CN103454494A - High-precision harmonic analysis method - Google Patents

Abstract

The invention discloses a high-precision harmonic analysis method which is improved and realized on the basis of quasi-synchronization DFT. The method includes the step that a frequency domain sampling position is changed along with drifting of signal frequency when the quasi-synchronization DFT is used for harmonic analysis, and namely, the frequency domain sampling position is mu 2pi/N, wherein mu is the drifting of the signal frequency, and mu is 1 when the drifting of the signal frequency does not exist. The method adopts the conception of variable grids, and namely, the frequency domain sampling position is changed along with the drifting of the signal frequency in harmonic analysis. The method is beneficial to acquiring information of amplitudes, initial phase angles, frequency and the like of each harmonic more accurately in the fields where harmonic analysis is applied, such as the fields of power quality monitoring, electronic product production testing and electrical equipment monitoring.

Description

A kind of high-precision harmonic analysis method

The application divides an application, the application number of original application: 201110245638.8, and invention and created name: a kind of harmonic analysis method, the applying date: 2011-8-24.

Technical field

The present invention relates to a kind of high-precision harmonic analysis method.

Background technology

The frequency analysis technology is widely used in various fields such as electric energy quality monitoring, electronic product production testing, electric equipment monitoring, is the important technical of carrying out power system monitor, quality inspection, monitoring of tools.The most widely used technology of frequency analysis is discrete Fourier transformation (DFT) and Fast Fourier Transform (FFT) (FFT) at present.The frequency analysis technology that accurate synchronous sampling technique and DFT technology combine can improve the precision of frequency analysis, and its formula is:

$\left\{\begin{array}{c}{a}_k=2F_{\mathrm{ak}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\cos \left(k\frac{2\mathrm{\π}}{N}i\right)\\ b_k=2F_{\mathrm{bk}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\sin \left(k\frac{2\mathrm{\π}}{N}i\right)\end{array}\right.,$

In formula: k is the number of times (as first-harmonic k=1,3 subharmonic k=3) that needs the harmonic wave of acquisition; Sin and cos are respectively sine and cosine functions; And a _kand b _kbe respectively real part and the imaginary part of k subharmonic; N is iterations; W determines by integration method, while adopting complexification trapezoidal integration method, and W=nN; γ _iit is a weighting coefficient; for all weighting coefficient sums; I the sampled value that f (i) is analysis waveform; N is sampling number in the cycle.

In the engineering application, frequency analysis is always carried out the sampling of finite point and is difficult to accomplish the synchronized sampling of stricti jurise.Like this, when the accurate synchronous DFT of application carries out frequency analysis, the short scope leakage that will exist the long scope caused due to truncation effect to leak and cause due to fence effect, make analysis result precision not high, even not credible.

Summary of the invention

The technical problem to be solved in the present invention is to provide a kind of high-precision harmonic analysis method, effectively to improve the analytical error of accurate synchronous DFT frequency analysis technology, obtain high-precision frequency analysis result, thereby improve the quality of the field instrument and equipments such as electric energy quality monitoring based on the frequency analysis theory, electronic product production testing, electric equipment monitoring and the validity of state judgement.

For solving the problems of the technologies described above, high-precision harmonic analysis method provided by the invention comprises: the position that the accurate synchronous DFT of application carries out the sampling of frequency analysis time-frequency domain changes according to the drift of signal frequency, be that described frequency domain sampling position is μ 2 π/N, wherein: the drift that μ is signal frequency, during without drift, μ is 1.

The thought of harmonic analysis method of the present invention based on variable fence, realize by 5 analytical procedures.

The thought of variable fence: the main cause of accurate synchronous DFT analytical error is position and the ideal position generation deviation that the drift of signal frequency causes spectrum peak to occur, if the analysis result still obtained to sample in frequency domain according to 2 π/N is extremely incorrect.Variable fence refers to: the position of frequency domain sampling be not 2 π that fix/N, but change according to the drift of signal frequency, the frequency sampling position is the drift that μ 2 π/N(μ is signal frequency).The frequency domain sampling fence changes along with the drift of signal frequency can accurately estimate the position that the each harmonic peak value occurs, and then obtains high-precision amplitude and phase angle information.

Frequency analysis step of the present invention is as follows:

(1) equal interval sampling W+2 sampling number is according to { f (i), i=0,1, w+1}(W is determined by selected integration method, the present invention does not specify a certain integration method, and integration method commonly used has complexification trapezoidal integration method W=nN, complexification rectangular integration method W=n (N-1), iterative Simpson integration method W=n (N-1)/2 etc., and the actual conditions that can apply according to the present invention are selected suitable integration method.Generally more satisfactory with complexification trapezoidal integration method effect.); (2) start the accurate synchronous DFT formula of application from sampled point i=0 $\left\{\begin{array}{c}{a}_k=2F_{\mathrm{ak}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\cos \left(k\frac{2\mathrm{\π}}{N}i\right)\\ b_k=2F_{\mathrm{bk}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\sin \left(k\frac{2\mathrm{\π}}{N}i\right)\end{array}\right.,$ Analyze W+1 data and obtain first-harmonic information

with

(3) from the accurate synchronous DFT formula of sampled point i=1 application $\left\{\begin{array}{c}{a}_k=2F_{\mathrm{ak}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f(i+1)\cos \left(k\frac{2\mathrm{\π}}{N}i\right)\\ b_k=2F_{\mathrm{bk}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f(i+1)\sin \left(k\frac{2\mathrm{\π}}{N}i\right)\end{array}\right.,$ Analyze W+1 data and obtain first-harmonic information

with

(4) application of formula $\mathrm{\μ}=N\frac{{\mathrm{tg}}^{-1}\left[\frac{F_{a0}^n\left(1\right)}{F_{b0}^n\left(1\right)}\right]-{\mathrm{tg}}^{-1}\left[\frac{F_{a0}^n\left(0\right)}{F_{b0}^n\left(0\right)}\right]}{2\mathrm{\π}}$ Calculate the frequency drift μ of signal;

(5) application of formula $\left\{\begin{array}{c}{a}_k=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\cos \left(k\frac{\mathrm{\μ}2\mathrm{\π}}{N}i\right)\\ b_k=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\sin \left(k\frac{\mathrm{\μ}2\mathrm{\π}}{N}i\right)\end{array}\right.$ Calculate amplitude and the phase angle of each harmonic.

Accurate synchronous DFT frequency analysis can effectively suppress long scope and leak, the main cause of its spectrum leakage is that the short scope that signal frequency drift causes is leaked, and signal frequency drift causes principal character that short scope leaks the is spectrum peak-to-peak value occurs position along with signal frequency drift synchronous change, so the variable fence frequency domain sample can effectively catch according to signal drift the position that the spectrum peak-to-peak value occurs, thereby obtains high-precision harmonic information.

Equal interval sampling is according to cycle T and the frequency f (as the power frequency component frequency f is 50Hz, the cycle is 20mS) of carrying out the ideal signal of frequency analysis, sampling N point in one-period, and sample frequency is f _s=Nf, and N>=64.

W+2 sampling number of described sampling is according to being to do corresponding selection according to selected integration method, if adopt complexification trapezoidal integration method, W=nN; If adopt complexification rectangular integration method, W=n (N-1); If adopt the iterative Simpson integration method, W=n (N-1)/2.Then according to sample frequency f _s=Nf, acquisition sampled point data sequence f (i), and i=0,1 ..., w+1}, n>=3, finally carry out frequency analysis to this data sequence.

An iteration coefficient γ _iby integration method, ideal period sampled point N and iterations n, determined, concrete derivation referring to document [Dai Xianzhong. the some problems [J] in accurate synchronized sampling application. electrical measurement and instrument, 1988, (2): 2-7.].

for all weighting coefficient sums.

A _kand b _kfor imaginary part and the real part of k subharmonic, according to a _kand b _kjust can obtain harmonic amplitude and initial phase angle.

The drift μ of signal frequency obtains according to the fixed relationship of sampling number N in neighbouring sample point first-harmonic phase angle difference and ideal period, and the drift μ of signal frequency also can be used for revising the frequency f of first-harmonic and higher hamonic wave ₁frequency f with higher hamonic wave _k.

Adopt above-mentioned high precision frequency analysis technology, the also i.e. frequency analysis technology based on variable fence thought has following technical advantage:

(1) high-precision frequency analysis result.No matter the analysis result that frequency analysis technology of the present invention obtains is that amplitude or phase angle error improve 4 more than the order of magnitude.

(2) frequency analysis technology of the present invention has fundamentally solved the low problem of accurate synchronous DFT analysis precision, and, without carrying out complicated inverting and correction, algorithm is simple.

(3) with respect to the synchronous DFT of standard, frequency analysis technology of the present invention only need to increase a sampled point and just solve the large problem of accurate synchronous DFT analytical error, is easy to realize.

(4) application the present invention improves existing instrument and equipment, and technical is feasible, and does not need to increase any hardware spending and just can make analysis result can improve 4 more than the order of magnitude.

(5) variable fence thought also is applicable to carry out repeatedly iteration and the frequency analysis process of non-once iteration too, now only need to resolve into repeatedly iteration to an iteration and realize just passable.Iteration is the same with iteration repeatedly in essence, just when calculating repeatedly iteration carry out substep calculating, and iteration is that the process of iteration is repeatedly merged in iteration coefficient γ i and once calculated, so the present invention is equally applicable to repeatedly iterative process.

Embodiment

A kind of high precision frequency analysis technology of the present invention comprises the following steps:

At first, an equal interval sampling W+2 sampled point, with the discrete series that obtains analyzed signal f (k), k=0,1 ..., w+1}.W is determined jointly by sampling number N in integration method, iterations n and ideal period.Equal interval sampling refers to according to the frequency f (as the power frequency component frequency is 50Hz, the cycle is 20mS) of carrying out the ideal signal of frequency analysis determines sample frequency f _s=Nf, at sample frequency f _seffect under the N point of sampling equably in one-period.Usually, periodic sampling point N=64 or abovely just can obtain frequency analysis result preferably, and iterations n=3-5 just can obtain comparatively ideal frequency analysis result.Integration method has complexification trapezoidal integration method W=nN, complexification rectangular integration method W=n (N-1), Simpson's integration method W=n (N-1)/2 etc. multiple, can be selected according to actual conditions.

Secondly, start the accurate synchronous DFT formula of application from sampled point k=0

$\left\{\begin{array}{c}{a}_k=2F_{\mathrm{ak}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\cos \left(k\frac{2\mathrm{\π}}{N}i\right)\\ b_k=2F_{\mathrm{bk}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\sin \left(k\frac{2\mathrm{\π}}{N}i\right)\end{array}\right.,$ Analyze W+1 data and obtain first-harmonic information

with

Wherein, an iteration coefficient γ _iby integration method, ideal period sampled point N and iterations n, determined, and

for all weighting coefficient sums.

Again, from the accurate synchronous DFT formula of sampled point k=1 application $\left\{\begin{array}{c}{a}_k=2F_{\mathrm{ak}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f(i+1)\cos \left(k\frac{2\mathrm{\π}}{N}i\right)\\ b_k=2F_{\mathrm{bk}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f(i+1)\sin \left(k\frac{2\mathrm{\π}}{N}i\right)\end{array}\right.,$ Analyze W+1 data and obtain first-harmonic information

Then, application of formula $\mathrm{\μ}=N\frac{{\mathrm{tg}}^{-1}\left[\frac{F_{a0}^n\left(1\right)}{F_{b0}^n\left(1\right)}\right]-{\mathrm{tg}}^{-1}\left[\frac{F_{a0}^n\left(0\right)}{F_{b0}^n\left(0\right)}\right]}{2\mathrm{\π}}$ Calculate the frequency drift μ of signal.After obtaining frequency drift μ, can be according to sample frequency f _scalculate the first-harmonic of acquisition analyzed signal and the frequency f of higher hamonic wave with sampling number N in ideal period.

Finally, application $\left\{\begin{array}{c}{a}_k=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\cos \left(k\frac{\mathrm{\μ}2\mathrm{\π}}{N}i\right)\\ b_k=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\sin \left(k\frac{\mathrm{\μ}2\mathrm{\π}}{N}i\right)\end{array}\right.$ Calculate the real part a of k subharmonic _kwith imaginary part information b _k, and then according to formula:

calculate amplitude

and according to formula:

calculate initial phase angle

Those skilled in the art will be appreciated that, above embodiment is only for the present invention is described, and not as limitation of the invention, the present invention can also be varied to more mode, as long as in connotation scope of the present invention, to variation, the modification of the above embodiment, all will drop in claims scope of the present invention.

Claims (1)

1. a harmonic analysis method is characterized in that comprising the following steps:

(1), an equal interval sampling W+2 sampling number certificate: f (i), i=0,1 ..., w+1};

(2), from sampled point i=0, start the accurate synchronous DFT formula of application:

$\left\{\begin{array}{c}{a}_k=2F_{\mathrm{ak}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\cos \left(k\frac{2\mathrm{\π}}{N}i\right)\\ b_k=2F_{\mathrm{bk}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\sin \left(k\frac{2\mathrm{\π}}{N}i\right)\end{array}\right.,$ Analyze W+1 data and obtain first-harmonic information

with

(3), from the accurate synchronous DFT formula of sampled point i=1 application:

$\left\{\begin{array}{c}{a}_k=2F_{\mathrm{ak}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f(i+1)\cos \left(k\frac{2\mathrm{\π}}{N}i\right)\\ b_k=2F_{\mathrm{bk}}^n=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f(i+1)\sin \left(k\frac{2\mathrm{\π}}{N}i\right)\end{array}\right.,,$

Analyze W+1 data and obtain first-harmonic information

(4), application of formula: $\mathrm{\μ}=N\frac{{\mathrm{tg}}^{-1}\left[\frac{F_{a0}^n\left(1\right)}{F_{b0}^n\left(1\right)}\right]-{\mathrm{tg}}^{-1}\left[\frac{F_{a0}^n\left(0\right)}{F_{b0}^n\left(0\right)}\right]}{2\mathrm{\π}},$

Calculate the frequency drift μ of signal;

(5) application of formula: $\left\{\begin{array}{c}{a}_k=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\cos \left(k\frac{\mathrm{\μ}2\mathrm{\π}}{N}i\right)\\ b_k=\frac{2}{Q}\underset{i=0}{\overset{W}{\mathrm{\Σ}}}{\mathrm{\γ}}_{i}f\left(i\right)\sin \left(k\frac{\mathrm{\μ}2\mathrm{\π}}{N}i\right)\end{array}\right.$

, amplitude and the phase angle of calculating each harmonic;

Described equal interval sampling is according to cycle T and the frequency f of carrying out the ideal signal of frequency analysis, sampling N point in one-period, and sample frequency is f _s=Nf, and N>=64;

The drift μ of signal frequency, be to obtain according to the fixed relationship of sampling number N in neighbouring sample point first-harmonic phase angle difference and ideal period, and the drift μ of signal frequency also can be used for revising the frequency f of first-harmonic and higher hamonic wave ₁frequency f with higher hamonic wave _k.