Low-Frequency Selective Amplifier Based on Semiconductor Triodes (PT)

A. A. Kuzmin, G. V. Kryukova

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

This article gives a description of a selective \(RC\)-amplifier based on semiconductor triodes (Fig. 1), using a double T-shaped bridge as negative feedback.

Fig. 1.

It is known [1] that the complex gain of a selective \(RC\)-amplifier is equal to

\[ \dot K_f=\frac{\dot K_0(1-\dot\beta)}{1-\dot K_0\dot\beta}, \tag{1} \]

where \(\dot K_0\) is the complex gain of the amplifier,
\(\dot\beta\) is the complex transmission coefficient of the feedback circuit.

In analyzing a selective \(RC\)-amplifier based on PT, it is necessary to take into account the comparatively small input resistance of the amplifier and the shunting of the load by the input resistance of the filter.

The complex transmission coefficient of the filter is equal to

\[ \dot\beta= \frac{-Y_{21}}{Y_{22}+\dfrac{1}{R_{\text{in}}}} = \frac{(1-v^2)R_{\text{in}}} {R_{\text{in}}(1-v^2+j4v)+2R(1+jv)}. \tag{2} \]

where \(R\) is the resistance of the filter, \(R_{\mathrm{in}}\) is the input resistance of the amplifier,

\[ \nu=\frac{\omega}{\omega_0}=\frac{\omega_0+\Delta\omega}{\omega_0}=1+\xi . \]

The transfer coefficient of the emitter follower is approximately equal to 1; therefore the gain is equal to the gain of the common-emitter stage,

\[ K_0= \frac{-\alpha \dot Z_n}{r_e+r_b(1-\alpha)} = -\frac{\alpha\displaystyle\frac{R_n Z_{\mathrm{in.f}}}{R_n+Z_{\mathrm{in.f}}}}{a}, \tag{3} \]

where \(r_e, r_b, \alpha\) are transistor parameters, \(R_n\) is the load resistance,

\[ Z_{\mathrm{in.f}}=-j\,\frac{R}{2\nu}(1+j\nu) \]

is the input complex impedance of the filter.

Substituting expressions (2) and (3) into (1) and carrying out the corresponding transformations, we obtain

\[ \dot K_f= \frac{2R^2K_0}{b}\cdot \frac{ \left(1+\dfrac{R_n}{R}\right)-j\dfrac{R_n}{R} }{ \left\{1-\dfrac{R_{\mathrm{in}}}{d}\,2\xi\left[\dfrac{R}{2}-\dfrac{2K_0R_nRR_{\mathrm{in}}}{b}\right]\right\} + } \]

\[ \frac{ \left(1+\dfrac{R_n}{R}\right)-j\dfrac{R_n}{R} }{ +j\,\dfrac{R_{\mathrm{in}}}{d}\,2\xi\left\{ \dfrac{K_0R_nR^2}{b} + \dfrac{2R_{\mathrm{in}}+R}{2} \left[1+\dfrac{2K_0R(R+R_n)}{b}\right] \right\} }, \tag{4} \]

where

\[ K_0=\frac{\alpha}{a}R_n,\qquad b=R^2+(2R_n+R)^2,\qquad d=R^2+(2R_{\mathrm{in}}+R)^2, \]

\[ R_{\mathrm{in}}\simeq \frac{ a+r_b\dfrac{R_n}{r_k} }{ (1-\alpha)\left[1-\alpha+\dfrac{R_n}{r_k}\right] }. \]

For a sufficiently high \(Q\), it may be assumed that the gain will have a maximum at \(\xi=0\), i.e.,

\[ \left|K_{f0}\right|= \frac{K_0}{ \sqrt{1+2\dfrac{R_n}{R}\left(1+\dfrac{R_n}{R}\right)} }. \tag{5} \]

The resonance curve calculated for \(R=8\ \text{k}\Omega\), \(C=0.5\ \mu\text{F}\), \(R_n=1.5\ \text{k}\Omega\), \(\alpha=0.98\), \(r_b=400\ \Omega\), \(r_e=5\ \Omega\), and \(r_k=500\ \text{k}\Omega\), and for a resonance frequency of 40 Hz, is shown in Fig. 2 by the dashed line.

From formula (4) we find an expression for the quality factor of the amplifier.

\[ Q = \frac{1}{ \left[ \left(\frac{R}{R_{\text{in}}}\right)^2 + \left(2+\frac{R}{R_{\text{in}}}\right)^2 \right] } \cdot \left\{ \frac{ K_0 \dfrac{R_n}{R_{\text{in}}} }{ 1+\left(2\dfrac{R_n}{R}+1\right)^2 } + \left(1+\frac{1}{2}\frac{R}{R_{\text{in}}}\right) \left[ 1+ \frac{ 2K_0\left(1+\dfrac{R_n}{R}\right) }{ 1+\left(1+2\dfrac{R_n}{R}\right)^2 } \right] \right\}. \tag{6} \]

Analyzing (6), it can be established that, first, when \(R\) and \(R_{\text{in}}\gg R_n\), the quality factor is

\[ Q=\frac{K_0+1}{4}, \]

i.e., the same as for \(RC\) amplifiers using electron tubes;

Fig. 2.

Fig. 3.

second, as the load resistance is increased, the quality factor first increases and then begins to fall owing to the slowing of the increase in the gain and the decrease in the input resistance of the amplifier. The dependence \(Q=f(R_n)\), calculated

for the above-indicated parameters of the triode and resonant frequency, is shown in Fig. 3 by the dashed line.

For an experimental verification of the theoretical results, a selective \(RC\) amplifier was assembled using P15-type triodes, as shown in Fig. 1. The input signal was supplied from a ZG-12 audio oscillator with an output resistance of 200 ohms. The input and output signals were measured with an LV-9 vacuum-tube voltmeter. The double T-shaped bridge was tuned to a frequency of 40 Hz.

The operating mode of the common-emitter stage was determined by the collector-circuit supply voltage, equal to \(-10\) V, and by the collector current, \(2\) mA. At this operating point the triode had the above-indicated parameters, calculated from the static characteristics.

The dependence of the figure of merit on the load resistance was measured (Fig. 3). This dependence has a maximum \(Q = 28\) at a load resistance \(R_n = 1.5\ \text{k}\Omega\). In Fig. 2 the solid line shows the experimental frequency response at the optimum load resistance.

As can be seen from these figures, the experimental and theoretical dependences agree to within 25%, which is quite acceptable for circuits using semiconductor triodes. The gain of the amplifier was 120.

As experiment and theory show, a selective \(RC\) amplifier based on semiconductor triodes possesses high selectivity and a high gain and can successfully replace a similar amplifier using electron tubes, which will substantially reduce its dimensions and increase the economy and durability of operation.

References

1. Kozina O. G., Frantsuzov A. A., On selective amplifiers. Radiotekhnika, 13, No. 12, 1958.

2. Saakov E. O., Theory and Calculation of Selective \(RC\) Systems. Gosenergoizdat, 1954.

3. Aseev B. N., Fundamentals of Radio Engineering. Svyazizdat, 1947.