Proceedings of the Tomsk Polytechnic Institute Named after S. M. Kirov, Holder of the Order of the October Revolution and the Order of the Red Banner of Labor

Volume 194
1972

On Measuring the Form Factor of a Curve

I. G. Leshchenko, A. N. Obergan

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

Attempts to create instruments for the direct measurement of the form factor of a voltage curve [1], using rectifiers with quadratic sections of the volt-ampere characteristics, did not yield positive results either with respect to measurement accuracy or to the technological feasibility of their mass production. At present, judging from published data, there are no instruments for the direct measurement of the form factor of a curve. It is determined by indirect methods, by measuring the effective and mean values of voltages and then calculating their ratio. In individual cases, for scientific research [2], \(K_{\phi}\) of current, voltage, emf, flux, etc. can be determined from oscillograms. However, these methods are cumbersome, have low productivity, and give a large measurement error.

The linear distortion meters in use cannot be employed to determine \(K_{\phi}\), since in the general case there is no mathematical relationship between \(K_{\text{ni}}\) and \(K_{\phi}\). In addition, nonlinear distortion meters manufactured by industry have a measurement error of 5--7% [3].

The necessity and advisability of developing and mass-producing instruments for direct, immediate measurement of \(K_{\phi}\) are currently determined by the following considerations.

First, in the calculation and design of electrical machines, apparatus, and electromagnetic automation devices, the formula for the effective value of voltage is used, expressed through the maximum magnetic flux (or induction) and the form factor of the curve, usually taken to be equal to 1.11, as for a sinusoid.

In real machines and apparatus, the curves of magnetic fluxes and induction differ from sinusoidal ones; consequently, in order to refine calculations, save materials, and conduct scientific research, it is advisable to determine a more accurate value of the form factor of induction curves.

As an example, one may cite scientific investigations of the spectral composition of induction and emf curves for various grades of steel at specified maximum inductions, with different sheet thicknesses and magnetic-field strengths, carried out at VNIIM [2].

The results of the investigation could be considerably broader both for a wider range of frequencies and magnetic-field strengths,

if we had at our disposal sufficiently accurate instruments for the direct measurement of the form factors of the curves of current, voltage, and emf.

Secondly, all rectifier instruments respond to the mean value of the current, while their calibration is carried out in rms values of sinusoidal current. Therefore, the measurement error of rectifier instruments depends on the form factor of the curve of the measured current and may prove to be quite large.

Thus, at \(K_{\phi}=2\), the error in measuring the rms value of the current will be

\[ \gamma_0=\frac{I_{\text{meas}}-I_{\text{rms}}}{I_{\text{rms}}} =\frac{1.11 I_{\text{mean}}-2I_{\text{mean}}}{2I_{\text{mean}}} =-45\%. \]

The availability of an instrument for determining \(K_{\phi}\) will make it possible to expand the field of application of rectifier instruments.

Thirdly, the availability of high-precision instruments for the direct measurement of the form factor of the curves of voltage, current, power, magnetic flux, and other quantities—especially automatic recording instruments for investigating complex dynamic processes in electromagnetic mechanisms and devices—will broaden the range of instruments of scientific inquiry.

Further, it appears possible to us to use such high-sensitivity instruments not only in testing ferromagnetic materials on alternating current, but also for quality control by electromagnetic methods through measurement of the \(K_{\phi}\) of the emf in the secondary winding of transformer transducers. It is known [5] that, upon deformation of ferromagnetic materials, the odd harmonics of the emf (induction under magnetization by sinusoidal current) of the secondary winding change sharply in amplitude and phase, and consequently so does the form factor of the curve.

The form factor of the current curve is also one of the initial parameters in the calculation and investigation of ballast devices for gas-discharge light sources [6].

Considering that the instruments proposed by us make it possible to detect small deviations from a specified form of the current or voltage curve \((0.01\div0.001)K_{\phi}\), the prospect of using \(K_{\phi}\)-meters in quality control of products by electromagnetic methods deserves attention.

In this respect it is characteristic to compare the appearance of phase-sensitive voltmeters, which have found wide application for purposes of investigation and quality control of products by changes in the real and imaginary components of secondary emfs.

The possibility of creating \(K_{\phi}\)-meters with manual and automatic balancing, with a measurement error of less than one percent and with high and low input resistance, among other things, may make them a serious competitor to meters of nonlinear distortions of symmetrical curve forms in alternating-current electrical circuits.

At the Department of Information and Measurement Technology, the authors have developed a number of circuits and instruments, and have carried out certain investigations of their prototypes for the direct measurement of the form factor of the curves of current and voltage. The principle of measuring the form factor of the curves of current and voltage and the electrical and magnetic quantities associated with them is based on comparison or division of the rms and mean values of the current, using a magnetoelectric quotient meter, known divider circuits, and a magnetoelectric galvanometer.

When a logometer is used, a direct current proportional to the rms value of the measured current and obtained with the aid of various converters is passed through one frame, while a preliminarily rectified current, obtained by semiconductor or electronic rectifiers with the minimum permissible distortions, is passed through the other frame. Thermoelectric converters of the TVB type were used as the converters, but heated resistors, photoelectric converters (lamp–photoresistor), and others may be used with equal success. A logometer responding to the ratio of currents can thus be calibrated in values of the form factor of the curve. Precise calibration of the instrument by means of deliberately known current-curve shapes considerably reduces its error due to the nonlinearity of the volt-ampere characteristics of the rectifiers, the rectification factor, the nonlinearity of the characteristics of the rms-value converters into dc, and other factors. The same calibration, though sufficiently accurate for only one value because of the linearity of the readout device for \(K_\phi\), also reduces the errors in comparison instruments using a magnetoelectric galvanometer as the null indicator, responding to the mean value of the measured quantity. This, the most promising, group of instruments with manual balancing is based on the zero or differential method of comparing a dc voltage proportional to the rms value of the measured quantity with a preliminarily rectified voltage. In the zero method, equality of the voltages (currents) being compared is established by moving the slider of a nonreactive potentiometer, which is calibrated in values of \(K_\phi\) and has a linear scale. Indeed, if we denote by \(E_{\mathrm{t}}\) and \(U_{\mathrm{cp}}\), respectively, the emf of the thermoconverter with conversion factor \(K_1\) for a given value of rms current \(I_{\mathrm{d}}\) and the mean value of the rectified voltage across the resistance \(R\), then at \(I_{\mathrm{g}}=0\), \(E_{\mathrm{t}}=U_{\mathrm{cp}}\) and \(K_1 \cdot I_{\mathrm{d}} = K_2 \cdot I_{\mathrm{cp}} \cdot R\), where \(K_2\) is the conversion factor of the rectifier bridge (see Fig. 1, b). Hence

\[ K_\phi=\frac{I_{\mathrm{d}}}{I_{\mathrm{cp}}}=\frac{K_2}{K_1}\cdot R. \]

In the differential circuit of a \(K_\phi\)-meter, the value of the measured quantity is judged from the deflection of the galvanometer (microammeter) pointer. The method may be recommended for measuring small and slowly varying changes in the curve shape, for quality control of products, etc.

Figure 1 shows the basic circuits of \(K_\phi\)-meters:

a) using a logarithmic measuring mechanism of a magnetoelectric system;

b) a voltage \(K_\phi\)-meter;

c) a current \(K_\phi\)-meter.

A more detailed description of one of the comparison instruments is given in this collection.

Automatic instruments can be comparatively easily made for measuring the form factor of a curve in a specified range and with a specified sensitivity, using known methods of tracking balancing of dc compensators with photoelectric converters of the F117 type or induction converters of the I31 type, responding to the mean value of current or voltage.

The error of the instruments is composed mainly of the nonlinearity of the characteristics of the converters of both comparison channels,

instability of the elements of the electrical circuit over time, temperature and frequency errors. The nonlinearity of one of the comparison (commutation) channels, for example, of the channel for converting the rms value of current, is eliminated by optimally selecting and setting, before measurement, a specified rms value of current at the transducer input, or by using a compensation method for comparing

Fig. 1

the thermo-emf and the emf of a standard cell. The nonlinearity of the characteristics of the rectifiers is reduced to a minimum by choosing the type of rectifiers and their operating conditions.

Conclusion

The proposed instruments for direct measurement of the form factors of current curves and other quantities are necessary for scientific research, are comparatively easy to implement, and are recommended for more complete development.

REFERENCES

1. A. Ya. Bezinkovich. Error of rectifier instruments caused by distortion of the form of the alternating-current curve and methods for determining it. Electrical Measurements, Proceedings of VNIIM, issue 14 (74), p. 134, Gostekhizdat, 1953.

2. N. G. Chernysheva. Distortions of magnetic induction curves and the emf. Magnetic Measurements and Research, Proceedings of VNIIM, issue 29 (89), pp. 58–82, Mashgiz, 1956.

3. K. D. Osipov, V. V. Pasynkov. Handbook of Radio-Measurement Instruments, Part III. Sovetskoe Radio Publishing House, 1959.

4. E. I. Langvagen. On the error in voltage measurement by detector devices in the presence of higher harmonics, reports of the VI Scientific-Technical Conference. Department of Technical Sciences of the Ust-Kamenogorsk Civil Engineering and Road Institute, 1966.

5. M. M. Shel. Investigation of the magnetoelastic method of measurements and defectoscopy by anisotropy sensors. Candidate dissertation, Irkutsk Branch of VNIIKhIMMASH, 1966.

6. E. N. Zharikov. Measurement of coefficients characterizing the form of current and voltage curves in circuits with gas-discharge lamps. Svetotekhnika, No. 11, 1960.