RU2377577C1 - Method of measuring harmonic signal vector - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: invention relates to electrical measurement techniques. The method can be used in apparatus for measuring passive and active complex values of alternating current, for example in alternating current bridges to measure parametres of multiple-element two-terminal networks by improving a branched bridge measuring circuit at the same time with several test effects with different frequencies, as well as when measuring in case of several harmonic interferences. The method involves obtaining projections

and

of a measured signal S_j(t)=A_jsin(2πt/T_j+φ_0j) on two orthogonal reference signals by adding the sample sum of synergetic signals σ(t), sampling instants of which make up sets {t_i'}_j and {T_i"=t_i'+T_j/4}, which are formed using a step-wise procedure from an initial set, which consists of one arbitrary initial time instant t₀, by obtaining at the first step an additional set which also consists of one instant, by shifting the initial set to an odd number of half-periods of the first signal, and then obtaining at the second step an additional set which consists of two instants, by shifting two instants of the overall set obtained at the first step, to an odd number of half-periods of the second signal and the procedure is continued with formation of an additional set on each step by shifting the overall set, formed at the previous step, to an odd number of half-periods of the next signal and with doubling on each step the number of instants for sampling the sum of the synergetic signals until the number of step equals M-1, and amplitude A_j phase shift φ_0j of signal S_j(t) is determined from the following expressions:

for

where t_i'=t_i+t₀, t₁=0, t₂=t₁+n₁T₁/2, t₃=t₁+n₂T₂/2, t₄=t₂+n₂T₂/2, t₅=t₁+n₃T₃/2, t₆=t₂+n₃T₃/2, t₇=t₃+n₃T₃/2, t₈=t₄+n₃T₃/2,……, t_2m+1=t₁+n_m+1T_m+1/2, t_2m+2=t₂+n_m+1T_m+1/2,…, t_2m+1=t_2m+n_m+1T_m+1/2,…, t_2m-1=t₂ ^m-2+τ_M,

m≠j.

EFFECT: possibility of accurate real time measurement of any synergetic harmonic signals with known periods, as well as possibility of simultaneous measurement of several signals under the condition that, the frequency of the measured signal is not a multiple of frequencies of synergetic signals.

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Description

The invention relates to the field of electrical engineering and can be used in measuring instruments for complex values of alternating current, for example, in bridges of alternating current.

There is a method of measuring the parameters of multi-element two-terminal networks by balancing a branched bridge measuring circuit with the simultaneous action of several test actions with different frequencies, shared using analog filters [Sheremet L.P. The principles of constructing bridge measuring circuits for simultaneous balancing at several frequencies. // Problems of technical electrodynamics, issue 54, Kiev: Naukova Dumka, 1975. - S.14-19].

The ability to perform measurements simultaneously at several frequencies provided by this method allows one to obtain information about rapidly changing complex measurement objects and the processes occurring in them. However, analog filters used to separate test actions with different frequencies, on whose selectivity the measurement accuracy depends, have inertia and complexity of implementation, increasing as their selectivity increases, which is a disadvantage of the method.

There is also a method of measuring the fundamental (first) harmonic of a periodic signal, adopted by the author as a prototype, according to which the measurement of the sine and cosine components of this harmonic, i.e. its projections onto two orthogonal ones, i.e. the reference signals shifted relative to each other by a 90 ° phase in real time are carried out in real time by uneven sampling of this signal at a multitude of time instants, which are linear combinations of half-periods of its harmonics with numbers equal to prime numbers: 1, 2, 3, 5, 7, 11, ..., and the summation of the set obtained on this set of time discrete [Pat. P.246351, Poland, MKI G01R, Sposob i uklad do pomiaru wektora harmonicznej podstawowej przebiegu odksztalconego / Sawicki J. - Warszawa: Biuletyn Urzedu Patentowego. - 1985. - Nr 19 (307). - P.39].

The disadvantage of this method is the ability to measure only one of the jointly acting signals, with a strictly specific type, namely the first harmonic of a periodic signal, which does not allow solving a number of measurement problems, for example, the above-mentioned problem of measuring complex objects with rapidly changing parameters, requiring simultaneous measurements on several frequencies, the task of measuring signals against a background of deterministic harmonic interference, etc.

The aim of the invention is to provide the ability to measure any of the M jointly acting harmonic signals if its frequency is not a multiple of the frequencies of other signals, moreover, in real time and with high accuracy, as well as the simultaneous measurement of several signals by increasing the number of measuring channels in proportion to the number of measured signals.

и

измеряемого сигналаThis goal is achieved by the fact that in the method of measuring the vector of a harmonic signal acting together with M-1 by other harmonic signals and having, like them, a known period T _j and unknown amplitude A _j and an initial phase shift φ _0j at which projections

and

measured signal

S _j (t) = A _j sin (2πt / T _j + φ _0j )

into two orthogonal vectors of reference signals associated with A _j and φ _0j , for example, by the relations

и

,

and

,

measured by sampling and summing the discrete discrete signal

using instant pulses acting at time instants, forming the sets {t _i ^' } and {t _i ^'' = t _i ^' + T _j / 4} _j , and {t _i ^' } _{j is} formed so that

и

and

where K _j - coefficient depending on the formation procedure

{t _i '} _j , a N is the number t _i ^' , any of the jointly acting signals is selected as the measured one, the repetition rate of which 1 / T _{j is} not a multiple of the frequencies of other signals, and the set {t _i ^' } _{j is} instantaneous discrete sampling pulse sum signal σ (t) is formed by a stepwise procedure from the initial set consisting of one arbitrary initial instant t _0, by preparing in the first step an additional set consisting also of a single point, by shifting the initial set of an odd Num about the half-periods of the first signal, then receiving at the second step an additional set consisting of two moments, by shifting the two moments of the total set obtained in the first step by an odd number of half-periods of the second signal and continue this procedure with the formation at each mth step of an additional sets by shifting the total set formed in the previous step by an odd number n _{m of} half-cycles of the next suppressed signal, and doubling the number of sampling times of the signal discrete σ (t), until the number of steps becomes equal to M-1, and the amplitude A _j and the phase shift φ _{0j of the} signal S _j (t) are determined by the relations

при

at

where t _i ^' = t _i + t ₀ , t ₁ = 0, t ₂ = t ₁ + n ₁ T _1/2 ,

m≠j. t ₃ = t ₁ + n ₂ T _2/2 , t ₄ = t ₂ + n ₂ T _2/2 , t ₅ = t ₁ + n ₃ T _3/2 , t ₆ = t ₂ + n ₃ T ₃ / 2, t ₇ = t ₃ t ₃ + n _3/2, t = t ₈ + n _{4 3 3} t / 2, ......,

m ≠ j.

The invention is illustrated by graphic material: figure 1 shows the procedure for the formation of many points in time {t _i '} _j samples discrete total signal

and figure 2 explains the method of obtaining the mathematical expression for K _j .

The essence of the invention lies in the fact that the proposed procedure for generating a set of time instants {t _i '} _j allows you to accurately and quickly (in real time) measure the projection onto the orthogonal vectors of the reference signals of any of the signals included in σ (t), in particular, the projection p ^' _j and p ^'' _{j of the} signal S _j (t), if its frequency is not a multiple of the frequencies of other signals, it is invariant with respect to the others, i.e. exclude the influence of all these signals on the measurement accuracy p ^' _j and p ^'' _j , and hence A _j and

φ _oj .

This is achieved in that the set of time points at which the instantaneous single pulses indicated by the arrows act, sampling the discrete signal σ (t), is formed using the procedure for step-by-step elimination of the influence (suppression) of all signals acting together with the measured one. As a result, at the mth step, m signals are suppressed, and at the (M-1) m, everything except the measured one. The degree of signal suppression is proportional to the accuracy of information about their frequencies.

According to this procedure, at each next step, the set of moments of time obtained at the previous step is bifurcated (and doubled) by shifting it relative to itself by a time interval equal to the odd number of half-periods of the signal suppressed at this step.

Let us explain it mathematically. Let there be an initial set {t _i } of sampling time points of the discrete (suppressed) harmonic signal generated at the (m-1) th step. And let an additional set {t _i *} be formed by shifting the original by a time interval equal to n _m T _m / 2, where n _m is an odd number. We show that the sum of the discrete harmonic signal with a period of T _m obtained on the total set of time instants, consisting of the original and additional sets, is zero. To do this, we number in arbitrary order the moments of time t _{i of the} original set. Then the time instants of the additional set will be t _i * = t _i + n _m T _m / 2. But the shift by the time interval n _m Т _m / 2, if n _m is an odd number, is equivalent to the phase shift by π, and sin (φ + π) = - sinφ and therefore the sum of all discrete pairs of the suppressed signal with the same time t _i and t _i * will be equal to zero, as required.

Note that the equality of the shift to an odd number of half-periods, and not one, is necessary to exclude the possible coincidence of sampling time points for discrete shifts of the sets {t _i } and {t _i *}, for example, in the case of T ₃ = T ₂ + T ₁ and similar exceptions.

So, regardless of which signals were suppressed in the previous steps, due to this procedure of generating sets of sampling time, a discrete harmonic signal will be suppressed at each next step.

It remains to prove that the final set of sampling time instants discrete the total signal allows to suppress, with the exception of the measured one, all signals.

For this, we consider the set formed at the next step, in particular, at the next step with respect to the one described above. It also consists of two identical sets, the main (initial) and the additional, shifted in time by n _m Т _{m + 1/2} , and due to this it allows to suppress a signal with a period

T _{m + 1} . But from the above proof it followed that the first of these sets (the original one) allows us to suppress a signal with a period T _m . The shift of this set by an arbitrary time interval Δt does not affect its property of suppressing this signal, since sin (φ + 2πΔt / T _m + π) = - sin (φ + 2πΔt / T _m ). Thus, the additional set, like the original one, also has the property of suppressing a signal with a period

T _m , and hence the total set will have this property. From this we obtain that the total set will have the property of suppressing signals with periods of both T _m and T _{m + 1} .

Applying the method of mathematical induction now, we find that the first set of sampling time instants discrete the total signal, which includes two instants of time, can suppress one of the sum of jointly acting signals

with a period of T ₁ , the second - two with periods

T ₁ and T ₂ , m-e - m with periods T ₁ , T ₂ , ..., T _m , and the final - (M-1) signals, respectively, with periods T ₁ , T ₂ , ..., T _m , T _M , where m ≠ j, aj is the number of the measured signal, as required.

We illustrate this procedure graphically using FIG. 1, corresponding to the case M = 5, where the initial (initial) set, which consists of one sample discrete shown by the arrow, initiates the formation of the entire set of discrete and corresponds to an arbitrary initial time moment t _{0, is} shown on the top line. Thus, the countdown of time t is relative to t ₀ , i.e. t _i '= t _i + t ₀ . The second row shows the first set consisting of two sampling pulses of a discrete corresponding to the set of two time instants spaced apart by the interval τ ₁ = n ₁ T _1/2 , which means that it allows to suppress a signal with a period T ₁ .

The third row shows the second set, consisting of two subsets, namely the one shown on the second row and shifted relative to it by τ ₂ = n ₂ T _2/2 of the same set. According to the foregoing, this set contains four discrete units corresponding to four time instants, i.e. twice the number of moments than the previous one, and allows you to suppress signals with periods T ₁ and

T ₂ . We confirm this by numerically substituting the occurring values of time instants into the mathematical expressions of the signals S ₁ (t) = A ₁ sin (2πt / T ₁ + φ ₀₁ ) and S ₂ (t) = A ₂ sin (2πt / T ₂ + φ ₀₂ ) and summing the resulting discrete, taking to simplify the calculations n _i = 1

The fourth row shows the third set, consisting of two sets shown on the third row, shifted relative to each other by τ ₃ = n ₃ T _3/2 , and allowing to suppress signals with periods T ₁ , T ₂ and T ₃ . This can be easily verified by substituting the instants of time t _i in the expressions for the signals S ₁ (t), S ₂ (t) and S ₃ (t) in the same way as it was done above for the third line.

Finally, the fifth row shows the final set of 2 ⁴ instants of time and the corresponding sampling pulses of the discrete, consisting of two sets depicted on the fourth row, shifted relative to each other by

τ ₄ = n ₄ T _4/2 , and allowing to suppress four signals with periods T ₁ , T ₂ , T ₃ and T ₄ . The pairs of sets formed at the next steps are marked by dashed lines above the arrows.

Having proved that the obtained sets of sample sampling times allow us to exclude the influence on the measurement accuracy of the signal S _j (f) of the signals acting with it, we note that a pair of sums

будет составлять пару его проекций (p^' _j и p^'' _j) на оси ортогональной системы координат, так как в силу периодичности синусоиды значения входящих в данные суммы дискрет приводятся к одному периоду S_j(t), a Δt_j=Т_j/4 означает фазовый сдвиг сигналов на 90°. Найдем теперь формулу для коэффициента K_j.at

will be a pair of its projections (p ^' _j and p ^'' _j ) on the axis of the orthogonal coordinate system, since due to the periodicity of the sine wave, the values of the discrete sums in these sums are reduced to one period S _j (t), a Δt _j = T _j / 4 means phase shift of the signals by 90 °. We now find the formula for the coefficient K _j .

To find the mathematical expression for K _j , we use the vector model of harmonic signals, and to simplify the calculations, we transform the signal representation form S _j (t) = A _j sin (2πt / T _j + φ _0j ) to a form convenient for analysis taking into account the invariance of K _j with respect to the amplitude A _j and the phase shift φ _{0j of the} signal S _j (t), setting A _j = 1 and φ _0j = 0

S _j (t) ⇒s (t) = sin2πt / T _j = sinω _j t,

and since the procedure for generating {t _i } _{j is the} same for all signals, we omit the index and write {t _i } _j in the form {t _i },

Where

and also put

K _j = K, T _j = T and ω _j = ω.

, где

- дробная часть частного от деления t_i на T=2π/ω, т.е. моменты времени выборки дискрет сигнала S_j(t) приводятся к одному периоду синусоиды S_j(t). В результате множество дискрет {s_i=s(t_i)} преобразуется в множество дискрет

, приведенных к одному периоду синусоиды s(t), а множество фазовых углов {φ(t_i)=ωt_i}, соответствующих дискретам S_j(t_i), преобразуется в множество

, приведенных к четырем квадрантам декартовой системы координат. При этом каждому из фазовых углов

соответствует дискрета

и наоборот.In addition, we take into account that, as noted above, due to the periodicity of the sinusoid, the set {t _i } is transformed into

where

is the fractional part of the quotient of dividing t _i by T = 2π / ω, i.e. the sampling time points of the discrete signal S _j (t) are reduced to a single period of the sine wave S _j (t). As a result, the discrete set {s _i = s (t _i )} is transformed into the discrete set

reduced to a single period of the sine wave s (t), and the set of phase angles {φ (t _i ) = ωt _i } corresponding to the discrete S _j (t _i ) is converted to the set

reduced to the four quadrants of the Cartesian coordinate system. Moreover, each of the phase angles

corresponds to discrete

and vice versa.

в полярной системе координат, а дискрета

- как проекция единичного вектора

на мнимую ось комплексной плоскости

.We now use the Euler formula, which in this case has the form: e ^jωt = cosωt + jsinωt, according to which the complex number acts as a unit vector

in the polar coordinate system, and discrete

- as a projection of a unit vector

on the imaginary axis of the complex plane

.

, где ω=2π/T_j, с действительными и мнимыми составляющими (проекциями на оси координат) s*_ix, s*_iy и их сумма

, полученная по правилам векторной алгебры. Поворот

на любой угол

не приводит к изменению его модуляThe constructions associated with the application of this formula are shown in FIG. 2, which shows the set of unit vectors

, where ω = 2π / T _j , with real and imaginary components (projections on the coordinate axis) s * _ix , s * _iy and their sum

obtained by the rules of vector algebra. Turn

to any angle

does not lead to a change in its module

where N = 2 ^M-1 , i.e. (unlike projections

S _Σx and S _Σy ) it is invariant with respect to

получено для случая единичного вектора, т.е. для синусоиды с единичной амплитудой (A_j=1), то выражениеFurther, since the mathematical expression for

obtained for the case of a unit vector, i.e. for a sinusoid with unit amplitude (A _j = 1), then the expression

where N = 2 ^M-1 acts as a normalizing factor.

Given the above simplifications of the notation of quantities, namely, ω _j = ω, we obtain

Thus, all relations between the parameters of the measured signal and the sums of the discrete are determined.

Claims (1)

A method for measuring the vector of a harmonic signal acting together with M-1 by other harmonic signals and having, like them, a known period T _j and unknown amplitude A _j , and an initial phase shift φ _0j at which projections

and

measured signal
S _j (t) = A _j sin (2πt / T _j + φ _0j ) into two orthogonal vectors of reference signals associated with A _j and φ _0j , for example, by the relations

and

measured by sampling and summing the discrete discrete signal

using instant pulses acting at time instants, the sets {t _i ^' } and {t _j ^'' = t _i ^' + T _j / 4} _j , and {t _j ^' } _j , are formed so that

and

where K _j is the coefficient depending on the formation procedure {t _i ^' }, and N is the number t _i ^' , characterized in that any of the jointly acting signals, the repetition rate of which 1 / T _{j is} not a multiple of the frequencies of other signals, is selected as the measured , and the set {t _i ^' } of time points of action of the instantaneous pulses of the sample, the discrete summary signal σ (t) is formed using a step-by-step procedure from the initial set consisting of one arbitrary initial moment t ₀ , by obtaining at the first step an additional set consisting also of e from one moment, by shifting the initial set by an odd number of half-periods of the first signal, then obtaining in the second step an additional set, consisting of already two moments, by shifting two moments of the total set obtained in the first step, by an odd number of half-periods of the second signal and continue this procedure with the formation at each mth step of an additional set by shifting the total set formed in the previous step by an odd number n _{m of} half-periods of the next of the suppressed signal and with doubling the number of sampling times, the signal discrete σ ( _t ) until the number of steps becomes equal to M-1, and the amplitude A _j and the phase shift φ _{0j of the} signal S _j (t) are determined by the relations:

at

where t _i ^' = t _i + t ₀ , t ₁ = 0, t ₂ = t ₁ + n ₁ T _1/2 ,
t ₃ = t ₁ + n ₂ T _2/2, t ₄ = t ₂ + n ₂ T _2/2, t ₅ = t ₁ + n _{3 3} T / 2,
t ₆ = t ₂ + n _{3 3} T / 2, t ₇ = t ₃ T ₃ + n _3/2, t = t ₈ + n _{4 3 3} T / 2, ......,

m ≠ j.

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