{out}\,x\,i-1}}, \tag{4} \]

where

\[ R_{\mathrm{out}\,x\,i-1} = \frac{r_{\kappa\,i-1}}{1+\beta_{0\,i-1}}\cdot \frac{a_{0\,i-1}}{a_{\infty\,i-1}} \]

is the output resistance at the mid-frequency of the \(i-1\)-th stage,

\[ C_{\mathrm{out}\,x\,i-1} = C_{\kappa\,i-1}(1+\beta_{0\,i-1}) \frac{a_{\infty\,i-1}}{a_{0\,i-1}} \]

is the output capacitance of the amplifier stage for \(\tau_i=0\).

Substituting (3) and (4) into (1), with

\[ \frac{R_{H\,i-1}}{R_{\mathrm{out}\,x\,i-1}}\ll 1, \]

we obtain

\[ Z_{2\,i}=R_{2\,i}\cdot \frac{1+j\Omega(m+q)+(j\Omega)^2 m\cdot q} {1+j\Omega(1+q)+(j\Omega)^2 m}, \tag{5} \]

where

\[ R_{2\,i}= \frac{R_{H\,i-1}\cdot R_{\mathrm{out}\,x\,i-1}} {R_{H\,i-1}+R_{\mathrm{out}\,x\,i-1}} \approx R_{H\,i-1}, \quad \Omega=\omega R_{H\,i-1}\cdot C_{\mathrm{out}\,x\,i-1}, \]

\[ q= \frac{\tau_{i-1}} {a_{\infty\,i-1}R_{H\,i-1}C_{\kappa\,i-1}(1+\beta_{0\,i-1})} \]

is the coefficient of relative inertialness of the transistor in the amplifier stage,

\[ m=\frac{L_{i-1}}{R_{n_{i-1}}^{2} C_{\mathrm{out}\,i-1}} \]
is the correction parameter of the \((i-1)\)-st stage.

The modulus of the generator resistance is equal to

\[ |Z_2|=R_2\cdot \sqrt{ \frac{ 1-\Omega^2(m^2-q^2)+\Omega^4 m^2 q^2 }{ 1+\Omega^2[1+2(q-m)+q^2]+\Omega^4 m^2 } }. \tag{6} \]

As is seen from expression (6), the generator resistance is purely active, i.e., \(Z_2=R_2\), if \(m=q=1\).

In the general case it may be considered that, in its character, the generator resistance in the operating frequency range \((\Omega \simeq 0 \div 1)\) is close to active if

\[ m^2=1+2(q-m), \]

i.e.,

\[ m=m_k=-1+\sqrt{2(1+q)}. \]

The dependence

\[ \frac{|Z_2|}{R_2}=f_1(\Omega) \]

for specified \(q\) at \(m=m_k\) is shown graphically in Fig. 1, from which it is seen that \(|Z_2|\) differs from \(R_2\)

Fig. 1.

even at \(\Omega_{\mathrm{op}}\simeq 1\) by less than 8%; moreover, in the case \(q>1\), by its character \(Z_2\) is inductive, and for \(q<1\) it is capacitive.

In Figs. 2 and 3, respectively, the dependences are shown

\[ \frac{|Z_2|}{R_2}=f_2(\Omega) \quad \text{for} \quad m=m_u=-1+\sqrt{1+(1+q)^2} \]

and

\[ \frac{|Z_2|}{R_2}=f_3(\Omega) \quad \text{for} \quad m=m_n \begin{cases} 0.25(1+q)^2, & \text{for } q<1,\\ q, & \text{for } q>1. \end{cases} \]

Here \(m_u\) is the correction parameter at which the optimum frequency response is ensured [1].

\(m_n\) is the correction parameter at which the transient response still has no overshoot.

It is seen from the graphs that, for \(m=m_u\) and \(m=m_n\), the modulus \(|Z_2|\) differs from \(R_2\) in the operating frequency range, even at \(q=0 \div 4\), by no more than 30%, and as a first approximation for practical calculations it may be assumed that in an intermediate stage of a transistor video ampli-

amplifier with a parallel correction circuit, the generator resistance is close to the active value, i.e. \(Z_2 \simeq R_2\).

Thus, our assumption that \(Z_{2i-1} \simeq R_{2i-1}\) may be considered admissible.

Consequently, the engineering formulas obtained in the analysis of a single-stage amplifier with a parallel correction circuit and an active generator resistance may also be considered acceptable for the intermediate stage of a video amplifier.

Fig. 2.

However, when designing multistage amplifiers it should be taken into account that, for \(q>1\), a noticeable gain in the overall gain area is obtained in the case where the first stages are somewhat undercorrected and the last is overcorrected. This is explained by the fact that, for \(q>1\), when \(m=m_{\mathrm{u}}\) and \(m=m_{\mathrm{p}}\), \(|Z_2|>R_2\), and the quality factor of the \(i\)-th stage becomes smaller than when \(|Z_2|<R_2\). An experimental check with a two-stage amplifier confirms this.

Fig. 3.

For example, at \(R_{g1}=R_{n1}=250\ \Omega\), a maximally flat frequency response of the first stage (using the triode P403-1 with an inductance \(L_1=15\ \mu\text{H}\) \((f_{\mathrm{gr}}=5.2\ \text{MHz})\)) was obtained. Separately for the second stage, with the same \(R_{g2}\) and \(R_{n2}\), the maximally flat frequency response using the triode P403-2 was obtained with \(L_2=17\ \mu\text{H}\) \((f_{\mathrm{gr}}=5\ \text{MHz})\).

For two stages, with \(L_1 = 15\ \mu\text{H}\) and \(L_2 = 17\ \mu\text{H}\), the frequency characteristic proved sufficiently uniform, with \(f_{\text{upper}} = 3.7\ \text{MHz}\). The flattest frequency characteristic was obtained with \(L_1 = 10\ \mu\text{H}\) and \(L_2 = 21\ \mu\text{H}\). In this case the upper limiting frequency was \(f_{\text{upper}} = 4.0\ \text{MHz}\).

References

1. Pustynskii I. I. A parallel circuit for correcting high-frequency distortions in transistor video amplifiers, Izvestiya TPI, vol. 105, 1960.