UDC 621.314.6 : 621.317.72

MEASURING ELEMENT FOR SYSTEMS STABILIZING THE EFFECTIVE VOLTAGE VALUE

A. V. LENYSHIN, E. G. EMELYANOV

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

The article presents an algorithm for piecewise-linear approximation of the quadratic current-voltage characteristic of a measuring transducer, which ensures minimal methodological error for a quasi-sinusoidal form of the voltage being converted. Expressions are obtained for the relative transfer coefficient of two variants of measuring-element circuits for the quasi-effective value of the voltage. A circuit for temperature compensation of the zero drift of the measuring element is presented, making it possible to reduce the drift to a level below \(10^{-4}\). It is shown that, in order to reduce the level of the alternating component of the signal at the transducer output, it is effective to use a passive notch filter tuned to the frequency of the second harmonic of the stabilized voltage.

6 illustrations, 6 references.

Precision sources of alternating voltage are finding ever broader application in various branches of the national economy, and in this connection the requirements imposed on their metrological and operating characteristics are continuously increasing.

One of the principal problems that must be solved when designing such sources is ensuring the specified stability of the effective value of the output voltage over a wide range of ambient temperatures and during continuous operation for hundreds and even thousands of hours.

As a first approximation, the problem reduces to developing a high-precision measuring transducer of the effective value of voltage (IPDZN) possessing high temperature and time stability. Among the most accurate IPDZN are thermoelectric and photoelectric ones [1, 2], which, however, have low time and temperature stability. The use of the automatic substitution method to eliminate the indicated shortcomings of thermoelectric and photoelectric IPDZN leads to a substantial complication of the IPDZN circuit and significantly reduces their speed of response. In this connection, measuring transducers of the quasi-effective value of voltage with sliding bias, constructed on the principle of piecewise-linear approximation of a quadratic current-voltage characteristic [3, 4, 5], are of interest.

The advantages of such measuring transducers (IP) are their simplicity, relatively high stability of the transfer coefficient, and wide bandwidth. Their principal shortcomings are a low transfer coefficient and the presence of a methodological error caused by

by the inaccuracy of approximating the quadratic volt-ampere characteristic, with the desire to reduce the systematic error leading to a reduction in the transfer coefficient.

For a given complexity of the converter, the systematic error for different methods of approximation is different and, moreover, depends not only on the harmonic coefficient, but also on the form of the signal being converted. Therefore, when designing such MEs, the choice of the approximation method is of great importance. The investigations carried out made it possible to establish that the smallest systematic error in converting quasi-sinusoidal voltages with \(K_{\mathrm{g}} \le 0.1\) is ensured if the approximation is performed by the least-squares method, i.e., when the approximating function (Fig. 1)

Fig. 1

\[ y_1(x)=y_1(x_i)+\frac{y_1(x_{i+1})-y_1(x_i)}{x_{i+1}-x_i}(x-x_i) \]

and the parabola \(y=x^2\) intersect in such a way that the integral

\[ S=\int_{x_i}^{x_{i+1}}\left[x^2-y_1(x_i)-\frac{y_1(x_{i+1})-y_1(x_i)}{x_{i+1}-x_i}(x-x_i)\right]^2 dx \]

has a minimum value. The approximation algorithm is carried out under the following conditions:

— the approximation range \([0, x_{\max}]\) is divided into equal intervals

\[ \Delta x=x_{i+1}-x_i=\mathrm{const}; \]

— the maximum negative absolute approximation errors at the points \(x_i, x_{i+1}\) are equal and are determined by the expression

\[ \Delta^-=\frac{1}{6}(x_{i+1}-x_i)^2; \]

— the maximum positive absolute approximation error at a point lying inside the interval \([x_i, x_{i+1}]\) is equal to

\[ \Delta^+=\frac{1}{12}(x_{i+1}-x_i)^2; \]

— the abscissa of the first break point of the approximating function \(y_1(x)\) is equal to

\[ x_1=\frac{13}{18}\cdot\Delta x. \]

Because the sections with a positive deviation of the approximating function from the parabola are longer than the sections with a negative ...

...with a positive deviation, and \(\Delta^+ < \Delta^-\), with this method of approximation a sufficiently good mutual compensation of errors is ensured. Calculation of the methodological error of a measuring converter (MC) with piecewise-linear approximation by the least-squares method for a quasi-sinusoidal input signal shows that the latter has a positive systematic error. However, if the abscissa \(x_1\) of the first break point of the current-voltage characteristic is determined from the condition of absence of methodological error for a sinusoidal signal, then, when converting signals of quasi-sinusoidal form with \(K_r \leq 0.1\), the actual error can be kept at the level of hundredths of a percent.

As already noted, the desire to reduce the methodological error of the MC leads to a decrease in its transmission coefficient. Let us show that there is a way that makes it possible, to some extent, to overcome the noted contradiction. Consider the MC whose circuit is shown in Fig. 2. The output voltage of this MC is equal to

Fig. 2

\[ U_{\text{out}} = iR_{\text{н}} - U_0 - \Delta U_0, \]

where

\[ i = \frac{S U^2}{U_{\text{ip}}} \]

is the current in the MC load;

\(U_{\text{ip}}\) is the voltage at the output of the MC;

\(S\) is a constant determined by the MC circuit;

\(U_0\) is the voltage of the reference source corresponding to the equilibrium state of the comparison circuit;

\(\Delta U_0\) is the unbalance voltage of the comparison circuit, necessary for biasing the multiplying device to the operating point;

\(U\) is the rms value of the input voltage.

Since \(iR_{\text{н}} = U_{\text{ip}}\), then \(U_{\text{ip}} = U\sqrt{SR_{\text{н}}}\). Thus we have

\[ U_{\text{out}} = U\sqrt{SR_{\text{н}}} - U_0 - \Delta U_0. \tag{1} \]

The relative transmission coefficient of the MC is determined as

\[ G_1 = \frac{dU_{\text{out}}/U_{\text{out}}}{dU_{\text{in}}/U_{\text{in}}} = U\sqrt{SR_{\text{н}}}/\Delta U_0. \]

At the nominal input voltage \(U\), the output voltage of the MC is \(U_{\text{ip}} \cong U_0\) (the relation is valid for \(U_0 \gg \Delta U_0\), which is always the case); therefore

\[ SR_{\text{н}}\frac{U^2}{U_{\text{ip}}} = U_0, \]

whence

\[ SR_{\text{н}} = \frac{U_{\text{ip}}^2}{U^2}. \]

Taking the last relation into account, the expression for the relative transfer coefficient can be written in the form

\[ G_1 = U_{\mathrm{in}}/\Delta U_0 . \tag{2} \]

For an ME constructed according to the circuit of Fig. 3, we similarly obtain

\[ U_{\mathrm{out}}=\frac{SR'_{\mathrm{n}}U^2}{U_{\mathrm{in}}}-U_{01}-U_{02}+\Delta U_0, \tag{3} \]

where

\[ U_0=-\frac{U_{01}}{2}+\sqrt{\left(\frac{U_{01}}{2}\right)^2+SR'_{\mathrm{n}}U^2}, \]

\[ SR'_{\mathrm{n}}=\frac{(U_{01}+U_{02})U_{\mathrm{in}}}{U^2}. \]

The relative transfer coefficient of the ME of the circuit in Fig. 3 is found as

\[ G_2=2\frac{U_{01}}{\Delta U_0}+2\frac{U_{\mathrm{in}}}{\Delta U_0} -\frac{2\left(U_{01}/\Delta U_0+2U_{\mathrm{in}}/\Delta U_0+U_{\mathrm{in}}^2/\Delta U_0\cdot U_{01}\right)} {\sqrt{1+4U_{\mathrm{in}}/U_{01}+4U_{\mathrm{in}}^2/U_{01}^2}} . \tag{4} \]

Denoting \(\dfrac{U_0}{U_{\mathrm{in}}}=m\) and taking into account that \(U_{01}=U_0-U_{\mathrm{in}}\), as well as (2), expression (4) can be reduced to the form

\[ G_2=\frac{2G_1}{1+1/m}. \tag{5} \]

From (5) it is evident that, for \(m \gg 1\),

\[ G_2 \cong 2G_1, \tag{6} \]

i.e., with an appropriate choice of \(m\), the ME according to the circuit of Fig. 3 provides a relative transfer coefficient twice as large in comparison with the ME of the circuit in Fig. 2. A disadvantage of the ME of the circuit in Fig. 3 is the presence

Fig. 3

Fig. 4

of two reference-voltage sources. This disadvantage, however, is easily eliminated (Fig. 4). It can be shown that for the ME of the circuit in Fig. 4 the relative transfer coefficient is equal to

\[ G_3=\frac{2G_1(n-1)}{n\left(\frac{1}{m}+1\right)-\frac{2}{m}}, \tag{7} \]

where

\[ n=\frac{R_{\mathrm{n}}+R_1}{R_2}. \]

In practice \(n \gg 1\), and expression (7) reduces to (5).

It should be noted that, as a result of increasing the relative transfer coefficient of the ME, in a number of cases it becomes possible to reduce the input voltage and, consequently, the consumed power as well, but, what is especially important, to use the nonli-

the nonlinearity of the volt-ampere characteristic of the ME diodes, in order to reduce the approximation error and, consequently, the methodological error of the ME.

As noted at the beginning of the article, it is very important to ensure the time and temperature stability of the ME parameters. The use of domestic precision resistors and semiconductor diodes makes it possible to provide the required stability of the transmission coefficient. The situation is somewhat worse with zero drift. Owing to the fact that the ME, used as the measuring element in the stabilization system, operates in the vicinity of a fixed level of the input voltage, it is possible to use a thermal-compensation circuit to reduce the temperature drift. Since the principal source of zero drift is the change in the voltage drop across the rectifier diodes (D1, D2) and the commutating diodes (D3, D4), thermal compensation can be provided by means of two semiconductor diodes (diodes D5 and D6 in the circuit of Fig. 5) connected into the reference-voltage source circuit. Thermal compensation is the more effective, the greater the correlation ensured between the characteristics of the diodes (achieved by the use

Fig. 5

of diode assemblies). Since the rectifier diodes and the commutating diodes operate in substantially different current modes, for better thermal compensation it is expedient to bring the current mode of diode D5, for example, close to the mode of diodes D1, D2, and that of diode D6 to the mode of diode D3. It is advisable to set the modes of diodes D5, D6 by adjusting resistors R9 and R10; however, in practice good results have been obtained when the modes were adjusted by a single resistor shunting both diodes D5 and D6. The relative zero drift of the converter according to the circuit of Fig. 5, referred to the ME input, when using S2-13A resistors and B94B diode assemblies in the range from 0 to \(+60^\circ\mathrm{C}\), did not exceed \(\pm 10^{-4}\).

An essential parameter of an ME used in a precision ac voltage source is the level and composition of the ac component of the output signal. Acting on the regulating element of the source stabilization system simultaneously with the useful signal, the ac component can cause parasitic amplitude modulation, be the cause of the appearance of combination components that reduce the stability of the source output voltage, or lead to a considerable increase in the level of higher harmonics (primarily the third) in the output voltage of the source. And in most cases, in the output signal of the ME the most dangerous is the ac component with the frequency of the second harmonic of the source output voltage.

It can be shown that, for a harmonic input signal of frequency \(\omega_g\), at the output of the ME there is present, in addition to the dc component—the useful signal, whose level is determined by (1) or (3)—also an ac ...

component with amplitude \(\dfrac{U_m}{2k_{\phi}}\) and frequency \(2\omega_r\), i.e., precisely the undesirable second harmonic. The filtering coefficient of the variable component \(k_{\phi}\) is uniquely determined by the permissible level of the third harmonic in the source output voltage [2]. It is practically impossible to ensure the required \(k_{\phi}\) solely by increasing the time constant \(\tau_p\) of the ME discharge circuit, since for a specified frequency \(\omega_r\) there exists a critical value \(\tau_{p\mathrm{cr}}\), upon exceeding which the ME begins to operate in a quasi-peak mode. In addition, as \(\tau_p\) increases, the inertia of the system for stabilizing the source output voltage increases. Increasing \(k_{\phi}\) by introducing additional filtering sections raises the order of the stabilization system, which sharply affects stability and the loop gain that can practically be realized in the system.

It is known, however, that significant loop gain in an automatic-control system whose order is higher than the second can be realized only when there is a large mutual separation of the time constants of the inertial sections. This circumstance is used in the ME whose circuit is shown in Fig. 6. Here the required filtering coefficient of the second harmonic \(k_{\phi}\)

Fig. 6

while ensuring a substantial separation of the time constants of the inertial sections is achieved by using a passive rejection filter tuned to the frequency \(2\omega_r\) as an additional filtering section. Indeed, for the circuit of Fig. 6 we have

\[ k_{\phi}= \frac{\sqrt{(1+\omega^2\tau_p^2)\cdot\left[1+q^2(\omega/\omega_0-\omega_0/\omega)^2\right]}} {q(\omega/\omega_0-\omega_0/\omega)}, \tag{8} \]

where
\(\tau_p\) is the time constant of the ME discharge circuit;
\(q\) is the quality factor of the rejection filter;
\(\omega_0\) is the rejection frequency of the filter;
\(\omega\) is the current frequency.

From (8) it is evident that at the frequency \(\omega=\omega_0=2\omega_r\) we have \(k_{\phi}=\infty\), i.e., theoretically complete suppression of the most dangerous component of frequency \(2\omega_r\) at the ME output can be obtained. In practice, \(k_{\phi}\gg 10^4\) can always be ensured, with \(10^3\) being due to the rejection filter.

The transfer function of the ME of the circuit in Fig. 6 has the form

\[ T(p)=\frac{G}{(1+p\tau_p)\cdot(1+p\tau_0)}, \]

i.e., for the signal envelope such a combination of filters is equivalent to the series connection of two integrating sections with time constants \(\tau_p\) and \(\tau_0\), where

\[ \tau_0=\frac{2q}{\omega_0} \]

[6]. For example, at \(f_r=400\ \text{Hz}\)

it is advisable to use an ME with $\tau_p \geq 0.01$ sec; in this case the time constant of the rejection filter with quality factor $q = 0.25$ will be

\[ \tau_0 = \frac{2q}{2\omega_{\mathrm{g}}} = 10^{-4}\ \text{sec}, \]

i.e., a difference in the time constants of no less than 100 times is ensured. In this case the loop gain realized in the stabilization system is practically determined by the ratio between the time constant of the discharge circuit $\tau_p$ and the time constant of the regulator $\tau_{\mathrm{reg}}$.

The ME according to the circuit of Fig. 6 was built and tested. At present, similar MEs are being used in a number of new developments carried out at the Department of Radio Engineering of TPI.

LITERATURE

1. T. B. Rozhdestvenskaya, Electrical Comparators for Accurate Measurements of Current, Voltage, and Power. Moscow, Standards Publishing House, 1964.

2. M. S. Roitman. Principles of Construction, Questions of the Theory and Practice of Precision AC Voltage Sources. Doctoral dissertation, Tomsk, 1971.

3. C. G. Wahrmann. A true RMS instrument. Brüel ja Kjar Technical Review, 1958, No. 3.

4. U. P. Tamm, R. R. Kharchenko. Estimation of the error of a detector of rms values with a sliding displacement from the waveform of the measured voltage. Avtometriya, 1969, No. 4.

5. L. I. Volgin. Linear Electrical Converters for Measuring Instruments and Systems. Moscow, “Soviet Radio,” 1971.

6. Yu. G. Kochinev. A selective low-frequency RC amplifier as an element of a control system. Avtomatika i telemekhanika, vol. XIX, 1958, No. 4.