Signals for Experimental Evaluation of the Error of Electronic Voltmeters Due to the Shape of Alternating Voltage
Abstract Generated abstract
This paper examines how the waveform of an alternating voltage affects the readings of electronic voltmeters calibrated in rms values for sinusoidal signals. It analyzes errors associated with amplitude, average-rectified, and rms converters through the crest factor and form factor, comparing sinusoidal, rectangular, triangular, pulse, and trapezoidal test signals with fixed or adjustable duty cycle. The authors show that varying the duty cycle extends the attainable range of waveform parameters, while trapezoidal voltage with adjustable duty cycle and edge durations provides the broadest and most useful regulation. The analysis also indicates that duty-cycle instability mainly changes signal energy measures rather than waveform factors, so the proposed experimental approach remains practically effective.
Full Text
UDC 621.317.725.621.385.621.317.75
SIGNALS FOR THE EXPERIMENTAL EVALUATION OF THE ERROR OF ELECTRONIC VOLTMETERS DUE TO THE WAVEFORM OF ALTERNATING VOLTAGE
G. P. TROFIMOV, A. A. PETRITSEV
(Presented by the scientific seminar of the Department of Radio Engineering)
The influence of the waveform of the measured signal on the readings of AC voltmeters can best be evaluated experimentally, by applying to the voltmeter signals of several typical, strictly known forms. The possibilities of a number of such signals for solving the stated problem are considered. It is shown that the widest range for regulating the crest factor and form factor—the principal characteristics of the signal waveform—is provided when trapezoidal alternating voltage is used, with adjustment of the duty cycle and of the durations of the edges.
Tables 3, references 4.
One of the important tasks in ensuring the unity of measurements in alternating current is associated with the need to take into account the influence of the waveform of the measured voltage on the readings of electronic voltmeters. A promising way to solve it consists in creating signal generators of typical idealized waveforms (sinusoidal, rectangular, triangular, etc.) and in experimentally determining the response of voltmeters to their action. This article considers the possibilities of a number of such signals for solving the stated problem.
Measurement of the intensity of alternating electrical oscillations is carried out by converting them into a constant voltage. In electronic voltmeters, converters of amplitude, rms, and average-rectified values are used; the transmission coefficient of these is usually calibrated in rms values of sinusoidal voltage (the scales of most voltmeters are graduated in rms values of the measured quantity)*. When the waveform of the converted signal deviates from sinusoidal, this leads to the appearance of an additional error of the converter and, consequently, of the voltmeter. For amplitude-value converters, according to [1], the error due to waveform is
\[ \delta_a = \frac{K_a}{K_{a0}} - 1, \tag{1} \]
where \(K_a\) and \(K_{a0}\) are the crest factors of the measured and calibration voltages.
Converters of average-rectified \((n = 1)\) and rms \((n = 2)\) values realize the dependence
* In what follows, precisely this case of conversion is meant.
\[ U_{\mathrm{av,eff}}=\left[\frac{1}{T}\int_{t_0}^{t_0+T}|U(t)|^n\,dt\right]^{1/n}, \tag{2} \]
where \(T\) is the period. Taking into account the deviation of the exponent \(\Delta n\) from the nominal value, the error of average-rectified-value converters is
\[ \delta_{\mathrm{av}}=\frac{K_{\phi0}}{K_{\phi}}\,[1+\Delta n(b+l_n K_{\phi})]-1, \tag{3} \]
where \(K_{\phi0}\) and \(K_{\phi}\) are the form factors of the calibration voltage and of the voltage being converted; \(b\) is a coefficient depending on the waveform of these same voltages. In rms-voltage converters, according to (2), the squaring operation is performed. Squarers based on electrothermal elements (thermistors, thermal converters, vacuum diodes with power introduced into the filament circuit) have no error due to the waveform of the applied signal.
For all other squarers, the waveform error
\[ \delta=\Delta n(a+l_n K_{\phi}) \tag{4} \]
occurs only when the transfer characteristics deviate from the quadratic law; moreover, expression (4) is valid for \(\Delta n \ll 1\). The parameter \(a\) depends on the waveform of the converted and calibration voltages.
Thus, the errors of voltmeters due to the voltage waveform, taking into account the deviation from the realized conversion law, determined by \(\Delta n\), depend on the peak factors or form factors.
A change in the peak factor and form factor can be implemented in two ways:
— by using a set of waveforms with known values of \(K_a\) and \(K_{\phi}\);
— by adjusting the duty factor of the calibration signal while keeping the pulse shape unchanged.
Table 1 gives several waveforms that are most easily realized in practice and the corresponding values of \(K_a\), \(K_{\phi}\). From a comparison of these oscillation forms it follows that the largest values of \(K_a\)
Table No. 1
| Signal type | \(U\) | \(U_{\mathrm{av}}\) | \(K_a=\dfrac{U_m}{U}\) | \(K_{\phi}=\dfrac{U}{U_{\mathrm{av}}}\) |
|---|---|---|---|---|
| Sinusoidal waveform diagram, period \(T\), time axis \(t\) | \(\dfrac{U_m}{\sqrt{2}}\) | \(\dfrac{2U_m}{\pi}\) | \(\sqrt{2}\) | \(\dfrac{\pi}{2\sqrt{2}}\) |
| Rectangular alternating waveform diagram, period \(T\), time axis \(t\) | \(U_m\) | \(U_m\) | \(1\) | \(1\) |
| Triangular alternating waveform diagram, period \(T\), time axis \(t\) | \(\dfrac{U_m}{\sqrt{3}}\) | \(\dfrac{U_m}{2}\) | \(\sqrt{3}\) | \(\dfrac{2}{\sqrt{3}}\) |
| Rectangular pulse waveform diagram, period \(T\), pulse duration \(t_{\mathrm{on}}\), time axis \(t\) | \(U_m\sqrt{1-2t/3}\) | \(U_m(1-t/2)\) | \(\dfrac{1}{\sqrt{1-2t/3}}\) | \(\dfrac{\sqrt{1-2t/3}}{1-t/2}\) |
and \(K_{\phi}\) are possessed by the triangular voltage (\(K_a=1.73;\ K_{\phi}=1.15\)), and the smallest by the rectangular voltage (\(K_a=K_{\phi}=1\)). Therefore these two waveforms make it possible to find the limiting value of voltmeter errors due to the waveform of alternating-sign signals.
It can be shown that the peak factors and form factors of smooth curves (parabolic, bell-shaped, cosine-squared, and other similar forms) occupy intermediate positions in the range of the indicated limiting values. Obtaining these curves at elevated frequencies (tens of kilohertz) is associated with great difficulties. Therefore, in order to expand the number of graduations of \(K_a\) and \(K_\phi\) in the range of limiting values formed by rectangular and triangular shapes, it is advisable to use a trapezoidal voltage with the adjustable parameter
\[ \gamma=\frac{t_\phi+t_{\mathrm{sp}}}{t_i}. \tag{5} \]
At present there are pulse generators with adjustable rise and fall times not only for the microsecond range, but also for the nanosecond range [2].
When estimating \(\delta_{\mathrm{sr}}\) and \(\delta\) by applying different curves, the latter must have different amplitudes. Otherwise, a change in \(U_{\mathrm{sr}}\) or \(U\) may conceal the influence of the shape of the calibration signal. The relations between the amplitudes of different curves are seen directly from the expressions in the second and third columns of Table 1. In the case of using a trapezoidal voltage, in order to maintain \(U=\mathrm{const}\), its amplitude must be varied in accordance with the dependence
\[ U_m(\gamma)=\frac{U_{m0}}{\sqrt{1-2\gamma/3}}, \tag{6} \]
and in order to maintain \(U_{\mathrm{sr}}=\mathrm{const}\), according to the expression
\[ U_m(\gamma)=\frac{U_{m0}}{1-\gamma/2}. \tag{7} \]
Here \(U_{m0}\) is the signal amplitude at \(\gamma=0\).
In practical situations, curves are encountered with much larger coefficients \(K_a\) and \(K_\phi\) than typical curves can provide, and the first method for estimating voltmeter errors due to signal shape proves insufficient.
A wider range of variation of \(K_a\), \(K_\phi\) is realized with the aid of the same curves with variable duty ratio (Table 2)*. As in the first method, to estimate the error of an electronic voltmeter due to signal shape it is necessary to vary \(K_\phi\) without changing \(U\) or \(U_{\mathrm{sr}}\). Then the deviation of the voltmeter readings will reflect only \(\delta\) or \(\delta_{\mathrm{sr}}\). It follows from Table 2 that, when the duty ratio is varied, in order to maintain \(U=\mathrm{const}\) it is necessary to vary the amplitude of the calibration signal in proportion to \(\sqrt{Q}\), and in order to maintain \(U_{\mathrm{sr}}=\mathrm{const}\), in proportion to \(Q\). In this case, the law of variation \(K_\phi(Q)\) remains the same in accordance with the expressions of Table 2.
The range of regulation of the duty ratio is determined by the allowable peak factor of the signal being converted (\(K_a \leq 5\)). As is seen from Table 2, the maximum duty ratio must be possessed by asymmetric (\(Q=26\)) and symmetric (\(Q=25\)) rectangular signals.
The minimum range of duty-ratio regulation is obtained for the triangular form (\(Q=8\)). Taking into account the advantages of rectangular voltage for solving a number of important problems in metrology and measurement technology [3, 4], it is desirable to have a rectangular signal in the set of generated functions. The widest range of problems is solved when a trapezoidal voltage with adjustable parameters is available.
* Questions of the practical implementation of generators of such signals should be considered in a separate article.
Instability of the duty cycle of signals leads to distortions of the results of evaluating waveform errors. Table 3 gives expres—
Table No. 2
| Signal type | \(U\) | \(U_{\mathrm{av}}\) | \(K_a\) | \(K_\phi\) |
|---|---|---|---|---|
| Sinusoidal bursts; \(Q=\dfrac{T}{2t_i}\) | \(\dfrac{U_m}{\sqrt{2}\sqrt{Q}}\) | \(\dfrac{2U_m}{\pi Q}\) | \(\sqrt{2}\sqrt{Q}\) | \(\dfrac{\pi}{2\sqrt{2}}\sqrt{Q}\) |
| Symmetric rectangular pulses; \(Q=\dfrac{T}{2t_i}\) | \(\dfrac{U_m}{\sqrt{Q}}\) | \(\dfrac{U_m}{Q}\) | \(\sqrt{Q}\) | \(\sqrt{Q}\) |
| Asymmetric rectangular pulses; \(Q=\dfrac{T}{t_i}\) | \(\dfrac{2U_m\sqrt{Q-1}}{Q}\) | \(\dfrac{4U_m(Q-1)}{Q^2}\) | \(\dfrac{Q}{2\sqrt{Q-1}}\) | \(\dfrac{Q}{2\sqrt{Q-1}}\) |
| Triangular pulses; \(Q=\dfrac{T}{2t_i}\) | \(\dfrac{U_m}{\sqrt{3}\sqrt{Q}}\) | \(\dfrac{U_m}{2Q}\) | \(\sqrt{3}\sqrt{Q}\) | \(\dfrac{2}{\sqrt{3}}\sqrt{Q}\) |
| Trapezoidal pulses; \(Q=\dfrac{T}{2t_i}\) | \(\dfrac{U_m}{\sqrt{Q}}\sqrt{1-\dfrac{2\gamma}{3}}\) | \(\dfrac{U_m(1-\gamma/2)}{Q}\) | \(\dfrac{\sqrt{Q}}{\sqrt{1-2\gamma/3}}\) | \(\dfrac{\sqrt{1-2\gamma/3}}{1-\gamma/2}\sqrt{Q}\) |
Table No. 3
| Signal type* | \(\dfrac{dU}{dQ}\) | \(\dfrac{dU_{\mathrm{av}}}{dQ}\) | \(\dfrac{dK_a}{dQ}\) | \(\dfrac{dK_\phi}{dQ}\) |
|---|---|---|---|---|
| Sinusoidal | \(-\dfrac{U_m}{\sqrt{8}\sqrt{Q^3}}\) | \(-\dfrac{2U_m}{\pi Q^2}\) | \(\dfrac{1}{\sqrt{2}\sqrt{Q}}\) | \(\dfrac{\pi}{4\sqrt{2}\sqrt{Q}}\) |
| Symmetric rectangular | \(-\dfrac{U_m}{2\sqrt{Q^3}}\) | \(-\dfrac{U_m}{Q^2}\) | \(\dfrac{1}{2\sqrt{Q}}\) | \(\dfrac{1}{2\sqrt{Q}}\) |
| Asymmetric rectangular** | \(\dfrac{U_m(2-Q)}{Q^2\sqrt{Q-1}}\) | \(\dfrac{2-Q}{Q^3}\,4U_m\) | \(\dfrac{Q-2}{4\sqrt{(Q-1)^3}}\) | \(\dfrac{Q-2}{4\sqrt{(Q-1)^3}}\) |
| Triangular | \(-\dfrac{U_m}{\sqrt{6}\sqrt{Q^3}}\) | \(-\dfrac{U_m}{2Q^2}\) | \(\dfrac{\sqrt{3}}{2\sqrt{Q}}\) | \(\dfrac{1}{\sqrt{3}\sqrt{Q}}\) |
| Trapezoidal | \(-\dfrac{U_m\sqrt{1-2\gamma/3}}{2\sqrt{Q^3}}\) | \(-\dfrac{U_m(1-\gamma/2)}{Q^2}\) | \(\dfrac{1}{2\sqrt{Q}\sqrt{1-2\gamma/3}}\) | \(\dfrac{\sqrt{1-2\gamma/3}}{2(1-\gamma/2)\sqrt{Q}}\) |
* Corresponding to Table 2.
** For this signal \(Q=\dfrac{T}{t_i}\).
sions for the derivatives of \(U\), \(U_{\mathrm{av}}\), \(K_a\), and \(K_\phi\) with respect to duty cycle. From these results it is evident that instability of the duty cycle changes \(U\) and \(U_{\mathrm{av}}\) more substantially than \(K_a\) and \(K_\phi\), i.e., deviations in voltmeter readings are due mainly to changes in the energy characteristics of the signal, and not to characteristics of the waveform. Consequently, even with
in the presence of some instability of the duty factor, the results of (1), (3), and (4) prove to be sufficiently reliable, and the method presented is sufficiently effective.
Conclusions
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For experimental determination of the error in the readings of electronic voltmeters due to the voltage waveform, it is advisable to use typical nonsinusoidal alternating-current signals with adjustable duty factor.
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The results of estimating voltmeter errors due to the shape of the voltage curve depend only weakly on fluctuations of the duty factor; i.e., the approach set forth in the article proves sufficiently effective.
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For determining voltmeter errors due to the shape of the voltage curve, a trapezoidal signal with adjustment of the duty factor and of the durations of the fronts appears preferable to others.
References
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N. I. Dokukina. Pulse generators with adjustable duration, rise time, and fall time. Surveys in Electronic Engineering. Series—Control and Measuring Apparatus. Issue 16 (85), VINITI, Moscow, 1969.
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M. S. Roitman, V. R. Tsibul’skii, G. P. Trofimov, A. V. Sokolov. Experimental determination of the transient error of thermal and photoelectric rms comparators. Metrologiya, 1973, No. 6.
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Yu. M. Tuz. A new method for calibrating amplitude, rms, and average-value millivoltmeters. Proceedings of Higher Educational Institutions of the USSR, Radio Engineering, 1964, No. 4.