ANALYTICAL DEPENDENCE OF THE INFLECTION FREQUENCY ON THE MACROSCOPIC PARAMETERS OF A TRANSISTOR

V. E. Stennikov

(Presented by the Scientific and Technical Seminar of the Scientific Research Institute of AEM)

Most of the formulas obtained below will be valid under the following assumptions:

the amplifier noise is determined by the noise of its first stage, in which the transistor is connected in a common-emitter circuit;

the internal resistance of the signal source is purely resistive and generates only thermal noise;

the \(1/f\)-type noise of the collector junction may be neglected [1, 2];

the noise of the input-stage load resistor and the shot noise of the reverse currents of the emitter and collector may be disregarded;

in the first stage of the amplifier, silicon bipolar transistors fabricated by planar technology are used as the amplifying element;

the transistor of the input low-noise stage operates in the microcurrent mode.

In the range of low and infralow frequencies, the noise properties of an amplifier in which, in addition to white noise, \(1/f\)-type noise is generated, are best characterized by the averaged noise factor (ANF). In [1] it was experimentally proved that \(1/f\)-type noise in silicon planar transistors is associated with processes in the volume of the emitter junction, and not with surface phenomena.

In [3], on the basis that the \(1/f\)-type noise in the emitter junction is determined by the recombination component of the current flowing through the forward-biased \(p\)—\(n\) junction, a formula is obtained for the equivalent noise current generator, namely

\[ \overline{i_\phi^{\,2}} \simeq 10^{-10}\cdot I_{\mathrm{e}}^{0}\cdot I_{\mathrm{e}}\cdot \frac{df}{f}, \tag{1} \]

where \(I_{\mathrm{e}}^{0}\) is the limiting emitter current, a parameter characterizing the transistor. The indicated noise generator in the low-frequency equivalent noise circuit of the transistor proposed by Giacoletto should be connected between the internal base point of the transistor and the emitter terminal. The limiting emitter current according to [3]

\[ I_{\mathrm{e}}^{0}\approx\left[2(\alpha_1-\alpha_2)/(1/\sqrt{I_{\mathrm{e}2}}-1/\sqrt{I_{\mathrm{e}1}})\right]^2, \tag{2} \]

where \(\alpha_1\) (\(\alpha_2\)) is the transistor ac current gain in the common-base circuit at emitter current \(I_{\mathrm{e}1}\) (\(I_{\mathrm{e}2}\)). Formulas (1), (2) were derived for low emitter injection levels (approximately for currents \(I_{\mathrm{e}}<0.1\text{–}1\ \mathrm{mA}^{*}\)).

The frequency separating the region of white noise from the region of \(1/f\)-type noise is called the inflection frequency. A more rigorous definition of the inflection frequency will be given below. In the region of \(1/f\)-type noise, the spectral density of transistor noise obeys a \(1/f^\lambda\) law. The exponent \(\lambda\) usually varies within the range \(0.9\text{–}1.4\). Experimental studies [4, 5] show that the inflection frequency for silicon planar transistors operating in the microcurrent regime depends very strongly on the emitter current, decreasing as \(I_{\mathrm{e}}\) is reduced. This regularity, which is very important for practice, is also indirectly reflected in (1).

In general form, the averaged noise factor is defined by the expression [6]

\[ \overline{F}=\int_{0}^{\infty}F\cdot G\,df\bigg/\int_{0}^{\infty}G\cdot df, \tag{3} \]

where \(F\) is the differential noise factor, or simply the noise factor;

\(G\) is the power gain of the amplifier.

Assume that \(G\) does not depend on frequency, and specify the limits of integration in (3); then

\[ \overline{F}=\frac{1}{f_{\mathrm{hi}}-f_{\mathrm{lo}}}\int_{f_{\mathrm{lo}}}^{f_{\mathrm{hi}}} F\cdot df, \tag{4} \]

where \(f_{\mathrm{hi}}\), \(f_{\mathrm{lo}}\) are, respectively, the upper and lower limiting frequencies of the amplifier noise passband. There is no theoretical expression in the known literature for \(F\) that covers both the white-noise region and the \(1/f\)-type noise region. We shall use Verster’s empirical formula [7]

\[ F-1=(F_0-1)\left[1+\left(\frac{f_c}{f}\right)^\lambda\right], \tag{5} \]

where \(F_0\) is the amplifier noise factor in the region of a uniform spectrum,

\(f_c\) is the inflection frequency.

According to (5), the inflection frequency can be defined as follows. If \(f=f_c\), then \(F-1=2(F_0-1)\). Thus the inflection frequency is the frequency, specified in the \(1/f\)-type noise region, at which the additional noise factor increases by \(3\ \mathrm{dB}\) (by a factor of two) in comparison with its value in the white-noise region. Solving (4) and (5) together, we obtain

\[ \frac{\overline{F}-1}{F_0-1} = 1+\frac{1}{f_{\mathrm{hi}}-f_{\mathrm{lo}}} \int_{f_{\mathrm{lo}}}^{f_{\mathrm{hi}}}\left(\frac{f_c}{f}\right)^\lambda\cdot df. \tag{6} \]

* More correctly, the measure of the injection level should be defined through the emitter-current density.

Let us find an expression for the inflection-frequency coefficient through the macroscopic parameters of the transistor. Figure 1,a shows the equivalent noise circuit of the input transistor stage. The transistor is connected in a common-emitter configuration, and its noise is represented by the Giacoletto equivalent circuit [8]. The expression for \(\overline{i_\phi^{\,2}}\) is given in (1). The capacitance \(C_\mathrm{e}\) is a charging capacitance; it does not participate in the transistor transfer mechanism [9], and therefore the generator \(a i_\mathrm{e}^{\,\prime}\) depends only on the current flowing through \(r_\mathrm{e}\).

From the signal graph (Fig. 1,b), after a number of transformations we determine the inflection-frequency coefficient:

\[ \overline{F} \approx 1+\frac{r_\mathrm{b}}{R_\mathrm{g}} +\frac{(R_\mathrm{g}+r_\mathrm{b})^2}{2R_\mathrm{g}r_\mathrm{e}\alpha} \cdot \left[ (1-\alpha)^2 \left( 1+\frac{2r_\mathrm{e}}{R_\mathrm{g}+r_\mathrm{b}} \right) + \frac{r_\mathrm{e}^{\,2}}{(R_\mathrm{g}+r_\mathrm{b})^2} \right] \cdot \left[ 1+\frac{\omega C_0^{\,2}r_\mathrm{e}^{\,2}}{2(1-\alpha)^2} \right] + R_\mathrm{g}I_\mathrm{e}\varepsilon', \tag{7} \]

Fig. 1. Equivalent noise circuit of the input stage (a) and its signal graph (b)

where

\[ C_0=C_\mathrm{e}+C_\mathrm{k}, \qquad \varepsilon'=6\cdot 10^9 \cdot I_\mathrm{e}^{0}\cdot \ln(f_{\mathrm{vsh}}/f_{\mathrm{nsh}})/(f_{\mathrm{vsh}}-f_{\mathrm{nsh}}). \tag{8} \]

The last term in (7) characterizes the \(1/f\)-type noise of the transistor. If the inequality

\[ r_\mathrm{e} \ll \sqrt{2}\,\frac{1-\alpha}{\omega C_0}, \tag{9} \]

is satisfied,

then in (7) only the last term depends on frequency. Since the value of the distributed base resistance is \(r_b=50\div 200\ \Omega\), and the optimum generator resistance in the microcurrent mode \((I_e \ll 100\ \mu\mathrm{A})\) is usually greater than one kilohm, it may be assumed that

\[ R_g \gg r_b . \tag{10} \]

Taking (9) and (10) into account, we transform (7) to the form

\[ \overline{F}\approx 1+\frac{r_b}{R_g}+\frac{1}{\beta} +\frac{r_e}{2R_g} +\frac{R_g}{2r_e\cdot \beta} +R_g\cdot I_e\cdot \varepsilon' . \tag{11} \]

Noting that the sum of the 2nd, 3rd, 4th, and 5th terms of the last expression characterizes the additional noise factor \((F_0-1)\) in the region of the uniform spectrum, we reduce (11) to the form

\[ \frac{\overline{F}-1}{F_0-1} = 1+ \frac{\varepsilon R_g/2r_e}{ \dfrac{1}{\beta_i} +\dfrac{R_g}{2\beta r_e} +\dfrac{r_e}{2R_g} +\dfrac{r_b}{R_g} }, \tag{12} \]

Fig. 2. Dependence of the inflection frequency on the emitter current of the input transistor; ⊙—experiment for transistor No. 2; △—No. 27; □—No. 36.

Fig. 3. Dependence of the inflection frequency on the emitter current of the input transistor; ⊙—experiment for transistor No. 4; □—No. 51.

where \(\varepsilon\), taking (8) into account, is equal to

\[ \varepsilon=2I_e\cdot r_e\cdot \varepsilon' \approx 31\cdot 10^7\cdot I_e^0\cdot \ln(I_{\mathrm{out}}/I_{\mathrm{in}})/(f_{\mathrm{out}}-f_{\mathrm{in}}). \]

Comparing equations (12) and (6), and taking \(\lambda=1\) in the latter, we find the inflection frequency:

\[ f_c\approx 31\cdot 10^7\cdot I_e^0 \left[ \frac{1}{\beta} +\frac{r_e^2}{R_g^2} +\frac{2r_e}{R_g} \left( \frac{1}{\beta}+\frac{r_b}{R_g} \right) \right], \tag{13} \]

Table

which is directly proportional to the limiting emitter current. The table gives the current gains of a number of silicon transistors and the values of the limiting emitter current calculated from (2). The current gains \(\beta\) were calculated from the measured values of \(\beta\) (the quasi-static current gain of a transistor in the common-emitter circuit) using the formulas given in [10].

The dependence of the inflection frequency on the emitter current in (13) is implicit: through the current gain \(\beta\) and the resistance \(r_e\). If the emitter current decreases, then \(\beta\) decreases and \(r_e\) increases. At constant \(R_g\) this leads to a decrease in the inflection frequency. Figure 2 shows calculated curves characterizing the dependence of the inflection frequency on the emitter current and constructed for unpackaged silicon planar transistors of type 2T319V. The results of calculations from experimental data are also shown there. For packaged transistors of types 2T306G, 2T312V, the experimental results differ more substantially from the calculated ones than for 2T319V, as is evident from Fig. 3.

Conclusions

1. The inflection frequency in the microcurrent regime depends substantially on the emitter current of the transistor, taking values down to units of hertz.

2. Experimental studies carried out for silicon planar transistors (types 2T319V, 2T306G, 2T312V) show that the theoretical dependences obtained are confirmed more satisfactorily for unpackaged transistors of type 2T319V.

3. For all the transistors studied, the discrepancy between the calculated and experimental results, as a rule, increases as the emitter current decreases.

References

1. Conti Mario. Surface and Bulk Effects in Low frequency noise in NPN Planar transistor, “Solid-state Electronics,” 1970, vol. 13, No. 14, November.

2. V. M. Pridorogin. Physical interpretation of some noise characteristics of transistors. In: Semiconductor Devices and Their Application, ed. by L. A. Fedotov, issue 21, Sovetskoe Radio Publishing House, 1969.

3. A. K. Naryshkin. Anti-noise corrections in wideband transistor amplifiers. “Svyaz,” 1969.

4. Hetchner D. Rauschkenwerte eines modernen Silizium—Planar—Transistor im Niederfrequenzgebiet, “Frequenz,” 1967, No. 2.

5. Wiggins M. J., An experimental study of 1/f noise in transistor, “IEE Intern. Convent. Rec.,” 1964, v. 12, No. 2.

6. Noise in Electronic Devices, ed. by L. D. Smullin and G. A. Haus, “Energiya,” 1964.

7. T. S. Verster. Low-frequency noise of UPT, “TIIER,” 1966, vol. 54, No. 9.

8. A. Van der Ziel. Fluctuation Phenomena in Semiconductors. Gosenergoizdat, 1961.

9. Transistors, ed. by I. G. Bergelson, Yu. A. Kamenetskii, I. F. Nikolaevskii. “Sovetskoe Radio,” 1968.

10. A. I. Setyukov. Some parameters of transistors in the microcurrent regime. In: Semiconductor Devices in Telecommunications Engineering, ed. by I. F. Nikolaevskii, issue 1, “Svyaz,” 1968.