CN104749433A - Method for accurately calculating and analyzing harmonic and electric energy quality - Google Patents

Abstract

The invention discloses a method for accurately calculating and analyzing harmonic and electric energy quality. The method comprises the following steps: selecting a detected signal sampling sequence; carrying out discrete Fourier transformation (DFT) on the sampling sequence; selecting a 3-order rapid attenuation window Nuttall; after windowing a signal, carrying out the discrete Fourier transformation; calculating to solve a phase phik; and finally, obtaining amplitude Ak, frequency fk and the phase phik of a detected signal to finish the analysis of the harmonic and the electric energy quality. According to the method, long-range leakage is eliminated by windowing, and an error, caused by the long-range leakage, is eliminated by using an interpolation algorithm; the frequency spectrum leakage caused by nonsynchronous sampling in the prior art is avoided by the method. A test comparing result shows that the frequency, amplitude and phase measurement errors of the algorithm are very small; compared with an actual measurement error, the errors can be completely ignored. With the adoption of the method, the equipment can be used as a harmonic measurement standard device capable of being used as a quality standard laboratory of a provincial-level technical supervision bureau or an electric power research institute and above.

Description

A kind of humorous method involving the quality of power supply of accurate computational analysis

Technical field

The invention belongs to electricity field, particularly relate to a kind of humorous method involving the quality of power supply of accurate computational analysis.

Background technology

In recent years, along with Electricity Demand increases fast, electric power industry development is progressively grown, and power grid construction paces are constantly accelerated, and harmonic wave more and more receives the concern of people to the impact of the quality of power supply.All nonlinear-loads can produce harmonic current, and the device type producing harmonic wave has: switched-mode power supply (SMPS), electronic ballast for fluorescent lamp, speed-regulating actuator, uninterrupted power source (UPS), magnetive cord equipment and some household electrical appliance are as televisor etc.Mains by harmonics comes from three aspects: one is energy source generation harmonic wave of low quality; Two is that electrical power trans mission/distribution system produces harmonic wave; Three is harmonic waves that consumer produces, and wherein the harmonic wave of consumer generation is maximum.

At present, conventional harmonic analyser is by discrete sampling measured signal, then the harmonic parameters that direct computation of DFT leaf analysis (DFT) obtains measured signal is carried out to sampled data, if sample frequency is not the integral multiple of harmonic frequency signal, namely not synchronized sampling, so can cause measuring error due to spectral leakage.Generally the humorous power quality analyzer that involves adopts phaselocked loop to ensure synchronized sampling, this can meet the measurement precise requirements of harmonic wave, but pin-point accuracy harmonic analysis instrument employing phaselocked loop is also improper, because phaselocked loop itself has error, and phaselocked loop can not follow the tracks of the signal of Severe distortion, and the calibrating signal artificially arranged may Severe distortion.

On the other hand, phaselocked loop is not adopted just to need extra measure to correct the measuring error of spectral leakage formation.Mainly contain following several method:

(1) time domain interpolation method.Carry out interpolation arithmetic according to selected interpolation algorithm to sampling number certificate, obtain new sampled data, its sampling rate meets synchronized sampling, then new sampled data is obtained to the harmonic parameters of measured signal by direct computation of DFT leaf analysis (DFT).First this algorithm calculates fundamental frequency with interpolation algorithm, then calculates the new sampled data meeting synchronized sampling, before calculating fundamental frequency, need the high fdrequency component of considering sampled signal, makes sampled signal close to sinusoidal wave.The shortcoming of this algorithm has: accuracy in computation depends on fundamental frequency, and calculates fundamental frequency and need link after filtering, and the accuracy in computation of fundamental frequency is affected; Because needs recalculate each sampled point, calculated amount is large.

(2) time domain is in conjunction with frequency domain correction algorithm.First calculate fundamental frequency with time domain interpolation or frequency domain algorithm, then calculate harmonic parameters by time domain in conjunction with frequency domain correction algorithm; The accuracy of this algorithm depends on the accuracy of fundamental frequency, when signal Severe distortion or serious asynchronous time, the accuracy of fundamental frequency can be affected.

(3) frequency domain interpolation is in conjunction with time domain interpolation algorithm.First calculate fundamental frequency with frequency interpolation algorithm, then calculate the new sampled data meeting synchronized sampling with time domain interpolation algorithm.The shortcoming of this algorithm is that calculated amount is large.

(4) frequency domain interpolation algorithm.Direct computation of DFT leaf analysis (DFT) is carried out to the sampled data of windowing and obtains frequency domain parameter, then calculate harmonic parameters accurately with frequency domain interpolation algorithm.As long as the window function selected, interpolation algorithm are suitable, the accuracy of this algorithm is very high, and its shortcoming needs long sample sequence and utilizes the attenuation characteristic of window function secondary lobe to reveal to eliminate long-range, and real-time is poor.

Summary of the invention

The object of the invention is to solve the problem, the humorous method involving the quality of power supply of a kind of accurate computational analysis provided by the invention, have pin-point accuracy, measurement range is wide, can measure the signal of Severe distortion.

The technical solution adopted in the present invention is: a kind of humorous method involving the quality of power supply of accurate computational analysis, comprises the following steps:

Step 1: measured signal sample sequence is formula (1):

A in formula (1) _k, f _k, the amplitude of frequency component, frequency, phase place respectively, f _ssample frequency, m=0,1 ..., M-1, M are sampling numbers, and being now equivalent to add a length to signal is the rectangular window of M;

Step 2: the discrete Fourier transformation (DFT) of this sequence is formula (2):

In formula (2), λ ∈ [0, M).Wherein W (λ) is the discrete Fourier transformation of rectangular window:

$W\left(\mathrm{\λ}\right)=e^{-\mathrm{i\π\λ}\frac{M-1}{M}}\frac{\sin \left(\mathrm{\λ\π}\right)}{\sin \left(\mathrm{\λ}\frac{\mathrm{\π}}{M}\right)}---\left(3\right)$

In formula (2), λ _k=f ^km/f _s, can prove to only have to work as λ _kfor integer and f _s> 2f _ktime, the amplitude A of signal just accurately may be obtained by formula (2) _k, frequency f _k, phase place work as λ _kwhen not being integer, can error be produced, form primarily of two parts:

A. by W (λ-λ in formula (2) _k) error that produces, | λ-λ _k| < 0.5, is called short distance spectral leakage;

B. by other frequency component and self mirror image W (λ+λ _k) part produce error, be called long-range spectral leakage;

In order to suppress this two kinds of errors, adopt window function and interpolation algorithm, the object of windowing significantly reduces long-range spectral leakage by the high fdrequency component in suppressed FCM (2), then accurately calculates W (λ-λ by interpolation algorithm _k) in λ _k.

Step 3: according to above-mentioned requirements, select 3 rank to decay the soonest window Nuttall window, this window can be expressed as formula (4):

$w\left(m\right)=0.375-0.5\cos \left(2\mathrm{\&pi;}\frac{m}{M}\right)+0.125\cos \left(4\mathrm{\&pi;}\frac{m}{M}\right)---\left(4\right)$

3 rank Nuttall windows are a kind of 3 rank Cosine Window, and its discrete Fourier transformation is formula (5):

W _N(λ)＝0.375W(λ)-0.25[W(λ-1)+W(λ+1)]+0.0625[W(λ-2)+W(λ+2)] (5)

As M>>1, there is formula (6): $W_N\left(\mathrm{\&lambda;}\right)\&cong;\frac{M\sin \left(\mathrm{\&pi;\&lambda;}\right)}{16\mathrm{\&pi;\&lambda;}}{e}^{-\mathrm{j\&pi;\&lambda;}\frac{M}{M-1}}\frac{4!}{\underset{h=1}{\overset{2}{\mathrm{\&Pi;}}}(h^2-{\mathrm{\&lambda;}}^2)}---\left(6\right)$

Although the side lobe peak of 3 rank Nuttall windows is-46.7db, not little, fast owing to decaying, reach 30db/octave, after 8 secondary lobes, secondary lobe is just lower than the secondary lobe of 4 rank Blackman-Harris windows; As long as therefore frequency resolution (f _s/ M) enough little, just can eliminate the interference between each harmonic wave.

Step 4: the discrete Fourier transformation after signal x (m) windowing is formula (7):

In order to eliminate short distance spectral leakage and reduce long-range spectral leakage further, adopt 5 point weight interpolation algorithms, make formula (8): λ _k=l _k+ δ _k(8);

Wherein l _kclosest to λ _kinteger ,-0.5≤δ _k< 0.5, definition α _kfor:

${\mathrm{\&alpha;}}_k=\frac{\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|X_w(l_k-i)\right|}{\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|X_w(l_k+i)\right|}---\left(9\right)$

Now can think that long-range spectral leakage is suppressed, therefore

$\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|X_w(l_k\&PlusMinus;i)\right|\&cong;\frac{A_k}{2}\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}|W_N\left(i\stackrel{\&OverBar;}{+}{\mathrm{\&delta;}}_k\right)---\left(10\right)$

Step 5: formula (9) and formula (10), can be changed into formula (11):

${\mathrm{\&alpha;}}_k=\frac{\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|W_N(i+{\mathrm{\&delta;}}_k)\right|}{\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|W_N(i-{\mathrm{\&delta;}}_k)\right|}---\left(11\right)$

Linear equation in two unknowns formula (12) can be obtained by formula (6) and formula (11):

$(1-{\mathrm{\&alpha;}}_k){\mathrm{\&delta;}}_k^2-7(1+{\mathrm{\&alpha;}}_k){\mathrm{\&delta;}}_k+12(1-{\mathrm{\&alpha;}}_k)=0---\left(12\right)$

Work as α _kclosely 1 time, δ _kclose to 0, therefore can ignore formula (12) becomes formula (13):

-7(1+α _k)δ _k+12(1-α _k)＝0,|1-α _k|≈0 (13)

Step 6: can δ be solved according to (12), (13) _k, can A be drawn by formula (14) _k:

$A_k\&cong;2\frac{\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{C}_4^{2-i}\left[\right|X_w(l_k-i)|+|X_w(l_k+i)\left|\right]+C_4^2\left|X_w\left(l_k\right)\right|}{\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{C}_4^{2-i}\left[\right|W_N(i-{\mathrm{\&delta;}}_k)|+|W_N(i+{\mathrm{\&delta;}}_k)\left|\right]+C_4^2\left|W_N\left({\mathrm{\&delta;}}_k\right)\right|}---\left(14\right)$

Because:

Phase place can be obtained by formula (15) finally, the amplitude A of measured signal is just obtained _k, frequency f _k, phase place to the analysis of harmonic wave and the quality of power supply.

The invention has the beneficial effects as follows,

The present invention compared with the existing technology, has the following advantages and beneficial effect:

Adopt power quality analyzer of the present invention have higher class of accuracy, measurement range wide, can the signal of Measurement accuracy distortion, reduce the requirement of real-time of conventional equipment to line voltage, current sample.Revealed by the long-range of eliminating of windowing, eliminate short distance by interpolation algorithm and reveal the error produced, overcome the spectral leakage that prior art produces because of non-synchronous sampling.

The display of test comparison result, the frequency of this algorithm, amplitude, phase measurement error are very little, relative to substantial measurement errors, can ignore completely.Harmonic measure error is far superior to GB (GB/T 14549-1993) requirement to A level harmonic analysis instrument, and frequency error measurement is also far smaller than GB (GB/T 15945-2008) to the measuring error requirement of frequency.Adopt power quality analyzer of the present invention can as the harmonic measure standard set-up economizing one-level and above Bureau of Technical Supervision or Electric Power Research Institute's power quality standard laboratory.

Below in conjunction with accompanying drawing, the present invention is described in further detail.

Accompanying drawing explanation

Fig. 1 is the test comparison structural drawing of the embodiment of the present invention.

Embodiment

In order to deepen the understanding of the present invention, below in conjunction with drawings and Examples, the present invention is further detailed explanation.Following examples only for technical scheme of the present invention is clearly described, and can not limit the scope of the invention with this.

The humorous method involving the quality of power supply of accurate computational analysis, comprises the following steps:

Step 1: measured signal sample sequence is formula (1):

A in formula (1) _k, f _k, the amplitude of frequency component, frequency, phase place respectively, f _ssample frequency, m=0,1 ..., M-1, M are sampling numbers, and being now equivalent to add a length to signal is the rectangular window of M.

Step 2: the discrete Fourier transformation (DFT) of this sequence is formula (2):

In formula (2), λ ∈ [0, M).Wherein W (λ) is the discrete Fourier transformation of rectangular window:

$W\left(\mathrm{\&lambda;}\right)=e^{-\mathrm{i\&pi;\&lambda;}\frac{M-1}{M}}\frac{\sin \left(\mathrm{\&lambda;\&pi;}\right)}{\sin \left(\mathrm{\&lambda;}\frac{\mathrm{\&pi;}}{M}\right)}---\left(3\right)$

In formula (2), λ _k=f ^km/f _s, can prove to only have to work as λ _kfor integer and f _s> 2f _ktime, the amplitude A of signal just accurately may be obtained by formula (2) _k, frequency f _k, phase place work as λ _kwhen not being integer, can error be produced, form primarily of two parts:

A. by W (λ-λ in formula (2) _k) error that produces, | λ-λ _k| < 0.5, is called short distance spectral leakage;

B. by other frequency component and self mirror image W (λ+λ _k) part produce error, be called long-range spectral leakage.

In order to suppress this two kinds of errors, adopt window function and interpolation algorithm, the object of windowing significantly reduces long-range spectral leakage by the high fdrequency component in suppressed FCM (2), then accurately calculates W (λ-λ by interpolation algorithm _k) in λ _k.

The requirement of window function and interpolation algorithm to window mainly contains:

A. for reducing long-range spectral leakage, require side lobe attenuation fast;

B. can obtain the Exact Solutions of interpolation equation, and calculated amount is more few better.

Step 3: according to above-mentioned requirements, select 3 rank to decay the soonest window Nuttall window, this window can be expressed as formula (4):

$w\left(m\right)=0.375-0.5\cos \left(2\mathrm{\&pi;}\frac{m}{M}\right)+0.125\cos \left(4\mathrm{\&pi;}\frac{m}{M}\right)---\left(4\right)$

3 rank Nuttall windows are a kind of 3 rank Cosine Window, and its discrete Fourier transformation is formula (5):

W _N(λ)＝0.375W(λ)-0.25[W(λ-1)+W(λ+1)]+0.0625[W(λ-2)+W(λ+2)] (5)

As M>>1, there is formula (6): $W_N\left(\mathrm{\&lambda;}\right)\&cong;\frac{M\sin \left(\mathrm{\&pi;\&lambda;}\right)}{16\mathrm{\&pi;\&lambda;}}{e}^{-\mathrm{j\&pi;\&lambda;}\frac{M}{M-1}}\frac{4!}{\underset{h=1}{\overset{2}{\mathrm{\&Pi;}}}(h^2-{\mathrm{\&lambda;}}^2)}---\left(6\right)$

Although the side lobe peak of 3 rank Nuttall windows is-46.7db, not little, fast owing to decaying, reach 30db/octave, after 8 secondary lobes, secondary lobe is just lower than the secondary lobe of 4 rank Blackman-Harris windows; As long as therefore frequency resolution (f _s/ M) enough little, just can eliminate the interference between each harmonic wave.Such as: if frequency resolution is 2Hz, fundamental frequency is 50Hz, then the secondary lobe number between harmonic wave is more than 20, and now secondary lobe has decayed to lower than-120db.

Step 4: the discrete Fourier transformation after signal x (m) windowing is formula (7):

In order to eliminate short distance spectral leakage and reduce long-range spectral leakage further, adopt 5 point weight interpolation algorithms, make formula (8): λ _k=l _k+ δ _k(8);

Wherein l _kclosest to λ _kinteger ,-0.5≤δ _k< 0.5, definition α _kfor:

${\mathrm{\&alpha;}}_k=\frac{\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|X_w(l_k-i)\right|}{\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|X_w(l_k+i)\right|}---\left(9\right)$

Now can think that long-range spectral leakage is suppressed, therefore

$\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|X_w(l_k\&PlusMinus;i)\right|\&cong;\frac{A_k}{2}\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}|W_N\left(i\stackrel{\&OverBar;}{+}{\mathrm{\&delta;}}_k\right)---\left(10\right)$

Step 5: formula (9) and formula (10), can be changed into formula (11):

${\mathrm{\&alpha;}}_k=\frac{\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|W_N(i+{\mathrm{\&delta;}}_k)\right|}{\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|W_N(i-{\mathrm{\&delta;}}_k)\right|}---\left(11\right)$

Linear equation in two unknowns formula (12) can be obtained by formula (6) and formula (11):

$(1-{\mathrm{\&alpha;}}_k){\mathrm{\&delta;}}_k^2-7(1+{\mathrm{\&alpha;}}_k){\mathrm{\&delta;}}_k+12(1-{\mathrm{\&alpha;}}_k)=0---\left(12\right)$

Work as α _kclosely 1 time, δ _kclose to 0, therefore can ignore formula (12) becomes formula (13):

-7(1+α _k)δ _k+12(1-α _k)＝0,|1-α _k|≈0 (13)

Step 6: can δ be solved according to (12), (13) _k, can A be drawn by formula (14) _k:

$A_k\&cong;2\frac{\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{C}_4^{2-i}\left[\right|X_w(l_k-i)|+|X_w(l_k+i)\left|\right]+C_4^2\left|X_w\left(l_k\right)\right|}{\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{C}_4^{2-i}\left[\right|W_N(i-{\mathrm{\&delta;}}_k)|+|W_N(i+{\mathrm{\&delta;}}_k)\left|\right]+C_4^2\left|W_N\left({\mathrm{\&delta;}}_k\right)\right|}---\left(14\right)$

Because:

Phase place can be obtained by formula (15) finally, the amplitude A of measured signal is just obtained _k, frequency f _k, phase place to the analysis of harmonic wave and the quality of power supply.

In order to verify correctness of the present invention, practicality, this programme adopts embodiment to be illustrated.

As shown in Figure 1, adopt power quality analyzer of the present invention and fluke 6140A to carry out test comparison, comprise and adopt power quality analyzer of the present invention, fluke 6140A, computing machine composition.Wherein, voltage input signal adopts same signal parallel connection input, and current input signal adopts same signal to connect and inputs.Test result is compared through fluke 6140A and is exported computing machine to.

Power quality analyzer of the present invention is adopted to be conventional equipment on market, after algorithm input equipment; Market conventional harmonic testing apparatus possesses measures 3 phase voltages, 6 phase currents simultaneously, and electric current can reach 30A, and voltage can reach 400V; The device current that the present embodiment adopts can reach 30A, and voltage can reach 600V, and sampling rate is fixed as 200kHz, and each sampling window width measured is 0.5 second.

Adopt the power quality analyzer of the inventive method and power quality standard signal source fluke 6140A to carry out 2 groups of matching measurements, often organize the voltage of measurement, current signal is first-harmonic and superposes multiple harmonic signal simultaneously; Wherein the harmonic signal of the 1st group of first-harmonic superposition has: 2,3,4,5,6,7,11,13,17,21,25,30,35,37,38,41,43,47,49,50,60,70,80,90,99,100, and the harmonic signal of the 2nd group of first-harmonic superposition has: 2 to 50,60,70,80,90,99,100.Signal parameter and measurement result are in table 1, table 2 and table 3.

Table 1 the 1st group of matching measurement result

Table 2 the 2nd group of matching measurement result

Table 3 harmonic frequency test comparison result

From the maximal value of the measurement error results table, can find out:

(1) amplitude error of harmonic voltage is+0.022%, is far smaller than GB (GB/T 14549-1993 " quality of power supply utility network harmonic wave ") requirement (5%) to A level harmonic analysis instrument.

(2) harmonic current error is+0.059%, is less than 0.1%, much smaller than the requirement (5%) of GB (GB/T 14549-1993) to A level harmonic analysis instrument.

(3) phase measurement error of harmonic voltage, electric current is less than 0.2 °, much smaller than GB (GB/T 14549-1993) to the requirement of A level harmonic analysis instrument (be not more than ± 5 ° or ± 1 ° h).

(4) the measurement maximum error of frequency is+0.00004%Hz, is far smaller than GB (GB/T 15945-2008) measuring error to frequency and requires (being not more than 0.01Hz).

(5) can show that mistake amplitude, phase place, frequency error are very little from result, relative to substantial measurement errors, the error that algorithm itself produces can be ignored.

Be noted that, the above embodiment is unrestricted to the explanation of technical solution of the present invention, the equivalent replacement of art those of ordinary skill or other amendments made according to prior art, as long as do not exceed thinking and the scope of technical solution of the present invention, all should be included within interest field of the presently claimed invention.

Claims (1)

1. the humorous method involving the quality of power supply of accurate computational analysis, is characterized in that, comprise the following steps:

Step 1: measured signal sample sequence is formula (1):

A in formula (1) _k, f _k, the amplitude of frequency component, frequency, phase place respectively, f _ssample frequency, m=0,1 ..., M-1, M are sampling numbers, and being now equivalent to add a length to signal is the rectangular window of M;

Step 2: the discrete Fourier transformation (DFT) of sample sequence is formula (2):

In formula (2), λ ∈ [0, M), wherein W (λ) is the discrete Fourier transformation of rectangular window:

$W\left(\mathrm{\&lambda;}\right)=e^{-\mathrm{i\&pi;\&lambda;}\frac{M-1}{M}}\frac{\sin \left(\mathrm{\&lambda;\&pi;}\right)}{\sin \left(\mathrm{\&lambda;}\frac{\mathrm{\&pi;}}{M}\right)}---\left(3\right)$

In formula (2), can prove to only have to work as λ _kfor integer and f _s> 2f _ktime, the amplitude A of signal just accurately may be obtained by formula (2) _k, frequency f _k, phase place

Step 3: according to above-mentioned requirements, select 3 rank to decay the soonest window Nuttall window, this window is expressed as formula (4):

$w\left(m\right)=0.375-0.5\cos \left(2\mathrm{\&pi;}\frac{m}{M}\right)+0.125\cos \left(4\mathrm{\&pi;}\frac{m}{M}\right)---\left(4\right)$

3 rank Nuttall windows are a kind of 3 rank Cosine Window, and its discrete Fourier transformation is formula (5):

W _N(λ)＝0.375W(λ)-0.25[W(λ-1)+W(λ+1)]+0.0625[W(λ-2)+W(λ+2)] (5)

As M>>1, there is formula (6): $W_N\left(\mathrm{\&lambda;}\right)\&cong;\frac{M\sin \left(\mathrm{\&pi;\&lambda;}\right)}{16\mathrm{\&pi;\&lambda;}}{e}^{-\mathrm{j\&pi;\&lambda;}\frac{M}{M-1}}\frac{4!}{\underset{h=1}{\overset{2}{\mathrm{\&Pi;}}}(h^2-{\mathrm{\&lambda;}}^2)}---\left(6\right)$

Although the side lobe peak of 3 rank Nuttall windows is-46.7db, not little, fast owing to decaying, reach 30db/octave, after 8 secondary lobes, secondary lobe is just lower than the secondary lobe of 4 rank Blackman-Harris windows; As long as therefore frequency resolution enough little, just can eliminate the interference between each harmonic wave;

Step 4: the discrete Fourier transformation after signal x (m) windowing is formula (7):

In order to eliminate short distance spectral leakage and reduce long-range spectral leakage further, adopt 5 point weight interpolation algorithms, make formula (8): λ _k=l _k+ δ _k(8);

Wherein l _kclosest to λ _kinteger ,-0.5≤δ _k< 0.5, definition α _kfor:

${\mathrm{\&alpha;}}_k=\frac{\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|X_w(l_k-i)\right|}{\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|X_w(l_k+i)\right|}---\left(9\right)$

Now can think that long-range spectral leakage is suppressed, therefore

Step 5: formula (9) and formula (10), can be changed into formula (11):

${\mathrm{\&alpha;}}_k=\frac{\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|W_N(i+{\mathrm{\&delta;}}_k)\right|}{\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{C}_2^{2-i}\left|W_N(i-{\mathrm{\&delta;}}_k)\right|}---\left(11\right)$

Linear equation in two unknowns formula (12) can be obtained by formula (6) and formula (11):

$(1-{\mathrm{\&alpha;}}_k){\mathrm{\&delta;}}_k^2-7(1+{\mathrm{\&alpha;}}_k){\mathrm{\&delta;}}_k+12(1-{\mathrm{\&alpha;}}_k)=0---\left(12\right)$

Work as α _kclosely 1 time, δ _kclose to 0, therefore can ignore formula (12) becomes formula (13):

-7(1+α _k)δ _k+12(1-α _k)＝0,|1-α _k|≈0 (13)

Step 6: can δ be solved according to (12), (13) _k, can A be drawn by formula (14) _k:

$A_k\&cong;2\frac{\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{C}_4^{2-i}\left[\right|X_w(l_k-i)|+|X_w(l_k+i)\left|\right]+C_4^2\left|X_w\left(l_k\right)\right|}{\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{C}_4^{2-i}\left[\right|W_N(i-{\mathrm{\&delta;}}_k)|+|W_N(i+{\mathrm{\&delta;}}_k)\left|\right]+C_4^2\left|W_N\left({\mathrm{\&delta;}}_k\right)\right|}---\left(14\right)$

Because:

Phase place can be obtained by formula (15) finally, the amplitude A of measured signal is just obtained _k, frequency f _k, phase place complete the analysis to harmonic wave and the quality of power supply.