RU2039360C1 - Method for determining phase shift between two sine-wave signals - Google Patents

Abstract

FIELD: measurement technology. SUBSTANCE: phase shift F₀ between two signals x(t) and y(t) filtered off d.c. component is determined by means of two instant values of one of signals assumed as measuring signal at time moments t₀₁ and t₁; first signal is selected when reference signal y(t) equals zero and second one, at half-wave center of signal y(t); then phase shift F₀ between signals is found from equation given in description of invention. EFFECT: reduced phase-shift measurement error at phase shifts approaching 0 or 180 deg even at infralow frequencies; permitted d c component in one of signals when measuring phase shift of value approaching 45 or 135 deg. 2 dwg

Description

 The invention relates to measuring equipment, in particular to methods for measuring the phase shift of two periodic electrical signals, and can be used for calibration of measuring channels, as well as for various types of phase processing of signals, mainly at infra-low frequencies.

 The method requires high precision measurements at small phase shifts between signals whose amplitudes vary in a large dynamic range.

 There is a simple method for determining the phase shift [1] in accordance with which the two studied signals are multiplied, the constant component of the signal obtained from the multiplication is isolated, and the voltage value of this constant component is measured, which is proportional to the absolute value of the phase shift.

 The method is characterized by insignificant measurement accuracy when isolating a constant component obtained from the multiplication of small signals, especially at infralow frequencies and small phase shifts.

 More sophisticated methods can improve measurement accuracy. The known method [2] in accordance with which the amplitudes of the sinusoidal signals are compared with the limit threshold value, while rectangular pulses are formed from the first signal with durations equal to the intervals between the intersection points of the signal half-waves with the limit threshold, two components are extracted from the second signal, and converted into bipolar pulses, determine the correlation coefficients between the generated pulse sequences from the first and second signals, and the desired phase shift is determined from the complex mathematical expression, taking into account the correlation coefficients.

 The method has a large number of operations, which reduces the accuracy of measurements, especially at infra-low frequencies with small phase shifts between signals with small amplitudes.

 The known method [3] in accordance with which three additional signals are formed to two studied signals: one of the studied signals is a reference signal, the first additional signal is phase-shifted with respect to the first signal under study, the other two additional signals are phase-shifted with fixed values with respect to reference signal under study, the values of the shifts of the additional signals are multiples of each other: the value of the phase shift between the signals under study is determined from the mathematical th expression, which includes the normalized values of the phase shifts chosen according to certain laws.

 The method is complicated in its implementation and has a low accuracy of determining the small phase shift between signals with small amplitudes, especially at infralow frequencies.

] signU21[arccosU22/

]
В соответствии с этим способом ведут измерение мгновенного значения фактически четырех сигналов, что при реализации измерений потребует четырех измерительных каналов. В результате погрешность измерения фазового сдвига будет велика из-за наличия четырех составляющих погрешностей от измерений четырех мгновенных значений, величины которых при различных значений разности фаз F_o будут изменяться, будут дополнительные погрешности от квадратурных сдвигов фаз на инфранизких частотах. Дополнительные погрешности появляются при измерении малых сдвигов фаз.The closest technical solution to the proposed one for a larger number of similar features is the method [4] according to which the sinusoidal signals from the DC component are filtered out, both signals are shifted by an angle of Π / 2 in the direction of advance and the instantaneous value of the first is measured at the same time signal U11, the instantaneous value of the first additional signal U12 shifted by Π / 2, the instantaneous value of the second signal U21 and the instantaneous value of the second additional signal U22 shifted by Π / 2, after which the phase value between the source signals is determined by the formula
F _o = signU11 [arccosU12 /

] signU21 [arccosU22 /

]
In accordance with this method, the instantaneous value of virtually four signals is measured, which, when implementing measurements, will require four measuring channels. As a result, the error in the measurement of the phase shift will be large due to the presence of four component errors from the measurements of four instantaneous values, the values of which for different values of the phase difference F _o will change, there will be additional errors from the quadrature phase shifts at infralow frequencies. Additional errors appear when measuring small phase shifts.

 The aim of the invention is to improve the accuracy of measurement.

The purpose of the method for determining the phase shift of two sinusoidal signals, according to which measure the instantaneous value of one of the signals X (t) and Y (t) filtered from the constant component, is achieved by measuring the second instantaneous value of the same signal X (t), taken as a measurement, and the first time t01 is selected when the other signal V (t), taken as the reference, is zero, the second time t1 is selected in the middle of the half-wave of the reference signal, and the phase shift value F _{o of the} signal X (t) relative to Y (t) is determined by mule
F _o = m (g + P _n ), where g = arctan [X (t01) / X (t1)] for the case when the measured signal X (t) outstrips the reference signal Y (t);
g = arctan [X (t01) / X (t1)] for the case when the measured signal X (t) is out of phase with the reference signal Y (t);
n = 0, m = 1 for common-mode signals, F _o | ≅ Π / 2;
m is -1; n = -1 for g> 0 or n = 1 for g <0 for antiphase signals, Π / 2 <F _o | ≅Π.

The phase shift is determined between two sinusoidal signals, which can be written as:
X (t) = A1sin (wt + F1) (1)
Y (t) A2 sin (wt + F2). (2) where A1 and A2 are the amplitudes of the studied signals;
F1 and F2 are the initial phases of the signals.

We write the ratio of these signals through the function f (t):
f (t) A1 sin (wt + F1) / A2 sin (wt + F2)
Transforming this expression and designating K = A1 / A2, we write the signal-quotient f (t) in the form
f (t) = K (sin wt cos F1 + sin F1 cos wt) / (sinx
x wt cos F2 + sin F2 cos wt) (3)
Dividing the numerator and denominator (3) by coswt ≠ 0, we obtain
f (t) K (tg wt cos F1 + sin F1) / (tg wt cosx
xF2 + sin F2) (4)
To determine the phase difference F _o between the signals X (t) and Y (t), the value of the initial phase shift F2 = 0 for F1> F2 is taken, then expression (4) is rewritten as
f (t) K [cos F _o + (sin F _o / tg wt)] (5)
Dividing the numerator and denominator in (5) by K, we get:
f (t) / K cos F _o + (sin F _o / tg wt) (6)
We consider expression (6) at time t1, when the value wt1 corresponds to the time corresponding to the middle of the half-wave of the divider signal, i.e., t1 = T / 4, where T / 4 = Π / 2. In this case, the denominator of the second term vanishes, since tg Π / 2 tends to infinity, therefore:
f (t1) / K cos F _o (7)
If we represent the function f (t) in terms of signal values, i.e. f (t) = X (t1) / Y (t1), then expression (7) is written as
[X (t1) / Y (t1)] / [X (t2) / Y (t1)] cosF _o (8)
After simplification
cos F _o X (t1) / X (t2) (9)
At time t01, corresponding to the point in time when the reference signal Y (t) is zero (Fig. 1), we can write the following expression:
X (t01) X (t2) sin F _o (10)
Expression (10) is rewritten as
sin F _o X (t01) / X (t2) (11)
Having divided the left and right sides of equations (11) and (9), respectively, we obtain
tg F _o X (t01) / X (t1) (12)
Therefore, the phase difference value F _o will be
F _o arctg [X (t01) / X (t1)] (13)
Formula (13) is valid for the first quadrant, i.e. with phase shifts 0 ≅ F _o ≅Π / 2. For phase shifts Π / 2 ≅F _o ≅ 0, the expression F _o = -arctg [X (t01) / X (t1)]
For phase shifts satisfying the expression Π ≅ F _o <Π / 2, the expression
F _o Π + arctg [X (t01) / X (t1)]
For phase shifts Π / 2 <F _o ≅Π, the expression
F _o Π arctan [X (t01) / X (t1)]
Therefore, for Π≅F _o ≅Π we can write
F _o m (g + Π _n ), (14) where g = arctan [X (t01) / X (t1)] for the case when the measured signal X (t) outpaces the reference signal Y (t);
g = -arctg [X (t01) / X (t1)] for the case when the measured signal X (t) is out of phase with the reference signal Y (t);
n = 0, m = 1 for common-mode signals, F _o | ≅Π / 2;
m = -1, n = -1 for g> 0 or n = 1 for g <0 for antiphase signals, Π / 2 <F _o | ≅ Π
From (9) it is possible to determine the phase shift F _o using the formula
F _o arccos [X (t1) / X (t2)] (15)
However, when using conventional technical means of measurement, when the values are measured using sixteen bit numbers, then for small values of phase shifts, the resolving ability for measuring the values of the arc cosine is limited, therefore, the measurement error is somewhat worsened.

From (10) it is possible to determine the phase shift F _o using the formula
F _o arcsin [X (t01) / X (t2)] (16)
Expression (16) allows to increase the accuracy of measurements of small phase shifts. However, in the analysis of signals with a small magnitude of amplitudes, the presence of a constant component in the measurement error at 0 ≅ | F _o | ≅ Π / 4.

Using for measuring arctg values from the proposed method allows to improve the method using arccos values when measuring small values of F _o and to improve the method using arcsin values when analyzing signals with small amplitudes. The following are examples of the implementation of the method.

 PRI me R 1. In FIG. 1 shows a simple device for implementing the method. The device contains two high-pass filters 1 and 2, respectively, and a two-beam oscilloscope 3. Filters 1 and 2 are connected to the sources of the first U1 (t) + U1 and second U2 (t) + U2 signals, respectively. The outputs of the filters 1 and 2 are connected to the first and second inputs of the two-beam oscilloscope 3, respectively. Filters 1 and 2 filter out the constant components of the signals (filters are needed if the signals have constant components), from their outputs the filtered signals U1 (t) and U2 (t) are fed to the first and second inputs of the two-beam oscilloscope 3, and the operator sees two sinusoidal signal (see Fig. 1).

 In accordance with the claims, their mutual position is out of phase or out of phase, and two values of one of the signals are measured at points t01 and t1, corresponding to the point in time when the reference signal is zero, and the point in time corresponding to the middle of the reference signal half-wave. Depending on the choice of the measured signal, taking into account the in-phase or out-of-phase signals, the value and sign of the phase difference of the studied signals are determined.

PRI me R 2. Filtered from a constant component, the signals are digitized in the ADC and recorded on a magnetic medium. After copying to a floppy disk, the recording is processed on an IBM PC / AT personal computer, a program using the proposed method. The measurement of the signal values and the calculation of the phase shift value F _o are carried out in accordance with the claims, taking into account the choice of the measured signal, in-phase or out-of-phase signals. As a result of the calculation, the value of the phase difference between the studied signals lying in the range from minus P to plus P appears on the display screen.

 The invention allows to determine the value of the phase shift between signals with any frequency. In particular, phase shifts were measured between signals of the infra-low-frequency range. A signal with a frequency of 0.1 Hz was supplied to the measuring channel, from the output of which a signal shifted in phase relative to the input signal was taken. The phase shift between the signals was determined in accordance with the proposed method depending on the options (according to example 1 or 2).

The accuracy of measurements is determined by the accuracy of measurements of the quantities included in formula (14). Since no auxiliary and preliminary transformations (except elimination of the constant components from the studied signals, if any) are carried out, the static maximum measurement error is determined by the sum of the measurement errors of only two parameters of the instantaneous values of one of the signals at time t01 and t1 (it is advisable inside one of half periods). Therefore, when processing the studied signals using a personal computer with the usual representation of numbers (for example, sixteen-bit personal computer IBM PC / AT), the resolution of the proposed method is increased. In the general case, it can be argued that with the principal angles of about 45 °, ^the proposed method has advantages over methods using the expressions for sin and cos, especially when measuring signals with small amplitudes (the influence of constant components in the signal is minimal). In real measurements at infra-low frequencies, it is not possible to completely get rid of the constant component, and then its presence will give an additional measurement error, especially for signals with small amplitudes. It should be noted that with increasing values of phase shifts from 0 to Π / 4, the influence of the constant component will decrease to zero, since when F _o | Π / 4 the equality X (t01) = X (t1) holds, the ratio of these values will be equal to "1". The presence of a constant component in the measuring signal in this case will not increase the error in determining the phase shift F _o .

The dynamic measurement error due to the error due to the final value of the sampling frequency can be determined by the ratio of the sampling interval to 1/4 of the period of the studied signals. To achieve dynamic error ^of not more than 0,001 is required to provide the ratio of the sampling interval to a quarter, a signal period of at least 1/57295, which corresponds to 2 to the power of minus 16.

The calculations was obtained by the maximum reduced error for measuring signal phase difference, measured in infranizko- frequency range about 1 Hz, which is close to ^about 0.001 when using 32-bit computers (e.g., IBM PC / RT), and the sampling frequency of about 250 kHz

A modern standard device F2-34 for measuring the phase shift between signals is characterized by a measurement error of 0.2 ^about , starting from 1 Hz and above, which is much worse than in the proposed method.

 All the considered methods that analyze the phase shift of the signals filtered from the constant component have a methodological error greater than the proposed method, since all other methods have either auxiliary transformations that reduce the error, or their measured parameters are larger in quantity than in the proposed method.

 The proposed method is acceptable in a wide frequency range and has a higher sensitivity when measuring small values of the phase difference than the prototype, and when the amplitude of one of the studied signals changes in the direction of decreasing in the proposed method, the measurement error is reduced due to the ability to measure instantaneous values of a larger amplitude signal.

Claims (1)

METHOD FOR DETERMINING PHASE SHIFT OF TWO SINUSOIDAL SIGNALS, according to which the instantaneous value of one of the signals X (t) and Y (t) filtered from the constant component is measured, characterized in that the second instantaneous value of the same signal X (t), taken as measuring, and the first time t _{0 1 is} selected when the other signal Y (t), taken as the reference, is zero, the second time t _{1 is} selected in the middle of the half-wave of the reference signal, and the phase shift value F _{0 of the} signal X (t) relative to signal Y (t) is determined by the formula
F _o = m (g + πn),
Where

for the case when the measured signal X (t) outpaces the reference signal Y (t);

;
for the case when the measured signal X (t) is out of phase with the reference signal Y (t);
n = 0, m = 1 for common-mode signals,

m = 1, n = -1 for g> 0 or n = 1 for g <0 for antiphase signals,

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