Two-Stage Instrumentation Amplifier
Abstract Generated abstract
This paper examines the design of transistor instrumentation amplifier sections using deep negative feedback without frequency response correction, with emphasis on stable, economical two-stage circuits suitable for production. It compares two direct-coupled circuit variants and argues for one arrangement because its feedback loop uses fewer elements and is less affected by supply decoupling filters, reducing instability of the feedback transfer coefficient. A calculation method is developed for selecting circuit elements that maximize the return difference at midrange frequencies, including expressions for the collector and feedback resistances under specified operating conditions. Experimental results for a gain-of-10 feedback amplifier report 0.5 percent frequency-response nonuniformity from 15 Hz to 300 kHz, gain instability below 0.2 percent for supply variation, and low temperature-related gain drift.
Full Text
TWO-STAGE INSTRUMENTATION AMPLIFIER
M. S. ROITMAN, V. M. SERGEEV
(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)
It is known that an effective means of obtaining high metrological characteristics of instrumentation amplifiers (IAs) is the introduction of deep negative feedback (NFB).
The permissible depth of negative feedback is limited by the requirement of ensuring a sufficient stability margin and is determined above all by the number of stages and the order of the system. This makes it necessary, when an IA with a large gain is required, to divide it into several elementary amplifier sections (AS), each of which is covered by maximally deep NFB. An analysis of the optimum number of stages of tube ASs is given in [1]. For transistor amplifiers, analogous expressions were obtained in [2] and, taking into account the nonminimum-phase properties of the transistor, in [3].
The optimum number of stages in an AS, when each stage has a Bode-ideal frequency response, for transistor amplifiers, according to [2] and [3], ranges from 2 to 3. However, carrying out such correction of the frequency response greatly complicates the amplifier and requires careful adjustment, which in mass production of ASs is entirely unacceptable. With an uncorrected response, the optimum number of AS stages does not exceed 2.
Constructing amplifiers from individual ASs requires a careful approach to the choice of circuit and a thorough calculation of the latter.
Each AS must make it possible to obtain sufficiently deep negative feedback while operating stably without correcting elements. This requirement predetermines both a rational construction of the entire amplifier as a whole and of the NFB circuit in particular (in the sense of reducing parasitic reactances), as well as the use of direct coupling between stages, which reduces phase shifts in the low-frequency region. In addition, in manufacturing ASs, an important factor becomes their reliability and economy, as well as their insensitivity to the spread of parameters of the active and passive elements of the circuit. Among the known circuit solutions for two-stage amplifiers satisfying the stated requirements, the most acceptable may be considered the circuits shown in Figs. 1 and 2. In them, the overall voltage NFB is effected by means of resistance $R_0$. In direct current, the stages are stabilized by local feedbacks ($R_{\mathrm{e}1}$, $R_{\mathrm{e}2}$) and by overall NFB through resistance $R_{\mathrm{b}2}$. With identical temperature stability, the two circuits are approximately equivalent in terms of the attainable value of the gain.
An important advantage that makes it possible to give preference to the circuit of Fig. 2 is the more successful construction of the negative-feedback (NFB) loop, requiring a minimal number of elements. Another merit is that the NFB loop is independent of the influence of decoupling filters, which are necessary when the amplifier is supplied from a single source. The indicated factors reduce the instability of the transfer coefficient of the NFB loop, which
Fig. 1. Fig. 2.
at deep NFB mainly determines the error of the amplifier. Thus, these positive qualities of the circuit of Fig. 2 make it possible to recommend it as an amplifier.
Calculation of the circuit elements providing the maximum depth of NFB at medium frequencies
To determine the depth of NFB at medium frequencies for the circuit of Fig. 2 by the classical method [1], it would be necessary to operate with fourth-order matrices, which, when the problem is solved in general form, leads to cumbersome results. A simpler method is set forth in [4].
On the basis of the procedure given in [4], when the condition
\[ R_{\text{in}} + R_i \gg (r_b + R_i)\cdot \frac{R_n\cdot(1+\beta)}{r_k}, \tag{1} \]
is fulfilled, where
\(R_{\text{in}}\) — input resistance of the transistor at medium frequencies,
\(R_i\) — resistance of the signal source,
\(R_n\) — load resistance,
\(r_b, \beta, r_k\) — parameters of the physical equivalent circuit of the transistor.
The value of the return difference for the circuit of Fig. 2 can be represented in the form
\[ F = A\cdot \frac{R_0\cdot R_{K_1}} {(R_{K_1}+R_{\text{in}_2})\cdot (R_n+R_{K_2}+R_0)}, \tag{2} \]
here
\[ A=\frac{\beta_1\cdot \beta_2\cdot R_n}{R_i' + R_{\text{in}_1}'}; \]
\[ R_i' = R_i \parallel R_{b_1}; \qquad R_{\text{in}_1}' = R_{\text{in}_1}\parallel R_{b_1}; \]
the indices 1 and 2 denote the parameters of transistors \(T_1\) and \(T_2\), respectively.
Expression 2 was derived while neglecting the local feedback in the first stage due to the resistance \(R_0\).
Using the dc coupling between the first and second stages, let us represent \(F\) as a function of \(R_{K_1}\) and constant coefficients. On the basis of the notation in Fig. 3 one may write
Fig. 3.
\[ E_K = U_{\mathrm{e}} + U_{K\mathrm{e}} + I_2 \cdot R_{K_2} + (I_1 + I_2 + I_3)\cdot R_0; \]
here
\[ \begin{aligned} I_1 &\text{— quiescent current of the first stage,}\\ I_2 &\text{— quiescent current of the second stage,}\\ I_3 &\text{— divider current,} \end{aligned} \]
but
\[ U_{\mathrm{e}} = I_1 \cdot R_{K_1} - U_{\mathrm{b e}}; \]
then
\[ E_K = U_{K\mathrm{e}} - U_{\mathrm{b e}} + I_1 \cdot R_{K_1} + I_2 \cdot R_{K_2} + (I_1 + I_2 + I_3)\cdot R_0 . \tag{3} \]
With deep feedbacks, the gain is determined as
\[ K_{\mathrm{fb}} = \frac{1}{\gamma}, \tag{4} \]
where \(\gamma\) is the feedback coefficient. For the circuit of Fig. 2,
\[ \gamma = \frac{R_0}{R_0 + R_{K_2}}, \tag{5} \]
taking (4) and (5) into account, one may write
\[ R_{K_2} = R_0 \cdot (K_{\mathrm{fb}} - 1); \tag{6} \]
substituting this value of \(R_{K_2}\) into (3), after transformations we obtain
\[ R_0 = R - \delta R_{K_1}; \tag{7} \]
here
\[ R = \frac{E_K - U_{K\mathrm{e}} + U_{\mathrm{b e}}}{I_1 + I_2 \cdot K_{\mathrm{fb}} + I_3}; \qquad \delta = \frac{I_1}{I_1 + I_2 \cdot K_{\mathrm{fb}} + I_3}. \]
Taking (7) into account, expression (2) may be written in the form
\[ F = A \cdot \frac{R_{K_1}\cdot (R - R_{K_1}\cdot \delta)} {(R_{K_1} + R_{\mathrm{in}_2}\cdot [R_{\mathrm{n}} + (R - R_{K_1}\cdot \delta)\cdot K_{\mathrm{fb}}])}. \tag{8} \]
Investigating expression (8) for a maximum with respect to \(R_{K_1}\), we obtain an equation for determining \(R_{K_1}\) that provides the maximum value of the return difference:
\[ R_{K_1} = n \pm \sqrt{n(n-m)}; \tag{9} \]
where
\[ n = \frac{R_{\mathrm{n}} + R \cdot K_{\mathrm{fb}}} {\delta \cdot K_{\mathrm{fb}} - \dfrac{R_{\mathrm{n}}}{R_{\mathrm{in}_2}}}; \qquad m = \frac{R}{\delta}; \]
at no load
\[ (R_{\mathrm{n}} = \infty)\quad n = -R_{\mathrm{in}_2}; \]
\[ \delta \cdot K_{\mathrm{fb}} = \frac{R_{\mathrm{n}}}{R_{\mathrm{in}}}. \]
the expression for \(R_{K_1}\) will take the form
\[ R_{K_1}=\frac{R}{2\delta}. \]
Having specified the dc operating conditions \((U_{KЭ_2}, I_1, I_2, I_3)\) for given \(E_K\) and \(K_{\text{fb}}\), on the basis of expressions (9), (7), and (6) it is possible to calculate all the elements of the circuit.
The experiment was carried out with an amplifier according to the circuit of Fig. 2, with a feedback gain factor \(K_{\text{fb}}=10\), loaded by a load resistance of \(2\ \text{k}\Omega\) (gain without feedback \(K=4000\)). The nonuniformity of the frequency response in the range \(15\ \text{Hz} \div 300\ \text{kHz}\) does not exceed 0.5%. The instability of the gain factor when the supply voltage is varied by \(\pm 10\%\) does not exceed 0.2%. The instability of the gain factor in the temperature range \(25—65^\circ\text{C}\) does not exceed 0.12% (the instability of the transfer coefficient of the divider \(R_{K_2}, R_0\), made with resistors of the BLP type, in this range is 0.08%); thus, when the feedback circuit is made of resistors with the same temperature coefficient, the instability of the amplifier in the indicated range will not exceed 0.04%.
The input resistance at midrange frequencies is \(3\ \text{k}\Omega\) (with compensation of the divider it can be increased to \(200\ \text{k}\Omega\)). The output resistance is \(60\ \Omega\).
In conclusion, the authors consider it their duty to thank engineer N. F. Denisova for assistance in carrying out the experimental work.
REFERENCES
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G. Bode. Network Theory and Feedback Amplifier Design. IL, 1948.
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M. S. Shmelevich. On the optimal number of stages in transistor amplifiers with negative feedback. Elektrosvyaz, No. 7, 1966.
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Report on the topic “Development of Certain Recommendations for the Design of Linear Transistor Amplifiers,” TsNII Svyazi, Moscow, 1965.
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B. Ya. Lur’e. Design of Transistor Amplifiers with Deep Feedback. Svyaz, 1965.