DEVICE FOR MEASURING THE FORM FACTOR OF A VOLTAGE WAVEFORM

A. N. OBERGAN, I. G. LESHCHENKO

(Presented by the Scientific and Technical Seminar of the Faculty of Automation and Computer Engineering)

In evaluating and analyzing periodic alternating currents, voltages, and other quantities that are symmetric with respect to the time axis, the concepts commonly used are the waveform form factor \(K_{\phi}\), the amplitude factor \(K_a\), and the nonlinear-distortion factor \(K_n\):

\[ K_{\phi}=\frac{A_{\mathrm{d}}}{A_{\mathrm{avg}}}, \qquad K_a=\frac{A_{\mathrm{m}}}{A_{\mathrm{d}}}, \qquad K_n=\frac{\sqrt{\sum_{n=1}^{\infty} A_n^2}}{A_1}, \]

where

\(A_{\mathrm{d}}\) is the rms value of the alternating quantity,
\(A_{\mathrm{avg}}\) is the average value of the alternating quantity over a period,
\(A_1\) is the rms value of the fundamental harmonic over a period.

In an article published in the present collection, the advisability and prospects were shown of developing and manufacturing instruments for direct measurement of the form factor of an alternating current, voltage, magnetic flux, and other quantities which, in the final analysis, when measuring \(K_{\phi}\), can be converted into a current or voltage.

At VNIIM, attempts were made to develop and create several circuit arrangements of instruments for measuring the waveform form factor, with the aim of investigating the errors of rectifier instruments [1]. Using a logometer of a magnetoelectric system as the measuring mechanism, it was proposed in the indicated circuits to use the linear and quadratic operating sections of the volt-ampere characteristics of various types of rectifiers. Thus, when the quadratic section of the rectifier characteristic was used in the rms-value circuit and the linear section in the average-current (voltage) circuit, the deflection of the moving part of the logometer proved to be proportional to the waveform form factor and to the rms value of the current (voltage). To eliminate the dependence of the instrument reading on the rms value of the voltage, additional circuits were used for squaring or extracting the square root. Naturally, the indicated instruments had many shortcomings, low accuracy, and therefore did not find application.

On the basis of a more thorough analysis of existing and possible methods for constructing \(K_{\phi}\)-meters and experimental results of investigation, including with a logometric measuring mechan-

Thus, the authors arrived at the conclusion that comparison instruments are expedient and promising for the indicated purposes. Below the operating principle of one such instrument and its technical data for measuring the shape of the voltage waveform in AC electrical circuits are described. The operating principle of the instrument is based on the compensation method of comparing the preliminarily rectified average value of the voltage with the constant emf of a vacuum thermoelectric converter, proportional to the rms value of the measured quantity.

The electrical circuit of the instrument (Fig. 1) is essentially a comparator of the rms and average values of the measured voltage. From the nonreactive divider \(R_5\), the voltage \(U\) is fed to

Fig. 1

two comparison channels. In the first channel the rms value of the current is converted into the direct thermoelectric emf \(E_t\) and is equal to

\[ I_d=\frac{U_d}{R_4+R_{ht}}, \]

where \(R_4\) is the additional nonreactive resistance;
\(R_{ht}\) is the resistance of the heater of the thermoconverter. In the second channel the measured voltage is rectified by the bridge circuit \(B\); the drop of the rectified voltage \(U_r\) across the resistance \(R_1+r\) is compared with the thermoelectric emf \(E_t\) by means of a highly sensitive magnetoelectric galvanometer responsive to the average value of the current. When \(E_t=U_r\), the galvanometer reading is zero and, taking into account that

\[ E_t=K_1\cdot I_d=\frac{K_1U_d}{R_4+R_{ht}},\qquad U_r=K_2\cdot U_{\mathrm{av}}\frac{R_1+r}{R_1+R+R_2}, \]

where

\(K_1\) is the conversion coefficient of the thermoconverter and, for the given value of \(I_d\), is constant,

\(K_2\) is the conversion coefficient of the rectifier bridge, loaded at the moment of measurement (\(I_g=0\)) by the resistance

\[ R_n=\frac{R_3(R_1+R+R_2)}{R_3+R_1+R+R_2}, \]

the form factor of the voltage waveform is

\[ K_f=\frac{U_d}{U_{\mathrm{av}}} =\frac{K_2}{K_1}\cdot \frac{(R_4+R_{ht})(R_1+r)}{R_1+R+R_2}. \]

Denoting all constant quantities of this expression by \(K\), we obtain

\[ K_f=K\cdot(R_1+r). \]

As we see, for \(K=\mathrm{const}\) the form factor of the curve is directly proportional to the resistance \(R_1+r\), and consequently also to the resistance \(r\), set by the potentiometer slider when the circuit is balanced. The smallest possible value \(K_{\mathrm{f}}=1\) is selected at \(r=0\); for this the circuit is calculated from the condition \(K\cdot R_1=1\). The largest possible value of \(K_{\mathrm{f}}\) for which the instrument is designed is established by choosing the resistance of the potentiometer \(R\), so that at \(R=R_1\) the limits of measurement of the form factor are from 1.00 to 2.00. Thus, the resolving power of the instrument, its multi-valuedness, and its sensitivity are determined by the choice of potentiometer \(R\) and by the sensitivity of the galvanometer (null indicator). If the resistance \(R\) is chosen in the form of a rheochord, then the resolving power can be made arbitrarily high and will be determined only by the sensitivity of the galvanometer. For this instrument, with \(R_1=240\ \Omega\), \(R=120\ \Omega\), and \(R_2=285000\ \Omega\) (\(K_{\mathrm{f}}=1.00 \div 1.50\)), with a potentiometer length of 250 mm, the division value is 0.002.

The accuracy and sensitivity of the \(K_{\mathrm{f}}\)-meter are determined by the stability of the coefficient \(K\) and by the sensitivities of the thermoconverter and the galvanometer. To ensure constancy of the coefficient \(K_1\), vacuum contactless thermoconverters of types TVB-3 and TVB-4 were investigated, and a TVB-4 thermoconverter was used in the instrument; it has higher sensitivity, stability of the conversion coefficient, and a volt-ampere characteristic in the operating region close to linear. In addition, to ensure high measurement accuracy, the operating point of the thermoconverter characteristic is set to be constant before measurement by the compensation method using a standard cell. For this purpose the double switch \(P\), shown in Fig. 1 as two keys, is placed in position 2, and by adjusting the resistance \(R_5\) a specified value of \(E_{\mathrm{t}}\) is set at \(I_{\mathrm{t}}=0\), equal in the instrument to 7 mV. The nonconstancy of the coefficient \(K_2\), caused by the nonlinearity of the rectifier characteristics, can be reduced to a minimum by selecting the rectifiers and choosing their operating mode. Studies carried out on various types of germanium and silicon diodes make it possible to conclude that they can be used over a broad frequency range, provided that the reverse current is small and the load characteristic of the rectifier circuit is linear. The latter condition is easily satisfied by choosing the resistance \(R_3\) much larger than the forward resistance of the bridge. \(R_3=7720\ \Omega\); the resistance of the rectifier bridge is 600 \(\Omega\) at a load current of 1.1 mA. All other elements of the circuit (resistors) are chosen to be highly stable manganin ones, which at the same time reduces errors due to temperature changes. The change in the emf of the standard cell with temperature can be allowed for by changing the resistance \(R_6\), analogously to how this is done in direct-current compensators. Protection of the galvanometer against overload by the alternating component of the rectified current can, if necessary, be accomplished by shunting the galvanometer with a capacitance.

The instrument is designed for a minimum voltage \(U_{\mathrm{d}}=9\ \mathrm{V}\). Transition to a lower voltage is easily achieved by changing the resistances \(R_4\) and \(R_3\) in order to provide the selected operating current of the thermoconverter. The lower limit of the voltage measured for this circuit is

\[ U_{\mathrm{dn}}=I_{\mathrm{d}}\cdot R_{\mathrm{t}}=5\cdot10^{-3}\cdot41\simeq0.2\ \mathrm{V}. \]

REFERENCES

1. A. Ya. Bezikovich, “The error of rectifier instruments caused by distortion of the alternating-current curve, and methods for determining it,” Electrical Measurements, Proceedings of VNIIM, no. 14 (74), p. 134, GEM, 1953.