SOME ISSUES IN THE THEORY AND PRACTICAL IMPLEMENTATION OF STRUCTURAL METHODS FOR IMPROVING THE ACCURACY OF MEASUREMENT AMPLIFIERS

M. S. ROITMAN, V. M. SERGEEV, V. A. BUTENKO

Introduction

The known limitations inherent in the method of negative feedback (NFB) in many cases no longer make it possible to ensure the necessary metrological characteristics of modern measurement amplifiers (MAs).

Structural methods for constructing MAs that implement the principle of invariance possess great potential capabilities for improving accuracy.

A necessary condition for achieving invariance is the principle of two-channel operation (multichannel operation), when the main channel (MC) performs the function of power amplification, while the additional channel is a correcting channel (CC).

Known two-channel MAs are conveniently classified, according to the method of summing the signals of the MC and CC, into systems with an external (internal) summer at the output and an external (internal) summer at the input.

Despite the high effectiveness of two-channel MA systems, they have not yet found wide practical application. This is explained by the fact that interest in the known ideas of the principle of invariance as applied to MAs arose comparatively recently, and designers still do not have sufficient information either on the theoretical foundations or on the issues of practical implementation of two-channel MAs.

Great credit for the development of the theory and practice of invariant systems as applied to the construction of high-precision, wide-range rms voltmeters and measuring transducers belongs to the staff of the Department of Information and Measurement Technology of the Kiev Polytechnic Institute (works by Yu. M. Tuz, L. M. Gapchenko, K. L. Serpilin, E. T. Volodarsky, and others).

The exceptionally high requirements imposed on high-voltage MAs that perform the functions of scaling devices in multivalued measures of alternating voltages*) led us to the need to clarify the possibilities of multichannel structures and to pay special attention to questions of their effective practical implementation.

In the present work, only methods for stabilizing the instantaneous value of the MA transfer coefficient are considered, when a substantial reduction of both linear and nonlinear distortions is possible.

*) With an output voltage of up to 300 V rms, the error of high-voltage MAs in the frequency range 20 Hz–200 kHz must not exceed 0.01–0.05% with a nonlinear distortion coefficient ≤ 0.03%.

Initial Premises for a Comparative Analysis of the Metrological Capabilities of Single-Channel and Two-Channel MAs

If the use of a two-channel structure is regarded as an additional measure for increasing the accuracy of the original MA, preliminarily stabilized by the introduction of extremely deep negative feedback (NFB), then it becomes necessary to determine a criterion for the effectiveness of two-channel systems.

In our view, the indicated criterion can be determined by comparing the error of the two-channel system with that of the original amplifier with NFB, whose amplitude-phase-frequency characteristics (APFC) of loop gain correspond to the ideal ones [1].

With ideally corrected APFC, the loop-gain modulus does not depend on frequency in the calculated operating frequency band. In this case, for a frequency-independent \(\beta\)-network, the gain modulus with NFB will be determined by the expression

\[ K_0(\omega)= \frac{K'_0}{\sqrt{1+2K'_0\beta\cos\varphi'_0(\omega)+(K'_0\beta)^2}}, \tag{1} \]

where \(K'_0\) is the gain modulus without NFB,
\(K'_0\beta=T_0\) is the loop-gain modulus,
\(\varphi'_0(\omega)\) is the phase of the loop gain.

The error of the modulus \(K_0(\omega)\) will be determined both by changes in \(K'_0(\omega)\) due to destabilizing factors and by the presence of the phase shift of the loop gain. However, while limitation of destabilizing factors is possible in principle, the phase shift in the case of ideal APFC is strictly determined by the adopted phase margin

\[ \varphi'_0(\omega)=2(1-y)\operatorname{arctg}\frac{x}{\sqrt{1-x^2}}. \]

Here \(y\) is the phase margin; \(x=\omega/\omega_0\) is the normalized frequency; \(\omega,\omega_0\) are, respectively, the current frequency and the upper frequency of the operating range. This circumstance makes it possible to regard the modulus error caused by the presence of phase as the minimum possible in the frequency range for an MA with NFB. For the limiting values of the phase, the error is determined from (1) as

\[ \delta(\varphi)= \frac{K(\omega)-K(0)}{K(0)} = \frac{1}{1-\dfrac{2T_0(1-\cos\varphi'_0)}{(1+T_0)^2}}-1. \tag{2} \]

Neglecting the \(T\)-error, for \(\varphi'_0\leq 60^\circ\) expression (2) can be simplified:

\[ \delta(\varphi)\simeq \frac{1-\cos\varphi'_0}{T_0} \simeq \frac{1}{2}\frac{\varphi_0^{\prime\,2}}{T_0}. \tag{3} \]

Expression (3) will be used for comparison with the error of two-channel systems.

System with an External Summing Unit at the Output

The functional diagram (FD) and the system graph (SG) are shown in Fig. 1.

The transfer of the system is determined by the expression

\[ K_{\Sigma}=K_0+\alpha K-\gamma K_0K, \tag{4} \]

where \(K_0\) is the transfer of the MC, \(K\) is the transfer of the CC. For the case of an inverting MC, the error-extraction device (EED) can be made in the form of a passive resistive divider \((R_1 - R_2)\). With an infinite input resistance of the CC, the coefficients \(\alpha\) and \(\gamma\) are determined by the expressions: \(\alpha = R_2/(R_1 + R_2)\); \(\gamma = R_1/(R_1 + R_2)\). When the tuning condition (error extraction) \(K_0\gamma = \alpha\) (5) and the invariance condition \(\alpha K = K_0\) (6) are satisfied, the system gain after introduction of the CC remains equal to the gain of the MC, i.e., \(K_{\Sigma}=K_0\), and the error of the system is determined by the product of the errors of both channels \(\delta_{\Sigma}=\delta_0\delta\) (7). In this case the input resistance of the system is equal to \(R_{11}=R_1 \parallel R_0\), where \(R_0\) is the input resistance of the MC, and the output resistance of the system is practically equal to the output resistance of the summator. Taking account of the finite value of the input resistance of the CC does not change the tuning condition (5) and only leads to the need to increase the gain \(K\): \(K=1+K_0+R_2/R'_{11}\), where \(R'_{11}\) is the input resistance of the CC. Exact fulfillment of conditions (5) and (6) is possible only at the frequency at which the phase shifts of both channels and of the EED are equal to zero. Since the use of the CC is regarded as an additional measure for increasing the accuracy of the measuring amplifier, then, as a rule, both the MC and the CC are enclosed in deep negative feedback. Assuming, as before, that the potential errors of each channel are due to phase shifts, we determine the error of the system for this case. From (4), taking into account the phase shift of the MC and CC with negative feedback, respectively \(\varphi_0,\ \varphi\), we obtain

\[ \delta_{\Sigma}(\varphi) = \sqrt{3-2\cos(\varphi_0-\varphi)-2\cos\varphi-2\cos\varphi_0-1}, \]

or, for \(\varphi \leqslant 60^\circ\), approximately

\[ \delta_{\Sigma}(\varphi)\simeq -\varphi_0\varphi . \tag{8} \]

It is known that the phase shift of an amplifier with negative feedback (with a frequency-independent \(\beta\)-network) is determined by the relation

\[ \varphi_0=\operatorname{arctg}\sin\varphi'_0(\cos\varphi'_0+T_0)^{-1} \]

or, for small values of \(\varphi'_0\),

\[ \varphi_0\simeq \varphi'_0(1+T_0)^{-1}\simeq \varphi'_0 T_0^{-1}. \]

Then, taking into account the values of the loop gain of the MC—\(T_0\), and of the CC—\(T\), expression (8) takes the form

\[ \delta_{\Sigma}(\varphi)\simeq -\varphi'_0\varphi'(T_0T)^{-1}. \tag{9} \]

The ratio of the errors determined by relations (3) and (9) shows the accuracy gain obtained by introducing the CC

\[ Q=\frac{\varphi'_0}{\varphi'}\cdot\frac{T}{2}. \tag{10} \]

It is obvious that the gain determined by expression (10) should be considered real only for the case when the optimum number of stages (according to Bode’s relations) has been used in the MC, so that the introduction of additional stages is impossible or gives an insignificant gain in the permissible value of loop gain. Otherwise, the useful gain of the CC could have been used in the MC for a considerable increase in \(T_0\). Thus, if it is assumed that \(T_0=T\), and in the original MC it is possible to include additional stages with a gain equal to the gain of the CC without negative feedback, then the resulting loop gain would become equal to \(T'_0=T_0K_0(1+T_0)\), and the ratio of the errors of such amplification and of the original two-channel system would be determined as \(Q'=0.5\varphi_0'^{\,2}/\varphi'K_0\).

Thus, obtaining a gain by distributing the optimum number of stages between the MC and the CC is practically impossible. However

It is known that, for a wide required frequency band, the optimal number of cascades in the OK is small and, as a rule, is no more than 3–4. Under these conditions, a substantial possibility for increasing the accuracy of the IU is the introduction of a KK.

Let us return to expression (10). Since the maximum output voltage of the KK is determined by the absolute value of the error of the OK, the power of the KK is small and it can be made with higher-quality (broadband) elements than the OK. In this case, evidently, the inequality \(\varphi' \ll \varphi'_0\) is valid. Then, even in the absence of OOS in the KK, the gain determined by the ratio \(Q'' = 0.5\varphi'_0/\varphi\) may be substantial. It is easy to show that, for small values of the phase shifts of the OK and KK, the resulting phase shift of the system is determined as \(\varphi_{\Sigma} = \varphi_0\delta + \varphi\delta_0\), which also testifies to the high effectiveness of the two-channel system.

All the preceding derivations were made under the assumption that the UVO and the adder are ideal. It is obvious that the error of the UVO fully enters into the resulting error of the system; in other words, the limiting accuracy of a two-channel system cannot be made higher than the accuracy of the UVO. Therefore, careful implementation of the UVO is one of the most important tasks in the practical realization of the system. The use of a UVO in the form of a divider \(R_1—R_2\) (see Fig. 1) does not make it possible to ensure its required accuracy over a wide frequency range. The desire to provide a high input resistance while preserving condition (5) leads to large values of \(R_2\), which, in the presence of the input capacitance of the KK, gives rise to a considerable phase shift of the UVO. A more acceptable construction of the UVO is shown in Fig. 2. In this case \(R_1 = R_2 = R\) is chosen, which, for any \(R\), ensures frequency independence of this circuit. The divider \(R_3, R_4\) is selected from the conditions \(R_4 \ll R\), \(K_c R_4 = R_3\). The dashed line in the figure denotes a correcting capacitor that equalizes the time constants of the resistors \(R_3\) and \(R_4\). With such an implementation of the UVO, the required gain of the KK must be doubled in comparison with expression (6).

An essential role in realizing the potential metrological capabilities of the system is played by the output adder. In the case of high-voltage IU, practically the only possible type of adder (when it is necessary to preserve information on the instantaneous value of the transfer coefficient of the OK) is a transformer (Fig. 3).

The presence of leakage inductance causes a significant additional error when the load changes, especially if it has a capacitive character. Interturn capacitance leads to deterioration of the amplitude-phase frequency response of the OK and to additional losses of useful power. In powerful high-voltage IU, the latter factor is sometimes decisive. An acceptable resolution of the known contradictions, when one wishes simultaneously to reduce leakage inductance and interturn capacitance, can be obtained either by using several transformers operating in a narrow frequency range or by reducing the transformation ratio, which leads to the need to increase the gain of the KK.

The schematic diagram of the high-voltage IU developed at the Department of Radio Engineering of Tomsk Polytechnic Institute according to the circuit of Fig. 3 is shown in Fig. 4. The dashed line denotes the OK, whose circuit is shown in Fig. 16.

The amplifier has the following characteristics: gain \((K_0) = 30\), output voltage \((U_2) = 300\ \text{V rms}\), output power \((P) = 15\ \text{W}\), output resistance \((R_{22}) = 10\ \Omega\), input resistance \((R_{11}) = 30\ \text{k}\Omega\), instability of the transfer coeffi-

Fig. 1

Fig. 2

Fig. 3

Fig. 5

Fig. 7

Fig. 8

Fig. 9

... ($\delta$) over 8 hours of operation $\le 0.01\%$, frequency error $<0.2\%$ up to 20 kHz; for the case of covering the entire frequency range (20 Hz–20 kHz) with a single summing transformer, the nonlinear distortion coefficient $(K_{\mathrm{g}}) \le 0.02\%$.

The circuit of Fig. 5 is free from the shortcomings inherent in the circuit with a summing transformer. However, the absence of a common conductor between the “grounds” of the MA and the load is not always permissible; nevertheless, in specific cases this circuit should be preferred. (The CC in such a system must be non-inverting.)

Fig. 4

Below are the technical characteristics of an MA implemented according to the circuit of Fig. 5: $K_0 = 30$, $U_2 = 300$ V, $P = 15$ W, $R_{22} \ll 20$ ohm, $R_{11} = 30$ kOhm, $\delta \le 0.01\%$, frequency error from 45 Hz to 20 kHz $\le 0.02\%$, $K_{\mathrm{g}} \le 0.02\%$.

The schematic diagram is shown in Fig. 6, where the CC, whose circuit is identical to that of Fig. 1b, is indicated by a dashed line.

If the CC is enclosed by negative feedback, then a two-channel system with an external summer at the output can be implemented without extracting the error in the EAF (Fig. 7). Here the role of the error is played by the static voltage of the CC. The transfer of the system is determined by the expression

\[ K_{\Sigma}=\frac{\gamma\left(K'_0+K\right)}{1+K'_0\beta}, \]

where $\gamma$ and $\beta$ have the previously defined values. When the invariance condition $K=\beta^{-1}$ is satisfied, the resulting error of the system is

\[ \delta_{\Sigma}=\frac{\delta}{1+K'_0\beta}, \tag{11} \]

Fig. 6.

Fig. 10

Fig. 11

Fig. 12 — labels visible in the diagram: \(K_1\), \(K_j\), \(K_n\), \(\beta_n\), \(\beta_j\).

Fig. 13 — labels visible in the diagram: \(R1\), \(R2\), \(K\), \(K_0\), \(\beta\), \(1\), \(\gamma\), \(\alpha\).

Fig. 14 — labels visible in the diagram: \(R1\), \(R2\), \(K\), \(K_0\), \(1\), \(\alpha\), \(\beta\).

Fig. 15 — labels visible in the diagram: \(R1\), \(R2\), \(K\).

Fig. 17 — labels visible in the diagram: \(R_0\), \(K\), \(K_0\), \(R1\), \(R2\).

Fig. 18 — labels visible in the diagram: \(R1\), \(R2\), \(K\), \(-K_0\), \(1\), \(\alpha\), \(\beta\).

i.e., the error of the OK is eliminated (for \(\varphi=0\)). In this case the \(T\)-error of the system is zero, and the gain is strictly determined by the ratio \(K_{\Sigma}=\gamma\beta^{-1}\). The value of \(\delta_y(\varphi)\) is equal to

Fig. 16

\[ \delta_y(\varphi)= \sqrt{ \frac{1-2A\left[1-\cos\left(\varphi'_0-\varphi\right)\right]} {1-2A\left(1-\cos\varphi'_0\right)} } -1. \]

Here \(A=T_0/(1+T_0)^2\simeq T_0^{-1}\).

For small values of the angles one can obtain

\[ \delta_y(\varphi)\simeq \frac{\varphi'_0\varphi}{T_0} - \frac{\varphi^2}{2T_0}. \]

If negative feedback with loop gain \(T\) is introduced into the KK, then

\[ \delta_y(\varphi)\simeq \frac{\varphi'_0\varphi'}{T_0T} - \frac{\varphi'^2}{2T_0T^2} \simeq \frac{\varphi'_0\varphi'}{T_0T} \]

(since the condition \(\varphi'<\varphi'_0\) is readily satisfied).

A comparison of the expressions for the errors of the system with error extraction and with compensation of statism indicates that, in terms of their potential capabilities, they are practically identical. At medium frequencies, greater

the first system is more effective, in which the errors of both channels are multiplied. In addition, the statism voltage exceeds the error voltage, which somewhat increases the required value of the maximum output voltage of the CC in the statism-compensation system. The only positive feature of such a system remains the combination of the functions of the \(\beta\)-circuit of the OK and the UVO, which reduces the number of precision elements.

In the case of an inverting OK, the UVO is implemented in the form of an active subtraction circuit, the simplest case of which is shown in Fig. 8. The error voltage is taken from point A and can be introduced into the OK similarly to the systems considered above. The difficulty of implementing an active subtracting device with high suppression of the in-phase component, and especially with high stability, makes such a method of extracting the error hardly acceptable, except in cases that will be specially stipulated.

A system with an internal summator at the output. The FS and GS are shown in Fig. 9. The main advantage of this structure over the one considered earlier consists in a significant reduction of the requirements imposed on the summator, which in this case is enclosed by negative feedback.

Let us consider the basic properties of the system:

\[ K_{\Sigma}=\frac{K_0+K^1\gamma}{1+K^1\beta}. \]

The invariance condition is determined by the relation \(\beta=\gamma K_0^{-1}\); in this case

\[ \delta_{\Sigma}=\delta_0(1+K^1\beta)^{-1}. \]

Comparison of the last expression with (11) shows the advantage of the system with an external summator, since the inequality \(\delta \ll \delta_0\) is satisfied,

\[ \delta_{\Sigma}(\varphi)= \sqrt{ \frac{1-2A[1-\cos(\varphi^1-\varphi_0)]} {1-2A(1-\cos\varphi^1)} } -1. \]

or, taking into account the simplifications adopted earlier,

\[ \delta_{\Sigma}(\varphi)\simeq \frac{\varphi^1\varphi_0}{T}-\frac{\varphi_0^2}{2T}. \]

Here \(A=T/(1+T)^2\simeq T^{-1}\); \(T=K^1\beta\).

For an OK with negative feedback we finally have

\[ \delta_{\Sigma}(\varphi)\simeq \frac{\varphi^1\varphi_0^1}{T T_0} -\frac{\varphi_0^{1\,2}}{2T T_0^2}. \]

In this case the last term may have a more substantial value than in the case of a statism-compensation system with an external summator, since in practice \(\varphi_0^1 \gg \varphi'\), which indicates somewhat smaller limiting errors of the latter system.

It should be emphasized that, whereas in systems with an external summator the CC may in principle also not be enclosed by negative feedback, in the system under consideration this is mandatory. It is obvious that, with considerable gain in the OK, certain difficulties may arise in the sense of ensuring a sufficiently large loop gain in the CC. Nevertheless, enclosing the summator by negative feedback in the circuit under consideration is a sufficiently weighty basis for its practical application.

A known implementation of the structure considered is given in [2], Fig. 10. Here the OK is loaded by a summing transformer, which is unacceptable in the case of powerful high-voltage measuring amplifiers, for which

one may recommend the circuit of Fig. 11. The structures considered, with statism compensation, are particular cases of iterative systems [3]. A substantial drawback of the systems considered above is the need for considerable gain in the correction channel.

Two-channel systems with an external summator at the input, in which positive feedback (PFB) is implemented, have high metrological characteristics at nearly unity gain of the correction channel.

Although the first works devoted to the application of PFB in measuring amplifiers appeared about 40 years ago [4, 5], and subsequently considerable attention was paid to these questions in many works by Soviet and foreign authors, measuring amplifiers with PFB have still not found practical application. In our opinion, the main reason for this situation is the absence of a clearly developed theory of stability for such devices. Thus, some authors [6, 7], while presenting practical circuits of measuring amplifiers with PFB, do not consider stability questions at all; others [8, 9] confine themselves to the assertion that the loop gain of the PFB channel must be less than unity. The invalidity of such an assertion was proved as early as in [1], where it was shown that a system unstable in the open-loop state (PFB with unit loop gain is introduced) becomes, under certain stability conditions, stable when the general negative-feedback loop is closed. One may point to practically the only work [10] in which the theory of amplifiers with PFB is considered at a high scientific level. In this work it is proved that the application of PFB gives a substantial gain in the accuracy of a measuring amplifier in a very narrow frequency range (in comparison with the frequency band of the amplifier with negative feedback designed according to Bode). But this work considers only the case of combined feedback, when PFB is introduced inside an amplifier encompassed by general negative feedback. Recently, measuring amplifiers with an external PFB channel have found practical application; in them, owing to the absence of correlation between the amplitude-frequency characteristics of the main channel and the correction channel, correction is possible that ensures stable operation of the system with a broadband PFB channel.

Let us consider the basic properties and possible variants of practical implementation of measuring amplifiers with PFB.

A system with an internal PFB channel of the feedback network is shown in Fig. 12. The transfer of the system is determined by the expression

\[ K_{\Sigma} = \frac{\prod_{i=1}^{n} K_i} {1 - K_1 \beta_{\text{p}} + \prod_{i=1}^{n} K_i \beta_0}. \]

Here \(\beta_0\) is the NFB coefficient, \(\beta_{\text{p}}\) is the PFB coefficient. If the feedback sensitivity is understood as the ratio

\[ D = \frac{dK_i}{K_i} \bigg/ \frac{dK_{\Sigma}}{K_{\Sigma}}, \]

then, when the condition \(K_1\beta_{\text{p}} = 1\), (12) is fulfilled for all cascades not encompassed by PFB, \(D = \infty\), and for the cascade encompassed by PFB,

\[ D_1 = \prod_{i=1}^{n} K_i \beta_0 = T_0, \]

i.e., it is equal to the loop gain of the system without PFB. If, as the cascade \(K_1\), a low-power preamplifier cascade is chosen, having a low level of nonlinear distortion and whose stabilization is easy to carry out, then the effect of reducing the error upon intro-

or POS may be considerable. The resulting error is then equal to \(\delta_\Sigma=\delta_1T_0^{-1}\), (13) where \(\delta_1\) is the error of the stage encompassed by POS. In this system \(T\)—the error—is equal to zero, and the system transmission, when condition (12) is satisfied, is equal to \(K_\Sigma=\beta_0^{-1}\). Exact fulfillment of condition (12) is possible only with zero phase shift of the stage encompassed by POS. To preserve the indicated condition over a wide frequency range, it is necessary to increase the bandwidth of the stage with POS, which, as will be shown, is limited by the stability condition.

As an example, consider the case of an amplifier consisting of three stages: \(K_1, K_2, K_3\), each of which is represented by a first-order link. The stage \(K\) is encompassed by POS.

Taking equality (12) into account, the characteristic equation of such a system, encompassed by an overall OOS with loop gain, takes the form

\[ p^3\tau_1\tau_2\tau_3+p^2\tau_1(\tau_2+\tau_3)+p\tau_1+T_0=0. \]

According to the Routh–Hurwitz criterion, the stability condition is determined by the inequality

\[ \tau_1>T_0\frac{\tau_2\tau_3}{\tau_2+\tau_3}, \]

from which it follows that the bandwidth of the stage with POS is correlated with the bandwidth of the remaining stages and, for large values of \(T_0\), is completely unsatisfactory.

System with an External Summator at the Input

The FS and GS are shown in Fig. 13. The transmission of the system is determined by the expression

\[ K_c=\frac{\gamma K_0^1}{1-\gamma K+K_0^1\beta}, \tag{14} \]

where \(K\) is the gain of the CC; \(\gamma,\beta\) are quantities defined earlier. When the condition \(K\gamma=1\) is fulfilled, the resulting error of the system is equal to

\[ \delta_\Sigma=\delta_0T_0^{-1}, \tag{15} \]

where \(\delta\) is the error of the CC. Although formulas (13) and (15) are formally identical, in a system with an external channel, owing to the practically unity gain of the CC (\(\gamma\simeq1\)), the latter may be encompassed by deep local OOS, so that the inequality \(\delta\ll\delta_1\) will hold, which indicates the higher efficiency of the scheme under consideration. Obviously, in this case as well the decisive importance is assumed by the phase component of the error, which can be obtained from (14), taking into account the equality \(K\gamma=1\),

\[ \delta_\Sigma(\varphi)= \frac{1}{ \sqrt{ 1+\frac{2}{T_0} \left[ \cos\varphi_0^1-\cos(\varphi_0^1-\varphi) +\frac{2(1-\cos\varphi)}{T_0} \right] } } -1 \]

The phase error is absent only when \(\varphi=0\). However, unlike the systems considered earlier, the minimum phase shift of the CC is limited by the stability condition. The lack of correlation between the frequency characteristics of the OC and the CC in this system nevertheless makes it possible to transform the characteristics of the CC in the required manner, while preserving the optimal characteristics of the OC.

As an example, a system was considered in which, for clarity, a classical amplifier optimally corrected according to Bode \([1,\ \text{p. }548]\) was chosen as the OC.

The cutoff frequency of amplifier 1 is 1 MHz, the phase margin is 30°, and the amplitude margin is 6 dB.

The correcting circuit is introduced in series with the CC. In this case the CC itself is taken as ideal in the band of the MC’s controlled frequencies, i.e., it is assumed that the AFC of the CC is determined only by the correcting circuit.

As the correcting circuit, an interstage correcting circuit optimal according to Bode was chosen, defined by the relations

\[ K=-20\lg\left[\sqrt{x^2-1}+x\right]; \qquad \varphi=\operatorname{arc\,tg}\frac{x}{\sqrt{1-x^3}}, \]

where \(x=\frac{\omega}{\omega_0}\), \(\omega_0\) is the cutoff frequency of the correcting circuit.

The limiting value of the loop gain of the MC with the indicated amplitude and phase margins is 29 dB. In order to preserve the former amplitude and phase margins of the resulting system with the CC, the phase margin of the MC was increased by 10° and amounted to 40°. In this case the limiting value of the loop gain, according to the known Bode relations, decreases to 25.4 dB. The cutoff frequency of the correcting circuit was taken equal to 6 MHz, i.e., 6 times higher than the cutoff frequency of the MC. The resulting hodograph of the system with the CC in this case remained the same as that of the original MC amplifier, i.e., the stability of the system did not deteriorate.

The values thus obtained for the error of the modulus due to the phase of the original amplifier and of the system with the CC are given in the table.

Table

The results obtained show that optimal correction makes it possible to preserve a considerable effectiveness of the system over the entire band of the optimally corrected MC (even taking into account the reduction in the limiting value of the loop gain of the latter due to the increase in phase margin).

Although the arguments presented do not claim rigor, they allow one to speak of the promising nature of using PFB with an external channel.

The system with an external summator at the input, with separation of the FS and GS errors, is shown in Fig. 14. The transfer of the system is determined by the expression

\[ K_\Sigma=\frac{K_0}{1-\alpha K+\gamma K_0 K}. \tag{16} \]

When the tuning conditions \(\gamma K_0=\alpha\) and invariance \(aK=1\) are satisfied, the gain upon introduction of the CC remains unchanged, i.e., \(K_\Sigma=K_0\), and the error (for \(\varphi=0\)) is \(\delta_\Sigma\simeq-\delta_0\delta\).

For equal depths of NFB in the MC, the error of this system is significantly smaller than that of the system in Fig. 13. In terms of stability, both systems are practically identical.

The error of the modulus due to the phase is determined by the expression

\[ \delta_\Sigma(\varphi)= \frac{1}{\sqrt{3-2[\cos\varphi-\cos(\varphi+\varphi_0)+\cos\varphi_0]}}-1, \]

which, for small angles, takes the form \(\delta_\Sigma(\varphi)\approx \varphi\varphi_0\).

Although, in principle, the minimum value of the phase of the CC, \(\varphi\), in this system is limited by stability conditions, in a potential sense this system is almost identical to a system with an external adder at the output, since in the latter, because of the large required gain, it is difficult to obtain a small phase shift in the CC. In practice, the unity gain of the CC and the absence of an output adder are an indisputable advantage of the two-channel system with PFB in the case of large gains in the MC and operation of the latter into a load with a large reactive component. Recommendations for choosing one of the two systems with an external adder are identical to those given earlier for systems with an external output adder, since they are mirror images.

The simplest implementation of the indicated circuit is shown in Fig. 15 [11]. With a large input resistance of the MC, the leakage inductance of the transformer has an insignificant effect; however, the interwinding capacitance greatly reduces the input resistance at high frequencies. Therefore, the indicated circuit is effective only when operating from signal sources with low output resistance. Otherwise, the introduction of a buffer follower is necessary. A high-voltage MA according to the circuit of Fig. 15, the basic circuit of which is shown in Fig. 16, has the following characteristics:

\(K_0=30,\ U_2=300\ \mathrm{V},\ P=15\ \mathrm{W},\ R_{22}\le 6\ \Omega\) (up to \(100\ \mathrm{kHz}\)), \(R_{11}=30\ \mathrm{k}\Omega\), frequency error up to \(100\ \mathrm{kHz}\le 0.1\%\), \(K_\Gamma\le 0.01\%\) (up to \(20\ \mathrm{kHz}\)), \(\delta\le 0.01\%\).

With an inverting MC, an original solution proposed in [7] can be used (Fig. 17). Here the amplifier simultaneously performs the role of an IOA and a CC. The CC may be enclosed by NFB and, in the simplest case, implemented as a follower, since the necessary gain of the CC in this circuit is strictly equal to unity. A disadvantage of such an implementation is the need for a “floating” source, whose capacitance with respect to ground also shunts the input resistance.

System with an internal adder at the input of the FN and SN in Fig. 18.

The transmission of the device is determined by the expression

\[ K_\Sigma=\frac{K_0(1+\alpha K)}{1+\gamma K_0K}. \]

When the tuning condition \(\gamma K_0=\alpha\) is satisfied, the gain of the system upon introduction of the CC remains equal to \(K_0\), and the error is determined by the expression

\[ \delta_\Sigma=\frac{\delta_0}{1+K}. \]

Error NFB is implemented in the system. It is not difficult to see that practically the same reduction in error could have been achieved by adding, in the MC path, an additional stage with gain \(K\), followed by introduction of such NFB, under which the original...

gain value \(K_0\). In this case, problems of ensuring stability arise that are identical for both variants.

REFERENCES

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