=Paper=
{{Paper
|id=Vol-3723/paper3
|storemode=property
|title=Diagnostic method development when weapons characteristics measuring based on spectral analysis for signals phase shift determination
|pdfUrl=https://ceur-ws.org/Vol-3723/paper3.pdf
|volume=Vol-3723
|authors=Vasyl Lytvyn,Victoria Vysotska,Sergey Tyshko,Oleksandr Lavrut,Tetiana Lavrut,Mariia Nazarkevych
|dblpUrl=https://dblp.org/rec/conf/modast/LytvynVTLLN24
}}
==Diagnostic method development when weapons characteristics measuring based on spectral analysis for signals phase shift determination==
https://ceur-ws.org/Vol-3723/paper3.pdf
Diagnostic method development when weapons
characteristics measuring based on spectral analysis for
signals phase shift determination
Vasyl Lytvyn1,†, Victoria Vysotska1,†, Sergey Tyshko2,∗,†, Oleksandr Lavrut3,†, Tetiana
Lavrut3,† and Mariia Nazarkevych1,†
1 Lviv Polytechnic National University, S. Bandera 12, 79013Lviv, Ukraine
2 State Scientific Research Institute of Armament and Military Equipment Testing and Certification, 18001
Cherkasy, Ukraine
3 Hetman Petro Sahaidachnyi National Army Academy, Heroes of Maidan 32, 79026Lviv, Ukraine
Abstract
The analysis of known methods that have found wide application in measuring technology and
are designed to control the technical and operational characteristics associated with the
measurement of phase shift during the development, manufacture and operation of weapons and
military equipment, has been carried out. Based on this analysis, it was determined that
measuring systems designed to determine the phase shift of two harmonic signals have two
information transmission channels. A measurement task was set to determine the phase shift of
two harmonic signals, using the spectral analysis of the signal obtained by summing the harmonic
signals after performing their two-semiperiod transformation. The assumptions necessary list
for the analytical ratios synthesis, which establishes the relationship between the phase spectra
and the amplitudes (power) of the signal obtained by summing harmonic signals after carrying
out their two-semiperiod transformation and phase shift of two harmonic signals, are
determined. Analytical relations are proposed that establish the relationship between the above-
mentioned characteristics. It is shown that the values of the spectrum of phases and amplitudes
calculated using the proposed expressions and ratios for calculating the Fourier series
coefficients differ by no more than 0.1%. The application of the proposed approach in the
artificial intelligence system to diagnose and determine the state of modern weapons and
military equipment will allow us to reduce the requirements for measuring equipment without
reducing the accuracy of measurements.
Keywords
decision-making system, complex system, phase shift, harmonic signal, spectral analysis1
MoDaST-2024: 6th International Workshop on Modern Data Science Technologies, May, 31 - June, 1, 2024, Lviv-
Shatsk, Ukraine
∗ Corresponding author.
† These authors contributed equally.
vasyl.v.lytvyn@lpnu.ua (V. Lytvyn); Victoria.A.Vysotska@lpnu.ua (V. Vysotska);
sergeytyshko57@gmail.com (S. Tyshko); alexandrlavrut@gmail.com (O. Lavrut); Lavrut_t_v@i.ua (T. Lavrut);
mariia.a.nazarkevych@lpnu.ua (M. Nazarkevych)
0000-0002-9676-0180 (V. Lytvyn); 0000-0001-6417-3689 (V. Vysotska); 0000-0003-3838-2027 (S.
Tyshko); 0000-0002-4909-6723 (O. Lavrut); 0000-0002-1552-9930 (T. Lavrut); 0000-0002-6528-9867 (M.
Nazarkevych)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
1. Introduction
Today, the Defense Forces of Ukraine actively use various types of weapons and military
equipment. An important component of solving the task of effective defence of the state is
ensuring the serviceable state of the anti-aircraft defence system, which is part of the units
and units of the defence forces, as well as its further modernization, as well as the
development of the latest types of weapons. The assessment of the technical condition of
the weapons and military equipment (WME) sample, or the decision on admission to the
supply of the latest weapons to the Armed Forces, is made based on tests, by analyzing its
technical characteristics for compliance with regulatory and technical documents (technical
task, technical conditions, operating instructions, repair documentation sets ). Defects are
an important technological operation during the capital repair works of WME. As you know,
the task of this technological operation is to determine the possibility of using components
from the composition of the sample subject to restoration. For mechanical components,
various parameters are measured, including geometric dimensions, and physical and
chemical properties of the material. To reduce the time and increase the accuracy of
measurements, predicting the state of the WME, it is necessary to use intelligent diagnostic
systems [16]. The use of artificial intelligence in such systems for processing measurement
results, and forecasting (simulation) of the future state of WME is an integral component of
the development of this field. The measurement of the above-mentioned characteristics of
small arms, artillery and missile-artillery weapons, and wheeled and tracked military
equipment is based on non-destructive control methods. Non-destructive control methods
include radiography, ultrasonic flaw detection, magnetic resonance research methods, and
others. Measuring systems that implement the specified measurement methods widely use
phaseometry methods [1, 2, 17, 34]. Also, phase measurement methods are widely used in
radar and radio navigation, aviation and space engineering, geodesy, mechanical
engineering, communication and many other fields [7, 21-27]. The phase-measuring
transformation of various physical processes into the phase shift of harmonic signals
ensures high metrological characteristics. Therefore, phaseometry, as a method of
transformation and measurement, has long gone beyond the traditional use in radio
engineering, navigation and communication and is successfully used in experimental
physics, radio physics, experimental medicine, modern fields of science and technology
when carrying out precision measurements [3-5, 13-15, 17-20, 37-39]. Based on the above,
in [2, 5], a list of WME parameters, which should be converted into a phase shift when
measured during sample manufacturing and testing, is defined. These parameters can
include: electrical and magnetic conductivity, permeability, geometric dimensions,
movement parameters, microdisplacement, capacity, element inductance, liquid level,
liquid or gas consumption, angle of rotation, speed of rotation, electric voltage and current,
temperature, distance, angle, delay time. Thus, conducting scientific research to find new
principles for determining the phase shift, which will allow to reduce the cost of performing
work on the control of the characteristics of the latest and modernized weapons at the
stages of development and production, is relevant.
2. Related works
For harmonic signals, such concepts as phase, initial phase, phase shift and delay time are
used in measuring technology. The most complete classification of methods for measuring
phase shifts of harmonic signals is given in works [2, 5, 6, 13, 14, 15].
According to the principle of measurement, the methods of phaseometry are divided into
compensatory methods and methods of converting the phase shift into other values -
voltage, time interval, and geometric parameters of oscillographic images of the
investigated signals. These methods differ from each other in technical implementation,
complexity and accuracy. Compensation methods [2, 5, 6, 13, 14] are based on the process
of balancing (compensating) the phase shift Δφ∈[0,2π) between the measured harmonic
signals, that is, reducing the phase shift to zero by adjusting the phase of one of the signals
using an adjustable phase shifter (measures of phase shift). This method ensures the
achievement of high measurement accuracy, close to the accuracy of a phase shifter.
Measurement methods based on the transformation of the phase shift into other signals [2,
5, 6, 13, 14, 15] allow determining the value of the phase shift of the signals after their
transformation into other intermediate values that are convenient to use for measurement.
These intermediate values include voltage, current, displacement of the electron beam of
the oscilloscope, and time values. The disadvantages of known methods include [2, 5, 6]:
a significant impact on the accuracy of measuring the phase shift of the error
component, which is caused by the phase asymmetry of the signal transmission
channels;
the presence of two channels for conducting analogue-to-digital conversion of input
signals, which leads to the need for mutual synchronization of the frequency of clock
generators for each of the channels;
significant impact on the accuracy of external and internal noise measurement;
non-linear nature of the scale.
At present, one of the effective and widespread ways of reducing the impact of external
and internal noise on the quality of solving problems of analyzing and processing signals of
various natures is the use of spectral analysis [35-36]. As is known, a periodic signal of any
form can be decomposed into harmonic signals whose frequencies are multiples of the
frequency (period) of the analyzed signal. A similar research method is called spectral
analysis, the mathematical basis of which is the Fourier series [8, 31-33].
Fourier series of arbitrary periodic signals can contain an infinitely large number of
terms. One of the advantages of the Fourier transform is that when limiting the Fourier
series to any finite number of its terms, it provides the best mean square error of
approximation to the original function (for a given number of terms). The convenience of
using the frequency representation of signals lies in the fact that harmonic functions are
eigenfunctions of operations of transfer, integration, differentiation and other linear
operations invariant in coordinates. They pass through linear systems without changing the
shape and frequency of the harmonics, only the initial phase and amplitude of the
oscillations change. In the general case, when expanding into a Fourier series of a periodic
Т Т
signal with a period 𝑇𝑠 , it is possible to use an interval [− с⁄2 , с⁄2].
2𝜋
If we denote the angular frequency through 𝜔с then since 𝜔с = Т , then for the function
с
Т Т
𝑓(𝑡) on the interval [− с⁄2 , с⁄2], the Fourier series has the form:
𝑛
𝑎0 (1)
𝑓(𝑡) = + ∑{𝑎𝑘 𝑐𝑜𝑠 𝜔𝑐 𝑘𝑡 + 𝑏𝑘 𝑠𝑖𝑛 𝜔𝑐 𝑘𝑡}.
2
𝑘=1
Expressions for determining the coefficients of the Fourier series 𝑎𝑘 , 𝑏𝑘 have the form:
𝑇𝑐
2 (2)
𝑎𝑘 = 𝑇 ∫ 2𝑇𝑐 𝑓(𝑡) 𝑐𝑜𝑠 𝜔𝑐 𝑘𝑡𝑑𝑡 , (𝑘 = 0,1,2, . . . , 𝑛)
𝑐 −
2
𝑇𝑐
2 (3)
𝑏𝑘 = 𝑇 ∫ 2𝑇𝑐 𝑓(𝑡) 𝑠𝑖𝑛 𝜔𝑐 𝑘𝑡𝑑𝑡 (𝑘 = 1,2, . . . , 𝑛).
𝑐 −
2
Series (1) represents the decomposition of the periodic signal f(t) into the sum of real
elementary harmonic functions (cosine and sine) with weighting coefficients, the geometric
sum of whose values (that is, the values of 𝑎𝑘 and 𝑏𝑘 are nothing but the real amplitudes of
the corresponding harmonic oscillations with frequencies 𝑘𝜔с. The set of amplitude values
of these harmonics forms a one-sided physically real (only for positive frequencies
𝑘𝜔с signal spectrum.
The goal of the work. To propose the scientific and technical basis of the alternative
principle of determining the phase shift, the mathematical basis of which is possible to
consider analytical relations that establish the relationship between the phase shift of two
harmonic signals and the spectral characteristics of the signal obtained when summing
them after carrying out a two-semiperiod transformation, which will allow to significantly
reduce the error component, due to the phase asymmetry of the signal transmission
channels and the influence of external and internal noise during monitoring of the
characteristics (parameters) of the WME.
3. Methods and models
At present, the determination of the phase shift is of greatest interest to phasometry. Phase
shift [9, 28-30] refers to the modulus of the difference between the initial phases of two
harmonic signals of the same frequency. As a rule, the measurement of phase shifts of
signals is based on the model of a harmonic signal, which is specified without changes in its
parameters over an infinite time interval. This model is ideal, and in practice, a model with
a finite time window is used, that is, measurements are carried out on a finite time interval.
Then, based on the above, we will formulate the problem of determining the phase shift
using the spectral analysis of the signal obtained as a result of the summation of two
harmonic signals, after carrying out their two-half-period transformation. Let's consider the
initial data necessary to solve the specified measurement problem.
Let there be two harmonic signals 𝑢1 (𝑡) and 𝑢2 (𝑡), which have a phase shift relative to
the other equal to 𝛥𝜑, in the interval from 0 to 2𝜋. Based on the fact that phase shift
measurements refer to relative measurements, the mathematical record of changes in
signals 𝑢1 (𝑡)and 𝑢2 (𝑡)can be represented in the form:
𝑢1 (𝑡) = 𝑈𝑚1 𝑐𝑜𝑠( 2𝜋𝑓𝑡), (4)
𝑢2 (𝑡) = 𝑈𝑚2 𝑐𝑜𝑠( 2𝜋𝑓𝑡 + 𝛥𝜑),
where 𝑈𝑚1 is signal amplitude 𝑢1 (𝑡); 𝑈𝑚2 is signal amplitude 𝑢2 (𝑡); 𝑓 = 1⁄Т is signal
frequency; Т is signal period.
The module is extracted from signals 𝑢1 (𝑡) and 𝑢2 (𝑡), as a result of which we get:
/ (5)
𝑢1 (𝑡) = |𝑢1 (𝑡)| = |𝑈𝑚1 𝑐𝑜𝑠( 2𝜋𝑓𝑡)|,
/
𝑢2 (𝑡) = |𝑢2 (𝑡)| = |𝑈𝑚2 𝑐𝑜𝑠( 2𝜋𝑓𝑡 + 𝛥𝜑)|,
/ /
After summing the signals 𝑢1 (𝑡) and 𝑢2 (𝑡), we get:
/
𝑢Σ (6)
𝑈1𝑚𝑖𝑛 − 𝑈2𝑚𝑖𝑛 / 𝑈1𝑚𝑖𝑛 + 𝑈2𝑚𝑖𝑛 2𝜋𝑓
𝑈1𝑚𝑖𝑛 + 𝑡 + (𝑈1𝑚𝑎𝑥 − ) sin ((2𝑓 − ) 𝑡) , 𝑡1 ≤ 𝑡 < 𝑡2
𝑡1.2 2 𝛥𝜑
=⟨
𝑈2𝑚𝑖𝑛 − 𝑈1𝑚𝑖𝑛 / 𝑈2𝑚𝑖𝑛 + 𝑈1𝑚𝑖𝑛 2𝜋𝑓
𝑈2𝑚𝑖𝑛 + 𝑡 + (𝑈2𝑚𝑎𝑥 − ) sin (( ) 𝑡) , 𝑡2 ≤ 𝑡 < 𝑡1
𝑡2.3 2 𝛥𝜑
Т
where: 𝑈1𝑚𝑖𝑛 and 𝑈2𝑚𝑖𝑛 are breakpoints of the function on the interval from 0 tо ;
2
𝛥𝜑 1 𝛥𝜑
𝑈1𝑚𝑎𝑥 = (𝑈𝑚1 + 𝑈𝑚2 ) 𝑐𝑜𝑠 2 is local maximum on the time interval 𝑡1.2 = 2𝑓 − 2𝜋𝑓 ;
′
𝛥𝜑 𝛥𝜑
′
𝑈2𝑚𝑎𝑥 = (𝑈𝑚1 + 𝑈𝑚2 ) 𝑠𝑖𝑛 2 is local maximum on the time interval 𝑡2.1 = 2𝜋𝑓 .
Figure 1: Structural diagram of the transformation of signals into a signal
Synthesis of time diagrams that describe the order of formation of the signal
/
𝑢∑ (𝑡) using relations (4) and (5) will be carried out in the Matlab environment using the
Simulink tool application. To do this, we synthesize the scheme shown in Fig. 1. In this
scheme, the synthesis of signal u1 (t) is carried out by the harmonic signal generator "Sine
Wave", and the signal u2 (t) by the generator "Sine Wave 1". From the "Sine Wave" output,
the signal enters the "Abs" block and the oscilloscope marked "Scope", which is designed to
form the time diagram of the 𝑢1 (𝑡) signal. According to the "Sine Wave 1" output, the signal
enters the "Abs 1" block and the oscilloscope marked "Scope 2", which is designed to form
/
the timing diagram of the u2 (t) signal. At the output of the "Abs" block, a signal 𝑢1 (𝑡) is
formed, which is fed to the first input of the summation device "Add" and "Scope 1". "Scope
/
1" forms a time diagram of the signal 𝑢1 (𝑡). Similarly, at the output of the "Abs 1" block, a
/
signal 𝑢2 (𝑡) is formed, which is fed to the second input of the summation device "Add" and
/
"Scope 3", which forms a time diagram of the signal 𝑢2 (𝑡). At the output of the "Add" block,
/
we will receive the signal 𝑢∑ (𝑡). The signal 𝑢∑ (𝑡) is fed to "Scope 4", which will form its
signal time diagram
4. Experiments, results and discussion
/ / /
Fig. 2 shows the time diagrams of the signals 𝑢1 (𝑡)/𝑢2 (𝑡), 𝑢1 (𝑡)/𝑢2 (𝑡) and 𝑢𝛴 (𝑡) at the
value 𝛥𝜑 = 0, as well as the "Sine Wave" generator settings tab ” and “Sine Wave1”.
Figure 2: Time diagrams of signals 𝑢1 (𝑡) and 𝑢2 (𝑡),, 𝑢1/ (𝑡) and 𝑢2/ (𝑡), and 𝑢𝛴/ (𝑡) at the value of
𝛥𝜑 = 0
According to the settings tabs, generators "Sine Wave" generates a sinusoidal signal with
an amplitude equal to 1, and "Sine Wave1" with an amplitude equal to 0.8, other settings of
the tabs are as follows: the constant component of the signals is 0, the frequency of the signal
is 1 rad/s, the initial phase of the signals is 0 rads, the quantization time is minimal for this
device, i.e. 0 on the tab. The timing diagram shown on "Scope" and "Scope 2" fully confirms
/
the settings on the tab. The time diagrams for the 𝑢∑ (𝑡) signal are shown on the "Scope 4"
/
tab. The analysis of this time diagram shows that the signal 𝑢∑ (𝑡) has a period equal to π
s, the maximum value is equal to 1.8, and the minimum value is equal to 0. The obtained
results correspond to the summation of two harmonic signals, after carrying out their two-
and-a-half period transformation.
/ / /
Fig. 3 shows the time diagrams of signals 𝑢1 (𝑡) and 𝑢2 (𝑡), 𝑢1 (𝑡) and 𝑢2 (𝑡) and 𝑢𝛴 (𝑡), at
the value of 𝛥𝜑 = 0,9 rad, as well as the generator settings tab "Sine Wave" and "Sine
Wave1". According to the configuration tabs, the "Sine Wave" generator generates a
sinusoidal signal with an amplitude equal to 1, the constant component of the signal is 0,
the frequency of the signal is 1 rad/s, the initial phase of the signal is 0 rad, the quantization
time is minimal for this device, i.e. 0 on the tab.
Figure 3: Time diagrams of signals 𝑢1 (𝑡) and 𝑢2 (𝑡),, 𝑢1/ (𝑡) and 𝑢2/ (𝑡), and 𝑢𝛴/ (𝑡) at the value of
𝛥𝜑 = 0.9 rad
According to the settings tabs, the "Sine Wave1" generator generates a sinusoidal signal
with an amplitude equal to 0.8, the constant component of the signal is 0, the frequency of
the signal is 1 rad/s, the initial phase of the signal is 0.9 rad, the quantization time is minimal
/ /
for this device, i.e. 0 at tab. The time diagrams for signals 𝑢1 (𝑡) and 𝑢2 (𝑡) are shown in the
"Scope 1" and "Scope 3" tabs. The analysis of the data of the time diagrams shows that they
/ /
have a period equal to π s, the maximum values of 𝑢1 (𝑡) are equal to 1, and 𝑢2 (𝑡) is 0.8, also
/ / /
the 𝑢2 (𝑡) signal precedes the 𝑢1 (𝑡) about 0.9. The time diagrams for the 𝑢∑ (𝑡) signal is
/
shown on the "Scope 4" tab. Analysis of this time diagram shows that the signal 𝑢∑ (𝑡) has
a period equal to π s. In the time interval from 0 to (π-0.9) s, the maximum value is
approximately 1.65, and the minimum value is approximately 0.78. In the time interval from
(𝜋 − 0,9) to π, the maximum value is approximately 0.8, and the minimum value is 0.6.
/ / /
Fig. 4 shows the time diagrams of the signals 𝑢1 (𝑡)/𝑢2 (𝑡), 𝑢1 (𝑡)/𝑢2 (𝑡), and 𝑢𝛴 (𝑡) at the
value of 𝛥𝜑 = 1.2 rad, as well as the tab for setting the generator "Sine Wave”/“Sine Wave1”.
Figure 4: Time diagrams of signals 𝑢1 (𝑡) and 𝑢2 (𝑡), 𝑢1/ (𝑡) and 𝑢2/ (𝑡), and 𝑢𝛴/ (𝑡) at Δφ=1.2 rad
According to the configuration tabs, the "Sine Wave" generator generates a sinusoidal
signal with an amplitude equal to 1, the constant component of the signal is 0, the frequency
of the signal is 1 rad/s, the initial phase of the signal is 0 rad, the quantization time is
minimal for this device, i.e. 0 on the tab. According to the settings tabs, the "Sine Wave1"
generator generates a sinusoidal signal with an amplitude equal to 0.8, the constant
component of the signal is 0, the frequency of the signal is 1 rad/s, the initial phase of the
signal is 1.2 rad, the quantization time is minimal for this device, i.e. 0 on the tab. The time
/ /
diagrams for signals 𝑢1 (𝑡) and 𝑢2 (𝑡) are shown in the "Scope 1" and "Scope 3" tabs. The
analysis of the data of the time diagrams shows that they have a period equal to 𝜋 s, the
/ / /
maximum values of 𝑢1 (𝑡) are equal to 1, and 𝑢2 (𝑡) is 0.8, also the 𝑢2 (𝑡)signal precedes the
/
𝑢1 (𝑡) about 1.2 rad. The time diagrams for the 𝑢∑ (𝑡) signal are shown on the "Scope 4"
/
tab. Analysis of this time diagram shows that the signal 𝑢∑ (𝑡) has a period equal to π s. In
the time interval from 0 to (𝜋 − 1.2) s, the maximum value is approximately 1.5, and the
minimum value is 0.95. In the time interval from (𝜋 − 1.2) to π, the maximum value is
approximately 1.05, and the minimum value is 0.75.
/
From Fig. 4, it can be seen, and it is also shown in [10, 11] that 𝑢∑ (𝑡) depending on the
value of Δφ of the input signals 𝑢1 (𝑡) and 𝑢2 (𝑡):
1 Т
is a periodic signal with a period of T == ;
2𝑓 2
has two local maxima 𝑈1𝑚𝑎𝑥 and 𝑈2𝑚𝑎𝑥 , which correspond to the moments of 𝑡1𝑚𝑎𝑥 ,
𝑡2𝑚𝑎𝑥 and two breaks 𝑈1𝑚𝑖𝑛 and 𝑈2𝑚𝑖𝑛 , which correspond to the moments of time 𝑡1 ,
𝑡2 , respectively;
a change in the value of the phase shift angle 𝛥𝜑 leads to a change in the values of
𝑈1𝑚𝑎𝑥 , 𝑈2𝑚𝑎𝑥 , 𝑈1𝑚𝑖𝑛 end 𝑈2𝑚𝑖𝑛 and time parameters 𝑡1𝑚𝑎𝑥 , 𝑡2𝑚𝑎𝑥 end 𝑡2 .
/
A change in the values of the above characteristics of the signal 𝑢∑ (𝑡) leads to a change
in its shape, which in turn will lead to a change in the values of the coefficients 𝑎𝑘 and 𝑏𝑘 of
the Fourier series depending on the change in the magnitude of the phase shift 𝛥𝜑.
When conducting spectral analysis, such concepts as amplitude spectrum, power
spectrum and phase spectrum are used. When conducting spectral analysis, such concepts
as amplitude spectrum, power spectrum and phase spectrum are used. According to [12],
the spectrum of amplitudes is understood as the set of absolute values of the coefficients 𝐶𝑘
(𝑘 = 0,1,2, . . . , 𝑛), which are determined from the known values of 𝑎𝑘 and 𝑏𝑘 by the ratio:
(7)
√𝑎𝑘2 + 𝑏𝑘2
|𝐶𝑘 | = .
2
The value |𝐶𝑘 | indicates the value of the amplitude of the k-th harmonic signal when
expanded into a Fourier series. The power spectrum is understood as a set of arguments of
the values |𝐶𝑘 |2. The spectrum of phases means the set of arguments ∠𝐶𝑘 (𝑘 = 0,1,2, . . . , 𝑛),
which are determined by the known values of 𝑎𝑘 and 𝑏𝑘 by the ratio:
𝑏𝑘 (8)
∠𝐶𝑘 = 𝑎𝑟𝑐𝑡𝑔 .
𝑎𝑘
The value of ∠𝐶𝑘 indicates the value of the initial phase of the k-th harmonic signal when
expanded into a Fourier series. Then, based on the above, we will formulate the task of
determining the phase shift using Fourier series decomposition, as a synthesis of analytical
relations that describe the relationship between the change in the spectrum of amplitudes
/
(power) and the spectrum of the phases of the signal 𝑢∑ (𝑡) depending on the change in
𝛥𝜑. Synthesis of relations that determine the relationship between the value of 𝛥𝜑 and the
characteristics of the spectrum of amplitudes (power) and the spectrum of phases of the
/
signal 𝑢∑ (𝑡) using the following assumptions. As can be seen from (4), the signal
/ / /
𝑢∑ (𝑡) is formed by summing two signals 𝑢1 (𝑡) and 𝑢2 (𝑡). These signals are formed by
signals 𝑢1 (𝑡) and 𝑢2 (𝑡), while the amplitude of 𝑈𝑚1 differs from the amplitude of 𝑈𝑚2 by v
times, and have a phase shift 𝛥𝜑.
Then it can be stated that the signal 𝑢1 (𝑡) precedes the signal 𝑢2 (𝑡)by a certain time
interval τ, i.e. 𝑢2 (𝑡) = 𝑣𝑢1 (𝑡 − 𝜏). According to the known values of 𝛥𝜑 and T, the time
interval τ is determined by the ratio:
𝛥𝜑 (9)
𝜏= Т.
2𝜋
As shown in [12], the shift of the signal in the time domain by some interval τ leads to a
change in the phase spectrum, but the spectrum of the signal amplitudes remains
unchanged. Taking into account expressions (7) and (8) provided that 𝑈𝑚1.𝑘 and 𝑈𝑚2.𝑘 are
/ /
the amplitude values of the 𝑘 -th harmonics of signals 𝑢1 (𝑡) and 𝑢2 (𝑡), respectively, and 𝜑1.к
/ /
and 𝜑2.к values of the initial phases of the 𝑘 -th harmonics of the signals 𝑢1 (𝑡) and 𝑢2 (𝑡),
/
respectively, the Fourier series for the signal 𝑢∑ (𝑡) will have appearance:
/
u (t ) 𝑈𝑚1.0 + ∑𝑛
𝑘=1[𝑈𝑚1.𝑘 𝑠𝑖𝑛( 2𝜔𝑘𝑡 + 𝜑1.𝑘 )] + 𝑈𝑚2.0 +
(10)
𝑛
+ ∑[𝑈𝑚2.𝑘 𝑠𝑖𝑛( 2𝜔𝑘𝑡 + 𝜑2.𝑘 )] =
𝑘=1
= (𝑈𝑚1.0 + 𝑈𝑚2.0 ) + [𝑈𝑚1.1 𝑠𝑖𝑛( 2𝜔𝑡 + 𝜑1.1 ) + 𝑈𝑚2.1 𝑠𝑖𝑛( 2𝜔𝑡 + 𝜑2.1 )] +
+[𝑈𝑚1.2 𝑠𝑖𝑛( 4𝜔𝑡 + 𝜑1.2 ) + 𝑈𝑚2.2 𝑠𝑖𝑛( 4𝜔𝑡 + 𝜑2.2 )] + ⋯ +
+[𝑈𝑚1.𝑛 𝑠𝑖𝑛( 2𝑛𝜔𝑡 + 𝜑1.𝑛 ) + 𝑈𝑚2.𝑛 𝑠𝑖𝑛( 2𝑛𝜔𝑡 + 𝜑2.𝑛 )]
/
To synthesize analytical relations for determining the values of the amplitude 𝑈𝑚 ∑ 1 and
/ /
the initial phase 𝜑𝑚 ∑ 1 of the first harmonic of the signal 𝑢∑ (𝑡), we will construct the vector
/
diagram of the first harmonics of the signals 𝑢1 (𝑡) and 𝑢2 (𝑡)in the presence of a phase shift
equal to 𝛥𝜑 of the signals 𝑢1 (𝑡) and 𝑢2 (𝑡), provided that the initial phase of the signal
𝑢1 (𝑡) is zero, taking into account the following remarks and assumptions:
it is known that the frequency of the first harmonic of the signal is equal to the
frequency of the signal;
/ /
it is shown in [10, 11] and it can be seen in Figs. 2...4 that 𝑢1 (𝑡), 𝑢2 (𝑡) and
/
𝑢∑ (𝑡)have a period that is two times shorter than 𝑢1 (𝑡) and 𝑢2 (𝑡);
/ /
it can be asserted that 𝑢2 (𝑡) = 𝑣𝑢1 (𝑡 − 𝜏) and 𝑢2 (𝑡) = 𝑣𝑢1 (𝑡 − 𝜏), that is, the time
/ /
interval of the shift τ between the signals 𝑢1 (𝑡) and 𝑢2 (𝑡), and the shift time interval
τ between signals 𝑢1 (𝑡) and 𝑢2 (𝑡) is the same;
based on relation (9) and the above remarks, it can be seen that the value of the
/ /
phase shift of the first harmonics of signals 𝑢1 (𝑡) and 𝑢2 (𝑡)will be equal to 2𝛥𝜑, if
there is a phase shift equal to Δφ of signals 𝑢1 (𝑡) and 𝑢2 (𝑡).
The above vector diagram is shown in Fig. 5. From Fig. 5, it can be seen that for the
calculation of the values of the characteristics of the vector of the first harmonic of the signal
/ / /
𝑢∑ (𝑡) according to the vectors of the first harmonics of the signals 𝑢1 (𝑡) and 𝑢2 (𝑡), the
parallelogram method was used.
Figure 5: Vector diagram for determining the value of the initial phase φm/ 1 of the first
harmonic of the signal u
/
(t )
/
Thus, the value of the amplitude 𝑈𝑚 ∑ 1 of the first harmonic of the signal
/
𝑢∑ (𝑡) according to the known values of the amplitudes of the first harmonics 𝑈𝑚1.1 and
/ /
𝑈𝑚2.1 = 𝑈 з 𝑚2.1 of the signals 𝑢1 (𝑡) and 𝑢2 (𝑡) in the presence of a phase shift equal to 𝛥𝜑 of
signals 𝑢1 (𝑡) and 𝑢2 (𝑡) is determined by the ratio:
/ (11)
𝑈𝑚 ∑ 1 = √𝑈 2 𝑚1.1 + 𝑈 2 𝑚2.1 + 2𝑈𝑚1.1 𝑈𝑚2.1 𝑐𝑜𝑠 2 𝛥𝜑
/
Then the value of the amplitude 𝑈𝑚 ∑ 𝑘 of the k-th harmonic of the signal
/
𝑢∑ (𝑡) according to the known values of the amplitudes 𝑈𝑚1.𝑘 and 𝑈𝑚2.𝑘 of the k-th
/ /
harmonics of the signals 𝑢1 (𝑡) and 𝑢2 (𝑡) at a phase shift value equal to 𝛥𝜑 of signals 𝑢1 (𝑡)
and 𝑢2 (𝑡) is calculated using the ratio:
/ (12)
𝑈𝑚 ∑ 𝑘 = √𝑈 2 𝑚1.𝑘 + 𝑈 2 𝑚2.𝑘 + 2𝑈𝑚1.𝑘 𝑈𝑚2.𝑘 𝑐𝑜𝑠 2 𝑘𝛥𝜑. (𝑘 = 0,1,2, . . . , 𝑛)
The synthesis of analytical ratios for calculating the values of 𝑈𝑚1.𝑘 will be carried out in
/
the following order. Let's determine the spectrum of amplitudes for the signal 𝑢1 (𝑡) under
the condition 𝑈𝑚1 = 1. The relations for calculating the coefficients of the Fourier series 𝑎1.𝑘
/
и 𝑏1.𝑘 of the signal 𝑢1 (𝑡) are as follows:
𝑇′ (13)
2
𝑎1.𝑘 = 𝑇 ′ ∫ 2𝑇′|𝑐𝑜𝑠( 𝜔𝑡)| 𝑐𝑜𝑠( 2𝜔𝑘𝑡)𝑑𝑡. (𝑘 = 0,1,2, . . . , 𝑛)
−
2
𝑇′ (14)
2 2
𝑏1.𝑘 = ∫ ′ |𝑐𝑜𝑠( 𝜔𝑡)| 𝑠𝑖𝑛( 2𝜔𝑘𝑡)𝑑𝑡, (𝑘 = 1,2, . . . , 𝑛)
𝑇 ′ −𝑇
2
Based on the known values of 𝑎1.𝑘 and 𝑏1.𝑘 , the value of 𝑈𝑚1(1).𝑘 is calculated by the ratio:
2 2 (15)
√𝑎1.𝑘 + 𝑏1.𝑘
𝑈𝑚1(1).𝑘 = .
2
The values of 𝑈𝑚1(1).𝑘 were calculated using the universal mathematical package
MathCAD, the results are presented in Fig. 6. Using the property of the linearity of the
/
Fourier series, the relationship for determining the value of the amplitude 𝑈𝑚 ∑ 𝑘 k-th
/
harmonic of the signal 𝑢∑ (𝑡) based on the known values of the amplitudes 𝑈𝑚1 and 𝑈𝑚2
at the value of the phase shift of the initial signals 𝑢1 (𝑡) and 𝑢2 (𝑡), which is equal to 𝛥𝜑, is
calculated using the relationship:
/ (16)
𝑈𝑚 ∑ 𝑘 = 𝑈𝑚1(1).𝑘 √𝑈 2 𝑚1. + 𝑈 2 𝑚2. + 2𝑈𝑚1. 𝑈𝑚2. 𝑐𝑜𝑠 2 𝑘𝛥𝜑, (𝑘 = 0,1,2, . . . , 𝑛)
/
To synthesize analytical relations to determine the values of the initial phase 𝜑𝑚 ∑ 1 of the
/
first harmonic of the signal 𝑢∑ (𝑡), we will use the theorem of sines for the triangle formed
/
by the sides 𝑈𝑚1.1 , 𝑈 р 𝑚2.1 and 𝑈𝑚 ∑ 1 on the vector diagram of the first harmonics of the
/ /
signals 𝑢1 (𝑡) and 𝑢2 (𝑡) in the presence of a phase shift equal to 𝛥𝜑 of the signals 𝑢1 (𝑡) and
𝑢2 (𝑡), we write
𝑈𝑚1.1 𝑈 𝑝 𝑚1.1
=
𝑠𝑖𝑛(𝛼) 𝑠𝑖𝑛 (𝜑/ )
𝑚∑1
/
Solving this equation relative to 𝜑𝑚 ∑ 1 based on the fact that
𝑈𝑚1.1 = 𝑈𝑚1(1).1 𝑈𝑚1 ,
𝑝
𝑈𝑚2.1 = 𝑈𝑚2.1 = 𝑈𝑚1(1).1 𝑈𝑚2 ,
/
𝛼 = 2𝛥𝜑 − 𝜑𝑚 ∑ 1 .
/
We obtain that the value of the initial phase 𝜑𝑚 ∑ 1 of the first harmonic of the signal
/
𝑢∑ (𝑡)depending on the value of Δφ is determined by the following relation, provided that
the initial phase of the signal u_1 (t) is zero
/ 𝑈𝑚1 (17)
𝜑𝑚 ∑ 1 = 𝑎𝑟𝑐𝑐𝑡𝑔 [𝑐𝑜𝑠 𝑒 𝑐(2𝛥𝜑) ⋅ ( + 𝑐𝑜𝑠( 2𝛥𝜑))]
𝑈𝑚2
𝑈𝑚1(1).𝑘 =
Figure 6: The value of 𝑈𝑚1(1).𝑘 for 𝑈𝑚1 = 1В
/
Taking into account (9), for the kth harmonic, the initial phase 𝜑𝑚 ∑ 𝑘 on the interval
[− 𝜋⁄2 , 𝜋⁄2] of the signal 𝑢∑ (𝑡)will be equal to:
/
/ 𝑈 (18)
𝜑𝑚 ∑ 𝑘 = 𝑎𝑟𝑐𝑐𝑡𝑔 [𝑐𝑜𝑠 𝑒 𝑐(2𝑘𝛥𝜑) ⋅ (𝑈𝑚1 + 𝑐𝑜𝑠( 2𝑘𝛥𝜑))], (𝑘 = 0,1,2, . . . , 𝑛)
𝑚2
Let us consider the possibility of using the obtained relations for the synthesis of the
/
spectrum of amplitudes and phases of the signal 𝑢∑ (𝑡). To do this, let's compare the
/
spectrum of amplitudes and phases of the signal 𝑢∑ (𝑡) obtained using relations (17) and
(18) and the following expressions:
𝑇′ (19)
2
𝑎𝑘 = 𝑇 ′ ∫ 2𝑇′[|𝑈𝑚1 𝑐𝑜𝑠( 𝜔𝑡)| + |𝑈𝑚2 𝑐𝑜𝑠( 𝜔𝑡 + 𝛥𝜑)|] 𝑐𝑜𝑠 2 𝜔𝑘𝑡𝑑𝑡, (𝑘 = 0,1,2, . . . , 𝑛)
−
2
𝑇′ (20)
2 2
𝑏𝑘 = ∫ ′ [|𝑈𝑚1 𝑐𝑜𝑠( 𝜔𝑡)| + |𝑈𝑚2 𝑐𝑜𝑠( 𝜔𝑡 + 𝛥𝜑)|] 𝑠𝑖𝑛 2 𝜔𝑘𝑡𝑑𝑡, (𝑘 = 1,2, . . . , 𝑛)
𝑇 ′ −𝑇
2
2 2 (21)
√𝑎1.𝑘 + 𝑏1.𝑘 𝑏𝑘
𝑈𝑚.𝑘 = , ∠𝐶𝑘 = 𝑎𝑟𝑐𝑡𝑔 .
2 𝑎𝑘
As an indicator of approximation of the spectrum of amplitudes, we use the difference
/
between the value of 𝑼𝒎 ∑ 𝒌 obtained using relation (16) and the value of 𝑼𝒎.𝒌 calculated
using relations (19-21). The calculations were performed in the MathCAD mathematical
package, and the comparison results for some values of 𝛥𝜑 , 𝑈𝑚1 and 𝑈𝑚2 are presented in
Fig. 7a. These results show that the relative deviation of the amplitude spectrum obtained
using (17) differs from the values of the amplitude spectrum obtained using (19 ... 21) less
than 0.1%. As an indicator of approximation of the phase spectrum, we use the difference
/
between the value of 𝜑𝑚 ∑ 𝑘 obtained using relation (18) and the value of ∠𝑪𝒌in the interval
[-π⁄2,π⁄2] calculated using relation (19, 20, 21).
The results of the comparison for some values of Δφ are presented in Fig. 7b. These
results show that the relative deviation of the phase spectrum obtained using (19) differs
from the values of the phase spectrum obtained using (19, 20, 21) by less than 0.1%.
5. Conclusions
An analysis of the known principles of measuring phase shifts was carried out. The analysis
showed that a significant contribution to the final measurement error of phase shifts is
made by the component caused by the phase asymmetry of signal transmission channels
and the influence of external and internal noise. As an alternative approach to determining
the phase shift, it is proposed to use a signal obtained as a result of summing harmonic
signals after carrying out their two-semiperiod transformation followed by its spectral
analysis. Analytical relations are proposed that establish the relationship between the
phase shift and the characteristics of the spectrum of amplitudes and phases of the
considered signal. The adequacy of the proposed analytical ratios was checked. As a result
of the verification, it was established that the relative discrepancy between the
characteristics of the spectrum obtained using the proposed ratios and using the ratios for
calculating the coefficients of the Fourier series does not exceed 0.1%.
a b
Figure 7: The difference between the value of a) the quantities U m/ k / U m.k and b) φк / Ck
These analytical relations can be considered as a mathematical basis for the synthesis of
methods for determining the phase shift of two harmonic signals using the spectral analysis
of the signal obtained as a result of their addition after carrying out their two-half-cycle
transformation. Further research should be directed to the development of intelligent
diagnostic systems, which will be built based on algorithms of artificial intelligence, which,
in turn, will reduce the time and increase the accuracy of measurements and forecasting the
state of modern WME.
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