EFFECT OF EMITTER POSITIVE FEEDBACK ON THE NOISE FIGURE OF A LOW-FREQUENCY AMPLIFIER

V. E. STENNIKOV

(Presented by the scientific-technical seminar of the Scientific Research Institute of Automation and Electromechanics)

In designing weak-signal amplifiers it is important to reduce the noise of the input stage. The literature contains information on the possibility of using positive feedback (PFB) for this purpose. For example, in [1, pp. 65–66] it is indicated that the effect of the “emitter noise source” in a transistor can be reduced by using PFB. However, neither a circuit nor an analytical calculation in support of this conclusion is given. A brief note [2] reports the results of an experimental investigation of a single-stage amplifier in which the noise is substantially reduced by PFB introduced through a transformer from the emitter circuit of the transistor into the base circuit.

The purpose of the present article is to carry out an analytical investigation of the possibility of reducing the noise figure of an amplifier by means of PFB implemented as indicated in [2]. We first make the following observation. Since PFB changes the frequency response of the open-loop amplifier, investigation of its noise figure with and without feedback must be carried out with the same noise passband [3], determined, for example, by a filter connected at the amplifier output.

Fig. 1a

Figure 1a shows the equivalent circuit of the amplifier under study. The transistor noise is represented by the low-frequency Giacoletto circuit [4]. The noise figure is calculated by the signal-flow-graph method [5, 6]. In the subsequent analysis the following restrictions are adopted:

Since the calculation is carried out for the low-frequency region, we shall not take into account the effect of the capacitances of the emitter and collector junctions of the transistor and the parasitic capacitance at the amplifier input;

we neglect the noise of the collector load;

the transformer is assumed to be ideal and noiseless.

Figure 1b shows the signal graph of Fig. 1a. The transformation ratios are denoted by \(n\) and \(n_1\): \(n=u_2/u_1\), \(n_1=u_3/u_1\). Determining the output responses from each independent noise source \((\bar e_r, \bar e_b, \bar i_b, \bar i_k)\), we find the expression for the noise figure of the amplifier

Fig. 1b

\[ F=1+\gamma\left\{ \frac{r_b}{R_g} + \frac{(1+\alpha)r_b^2\left[1+\frac{R_g}{r_b}\cdot\frac{1}{\gamma}\right]^2}{2r_eR_g} + \frac{r_b^2}{2\alpha R_g r_e} \left[ \frac{r_e}{r_b}+(1-\alpha) \left(1+\frac{R_g}{r_b}\cdot\frac{1}{\gamma}\right)^2 \right] \right\}, \tag{1} \]

where

\[ \gamma=1/(n-n_1)^2. \tag{2} \]

A preliminary study of equations (1) and (2) makes it possible to state that, in the amplifier under consideration, no noise source can be suppressed by positive feedback, since expression (1) contains no terms with a negative sign. The minimum value of the noise figure is obtained by choosing the optimum transformation ratios \(n\) and \(n_1\). For this purpose it is necessary to investigate the expression for \(F\) for a minimum with respect to the parameter \(\gamma\). Omitting the intermediate transformations, we give the results of the analysis:

\[ \gamma_{\mathrm{opt}}= R_g/\sqrt{\frac{r_e^2\beta+(r_e+r_b)^2+2r_e\cdot r_b\cdot\beta}{\beta}}, \]

\[ F_{\min}=1+\frac{r_b+r_e}{r_e\beta} + \sqrt{ \frac{1}{\beta} +\frac{2r_b}{r_e\beta} +\frac{(r_b+r_e)^2}{r_e^2\beta^2} }. \tag{3} \]

To draw final conclusions, one should compare (3) with the expression for the minimum noise figure calculated for the amplifier of Fig. 1a without a transformer. Setting in (1) \(n=1\), \(n_1=0\), after a series of transformations we arrive at the well-known expression obtained by Nielsen [7],

\[ F=1+\frac{r_b}{R_g}+\frac{r_e}{2R_g}+\frac{(r_e+r_b+R_g)^2}{2r_e\cdot R_g\cdot \beta}. \tag{4} \]

Examining (4) for a minimum with respect to \(R_g\), we find that the resulting expression for \(F_{\min}\) coincides exactly with (3). In other words, positive feedback (implemented according to the circuit of Fig. 1a) gives no noise advantage.

Conclusions

1. The use of positive feedback introduced through a transformer from the emitter circuit of the transistor into the base circuit does not make it possible to reduce the noise figure of the amplifier.

2. The reduction in the noise level obtained experimentally [2] can be explained both by a narrowing of the noise passband as the depth of positive feedback is increased, and by an improvement in noise matching.

References

1. Reinfelder, William A. Design of Low-Noise Input Circuits Using Transistors. “Svyaz,” 1967.

2. Positive, Negative Feedback Combine to Reduce Noise, “Electronic Design,” 1962, v. 10, V, No. 11.

3. Faulkner E. A. The Design of Low-noise Audio-frequency Amplifiers, “Radio and Electron. Engr.,” 1968, v. 36, No. 1.

4. Van der Ziel A. Fluctuation Phenomena in Semiconductors. Moscow, IL, 1961.

5. Mason S. J., Zimmerman G. J. Electronic Circuits, Signals, and Systems. Moscow, IL, 1963.

6. Abrahams J., Coverly J. Analysis of Electrical Circuits by the Graph Method. “Mir,” 1967.

7. Nielsen E. G. Behavior of Noise Figure in Junction Transistors, “Proc. IRE,” July, 1957.