Analysis of Third- and Fourth-Order Amplifiers in the Low-Frequency Region
Abstract Generated abstract
This paper analyzes third- and fourth-order low-frequency amplifier circuits with parallel negative feedback, intended for very low frequency and long-duration pulse signal amplification using microcurrent silicon bipolar transistors. Using signal-graph methods, it derives a general complex gain expression, amplitude-frequency characteristics, coefficient formulas, and Hurwitz stability conditions for several circuit variants with different finite or infinite coupling and feedback capacitances. The analysis shows that dc negative feedback can reduce the sensitivity of stability and cutoff-frequency behavior to unstable parameters such as return difference and input resistance, while frequency-dependent ac feedback of the considered form gives only limited improvement in lower cutoff frequency. The fourth-order amplifier and the third-order version with infinite output coupling capacitance are identified as the more suitable configurations for long pulse amplification and correction of pulse-top droop.
Full Text
Analysis of Third- and Fourth-Order Amplifiers in the Low-Frequency Range
V. E. Stennikov
(Presented by the scientific seminar of the Research Institute of Automation and Electromechanics)
In [1] an analysis was carried out of second-order amplifiers with parallel feedback, intended for amplifying signals in the frequency band of the order of \(10^{-1} \div 10^{4}\) Hz. In amplifiers, in order to reduce weight, dimensions, and power consumption, silicon planar bipolar transistors are used, operating in the microcurrent mode \((I_{\mathrm{e}} = 1 \div 100\ \mu\text{A})\). The present article analyzes third- and fourth-order amplifiers in the low-frequency region (large times).
Fig. 1. Circuit of a fourth-order amplifier.
The circuit of a fourth-order amplifier is shown in Fig. 1. The chain \(R_2, C_2, R_3\) stabilizes the operating mode of the amplifier with galvanic coupling of stages—\(K_0\)—with respect to direct current, while the resistor \(R_c\) and capacitor \(C_4\) provide negative feedback (NFB) with respect to alternating current. This chain includes the input \((C_1)\) and output \((C_3)\) capacitors. The input resistance \((R_{\text{in}})\) of the amplifier with galvanic coupling of stages (GCS) is externalized, and the output circuit of the GCS is represented, according to the theorem on an equivalent generator, by a source of electromotive force \(e\) and an output resistance \(R_{\text{out}}\). The resistance \(R_g\) is the sum of the resistances of the signal source and an additional resistor.
As the method of analysis, the method of signal graphs [2] is used. Figure 2 shows the signal graph of the amplifier circuit in Fig. 1. The determinant of the signal graph was found by the method of double expansion over the branches \(1/Z_c\) (shown by one cross) and \(R_{\text{in}}\) (shown by two crosses).
Fig. 2. Signal graph of the circuit in Fig. 1.
The relative complex gain of the amplifier is, in general form, expressed by the formula
\[ M(p)=\frac{K(p)}{K_c} = \frac{ 1+\dfrac{a_1}{p\tau_1}+\dfrac{a_2}{(p\tau_1)^2} }{ 1+\dfrac{b_1}{p\tau_1}+\dfrac{b_2}{(p\tau_1)^2}+\dfrac{b_3}{(p\tau_1)^3}+\dfrac{b_4}{(p\tau_1)^4} }, \tag{1} \]
where
\[ \begin{aligned} K(p)&\text{ is the complex gain, } p=j\omega,\\ \omega&\text{ is the angular frequency, } \tau_1=C_1R_g,\\ K_c&\text{ is the gain as } \omega\to\infty,\\ a_1,\ a_2,\ b_1,\ b_2,\ b_3,\ b_4&\text{ are coefficients independent of }p. \end{aligned} \]
The values of these coefficients and other parameters of the amplifier in Fig. 1 are presented in the table. All time constants of the amplifier are normalized by \(\tau_1\). The parameter \(F_0\) given in the table denotes the feedback difference of the amplifier at \(R_g=0\), where
\[ F_0=K_0/K_c, \]
where
\[ K_0 \]
is the gain of the amplifier at midrange frequencies with the AC feedback loop closed, for \(R_g=0\) and \(R_H=\infty\). For finite \(R_g\), the feedback difference of the amplifier at midrange frequencies is determined by the formula
\[ F=F_0(R_{\mathrm{in}}\parallel R_2)/\left[R_g+(R_{\mathrm{in}}\parallel R_2)\right]. \]
According to the table, the coefficients \(b_1,b_2,b_3\) can be represented in the form
\[ b_i=b_{i0}+b_{i1}\frac{1}{F_0}\quad (i=1,2,3). \]
If the amplifier has dc negative feedback \((x\ne\infty)\), then it is possible to stabilize the coefficients \(b_i\), i.e., to make them weakly dependent on the unstable parameters \(F_0\) and \(R_{\mathrm{in}}\), and consequently the lower cutoff frequency will have low sensitivity to changes in these parameters. Analyzing formulas (3–5) from the table, we note that the coefficients \(b_i\) will be stabilized when
\[ b_{i0}\gg b_{i1}F_0. \tag{2} \]
In practice, the last relation is not difficult to realize, increasing \(F_0\).
From (1) it is not difficult to obtain an expression for the coefficient of frequency distortion of the amplifier (the generalized amplitude-frequency characteristic (AFC) in the low-frequency region)
\[ M(x)= \sqrt{ \frac{ \left(1-\dfrac{a_2}{x_2}\right)^2+\dfrac{a_1^2}{x^2} }{ \left(1-\dfrac{b_2}{x^2}+\dfrac{b_4}{x^4}\right)^2 +\dfrac{1}{x^2}\left(b_1-\dfrac{b_3}{x^2}\right)^2 } }, \tag{3} \]
where
\(x\) is the normalized dimensionless frequency,
\[ x=\omega\tau_1 . \]
The amplifier will be stable (by Hurwitz) if
\[ b_2>b_4\frac{b_1}{b_3}+\frac{b_3}{b_1}. \tag{4} \]
Introduce the notation
\[ b_2 \Big/ \left(b_4\frac{b_1}{b_3}+\frac{b_3}{b_1}\right)=L>1, \tag{5} \]
where
\(L\) is the stability parameter. For \(L=1\) the amplifier is at the stability boundary.
Let us consider the characteristics of amplifiers that are particular variants of the amplifier in Fig. 1.
Variant 1, \(\chi_2=\infty\) \((C_4=\infty,\ a_2=b_4=0)\)
The stability condition, by virtue of (4) and taking (2) into account:
\[ \chi_1>\left[\chi\Big/\left(1+\frac{R_2}{R_3}+\frac{R_C}{R_3}\right)\right]-1. \tag{6} \]
When \(C_3\) is varied and the other parameters entering into (6) are unchanged, stability deteriorates as \(C_3\) decreases. Figure 3 shows AFCs constructed from (3) with (6) taken into account. In this and the following graphs the notation used is \(\sigma=R_2/R_3\), \(\sigma_2=R_C/R_3\), \(\gamma_1=R_r/R_{\mathrm{in}}\), \(\gamma_2=R_{\mathrm{out}}/R_n\). The graphs in Fig. 3 also give the values of the relative correction coefficient \(\nu\), which characterizes the relative gain in the lower cutoff frequency when AC negative feedback is introduced. Moreover,
Fig. 3. Amplitude-frequency characteristics of the amplifier (Fig. 1, \(\chi_2=\infty\)) for \(K_C=100\), \(F_0=50\), \(\sigma=1\), \(\chi=10\), \(\gamma_1=0.5\); \(\gamma_2=0.1\); \(\sigma_2=10\).
\[ \nu=x_n/x_{nc}\cdot F, \]
where \(x_{\mathrm{nc}}, x_{\mathrm{n}}\) are the normalized lower cutoff frequencies with NFB and without it, respectively.
It is seen that \(\nu\) decreases rather rapidly in the region of rises in the AFC. The characteristic for \(\chi_1=\infty\) corresponds approximately to the optimal AFC, that is, to the case when \(\nu\) reaches a maximum.
Variant 2. \(\chi_1=\infty\) \((C_3=\infty,\ b_4=0)\).
The approximate stability condition is:
\[ \chi_2>\left[\chi \bigg/ \left(1+\frac{R_2}{R_3}+\frac{R_c}{R_3}\right)\right]-1 . \tag{7} \]
Expression (7) is similar to (6), except that \(\chi_1\) is replaced by \(\chi_2\).
Figure 4 gives the AFCs of the amplifier. In the plots, as the second parameter (in addition to \(\chi_2\)), the values of the stability parameter \(L\), calculated from (5), are indicated.
Fig. 4. Amplitude-frequency characteristics of the amplifier (Fig. 1, \(\chi_1=\infty\)) for \(R_c=100,\ F_0=50,\ \sigma=1,\ \chi=10,\ \gamma_1=0.5;\ \gamma_2=0.1;\ \sigma_2=10\).
From Fig. 4 we note an interesting regularity: the lower cutoff frequency is practically independent of \(\chi_2\), that is, frequency-dependent NFB gives practically no gain in the lower cutoff frequency in comparison with feedback. If the AFC for \(\chi_2=\infty\) is taken in the monotonic region and away from the optimum, the gain will be somewhat greater, but, as calculations show, by no more than a factor of two. In practice it is far more advantageous to achieve the optimal AFC by increasing the magnitude of the feedback difference at \(\chi_2=\infty\) than to do this by means of frequency-dependent NFB of the type indicated in Fig. 1 \((C_4R_c)\). However, the use of frequency-dependent NFB is advantageous in that it is possible to correct completely, and, if necessary, overcorrect, the linear component of the droop of pulse tops. Indeed, according to [3], the linear component of the droop of pulse tops will be corrected if \(a_1=b_1\), or
\[ C_2 R_2 R_3 = C_4 R_c^2 . \tag{8} \]
When (8) is implemented, the permissible duration of the amplified pulses according to [4] is determined by the formula
\[ \tau_{\mathrm{u}}=\tau_1\cdot q\cdot \sqrt[3]{\frac{0.06\cdot \Delta_3\%}{b_1(b_2-a_2)-b_3}}, \tag{9} \]
where \(\Delta_3\%\) is the error in determining the droop, determined by the third term of the expansion of the transient characteristic in a power series,
\(q=1\) for single pulses, \(q=1.59\) for amplification of rectangular periodic pulses with a duty factor equal to two. Substituting the values of the coefficients \(b_i\) from the table into formula (9) and, taking (2) into account, we obtain
\[ \tau_{\mathrm{n}}=\tau_1 q\sqrt[3]{ \frac{0.06\,\Delta_3\%\,\chi^2} {\left(1+\dfrac{R_2+R_c}{R_3}\right)\dfrac{1}{\chi_2} \left[\dfrac{R_c}{R_3}(1+\chi_2)-\left(1+\dfrac{R_2}{R_3}\right)\right]-1} }. \]
This third-order amplifier is convenient to use for amplifying pulses of long duration (on the order of hundreds of milliseconds).
Variant 3. \(\chi=\infty\) \((C_2=\infty,\ a_2=b_4=0)\).
Stability condition:
\[ F_0< \left[\left(\frac{1}{\chi_1}+\frac{1}{\chi_2}\right)(n+1)+n\right] \left(1+\chi_1+\chi_2+\frac{1}{n}\right), \tag{10} \]
where \(n\) is given by formula (7b) in the table.
Condition for correcting the linear component of the pulse tops:
\[ \frac{1}{\chi_2} = \frac{1}{F_0} \left[ (1+n)\left(\frac{1}{\chi_1}+\frac{1}{\chi_2}\right)+n \right]. \tag{11} \]
The last two relations are valid for \(R_{\mathrm{p}}\gg R_{\mathrm{out}}\). The absence of direct-current negative feedback has led to the fact that conditions (10) and (11) proved sensitive to changes in the unstable parameters \(F_0\) and \(R_{\mathrm{in}}\). For the indicated reason, this amplifier variant should be used after preliminary stabilization of \(F_0\) and \(R_{\mathrm{in}}\) (for example, by introducing a series negative feedback into the DC amplifier).
Variant 4, all capacitances are finite.
We have a fourth-order amplifier, which will be stable according to (4) and taking (2) into account if
\[ \chi< -\frac{\chi_1\chi_2}{1+\chi_1+\chi_2} \left(1+\frac{R_2+R_c}{R_3}\right) \left\{ \left(1+\frac{1}{\chi_1}+\frac{1}{\chi_2}\right) - \frac{1+\left[(R_2+R_3)/R_c\right]}{1+\chi_1+\chi_2} \right\}. \tag{12} \]
Condition (12) imposes a strong limitation on lowering the lower cutoff frequency of the amplifier with negative feedback. Figure 5 gives the frequency-response curves of the amplifier. The curve for \(\chi_2=\infty\) was deliberately taken in the region of monotonicity, sufficiently far from the optimal frequency response. From the graph it is seen—
Fig. 5. Amplitude–frequency characteristics of the amplifier (Fig. 1) for
\(\sigma=\sigma_2=10,\ \chi=10,\ \chi_1=1;\ F_0=50,\ R_c=100,\ \gamma_1=0.5;\ \gamma_2=0.1.\)
Table IIa
Amplifier parameters
| Formula No. | Parameter | Approximate expression (exact for \(z_1, z_2, \tau_2, \tau_3, \tau_4\)) | Conditions for the approximation |
|---|---|---|---|
| 1 | \(a_1\) | \(\displaystyle \frac{1}{z}\left(1+\frac{R_2}{R_3}+\frac{z}{z_2}\right)\) | \(\displaystyle R_3 \gg R_{\mathrm{out}},\quad R_c \gg R_n\) |
| 2 | \(a_2\) | \(\displaystyle \frac{1}{z\,z_2}\left(1+\frac{R_c}{R_3}\right)\) | \(\displaystyle R_c > R_{\mathrm{in}},\quad R_c \gg R_n\) |
| 3 | \(b_1\) | \(\displaystyle \frac{1}{z}\left(1+\frac{R_2}{R_3}+\frac{R_c}{R_3}\right)+\frac{1}{F_0}\left[\frac{1}{z_2}\left(1+\frac{R_g}{R_{\mathrm{in}}\parallel R_2}\right)\times\left(1+\frac{R_{\mathrm{out}}}{R_n}\right)+\frac{z_2}{z_1}\left(1+\frac{R_{\mathrm{out}}}{R_n}\right)\frac{R_g}{R_{\mathrm{in}}\parallel R_2}\right]\) | \(\displaystyle R_g \gg R_{\mathrm{out}}\) |
| 4 | \(b_2\) | \(\displaystyle \frac{1}{z\,z_2}\left\{\frac{R_c}{R_3}\left(1+\frac{z_2}{z_1}\right)+z_2+\frac{z_2}{z_1}+\frac{z_2}{F_0}\left[\left(1+\frac{R_{\mathrm{out}}}{R_n}\right)\frac{R_g}{R_{\mathrm{in}}\parallel R_2}+\frac{1}{z_2}\right]\times\left(1+\frac{R_g}{R_{\mathrm{in}}\parallel R_2}\right)\right\}+\left(1+\frac{R_g}{R_{\mathrm{in}}\parallel R_2}\right)\frac{R_g}{R_{\mathrm{in}}\parallel R_2}\) | \(\displaystyle k_0 \gg \left(1+\frac{R_{\mathrm{out}}}{R_g}\right)\left(\frac{R_2}{R_{\mathrm{in}}}+\frac{R_3}{R_g}\right)\times\left(1+\frac{R_{\mathrm{out}}}{R_n}\right),\) \(\displaystyle F_0\,\frac{R_{\mathrm{out}}}{R_g}\gg \left(1+\frac{R_{\mathrm{out}}}{R_n}\right)\times\left(1+\frac{R_{\mathrm{out}}}{R_g\parallel R_2}\right),\) \(\displaystyle k_0\,\frac{R_{\mathrm{out}}}{R_3}\gg \frac{R_n}{R_c}\left(1+\frac{R_{\mathrm{out}}}{R_2}\right)\) |
| 5 | \(b_3\) | \(\displaystyle \frac{1}{z\,z_1}\left[\frac{R_c}{R_3}\left(1+\frac{1}{z_2}+\frac{z_1}{z_2}\right)+\frac{1}{F_0}\frac{z_1}{z_2}\frac{R_g}{R_{\mathrm{in}}\parallel R_2}\right]\) | |
| 6 | \(b_4\) | \(\displaystyle \frac{R_c}{R_3\,z\,z_1\,z_2}\) | |
| 7 | \(k_c,\ n\) | \(\displaystyle \frac{R_c}{R_g}\ (7a),\quad n=\frac{R_g}{R_{\mathrm{in}}\parallel R_2}\ (7b)\) | |
| 8 | \(z,\ z_1,\ z_2,\ \tau_1,\ \tau_2,\ \tau_4\) | \(\displaystyle z_2;\ \tau_1\ (8a),\quad \tau_3;\ \tau_1\ (8b)\) | |
| 9 | \(z_2,\ z_3,\ \tau_4\) | \(\displaystyle C_2R_2\ (9a),\quad C_3R_n\ (9b),\quad C_4R_c\ (9c)\) |
but that the frequency-dependent negative feedback has little effect on the lower cutoff frequency of the amplifier. The condition for correcting the linear component of the tops of the pulses coincides with (8).
Conclusions
-
For amplifying pulse signals of long duration, the most suitable are the fourth-order amplifier and the third-order amplifier with \(C_3=\infty\). The stability conditions (7), (12) and the conditions for correcting the linear component of the droop of the pulse tops (8) for these amplifiers have low sensitivity to changes in the most unstable amplifier parameters—the return difference \(F_0\) and the input resistance \(R_{\mathrm{in}}\). These same conditions prove to be strongly dependent on \(F_0\) and \(R_{\mathrm{in}}\) in the absence of DC negative feedback \((C_2=\infty)\).
-
Frequency-dependent AC negative feedback of the type indicated in Fig. 1 \((C_4, R_c)\) makes it possible to obtain an improvement in the lower cutoff frequency by no more than a factor of two in comparison with the case when \(C_4=\infty\), for equal values of the return difference at midrange frequencies in both cases.
References
-
V. E. Stennikov. Low-frequency amplifiers using bipolar transistors. In: Electrical Engineering Apparatus, Vol. 1, “Control and Monitoring Devices.” Energiya, 1971.
-
S. J. Mason, G. J. Zimmerman. Electronic Circuits, Signals, and Systems. Moscow: IL, 1963.
-
T. M. Agakhanyan. Linear Pulse Amplifiers. Svyaz, 1970.
-
R. A. Smirnov. Two-stage Pulse Amplifiers with Feedback. Tomsk: Tomsk University Press, 1965.