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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 128368 11 pageshttpdxdoiorg1011552013128368

Research ArticlePractical Aspects of Broken Rotor Bars Detection in PWMVoltage-Source-Inverter-Fed Squirrel-Cage Induction Motors

Hong-yu Zhu1234 Jing-tao Hu123 Lei Gao123 and Hao Huang13

1 Shenyang Institute of Automation Chinese Academy of Sciences Shenyang 110016 China2University of Chinese Academy of Sciences Beijing 100049 China3 Key laboratory of Industrial Information Technology Chinese Academy of Sciences Shenyang 110016 China4 School of Electronic and Information Engineering University of Science and Technology Liaoning Anshan 114044 China

Correspondence should be addressed to Hong-yu Zhu zhuhongyusiacn

Received 5 June 2013 Revised 18 September 2013 Accepted 23 October 2013

Academic Editor Xianxia Zhang

Copyright copy 2013 Hong-yu Zhu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Broken rotor bars fault detection in inverter-fed squirrel cage induction motors is still as difficult as the dynamics introducedby the control system or the dynamically changing excitation (stator) frequency This paper introduces a novel fault diagnosistechniques using motor current signature analysis (MCSA) to solve the problems Switching function concept and frequencymodulation theory are firstly used to model fault current signal The competency of the amplitude of the sideband componentsat frequencies (1 plusmn 2119904)119891

119904as indices for broken bars recognition is subsequently studied in the controlled motor via open-

loop constant voltagefrequency control method The proposed techniques are composed of five modules of anti-aliasing signalacquisition optimal-slip-estimation based on torque-speed characteristic curve of squirrel cage motor with different load typesfault characteristic frequency determination nonparametric spectrum estimation and fault identification for achieving MCSAefficiently Experimental and simulation results obtained on 3 kW three-phase squirrel-cage induction motors show that the modeland the proposed techniques are effective and accurate

1 Introduction

Squirrel-cage induction motors have dominated the field ofelectromechanical energy conversion They consume morethan 60 of the electrical energy produced and are presentin the main industrial applications [1 2] Although stillconsidered proverbially robust all components of inductionmachines are subject to increased stress particularly whenoperated in a controlled mode and with repeated load cycles[3] According to studies broken rotor bars and crackedend-ring faults in the rotor cage are responsible for about5ndash10 of all breakdowns and incipient detection of theseevents remains a key issue [4 5] The main reason why earlydetection is important is that although broken rotor bars maynot cause immediate failure there can be serious secondaryeffects associated with their presence [6 7] If faults areundetected theymay lead to potentially catastrophic failuresThus health-monitoring techniques to prevent the inductionmotor failures are of great concern in the industry and aregaining an increasing attention

Motor current signature analysis (MCSA) has beensuccessfully applied to detect broken-rotor bar faults byinvestigating the sideband components around the suppliedcurrent fundamental frequency (ie the line frequency) 119891

119891119887= (1 plusmn 2119904) 119891 (1)

where 119891119887are the sideband frequencies associated with the

broken rotor bars and 119904 is the per unit motor slip [4 8 9]Some quantitative conclusion based on the left sidebandvalues (LSB) of the amplitude-frequency spectrum plot of amotor phase current even recommends minus50 dBsim minus35 dB ofthe sideband harmonics as the threshold level for generatinga warning or alarm [10] In addition MCSA has been testedinmany industrial cases since the 1980s with good results [11ndash16]

Even though numerous successful main-fed motor faultdetectionmethods are reported in the literature bibliographyrelative to inverter-fed motors on this topic seems to be poorWith the increased emphasis on energy conservation and

2 Journal of Applied Mathematics

lower cost induction machines are supplied and controlledby inverters and the use of inverter drives increases dayby day As a result dynamic performance is excellent dueto all kinds of control methods and the next steps in drivedevelopment are going to be driven by attempts to increasereliability and reducemaintenance costs By using the currentsensor feedback and microprocessing unit the new tread forlow-cost protection applications seems to be drive-integratedfault diagnosis systems without using any external hardwareThus it becomes more demanding to detect faults by usingMCSA in these drives

The introduction of voltage-source-inverter-fed (VSI-fed) motors has produced significant changes in the fieldof diagnosis and control needing further research in orderto overcome various challenges Contrary to the motor linecurrent taken directly from the main the inverter-fed motorline current includes remarkable noise (inherent floor noisewhich reduces the possibility of true fault pattern recognitionusing line current spectrum) due to the high-frequencyswitching devices EMI and EMC effects and so forth Thecurrent signal for rotor fault diagnosis needs precise and highresolution information which means the diagnosis systemdemands additional hardware such as a low-pass filter high-precision AD converter and additional software Moreoverclosed-loop control in IM drives introduces new difficultiesin broken rotor bar detection as traditional fault indicatorstend to be masked by a control algorithm Therefore themethods developed for an open-loop operation are not ableto produce adequate and reliable information on the extent ofthe fault and have to be adapted [4 17 18] All these influencescomplicate the utilization of frequency analysis methodsVSI-fed motor faults have been analyzed and the initialresults are given in the literature but further investigation isstill required

The classic MCSA method works well under constantload torque and with high-power motors and it has mainlybeen designed for fixed frequency supply such as formachines connected to the electrical grid To obtain satis-factory test results in [7] recommendations are given fromthe authorrsquos experience As stated in this paper literally ldquotheload on the motor should be sufficiently high to produce atleast 35 of rated full load rotor slip for a reliable single-testbroken bar analysisrdquo and ldquoif themotor to be tested is fed froma variable-speed converter drive the frequency of the driveshould be locked at the 5060 Hz power supply frequency forthe testrdquo Yet difficulties emergewhen inverted-fedmotors areapplied to drive fans pumps or other mechanisms involvingspeed control for energy-saving purpose In these casesthe excitation frequency will truly depend on the speedreference and the load applied to the systemTherefore unlikethe utility-driven case the stator excitation frequency willdynamically change and the position of the current harmon-ics appearing on the stator-current spectrum due to electricalfaults is highly dependent on the mechanical motor load andexcitation frequency which affects the slip frequency As aconsequence the conventional MCSA must be amended toaccommodate the new scenarios Unfortunately to the bestof our knowledge in the published literatures there is noresearch work on this subject

id

Sia Sib SicSua Sub Suc

ia998400

ib998400

ic998400

ua998400

ub998400

uc998400

uaub uc

ud Z

Z

Ziaibic

Msim

+

minus

Figure 1 Schematic diagram of the PWMVSI-fed adjustable speeddrive

In this context following explicitly derivation from asimplified fault signal model a new online fault diagnosistechnique based on MCSA for inverter-fed squirrel-cageinduction motors is present Compared with traditionalMCSA the novelty of the proposed method is that brokenrotor bars fault in the controlled motor via open-loop con-stant voltagefrequency control method could be identifiedeven if the motor operates at different excitation frequenciesTo do this oversampling signal acquisition technique is usedto suppress significantly noise contained in the inverted-fed motor line current and fault-indicative frequencies withvariable excitation frequency are determined by torque-speedcharacteristic curve of squirrel-cage motor with differentload types To obtain satisfactory results nonparametricspectrum estimation algorithm and fault identification aresubsequently presented Including this introductory sectionthis paper is organized into six sections Section 2 presentsa theoretical analysis model of stator current of inverter-fedsquirrel-cage motor which is based on switching functionconcept and modulation theory A detailed description ofthe harmonic components contained in current is givenSection 3 elaborates the new broken rotor bars fault diagnosistechniques In Section 5 the proposed techniques are vali-dated by laboratory tests the method is applied to differentstator currents obtained from healthy and faulty machinesExperimental and simulation results as well as the corre-sponding analysis and discussion are presented in Section 4Finally conclusions and recommendations are presented inthe last section

2 Analytic Model of Stator Current Signaturefor Squirrel-Cage Induction Motor withConstant Volt-per-Hertz Control Technique

21 Analytic Model of Stator Current Signature for No-FaultSquirrel-Cage Induction Motor Switching function conceptis a powerful tool in understanding and optimizing theperformance of the converter [19 20] In [20] an analyticalapproach for characterizing the current harmonics and inter-harmonics of theVSI-fedASD injected into the supply systemin steady is presented using the switching function conceptapplicable to PWM VSI that is studied in this paper Figure 1shows the schematic diagram of a typical PWM VSI-fedadjustable speed drive where 119878

119894119886 119878119894119887 and 119878

119894119888represent the

rectifier converter current switching functions that correlate

Journal of Applied Mathematics 3

the three-phase AC source currents 1198941015840

119886 1198941015840119887 and 119894

1015840

119888and the

inverter input DC current 119894119889 119878119906119886 119878119906119887 and 119878


inverter voltage switching functions that correlate inverter dcinput voltage and output phase voltages 119906

119886 119906119887and 119906119888119885 is the

impedance operator seen from the inverter output terminalscorresponding to the neutral point of induction motorIn order to calculate and analyse the harmonic currentsgenerated by the VSI-fed ASD an analytical model basedon the modulation theory and switching function concept isproposed and expressed by (2)sim(6) After obtaining the phasecurrents of PWM VSI-fed motor the harmonic componentsof the current might be extracted by the use of a certainspectrum analysis method as follows

(1198941015840

119886 1198941015840

119887 1198941015840

119888) = 119894119889(119878119894119886 119878119894119887 119878119894119888) (2)

119894119889= (119894119886 119894119887 119894119888) (119878119906119886 119878119906119887 119878119906119888)1015840

(3)

119906119889= (1199061015840

119886 1199061015840

119887 1199061015840

119888) (119878119894119886 119878119894119887 119878119894119888)1015840

(4)

(119906119886 119906119887 119906119888) = 119906119889(119878119906119886 119878119906119887 119878119906119888) (5)

(119894119886 119894119887 119894119888) =

(119906119886 119906119887 119906119888)

119885 (6)

As shown in Figure 1 for the two-level natural sampledPWMwith a triangular wave of carrier signal the three-phasevoltage switching functions 119878

119906119886 119878119906119887and 119878119906119888can be expressed

as the following double Fourier series as follows

119878119906119886

(119905) =119872

2sin (120596

119904119905) +

2

120587

infin

sum

119898=1

1198690(119898119872120587

2)

sdot sin(119898120587

2) sdot sin (119898120596

119888119905)

+2

120587

infin

sum

119898=1

infin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sin [

(119898 + 119899) 120587

2]

times sin (119898120596119888119905 + 119899120596

119904119905)

119878119906119887

(119905) =119872

2sin(120596

119904119905 minus

2120587

3) +

2

120587

infin

sum

119898=1

1198690(119898119872120587

2)

sdot sin(119898120587

2) sdot sin (119898120596

119888119905)

+2

120587

infin

sum

119898=1

infin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sin [

(119898 + 119899) 120587

2]

times sin [119898120596119888119905 + 119899 (120596

119904119905 minus

2120587

3)]

119878119906119888

(119905) =119872

2sin(120596

119904119905 +

2120587

3) +

2

120587

infin

sum

119898=1

1198690(119898119872120587

2)

sdot sin(119898120587

2) sdot sin (119898120596

119888119905)

+2

120587

infin

sum

119898=1

infin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sin [

(119898 + 119899) 120587

2]

times sin [119898120596119888119905 + 119899 (120596

119904119905 +

2120587

3)]

(7)

For a hypothetical ideal motor supplied from a balancedthree-phase source of sinusoidal voltages and driving aconstant load the following waveform of selected phase-astator current may be assumed by substituting (7) into (5)and (6) where the commutation effect is ignored and properorigin of coordination selected for convenience of analysis isas follows

119894119886=

119872119864119889

21003816100381610038161003816119885 (120596119904)1003816100381610038161003816

cos [120596119904119905 minus 120593 (119885 (120596

119904))]

+2119864119889

1205871003816100381610038161003816119885 (119898119873120596

119904)1003816100381610038161003816

infin

sum

119898=1

1198690(119898119872120587

2) sin(

119898120587

2)

times cos [119898119873120596119904119905 minus 120593 (119885 (119898119873120596

119904))]

+2119864119889

1205871003816100381610038161003816119885 ((119898119873 + 119899) 120596

119904)1003816100381610038161003816

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898

sin [(119898 + 119899) 120587

2] sdot cos [ (119898119873 + 119899) 120596

119904119905

minus120593 (119885 ((119898119873 + 119899) 120596119904))]

= 1198681cos (120596

119904119905 minus 1205931) + 1198682

infin

sum

119898=1

1198690(119898119872120587

2) sin(

119898120587

2)

times cos (119898119873120596119904119905 minus 1205932)

+ 1198683

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sdot sin [

(119898 + 119899) 120587

2]

times cos [(119898119873 + 119899) 120596119904119905 minus 1205933]

(8)

where119872 is amplitude modulation index119872 = 119880119904119880119888le 1 120596

119904

is frequency of the modulation waveform (reference) (rads)120596119904

= 2120587119891119904 120596119888is frequency of the carrier signal (rads) 119873

ratio of the carrier frequency to the modulation frequency119873 = 120596

119888120596119904ge 1 and 119869

0 119869119899are Bessel functions of the first

kind with order 0 and order 119899 respectively

22 Analytic Model of Stator Current Signature for FaultySquirrel-Cage Induction Motor When broken rotor bar orcracked end-ring faults develop in the rotor cage the currenttorque and speed of the motor are affected typically in aperiodic manner In the case of periodic disturbances allthree line currents 119894

119886 119894119887 and 119894

119888are simultaneously modulated

with the fundamental frequency 119891119900of the fault-induced

oscillation of motor variables If only amplitude modulationand fundamental frequency 119891

119900are considered current in

phase-a of the supply line may now be expressed as

119894119891= 119894119886[1 + 119886 cos (120596

119900119905)] (9)


where 119886 denotes the modulation depth (modulation index)and 120596

119900= 2120587119891

119900= 2119904120596

119904= 4120587119904119891

119904 The value of the modulation

index depends on the severity of the abnormality and motorloads

Substituting (8) in (9) yields

119894119891= 1198681cos (2120587119891

119904119905 minus 1205931) + 1198682

infin

sum

119898=1

1198690(119898119872120587

2) sin(

119898120587

2)

times cos (2120587119898119873119891119904119905 minus 1205932)

+ 1198683

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sdot sin [

(119898 + 119899) 120587

2]

times cos [2120587 (119898119873 + 119899) 119891119904119905 minus 1205933]

+ 1198861198681cos [2120587 (1 + 2119904) 119891

119904119905 minus 1205931]

+ cos [2120587 (1 minus 2119904) 119891119904119905 minus 1205931]

+ 1198861198682

infin

sum

119898=1

1198690(119898119872120587

2) sdot sin(

119898120587

2)

sdot cos [2120587 (119898119873 + 2119904) 119891119904119905 minus 1205932]

+ cos [2120587 (119898119873 minus 2119904) 119891119904119905 minus 1205933]

+ 1198861198683

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sdot sin [

(119898 + 119899) 120587

2]

sdot cos [2120587 (119898119873 + 119899 + 2119904) 119891119904119905 minus 1205933]

+ cos [2120587 (119898119873 + 119899 minus 2119904) 119891119904119905 minus 1205933]

(10)

indicating that the spectrum of stator current for inverter-fedhealthy squirrel-cage motor contains apart from the funda-mental 119891

119904equal to the inverter excitation (stator) frequency

119898119873119891119904harmonics at the carrier frequency andmultiples of the

carrier frequency and (119872119898+119899)119891119904harmonics in the sidebands

around each multiple of the carrier frequency When a barbreaks a rotor asymmetry occursThe result is that it inducesin the stator current additional frequency at 119891

119887= (1 plusmn 2119904)119891

119904

1198911198871

= (119898119873 plusmn 2119904)119891119904 and 119891

1198872= (119898119873 + 119899 plusmn 2119904)119891

119904around

harmonics frequency depicted aboveThe amplitude of fault-indicative frequencies 119891

119887= (1 plusmn 2119904)119891

119904 1198911198871

= (119898119873 plusmn 2119904)119891119904

and 1198911198872

= (119898119873 + 119899 plusmn 2119904)119891119904depends on faulty severity loads

and excitation (stator) frequency and the distance between119891119887 1198911198871 and 119891

1198872and corresponding harmonics frequency 119891

119904

119898119873119891119904 (119872119898 + 119899)119891119904is equal to 2119904119891

119904 The amplitude of 119891

1198871

and 1198911198872can be considered negligible compared to that of119891

119887

as a result 119891119887

= (1 plusmn 2119904)119891119904are adopted as fault-indicative

frequenciesIn practice the current is modulated with respect to not

only the amplitude but also the phase and the modulationprocess is more complex than that described by (9) butthe derived equation (10) provides an adequate basis forqualitative assessment of diagnostic media

3 Diagnosis Techniques of BrokenRotor Bars for Squirrel-Cage InductionMotor with Constant VoltageFrequencyControl Method

31 Principles of Diagnosis A motor diagnosis techniquewhich contains the five processing modules illustrated inFigure 2 is presented based onmodel of squirrel-cage induc-tion motor stator current signature depicted in Section 2The current flowing in single phase of the induction motoris sensed by anti-aliasing signal acquisition module andsent to spectrum estimation module where the obtainedtime-domain signal can be decomposed into components ofdifferent frequency using Welchrsquos periodogram method Inoptimal-slip-estimation module based on the real speed theinverter excitation frequency and torque-speed characteristiccurve of squirrel-cage motor with different load types themotor slip is calculated and the consequent optimal slip esti-mation value is transmitted to fault characteristic frequencydetermination module where characteristic frequencies ofbroken rotor bars are calculated Depending on whether thecharacteristic frequencies of 119891

119887= (1 plusmn 2119904)119891

119904could be found

in power spectrum obtained in spectrum estimationmodulethe conclusion of failure or not could be drawn this work isdone in fault identification module

32 Method of Antialiasing Signal Acquisition When carry-ing out diagnostic analysis one of the key elements to obtaingood results is to properly choose acquisition parametersthe sampling frequency and the number of samples Thereare two different constraints analysis signal bandwidth andfrequency resolution for the spectrum analysis

It must be considered when capturing the stator currentsignal that the sampling frequency119891sample plays an importantrole Generally statistical bands of fault-indicative frequen-cies can be ascertained from theory analysis andmanyMCSAexperiments [2 8 21ndash23] Table 1 summarizes the range ofsideband frequencies in terms of faulty types We see inTable 1 that the spectral analysis of stator currents mightbe done in low frequency range (typically between 0 and2 kHz) to focus on the significant phenomena [24] andin the case of broken rotor bars fault the fault-indicativefrequencies are under 400Hz Thus taking into account theNyquist criterion a very high sampling frequency value is notmandatory in case that themotors are supplied by the ac gridSampling frequencies of 2k or 5k sampless (standard in dataacquisition devices) enable good resolution analysis

In contrast for the inverter-fed motors sampling pro-cedure is more demanding As it is known stator currentcontains high-order frequency harmonics in this case dueto switching frequencies in modern inverter typically above10 kHz Thus aliasing may occur due to wrong samplingAntialiasing techniques have to be used in order to have acorrect current spectrum and prevent a false failure alert

For decreasing alias to an acceptable level it is common todesign a sharp-cutoff low-pass anti-aliasing filter and samplethe signal at a frequency lower than the dominant frequencycomponents such as the fundamental and the switching


3 phase AC supply

SpeedRe

fere

nce f

requ

ency

Optimal slip estimation

Signal acquisitionDown-sampling

AD over-sampling

Anti-aliasing filter

Spectrum estimation

Fault identification

Characteristic frequency determination

Msim

nfs

fb = (1 plusmn 2s)fs

(f P(f)) = (f 1K

Kminus1sumK=0

Pk(f))sim

sfans pump (1 minus sfans pump )2 = [se(1 minus se)2](fsf1)

sconstant = se(f1fs)

f = fb faulty f ne fb healthy

Figure 2 Schematic diagram of fault diagnosis

Table 1 Range of sideband frequencies in terms of faulty types

Fault types Theoretical formula Range Frequencies

Air-gap eccentricity 119891ecc = 119891119904[1 plusmn 119898(

1 minus 119904

119901)] Low high 0sim900Hz

Broken rotor bars 119891119887= 119891119904[119896(

1 minus 119904

119901) plusmn 119904] Low 0sim400Hz

Bearing failure 119891119894119900

=119899119887

2119891119903[1 plusmn

119887119889

119901119889

cos120573] High 0sim2000Hz

Interturn short circuit 119891stl = 119891119904[119898

119901(1 minus 119904) plusmn 119896] 119891sth = 119891

119904[1 plusmn 119898119885

2(1 minus 119904


frequencies [25] However such sharp-cutoff analog filtersare difficult and expensive to implement and if the systemis to operate with a variable sampling rate adjustable filterswould be required Furthermore sharp-cutoff analog filtersgenerally have a highly nonlinear phase response particularlyat the pass-band edge In our proposedmethod oversamplingas an alternative technique is used The principle of over-sampling is briefly reviewed as follows (see [26] for details) Avery simple anti-aliasing filter that has a gradual cutoff withsignificant attenuation is firstly applied to prefilter the motorstator phase-a current Next implement the AD conversionat a higher sampling rate After that downsampling theobtained signal with a lower sampling frequency is imple-mented using a digital low-pass filter

As for the inverter supply several harmonics could bemixed up in case that low resolution of the band side waschosen In general one can take the following steps to selectdata acquisition parameters in order to achieve the correctresolution needed

(1) Definition of sampling frequency For anti-aliasingpurpose 119891sample has limitation as 119891sample ⩾ 2119891

119888 where

119891119888is the highest fault-indicative frequency

(2) Selection of required frequency resolution Δ119891(3) Specification of the number of samples 119873 =

119891sampleΔ119891(4) Determination of sampling time 119879 = 119873119891sample =

119873119879119904

33 Calculation of Slip and Fault-Indicative Frequencies withVariable Excitation Frequency From Section 2 broken rotorbars can be detected bymonitoring the stator current spectralcomponents 119891

119887= (1 plusmn 2119904)119891

119904 where harmonic frequencies119891

119887

are a function of slip 119904 and excitation frequency 119891119904 In

case of broken rotor bars fault excitation frequency 119891119904

and the corresponding slip 119904 must be firstly determinedin order to find harmonic frequencies119891

119887 The result of a

motor diagnosis using MCSA is incorrect if the detectedslip has an error [8 27] One of the most popular waysto obtain the information of the slip frequency is to usespeed sensor In our proposed method slip 119904 is calculated inoptimal-slip-estimation module based on the real measuredspeed and torque-speed characteristic curve of squirrel-cagemotor

The torque-speed characteristic curve of different loadtypes is shown in Figure 3 Curves (1) (2) and (3) are inherentcharacteristic curves of squirrel-cage motor correspondingto different excitation frequency 119891

119904 For ease of analysis

it is assumed that torque characteristic curves (1) (2) and(3) intersect mechanical characteristic curve119891

1at nominal

operating point 119860 and only excitation frequency 119891119904lt 1198911is

considered (if 119891119904gt 1198911 motor operates at point119860

3of curve

(3)) In Figure 3 the value of slip 119904 corresponding to point119860 is 119904119890= (11989901

minus 119899119890)11989901 The no-load speedes corresponding

to different excitation frequency can be expressed as 11989901

=

601198911119901 and 119899

02= 60119891

119904119901 where 119901 denotes the number of

pole-pairs


120572 = 0 constant torch load120572 = 1 load proportional to speed

120572 = 2 fanspump loadA3

A2

A

A1

0 T Te TL

nn03n3

n02n2n1

n01ne (3) fs gt f1

(1) f1

(2) fs lt f1

Figure 3 Torque-speed characteristic curve of squirrel-cage motorwith different load types

If motor works with constant torque load the operatepoint shifts from 119860 to 119860

1 the slip 119904 corresponding to 119860

1is

equal to 119904constant = (11989902

minus 1198991)11989902 Considering the congruent

relationship between Δ11989901119899119890119860 and Δ119899

0211989911198601 we can deduce

the specific slip formula as follows 119904constant = 119904119890(1198911119891119904)

If motor works with fanspump load the operating pointshifts from119860 to119860

2and the slip 119904 corresponding to119860

2is equal

to 119904fanspump = (11989902

minus 1198992)11989902 Considering the similarity of

Δ11989901119899119890119860 to Δ119899

0211989911198601and relationship of 119879119879

119890= (1198992119899119890)2

between speed and torque of fanspump we can deduce thespecific slip formula as follows 119904fanspump(1 minus 119904fanspump)

2

=

[119904119890(1 minus 119904

119890)2

](1198911199041198911)

34 Nonparametric Spectrum Estimation Algorithms MCSAtechniques include parametric and nonparametric spectrumanalysis of the motor current in general [28] Among thenonparametric algorithms we useWelchrsquos periodogram algo-rithms based on DFT to compute the power spectrum of thephase-a motor current data

Let 119894119873[119899] = 119894[0] 119894[1] 119894[119873 minus 1] be a discrete time

signal which is obtained by sampling the phase-a motorcurrent signal 119894(119905) for a duration of sampling time 119879 Toreduce the variance of power spectrum estimate the 119873-point data sequence 119894

119873[119899] = 119894[0] 119894[1] 119894[119873 minus 1] is

first partitioned into 119870 overlapping segments The length ofeach segment consists of 119871samples and these segments can beoverlapping on each other with (119871 minus 119878) overlapping sampleswhere 119878 is the number of points to shift between segmentsThereafter the periodogram of each segment is calculatedand the obtained periodograms are then averaged to give thepower spectrum estimate

The length of segment 119871 is dependent on the requiredresolution In order to increase the quality of power spectrumestimates the signal segments can be windowed before calcu-lating FFTTheproposedmethods permit reduce the varianceof the estimate at the expense of a decreased frequencyresolution However it is difficult to trade off between thefrequency resolution and the estimate variance It has beennoted that the use of 50 overlapping percentage among thepartitioned segments leads to efficient implementation of the

fast Fourier transform (FFT) algorithm and in this case therelationship between 119870 and 119871 of segment as follows 119870 =

(119873 minus 1198712)(1198712)

Algorithm 1

Step 1 Subdividing 119873-point sampled data sequence 119894119873[119899] =

119894[0] 119894[1] 119894[119873 minus 1] into 119870 overlapping segments the 119896thsegment data 119909[119899 + 119896119878] 0 ⩽ 119899 ⩽ 119871 minus 1 and 0 ⩽ 119896 ⩽ 119870 minus 1 isas follows

Segment 1 119909[0] 119909[1] 119909[119871 minus 1]Segment 1 119909[119878] 119909[119878 + 1] 119909[119871 + 119878 minus 1]Segment119870 119909[119873 minus 119871] 119909[119873 minus 119871 + 1] 119909[119873 minus 1]

Step 2 Weighting 119896th segment 119889[119899] denote rectangularwindow function

119909119896

[119899] = 119889 [119899] 119909 [119899 + 119896119878] 0 ⩽ 119899 ⩽ 119871 minus 1 0 ⩽ 119896 ⩽ 119870 minus 1

(11)

Step 3 Calculating power spectrum of the 119896th segment data

119875119896

(119891) =1

ULT119883(119891) [119883 (119891)]

lowast

=1

ULT1003816100381610038161003816119883 (119891)

1003816100381610038161003816

2

(12)

where 119883(119891) = 119879sum119871minus1

119899=0119909(119896)

[119899]119890minus1198952120587119891119899119879denote DFT of the 119896th

segment data and 119880 = 119879sum119871minus1

119899=01198892

[119899] denote normalizationfactor

Step 4 Averaging all segments power spectrum

(119891) =1

119870

119870minus1

sum

119896=0

119875119896

(119891) (13)

4 Experiment Setup and SignalAcquisition Methods

41 Experiment Setup Schematic diagram of the experimentsetup is shown in Figure 4This system can be used to sampleline current 119894

119886and line voltage 119906

119886119887(if necessary it can be

arranged to sample the other line currents 119894119887and 119894119888and

line voltages 119906119887119888and 119906

119888119886 too) and speed signals The main

components of the experiment setup are as follows

(1) Three-phase 3 kW SCIM (see Table 2 for details)(2) Digital stroboscope coupled with the shaft of the

SCIM as angular-speed sensor to measure and recordthe time variation of the speed

(3) DC generator coupled with the SCIM to provide itsadjustable load

(4) Mechanical coupling between SCIM and dc genera-tor

(5) Variable resistor bank as a variable load of the gen-erator the load of the generator and consequentlythe induction motor can be adjusted by varying this


DCgenerator

Conditioning circurt

Excitation circurt

Resistor bank

Squirrel cage motor

Dataacquisition card

Inverter PC

sim3-phase source

Figure 4 Schematic diagram of the experiment

resistance andor regulating the excitation current ofthe generator by relevant variable resistor The resis-tance of this variable-resistor bank can be selectedstep by step by a selector on the bank In the operatingmotor a suitable position of the selector is selectedand consequently the induction motor is loadedBy regulating the output voltage of the generatorinserted in the excitation current path the load levelis regulated precisely

(6) Induction-motor drive system type Simens440 withrating values in accordancewith that of the SCIM thisdrive has been mainly designed and built for open-loop scalar controller in constant voltagefrequency(CV119891) method

(7) Three-phase change-over switch to exchange themotor connections from the mains for the driveoutput

(8) Signal conditioning circuits since the used DAQ cardaccepts only voltage signals with maximum ampli-tudes of plusmn10V the type and amplitude of the signalsare prepared before connection to the card At the firststage TBC300LTP Hall-effect current transformer isused to prepare the current signal and isolate it fromthe power circuit Secondary side current is thenconverted to proportional voltage signals by currentshunts Then all signals are transmitted to the DAQcard using a special shielded cable

(9) PC equipped with a self-made data acquisition cardfor sampling the electrical data at a certain adjustablefrequency and storing them in the memory

Experiment setup to collect motor data and broken rotorused in the tests are illustrated in Figure 5

42 Signal Acquisition Requirements The experimentsinvolved collecting the phase-a stator current and speeddata of the induction motor for different load conditionsand different excitation frequencies of 20Hz 32Hz 40Hzand 50Hz with three broken-rotor-bar fault and withoutany fault The load conditions of the motor are 2550 75 and 94 full load respectively These load

Table 2 Induction motor technical data

Parameter Value UnitNominal power 30 kWNominal voltage 380 VNominal current 68 ANominal frequency 50 HzConnection Δ

Number of poles 4Rotor slots number 28

condition percentages are determined according to themotor nameplate information given in Table 2

Signal over-sampling method has been chosen in orderto avoid aliasing The stator current of motor was sampledwith a frequency of 2 kHz for main-fed and 40 kHz forinverter-fed case In the inverter-fed case software filtershave been implemented in order to avoid aliasing Morespecifically an 8-order anti-aliasing digital butter-worth filterwas implemented and resampling of the signal has been doneat 2 kHz In our case thirty seconds long data is acquiredfrom all twomotors for each load condition at each frequencymentioned above Thus the analyzed frequencies vary from0 to 1 kHz with a resolution of 003Hz For the featureextraction and discriminant analysis starting with the firstsample the acquired data is processed with a sliding windowsize of 30000 samples at a slide amount of 10000 resultingin 60 different data sets all together

Figure 6 illustrates the intercept parts of 50Hz statorcurrent signal from the (a) main-fed and (b) inverter-fed healthy squirrel-cage motors An expert inspection ofthese waveforms reveals that the inverter-fed motor currentwaveform is heavily contaminated by the noise-like additivewaveform due to PWM switching of the voltage sourceinverter If we were to use this motor current waveformdata to extract necessary features for fault detection andclassification we need to preprocess the data

5 Experimental Results

51 Experiment 1 A nominally healthy squirrel-cage motorwas firstly tested and the power spectrum of the statorcurrent centered on the fundamental component supplied by(a) main and (b) inverter is shown in Figure 7 The resultsconfirm that themotor rotor is healthy since the sidebands119891

119887

are not present Comparisons of Figures 7(a) and 7(b) showsthat a large amount of harmonic components is included inFigure 7(b) and the inverter supply does affect the spectrumi-dentifiability Figure 8 indicates power spectrum of the statorcurrent around the fundamental component for (a) main-fed and (b) inverter-fed fault squirrel-cage motors with threebroken bars and full load The annotations appearing in thefigure denote themain sideband components around the sup-ply frequency and corresponding amplitudes Comparison ofFigures 7 and 8 indicates that sideband components appearwhich demonstrates the broken bars occurred Althoughmore frequency information appears one could still identify


(a) (b)

Figure 5 Experiment setup to collect motor data and broken rotor used in the tests

0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(a)

0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(b)

Figure 6 50Hz stator current signal from the (a) main-fed and (b) inverter-fed healthy squirrel-cage motor

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Healthy

Frequency (Hz)

n = 1425 rmin f = 50Hz

(a)

Am

plitu

de (d

B)

Frequency (Hz)35 40 45 50 55 60 65

minus80

minus60

minus40

minus20

0Healthy

n = 1425 rmin fs = 50Hz

(b)

Figure 7 Power spectrumof the stator current around the fundamental component for (a)main-fed and (b) inverter-fed healthy squirrel-cagemotor

Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0Faulty

Frequency (Hz)

45Hz minus34dB n = 1425 rmin f = 50Hz

(a)

Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Frequency (Hz)

Faulty45Hz minus34dB

55Hz minus40dBn = 1425 rmin fs = 50Hz

(b)

Figure 8 Power spectrum of the stator current around the fundamental component for (a) main-fed and (b) inverter-fed fault squirrel-cagemotor


20 30 40 50 60 70minus80

minus60

minus40

0A

mpl

itude

(dB)

Frequency (Hz)

Faulty

41Hz minus38dB 49Hz minus39dBminus20 n = 1290 rmin fs = 45Hz

(a)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

25 30 35 40 45 50 55minus80

minus60

minus40

minus20

0

434Hz minus38dB366Hz minus375dB n = 1150 rmin fs = 40Hz

(b)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

294Hz minus40dB346Hz minus40dB

n = 921 rmin fs = 32Hz

(c)A

mpl

itude

(dB)

Frequency (Hz)

18Hz minus42dB 22Hz minus42dB

Faulty

0 10 20 30 40minus80

minus60

minus40

minus20

0


(d)

Figure 9 Power spectrum of the stator current around different excitation frequency of (a) 45Hz (b) 40Hz (c) 32Hz and (d) 20Hz

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(a)

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

337Hz minus43dB303Hz minus41dBn = 935 rmin fs = 32Hz

(b)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(c)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(d)

Figure 10 Power spectrum of the stator current around (left) 25 and (right) 50 full loads Top row 32Hz and bottom row 50Hz referencefrequency

in Figure 8(b) broken rotor bars harmonics at 45Hz and55Hz using our proposed method Comparison of Figures8(a) and 8(b) also indicates that for the same level of damageat the same load the spectrum sidebands have the sameamplitude for different supply

52 Experiment 2 The second experiment involved faultsquirrel-cage motor operating at different excitation fre-quency of inverter with three broken basr and full load

Figure 9 gives the power spectrum of the stator currentaround the excitation frequencies of 45Hz 40Hz 32Hz and20Hz As expected the sideband components119891

119887= (1plusmn2119904)119891

119904

depending on excitation frequency and load are presentComparison of Figures 9(a) 9(b) 9(c) and 9(d) indicates thatthe left sideband harmonic component (1 minus 2119904)119891

119904varies from

(41Hzminus36 dB) to (18Hzminus42 dB) and right harmonic (1 +

2119904)119891119904from (49Hzminus39 dB) to(22Hzminus42 dB) when excitation

frequency varies from45Hz to 20HzAn expert inspection of


these spectrums reveals that119891119887= (1plusmn2119904)119891

119904are always located

around the excitation frequency and the distance from theexcitation frequency is 2119904119891

119904 It also can be seen from Figure 9

that the amplitude of left sideband harmonic component (1minus2119904)119891119904decreases with the excitation frequency which is the

result that when excitation frequency decreases the outputvoltage becomes lower andmuch smaller current is drawn bymotor

53 Experiment 3 In Experiment 3 additional experimentswere performed with the faulty motor at different excitationfrequencies (32Hz 50Hz) under two different loads (2550) to assess the performance of the proposed method overthe full range of motor loads Using the proposed diagnosismethod the collected phase current data of the fault motorwith three broken bars were analyzed Figures 10(a) and 10(b)show the results with 32Hz excitation frequency at differentloads and it can be seen that the fault characteristic frequencyof broken rotor bars is exposed clearly With the increaseof load the components of characteristic frequency becomemore andmore significant such as in Figure 10(a) of 25 loadand Figure 10(b) of 50 load The similar results could bereceived for fault motor with 50Hz excitation frequency atdifferent loads see Figure 10(c) of 25 load and Figure 10(d)of 50 load

6 Conclusion

Open-loop voltage-source-inverter-fed squirrel-cage motorswith constant voltagefrequency control method are widelyused to drive fans pumps or other mechanisms involvingspeed control for energy-saving purpose In these casesthe motor operates steadily with different excitation fre-quencies Unlike the utility-driven case the position of thecurrent harmonics appearing on the stator-current spec-trum due to broken rotor bar faults is highly dependenton the mechanical motor load and excitation frequencywhich affects the slip frequency As a consequence thereliable identification and isolation of faults remains an openissue

In this paper a simplified fault current signal modelis firstly established using switching function concept andfrequency modulation theory It is demonstrated that theinverter-fed motor current is heavily contaminated due toPWM switching of the voltage source inverter Howeverthe broken rotor bars fault characteristic frequency 119891

119887=

(1 plusmn 2119904)119891119904 depending on faulty severity loads and excitation

frequency is always located around the excitation frequencyand the distance from the excitation frequency is 2119904119891

119904 Novel

broken rotor bar fault diagnosis techniques using motorcurrent signature analysis (MCSA) for open-loop voltage-source-inverter-fed squirrel-cage inductionmotors with con-stant voltagefrequency control method are subsequentlyproposed Experimental results obtained on self-made 3 kWthree-phase squirrel-cage induction motors are discussedIt is shown that experimental and simulation results areconsistent with those of the model revealed and the proposedtechniques are effective and accurate

The method described works well under constant loadtorque but some difficulties appear with regard to closed-loop control-operated machines when 119891

119904and 119904 vary almost

simultaneously and it is impossible to employ the proposedmethod to diagnose broken rotor bar fault At the momentfurther research is carried out for the features advantageslimitations and improvements of the proposed techniques

Acknowledgment

This work was supported by the Chinese academy of SciencesKey Deployment Project under Grant no KGZD-EW-302

References

[1] M E H Benbouzid and G B Kliman ldquoWhat stator currentprocessing-based technique to use for induction motor rotorfaults diagnosisrdquo IEEE Transactions on Energy Conversion vol18 no 2 pp 238ndash244 2003

[2] J Cusido L Romeral J A Ortega J A Rosero and A GEspinosa ldquoFault detection in induction machines using powerspectral density in wavelet decompositionrdquo IEEE Transactionson Industrial Electronics vol 55 no 2 pp 633ndash643 2008

[3] T M Wolbank P Nussbaumer H Chen and P E MacheinerldquoMonitoring of rotor-bar defects in inverter-fed inductionmachines at zero load and speedrdquo IEEE Transactions on Indus-trial Electronics vol 58 no 5 pp 1468ndash1478 2011

[4] A Bellini F Filippetti C Tassoni and G A CapolinoldquoAdvances in diagnostic techniques for induction machinesrdquoIEEE Transactions on Industrial Electronics vol 55 no 12 pp4109ndash4126 2008

[5] A H Bonnett and C Yung ldquoIncreased efficiency versusincreased reliabilityrdquo IEEE Industry Applications Magazine vol14 no 1 pp 29ndash36 2008

[6] J Cusido L Romeral J A Ortega A Garcia and J R RibaldquoWavelet and PDD as fault detection techniquesrdquo Electric PowerSystems Research vol 80 no 8 pp 915ndash924 2010

[7] I Culbert and W Rhodes ldquoUsing current signature anal-ysis technology to reliably detect cage winding defects insquirrel cage induction motorsrdquo in Proceedings of the 52ndAnnual Petroleum and Chemical Industry Conference pp 95ndash101 September 2005

[8] J H Jung J J Lee and B H Kwon ldquoOnline diagnosis ofinduction motors using MCSArdquo IEEE Transactions on Indus-trial Electronics vol 53 no 6 pp 1842ndash1852 2006

[9] S H Kia H Henao andG A Capolino ldquoA high-resolution fre-quency estimation method for three-phase induction machinefault detectionrdquo IEEE Transactions on Industrial Electronics vol54 no 4 pp 2305ndash2314 2007

[10] D Matic F Kulic M Pineda-Sanchez and I KamenkoldquoSupport vector machine classifier for diagnosis in electricalmachines application to broken barrdquo Expert Systems withApplications vol 39 no 10 pp 8681ndash8689 2012

[11] M E H Benbouzid ldquoA review of induction motors signatureanalysis as a medium for faults detectionrdquo IEEE Transactions onIndustrial Electronics vol 47 no 5 pp 984ndash993 2000

[12] M F Cabanas F Pedrayes M G Melero et al ldquoUnambiguousdetection of broken bars in asynchronous motors by means ofa flux measurement-based procedurerdquo IEEE Transactions onInstrumentation and Measurement vol 60 no 3 pp 891ndash8992011


[13] R A Gupta A K Wadhwani and S R Kapoor ldquoEarly esti-mation of faults in induction motors using symbolic dynamic-based analysis of stator current samplesrdquo IEEE Transactions onEnergy Conversion vol 26 no 1 pp 102ndash114 2011

[14] W T Thomson and M Fenger ldquoCase histories of currentsignature analysis to detect faults in induction motor drivesrdquoin Proceedings of the IEEE International Electric Machines andDrives Conference (IEMDC rsquo03) 2003

[15] S Choi B Akin M M Rahimian and H A ToliyatldquoImplementation of a fault-diagnosis algorithm for inductionmachines based on advanced digital-signal-processing tech-niquesrdquo IEEE Transactions on Industrial Electronics vol 58 no3 pp 937ndash948 2011

[16] B Akin S B Ozturk H A Toliyat and M Rayner ldquoDSP-based sensorless electric motor fault diagnosis tools for electricand hybrid electric vehicle powertrain applicationsrdquo IEEETransactions on Vehicular Technology vol 58 no 5 pp 2150ndash2159 2009

[17] A Bellini ldquoClosed-loop control impact on the diagnosis ofinduction motors faultsrdquo IEEE Transactions on Industry Appli-cations vol 36 no 5 pp 1318ndash1329 2000

[18] S M A Cruz and A J M Cardoso ldquoDiagnosis of rotor faults inclosed-loop induction motor drivesrdquo in Proceedings of the IEEE41st IAS Annual Meeting on Industry Applications pp 2346ndash2353 October 2006

[19] B K Lee and M Ehsani ldquoA simplified functional simulationmodel for three-phase voltage-source inverter using switchingfunction conceptrdquo IEEE Transactions on Industrial Electronicsvol 48 no 2 pp 309ndash321 2001

[20] G W Chang and S K Chen ldquoAn analytical approach forcharacterizing harmonic and interharmonic currents generatedbyVSI-fed adjustable speed drivesrdquo IEEE Transactions on PowerDelivery vol 20 no 4 pp 2585ndash2593 2005

[21] S Nandi H A Toliyat and X D Li ldquoCondition monitoringand fault diagnosis of electrical motorsmdasha reviewrdquo IEEE Trans-actions on Energy Conversion vol 20 no 4 pp 719ndash729 2005

[22] M Eltabach and A Charara ldquoComparative investigation ofelectric signal analyses methods for mechanical fault detectionin induction motorsrdquo Electric Power Components and Systemsvol 35 no 10 pp 1161ndash1180 2007

[23] M Eltabach J Antoni and M Najjar ldquoQuantitative analysis ofnoninvasive diagnostic procedures for induction motor drivesrdquoMechanical Systems and Signal Processing vol 21 no 7 pp2838ndash2856 2007

[24] A Yazidi H Henao G A Capolino M Artioli and FFilippetti ldquoImprovement of frequency resolution for three-phase induction machine fault diagnosisrdquo in Proceedings of theIEEE 40th IAS Annual Meeting on Industry Applications vol 1pp 20ndash25 October 2005

[25] I P Georgakopoulos E D Mitronikas and A N SafacasldquoDetection of induction motor faults in inverter drives usinginverter input current analysisrdquo IEEE Transactions on IndustrialElectronics vol 58 no 9 pp 4365ndash4373 2011

[26] A V Oppenheim R W Schafer and J R Buck Discrete-TimeSignal Processing Tsinghua University Press Beijing China2nd edition 2005

[27] J Faiz and B M Ebrahimi ldquoA new pattern for detectingbroken rotor bars in induction motors during start-uprdquo IEEETransactions on Magnetics vol 44 no 12 pp 4673ndash4683 2008

[28] B Ayhan M Y Chow and M H Song ldquoMultiple discriminantanalysis and neural-network-based monolith and partitionfault-detection schemes for broken rotor bar in inductionmotorsrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1298ndash1308 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


lower cost induction machines are supplied and controlledby inverters and the use of inverter drives increases dayby day As a result dynamic performance is excellent dueto all kinds of control methods and the next steps in drivedevelopment are going to be driven by attempts to increasereliability and reducemaintenance costs By using the currentsensor feedback and microprocessing unit the new tread forlow-cost protection applications seems to be drive-integratedfault diagnosis systems without using any external hardwareThus it becomes more demanding to detect faults by usingMCSA in these drives

The introduction of voltage-source-inverter-fed (VSI-fed) motors has produced significant changes in the fieldof diagnosis and control needing further research in orderto overcome various challenges Contrary to the motor linecurrent taken directly from the main the inverter-fed motorline current includes remarkable noise (inherent floor noisewhich reduces the possibility of true fault pattern recognitionusing line current spectrum) due to the high-frequencyswitching devices EMI and EMC effects and so forth Thecurrent signal for rotor fault diagnosis needs precise and highresolution information which means the diagnosis systemdemands additional hardware such as a low-pass filter high-precision AD converter and additional software Moreoverclosed-loop control in IM drives introduces new difficultiesin broken rotor bar detection as traditional fault indicatorstend to be masked by a control algorithm Therefore themethods developed for an open-loop operation are not ableto produce adequate and reliable information on the extent ofthe fault and have to be adapted [4 17 18] All these influencescomplicate the utilization of frequency analysis methodsVSI-fed motor faults have been analyzed and the initialresults are given in the literature but further investigation isstill required

The classic MCSA method works well under constantload torque and with high-power motors and it has mainlybeen designed for fixed frequency supply such as formachines connected to the electrical grid To obtain satis-factory test results in [7] recommendations are given fromthe authorrsquos experience As stated in this paper literally ldquotheload on the motor should be sufficiently high to produce atleast 35 of rated full load rotor slip for a reliable single-testbroken bar analysisrdquo and ldquoif themotor to be tested is fed froma variable-speed converter drive the frequency of the driveshould be locked at the 5060 Hz power supply frequency forthe testrdquo Yet difficulties emergewhen inverted-fedmotors areapplied to drive fans pumps or other mechanisms involvingspeed control for energy-saving purpose In these casesthe excitation frequency will truly depend on the speedreference and the load applied to the systemTherefore unlikethe utility-driven case the stator excitation frequency willdynamically change and the position of the current harmon-ics appearing on the stator-current spectrum due to electricalfaults is highly dependent on the mechanical motor load andexcitation frequency which affects the slip frequency As aconsequence the conventional MCSA must be amended toaccommodate the new scenarios Unfortunately to the bestof our knowledge in the published literatures there is noresearch work on this subject

id

Sia Sib SicSua Sub Suc

ia998400

ib998400

ic998400

ua998400

ub998400

uc998400

uaub uc

ud Z

Z

Ziaibic

Msim

+

minus

Figure 1 Schematic diagram of the PWMVSI-fed adjustable speeddrive

In this context following explicitly derivation from asimplified fault signal model a new online fault diagnosistechnique based on MCSA for inverter-fed squirrel-cageinduction motors is present Compared with traditionalMCSA the novelty of the proposed method is that brokenrotor bars fault in the controlled motor via open-loop con-stant voltagefrequency control method could be identifiedeven if the motor operates at different excitation frequenciesTo do this oversampling signal acquisition technique is usedto suppress significantly noise contained in the inverted-fed motor line current and fault-indicative frequencies withvariable excitation frequency are determined by torque-speedcharacteristic curve of squirrel-cage motor with differentload types To obtain satisfactory results nonparametricspectrum estimation algorithm and fault identification aresubsequently presented Including this introductory sectionthis paper is organized into six sections Section 2 presentsa theoretical analysis model of stator current of inverter-fedsquirrel-cage motor which is based on switching functionconcept and modulation theory A detailed description ofthe harmonic components contained in current is givenSection 3 elaborates the new broken rotor bars fault diagnosistechniques In Section 5 the proposed techniques are vali-dated by laboratory tests the method is applied to differentstator currents obtained from healthy and faulty machinesExperimental and simulation results as well as the corre-sponding analysis and discussion are presented in Section 4Finally conclusions and recommendations are presented inthe last section

2 Analytic Model of Stator Current Signaturefor Squirrel-Cage Induction Motor withConstant Volt-per-Hertz Control Technique

21 Analytic Model of Stator Current Signature for No-FaultSquirrel-Cage Induction Motor Switching function conceptis a powerful tool in understanding and optimizing theperformance of the converter [19 20] In [20] an analyticalapproach for characterizing the current harmonics and inter-harmonics of theVSI-fedASD injected into the supply systemin steady is presented using the switching function conceptapplicable to PWM VSI that is studied in this paper Figure 1shows the schematic diagram of a typical PWM VSI-fedadjustable speed drive where 119878

119894119886 119878119894119887 and 119878


rectifier converter current switching functions that correlate



119886 1198941015840119887 and 119894

1015840

119888and the




119886 119906119887and 119906119888119885 is the


(1198941015840

119886 1198941015840

119887 1198941015840

119888) = 119894119889(119878119894119886 119878119894119887 119878119894119888) (2)

119894119889= (119894119886 119894119887 119894119888) (119878119906119886 119878119906119887 119878119906119888)1015840

(3)

119906119889= (1199061015840

119886 1199061015840

119887 1199061015840

119888) (119878119894119886 119878119894119887 119878119894119888)1015840

(4)

(119906119886 119906119887 119906119888) = 119906119889(119878119906119886 119878119906119887 119878119906119888) (5)

(119894119886 119894119887 119894119888) =

(119906119886 119906119887 119906119888)

119885 (6)




119878119906119886

(119905) =119872

2sin (120596

119904119905) +

2

120587

infin

sum

119898=1

1198690(119898119872120587

2)

sdot sin(119898120587

2) sdot sin (119898120596

119888119905)

+2

120587

infin

sum

119898=1

infin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sin [

(119898 + 119899) 120587

2]

times sin (119898120596119888119905 + 119899120596

119904119905)

119878119906119887

(119905) =119872

2sin(120596

119904119905 minus

2120587

3) +

2

120587

infin

sum

119898=1

1198690(119898119872120587

2)

sdot sin(119898120587

2) sdot sin (119898120596

119888119905)

+2

120587

infin

sum

119898=1

infin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sin [

(119898 + 119899) 120587

2]

times sin [119898120596119888119905 + 119899 (120596

119904119905 minus

2120587

3)]

119878119906119888

(119905) =119872

2sin(120596

119904119905 +

2120587

3) +

2

120587

infin

sum

119898=1

1198690(119898119872120587

2)

sdot sin(119898120587

2) sdot sin (119898120596

119888119905)

+2

120587

infin

sum

119898=1

infin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sin [

(119898 + 119899) 120587

2]

times sin [119898120596119888119905 + 119899 (120596

119904119905 +

2120587

3)]

(7)


119894119886=

119872119864119889

21003816100381610038161003816119885 (120596119904)1003816100381610038161003816

cos [120596119904119905 minus 120593 (119885 (120596

119904))]

+2119864119889

1205871003816100381610038161003816119885 (119898119873120596

119904)1003816100381610038161003816

infin

sum

119898=1

1198690(119898119872120587

2) sin(

119898120587

2)

times cos [119898119873120596119904119905 minus 120593 (119885 (119898119873120596

119904))]

+2119864119889

1205871003816100381610038161003816119885 ((119898119873 + 119899) 120596

119904)1003816100381610038161003816

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898

sin [(119898 + 119899) 120587

2] sdot cos [ (119898119873 + 119899) 120596

119904119905

minus120593 (119885 ((119898119873 + 119899) 120596119904))]

= 1198681cos (120596

119904119905 minus 1205931) + 1198682

infin

sum

119898=1

1198690(119898119872120587

2) sin(

119898120587

2)

times cos (119898119873120596119904119905 minus 1205932)

+ 1198683

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sdot sin [

(119898 + 119899) 120587

2]

times cos [(119898119873 + 119899) 120596119904119905 minus 1205933]

(8)


119904




119888120596119904ge 1 and 119869




119886 119894119887 and 119894






119894119891= 119894119886[1 + 119886 cos (120596

119900119905)] (9)



119900= 2120587119891

119900= 2119904120596

119904= 4120587119904119891




119894119891= 1198681cos (2120587119891

119904119905 minus 1205931) + 1198682

infin

sum

119898=1

1198690(119898119872120587

2) sin(

119898120587

2)

times cos (2120587119898119873119891119904119905 minus 1205932)

+ 1198683

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sdot sin [

(119898 + 119899) 120587

2]

times cos [2120587 (119898119873 + 119899) 119891119904119905 minus 1205933]

+ 1198861198681cos [2120587 (1 + 2119904) 119891

119904119905 minus 1205931]

+ cos [2120587 (1 minus 2119904) 119891119904119905 minus 1205931]

+ 1198861198682

infin

sum

119898=1

1198690(119898119872120587

2) sdot sin(

119898120587

2)

sdot cos [2120587 (119898119873 + 2119904) 119891119904119905 minus 1205932]

+ cos [2120587 (119898119873 minus 2119904) 119891119904119905 minus 1205933]

+ 1198861198683

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sdot sin [

(119898 + 119899) 120587

2]

sdot cos [2120587 (119898119873 + 119899 + 2119904) 119891119904119905 minus 1205933]

+ cos [2120587 (119898119873 + 119899 minus 2119904) 119891119904119905 minus 1205933]

(10)






119887= (1 plusmn 2119904)119891

119904

1198911198871

= (119898119873 plusmn 2119904)119891119904 and 119891

1198872= (119898119873 + 119899 plusmn 2119904)119891

119904around


119887= (1 plusmn 2119904)119891

119904 1198911198871

= (119898119873 plusmn 2119904)119891119904

and 1198911198872




119904

119898119873119891119904 (119872119898 + 119899)119891119904is equal to 2119904119891


1198871


119887

as a result 119891119887






119887= (1 plusmn 2119904)119891








3 phase AC supply

SpeedRe

fere

nce f

requ

ency



AD over-sampling


Spectrum estimation



Msim

nfs


(f P(f)) = (f 1K

Kminus1sumK=0

Pk(f))sim








1 minus 119904



1 minus 119904

119901) plusmn 119904] Low 0sim400Hz


=119899119887

2119891119903[1 plusmn

119887119889

119901119889



119901(1 minus 119904) plusmn 119896] 119891sth = 119891

119904[1 plusmn 119898119885

2(1 minus 119904





119888 where




119873119879119904


119887= (1 plusmn 2119904)119891


119887









1at nominal



3of curve




=

601198911119901 and 119899

02= 60119891


pole-pairs




A2

A

A1

0 T Te TL

nn03n3

n02n2n1

n01ne (3) fs gt f1

(1) f1

(2) fs lt f1




1is




0211989911198601 we can deduce




2is equal

to 119904fanspump = (11989902


Δ11989901119899119890119860 to Δ119899


119890= (1198992119899119890)2


2

=

[119904119890(1 minus 119904

119890)2

](1198911199041198911)




119873[119899] = 119894[0] 119894[1] 119894[119873 minus 1] is




(119873 minus 1198712)(1198712)

Algorithm 1





119909119896

[119899] = 119889 [119899] 119909 [119899 + 119896119878] 0 ⩽ 119899 ⩽ 119871 minus 1 0 ⩽ 119896 ⩽ 119870 minus 1

(11)


119875119896

(119891) =1

ULT119883(119891) [119883 (119891)]

lowast

=1

ULT1003816100381610038161003816119883 (119891)

1003816100381610038161003816

2

(12)

where 119883(119891) = 119879sum119871minus1

119899=0119909(119896)



119899=01198892



(119891) =1

119870

119870minus1

sum

119896=0

119875119896

(119891) (13)















DCgenerator


Excitation circurt

Resistor bank

Squirrel cage motor


Inverter PC

sim3-phase source

















119887



(a) (b)


0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(a)

0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(b)


35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Healthy

Frequency (Hz)

n = 1425 rmin f = 50Hz

(a)

Am

plitu

de (d

B)

Frequency (Hz)35 40 45 50 55 60 65

minus80

minus60

minus40

minus20

0Healthy

n = 1425 rmin fs = 50Hz

(b)


Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0Faulty

Frequency (Hz)


(a)

Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Frequency (Hz)



(b)



20 30 40 50 60 70minus80

minus60

minus40

0A

mpl

itude

(dB)

Frequency (Hz)

Faulty


(a)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

25 30 35 40 45 50 55minus80

minus60

minus40

minus20

0


(b)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0



(c)A

mpl

itude

(dB)

Frequency (Hz)


Faulty

0 10 20 30 40minus80

minus60

minus40

minus20

0


(d)


26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(a)

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(b)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(c)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(d)





119887= (1plusmn2119904)119891

119904


119904varies from












6 Conclusion



119887=



119904 Novel





Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119886 1198941015840119887 and 119894

1015840

119888and the




119886 119906119887and 119906119888119885 is the


(1198941015840

119886 1198941015840

119887 1198941015840

119888) = 119894119889(119878119894119886 119878119894119887 119878119894119888) (2)

119894119889= (119894119886 119894119887 119894119888) (119878119906119886 119878119906119887 119878119906119888)1015840

(3)

119906119889= (1199061015840

119886 1199061015840

119887 1199061015840

119888) (119878119894119886 119878119894119887 119878119894119888)1015840

(4)

(119906119886 119906119887 119906119888) = 119906119889(119878119906119886 119878119906119887 119878119906119888) (5)

(119894119886 119894119887 119894119888) =

(119906119886 119906119887 119906119888)

119885 (6)




119878119906119886

(119905) =119872

2sin (120596

119904119905) +

2

120587

infin

sum

119898=1

1198690(119898119872120587

2)

sdot sin(119898120587

2) sdot sin (119898120596

119888119905)

+2

120587

infin

sum

119898=1

infin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sin [

(119898 + 119899) 120587

2]

times sin (119898120596119888119905 + 119899120596

119904119905)

119878119906119887

(119905) =119872

2sin(120596

119904119905 minus

2120587

3) +

2

120587

infin

sum

119898=1

1198690(119898119872120587

2)

sdot sin(119898120587

2) sdot sin (119898120596

119888119905)

+2

120587

infin

sum

119898=1

infin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sin [

(119898 + 119899) 120587

2]

times sin [119898120596119888119905 + 119899 (120596

119904119905 minus

2120587

3)]

119878119906119888

(119905) =119872

2sin(120596

119904119905 +

2120587

3) +

2

120587

infin

sum

119898=1

1198690(119898119872120587

2)

sdot sin(119898120587

2) sdot sin (119898120596

119888119905)

+2

120587

infin

sum

119898=1

infin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sin [

(119898 + 119899) 120587

2]

times sin [119898120596119888119905 + 119899 (120596

119904119905 +

2120587

3)]

(7)


119894119886=

119872119864119889

21003816100381610038161003816119885 (120596119904)1003816100381610038161003816

cos [120596119904119905 minus 120593 (119885 (120596

119904))]

+2119864119889

1205871003816100381610038161003816119885 (119898119873120596

119904)1003816100381610038161003816

infin

sum

119898=1

1198690(119898119872120587

2) sin(

119898120587

2)

times cos [119898119873120596119904119905 minus 120593 (119885 (119898119873120596

119904))]

+2119864119889

1205871003816100381610038161003816119885 ((119898119873 + 119899) 120596

119904)1003816100381610038161003816

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898

sin [(119898 + 119899) 120587

2] sdot cos [ (119898119873 + 119899) 120596

119904119905

minus120593 (119885 ((119898119873 + 119899) 120596119904))]

= 1198681cos (120596

119904119905 minus 1205931) + 1198682

infin

sum

119898=1

1198690(119898119872120587

2) sin(

119898120587

2)

times cos (119898119873120596119904119905 minus 1205932)

+ 1198683

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sdot sin [

(119898 + 119899) 120587

2]

times cos [(119898119873 + 119899) 120596119904119905 minus 1205933]

(8)


119904




119888120596119904ge 1 and 119869




119886 119894119887 and 119894






119894119891= 119894119886[1 + 119886 cos (120596

119900119905)] (9)



119900= 2120587119891

119900= 2119904120596

119904= 4120587119904119891




119894119891= 1198681cos (2120587119891

119904119905 minus 1205931) + 1198682

infin

sum

119898=1

1198690(119898119872120587

2) sin(

119898120587

2)

times cos (2120587119898119873119891119904119905 minus 1205932)

+ 1198683

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sdot sin [

(119898 + 119899) 120587

2]

times cos [2120587 (119898119873 + 119899) 119891119904119905 minus 1205933]

+ 1198861198681cos [2120587 (1 + 2119904) 119891

119904119905 minus 1205931]

+ cos [2120587 (1 minus 2119904) 119891119904119905 minus 1205931]

+ 1198861198682

infin

sum

119898=1

1198690(119898119872120587

2) sdot sin(

119898120587

2)

sdot cos [2120587 (119898119873 + 2119904) 119891119904119905 minus 1205932]

+ cos [2120587 (119898119873 minus 2119904) 119891119904119905 minus 1205933]

+ 1198861198683

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sdot sin [

(119898 + 119899) 120587

2]

sdot cos [2120587 (119898119873 + 119899 + 2119904) 119891119904119905 minus 1205933]

+ cos [2120587 (119898119873 + 119899 minus 2119904) 119891119904119905 minus 1205933]

(10)






119887= (1 plusmn 2119904)119891

119904

1198911198871

= (119898119873 plusmn 2119904)119891119904 and 119891

1198872= (119898119873 + 119899 plusmn 2119904)119891

119904around


119887= (1 plusmn 2119904)119891

119904 1198911198871

= (119898119873 plusmn 2119904)119891119904

and 1198911198872




119904

119898119873119891119904 (119872119898 + 119899)119891119904is equal to 2119904119891


1198871


119887

as a result 119891119887






119887= (1 plusmn 2119904)119891








3 phase AC supply

SpeedRe

fere

nce f

requ

ency



AD over-sampling


Spectrum estimation



Msim

nfs


(f P(f)) = (f 1K

Kminus1sumK=0

Pk(f))sim








1 minus 119904



1 minus 119904

119901) plusmn 119904] Low 0sim400Hz


=119899119887

2119891119903[1 plusmn

119887119889

119901119889



119901(1 minus 119904) plusmn 119896] 119891sth = 119891

119904[1 plusmn 119898119885

2(1 minus 119904





119888 where




119873119879119904


119887= (1 plusmn 2119904)119891


119887









1at nominal



3of curve




=

601198911119901 and 119899

02= 60119891


pole-pairs




A2

A

A1

0 T Te TL

nn03n3

n02n2n1

n01ne (3) fs gt f1

(1) f1

(2) fs lt f1




1is




0211989911198601 we can deduce




2is equal

to 119904fanspump = (11989902


Δ11989901119899119890119860 to Δ119899


119890= (1198992119899119890)2


2

=

[119904119890(1 minus 119904

119890)2

](1198911199041198911)




119873[119899] = 119894[0] 119894[1] 119894[119873 minus 1] is




(119873 minus 1198712)(1198712)

Algorithm 1





119909119896

[119899] = 119889 [119899] 119909 [119899 + 119896119878] 0 ⩽ 119899 ⩽ 119871 minus 1 0 ⩽ 119896 ⩽ 119870 minus 1

(11)


119875119896

(119891) =1

ULT119883(119891) [119883 (119891)]

lowast

=1

ULT1003816100381610038161003816119883 (119891)

1003816100381610038161003816

2

(12)

where 119883(119891) = 119879sum119871minus1

119899=0119909(119896)



119899=01198892



(119891) =1

119870

119870minus1

sum

119896=0

119875119896

(119891) (13)















DCgenerator


Excitation circurt

Resistor bank

Squirrel cage motor


Inverter PC

sim3-phase source

















119887



(a) (b)


0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(a)

0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(b)


35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Healthy

Frequency (Hz)

n = 1425 rmin f = 50Hz

(a)

Am

plitu

de (d

B)

Frequency (Hz)35 40 45 50 55 60 65

minus80

minus60

minus40

minus20

0Healthy

n = 1425 rmin fs = 50Hz

(b)


Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0Faulty

Frequency (Hz)


(a)

Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Frequency (Hz)



(b)



20 30 40 50 60 70minus80

minus60

minus40

0A

mpl

itude

(dB)

Frequency (Hz)

Faulty


(a)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

25 30 35 40 45 50 55minus80

minus60

minus40

minus20

0


(b)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0



(c)A

mpl

itude

(dB)

Frequency (Hz)


Faulty

0 10 20 30 40minus80

minus60

minus40

minus20

0


(d)


26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(a)

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(b)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(c)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(d)





119887= (1plusmn2119904)119891

119904


119904varies from












6 Conclusion



119887=



119904 Novel





Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119900= 2120587119891

119900= 2119904120596

119904= 4120587119904119891




119894119891= 1198681cos (2120587119891

119904119905 minus 1205931) + 1198682

infin

sum

119898=1

1198690(119898119872120587

2) sin(

119898120587

2)

times cos (2120587119898119873119891119904119905 minus 1205932)

+ 1198683

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sdot sin [

(119898 + 119899) 120587

2]

times cos [2120587 (119898119873 + 119899) 119891119904119905 minus 1205933]

+ 1198861198681cos [2120587 (1 + 2119904) 119891

119904119905 minus 1205931]

+ cos [2120587 (1 minus 2119904) 119891119904119905 minus 1205931]

+ 1198861198682

infin

sum

119898=1

1198690(119898119872120587

2) sdot sin(

119898120587

2)

sdot cos [2120587 (119898119873 + 2119904) 119891119904119905 minus 1205932]

+ cos [2120587 (119898119873 minus 2119904) 119891119904119905 minus 1205933]

+ 1198861198683

infin

sum

119898=1

plusmninfin

sum

119899=plusmn1

119869119899(1198981198721205872)

119898sdot sin [

(119898 + 119899) 120587

2]

sdot cos [2120587 (119898119873 + 119899 + 2119904) 119891119904119905 minus 1205933]

+ cos [2120587 (119898119873 + 119899 minus 2119904) 119891119904119905 minus 1205933]

(10)






119887= (1 plusmn 2119904)119891

119904

1198911198871

= (119898119873 plusmn 2119904)119891119904 and 119891

1198872= (119898119873 + 119899 plusmn 2119904)119891

119904around


119887= (1 plusmn 2119904)119891

119904 1198911198871

= (119898119873 plusmn 2119904)119891119904

and 1198911198872




119904

119898119873119891119904 (119872119898 + 119899)119891119904is equal to 2119904119891


1198871


119887

as a result 119891119887






119887= (1 plusmn 2119904)119891








3 phase AC supply

SpeedRe

fere

nce f

requ

ency



AD over-sampling


Spectrum estimation



Msim

nfs


(f P(f)) = (f 1K

Kminus1sumK=0

Pk(f))sim








1 minus 119904



1 minus 119904

119901) plusmn 119904] Low 0sim400Hz


=119899119887

2119891119903[1 plusmn

119887119889

119901119889



119901(1 minus 119904) plusmn 119896] 119891sth = 119891

119904[1 plusmn 119898119885

2(1 minus 119904





119888 where




119873119879119904


119887= (1 plusmn 2119904)119891


119887









1at nominal



3of curve




=

601198911119901 and 119899

02= 60119891


pole-pairs




A2

A

A1

0 T Te TL

nn03n3

n02n2n1

n01ne (3) fs gt f1

(1) f1

(2) fs lt f1




1is




0211989911198601 we can deduce




2is equal

to 119904fanspump = (11989902


Δ11989901119899119890119860 to Δ119899


119890= (1198992119899119890)2


2

=

[119904119890(1 minus 119904

119890)2

](1198911199041198911)




119873[119899] = 119894[0] 119894[1] 119894[119873 minus 1] is




(119873 minus 1198712)(1198712)

Algorithm 1





119909119896

[119899] = 119889 [119899] 119909 [119899 + 119896119878] 0 ⩽ 119899 ⩽ 119871 minus 1 0 ⩽ 119896 ⩽ 119870 minus 1

(11)


119875119896

(119891) =1

ULT119883(119891) [119883 (119891)]

lowast

=1

ULT1003816100381610038161003816119883 (119891)

1003816100381610038161003816

2

(12)

where 119883(119891) = 119879sum119871minus1

119899=0119909(119896)



119899=01198892



(119891) =1

119870

119870minus1

sum

119896=0

119875119896

(119891) (13)















DCgenerator


Excitation circurt

Resistor bank

Squirrel cage motor


Inverter PC

sim3-phase source

















119887



(a) (b)


0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(a)

0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(b)


35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Healthy

Frequency (Hz)

n = 1425 rmin f = 50Hz

(a)

Am

plitu

de (d

B)

Frequency (Hz)35 40 45 50 55 60 65

minus80

minus60

minus40

minus20

0Healthy

n = 1425 rmin fs = 50Hz

(b)


Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0Faulty

Frequency (Hz)


(a)

Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Frequency (Hz)



(b)



20 30 40 50 60 70minus80

minus60

minus40

0A

mpl

itude

(dB)

Frequency (Hz)

Faulty


(a)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

25 30 35 40 45 50 55minus80

minus60

minus40

minus20

0


(b)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0



(c)A

mpl

itude

(dB)

Frequency (Hz)


Faulty

0 10 20 30 40minus80

minus60

minus40

minus20

0


(d)


26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(a)

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(b)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(c)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(d)





119887= (1plusmn2119904)119891

119904


119904varies from












6 Conclusion



119887=



119904 Novel





Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3 phase AC supply

SpeedRe

fere

nce f

requ

ency



AD over-sampling


Spectrum estimation



Msim

nfs


(f P(f)) = (f 1K

Kminus1sumK=0

Pk(f))sim








1 minus 119904



1 minus 119904

119901) plusmn 119904] Low 0sim400Hz


=119899119887

2119891119903[1 plusmn

119887119889

119901119889



119901(1 minus 119904) plusmn 119896] 119891sth = 119891

119904[1 plusmn 119898119885

2(1 minus 119904





119888 where




119873119879119904


119887= (1 plusmn 2119904)119891


119887









1at nominal



3of curve




=

601198911119901 and 119899

02= 60119891


pole-pairs




A2

A

A1

0 T Te TL

nn03n3

n02n2n1

n01ne (3) fs gt f1

(1) f1

(2) fs lt f1




1is




0211989911198601 we can deduce




2is equal

to 119904fanspump = (11989902


Δ11989901119899119890119860 to Δ119899


119890= (1198992119899119890)2


2

=

[119904119890(1 minus 119904

119890)2

](1198911199041198911)




119873[119899] = 119894[0] 119894[1] 119894[119873 minus 1] is




(119873 minus 1198712)(1198712)

Algorithm 1





119909119896

[119899] = 119889 [119899] 119909 [119899 + 119896119878] 0 ⩽ 119899 ⩽ 119871 minus 1 0 ⩽ 119896 ⩽ 119870 minus 1

(11)


119875119896

(119891) =1

ULT119883(119891) [119883 (119891)]

lowast

=1

ULT1003816100381610038161003816119883 (119891)

1003816100381610038161003816

2

(12)

where 119883(119891) = 119879sum119871minus1

119899=0119909(119896)



119899=01198892



(119891) =1

119870

119870minus1

sum

119896=0

119875119896

(119891) (13)















DCgenerator


Excitation circurt

Resistor bank

Squirrel cage motor


Inverter PC

sim3-phase source

















119887



(a) (b)


0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(a)

0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(b)


35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Healthy

Frequency (Hz)

n = 1425 rmin f = 50Hz

(a)

Am

plitu

de (d

B)

Frequency (Hz)35 40 45 50 55 60 65

minus80

minus60

minus40

minus20

0Healthy

n = 1425 rmin fs = 50Hz

(b)


Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0Faulty

Frequency (Hz)


(a)

Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Frequency (Hz)



(b)



20 30 40 50 60 70minus80

minus60

minus40

0A

mpl

itude

(dB)

Frequency (Hz)

Faulty


(a)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

25 30 35 40 45 50 55minus80

minus60

minus40

minus20

0


(b)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0



(c)A

mpl

itude

(dB)

Frequency (Hz)


Faulty

0 10 20 30 40minus80

minus60

minus40

minus20

0


(d)


26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(a)

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(b)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(c)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(d)





119887= (1plusmn2119904)119891

119904


119904varies from












6 Conclusion



119887=



119904 Novel





Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












A2

A

A1

0 T Te TL

nn03n3

n02n2n1

n01ne (3) fs gt f1

(1) f1

(2) fs lt f1




1is




0211989911198601 we can deduce




2is equal

to 119904fanspump = (11989902


Δ11989901119899119890119860 to Δ119899


119890= (1198992119899119890)2


2

=

[119904119890(1 minus 119904

119890)2

](1198911199041198911)




119873[119899] = 119894[0] 119894[1] 119894[119873 minus 1] is




(119873 minus 1198712)(1198712)

Algorithm 1





119909119896

[119899] = 119889 [119899] 119909 [119899 + 119896119878] 0 ⩽ 119899 ⩽ 119871 minus 1 0 ⩽ 119896 ⩽ 119870 minus 1

(11)


119875119896

(119891) =1

ULT119883(119891) [119883 (119891)]

lowast

=1

ULT1003816100381610038161003816119883 (119891)

1003816100381610038161003816

2

(12)

where 119883(119891) = 119879sum119871minus1

119899=0119909(119896)



119899=01198892



(119891) =1

119870

119870minus1

sum

119896=0

119875119896

(119891) (13)















DCgenerator


Excitation circurt

Resistor bank

Squirrel cage motor


Inverter PC

sim3-phase source

















119887



(a) (b)


0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(a)

0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(b)


35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Healthy

Frequency (Hz)

n = 1425 rmin f = 50Hz

(a)

Am

plitu

de (d

B)

Frequency (Hz)35 40 45 50 55 60 65

minus80

minus60

minus40

minus20

0Healthy

n = 1425 rmin fs = 50Hz

(b)


Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0Faulty

Frequency (Hz)


(a)

Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Frequency (Hz)



(b)



20 30 40 50 60 70minus80

minus60

minus40

0A

mpl

itude

(dB)

Frequency (Hz)

Faulty


(a)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

25 30 35 40 45 50 55minus80

minus60

minus40

minus20

0


(b)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0



(c)A

mpl

itude

(dB)

Frequency (Hz)


Faulty

0 10 20 30 40minus80

minus60

minus40

minus20

0


(d)


26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(a)

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(b)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(c)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(d)





119887= (1plusmn2119904)119891

119904


119904varies from












6 Conclusion



119887=



119904 Novel





Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










DCgenerator


Excitation circurt

Resistor bank

Squirrel cage motor


Inverter PC

sim3-phase source

















119887



(a) (b)


0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(a)

0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(b)


35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Healthy

Frequency (Hz)

n = 1425 rmin f = 50Hz

(a)

Am

plitu

de (d

B)

Frequency (Hz)35 40 45 50 55 60 65

minus80

minus60

minus40

minus20

0Healthy

n = 1425 rmin fs = 50Hz

(b)


Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0Faulty

Frequency (Hz)


(a)

Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Frequency (Hz)



(b)



20 30 40 50 60 70minus80

minus60

minus40

0A

mpl

itude

(dB)

Frequency (Hz)

Faulty


(a)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

25 30 35 40 45 50 55minus80

minus60

minus40

minus20

0


(b)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0



(c)A

mpl

itude

(dB)

Frequency (Hz)


Faulty

0 10 20 30 40minus80

minus60

minus40

minus20

0


(d)


26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(a)

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(b)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(c)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(d)





119887= (1plusmn2119904)119891

119904


119904varies from












6 Conclusion



119887=



119904 Novel





Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)


0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(a)

0 200 400 600 800minus10

0

10

Samples

Am

plitu

de

(b)


35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Healthy

Frequency (Hz)

n = 1425 rmin f = 50Hz

(a)

Am

plitu

de (d

B)

Frequency (Hz)35 40 45 50 55 60 65

minus80

minus60

minus40

minus20

0Healthy

n = 1425 rmin fs = 50Hz

(b)


Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0Faulty

Frequency (Hz)


(a)

Am

plitu

de (d

B)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Frequency (Hz)



(b)



20 30 40 50 60 70minus80

minus60

minus40

0A

mpl

itude

(dB)

Frequency (Hz)

Faulty


(a)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

25 30 35 40 45 50 55minus80

minus60

minus40

minus20

0


(b)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0



(c)A

mpl

itude

(dB)

Frequency (Hz)


Faulty

0 10 20 30 40minus80

minus60

minus40

minus20

0


(d)


26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(a)

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(b)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(c)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(d)





119887= (1plusmn2119904)119891

119904


119904varies from












6 Conclusion



119887=



119904 Novel





Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20 30 40 50 60 70minus80

minus60

minus40

0A

mpl

itude

(dB)

Frequency (Hz)

Faulty


(a)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

25 30 35 40 45 50 55minus80

minus60

minus40

minus20

0


(b)

Am

plitu

de (d

B)

Frequency (Hz)

Faulty

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0



(c)A

mpl

itude

(dB)

Frequency (Hz)


Faulty

0 10 20 30 40minus80

minus60

minus40

minus20

0


(d)


26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(a)

26 28 30 32 34 36 38minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(b)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(c)

35 40 45 50 55 60 65minus80

minus60

minus40

minus20

0

Am

plitu

de (d

B)

Frequency (Hz)

Faulty


(d)





119887= (1plusmn2119904)119891

119904


119904varies from












6 Conclusion



119887=



119904 Novel





Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















6 Conclusion



119887=



119904 Novel





Acknowledgment


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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