On High-Frequency Correction in Transistor Video Amplifiers

I. N. Pustynskii, F. M. Usoltsev

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

Introduction

It is known that in transistor video amplifiers the reduction of gain at high frequencies is due not only to the parasitic capacitances of the circuit (as in tube amplifiers), but also to the comparatively low limiting frequency of the semiconductor triodes (PT) themselves, especially in the common-emitter circuit.

The methods of high-frequency correction in video amplifiers using transistors and tubes are analogous. Most often, for correction of high frequencies in video amplifiers using PT, \(RC\)-feedback in the emitter circuit is employed, since it has appreciable advantages in comparison with inductive correction circuits. These advantages include:

a) more stable operation of the stage because of the presence of feedback and the absence of correcting inductances;

b) smaller dimensions of the stage, since, with sufficiently deep feedback for temperature stabilization, there is no need to use an \(RC\) network with an electrolytic capacitor in the emitter circuit;

c) more stable properties of the stage over time, with changes in the supply voltage and when using transistors with noticeably different parameters;

d) smaller nonlinear distortions, etc.

There are works [1, 2] in which high-frequency correction by means of feedback is considered for two special cases: \(C_k = 0,\ \beta \ne \beta_0\) and \(C_k \ne 0,\ \beta = \beta_0\).

Of interest is the general case, when \(C_k \ne 0\) and \(\beta \ne \beta_0\), since it is the one most often realized in practice.

This case is considered in the present work.

Figure 1 shows a single-stage video amplifier using a PT and its equivalent circuit, valid under the following assumptions:

1. The stage operates in the mode of amplification of small signals.

2. We neglect effects associated with base modulation.

3. The emitter resistance is equal to
   \[ \frac{r_e}{1 + j \dfrac{\omega}{\omega_\alpha}} \simeq r_e, \]
   since, when using-

when high-frequency triodes are used (types P401–P403, etc.), usually

\[ \frac{\omega}{\omega_{\alpha}} \ll 1 . \]

Here

\[ \beta=\frac{\beta_0}{1+j\omega\tau_\beta} \]

is the base-current transfer coefficient,

\[ R_b=R_2 \parallel R_1 \parallel R_2' \]

is the equivalent generator resistance,

\[ \tau_\beta \]

is the base-current transfer time constant,

and \(R_0 C_0\) is the correcting network, whose time constant is equal to \(\tau_0\).

Fig. 1. a) Circuit of a single-stage amplifier. b) Equivalent circuit of an amplifier stage.

Calculation of a Stage from Frequency and Transient Characteristics

From feedback theory [3] it is known that the voltage gain of a stage in the presence of feedback is equal to

\[ K_u=\frac{K_u^0}{A}, \]

where

\[ K_u^0 \]

is the voltage gain of the stage without feedback,

\[ A=\frac{F}{M} \]

is the feedback coefficient,

\[ F=\frac{\Delta}{\Delta^0} \]

is the return difference,

\[ M=1-\frac{Z_e(1+\beta)}{\beta Z_k} \]

is a coefficient indicating the influence of feedback on the transfer parameter,

and \(\Delta, \Delta^0\) are the determinants of the matrix of \(z\)-parameters with feedback elements and without them, respectively.

Taking into account that usually

\[ \frac{z_e(1+\beta)}{\beta z_k}\ll 1, \]

with an active load we obtain

\[ K_u \simeq \frac{K_u^0}{F} =K_{u0}\frac{1+j\omega a_1}{1+j\omega b_1+(j\omega)^2 b_2}, \tag{1} \]

where

\[ a_1=m\tau_{nb}, \tag{1a} \]

\[ b_1=\frac{\tau_{nb}}{\alpha_{nb}}\left[1+m\alpha_{nbe}+m_n(\alpha_{nb}-1)\right], \tag{1b} \]

\[ b_2=\frac{\tau_{nb}}{\alpha_{nb}}\,m\left[1+m_n(\alpha_{nbe}-1)\right], \tag{1v} \]

where, in turn,

\[ m=\frac{\tau_0}{\tau_{nb}}, \qquad m_n=\frac{\tau_n}{\tau_{nb}}, \qquad \tau_n=C_k(R_n+R_b+r_b), \]

\[ \tau_{nb}= \frac{\tau_\beta+R_n C_k(1+\beta_0)} {1+\dfrac{R_n}{r_k}(1+\beta_0)} \quad\text{is the equivalent time constant of the amplifier-stage circuit,} \]

\[ \alpha_{nbe}=1+ \frac{r_e(1+\beta_0)} {(R_b+r_b)\left[1+\dfrac{R_n}{r_k}(1+\beta_0)\right]} \quad\text{is the coefficient of internal feedback of the stage for midrange frequencies,} \]

\[ \alpha_{nb}=1+ \frac{(R_0+r_e)(1+\beta_0)} {(R_b+r_b)\left[1+\dfrac{R_n}{r_k}(1+\beta_0)\right]} \quad\text{is the coefficient of total (internal and external) feedback of the stage for midrange frequencies,} \]

\[ K_{uo}= \frac{\beta_0 R_n} {(R_b+r_b)\left[1+\dfrac{R_n}{r_k}(1+\beta_0)\right]\alpha_{nb}} \quad\text{is the gain of the stage for midrange frequencies.} \]

If, in parallel with \(R_n\), there is a capacitance \(C_n\) of units or tens of picofarads, then, with accuracy sufficient for practice, the voltage gain may be written in the same form as for \(C_n=0\), but under the condition that here already

\[ \tau_n=C_nR_n+C_k(R_n+R_b+r_b), \quad \text{and} \quad \tau_{nb}= \frac{\tau_\beta+R_n\left[C_k(1+\beta_0)+C_n\right]} {1+\dfrac{R_n}{r_k}(1+\beta_0)}. \]

The magnitude of the frequency-distortion coefficient is equal to

\[ \left|\frac{K_u}{K_{uo}}\right|= \sqrt{ \frac{1+\omega^2 a_1^2} {1+\omega^2(b_1^2-2b_2)+\omega^4 b_2^2} }. \tag{2} \]

Investigating this expression by known methods (in the simplest case, by equating the coefficients of \(\omega^2\)), we obtain the condition for optimal correction of the frequency response

\[ a_1^2=b_1^2-2b_2. \tag{3} \]

Substituting into (3) the values \(a_1\), \(b_1\), and \(b_2\) from expressions (1a), (1b), and (1v), we obtain the correction parameter for which the optimal frequency response is ensured.

\[ m_c = \frac{ -(1-m_n)+ \sqrt{(1-m_n)^2+\left[1+m_n(\alpha_{nb}-1)\right]^2 \frac{\alpha_{nb}+\alpha_{nbe}}{\alpha_{nb}-\alpha_{nbe}}} }{ \alpha_{nb}+\alpha_{nbe} }. \tag{4} \]

In engineering calculations, using formula (4) is, of course, inconvenient; it is therefore advisable to represent it graphically. In the general case this cannot be done, since there are two variable parameters, \(m_n\) and \(\alpha_{nbe}\).

Since in real circuits \(\alpha_{nbe}\) is most often only slightly greater than unity, we shall confine ourselves to two values: \(\alpha_{nbe}=1\) and \(\alpha_{nbe}=2\) (Fig. 2).

Fig. 2. Dependence of the correction parameter (providing the optimal frequency response) on the feedback coefficient: solid line — \(\alpha_{nbe}=1\); dashed line — \(\alpha_{nbe}=2\).

For \(\alpha_{nb}\gg \alpha_{nbe}\), which occurs with deep external negative feedback, expression (4) takes the form

\[ m_c \simeq m_n, \tag{4a} \]

and in the case \(m_n\alpha_{nb}\ll 1\),

\[ m_c=\frac{0.41}{\alpha_{nb}}. \tag{4b} \]

The transient response of the amplifier stage (Fig. 1) is written in the form

\[ h(t)=1+\gamma\,\frac{\tau_0\alpha-1}{\gamma-\alpha}e^{-\alpha t} -\alpha\,\frac{\tau_0\gamma-1}{\gamma-\alpha}e^{-\gamma t}, \tag{5} \]

where

\[ \alpha=\frac{b_1-\sqrt{b_1^2-4b_2}}{2b_2}, \tag{5a} \]

\[ \gamma=\frac{b_1+\sqrt{b_1^2-4b_2}}{2b_2}. \tag{5b} \]

Analysis of expression (5) shows that the transient response may be monotonic (when \(\tau_0\alpha\le 1\)), and may have an aperiodic

overshoot (for \(b_1^2 \gg 4b_2\)) or have an oscillatory character (for \(b_1^2 < 4b_2\)).

From the condition \(\tau_0 \alpha = 1\) we obtain that the correction parameter corresponding to the boundary of a monotonic transient response is

\[ m=m_n,\quad \text{i.e.}\quad C_0=\frac{C_k(R_b+R_n+r_b)+C_nR_n}{R_0}. \]

In this case the transient response of the stage is written in the form

\[ h(t)=1-e^{-\gamma t}. \tag{6} \]

The rise time of the leading edge of the pulse is equal to

\[ t_{nk}=\frac{2.2}{\gamma} =2.2\,\frac{\tau_{nb}}{\alpha_{nb}}\,[1+m_n(\alpha_{nbe}-1)], \tag{7} \]

and the rise time of the circuit without correction is

\[ t_{nb}=2.2\,\frac{\tau_{nb}}{\alpha_{nbe}}\,[1+m_n(\alpha_{nbe}-1)]. \tag{8} \]

For \(\tau_0 \alpha=1\), the figure of merit of the stage with correction is equal to its figure of merit without correction, since

\[ \frac{K_{shob}}{K_{shok}}=\frac{t_{nk}}{t_{nb}}. \]

The boundary between the aperiodic and oscillatory character of the peak of the transient response is determined by the condition

\[ b_1^2=4b_2, \tag{9} \]

from which, substituting the values of \(b_1\) and \(b_2\) from (1б) and (1в), we find that the correction parameter is equal to

\[ m_{nk}= \left[ \frac{\sqrt{\alpha_{nb}[1+m_n(\alpha_{nbe}-1)]}}{\alpha_{nbe}} - \frac{\sqrt{\alpha_{nb}[1+m_n(\alpha_{nbe}-1)]-\alpha_{nbe}[1+m_n(\alpha_{nb}-1)]}}{\alpha_{nbe}} \right]^2. \tag{10} \]

To speed up engineering calculations, formula (10) is represented graphically in Fig. 3.

For \(b_1^2=4b_2\), as is seen from (5a) and (5b), \(\alpha=\gamma=\dfrac{2}{b_1}\), and the operational expression of the transient response takes the form

\[ h(p)=\frac{1+pa_1}{(p+2/b_1)^2}. \tag{11} \]

This expression corresponds to the transient response [4]

\[ h(t)=1+\left(\frac{2a_1-b_1}{b_1^2}\,2t-1\right)e^{-2t/b_1}. \tag{12} \]

Analysis of this formula shows that the time corresponding to the maximum overshoot is

\[ t_{\max}=\frac{a_1 b_1}{2a_1-b_1}, \tag{13} \]

and the overshoot is

\[ \delta=\frac{2a_1-b_1}{b_1}\,e^{-\frac{2a_1}{2a_1-b_1}} . \tag{14} \]

Fig. 3. Dependence of the correction parameter (which provides the boundary between the aperiodic and oscillatory character of the transient response) on the feedback coefficient: solid line, \(\alpha_n b e=1\); dashed line, \(\alpha_n b e=2\).

Of interest is the transient response of the stage when conditions (3) and (9) are simultaneously satisfied, i.e., when \(m=m_{pk}=m_{ch}\). In this case \(b_1=\sqrt{2}\,a_1\), and expressions (12), (13), and (14) respectively take the form

\[ h(t)=1+\left(\frac{2-\sqrt{2}}{a_1}t-1\right)e^{-\frac{\sqrt{2}}{a_1}t}, \tag{12a} \]

\[ t_{\max}=\frac{\sqrt{2}}{2-\sqrt{2}}\,a_1, \tag{13a} \]

\[ \delta=(\sqrt{2}-1)e^{-\frac{2}{2-\sqrt{2}}}\simeq 1.4\%. \tag{14a} \]

In this case the rise time of the leading edge of the pulse is

\[ t_{npk}=0.955\,\tau_{nf}m . \tag{15} \]

From (3) and (9) we find that

\[ m=\frac{1+m_p(\alpha_{nbt}-1)}{\sqrt{2}\alpha_{nbt}-\alpha_{nbe}} =\frac{2}{\alpha_{nbt}}\,[1+m_p(\alpha_{nbe}-1)]. \tag{16} \]

Whence the required feedback coefficient is

\[ \alpha_{nbt}= \frac{[(2\sqrt{2}-1)(1-m_p)+2\sqrt{2}\alpha_{nbe}m_p+1]}{2m_p} \]

\[ +\frac{ \sqrt{\left[(2\sqrt{2}-1)(1-m_p)+2\sqrt{2}\alpha_{nbe}m_p\right]^2 -8\alpha_{nbe}m_p[1+m_p(\alpha_{nb}-1)]} }{2m_p}. \tag{17} \]

Formula (17) is shown graphically in Fig. 4.

Fig. 4. Relationship between the feedback coefficient and the parameter \(m_p\) under the condition \(m=m_{pk}=m_t\).

For \(\alpha_{nbe}m_p \ll 1\), which occurs rather often,

\[ \alpha_{nbt}=\frac{2\sqrt{2}-1}{m_p}. \tag{18} \]

For \(\alpha_{nbe}\simeq 1\), expression (16) is written in the form

\[ m=\frac{2}{\alpha_{nbt}}. \tag{19} \]

The gain in rise time under the condition \(m=m_{pk}=m_t\), in comparison with the condition \(m=m_p\), is found to be equal to 15%.

As is seen from Fig. 4, for \(m_p \ll 1\) simultaneous fulfillment of conditions (3) and (9) is difficult, since this requires excessively deep negative feedback \((\alpha_{nbt}>20)\).

Experiment

In order to verify the theoretical results, an experiment was carried out.

Table 1

The circuit of the experimental setup is shown in Fig. 5. The 103-I amplifier makes it possible to provide sufficient beam deflection on the screens of the IPKh-1 and IChKh-1 when the stage operates in the small-signal amplification mode \((U_{\text{out max}} \approx 0.5\ \text{V})\). The input capacitance of the 103-I amplifier, together with the wiring capacitance, is about \(40\ \text{pF}\), i.e., \(C_{\text{n}} = 40\ \text{pF}\).

Fig. 5. Circuit of the experimental setup.

For specified \(R_2(R_b)\), \(R_{\text{n}}\), and \(R_0\), the correcting capacitance \(C_0\) was selected so as to provide the widest passband without overshoot on the IChKh-1, or an overshoot in the transient response \((1 \div 3\%)\) on the IPKh-1.

The experiment was carried out with a triode of type P403, which had the following parameters: \(\beta_0 = 62\), \(r_b = 100\ \Omega\), \(r_e = 12\ \Omega\), \(r_k = 157\ \text{k}\Omega\), \(C_k = 5\ \text{pF}\), \(\tau_z = 0.247\ \mu\text{s}\).

The parameters were measured with the triode operating at \(I_k = 2\ \text{mA}\) and \(U_{ke} = -5\ \text{V}\). This same dc operating mode of the triode was maintained during the experiment as well.

The results of the calculation and the experiment are given in Table 1. As can be seen from the table, the maximum discrepancy between the calculated and experimental results does not exceed 20%.

For clarity, calculated values of the correction parameter \(m_i\) for a purely active load \((C_{\text{n}} = 0)\) are also given here. In the calculation, the correction parameter \(m_i\) was found from formula (4).

REFERENCES

1. T. M. Agakhanyan, “Radio engineering, reduction of distortion of pulse fronts in video amplifiers using planar triodes,” 11, 9, 46, 1956.

2. K. S. Rzhevkin, E. S. Andrianov, “Radio engineering and electronics. Correction of amplifiers using semiconductor triodes,” 2, 9, 1157, 1957.

3. Ya. K. Trokhimenko, “Radio engineering. Feedback in circuits with crystal triodes,” 11, 9, 46, 1956.

4. V. L. Ditkin, P. I. Kuznetsov, Handbook of Operational Calculus. Gostekhizdat, 1951.