Method for Measuring Phase Angles with Continuous Photographic Recording
Abstract Generated abstract
This paper describes a two-channel method for measuring phase differences between alternating-current signals while permitting continuous photographic recording from a cathode-ray indicator. The method forms sum and difference voltages from the input signals, uses automatic gain control to hold the sum-channel output constant, and derives the measured phase from the difference-channel output, which is proportional to the tangent of half the phase angle. It also outlines a comparison circuit for placing an angular mark on a linear sweep, a supplementary arrangement for determining the sign of the phase angle, and a simple error analysis. The proposed approach reduces sensitivity to phase unbalance in the main amplifier channels and supports continuous recording, though it has dead zones in sign determination and reduced accuracy at low signal-to-noise ratios.
Full Text
A Method for Measuring Phase Angles with Continuous Photographic Recording
E. I. Fialko
(Presented by the scientific seminar of the Radio Engineering Faculty)
Preliminary Remarks
A number of methods are known for measuring phase differences of alternating currents using two-channel amplification. To ensure high measurement accuracy, in some cases the phase-frequency characteristics of the channels must be identical. Ensuring identical phase-frequency characteristics of amplifier channels when there is a large number of stages is associated with considerable technical difficulties. Therefore, methods for measuring phase angles that are not associated with stringent requirements on the amplifier channels from the standpoint of identical phase-frequency characteristics are of definite interest.
In a number of cases it is necessary to record measurement results continuously. This is conveniently accomplished by photographing the indicator screen on continuously moving film. In this case it is desirable to have a linear (diametral) sweep and an angular mark in the form of a point (the film moves perpendicular to the sweep line).
Below is a brief description of a simple method for measuring the phase differences of two signals, based on the use of two-channel amplification.
Basic Relations
The block diagram of a phase meter implementing the indicated method is shown in Fig. 1.
Fig. 1. Simplified block diagram of the phase meter. 1—“sum” device, 2—“difference” device, 3, 4—main amplifier channels, 5—automatic gain-control stage.
The voltages under investigation
\[ u_1 = U \cdot \sin \omega t \quad \text{and} \quad u_2 = U \cdot \sin(\omega t + \varphi) \]
are applied to the corresponding input devices, in which the sum and difference voltages are formed,
\[ u_{s.\,in}=u_1+u_2=2\cdot U\cdot \sin\left(\omega t+\frac{\varphi}{2}\right)\cdot \cos\frac{\varphi}{2}, \]
\[ u_{d.\,in}=u_1-u_2=2\cdot U\cdot \cos\left(\omega t+\frac{\varphi}{2}\right)\cdot \sin\frac{\varphi}{2}, \]
which are amplified in different channels of the receiver.
The output sum voltage is applied to an automatic gain-control stage (AGC), which controls the gain of both the sum and the difference channels. Since the voltage amplitude at the output of the sum channel is maintained constant,
\[ U_{s.\,out}=U_{s.\,in}\cdot k_s=U_0, \]
the gain of the sum channel is inversely proportional to the amplitude of the input sum voltage,
\[ k_s=\frac{U_0}{U_{s.\,in}}. \]
If the amplifier channels of the receiver are identical with respect to amplitude and amplitude-frequency distortions, i.e., \(k_s=k_d\), then the voltage amplitude at the output of the difference channel will be equal to
\[ U_{d.\,out}=U_{d.\,in}\cdot k_d=\frac{U_{d.\,in}}{U_{s.\,in}}\cdot U_0. \]
It is not difficult to see that
\[ \frac{U_{d.\,in}}{U_{s.\,in}}=\operatorname{tg}\frac{\varphi}{2}. \]
Thus,
\[ U_{d.\,out}=U_0\cdot \operatorname{tg}\frac{\varphi}{2} \quad \text{or} \quad \varphi=2\cdot \operatorname{arctg}\left(\frac{U_{d.\,out}}{U_0}\right), \]
that is, the voltage at the receiver output is proportional to the tangent of one half of the phase angle being measured. This voltage is applied to the peak detector of the indicator.
A comparison circuit may be used for measuring the angle. The constant voltage taken from the peak detector biases, via the control grid, the tube of the angle-marking unit. A sawtooth voltage (from the sawtooth-voltage generator), which is the sweep voltage and is applied to the horizontal-deflection plates of the cathode-ray tube, is also applied to the control grid of this tube.
When the “sawtooth” voltage compensates the voltage taken from the peak detector, a pulse is produced in the anode of the tube, which is then differentiated and amplified. A narrow positive im-
a pulse is applied to the modulating electrode of the tube. A bright point will appear on the sweep line, the distance of which from the beginning of the sweep will correspond to the value of the measured angle \(\varphi\).
Scale marks are also applied periodically or continuously to the modulating electrode of the tube, facilitating the reading of the angle \(\varphi\). A scale-mark generator serves this purpose.
To determine the sign of the angle, the sum and difference voltages are first additionally amplified and then rotated in “opposite directions” by \(45^\circ\). Thus, depending on the “sign of the angle,” the difference voltage will be either in phase or in antiphase with the sum voltage. Then the sum and difference voltages are added and subtracted. The resulting sum will be greater or less than the resulting difference, depending on the sign of the phase angle. By applying both of these voltages to the input of a balanced circuit, we obtain at its output a voltage whose magnitude and sign correspond to the magnitude and sign of the phase angle. This output voltage can be used to control a relay that switches on a lamp producing a mark beyond the perforation of the motion-picture film (for example, for a positive angle the mark is applied, and for a negative one it is absent).
Instrumental errors
With amplitude and phase unbalances of the channels of the receiver and indicator, errors may arise in the determination of phase angles. As a simple analysis shows, the instrumental error produced by the input system and the receiver will be equal to
\[ \Delta \varphi = \sqrt{\Delta \varphi_{\mathrm{in}}^{2} + \Delta \xi^{2} + \Delta \psi^{2}}, \]
where \(\Delta \varphi_{\mathrm{in}}\) is the difference of the phase distortions in the input section (before the amplifier channels);
\[ \Delta \xi = \frac{k_c}{k_p} - 1, \]
\(k_c\) and \(k_p\) are the gain factors of the receiver channels; \(\Delta \psi\) is the coefficient characterizing the deviation of the AGC from an ideal effective AGC:
\[ \Delta \psi = \frac{U_{c\,\mathrm{out}\,\max}}{U_0} - 1. \]
Thus, the amplitude unbalance of the amplifier channels, the phase unbalance of the input section, and the ineffectiveness of the AGC substantially affect the instrumental error. The amplitude unbalance (small) of the input circuit and the phase unbalance (of any magnitude) of the amplifier channels are insignificant. The influence of the indicator circuits on the instrumental error is not considered here.
The described method for measuring phase angles was proposed by the author in 1950.
Conclusions
- The advantages of the proposed method for measuring phase angles are: the admissibility of considerable phase unbalance of the main amplifier channels; the possibility of continuous photorecording-
of the measured phase difference; the simplicity of the circuit in the case where it is required to determine only the modulus of the phase difference.
- Disadvantages of the method: the presence of “dead zones” associated with the ineffective operation of the angle-sign determination unit; a reduction in measurement accuracy when the signal-to-noise ratio decreases.