PRECISION PHASE INDICATOR

M. S. ROITMAN, A. I. KRAMNYUK, Yu. V. PIKALKIN

(Presented by the scientific and technical seminar of the Faculty of Automation and Computer Engineering)

Studies of photoelectric converters carried out at the Department of Radio Engineering have shown that there is a real possibility of comparing, by effective value, two alternating voltages over a wide frequency range with an error of less than 0.01% [1].

The availability of precision comparison circuits makes it possible, in turn, to proceed to the creation of accurate phase indicators. Since the error of a differential indicator can be reduced to 0.001–0.003%, the potential accuracy of a phase indicator based on the sum-and-difference method is 0.0007–0.002. The practically attainable accuracy is limited by three factors: the error of the circuit forming the sum and difference of two voltages, harmonic components, and the error of the comparison circuit.

The present work is devoted to consideration of these factors, as well as to questions of the practical implementation of an accurate 90° phase indicator.

The operating principle of the instrument is as follows. The voltages $U'$ and $U''$ (Fig. 1) are fed, through two identical amplifiers with the same gain, to a circuit for forming the sum and difference, implemented on a transformer. Voltages equivalent to the vector sum and difference of the input voltages feed incandescent lamps that illuminate photoresistors connected in the arms of a bridge. When the phase shift between $U'$ and $U''$ is not equal to 90°, the illumination of the photoresistors is different, which leads to unbalance of the bridge and is recorded by a galvanometer.

As is evident from the schematic diagram of the phase indicator, nonidentity of the phase characteristics of the amplifiers will enter directly as an error into the 90° indication error. To reduce the amplifiers’ own phase distortions, a number of measures were taken. Since the amplifier is provided with negative feedback ($K\beta = 30$), its own distortions in the working frequency band are small; the number of elements introducing phase distortions has been reduced to a minimum, and the influence of the load is eliminated by a complex cathode follower. The latter makes it possible to reduce the phase distortions introduced by the transformer at low frequencies. An experimentally obtained dependence of phase distortions on frequency showed that it does not exceed 0.3° in the working frequency range, while the amount of nonidentity can be reduced to a value on the order of 0.03°. In especially precise measurements, an additional correction of phase shifts can be introduced.

amplifiers. And since the indication error due to the nonidentity of the phase characteristics of the amplifiers is static, it can be taken into account during measurement.

Let us estimate the indication error, taking into account the error of the circuit for forming the sum and difference and the presence of harmonic components in the input signals.

The effective value of the sum and difference at the output of the circuit is determined as

\[ U' + U'' = \sqrt{\frac{1}{T}\int [U'(t)+U''(t)]^2} = \frac{1}{\sqrt{2}} \sqrt{[U']^2+[U'']^2+2U'U''\cos(\varphi'-\varphi'')} \tag{1} \]

\[ U' - U'' = \sqrt{\frac{1}{T}\int [U'(t)-U''(t)]^2} = \frac{1}{\sqrt{2}} \sqrt{[U']^2+[U'']^2-2U'U''\cos(\varphi'-\varphi'')} \tag{2} \]

In the general case the input voltages represent a sum of harmonic oscillations. Taking the latter into account and on the basis of (1), (2), the voltage at the output of the circuit is equal to

\[ U_{\text{out}}= \frac{1}{\sqrt{2}} \sqrt{ \sum_{k=1}^{k=m} \left[(U'_k)^2+(U''_k)^2+2U'_kU''_k\cos(\varphi'_k-\varphi''_k)\right] } - \frac{1}{\sqrt{2}} \sqrt{ \sum_{k=1}^{k=m} \left[(U'_k)^2+(U''_k)^2-2U'_kU''_k\cos(\varphi'_k-\varphi''_k)\right] }. \tag{3} \]

For a phase shift close to \(90^\circ\), the magnitude of the output voltage and the sensitivity of the indicator \((S)\) are related by

\[ S=\frac{U_{\text{out}}}{\sqrt{(U'_1)^2+(U''_1)^2}}, \tag{4} \]

\[ U_{\text{out}}=S\sqrt{(U'_1)^2+(U''_1)^2}. \tag{5} \]

Since the values of the input voltage are applied to the summing and subtracting circuits with different transfer coefficients (because of the asymmetry of the transformer), we shall take this factor into account by introducing the asymmetry coefficient \(\delta\).

Calculating the values of the radicals in (3) by approximate formulas, taking into account expression (5) and the asymmetry coefficient, we obtain

\[ \left[ \sqrt{(U'_1)^2+(U''_1)^2} + \frac{U'_1U''_1}{\sqrt{(U'_1)^2+(U''_1)^2}} \cos(\varphi'_1-\varphi''_1) \right] + \]

\[ + \frac{1}{2\sqrt{(U'_1)^2+(U''_1)^2}} \sum_{k=2}^{k=m} \left[(U'_k)^2+(U''_k)^2+2U'_kU''_k\cos k(\varphi'_k-\varphi''_k)\right] - \]

\[ -(1\pm\delta) \left[ \sqrt{(U'_1)^2+(U''_1)^2} - \frac{U'_1U''_1}{\sqrt{(U'_1)^2+(U''_1)^2}} \cos(\varphi'_1-\varphi'_1) \right] - \]

\[ - \frac{1\pm\delta}{2\sqrt{(U'_1)^2+(U''_1)^2}} \sum_{k=2}^{k=m} \left[(U'_k)^2+(U''_k)^2 -2U'_kU''_k\cos k(\varphi'_k-\varphi''_k)\right] = U_{\text{out}}. \tag{6} \]

We are interested in the magnitude of the phase shift between the first harmonics of the input voltages, i.e.,

\[ \varphi = \varphi'_1 - \varphi''_1 . \tag{7} \]

After simple transformations from (6) we find

\[ \varphi=\arccos\left\{\sqrt{2}S+ \frac{\delta}{2U'_1U''_1}\sum_{\kappa=1}^{\kappa=m}\left[(U'_\kappa)^2+(U''_\kappa)^2\right]+ \right. \]

\[ \left. +\sum_{\kappa=2}^{\kappa=m} n'_\kappa n''_\kappa \cos \kappa(\varphi'_\kappa-\varphi''_\kappa) \right\}, \tag{8} \]

where \( n'_\kappa=\dfrac{U'_\kappa}{U'_1},\quad n''_\kappa=\dfrac{U''_\kappa}{U''_1} \) are the corresponding harmonic components of the input voltages.

Denoting the ratio of amplitudes by \( \dfrac{U'_1}{U''_1}=P \) and neglecting, in the second term, the higher harmonics because of their smallness, we finally obtain

\[ \varphi=\arccos\left[\sqrt{2}S+\frac{\delta}{2}\left(P+\frac{1}{P}\right)+ \sum_{\kappa=2}^{\kappa=m} n'_\kappa n''_\kappa \cos \kappa(\varphi'_\kappa-\varphi''_\kappa)\right]. \tag{9} \]

Accordingly, the deviation from \(90^\circ\) is

\[ \Delta\varphi=\pm\arcsin\left[\sqrt{2}S+\frac{\delta}{2}\left(P+\frac{1}{P}\right)+ \sum_{\kappa=2}^{\kappa=m} n'_\kappa n''_\kappa \cos \kappa(\varphi'_\kappa-\varphi'_\kappa)\right]. \tag{10} \]

The differential comparators developed at the radio-engineering department make it possible to compare two voltages with an accuracy of \(0.002\%\). For the present case \(S=\delta=0.00002\).

If the amplitudes of the input voltages are equal, \(p=1\). If the amplitudes of the 2nd and 3rd harmonics in the input voltages are, respectively, 2 and 1%, then the maximum indication error is

\[ \Delta\varphi \ll 1'. \]

By graduating the indicator scale, it is possible to record not only the presence of a deviation from \(90^\circ\), but also the sign and magnitude of this deviation.

Conclusions

1. The investigations carried out have shown the expediency of using photoelectric converters in phase indicators. The potential indication accuracy can reach hundredths of a degree.

2. An increase in indication accuracy can be achieved by increasing the sensitivity of the indicators and by creating broadband amplifiers with small phase distortions.

REFERENCES

1. B. A. Perminov, M. S. Roitman, E. I. Tsimbalist. An alternating-current comparator using photoelectric converters. Avtometriya, No. 5, 1965.