UDC 621.317.7.001.6

RMS Value Comparators

M. S. Roitman, Yu. G. Svinolupov, V. P. Zibalov, N. P. Kalinichenko

(Presented by the scientific seminar of the Department of Radio Engineering)

A brief analysis is given of the comparison error in comparators implementing the method of periodic substitution, in which, from the entire spectrum of error components, only the constant and slowly varying components—whose frequencies are much lower than the commutation frequency—are excluded. The post-commutation transient processes that arise in a measuring converter when comparing an alternating voltage with a constant one are analyzed. Ways of eliminating the influence of transient processes on comparison accuracy are shown.

Certain modifications of the previously proposed method of periodic automatic correction are considered, making it possible to significantly expand its capabilities.

8 illustrations, 7 references.

RMS value comparators are finding increasingly wide application in measurement technology. This is explained above all by the fact that the method of voltage measurement based on the comparison of rms values is the most accurate at the present time.

However, as was noted in [1], all rms value comparators with a wide frequency range have comparatively low temporal stability, and one possible way of ensuring high long-term stability of a comparator is to apply the method of periodic conversion. The resulting measurement error in comparators of this type is determined as

\[ \Delta H_{\mathrm{res}} \sim \frac{d\,\Delta H_{s}(x,t)}{dt}\,\Delta T + \Delta H_{\mathrm{com}}\left(\frac{1}{T}\right)\ldots, \tag{1} \]

where \(T\) is the commutation period;

\(x\) is the measured quantity;

\[ \Delta H_{\mathrm{com}}\left(\frac{1}{T}\right) \]
is the error component due to post-commutation transient processes.

In the general case,
\[ \Delta H_{\mathrm{com}}\left(\frac{1}{T}\right) \]
may be a random quantity, and \(\Delta H_{\mathrm{res}}\) can be reduced by increasing the time constant of the filter at the output of the synchronous detector SD [1]. However, this ultimately leads to an increase in the inertia of the comparator and can be justified only when measuring rms values of voltages that do not vary with time or that vary slowly. In any case, when comparing voltages by the indicated method.

from the entire spectrum of error components only the constant and slowly varying components are excluded, whose frequencies are much lower than the commutation frequency. It is inadvisable, however, to increase the latter for the following reasons: first, the commutation period must be much greater than the equivalent time constant of the measuring converter (MC); second, increasing the commutation frequency leads to an increase in additional comparison errors caused by transient processes,

\[ \Delta H_{\mathrm{com}}\left(\frac{1}{T}\right). \]

Particular attention should be paid to the latter in the low-frequency region of the measured voltages, i.e., where the converter time constant is commensurable with the period of the measured voltage and the measuring converter “tracks” the instantaneous value of the power. It is obvious that the change in instantaneous power at the moment of commutation will depend to a large extent on the phase relations of the controlling and commutated voltages [1], which ultimately will lead to a change in the character and amplitude of the transient process during switching.

Of interest is switching from a dc voltage to an ac voltage, and conversely, as the case most frequently encountered in practice. Below an attempt is made to analyze the transient processes occurring in a photoelectric comparator of rms values under such switching, and practical results are presented.

The analysis is divided into two stages. The first is the transition from a dc voltage to an ac voltage; the second, from an ac voltage to a dc voltage. In both cases the photoelectric converter with an incandescent lamp and a photoresistor is interpreted, for small increments of the input signal, as series-connected blocks implementing a quadratic nonlinearity and a second-order inertial element. Then the output quantity of the converter (the conductance of the photoresistor) is determined as

\[ h(t) \doteq F(p)=K_{\mathrm{p}}P(p)\cdot G(p), \]

where \(K_{\mathrm{p}}\) is the transfer coefficient of the photoelectric converter, and \(P(p)\) is the image of the change in input power.

In the first case (when switching from a dc voltage to an ac voltage) \(P(p)\doteq h_{1}(t)\) has the form

\[ P(p)\doteq h_{1}(t)=h_{0}\sin(\Omega t+\varphi), \]

\[ \Omega=2\omega \]

is twice the frequency of the ac voltage;

\[ h_{0}=1. \]

\[ \varphi_{0}=\frac{\varphi}{2}+\frac{\pi}{4} \]

is the initial phase angle of the ac voltage at the point of commutation.

\[ h_{2}(t)\doteq G(p)\cdot K_{\mathrm{p}} =\frac{K_{\mathrm{p}}}{(1+T_{1}p)(1+T_{2}p)}, \]

where \(T_{1}\) and \(T_{2}\) are the equivalent time constants of the incandescent lamp and the photoresistor, respectively.

Then

\[ h_{2}(t)= \frac{T_{1}T_{2}\left(e^{-\frac{t}{T_{1}}}-e^{-\frac{t}{T_{2}}}\right)} {T_{2}-T_{1}}\cdot K_{\mathrm{p}}. \]

Hence \(h(t)\doteq G(p)\cdot P(p)\) will be determined as (using the convolution theorem)

\[ h(t)=\int_0^t \left[h_1(t)\cdot h_2(t-\tau)\right]\,d\tau = \]

\[ =\frac{K_{\mathrm p}T_1T_2}{T_2-T_1} \int_0^t\left(e^{-\frac{\tau}{T_1}}-e^{-\frac{\tau}{T_2}}\right)\sin(\Omega t-\Omega\tau+\varphi)\,d\tau = \]

\[ =\frac{K_{\mathrm p}T_1T_2}{T_2-T_1} \left[ \int_0^t e^{-\frac{\tau}{T_1}}\sin(\Omega t-\Omega\tau+\varphi)\,d\tau - \right. \]

\[ \left. -\int_0^t e^{-\frac{\tau}{T_2}}\sin(\Omega t-\Omega\tau+\varphi)\,d\tau \right]. \]

After expanding the brackets and carrying out the corresponding transformations, the first part of the expression \(h(t)\) will be equal to

\[ h'(t)= \frac{T_1T_2K_{\mathrm p}} {T_1(T_2-T_1)\left(\dfrac{1}{T_1^2}-\Omega^2\right)} \left[ e^{-\frac{t}{T_1}}(\Omega T_2\cos\varphi-\sin\varphi)-\cos(\Omega t+ \right. \]

\[ \left. +\varphi)\Omega T_1+\sin(\Omega t+\varphi) \right]\ldots \tag{2} \]

and, correspondingly, the second part

\[ h''(t)= \frac{T_1T_2K_{\mathrm p}} {T_2(T_2-T_1)\left(\dfrac{1}{T_2^2}-\Omega^2\right)} \left[ e^{-\frac{t}{T_2}}(\Omega T_2\cos\varphi-\sin\varphi)-\cos(\Omega t+ \right. \]

\[ \left. +\varphi)\Omega T_2+\sin(\Omega t+\varphi) \right]\ldots \tag{3} \]

\(h(t)=h'(t)+h''(t)\) gives a clear picture of the character of the transient process when switching a photoelectric converter (PC) from a constant voltage to an alternating one equal to the former in rms value. When changing from an alternating voltage to a constant input action, the inertial element should be assigned the relative change in instantaneous power supplied to it at the moment of switching, i.e., as a nonunit step whose amplitude varies from \(-1\) to \(+1\), depending on the phase of the alternating voltage at which the switching is carried out.

Then, knowing

\[ F(p)=\frac{1}{p(p+a)(p+b)} \doteq f(t)=\frac{1}{ab}+\frac{1}{b-a}\left(\frac{e^{-bt}}{b}-\frac{e^{-at}}{a}\right), \]

one can, by analogy, determine \(h(t)\) for such switching:

\[ h(t)=A\cdot K_{\mathrm p} \left[ \frac{1}{T_1T_2}+\frac{1}{T_2-T_1} \left( \frac{e^{-\frac{t}{T_1}}}{T_1} - \frac{e^{-\frac{t}{T_2}}}{T_2} \right) \right]\ldots, \tag{4} \]

where

\[ A=\sin\left(2\varphi_0-\frac{\pi}{2}+\operatorname{arc\,tg}\Omega T_1+\operatorname{arc\,tg}\Omega t_2\right). \]

From expressions (4, 2, 3) it is evident that the smallest amplitude of the transient process when switching the PC from a constant voltage to an alternating one is observed when the alternating voltage is commutated at the phase

\[ \varphi_0 = \frac{1}{2}\left(\frac{\pi}{2} - \operatorname{arc\,tg}\frac{\Omega T_1 - \Omega T_2}{1 - \Omega^2 T_1 T_2}\right), \]

and when switching the PC from alternating voltage to direct voltage, in phase,

\[ \varphi_0 = \frac{\pi}{2}\cdot n,\quad n = 0,\; 1,\; 2, \]

which, generally speaking, is difficult to implement in practice.

A more detailed observation of the transient process is provided by analysis of the mathematical model of the PC with the aid of a computer. In [2] a system of differential equations is given that describes an incandescent lamp. It does not appear possible to obtain a general solution of such a system of equations, since the system is nonlinear, and moreover the nonlinearity enters in a rather complicated way. Therefore, a simplified system of differential equations describing the PC [2] was solved stage by stage by the Runge—Kutta method. At the first stage, the conductance of the photoresistor was determined when the incandescent lamp was supplied with alternating voltage,

\[ \sigma_{\mathrm{av}} = \frac{1}{T}\int_0^T \sigma_{\mathrm{av}}\,dt, \]

where \(T\) is the period of the supply sinusoidal voltage. We note that \(\sigma_{\mathrm{av}}\) was found after completion of the transient process, a criterion for which may be, for example,

\[ \frac{\sigma_{\max/\mathrm{prev}} - \sigma_{\max/\mathrm{cur}}}{\sigma_{\max/\mathrm{cur}}} < \varepsilon . \]

At the second stage, the conductance of the photoresistor was found when the lamp was supplied with a constant voltage,

\[ U_{\mathrm{in}}(t) = \frac{U_m}{\sqrt{2}} + \delta_{\mathrm{lf}}, \]

where \(\delta_{\mathrm{lf}}\) is the low-frequency comparison error. The magnitude of the latter was determined from the condition that \(\sigma = \sigma_{\mathrm{av}}\), where \(\sigma_{\mathrm{av}}\) is the value of the conductance of the photoresistor in the steady state.

After this, switching of L was carried out from direct voltage to alternating voltage and vice versa; in doing so, various phases of the alternating voltage were specified. Analysis of the results obtained shows that the minimum amplitude of the transient process is observed when the alternating voltage is switched in a phase close to \(0\) or \(\pi/2\).

Figure 1 shows an experimental circuit that makes it possible to observe the transient processes occurring in the PC on the screen of an oscilloscope, where G is a sinusoidal-voltage generator with low output resistance;

K is an electromechanical commutator;

CU is the commutator control circuit;

S is the circuit for synchronizing the generator voltage and CU;

L\(_1\), \(R_l\), \(R\), \(R_\phi\) are the photoelectric transducer.

A mass-produced OEP-1-type optocoupler was used as the PC. In Figs. 2,a and 2,b the oscillograms of the output voltage of the PC and the results obtained on a computer when switching the alternating voltage in the corresponding phases are shown. In Fig. 2,c the oscillograms of the transient processes are shown, as well as the oscillogram of the input voltage with the instant of its switching indicated. Figure 3 shows intermediate results of solving the system of proposed equations [2],

Fig. 1

as well as the transient process at the output of the FC for various phases of switching of the alternating voltage.

The magnitude of the comparison error caused by transient processes is determined as

\[ I_{\mathrm{avg}}=\frac{1}{t}\int_{0}^{t} U(t)\cdot Y(t)\,dt, \]

where \(I_{\mathrm{avg}}\) is the magnitude of the average current of the synchronous detector when the rms values of the voltages at the comparator input are equal;

\(Y(t)\) is a description of the change in the conductance of the SD elements;

\(U(t)\) is the function describing the change in the input signal of the SD.

The value of this quantity can be reduced by using, as noted earlier, synchronization or strobing. Practical results indicate that strobing is a more effective measure for eliminating the influence of transient processes. Figure 4 shows the circuit of a comparator implementing this method. Moreover, the use in it of a storage element makes it possible practically to eliminate the low-frequency filter from the structure of the comparator, thereby significantly increasing its speed of response. The device operates as follows.

The signal from the FC output, through amplifier У, is applied to the input of the synchronous detector (SD), and from its output, through interrupter \(K_c\), it is applied simultaneously to the input of the matching device (MD), with a high input resistance, and to the input of the storage element SE. The function of the latter may be performed by a capacitor of type K50-7, while the MD may be made on the basis of an insulated-gate transistor of type KP305Zh. On a signal from the MD control circuit, interrupter \(K_c\), for the duration of the strobe, breaks the connection of the SD with the MD, thereby eliminating the influence of transient processes on the indication of the instrument NI; the voltage at the MD input during this time is maintained constant by means of the storage element. The control circuit consists of a master generator MG, which forms the strobe signal, and a trigger, which is started by the leading edge of the strobe and controls the commutator and the synchronous detector. Such a comparator can be used for stabilizing slowly varying voltages.

It should be noted that the ultimately achievable comparison accuracy in the low-frequency region of the measured voltages is limited

Fig. 2, a

Fig. 2b

Fig. 2c

Fig. 3

low-frequency error of the FP, \(\sigma_{f\ell}\), and by the level of the second harmonic at the converter output, \(K_{\phi2}\).

\[ \delta_{f\ell}= \frac{1}{16\,[1+(2\omega T_\ell)^2]} \left[ K_{\mathrm{p}}-K_\ell a\,\frac{R_\ell-R}{R_\ell+R} \right]. \]

Here \(K_{\mathrm{p}}, K_\ell\) are the transfer coefficients of the FP and of L, respectively,

\[ a \approx 1.2; \]

\(T_\ell\) is the time constant of the filament body of L.

\[ K_{\phi2}=\frac{\sigma_{\phi2}}{\sigma_{\mathrm{cp}}} = \frac{ K_{\mathrm{p}}\cos(2\omega t+\pi+\varphi_\ell+\varphi_\phi) }{ 2\sqrt{1+(2\omega T_\ell)^2}\sqrt{1+(2\omega T_\phi)^2} }, \]

Fig. 4

where \(T_\phi\) is the equivalent time constant of the photoresistor,

\(\omega\) is the frequency of the measured voltage.

As for \(\delta_{f\ell}\), its value may, in the limiting case, be ignored and treated as a systematic error; however, the second harmonic arising at the FP output may lead to considerable distortion of the comparator readings, and, in order to eliminate this influence, a filter should be placed at the FP output; it may be of any order. It should be borne in mind, however, that in voltage-stabilization systems with such comparators the filter must be single-valued.

As noted earlier [1], in comparators implementing the method of automatic substitution (periodic conversion), very stringent requirements are imposed on the commutator, since the errors introduced by it enter completely into the resulting comparison error; the principal requirements include:

— identity of the transfer coefficients of the commutator channels for direct and alternating current;

— the required stability of the transfer coefficients;

— high reliability indices.

Unfortunately, at present not a single commutator based on electron tubes or semiconductor elements satisfies this far from complete list of requirements. Additional matching devices that make it possible to reduce the requirements imposed on commutators are so far feasible only for alternating current. All this makes it necessary to use, in high-frequency comparators, either hybrid electromechanical commutators [1], or to give preference to magnetically controlled sealed contacts. The reliability of comparators with commutators of this type can be increased only by reducing the commutation frequency. However, this greatly complicates the amplification and filtering of the alternating component. On the other hand, reducing the commutation frequency is inadvisable, as was noted earlier.

The method of periodic automatic correction is to a considerable extent free of the indicated shortcomings; in essence, it is a distinctive combination of the methods of automatic substitution and continuous automatic correction. A comparator constructed according to

this principle, is described in [1]. Here we shall consider modifications of this method that make it possible to expand its capabilities considerably.

A disadvantage of the comparator with automatic correction described in [1] is that, when the comparator is being corrected, both of its inputs are connected to the reference voltage and the measurement process is interrupted. When such a comparator is used in voltage-stabilization systems, this leads to undesirable breaks in the feedback loop, as a result of which a change is observed in the output voltage of the stabilized source during correction and after the loop is closed [7].

In the rms-value comparator proposed below, the indicated disadvantages can be eliminated by introducing into the circuit an additional measuring converter that performs the functions of the correcting element; this made it possible, without interrupting the measurement process, to correct the comparator parameters and thereby ensure its high long-term stability.

The comparator circuit is shown in Fig. 5, a.

\(L_1, R_{l1}, R_1, R_2, R_3, T_1, R_4\) constitute MC\(_1\); \(L_2, R_{l2}, R_5, R_6\)—MC\(_2\); \(L_3, R_{l3}, R_7, R_8, R_9, T_2, R_{10}\)—MC\(_3\), which performs the role of the correcting element.

Fig. 5, a

The amplifier \(K_1\), whose inputs are connected to points \(a\) and \(b\), forms, together with MC\(_1\) and MC\(_2\), the differential comparator proper. The amplifier \(K_2\), whose inputs are connected to points \(b\) and \(c\), and contact 1 of relay I serve to balance the bridge \(R_8, R_9, R_5, R_6\); the amplifier \(K_3\), whose inputs are connected to points \(a\) and \(c\), and contact 2 of relay I serve to balance the bridge \(R_2, R_3, R_5, R_6\). Contact group 1P2 serves to connect MC3 either to the reference source \(U_0\) or to the measured voltage \(U_x\). The device operates as follows.

In the initial state the contact group K1P2 is in position 1, contact 1P1 is closed, and contact 2P1 is open. The filament lamp \(L_3\), through 1KP2 and resistor \(R_{l3}\), is connected to voltage \(U_0\). The bridge unbalance voltage between points \(b\) and \(c\), arising

as a result of the nonuniform drift of the photoresistors, is applied to the inputs of amplifier K and from its output, through contact 1P1, to storage capacitor \(C_2\) and the gate of field-effect transistor \(T_2\) with an insulated gate. Thus, the potentials between points \(b\) and \(c\) are equalized, with the static-error accuracy, by changing the resistance \(R_{\mathrm{T}2}+R_{10}\). After a certain time of continuous operation, on a signal from the control circuit (CC), group 1P2 is switched to position 2. At the same time contact 1P1 opens, and contact 2P1 closes. Measuring converters \(IP_1\) and \(IP_3\) are connected to voltage \(U_0\). The potentials between points \(a\) and \(c\) are compared; if \(U_{ac}\ne 0\), this voltage, through amplifier \(K_3\) and contact 2P1, is applied to gate \(T_1\) and storage capacitor \(C_1\).

Thus, by means of the two operations described above, the drift of the differential comparator (\(IP_2\) and \(IP_2\)) is corrected, while the continuity of measurement is not disturbed. In the intervals between switchings, the voltages at the gates of the field-effect transistors are maintained by the charges on capacitors \(C_1\) and \(C_2\), which have high leakage resistances. Capacitors of the K77-1 type were used as the storage capacitors; as the controlled resistor, a KP305Zh field-effect transistor was used, which in this case is connected “partially,” by means of the divider \(R_1, R_4\), \((R_7, R_{10})\), in order to reduce the effect of its own fluctuations on the comparison accuracy. Here the attenuation factor is determined as

\[ K=\frac{R_1R_3}{(R_4+R_{\mathrm{T}}+R_1)(R_2+R_3)} . \]

As experimental studies have shown, with this connection the comparator drift referred to the input, caused by fluctuations of the field-effect transistor, does not exceed \(0.001\%\) in 5 min. With careful adjustment of the comparator, this makes it possible to increase significantly the time interval between comparator corrections.

Fig. 5, b

The disadvantages of this correction method include doubled power consumption due to \(IP_3\), and complication of the CC in order to eliminate transient processes during switching. As practical results show, the correction time lies within \(1\)—\(1.5\) s, and this makes it possible, by correcting the comparator after a time \(t>3\) min, to successfully apply the method of automatic comparator correction [1], but with storage of the measurement result during the correction time.

The circuit of such a comparator is shown in Fig. 5,b. The unbalance amplifier K is connected, through the normally closed contact 1P1, to the gate of a field-effect transistor that performs the functions of a source follower and of the storage capacitor \(C_2\). During correction, contact 1P1 opens and closes only after the transients associated with switching NP\(_2\) have ended. During this time (1.5–2 sec), the voltage at the output of the source follower changes only slightly. Such a circuit forms the basis of the wide-range rms-value comparator being developed.

REFERENCES

1. M. S. Roitman, Yu. G. Svinolupov, Yu. A. Bulatov. Comparators of alternating voltages with high long-term stability. Izvestiya TPI, vol. 270, Tomsk, 1973.

2. M. S. Roitman. Theory of photoelectric converters. Avtometriya, 1972, No. 1.

3. Electrical methods of automatic control. Edited by K. B. Karandeev. Energiya, 1965.

4. V. I. Anisimov, A. P. Golubev. Transistor modulators. Energiya, 1964.

5. M. S. Roitman, Yu. M. Fomichev, Yu. A. Bulatov, Yu. G. Svinolupov. GK-12 generator-calibrator. Izv. TPI, vol. 270, Tomsk, 1973.

6. Yu. P. Volkov, B. A. Kalinchuk, E. D. Koltik, O. A. Pichugin, V. P. Piastro. Optoelectronic modulators. Izmeritel’naya tekhnika, 1973, No. 8.

7. Yu. M. Fomichev, Yu. A. Varakuta, Yu. G. Svinolupov. GK-12A generator-calibrator. (In the present collection).