RU2090895C1 - Method measuring root-mean-square value of signal - Google Patents

Abstract

FIELD: measurement technology, measurement of root-mean-square value of signal on infralow frequencies when fast action and measurement accuracy during processing of signals changing in large dynamic range is required. SUBSTANCE: method is based on isolation of first harmonic and higher harmonics from spectrum of input signal, on determination of their root-mean-square values. In this case frequency of first harmonic of input signal is determined, reference sinusoidal signal with frequency of first harmonic of input signal is formed, first time interval is selected in which input signal does not change its sign, center of first time interval is found, second time interval is selected in which reference sinusoidal signal does not change its sign, center of second time interval is determined, momentary values of input and reference signals are measured correspondingly from centers of first and second time intervals. Modulus of ratio of momentary values of input and reference signals is found, moduli of relations of momentary values of input and reference signals are determined many times for each pair of time moments equally distant from centers of first and second time intervals correspondingly. Averaged value of moduli of relations of measured momentary values of input and reference signal is found. Root-mean-square value of input signal is determined as square root from sum of squares of two values - product of root-mean-square value of reference sinusoidal signal by averaged value of moduli of relations of measured momentary values of input and reference signals and root-mean-square value of product of momentary values of reference sinusoidal signal by square root from difference of squares of calculated values of moduli of relations of momentary values of input and reference signals and averaged value of moduli of measured momentary values of input and reference signals. EFFECT: fast action and accuracy of measurements when processing signals changing in large dynamic range. 2 dwg

Description

 The invention relates to measuring equipment and is intended to measure the root mean square value of the signal for predominant use at infra-low frequencies, when measurement accuracy is required when processing signals that vary in a large dynamic range.

 A known method of measuring the rms (effective) value of an alternating current voltage of arbitrary shape [1] by linear half-wave rectification and measuring the effective value of the variable component of the rectified voltage, which is summed with its average rectified value.

 This method has inherent disadvantages of large errors when processing signals of various shapes, especially at infralow frequencies.

 There is another method of measuring the effective AC voltage [2] based on measuring the rectified voltage, followed by squaring all the rectified values and extracting the square root of the sum of the obtained squares, which is used to judge the value of the effective voltage value.

 The method has inherent disadvantages associated with insufficient conversion accuracy, especially at infrared frequencies, and the need to calculate the square root of the sum of squares of many quantities, which increases the instrumental error.

 There is another method for measuring the effective value of an arbitrary voltage AC voltage [3] according to which the voltage is rectified many times, the voltage proportional to the reference voltage is subtracted from the rectified measured voltage and rectified again, the proportionality coefficients being set equal to the members of the geometric progression by summing n rectified voltages with their weight coefficients, the values of which also correspond to a geometric progression, but the constant composition is distinguished from the sum which is used as a reference voltage and whose value is used to judge the actual value of the measured voltage.

 The disadvantages of the method are the difficulty of maintaining high accuracy in measuring voltage at infra-low frequencies, provided that the values of these voltages can vary in a large dynamic range, since it is necessary to average the voltages, which also vary in a large dynamic range.

 There is another method of measuring the rms value of the alternating voltage [4] based on the squaring of the instantaneous values of the alternating voltage, averaging the converted signal and extracting the square root from the obtained voltage, which is used to judge the rms value of the signal.

 The disadvantages are related to the need to average with high accuracy the signals that vary in a large dynamic range, which is difficult in the infra-low-frequency region.

 There is another method of measuring the rms value of the alternating voltage [5] according to which the constant components are sequentially extracted from the rectified values of the measured voltages for their conversion, additional harmonic oscillations are generated, shifted in phase by P / 2, alternately model the selected components, summarize these modulated pairs signals, as a result, receive a total oscillation, the amplitude of which corresponds to the square root of the sum of the squares of the selected state vylyayuschih.

 The disadvantages of the method are associated with the need to isolate with great accuracy the constant components of the signals that vary in a large dynamic range, which is difficult in the infra-low frequency region.

 The closest to the invention method for similar technical features used, taken as a prototype, is a method of measuring the rms value of the signal [6] based on the selection of the first harmonic and higher-order harmonics from the signal spectrum using narrow-band filtering, converting the selected harmonics into constant values proportional to RMS values, and the subsequent calculation of the square root of the sum of the squares of the values obtained.

 The method has a low accuracy, since the extraction of the first harmonic of the signal at infralow frequencies, when the signal changes in a large dynamic range, requires an increase in integration time, which reduces speed.

 The aim of the invention is to increase the accuracy of measurements with increasing speed.

The goal in the method of measuring the rms value of the signal is achieved by extracting the first harmonic and higher-order harmonics from the input signal spectrum, determining their rms values, determining the frequency of the first harmonic of the input signal, generating a reference sinusoidal signal with the frequency of the first harmonic of the input signal, selecting the first time the interval in which the input signal does not change its sign, determine the middle of the first time interval, choose the second time interval in which the sinusoidal signal does not change sign, determine the middle of the second time interval, measure the instantaneous values of the input signal at time t ₁ , and the reference signal at time t ₂ , and the time t ₁ and t ₂ are chosen equal to the midpoints of the first and second time intervals, determine the modulus of the relationship of the instantaneous values of the input and reference signals, repeatedly determine the modules of the relationship of the instantaneous values of the input and the reference signals for each pair of time instants, equally from the midpoints of the first and second time intervals, respectively, determine the average value of the ratio moduli of the measured instantaneous values of the input and reference signals, and the root-mean-square value of the input signal is determined as the square root of the sum of the squares of two quantities - the product of the root-mean-square of the reference sinusoidal signal and the average value of the ratio modules measured instantaneous values of the input and reference signals and the root mean square value of the product of instantaneous s values of the reference sine wave to square root of the difference of the squares of the calculated values of the moduli relationships instantaneous values of the input and reference signals and the average value of the measured moduli relationships instantaneous values of the input and reference signals.

где U_оск среднеквадратическое значение опорного синусоидального сигнала;
Т время интегрирования;
A_osin ωt₂ мгновенные значения опорного синусоидального сигнала с частотой первой гармоники ω и амплитудой А_o в моменты времени t₂;
K(ωt) вычисленные значения модулей отношений мгновенных значений входного и опорного сигналов;
К_с усредненное значение модулей отношений измеренных мгновенных значений входного и опорного сигналов, для определения которого формируют опорный синусоидальный сигнал с частотой первой гармоники входного сигнала, многократно определяют модули отношений мгновенных значений входного и опорного сигналов для каждой пары моментов времени, равноотстоящих от середин выбранных временных интервалов, внутри которых соответственно входной и опорный сигналы не изменяют свои знаки.The essence of the method lies in the fact that the rms value U _{ck of the} input signal is determined using a mathematical expression

where U _{Osk is the} rms value of the reference sinusoidal signal;
T integration time;
A _o sin ωt ₂ instantaneous values of the reference sinusoidal signal with a frequency of the first harmonic ω and amplitude A _o at time t ₂ ;
K (ωt) calculated values of the moduli of relations of instantaneous values of the input and reference signals;
K _{c, the} average value of the moduli of the ratios of the measured instantaneous values of the input and reference signals, for the determination of which a reference sinusoidal signal is generated with the frequency of the first harmonic of the input signal, the moduli of the ratios of the instantaneous values of the input and reference signals for each pair of time instants equally spaced from the middle of the selected time intervals are repeatedly determined , inside which, respectively, the input and reference signals do not change their signs.

The input quasi-sinusoidal signal U _x (t) and the reference sinusoidal signal U _y (t) with a frequency ω of the first harmonic of the input signal can be represented as functions on selected time intervals bj that do not contain signals equal to zero:
U _x (t) U _x (bj), (1)
U _y (t) U _y (bj), (2)
where t is time;
U _x (bj), U _y (bj) corresponding signals at selected time intervals bj.

где A_x, A_y значения амплитуд аппроксимирующих сигналов;
ω круговая частота первой гармоники входного сигнала;
F_x, F_y начальные фазы входного и опорного сигналов.The signals U _x (bj) and U _y (bj) at the selected time intervals bj will be approximated in the form of fragments of sinusoids for which

where A _x , A _y the amplitudes of the approximating signals;
ω is the circular frequency of the first harmonic of the input signal;
F _x , F _{y are the} initial phases of the input and reference signals.

We divide the instantaneous values of the input and reference signals to each other and consider the partial function f (bj):
f (bj) = K _a [sin (ωt + F _x )] / sin (ωt + F _y ), (5)
where f (bj) is a function on the time interval bj, determined by the ratio of the instantaneous values of the signals U _x (bj) and U _y (bj);
K _a (A _x / A _y ) is the ratio of the amplitudes of the signals from (3) and (4).

Разность фаз F₀ между сигналами U_x(bj) и U_y(bj) равна
F₀ F_x F_y. (8)
Если, к примеру, F_x > F_y, то можно принять F_y 0, и после преобразования из выражения (7) получим
L=cosF_o+(sinF_o)/(tgωt) (9)
Если F_x < F_y, то можно принять F_x 0, и после преобразования из выражения (7) получим
L=1/[cosF_o+(sinF_octgωt)] (10)
Выполнение условия (6) сводится к выполнению условия
cosF_o+sinF_o/[tg(2π/T)t_o]=1, (11)
где t_o соответствует искомому моменту времени;
Т период первой гармоники входного сигнала.For time t _o on the interval bj, when the value of the function f (bj) f (t _o ) K _a , the condition
[sin (ωt _o + F _x )] / [sin (ωt _o + F _y )] = 1 (6)
Denote the left side of equation (6) by L, then

The phase difference F ₀ between the signals U _x (bj) and U _y (bj) is
F ₀ F _x F _y . (eight)
If, for example, F _x > F _y , then we can take F _y 0, and after conversion from expression (7) we obtain
L = cosF _o + (sinF _o ) / (tgωt) (9)
If F _x <F _y , then we can take F _x 0, and after the conversion from expression (7) we obtain
L = 1 / [cosF _o + (sinF _o ctgωt)] (10)
The fulfillment of condition (6) is reduced to the fulfillment of the condition
cosF _o + sinF _o / [tg (2π / T) t _o ] = 1, (11)
where t _o corresponds to the desired point in time;
T is the period of the first harmonic of the input signal.

Let (2π / T) t _o = β be the value of the angle determined by the position t _o on the time interval of period T. Then, after permutations, we rewrite expression (11) in the following form:
tgβ = sinF _o / (1-cosF _o ). (12)
After the transformations we get
sinF _o / (1-cosF _o ) = ctg (F _o / 2). (thirteen)
From (12) and (13) it follows
tgβ = ctg (F _o / 2). (14)
From expression (14) we obtain
tgβ = tg [(π / 2) - (F _o / 2)] = tg [π-F _o ) / 2] (15)
From equality (15) we obtain the expression for β
b = (π-F _o ) / 2. (16)
Since β = (2π / T) t _o corresponds to the time moment when condition (6) is fulfilled, from (16) we determine the position of the point t _o on the interval bj. The angle π corresponds to a half-period, that is, to such a time interval when the value of the input or reference signal does not change its sign. The position of the point t _o on the interval bj corresponds to the middle of the time interval lying inside one of the half-periods, from which the time interval corresponding to the phase shift between the signals is excluded. This time t _o is at the same distance from the midpoints of the selected time intervals, where the sign of the input or reference signal does not change, see FIG. one.

If the signals are sinusoidal, then at the time t _o in the middle of the interval equal to the phase shift F _o , the signals will be equal to A _x sin (F _o / 2) and A _y sin (F _o / 2), respectively, and the ratio of the instantaneous values of the signals will be defined as K _a A _x / A _y .

The instantaneous value of the input signal can be measured at time t ₁ , which is far from the middle of the selected first time interval of this signal, for example, by the time interval corresponding to the phase shift F _o or [((π / 2)) F _o ] and the reference measurement signal at time t ₂ , which is spaced from the middle of the selected second time interval of its signal by the time interval, respectively, F _o or [((π / 2)) F _o ] For time t _{1, the} instantaneous value of the input signal can be represented as A _x sinF ₀ A _y cos [((π / 2)) F ₀ ] and the instantaneous value the reference signal at time t ₂ can be represented as A _y sinF ₀ A _y cos [((π / 2)) F ₀ ] The modulus of the ratio of the instantaneous values of the input and reference sinusoidal signals will be K _a A _x / A _y .

будет равно A_xcosF₀, другого сигнала в момент времени

будет равно A_ycosF₀. Модуль их отношений будет равен K_a A_x/A_y.Instantaneous value of one signal at a time

will be equal to A _x cosF ₀ , another signal at time

will be equal to A _y cosF ₀ . The modulus of their relationship will be K _a A _x / A _y .

где K(ωt) вычисленные значения модулей отношений мгновенных значений входного и опорного сигналов;
U_x(t₁) мгновенное значение входного сигнала в момент времени t₁;
U_y(t₂) мгновенное значение опорного сигнала в момент времени t₂.The moments of time t ₁ , t ₂ relate respectively to the time moments of measuring the instantaneous values of the input and reference signals. The relationship modules of the instantaneous values of the input U _x (t) and the reference U _y (t) signals at certain points in time can be represented as follows:

where K (ωt) are the calculated values of the moduli of the relations of the instantaneous values of the input and reference signals;
U _x (t ₁ ) the instantaneous value of the input signal at time t ₁ ;
U _y (t ₂ ) the instantaneous value of the reference signal at time t ₂ .

The time moments t ₁ and t ₂ , when measurements are taken, are equally spaced from the midpoints of the selected first and second time intervals, in which the input and reference signals, respectively, do not change their sign.

Each pair of U _x (t ₁ ) and U _y (t ₂ ) corresponds to its own value of the modulus of the ratio of the instantaneous values of the input and reference signals.

 The midpoints of the selected first and second time intervals, in which respectively the input and reference signals do not change their sign, can be considered as the midpoints of the selected half-waves.

Therefore, if we measure the ratios of the instantaneous values of the sinusoidal waveforms at different points in time that are equally spaced from the midpoints of the selected half-wave time intervals of their signals, then the modules K (ωt) of the ratios of the instantaneous values of the input and reference signals will be K (ωt) K _a A _x / A _y

 If there are distortions in the input signal due to the presence of harmonic components of the input signal, deviations will be observed in the obtained values of the moduli of the relations of the instantaneous values of the input and reference signals.

и так далее, A_xsinF_o/A_ysinF_o K_a для измерений в моменты времени t₁ и t₂; A_xcosF_o/A_ycosF_o для измерений в моменты времени

и так далее.In FIG. Figure 1 shows an example of the definitions of time instants t ₀ , t ₁ , t ₂ equally spaced from the midpoints of the selected half-wave time intervals of the input and reference signals for an arbitrary phase shift between the signals. For the phase shift F _o receive several pairs of time points t ₁ , t ₂ ;

and so on, A _x sinF _o / A _y sinF _o K _a for measurements at time t ₁ and t ₂ ; A _x cosF _o / A _y cosF _o for measurements at time instants

and so on.

Measurements at time t _o can be considered as a special case of the selection of the corresponding pair, when t ₁ t ₂ .

где U(t) напряжение входного сигнала.The root mean square value U _{ck of the} input signal U (t) is determined from the known relation

where U (t) is the input voltage.

We write the input voltage signal U (t) in the form
U (t) = βsinωt + g (t) (19)
where β is the amplitude of the sinusoidal voltage of the first harmonic of the input signal;
w circular frequency of the first harmonic of the input signal;
g (t) is a function whose values change in time so that equality (19) holds.

Suppose that the reference sinusoidal signal U (y) U '(t) has a frequency w of the first harmonic, then
U ′ (t) = A _o sin (ωt + F _o ) (20)
where A _{o the} amplitude of the reference sinusoidal signal;
F _o phase shifts between the input and reference signals.

Усредним вычисленные значения модулей отношений измеренных мгновенных значений входного и опорного сигналов и получим их усредненное значение K_с.For each of the phase shifts F _o we will determine the modules K (ωt) of the ratios of the instantaneous values of the input U (t ₁ ) and the reference U '(t ₂ ) signals. Then the input signal U (t) can be approximated as follows:
U (t) = [A _o sin (ωt ₂ )] K (ωt) (21)
where A _about sin (ωt ₂ ) instantaneous values of the reference sinusoidal signal at time t ₂ ;
K (ωt) calculated values of the moduli of relations of instantaneous values of different pairs of input U (t ₁ ) and reference U '(t ₂ ) signals for the corresponding time instants t ₁ and t ₂ equally spaced from the midpoints of the first and second selected time intervals. You can write

We average the calculated values of the moduli of the ratios of the measured instantaneous values of the input and reference signals and obtain their average value K _s .

If the input signal is a sinusoid, that is, g (t) 0 in (19), then with different phase shifts, averaging the values of the moduli K (ωt) of the ratios of the instantaneous values of the input and reference signals, they get the same value of K _c , which will be numerically equal K _c B / A _o , and then the input signal U (t) will be approximated in the following form:
U (t) = (A _o K _c ) sin (ωt ₂ ) + g (t), (23)
where K _{c is the} average value of the moduli of the ratios of the measured instantaneous values of the input and reference signals.

Подставляя (25) в (19), получим для входного сигнала
U(t)=A_oK_csin(ωt₂)+A_osin(ωt₂)[K(ωt)-K_c (26)
Из (26) можно определить значение квадрата входного сигнала:

Запишем выражение квадрата среднеквадратического значения входного сигнала в следующем виде:

После преобразования выражение (29) будет иметь вид

Из (30) среднеквадратическое значение входного сигнала можно представить в следующем виде:

Первое слагаемое правой части выражения (31) представляет собой квадрат произведения среднеквадратического значения опорного синусоидального сигнала на усредненное значение K_c модулей отношения мгновенных значений входного и опорного сигналов, что соответствует составляющей первой гармоники входного сигнала, поэтому (31) можно записать в следующем виде:

где U_оск среднеквадратическое значение опорного синусоидального напряжения;
K(ωt) вычисленные значения модулей отношения мгновенных значений входного и опорного сигналов;
K_c усредненное значение модулей отношения измеренных мгновенных значений входного и опорного сигналов.From (21) and (23) we define the expression for the function g (t):

Substituting (25) into (19), we obtain for the input signal
U (t) = A _o K _c sin (ωt ₂ ) + A _o sin (ωt ₂ ) [K (ωt) -K _c (26)
From (26) we can determine the value of the square of the input signal:

We write the expression of the square of the mean square value of the input signal in the following form:

After the transformation, expression (29) will have the form

From (30), the rms value of the input signal can be represented as follows:

The first term on the right-hand side of expression (31) is the square of the product of the rms value of the reference sinusoidal signal by the average value K _c of the ratio of the instantaneous values of the input and reference signals, which corresponds to the component of the first harmonic of the input signal, therefore (31) can be written in the following form:

where U _{Osc is the} rms value of the reference sinusoidal voltage;
K (ωt) calculated values of the moduli of the ratio of the instantaneous values of the input and reference signals;
K _{c the} average value of the modules of the ratio of the measured instantaneous values of the input and reference signals.

Первое слагаемое под общим корнем в правой части выражения (33) представляет собой квадрат среднеквадратического значения составляющей первой гармоники входного сигнала, а второе слагаемое равно квадрату среднеквадратического значения гармонических составляющих входного сигнала.The difference of the squares of the two quantities K (ωt) and K _c in expression (32) can be represented as the square of the square root of the difference of the squares of the same quantities, therefore (32) can be represented in the following form:

The first term under the common root on the right side of expression (33) is the square of the rms value of the component of the first harmonic of the input signal, and the second term is equal to the square of the rms value of the harmonic components of the input signal.

 Thus, the rms value of the input signal is equal to the square root of the sum of the squares of two values of the product of the rms value of the reference sinusoidal signal and the average value of the moduli of the ratios of the measured instantaneous values of the input and reference signals and the rms value of the product of the instantaneous values of the reference sinusoidal signal and the square root of the difference of the squares of the calculated values modules of relations of instantaneous values of the input and reference signals and srednennogo moduli relationships measured instantaneous values of the input and reference signals.

 It should be noted that in the proposed method, it is not necessary to separately isolate the first harmonic of the input signal or to isolate the constant component from the input signal, which varies in a large dynamic range, which increases the accuracy of measurements. At the same time, the integration time during processing of selected only higher harmonic components can be reduced, which increases the speed of the method during its implementation.

In FIG. 2 shows a structural diagram of a device that implements the method. The input signal U _x (t) is input to the shaper 1, the output of which is formed by the voltage U ₁ proportional to the period T of the first harmonic of the input signal. This voltage U _{1 is} supplied to the input of the controlled reference generator 2, the output of which generates sinusoidal oscillations, the period of which depends on the controlled voltage U ₁ and is equal to the period T of the first harmonic of the input signal.

The sinusoidal voltage U _{2 with the} amplitude A _y from the output of the reference generator 2 is fed to the input of the phase shifter 3, the output of which receives a sinusoidal voltage U _y (t) of the same amplitude, which does not change with changes in phase shifts. Thus, the input voltage signal U _x (t) and the reference sinusoidal voltage signal U _y (t) having a phase shift between themselves, for example, as shown in Fig. 2, are supplied to the two inputs of the two-beam oscilloscope 4. one.

For each phase shift F _o measure the instantaneous values of the signals at a selected pair of time instants t ₁ and t ₂ , calculate the values of the absolute value of the ratio of the instantaneous values of the input and reference signals, average the calculated values of the absolute value of the ratio of the instantaneous values of the input and reference signals, determine the square root of the difference of the squares of two values of the calculated values of the modules K (ωt) of the ratios of the instantaneous values of the input and reference signals when changing the phase shift and the average value of the modules from wearing the measured instantaneous values of the input and reference signals, after which the rms value of the product of the instantaneous values of the reference sinusoidal signal at times t ₂ is determined by the obtained root square root of the difference of the squares of the two quantities K (ωt) and K _c , and the rms value of the input signal is obtained after calculating the square root of the sum of the squares of two quantities according to the claims in accordance with expression (33).

To increase the resolution, it is necessary to increase the number of selected pairs of time instants t ₁ , t ₂ for measurements.

 When using a precision reference generator and a small step when changing the phase shift, the method has a high resolution, while the method does not require the use of filters to process the first harmonic of the input signal, which increases the accuracy of measurement at infra-low frequencies.

 Used sources of information.

 1. USSR author's certificate N 439763, cl. G 01R 19/02, 1973.

 2. Auth. testimonial. USSR N 495612, class G 01R 19/02, 1974.

 3. Auth. testimonial. USSR N 789788, class G 01R 19/02, 1977.

 4. Volgin L. I. Measuring transformations of an alternating voltage into a constant. M. Sov. radio, 1977, p. 139.

 5. Auth. testimonial. USSR N 920538, class G 01R 19/02, 1980.

 6. Ed. Sheingolda, Handbook of nonlinear circuits. M. Mir, 1977, p. 114-121.

Claims (1)

A method for measuring the rms value of the signal, which consists in extracting the first harmonic and higher-order harmonics from the spectrum of the input signal, determining their rms values, characterized in that they determine the frequency of the first harmonic of the input signal, form a reference sinusoidal signal with the frequency of the first harmonic of the input signal, choose the first time interval in which the input signal does not change its sign, determine the middle of the first time interval, select the second time interval a shaft, wherein the reference sine wave does not change its sign, determining the middle of the second time interval, the measured instantaneous value of the input signal at time t _1, and the reference signal at time t _2, the times t ₁ and t ₂ are selected equidistant, respectively, from the midpoints of the first and second time intervals, determine the modulus of the ratio of the instantaneous values of the input and reference signals, repeatedly determine the modules of the ratio of the instantaneous values of the input and reference signals for each pair of moments of time equidistant from the midpoints of the first and second time intervals, determine the average value of the modulus of the ratio of the measured instantaneous values of the input and reference signals, and the root mean square value of the input signal is determined as the square root of the sum of the squares of two quantities - the product of the mean square value of the reference sinusoidal signal by the average value of the modules the ratio of the measured instantaneous values of the input and reference signals and the rms value of pr the instantaneous values of the reference sinusoidal signal are reduced to the square root of the difference of the squares of the calculated values of the moduli of relations of the instantaneous values of the input and reference signals and the average value of the moduli of the ratios of the measured instantaneous values of the input and reference signals.

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