Proceedings
of the Tomsk Order of the Red Banner of Labor Polytechnic Institute
named after S. M. Kirov

Vol. 105　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1960

Correction of Distortions of Flat Tops of a Periodic Pulse Sequence in Video Amplifiers Based on Planar Semiconductor Triodes (PST)

I. N. Pustynskii

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

Introduction

Figure 1 shows the equivalent circuit of an amplifier stage based on a PST with a common emitter. Here the circuit \(R_{\phi} C_{\phi}\) is intended for correcting distortions of the pulse top caused by the presence of the transition capacitance \(C_p\),

\[ R_1=\frac{R_b \cdot R_{v.x}}{R_b+R_{v.x}}, \]

where \(R_b\) is the resistance providing bias to the base of the triode of the following stage, whose input resistance at low frequency is equal to \(R_{v.x}\).

It is assumed that capacitance in the emitter circuit of the subsequent stage is absent, since in amplifying pulses of long duration it is inexpedient to include it because of the considerable size of the capacitor. If it is necessary to include this capacitance, its influence can be taken into account by introducing an equivalent value of the time constant of the transition circuit [2].

In contrast to tube amplifiers, in amplifiers based on semiconductor triodes (ST) the values of the resistances \(R_1\) and \(R_n\) are commensurable, as a result of which the correction formulas for tube amplifiers lose their validity here.

Expression for the Shape of Pulses of a Periodic Sequence at the Amplifier Output

The normalized transient characteristic for the circuit in Fig. 1 has the form

\[ h(t)=\frac{1}{\gamma-\alpha}\left[(a_0-\alpha)e^{-\alpha t}-(a_0-\gamma)e^{-\gamma t}\right], \tag{1} \]

where, for \(R_{vy.x} \gg R_n+R_{\phi}\) and \(R_{vy.x} \gg R_1\) (which is usually the case)

\[ a_0=\frac{1+d}{d\cdot \tau_H} \tag{1а} \]

\[ \tau_H=R_H\cdot C_\phi \tag{1б} \]

\[ \alpha=\frac{a_1-\sqrt{a_1^2-4a}}{2a} \tag{1в} \]

\[ \gamma=\frac{a_1+\sqrt{a_1^2-4a}}{2a} \tag{1г} \]

\[ a=d\tau_H\cdot\tau_n\cdot(1+\kappa_1) \tag{1д} \]

\[ \tau_n=C_n\cdot R_1 \tag{1е} \]

\[ d=\frac{R_\phi}{R_H} \tag{1ж} \]

\[ \kappa_1=-\frac{R_H}{R_1} \tag{1з} \]

\[ a_1=d\tau_H+\tau_n+\tau_n\cdot\kappa_1(1+d) \tag{1и} \]

Fig. 1.

In accordance with the superposition principle [1], for the output signal we obtain the expression

\[ i_2(t)=K_{io}\cdot\frac{1}{\gamma-\alpha}\cdot \left\{ (a_0-\alpha)e^{-\alpha t} \left[ (1-e^{\alpha t_u})\cdot(1+e^{\alpha T}+e^{2\alpha T}+\ldots+ \right.\right. \]

\[ \left. +e^{(n-1)\alpha T})+e^{n\alpha T} \right] -(a_0-\gamma)e^{-\gamma t} \left[ (1-e^{\gamma t_u})\cdot(1+ \right. \]

\[ \left.\left. +e^{\gamma T}+e^{2\gamma T}+\ldots+e^{(n-1)\gamma T})+e^{n\gamma T} \right] \right\}, \tag{2} \]

where

\(t_u\) is the pulse duration,
\(T\) is the pulse repetition period,

\[ K_{io}=\frac{\beta R_H}{R_1+R_H} \]

is the current gain of the stage at medium frequencies.

After \(n\) periods, i.e., in the interval from \(t=nT\) to \(t'=nT+t_u\), the output signal is equal to

\[ i_2(t_1)=K_{io}\cdot\frac{1}{\gamma-\alpha}\cdot \left\{ (a_0-\alpha)\cdot \left[ \frac{1-e^{\alpha t_u}}{e^{\alpha T}-1}\cdot \left(e^{-\alpha t_1}-e^{-\alpha(nT+t_1)}\right)+e^{-\alpha t_1} \right] \right. \]

\[ \left. -(a_0-\gamma)\cdot \left[ \frac{1-e^{\gamma t_u}}{e^{\gamma T}-1}\cdot \left(e^{-\gamma t_1}-e^{-\gamma(nT+t_1)}\right)+e^{-\gamma t_1} \right] \right\}, \tag{3} \]

where \(t_1\) varies from 0 to \(t_u\).

In the steady-state regime, since \(e^{-\alpha(nT+t_1)}\simeq e^{-\gamma(nT+t_1)}\simeq 0\), the output signal is equal to

\[ i_2(t)=K_{io}\frac{1}{\gamma-\alpha}\cdot \left[ (a_0-\alpha)\cdot e^{-\alpha t}\cdot \frac{e^{\alpha T}-e^{\alpha t}}{e^{\alpha T}-1} - (a_0-\gamma)\cdot e^{-\gamma t}\cdot \frac{e^{\gamma T}-e^{\gamma t}}{e^{\gamma T}-1} \right]. \tag{4} \]

Here, for convenience, we omit the subscript on \(t_1\).

Since the maximum distortions of the top of the pulse are observed at \(t_u=0.5T\) [1], this case is of greatest interest. For \(t_u=0.5T\), formula (4) may be written in the form

\[ i_2(t)=K_{i0}\frac{1}{\gamma-\alpha}\left[(a_0-\alpha)\frac{e^{-\alpha t}}{1+e^{-\alpha t_u}}-(a_0-\gamma)\frac{e^{-\gamma t}}{1+e^{-\gamma t_u}}\right], \tag{5} \]

where \(0\leq t\leq t_u\).

Optimal correction parameters

We shall consider as optimal those parameters of the amplifier circuit which ensure at its output a signal symmetrical in height (Fig. 2) with the minimum possible top distortions.

Fig. 2.

In practical calculations, the problem of determining the optimal parameters is reduced mainly to finding the values of the capacitances \(C_p\) and \(C_\phi\). The values of the resistances \(R_1\), \(R_n\), and \(R_\phi\) are usually known; however, the value \(R_\phi\) may be varied within certain limits.

The first equation for determining \(C_p\) and \(C_\phi\) is obtained from expression (5), if one requires that the equality \(i_2(0)=i_2(t_u)\) be satisfied. This equation has the form

\[ a_0= \frac{ \gamma\,\operatorname{th}\dfrac{\gamma t_u}{2} - \alpha\,\operatorname{th}\dfrac{\alpha t_u}{2} }{ \operatorname{th}\dfrac{\gamma t_u}{2} - \operatorname{th}\dfrac{\alpha t_u}{2} }. \tag{6} \]

Since in real cases usually \(\dfrac{\alpha t_u}{2}<1\) and \(\dfrac{\gamma t_u}{2}<1\), equation (6), after expanding the hyperbolic tangents in power series, may be simplified. Then

\[ a_0\simeq(\gamma+\alpha) \frac{ 1-\dfrac{t_u^2}{12}(\gamma^2+\alpha^2) }{ 1-\dfrac{t_u^2}{12}(\gamma^2+\alpha\gamma+\alpha^2) }. \]

or, since \(\dfrac{t_u^2}{12}(\gamma^2+\alpha\gamma+\alpha^2)\ll1\),

\[ a_0\simeq(\gamma+\alpha)\left(1+\frac{t_u^2}{12}\gamma\alpha\right). \tag{6a} \]

After substituting the circuit parameters into equation (6a), it assumes the form

\[ \frac{\tau_H}{\tau_n}= \frac{ d(1+k_1)-\frac{1}{12}\left(\frac{t_u}{\tau_n}\right)^2 + \sqrt{ \left[d(1+k_1)-\frac{1}{12}\left(\frac{t_u}{\tau_n}\right)^2\right]^2 -\frac{1}{3}\left(\frac{t_u}{\tau_n}\right)^2(1+k_1)\left[1+k_1(1+d)\right] } }{ 2d(1+k_1) }. \tag{7} \]

For \(\dfrac{t_u}{\tau_n}\ll d\), formula (7) can be considerably simplified:

\[ \frac{\tau_H}{\tau_n}\simeq 1-\frac{1}{12}\left(\frac{t_u}{\tau_n}\right)^2\frac{1+d}{d^2}. \tag{7a} \]

Using formulas (7) and (7a), curves have been plotted in Fig. 3 for the most frequently used values of \(k_1\), \(d\), and \(\dfrac{t_u}{\tau_n}\). It is evident from the curves that, for

\[ \frac{t_u}{\tau_n}\ll d, \]

the maximum error in determining \(\dfrac{\tau_H}{\tau_n}\) by the approximate formula (7a), as compared with the more exact formula (7), does not exceed 3--10 percent.

The curves of Fig. 3 can be used for engineering calculations.

Hereafter, condition (7) will be called the correction condition.

The second equation for determining \(C_n\) and \(C_\phi\) can be obtained from the condition that the permissible distortions be present, as determined by the quantity \(\Delta i=i_{\max}-i_0\) (Fig. 2). The magnitude of the relative distortion of the pulse top, measured with respect to the mean line, is equal to

\[ \delta=\frac{\Delta i}{i_0}=2i_{\max}-1, \tag{8} \]

since \(i_0=0.5\).

The quantity \(i_{\max}\) is readily found from expression (5) if, in place of \(t\), one substitutes \(t_{\max}\), the time corresponding to the maximum distortion of the top.

As in tube amplifiers [1], without appreciable error it may be assumed that \(t_{\max}=t_u/2\). Then

Fig. 3. Dependence of \(\dfrac{\tau_H}{\tau_n}\) on \(\dfrac{t_u}{\tau_n}\) for various \(k_1\) and \(d\).

\[ \begin{array}{ll} \text{solid curves} & k_1=0.1,\\ \text{dashed curves} & k_1=2.0, \end{array} \left\} \begin{array}{l} \text{curves calculated}\\ \text{from formula (7);} \end{array} \right. \]

\[ \text{dash-dot curves calculated from formula (7a).} \]

\[ i_{\max}=K_{i0}\, \frac{ \operatorname{sh}\dfrac{\gamma t_u}{2} - \operatorname{sh}\dfrac{\alpha t_u}{2} }{ 2\operatorname{sh}\left(\dfrac{\gamma t_u}{2}-\dfrac{\alpha t_u}{2}\right) } \tag{9} \]

and

\[ \delta= \frac{ \operatorname{sh}\dfrac{\gamma t_u}{2} - \operatorname{sh}\dfrac{\alpha t_u}{2} }{ \operatorname{sh}\left(\dfrac{\gamma t_u}{2}-\dfrac{\alpha t_u}{2}\right) } -1. \tag{10} \]

After expansion of the hyperbolic sines, formula (10) takes the form

\[ \delta \approx \frac{ 1+\dfrac{t_u^2}{24}\left(\gamma^2+\alpha\gamma+\alpha^2\right) }{ 1+\dfrac{t_u^2}{24}(\gamma-\alpha)^2 } -1 \]

or, since

\[ \frac{t_u^2}{24}(\gamma-\alpha)^2 \ll 1, \]

\[ \delta \approx \frac{1}{8}\,t_u^2\gamma\alpha . \tag{10a} \]

After expressing \(\alpha\) and \(\gamma\) through the circuit parameters, formula (10a) is written in the form

\[ \delta \approx \frac{1}{8}\cdot \left(\frac{t_u}{\tau_n}\right)^2 \frac{1}{ \dfrac{\tau_H}{\tau_n}\cdot d(1+k_1) } \]

or

\[ \delta \approx \left(\frac{t_u}{\tau_n}\right)^2 \cdot \frac{12.5\%}{ \dfrac{\tau_H}{\tau_n}\cdot d(1+k_1) }, \tag{11} \]

i.e., in amplifiers using PTs the distortion \(\delta\) is \((1+k_1)\) times smaller than in tube amplifiers [1].

Fig. 4a.

To speed up engineering calculations using formula (11), graphs were also constructed (Fig. 4). In Fig. 4b, moreover, for a visual assessment of the correction effect, curves are given

\[ \Delta = 200\%\, \operatorname{th}\frac{t_u}{2\tau_n(1+\kappa_1)} \simeq 100\%\,\frac{t_u}{\tau_n(1+\kappa_1)}, \tag{12} \]

where \(\Delta\) is the relative droop of the pulse top in the absence of the \(R_{\phi}C_{\phi}\) network.

Fig. 4b.

Fig. 4v.

Quantitatively, the effect achieved by the correction can be estimated from the magnitude of the ratio of the distortion without correction to the distortion with correction, i.e.,

\[ \varepsilon_k=\frac{\Delta}{\delta}=8\,\frac{d\tau_n}{t_u}. \tag{13} \]

From (13) it is evident that the correction is more effective for smaller \(t_i\) and for larger \(d\) and \(\tau_n\). For example, at \(d = 1.5\), \(\dfrac{t_i}{\tau_n} = 1\), \(\dfrac{\tau_n}{\tau_p} = 0.9\) (see Fig. 3), \(\sigma_k = 10.8\), whereas at \(d = 0.75\), \(\dfrac{t_i}{\tau_n} = 1\), \(\dfrac{\tau_n}{\tau_p} = 0.65\), \(\sigma_k = 3.9\).

Experiment

To verify the theoretical results, an experiment was carried out. The circuit of the experimental setup is shown in Fig. 5. The bias to the base and collector of the amplifier triodes was supplied from separate sources \(E_b\) and \(E_k\), so that the voltages could be varied independently.

From a GIP-1 rectangular-pulse generator, pulses with duty factor \(Q = 2\) and duration \(t_i = 8\ \mu\text{s}\) were applied to the base of the first amplifier stage through a resistance \(R = 10\ \text{k}\Omega\). From the amplifier the pulses were fed to an EO-7 oscilloscope. For specified \(R_{\text{in}}\), \(C_p\), \(k_1\), and \(d\), the capacitance \(C_{\phi k}\) was selected so as to satisfy the correction condition, and the magnitude of the top distortion was measured. The results obtained by calculation and experimentally are given in Table 1. As is evident from the table, the discrepancy between calculation and experiment is insignificant.

Fig. 5. Circuit of the experimental setup. Circuit parameters: \(R = 10\ \text{k}\Omega\), \(R_{b1} = 130\ \text{k}\Omega\), \(R_{b2} = 75\ \text{k}\Omega\), \(R_1 = 12\ \text{k}\Omega\), \(R_e = 680\ \Omega\), \(C_1 = 50\ \mu\text{F}\), \(E_{b\,\text{avg}} = 4\ \text{V}\), \(E_{k\,\text{avg}} = 10\ \text{V}\).

The input resistance of the second stage \(R_{\text{in}}\) was varied by changing the resistance \(R_e\). In this case \(R_{\text{in}}\) was measured as follows.

A specified resistance \(R_1\) was connected, and the pulse voltage at the base of the triode was measured with the emitter circuit open. Then the emitter circuit was closed and, by varying \(R_e\), the voltage at the base of the triode was reduced by a factor of two in comparison with the voltage with the circuit open. In this case \(R_{\text{in}} = R_1\).

The capacitance \(C_1\) was intended to prevent any change in the dc operating mode of the triode when \(R_1\) was connected.

When \(E_b\) was increased, a certain droop of the pulse top was observed, evidently due to a decrease in \(R_{\text{in}}\). In the presence of the resistance \(R_e\) (even of the order of \(100\ \Omega\)), because of negative feedback, the disturbance of the correction conditions was substantially reduced both when replacing triodes and when changing \(E_b\).

When \(E_k\) was changed, the correction conditions were not noticeably disturbed, evidently because a change in \(U_k\) has little effect on the input resistance of the triode.

Procedure for Engineering Calculation of the Correction Parameters

Calculation from graphs

1. From the known values \(k_1\), \(d\), and \(\delta\), using the graphs in Fig. 4, determine the permissible value

\[ \frac{t_i}{\tau_n} = l_1 . \]

Table 1

1. We calculate the required value of the coupling capacitance

\[ C_p=\frac{\tau_p}{R_1}=\frac{t_u}{l_1 R_1} \]

and take the nearest larger standard value \(C_{p\,n}\).

1. We determine the actual ratio

\[ \frac{t_u}{\tau_p}=\frac{t_u}{R_1 C_{p\,n}}=l_{1n}. \]

1. From the graphs in Fig. 3, for known values of \(\dfrac{t_i}{\tau_p}\), \(d\), and \(\kappa_1\), we find the ratio

\[ \frac{\tau_n}{\tau_p}=l_2. \]

1. We calculate the capacitance \(C_{\phi k}\) that satisfies the correction condition,

\[ C_{\phi k}=\frac{\tau_n}{R_n}=\frac{l_2\tau_p}{R_n}, \]

and take the nearest standard value \(C_{\phi n}\). If \(C_{\phi n}\) differs appreciably from \(C_{\phi k}\), then it is desirable to change \(R_\phi\) somewhat (i.e., \(d\)) so that \(C_{\phi n}\simeq C_{\phi k}\), or, in the remaining stages of the amplifier, when selecting \(C_{\phi n}\), to allow for the overcorrection (or, correspondingly, undercorrection) in this stage.

Analytical Calculation

When

\[ \frac{t_i}{\tau_p}\ll d, \]

there is no need to use the graphs to determine the correction parameters. The calculation procedure is then as follows.

1. From the known \(\kappa_1\), \(d\), and \(\delta\) (%) we determine the ratio

\[ \frac{t_i}{\tau_p}=\sqrt{\frac{(1+\kappa_1)\delta\cdot d}{12.5}}=l_1. \tag{14} \]

Formula (14) is obtained by solving (7a) and (11) jointly. If \(l_1\) is greater than \(d\), then it is not advisable to use analytical calculation because of the large error.

1. From (14) we calculate

\[ C_p=\frac{\tau_n}{R_1}=\frac{t_i}{l_1R_1} \]

and take the nearest larger standard value \(C_{pn}\).

1. We determine the actual ratio

\[ \frac{t_i}{\tau_p}=\frac{t_i}{R_1C_{pn}}=l_{1n}. \]

1. We find the value of \(\dfrac{\tau_n}{\tau_p}\) from formula (7a). Subsequently we proceed as in item 5 of the calculation from the graphs.

Conclusions

1. Analysis of the graphs in Fig. 3 shows that, in order to ensure greater stability of correction when \(\tau_p\) changes, it is necessary to choose smaller \(\kappa_1\) and larger \(d\), while when \(\tau_n\) changes, the opposite is true.

Since the most unstable quantity is \(\tau_p\) (owing to the considerable change of \(R_{\mathrm{in}}\) with temperature), it is desirable to choose larger \(d\) and smaller \(\kappa_1\); in addition, it is expedient to stabilize \(R_{\mathrm{in}}\) [2].

1. The change in distortions \(\delta\) due to any factors is less noticeable at larger \(\kappa_1\) and \(d\) (Fig. 4).

2. The effect achieved by the correction is greater the larger \(d\) and \(\tau_n\) are and the smaller \(t_i\) is (see (13)).

3. Distortions of the pulse top are not critical with respect to changes in the voltage at the triode collector.

References

1. O. B. Lur’e, Video-Frequency Amplifiers, Sovetskoe Radio, 1955.

2. I. N. Pustynskii, “On correction of distortions of the flat top of a pulse in video amplifiers based on planar semiconductor triodes” (the article is included in this collection).