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Nowadays the root-mean-square (RMS) is one of the most informative parameters of electrical power grid signals. By definition, the RMS measurement technique involves averaging the square of the input signal over time multiple of the input signal period. However, for digital measurement methods, the input signal is represented as a sequence of discrete samples obtained by applying analog-to-digital converter (ADC). The paper discusses ways of approximating of the signal samples by applying polynomial functions of zero, first and second order. Analytical expressions are obtained for the RMS measurement error for the case of applying of different order approximation polynomials. The influence of the input signal amplitude, signal frequency, initial phase, sampling frequency and total measurement time on the RMS measurement error for the case of applying these measurement approaches is considered. The influence of the input signal frequency deviation on the RMS measurement error has been especially thoroughly investigated. The methods of reducing the RMS measurement error for the case of sinusoidal and polyharmonic input signals are proposed. By application of Matlab and Simulink software packages, a simulation mathematical model for all considered approaches is performed.

Currently, instrumentation amplifiers are widely used to construct measurement channels of measurement and control equipment. This amplifiers are differential, which makes it possible to obtain an output signal proportional to the difference between the input voltages. The value of the voltage gain, depending on the design features, can be set by using an additional external resistance, external jumpers or by applying an external control digital code. Like any measuring transducer, an instrumentation amplifier is characterized by an additive, multiplicative, and nonlinear error component. Accounting for each of the listed error components in the total measurement error of the considered signal parameter is often a difficult task. For this reason, simulation modeling is widely applied. This paper discusses the principles of constructing a model of an instrumental amplifier, taking into account the imperfection of its transfer characteristic. The presence of additive and multiplicative components of the error, transfer function nonlinearity, magnitude response, finite input and output ranges are taken into account. An existing model obtained by using the Spice-model in Tina-TI program (developed by Texas Instruments company) is considered and a model was performed by using the Simulink software program. The resulting model makes it possible to simulate instrumental amplifiers of all known types, as well as decisive amplifiers, dividers and other analog and digital measurement converters.

Among the digital methods for frequency measurement in electrical power systems the method based on zero crossings of the input signal (zero crossing technique) is the most popular. This method is easy to implement and characterizes a high measurement accuracy. The main reason for the error is the influence of input noise. One way to improve measurement accuracy is application of a digital filter. This paper is devoted to the application of a sliding discrete Fourier transform (DFT) as an input bandpass filter. This approach allows to effectively suppress noise and non-fundamental spectral components of the signal, which increase the sensitivity of the measurement method to input noise. An approach to the implementation of the sliding DFT (based on the direct implementation of the real part of the DFT) is proposed, which significantly reduces its complexity in the problem being solved. A method is proposed to reduce the spectrum leakage effect, which makes it possible to increase the frequency measurement accuracy. Dependencies are obtained empirically, which make it possible to estimate the frequency measurement error for a given input noise level and parameters of applied filter (sliding DFT). By applying Simulink software package, a simulation model of the considered measurement method and represented modification has been developed. The proposed model makes it possible to estimate the static and dynamic errors of frequency measurement for the input signals of arbitrary form.

The possibility for calibrating Power Quality Analyzer by harmonics is presented by using a reference square waveform signal. Based on the calibration model, a control algorithm is synthesized of the calibration measurement procedures and the processing of the uncertainty of the calibration results. Results from the calibration of the Fluke 435 analyzer with the Metrix CX1651 calibrator on voltage and current harmonics are presented, which confirm the possibilities of using the reference square waveform signal.

Frequency measurement is one of the most important tasks in measurement technique today. The method based on quadrature demodulation is widely used to measure frequency of the electric power networks. The frequency measurement error by using this method contains two components: instrumental (associated with the imperfection of the used chips) and methodical (associated with the applied measurement method). The instrumental component associated with used chips imperfection (primarily the analog-to-digital converter – ADC) is discussed in this paper. The additive, multiplicative components error components nonlinearity and quantization error are considered. It is shown that the additive and multiplicative components practically do not affect to the frequency measurement error by the method under consideration. Particular attention is paid to ADC nonlinearity, since for most ADCs it prevails over quantization error and cannot be compensated for by approaches such as zero adjusting and calibration. To represent nonlinearity, the “worst case” method, approximation by polynomial functions, and representation of nonlinearity as random (stochastic) function are considered. Analytical expressions were obtained for estimating the frequency measurement error by the quadrature demodulation technique when it is represented by the discussed approaches. It is shown that the largest error occurs for nonlinearity, which form is close to pseudorandom. A simulation model of frequency measurement by this method was designed by Matlab and Simulink software packages, taking into account the considered ADC imperfections.

For calibrating the Power Quality Analyzer by Total Harmonic Distortion of voltage and current, a calibration approach is proposed, by using a reference square waveform signal. A mathematical model for Total Harmonic Distortion calibration is presented and an algorithm is implemented for control the calibration process, collection and processing of results. The experimental results obtained from the performed Total Harmonic Distortion calibrations, confirm the applicability of the proposed approach.

The most popular method for the frequency measurement of electrical power grid signals is the zero crossing technique. The main reason for the measurement error of this method is the presence of noise. In the presence of harmonics and flicker, the sensitivity of the method to flicker may increase. One method to reduce the effects of noise and harmonics is the input signal pre-filtration by a low-pass filter. The paper discusses the application of moving average filter (MAF) to solve this problem. A variant of constructing a band-pass filter (BPF) based on a recursive MAF is proposed. Approaches to adaptive adjustment of the order of recursive and non-recursive MAF filter to the current (measured) value of the input signal frequency are considered. It is shown that the synchronization of the moment when the filter order changes and the moment when the output signal crosses zero allows to reduce the duration of the transient process to one sample time. The influence of the number of MAF stages on the frequency response of the resulting BPF and settling time is analyzed. By application of the Simulink software package, a BPF simulation model was performed on the basis of a recursive MAF.

Active power is one of the most important parameters of electrical power grids. To measure active power the digital measurement methods are currently applied, which are characterizes high accuracy and suitable for the measurement of sinusoidal and polyharmonic signals. When applying digital measurement methods, analog to digital converters are used to obtain voltage and current samples, the conversion function of which is nonlinear. This introduces a measurement error that cannot be eliminated by zero-adjustment and calibration technique. The nonlinearity shape depends on the analog to digital converter (ADC) architecture. Sigma-delta ADCs are characterized by a relatively smooth nonlinearity shape, while successive-approximation register (SAR) ADCs characterizes a nonlinearity close to a stochastic function. In this paper, to approximate nonlinearity and estimate the measurement error of active power, the application of harmonic function is considered. By changing the number of nonlinear oscillations, this approach can be applied to approximate the nonlinearity of sigma-delta, SAR and pipeline ADC architectures. Analytical expressions are obtained that make it possible to estimate the active power measurement error by nonlinearity parameters, input signal parameters and ADC architecture. The analysis of the influence of integral nonlinearity, the number of oscillations and the input signals amplitude value on the active power error is carried out. The influence of the sampling frequency on the active power measurement error is analyzed in detail. An algorithm is proposed for choosing the number of nonlinearity oscillations which is based on analysis of the typical shape of ADC nonlinearity. The reliability of analytical expressions is confirmed by the results of simulation modeling performed by Matlab.

The absolute majority of power quality parameters are measured indirectly through the signal complex spectrum. For this reason, amplitude and phase spectrum accurate measurement is of the utmost importance. Discrete Fourier transform is most commonly used to measure the complex spectrum. The complex spectrum measurement error is associated with so-called “spectrum leakage” effect. The paper proposes a method to compensate of the “spectrum leakage” effect, based on additional frequency measurement and calculation of tuning coefficients for complex spectrum. The features of the application of the proposed method for the case of sinusoidal and polyharmonic signals are considered. Modifications are proposed for the case of applying the fast Fourier transform algorithm (FFT) and Goertzel algorithm. Analytical relations are obtained, which make it possible to estimate the amplitude and phase spectrum measurement error. By applying Matlab, simulation modeling has been performed, which makes it possible to evaluate the effectiveness of proposed technique.

For the modern measurement transducers of root mean square (RMS), digital measurement methods are usually implemented. This paper discusses a digital RMS measurement method based on application of a low-pass digital filter. The high measurement speed and the possibility of application for sinusoidal and non-sinusoidal periodic input signals are main advantages of this technique. The finite implementation of all RMS digital measurement techniques involves the application of an analog to digital converter (ADC). Real ADCs have a large number of error sources including offset (additive component), conversion coefficient deviation (multiplicative component), nonlinearity, and quantization error. It is shown that for the considerate measurement method the offset error approximately does not affect to the RMS error. It was researched that the multiplicative component is proportional to the RMS error for the considerate measurement method. The nonlinearity and quantization error of applied ADC influence on the RMS error is discussed. This influence can be calculated by signal spectral components which are not equal to zero. For the nonlinearity influence calculation, the analytical expressions are obtained. The case of nonlinearity representation by second and third order polynomials, the “worst case” method and stochastic function is considerate. The model of RMS digital measurement transducer is designed by Simulink software.

Currently, there are a large number of frequency measurement methods. Most of it can only be applied to a sinusoidal input signal. Other methods are limited in practical application due to the significant measurement time. The measurement method based on the application of quadrature demodulation technique is devoid of these drawbacks. This method is promising for frequency measurement tasks in modern power grids. The basic principles of frequency measurement method based on a quadrature demodulator are described. By using the simulation modeling which is developed in Simulink software, the frequency measurement error of this measurement method was investigated. The application of Butterworth and Chebyshev infinite impulse response (IIR) digital filters is considered. An analytical dependence is obtained that relates the frequency measurement error and filter parameters. Approaches based on the application of post-filtration and averaging are proposed, which allows to significantly reduce the frequency measurement error by the considered method. The relationship between the parameters of the post-filter and the final error of frequency measurement is investigated. A method for choosing the post-filter parameters is proposed. The reliability of the obtained analytical expressions is confirmed by the results of simulation modeling which is performed in Simulink software package.

Currently, the quadrature demodulation technique is applied to measure the amplitude, phase spectra and frequency of periodic nonsinusoidal signals. Compared to the discrete Fourier transform, this approach allows for a significantly smaller measurement error. The input signal of the quadrature demodulator comes from the analog-to-digital converter (ADC) output. The conversion function of a real ADC is not ideal: there is an offset, multiplicative and nonlinear components of the error, quantization error. All of these features affect the measurement errors of the amplitude spectrum obtained by applying the quadrature demodulator. This article discusses the influence of each of the listed error components on the measurement error of the amplitude spectrum. The ADC nonlinearity is represented as a polynomial function of the third order, in the form of a random function and by the “worst case” method. Analytical expressions have been obtained that make possible to estimate the amplitude measurement error for the considered forms of ADC nonlinearity representation. By using software package Simulink, simulation models of a quadrature demodulator and ADC are performed. A quadrature demodulator with a third-order Butterworth IIR-filter is considered. The reliability of the analytical relationships is confirmed by the coincidence of the results which are obtained by using analytical expressions and the results of simulation at the check points.

In modern digital measurement devices to obtain samples in the signal measurement channel, analog-to-digital converters (ADCs) are applied. Real ADCs have finite resolution, which results in time domain to the quantization error effect (so called quantization noise). Regardless of the applied measurement method, the quantization noise results to measurement error of RMS, active power, complex spectrum and other. A research of influence on the ADC parameters (resolution, sampling frequency, input range, ADC architecture type) to the RMS measurement error is carried out. It is established that the most important ADC parameters are resolution and sampling frequency. The proposed RMS error estimating method is based on the application of a parabolic interpolation polynomial function (by the sampling frequency as a parameter). The interpolation polynomial coefficients are determined as a result of simulation modeling which is performed in Matlab 7. Random functions were applied to simulate the quantization noise. Analytical expressions are obtained that allows to estimate the RMS measurement error from the signal parameters and ADC parameters by using the proposed error estimation method. The simulation results show that the proposed method of the error calculation allows at least to reduce the RMS error estimate (in comparison with “worst case” method) at least to 30 %. The proposed approach for the error estimation can be used for sinusoidal signals or polyharmonic signals the shape of which is close to sinusoidal. The application of the proposed approach can be extended to the problem of active power and complex spectrum measurement.

Nowadays active power refers to the power quality parameters. That’s why, measurement of active power parameters is required in power quality measuring instruments. This paper describes a technique for active power measurement which apply a discrete Fourier transform of instantaneous power. The conveniences and drawbacks of this measurement method are considered in comparison with the accumulation of instantaneous power method and the application of low-pass filtration method. Analytical expressions are obtained that make it possible to estimate the active power measurement error caused by “spectrum leakage” effect for one-harmonic and polyharmonic measured signals situations. Approaches to reducing the methodological measurement error without significant changes of the original measurement algorithm are considered. Methods for decreasing the active power measurement error as a result of finite frequency deviation are proposed. The metrological and implementation features of discrete Fourier transform implementation for solving problem are considered. The simulation model is performed by the implementation of the Goertzel algorithm. With the application of a simulation model, performed by Simulink package, the accuracy and credibility of the obtained analytical formulas are estimated.

Currently, the leading manufacturers of integrated circuits produce Sigma-Delta ADCs, focused on the application as a part of a measurement channels of power quality (PQ) instruments. The magnitude response of such an ADC is determined by a built-in digital filter, usually with an cascaded integrator-comb (CIC) filter structure. However, the magnitude response of this filter (from the point of view of the low-pass filter function) do not have the required parameters: the filter has a slope of the magnitude response (MR) in the passband and a relatively large bandwidth between passband and stopband. In this article, two variants of the method for correcting the Sigma-Delta ADC magnitude response which is a part of PQ instrument measurement channel are considered. The first approach involves the application of a single corrective digital filter that compensates the Sigma-Delta ADC passband ripple and reduces the bandwidth between passband and stopband. The second approach involves the application of two corrective digital filters, one of which compensates the ADC passband ripple, and the other one reduces the stopband frequency. Both options considered assume the application of Matlab built-in functions. The methods for calculating the coefficients of the correcting filters are considered for both approaches. By the application of simulation modeling, the characteristics of the PQ instrument measurement channel, which contains the CIC filter and the compensating filter, are obtained. The possibility of modeling the PQ instrument measuring channel in the Simulink software package is considered. The features of modeling of the blocks of the CIC-filter, filter-compensator and data format conversion for the case of the PQ measurement channel are analyzed.

Currently, frequency measurement is one of the most popular tasks in power engineering. The voltage and current of real power networks are different from sinusoidal ones: there are harmonics, supraharmonics, noise and flicker. For this reason, the application of measurement methods designed to measure the frequency of a sinusoidal voltage results in larger measurement errors. This paper discusses two most popular frequency measurement methods applied in power engineering: zero crossing technique and a method based on the amplitude spectrum analysis. The paper provides a description of each method, indicates the main advantages and disadvantages, considers the features for the practical implementation. Analytical relations are obtained which allows to estimate the methodological error of all the methods under consideration. In the paper main modification directions for the considered methods are shown, aimed mainly at simplifying the implementation and reducing the methodological error. The influence of the parameters of the input signals, namely the sample time and the measurement time, on the frequency measurement error is considered. A comparative analysis of the methodological error of the considered methods for the cases of sinusoidal signal, polyharmonic signal and combined signals (sinusoidal signal and sinusoidal flicker) was made by simulation modeling in Matlab and Simulink software. The influence of supraharmonics on the measurement error for the case of considered methods is researched. The reliability of the obtained analytical relations is confirmed by comparison of the results of analytical and simulation modeling at check points.

Currently, the most popular method of the frequency measurement of electric power grids signals is the zero-crossing technique. The main disadvantage of this method is associated with the sensitivity of this method to harmonics, noise and flicker of the measured signal. The existing methods of reducing the measurement error results to an increase of the measurement time, which is a significant disadvantage for a number of practical tasks. This paper discusses a method of reducing the frequency measurement error when applying the zero-crossing technique. The proposed method is based on the determination of the moments of signal transition through the level which corresponds to signal maximum derivative. To clarify the moments of signal transition through a given level, a piecewise-linear approximation of the signal in the vicinity of its transition through a given level was applied. To determine the value of the derivative, a seventh-order digital differentiator is proposed. Analytical expressions for the frequency measurement error estimation for the proposed measurement method are obtained. The effectiveness of the proposed method of reducing the measurement error in comparison with the “traditional” zero-crossing technique has been evaluated. By using the Simulink software package, a model of simulation modeling of a frequency measurement transducer that implements the proposed measurement method has been proposed. The reliability of the obtained analytical expressions is confirmed by the coincidence of the results of simulation modeling and analytical expressions at check points.

The possibility of using standard periodic non-harmonic signals is considered instead of the traditionally used set of harmonic signals when calibrating by a total harmonic distortion of power quality analyzers. The use of squarewave, triangular, sawtooth signals and a truncated sine wave signal is proposed, which presented in Fourier order contain specific harmonic components. Thus, by selecting a specific standard periodic signal, a corresponding specific reference value of the total harmonic distortion is set. Results of calibration of Fluke 435 power quality analyzer using Metrix CX1651 calibrator are presented, which give an experimental estimate of the ability to calibrate by a total harmonic distortion using standard periodic non-harmonic signals.

Currently, the method based on averaging of the squares of samples is most often applied to measure the RMS in a given frequency band. This method has several advantages: the simplicity of implementation, high accuracy and simplicity of analysis of measurement error. To ensure a low measurement error values, adjustment of the number of averaged samples or adjustment of the measurement start time can be applied. Another important advantage of the considerate method is the possibility of its application for both a sinusoidal and a polyharmonic input signal. The article discusses the implementation features of the considerate measurement method for the case of an input non-sinusoidal periodic signal. Analytical dependences are obtained for the RMS measurement error for the case of a non-sinusoidal input signal. Methods of reducing the measurement error are proposed and estimates of their effectiveness are obtained. Analytical expressions are obtained that allows to evaluate the effect of the additive, multiplicative, nonlinear components of the error of the signal samples on the RMS measurement error. The reliability of the analytical relationships is confirmed by the coincidence of the results which are obtained by applying the analytical expressions and the results of simulation at check points.

Currently, for reactive power measurement the digital measurement methods are applied both in time and in frequency domain. When determination of the voltage and current samples, the errors of the measurement channels exists that results to the reactive power measurement error. The most important case of the occurrence of the voltage and current samples measurement errors is an analog-to-digital converter (ADC). The measurement errors of voltage and current samples can be divided into the following components: additive, multiplicative, linearity and quantization. The paper consider the influence of each of the listed error components on the reactive power measurement error for the time domain and frequency domain measurement method. Analytical expressions are obtained that relate each component of the samples error and the reactive power measurement error. The methods for representing the ADC nonlinearity and their influence on the reactive power measurement error are considered. By using the Simulink and Matlab software package, a simulation model of voltage and current measurement channels and the influence on the reactive power measurement error was obtained. The reliability of the obtained analytical expressions is confirmed by the coincidence of the simulation results and analytical expressions at check points.

Currently, the measurement method of root mean square value (RMS), based on samples accumulation, is the most popular. The advantages of this method include ease of implementation, high accuracy and additional opportunities to reduce the methodological component of the total measurement error. Studies show that increasing measurement time also contributes to a reduction in error. However, in a number of measurement and automatic control problems, one of the main requirements is to reduce the measurement time. This problem can be solved by a method based on moving averaging, that includes the advantages of the "classical" method, but is largely devoid of its shortcomings. An algorithm of RMS measurement for implementation on the basis of recursive and non-recursive moving average filters is proposed. The influence of the sinusoidal signal parameters on the RMS measurement error is analyzed. An analytical expression is obtained that allows to estimate the RMS measurement error from the signal parameters and the number of samples. Approaches are proposed for reducing the RMS measurement error when applying the measurement method under consideration. It is shown that the application of post-filtration of RMS measurement results can reduce the final RMS measurement error. The veracity of the analytical expressions obtained in the paper is confirmed by the results of simulation at check points.

In this article, a comparative analysis of digital methods for the measurement of active power with the application of approximation polynomials is performed. Zero-order and first-order polynomials are considered. For each of the applied approximating polynomials, a measurement algorithm is proposed; the methodological error for the case of sinusoidal current and voltage is analyzed; recommendations to reduce the measurement error have been developed. Analytical relations are obtained allows to choose the values of the sampling frequency and the measurement time. It is shown that approximation by zero-order polynomials is much more efficient than approximation by first-order polynomials. For the case of applying the approximation polynomial of the zero-order, the methods for reducing the methodological measurement error are proposed. These methods are based on the adjustment of the number of averaged samples, the adjustment of the initial phase of the signal and the combined adjustment of the number of averaged samples and the value of the initial phase (combined technique). To apply the proposed methods, an additional measurement frequency converter is required. The reliability of the analytical relationships is confirmed by the coincidence of the analytical results and the results of simulation modeling at control points. By using software package Simulink, simulation modeling was performed to evaluate the effectiveness of the considered methods for the active power measurement.