research article stator flux observer for induction motor

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 927582 8 pageshttpdxdoiorg1011552013927582

Research ArticleStator Flux Observer for Induction Motor Based onTracking Differentiator

Dafang Wang1 Zhenfei Hu1 Cheng Zhu2 Chuanwei Zhou1 and Yajing Xie1

1 School of Automotive Engineering Harbin Institute of Technology Weihai 264209 China2 China Automotive Technology amp Research Center Tianjin 300300 China

Correspondence should be addressed to Dafang Wang wdflcjl163com

Received 23 September 2013 Revised 21 October 2013 Accepted 4 November 2013

Academic Editor Rongni Yang

Copyright copy 2013 Dafang Wang 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

Voltage model is commonly used in direct torque control (DTC) for flux observing of asynchronous motor In order to improvelow-speed and dynamic performance of the voltage model a modified low-pass filter (LPF) algorithm is proposed Firstly thetracking differentiator is brought in to modulate the measured stator current which suppresses the measurement noise and thenamplitude and phase compensation is made towards the stator electromotive force (EMF) after which the stator flux is obtainedthrough a low-pass filterThis method can eliminate the dynamic error of flux filtered by LPF and improve low-speed performanceExperimental results demonstrate effectiveness and improved dynamic performance of such method

1 Introduction

The direct torque control technology based on stator fluxorientation has beenwidely used in high-performance induc-tion motor control system It has the advantages of simplestructure not being sensitive to motor parameters and soforth [1 2] The key of achieving direct torque control ofasynchronous motor effectively lies in the accurate obtainingof stator flux information especially observing motor fluxexactly at a low stator frequency is even a big issue of ACspeed regulation [3ndash5]

In the aspect of robust estimator scholars have done alot of research work Reference [6] considers the modelingand adaptive output tracking of an FCTFPM as a nonlinearsystem with unknown nonlinearities by utilizing HGO andRBF neural networks A fuzzy reliable control strategy hasbeen presented for the tracking problem of the longitudinaldynamics of FAHVs model with actuator or sensor faultsand external disturbance Based on the T-S fuzzy modelingtechnology a T-S fuzzy model has been constructed torepresent the nonlinear dynamics of the FAHVs [7] Refer-ence [8] presents a fault tolerant tracking controller for aVTOL aircraft flight in uncertain conditions The consideredsystem contains structured uncertainties which affect themechanical parameters of the air vehicle Reference [9]

investigates the energy-to-peak filtering problem for nonuni-formly sampled nonlinear systems The sampled nonlinearsystems aremodeled by T-S fuzzy systems under the discrete-time framework Reference [10] is devoted to the ammoniacoverage ratio estimation problem in SCR systems Reference[11] studies the state estimation problem for discrete-timesystems subject to network-induced delay By considering theoccurrence probability for the delay the exponential mean-square stability and the119867

infinperformance are exploited for the

estimation error system Reference [12] investigates the 119867infin

filtering problem for T-S fuzzy systems under the discrete-time framework By using Finslerrsquos lemma a new 119867

infincri-

terion for discrete-time fuzzy systems is obtained With thepartition technique the Lyapunovweightingmatrices and theparameters to be designed are decoupled

Voltage model is the basic method of the following statorflux It uses an ideal integrator described in formula (1) Thealgorithm of this model is simple which only need to knowthe stator resistance which is why the flux observing methodwhich is based on voltagemodel has always been given specialimportance [13 14] Consider

120595119904= int (u

119904minus i119904119877119904) 119889119905 (1)

2 Mathematical Problems in Engineering

Although the voltage model is quite simple there are stillsome problems in practical application [15] (1) A small DCbias or drift in the current measurement channel will causethe integrator saturation (2) The stator resistance variationin low stator frequency makes the stator flux amplitude andphase observations have a big error (3)The initial value of theintegral produces the dc bias in the observed flux amplitude

In order to eliminate the effect ofDCbias it is proposed in[16] that the pure integrator should be replaced with low-passfilter but amplitude and phase error of flux can be introducedto it Adopting a programmable cascaded low-pass filter canovercome the effect of zero drift theoretically [17] but thecutoff frequency is of high precision It is difficult to getthe desired effect in practical application and the dynamicperformance is poor Using improved PLL to observe thestator flux is proposed in [18] but this method needs to usethe motor speed information based on permanent magnetsynchronous motor and the system has two convergencepoints meanwhile

The biggest reason why flux observation is inaccurate isthat the DC bias and unbalanced gain exist in current mea-surement channel hence this paper reduces the current mea-surement interference by using tracking differentiator [19 20]to filter the measuring stator current it also restrains the DCcomponent of stator current by using low-pass filter instead ofpure integral and it eliminates the amplitude attenuation andphase error brought by low-pass filter by using the back EMFcompensation algorithm Experimental results demonstrateeffectiveness and improved dynamic performance of suchmethod

2 Traditional Method of StatorFlux Estimation

The voltage model for flux observation is obtained accordingto the stator voltage equation Stator voltage equation and fluxequation are expressed as

u119904= 119877119904i119904+119889120595119904

119889119905 (2)

120595119904= int (u

119904minus i119904119877119904) 119889119905 (3)

where u119904is the stator voltage119877

119904is the stator resistant i

119904is the

stator current and 120595119904is the stator flux

Equation (3) is called the U-I model or the voltage modelof flux estimator Since the formula contains a pure integratorsmall DC bias can cause integral saturation which will resultin flux estimation error So it is usual to replace pure integralwith the first-order low-pass filter in voltage model namelylet the integral input signal go through a high-pass filter firstlyto filtrate the DC component

The stator flux has invariant amplitude in a steady staterevolving in the synchronous frequency which can be ex-pressed as

120595119904=10038161003816100381610038161205951199041003816100381610038161003816 119890119895120596119890119905

=10038161003816100381610038161205951199041003816100381610038161003816 ang120596119890119905

(4)

where 120596119890is the synchronous frequency of the motor

Equations (2) and (4) become

119889120595119904

119889119905= 119895120596119890lowast 120595119904= u119904minus i119904119877119904

120595119904=u119904minus i119904119877119904

119895120596119890

(5)

Equation (5) is the expression of actual stator fluxWhen observing stator flux through low-pass filter the

following equation can be written

119889119904

119889119905+ 120596119888119904= 119895120596119890lowast 119904+ 120596119888119904= u119904minus i119904119877119904

119904=u119904minus i119904119877119904

119895120596119890+ 120596119888

(6)

where 119904is the stator flux which is observed by low-pass filter

and 120596119888is the cutoff frequency of the low-pass filter

The connection between the estimated stator flux 119904and

the actual stator flux 120595119904can be concluded from (5) and (6)

which can be expressed as

119904=

120596119890

radic1205962119890+ 1205962119888

120595119904ang120579

120579 =120587

2minus arctan

120596119890

120596119888

(7)

From (7) it is clear that errors of flux in the amplitudeand the phase result from the replacement of pure integralwith a low-pass filter The higher the cutoff frequency is themore serious the distortion is shown in the flux amplitude andphase

According to the principle of the direct torque control theflux error can affect the steady state and dynamic operationof asynchronous motor directly Direct torque control selectsthe appropriate voltage vector according to the observed fluxand torque Meanwhile the low-pass filter cuts down the fluxamplitude so when the observing flux reached a given valuethe actual flux has already been far beyond that which leadsto the saturation of motor magnetic field Phase shift of theobserved flux can influence the accurate selection of voltagevector as well as the control characteristic of the motorBesides the estimation of torque in direct torque control isalso affected by flux value directly Therefore it is essential tohave amplitude and phase compensation for the result of low-pass filter Traditional method of flux observing is shown inFigure 1

3 Improved Method of Flux Observing

31 Modified Low-Pass Filter (LPF) Compensation AlgorithmMake amplitude and phase compensation for the result of thelow-pass filter as follows

119904G = 120595

119904 (8)

where 119904is the stator fluxwhich is observed by low-pass filter

G is the penalty function and 120595119904is the stator flux

Mathematical Problems in Engineering 3

Amplitude and phase compensation


+minus

+minus

us120573

es120573

120595998400s120573

120595s120573

1

j120596e + 120596c

1

j120596e + 120596c

120596e120595998400s120572

120595s120572

us120572

es120572

is120572Rs is120573Rs

Figure 1 Traditional method of flux observing

According to (7) the penalty functionG can be expressedas

G =

radic1205962119890+ 1205962119888

120596119890

119890119895(arctan(120596

119890120596119888minus1205872))

=

radic1205962119890+ 1205962119888

120596119890

119890119895120588(120596119890)

119890119895120588(120596119890)

= cos [120588 (120596119890)] + 119895 sin [120588 (120596

119890)]

cos [120588 (120596119890)] =

120596119890

radic1205962119890+ 1205962119888

sin [120588 (120596119890)] =

120596119888

radic1205962119890+ 1205962119888

(9)

where 120596119890is the synchronous frequency of the motor 120596

119888is the

cutoff frequency of the low-pass filterBy choosing appropriate cutoff frequency this compen-

sation algorithm can make the flux observer have a betterability in restraining DC drift and also have a strong abilityof anti-interference But this algorithm has a poor dynamicperformance The estimation of flux value will have big errorwhen stator current frequency has a sudden change [21] Forthis reason the sequence of applying the low-pass filteringalgorithm and the flux compensation can be exchangedmaking compensation for the back electromotive force firstlyas follows

E119904= u119904minus i119904119877119904 E

119904G = E

119904 (10)

In the 120572 minus 120573 coordinate system back electromotive forcecomponents 119890

119904120572and 119890119904120573are at 1205872 space angle Assuming that

the stator flux is in counterclockwise rotation it passes 120572 axisfirstly and then 120573 axis Therefore in a constant flux controlmode 119890

119904120572and 119890

119904120573have the same amplitude and different

phase which can be rewritten as

119890119904120572= 119895119890119904120573 119890

119904120573= minus119895119890

119904120572 (11)

Combining (9)sim(11) leads to the following expression

119890119904120572= 119890119904120572+120596119888

120596119890

119890119904120573

119890119904120573= 119890119904120573minus120596119888

120596119890

119890119904120572

(12)

Practice shows that the optimal cutoff frequency for low-pass filter should be 20sim30 of the synchronous frequency120596119890[16] It is hard to estimate synchronous frequency of

the motor when it is running whereas the stator currentfrequency can be obtained through the detected currentsignal so it can replace the synchronous frequencyThereforethe cutoff frequency of low-pass filter can be calculated asfollows

120596119888= 1205960+ 119896120596119894 (13)

where 1205960is the initial value It ensures that when rotating

speed is close to zero the cutoff frequency will not be toolow 120596

119894is the stator current frequency and 119896 is coefficient of

proportionality (02-03)According to the above in order to achieve this modified

low-pass filter algorithm it is necessary to settle on the statorcurrent frequency120596

119894The space situation of the stator current

in the 120572-120573 coordinate system can be expressed as

120579119894= arctan(

119894119904120573

119894119904120572

) (14)

where the stator current frequency can be obtained by differ-entiating 120579

119894 the discretization formula is shown as follows

120596119894(119896)

=120579(119896)minus 120579(119896minus1)

Δ119879 (15)

32TrackingDifferentiator Because of themeasurement noisethe stator current can affect the precision of flux observationtherefore stator current needs to be filtered Tracking differ-entiator (TD) is the essential part of ADRC [22] The initialpurpose of TD is to rationally extract continuous signal anddifferential signal from discontinuous or band random noisesignal when it comes to the practical engineering problemsAfter a further research on TD discretization form of TDwasproposed making it easier for computer calculation and bet-ter in filtering

TD discretization formula can be written as

1199091(119896 + 1) = 119909

1(119896) + ℎ lowast 119909

2(119896)

1199092(119896 + 1)

= 1199092(119896) + ℎ lowast 119891119904119905 (119909

1(119896) minus V (119896) 119909

2(119896) 119903 ℎ

1)

(16)


Band-limitedwhite noise 1

Subsystem 1

Unit delay

Unit delay

Scope 1

Gain 1

0001

Add 1

Sine wave 1

+

+ Add 2+

+1

z

1

z

x1(k)

x1(k)

x2(k)

x2(k)x2(k + 1)

x2(k

+1)

x1(k + 1)

Vin

Figure 2 TD modeling in Simulink

0 01 02 03 04 05 06 07 08 09 1minus2

minus1

0

1

2

t (s)

(a) Signal before filtering

0 01 02 03 04 05 06 07 08 09 1minus2

minus1

0

1

2

t (s)

(b) Signal after filtering

Figure 3 Filtering result of TD

where V(119896) is input signal1199091is the tracking signal of V(119896) and

1199092is the derivative of 119909

1which can be seen as the derivative

of input signal Consider

119891119904119905 (1199091(119896) minus V (119896) 119909

2(119896) 119903 ℎ

1) = minus119903 lowast sat (119892 (119896) 120575)

120575 = ℎ1lowast 119903 120575

1= ℎ1lowast 120575

119890 (119896) = 1199091(119896) minus V (119896)

119910 (119896) = 119890 (119896) + ℎ1lowast 1199092(119896)

119892 (119896) =

1199092(119896) + sign (119910 (119896))

lowast

radic81199031003816100381610038161003816119910 (119896)

1003816100381610038161003816 + 1205752 minus 120575

2

1003816100381610038161003816119910 (119896)1003816100381610038161003816 ge 1205751

1199092(119896) +

119910 (119896)

ℎ1

1003816100381610038161003816119910 (119896)1003816100381610038161003816 le 1205751

sat (119909 120575) =

sign (119909) |119909| ge 120575

119909

120575|119909| le 120575

(17)

where ℎ is integration step and 119903 is a parameter that deter-mines the tracking speed

In order to validate the filtering performance of trackingdifferentiator as shown in Figure 2 a Simulink simulationmodel is built with an interference input signal as follows

V (119905) = sin 119905 + 119889 (119905) (18)

where 119889(119905) is the uniformly distributed random disturbancesignal with the amplitude 1 and is used to simulate themeasurement noise of the current sampling

The simulation results are shown in Figure 3 it can beseen that the tracking differentiator restores the contaminatedoriginal signal Hence this paper tries to introduce trackingdifferentiator to filter the stator current

33 Build Up Complete Flux Observing Model Based on theabove analysis the complete illustrative diagram of stator fluxobserving model is shown in Figure 4 The current whichis used to count the back electromotive force is filtered bytracking differentiator and then compensate the back electro-motive force and then get the stator flux through the low-passfilter Use the stator flux signal to calculate angular frequencyand cutoff frequency which is fed back to compensationalgorithm and low-pass algorithm

4 Experimental Verification

41 Experimental Platform In order to verify the perfor-mance of the flux observing model in this paper an exper-imental platform is established for the asynchronous motordirect torque control system as shown in Figure 5The exper-imental platform is powered by programmable DC supplythe development suite is the high-voltage motor control andPFC Development Suite v20 from TI Company the MCUis TMS320F28335 parameter of the triphase asynchronousmotor is shown in Table 1 The proposed method involves


+ +

Formula (13) (15)

TD

is120572

TD Rs Rs

+ +

+

minus minus

minus

us120572

es120572

120596c120596e

120596c120596e

1

j120596e + 120596c

1

j120596e + 120596c

120596e 120596c

120595s120573120595s120572

us120573

es120573

is120573

Figure 4 Model of the proposed stator flux observer

Developmentsuite Oscilloscope

Programmable DCLoad experimental

platform

Asynchronousmotor

Figure 5 The experimental platform

Table 1 Motor parameters

Rated value Parameter valueRated speed 1725 rmin Stator resistance 1105ΩRated power 184W Rotor resistance 611ΩRated voltage 220V Self-inductance 0316423HRated torque 1Nsdotm Mutual inductance 0293939HRated current 13 A Number of pole-pairs 2

some division operations requiring a higher speed processorIn this paper we chose DSP28335 up to 150MHz which canmeet the requirements

In the control procedure of experiment the control pe-riod is 100120583119904 The two flux observing methods are comparedin the experiment the result of the traditional method isshown in Figure 1 whereas the result of themethod proposedin this paper is shown in Figure 4 Except the flux observingmethod other conditions are all the same in this experiment

42 Experimental Results Figures 6 and 7 give the comparedexperimental waveforms of the two methods where the tar-geted motor speeds are both 150 rmin It can be seen fromFigure 6 that when the targeted motor speed is higher

the speed waveforms are basically the same of the two meth-ods From Figure 7 it can be seen that the current waveformsare basically the same of the twomethods as well but currentharmonic wave is smaller in the method proposed in thispaper which illustrates that the use of tracking differentiatorin filtering has improved the current fluctuation

With the targeted speed getting slower the traditionalmethod can barely guarantee performance of the motor con-trol Figures 8 and 9 give the compared experimental wave-forms of the two methods where the targeted motor speedsare both 50 rmin It can be seen from Figure 8 that in thetraditional method the motor speed has a huge fluctuationthat is to say the DTC has already become invalid themotor speed is kept around the targeted one and has a smallfluctuation by using the flux observing method of this paperFrom the current waveforms shown in Figure 9 it is clear thatthere are a lot of current harmonic waves and the waveformalso has distortions in the traditional method but currentwaveform in the method of this paper is still in good state

When the targeted speed is set to be 25 rmin the exper-imental result of DTC system using traditional method hasa poor performance the motor operates intermittently andthe control is totally invalid whereas the DTC system basedon flux observing model of this paper can still run smoothlyUsing the method of this paper the speed and the currentwaveforms which are shown in Figure 10 aimed at a speedof 25 rmin The experimental results show that the fluxobservation method proposed in this paper can improve thelow-speed performance of DTC

Under the experimental condition that the speed changessharply from 150 rmin to 50 rmin in the 45 s Figures 11and 12 give the current waveforms of two methods Whenthe speed turns sharply it is obvious that the load andcurrent fluctuation are smaller in the method of this paperWhen the targeted speed is 50 rmin under the experimentalcondition that the load torque changes sharply from 0Nsdotm to03Nsdotm in the 45 s Figures 13 14 and 15 show the speed andstator current and torque waveforms of two methods Whenthe load torque changes sharply the torque of traditionalmethod has a huge fluctuation whereas in the method of thispaper the torque has a small fluctuation Experimental resultsdemonstrate the improved dynamic performance of such fluxobserving method mentioned in this paper

5 Conclusion

To solve the stator flux observation problem of asynchronousmotor the observation scheme that compensates for backEMF firstly and then filters through low-pass filter is pro-posed at themean time tracking differentiator is used to filterthe stator current and the simple voltage model is retainedand all the above lead to the improvement of the dynamicprecision of flux observing This scheme can improve thedynamic and low-speed performance of the DTC system ofinduction motors The accuracy of flux observation is lessinfluenced by the stator frequency mutation and there isless current harmonic waves with efficiently restrained torquefluctuation


2 3 4 5 60

50

100

150

200Sp

eed

(rm

in)

t (s)

(a) Traditional method

2 3 4 5 60

50

100

150

200

Spee

d (r

min

)

t (s)

(b) Method in this paper

Figure 6 Motor speed waveform when target speed is 150 rmin

2 3 4 5 6t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)


2 3 4 5 6t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)


Figure 7 Stator current waveform when target speed is 150 rmin

2 3 4 5 6 70

20406080

100

t (s)

Spee

d (r

min

)


2 3 4 5 6 70

20406080

100

Spee

d (r

min

)

t (s)



2 3 4 5 6 7t (s)

minus1

minus05

0

05

1

Curr

ent (

A)


2 3 4 5 6 7t (s)

minus1

minus05

0

05

1

Curr

ent (

A)



2 3 4 5 6 7 8 9 100

25

50

Spee

d (r

min

)

t (s)

(a) Motor speed

2 3 4 5 6 7 8 9 10minus03

minus02

minus01

0

01

02

03

t (s)

Curr

ent (

A)

(b) Stator current

Figure 10 Waveforms with the proposed method when target speed is 25 rmin


0

50

100

150

200Sp

eed

(rm

in)

4 45 5 55 6

t (s)


4 45 5 55 60

50

100

150

200

Spee

d (r

min

)

t (s)


Figure 11 Motor speed waveform with speed step input from 150 rmin to 50 rmin

4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


Figure 12 Stator current waveform with speed step input from 150 rmin to 50 rmin

3 4 5 6 70

255075

100125

Spee

d (r

min

)

t (s)


3 4 5 6 70255075

100125

Spee

d (r

min

)

t (s)


Figure 13 Motor speed waveform with torque input from 0Nsdotm to 03Nsdotm

3 4 5 6 7minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


3 4 5 6 7t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)


Figure 14 Stator current waveform with torque input from 0Nsdotm to 03Nsdotm

3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)


3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)


Figure 15 Torque waveform with torque input from 0Nsdotm to 03Nsdotm


Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgments

The work is sponsored by the Aerospace Support Technol-ogy Fund (2013-HT-HGD09) the National Laboratory forElectric Vehicles Foundations (NELEV-2013-004) ShandongProvince Outstanding Young Scientists Research AwardFunds (BS2012NJ001) and Beijing Science and TechnologyProject (Z121100005612001)

References

[1] Q-M Cheng Y-M Cheng Y-F Wang and M-M WangldquoOverview of control strategies for AC motorrdquo Power SystemProtection and Control vol 39 no 9 pp 145ndash154 2011

[2] W Liu H Shen L-G Gao and H-Z Rong ldquoStudy on directtorque control of brushless doubly-fed machines used for windpower generationrdquo Power SystemProtection andControl vol 38no 5 pp 77ndash81 2010

[3] Q Wu and C Shao ldquoApplication of tracking-differentiator onstator flux estimation for induction motorrdquo Chinese Journal ofMechanical Engineering vol 44 no 12 pp 291ndash295 2008

[4] C Zhenfeng Z Yanru L Jie et al ldquoSpeed identification forinduction motor based on improved flux observerrdquo Transac-tions of China Electrotechnical Society vol 27 no 4 pp 42ndash472012

[5] L-Z Wang ldquoSimulation of improved direct torque controlsystem for permanent magnet synchronous motorrdquo PowerSystem Protection and Control vol 37 no 19 pp 65ndash68 2009

[6] H R Karimi and A Babazadeh ldquoModeling and output trackingof transverse flux permanent magnet machines using high gainobserver and RBF Neural networkrdquo ISA Transactions vol 44no 4 pp 445ndash456 2005

[7] X Hu H Gao H R Karimi L Wu and C Hu ldquoFuzzy reliabletracking control for flexible air-breathing hypersonic vehiclesrdquoInternational Journal of Fuzzy Systems vol 13 no 4 pp 323ndash334 2011

[8] M Chadli S Aouaouda H R Karimi et al ldquoRobust faulttolerant tracking controller design for a VTOL aircraftrdquo Journalof the Franklin Institute vol 350 no 9 pp 2627ndash2645 2013

[9] H Zhang Y Shi and J Wang ldquoOn energy-to-peak filtering fornonuniformly sampled nonlinear systems a Markovian jumpsystem approachrdquo IEEE Transactions on Fuzzy Systems 2013

[10] H Zhang J Wang and Y Y Wang ldquoRobust filtering forammonia coverage estimation in Diesel engine selective cat-alytic reduction (SCR) systemsrdquo ASME Transactions Journalof Dynamic Systems Measurement and Control vol 135 no 6Article ID 064504 7 pages 2013

[11] H Zhang Y Shi and M X Liu ldquo119867infin

switched filtering fornetworked systems based on delay occurrence probabilitiesrdquoASME Transactions Journal of Dynamic Systems Measurementand Control vol 135 no 6 Article ID 061002 5 pages 2013

[12] H Zhang Y Shi and A Saadat Mehr ldquoOn 119867infin

filtering fordiscrete-time takagi-sugeno fuzzy systemsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 396ndash401 2012

[13] Z-M He Y Liao and D-W Xiang ldquoImprovement of low-pass filter algorithm for stator flux estimatorrdquo Proceedings of

the Chinese Society of Electrical Engineering vol 28 no 18 pp61ndash65 2008

[14] Y Zhang J Wang and H Li ldquoA method of the stator flux EXFestimation for induction motors based on genetic algorithmoptimizingrdquo Transactions of China Electrotechnical Society vol24 no 9 pp 64ndash70 2009

[15] E Zhou X Fu XWu and PDai ldquoFlux observer for eliminationof current measurement error at low stator frequencyrdquo Transac-tions of China Electrotechnical Society vol 26 no 6 pp 67ndash722011

[16] Z Xu and Q Wenlong ldquoA novel compensation of stator fluxestimating in low speedrdquo Advanced Technology of ElectricalEngineering and Energy vol 22 no 3 pp 50ndash54 2003

[17] T-W Chun M-K Choi and B K Bose ldquoA novel start-upscheme of stator flux oriented vector controlled inductionmotor drive without torque jerkrdquo in Proceedings of the 36th IASAnnual Meeting Conference Record of the Industry Applications(IAC rsquo01) pp 148ndash153 Chicago Ill USA October 2001

[18] J-W Gao X-H Wen J-W Chen and F Zhao ldquoNovel motorstator flux observer based on PLLrdquo Proceedings of the ChineseSociety of Electrical Engineering vol 27 no 18 pp 41ndash47 2007

[19] J Q Han and L L Yuan ldquoThe discrete form of a tracking-differentiatorrdquo Journal of Systems Science and MathematicalSciences vol 19 no 3 pp 268ndash273 1999

[20] L-Q Wu H Lin and J-Q Han ldquoStudy of tracking differentia-tor on filteringrdquo Journal of System Simulation vol 16 no 4 pp651ndash652 2004

[21] L Gang L Du R Yifeng et al ldquoImprovement research ofinductionmotor stator flux observationmethodrdquoElectric Drivevol 40 no 8 pp 28ndash30 2010

[22] L Hongbo Z Kai Z Hui et al ldquoAn improved close-looprotor flux observer and speed estimation of induction motorbased on active disturbance rejectionrdquo Transactions of ChinaElectrotechnical Society vol 27 no 4 pp 59ndash64 2012

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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


Although the voltage model is quite simple there are stillsome problems in practical application [15] (1) A small DCbias or drift in the current measurement channel will causethe integrator saturation (2) The stator resistance variationin low stator frequency makes the stator flux amplitude andphase observations have a big error (3)The initial value of theintegral produces the dc bias in the observed flux amplitude

In order to eliminate the effect ofDCbias it is proposed in[16] that the pure integrator should be replaced with low-passfilter but amplitude and phase error of flux can be introducedto it Adopting a programmable cascaded low-pass filter canovercome the effect of zero drift theoretically [17] but thecutoff frequency is of high precision It is difficult to getthe desired effect in practical application and the dynamicperformance is poor Using improved PLL to observe thestator flux is proposed in [18] but this method needs to usethe motor speed information based on permanent magnetsynchronous motor and the system has two convergencepoints meanwhile

The biggest reason why flux observation is inaccurate isthat the DC bias and unbalanced gain exist in current mea-surement channel hence this paper reduces the current mea-surement interference by using tracking differentiator [19 20]to filter the measuring stator current it also restrains the DCcomponent of stator current by using low-pass filter instead ofpure integral and it eliminates the amplitude attenuation andphase error brought by low-pass filter by using the back EMFcompensation algorithm Experimental results demonstrateeffectiveness and improved dynamic performance of suchmethod

2 Traditional Method of StatorFlux Estimation

The voltage model for flux observation is obtained accordingto the stator voltage equation Stator voltage equation and fluxequation are expressed as

u119904= 119877119904i119904+119889120595119904

119889119905 (2)

120595119904= int (u

119904minus i119904119877119904) 119889119905 (3)

where u119904is the stator voltage119877

119904is the stator resistant i

119904is the

stator current and 120595119904is the stator flux

Equation (3) is called the U-I model or the voltage modelof flux estimator Since the formula contains a pure integratorsmall DC bias can cause integral saturation which will resultin flux estimation error So it is usual to replace pure integralwith the first-order low-pass filter in voltage model namelylet the integral input signal go through a high-pass filter firstlyto filtrate the DC component

The stator flux has invariant amplitude in a steady staterevolving in the synchronous frequency which can be ex-pressed as

120595119904=10038161003816100381610038161205951199041003816100381610038161003816 119890119895120596119890119905

=10038161003816100381610038161205951199041003816100381610038161003816 ang120596119890119905

(4)

where 120596119890is the synchronous frequency of the motor

Equations (2) and (4) become

119889120595119904

119889119905= 119895120596119890lowast 120595119904= u119904minus i119904119877119904

120595119904=u119904minus i119904119877119904

119895120596119890

(5)

Equation (5) is the expression of actual stator fluxWhen observing stator flux through low-pass filter the

following equation can be written

119889119904

119889119905+ 120596119888119904= 119895120596119890lowast 119904+ 120596119888119904= u119904minus i119904119877119904

119904=u119904minus i119904119877119904

119895120596119890+ 120596119888

(6)

where 119904is the stator flux which is observed by low-pass filter

and 120596119888is the cutoff frequency of the low-pass filter

The connection between the estimated stator flux 119904and

the actual stator flux 120595119904can be concluded from (5) and (6)

which can be expressed as

119904=

120596119890

radic1205962119890+ 1205962119888

120595119904ang120579

120579 =120587

2minus arctan

120596119890

120596119888

(7)

From (7) it is clear that errors of flux in the amplitudeand the phase result from the replacement of pure integralwith a low-pass filter The higher the cutoff frequency is themore serious the distortion is shown in the flux amplitude andphase

According to the principle of the direct torque control theflux error can affect the steady state and dynamic operationof asynchronous motor directly Direct torque control selectsthe appropriate voltage vector according to the observed fluxand torque Meanwhile the low-pass filter cuts down the fluxamplitude so when the observing flux reached a given valuethe actual flux has already been far beyond that which leadsto the saturation of motor magnetic field Phase shift of theobserved flux can influence the accurate selection of voltagevector as well as the control characteristic of the motorBesides the estimation of torque in direct torque control isalso affected by flux value directly Therefore it is essential tohave amplitude and phase compensation for the result of low-pass filter Traditional method of flux observing is shown inFigure 1

3 Improved Method of Flux Observing

31 Modified Low-Pass Filter (LPF) Compensation AlgorithmMake amplitude and phase compensation for the result of thelow-pass filter as follows

119904G = 120595

119904 (8)

where 119904is the stator fluxwhich is observed by low-pass filter

G is the penalty function and 120595119904is the stator flux




+minus

+minus

us120573

es120573

120595998400s120573

120595s120573

1

j120596e + 120596c

1

j120596e + 120596c

120596e120595998400s120572

120595s120572

us120572

es120572

is120572Rs is120573Rs



G =

radic1205962119890+ 1205962119888

120596119890

119890119895(arctan(120596

119890120596119888minus1205872))

=

radic1205962119890+ 1205962119888

120596119890

119890119895120588(120596119890)

119890119895120588(120596119890)

= cos [120588 (120596119890)] + 119895 sin [120588 (120596

119890)]

cos [120588 (120596119890)] =

120596119890

radic1205962119890+ 1205962119888

sin [120588 (120596119890)] =

120596119888

radic1205962119890+ 1205962119888

(9)


119888is the



E119904= u119904minus i119904119877119904 E

119904G = E

119904 (10)




119904120572and 119890



119890119904120572= 119895119890119904120573 119890

119904120573= minus119895119890

119904120572 (11)


119890119904120572= 119890119904120572+120596119888

120596119890

119890119904120573

119890119904120573= 119890119904120573minus120596119888

120596119890

119890119904120572

(12)



120596119888= 1205960+ 119896120596119894 (13)








120579119894= arctan(

119894119904120573

119894119904120572

) (14)



120596119894(119896)

=120579(119896)minus 120579(119896minus1)

Δ119879 (15)



1199091(119896 + 1) = 119909

1(119896) + ℎ lowast 119909

2(119896)

1199092(119896 + 1)

= 1199092(119896) + ℎ lowast 119891119904119905 (119909

1(119896) minus V (119896) 119909

2(119896) 119903 ℎ

1)

(16)



Subsystem 1

Unit delay

Unit delay

Scope 1

Gain 1

0001

Add 1

Sine wave 1

+

+ Add 2+

+1

z

1

z

x1(k)

x1(k)

x2(k)

x2(k)x2(k + 1)

x2(k

+1)

x1(k + 1)

Vin


0 01 02 03 04 05 06 07 08 09 1minus2

minus1

0

1

2

t (s)


0 01 02 03 04 05 06 07 08 09 1minus2

minus1

0

1

2

t (s)







119891119904119905 (1199091(119896) minus V (119896) 119909

2(119896) 119903 ℎ

1) = minus119903 lowast sat (119892 (119896) 120575)

120575 = ℎ1lowast 119903 120575

1= ℎ1lowast 120575

119890 (119896) = 1199091(119896) minus V (119896)

119910 (119896) = 119890 (119896) + ℎ1lowast 1199092(119896)

119892 (119896) =

1199092(119896) + sign (119910 (119896))

lowast

radic81199031003816100381610038161003816119910 (119896)

1003816100381610038161003816 + 1205752 minus 120575

2

1003816100381610038161003816119910 (119896)1003816100381610038161003816 ge 1205751

1199092(119896) +

119910 (119896)

ℎ1

1003816100381610038161003816119910 (119896)1003816100381610038161003816 le 1205751

sat (119909 120575) =

sign (119909) |119909| ge 120575

119909

120575|119909| le 120575

(17)



V (119905) = sin 119905 + 119889 (119905) (18)







+ +

Formula (13) (15)

TD

is120572

TD Rs Rs

+ +

+

minus minus

minus

us120572

es120572

120596c120596e

120596c120596e

1

j120596e + 120596c

1

j120596e + 120596c

120596e 120596c

120595s120573120595s120572

us120573

es120573

is120573




platform

Asynchronousmotor











5 Conclusion



2 3 4 5 60

50

100

150

200Sp

eed

(rm

in)

t (s)


2 3 4 5 60

50

100

150

200

Spee

d (r

min

)

t (s)



2 3 4 5 6t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)


2 3 4 5 6t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)



2 3 4 5 6 70

20406080

100

t (s)

Spee

d (r

min

)


2 3 4 5 6 70

20406080

100

Spee

d (r

min

)

t (s)



2 3 4 5 6 7t (s)

minus1

minus05

0

05

1

Curr

ent (

A)


2 3 4 5 6 7t (s)

minus1

minus05

0

05

1

Curr

ent (

A)



2 3 4 5 6 7 8 9 100

25

50

Spee

d (r

min

)

t (s)

(a) Motor speed

2 3 4 5 6 7 8 9 10minus03

minus02

minus01

0

01

02

03

t (s)

Curr

ent (

A)

(b) Stator current



0

50

100

150

200Sp

eed

(rm

in)

4 45 5 55 6

t (s)


4 45 5 55 60

50

100

150

200

Spee

d (r

min

)

t (s)



4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)



3 4 5 6 70

255075

100125

Spee

d (r

min

)

t (s)


3 4 5 6 70255075

100125

Spee

d (r

min

)

t (s)



3 4 5 6 7minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


3 4 5 6 7t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)



3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)


3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












+minus

+minus

us120573

es120573

120595998400s120573

120595s120573

1

j120596e + 120596c

1

j120596e + 120596c

120596e120595998400s120572

120595s120572

us120572

es120572

is120572Rs is120573Rs



G =

radic1205962119890+ 1205962119888

120596119890

119890119895(arctan(120596

119890120596119888minus1205872))

=

radic1205962119890+ 1205962119888

120596119890

119890119895120588(120596119890)

119890119895120588(120596119890)

= cos [120588 (120596119890)] + 119895 sin [120588 (120596

119890)]

cos [120588 (120596119890)] =

120596119890

radic1205962119890+ 1205962119888

sin [120588 (120596119890)] =

120596119888

radic1205962119890+ 1205962119888

(9)


119888is the



E119904= u119904minus i119904119877119904 E

119904G = E

119904 (10)




119904120572and 119890



119890119904120572= 119895119890119904120573 119890

119904120573= minus119895119890

119904120572 (11)


119890119904120572= 119890119904120572+120596119888

120596119890

119890119904120573

119890119904120573= 119890119904120573minus120596119888

120596119890

119890119904120572

(12)



120596119888= 1205960+ 119896120596119894 (13)








120579119894= arctan(

119894119904120573

119894119904120572

) (14)



120596119894(119896)

=120579(119896)minus 120579(119896minus1)

Δ119879 (15)



1199091(119896 + 1) = 119909

1(119896) + ℎ lowast 119909

2(119896)

1199092(119896 + 1)

= 1199092(119896) + ℎ lowast 119891119904119905 (119909

1(119896) minus V (119896) 119909

2(119896) 119903 ℎ

1)

(16)



Subsystem 1

Unit delay

Unit delay

Scope 1

Gain 1

0001

Add 1

Sine wave 1

+

+ Add 2+

+1

z

1

z

x1(k)

x1(k)

x2(k)

x2(k)x2(k + 1)

x2(k

+1)

x1(k + 1)

Vin


0 01 02 03 04 05 06 07 08 09 1minus2

minus1

0

1

2

t (s)


0 01 02 03 04 05 06 07 08 09 1minus2

minus1

0

1

2

t (s)







119891119904119905 (1199091(119896) minus V (119896) 119909

2(119896) 119903 ℎ

1) = minus119903 lowast sat (119892 (119896) 120575)

120575 = ℎ1lowast 119903 120575

1= ℎ1lowast 120575

119890 (119896) = 1199091(119896) minus V (119896)

119910 (119896) = 119890 (119896) + ℎ1lowast 1199092(119896)

119892 (119896) =

1199092(119896) + sign (119910 (119896))

lowast

radic81199031003816100381610038161003816119910 (119896)

1003816100381610038161003816 + 1205752 minus 120575

2

1003816100381610038161003816119910 (119896)1003816100381610038161003816 ge 1205751

1199092(119896) +

119910 (119896)

ℎ1

1003816100381610038161003816119910 (119896)1003816100381610038161003816 le 1205751

sat (119909 120575) =

sign (119909) |119909| ge 120575

119909

120575|119909| le 120575

(17)



V (119905) = sin 119905 + 119889 (119905) (18)







+ +

Formula (13) (15)

TD

is120572

TD Rs Rs

+ +

+

minus minus

minus

us120572

es120572

120596c120596e

120596c120596e

1

j120596e + 120596c

1

j120596e + 120596c

120596e 120596c

120595s120573120595s120572

us120573

es120573

is120573




platform

Asynchronousmotor











5 Conclusion



2 3 4 5 60

50

100

150

200Sp

eed

(rm

in)

t (s)


2 3 4 5 60

50

100

150

200

Spee

d (r

min

)

t (s)



2 3 4 5 6t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)


2 3 4 5 6t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)



2 3 4 5 6 70

20406080

100

t (s)

Spee

d (r

min

)


2 3 4 5 6 70

20406080

100

Spee

d (r

min

)

t (s)



2 3 4 5 6 7t (s)

minus1

minus05

0

05

1

Curr

ent (

A)


2 3 4 5 6 7t (s)

minus1

minus05

0

05

1

Curr

ent (

A)



2 3 4 5 6 7 8 9 100

25

50

Spee

d (r

min

)

t (s)

(a) Motor speed

2 3 4 5 6 7 8 9 10minus03

minus02

minus01

0

01

02

03

t (s)

Curr

ent (

A)

(b) Stator current



0

50

100

150

200Sp

eed

(rm

in)

4 45 5 55 6

t (s)


4 45 5 55 60

50

100

150

200

Spee

d (r

min

)

t (s)



4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)



3 4 5 6 70

255075

100125

Spee

d (r

min

)

t (s)


3 4 5 6 70255075

100125

Spee

d (r

min

)

t (s)



3 4 5 6 7minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


3 4 5 6 7t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)



3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)


3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Subsystem 1

Unit delay

Unit delay

Scope 1

Gain 1

0001

Add 1

Sine wave 1

+

+ Add 2+

+1

z

1

z

x1(k)

x1(k)

x2(k)

x2(k)x2(k + 1)

x2(k

+1)

x1(k + 1)

Vin


0 01 02 03 04 05 06 07 08 09 1minus2

minus1

0

1

2

t (s)


0 01 02 03 04 05 06 07 08 09 1minus2

minus1

0

1

2

t (s)







119891119904119905 (1199091(119896) minus V (119896) 119909

2(119896) 119903 ℎ

1) = minus119903 lowast sat (119892 (119896) 120575)

120575 = ℎ1lowast 119903 120575

1= ℎ1lowast 120575

119890 (119896) = 1199091(119896) minus V (119896)

119910 (119896) = 119890 (119896) + ℎ1lowast 1199092(119896)

119892 (119896) =

1199092(119896) + sign (119910 (119896))

lowast

radic81199031003816100381610038161003816119910 (119896)

1003816100381610038161003816 + 1205752 minus 120575

2

1003816100381610038161003816119910 (119896)1003816100381610038161003816 ge 1205751

1199092(119896) +

119910 (119896)

ℎ1

1003816100381610038161003816119910 (119896)1003816100381610038161003816 le 1205751

sat (119909 120575) =

sign (119909) |119909| ge 120575

119909

120575|119909| le 120575

(17)



V (119905) = sin 119905 + 119889 (119905) (18)







+ +

Formula (13) (15)

TD

is120572

TD Rs Rs

+ +

+

minus minus

minus

us120572

es120572

120596c120596e

120596c120596e

1

j120596e + 120596c

1

j120596e + 120596c

120596e 120596c

120595s120573120595s120572

us120573

es120573

is120573




platform

Asynchronousmotor











5 Conclusion



2 3 4 5 60

50

100

150

200Sp

eed

(rm

in)

t (s)


2 3 4 5 60

50

100

150

200

Spee

d (r

min

)

t (s)



2 3 4 5 6t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)


2 3 4 5 6t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)



2 3 4 5 6 70

20406080

100

t (s)

Spee

d (r

min

)


2 3 4 5 6 70

20406080

100

Spee

d (r

min

)

t (s)



2 3 4 5 6 7t (s)

minus1

minus05

0

05

1

Curr

ent (

A)


2 3 4 5 6 7t (s)

minus1

minus05

0

05

1

Curr

ent (

A)



2 3 4 5 6 7 8 9 100

25

50

Spee

d (r

min

)

t (s)

(a) Motor speed

2 3 4 5 6 7 8 9 10minus03

minus02

minus01

0

01

02

03

t (s)

Curr

ent (

A)

(b) Stator current



0

50

100

150

200Sp

eed

(rm

in)

4 45 5 55 6

t (s)


4 45 5 55 60

50

100

150

200

Spee

d (r

min

)

t (s)



4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)



3 4 5 6 70

255075

100125

Spee

d (r

min

)

t (s)


3 4 5 6 70255075

100125

Spee

d (r

min

)

t (s)



3 4 5 6 7minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


3 4 5 6 7t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)



3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)


3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ +

Formula (13) (15)

TD

is120572

TD Rs Rs

+ +

+

minus minus

minus

us120572

es120572

120596c120596e

120596c120596e

1

j120596e + 120596c

1

j120596e + 120596c

120596e 120596c

120595s120573120595s120572

us120573

es120573

is120573




platform

Asynchronousmotor











5 Conclusion



2 3 4 5 60

50

100

150

200Sp

eed

(rm

in)

t (s)


2 3 4 5 60

50

100

150

200

Spee

d (r

min

)

t (s)



2 3 4 5 6t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)


2 3 4 5 6t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)



2 3 4 5 6 70

20406080

100

t (s)

Spee

d (r

min

)


2 3 4 5 6 70

20406080

100

Spee

d (r

min

)

t (s)



2 3 4 5 6 7t (s)

minus1

minus05

0

05

1

Curr

ent (

A)


2 3 4 5 6 7t (s)

minus1

minus05

0

05

1

Curr

ent (

A)



2 3 4 5 6 7 8 9 100

25

50

Spee

d (r

min

)

t (s)

(a) Motor speed

2 3 4 5 6 7 8 9 10minus03

minus02

minus01

0

01

02

03

t (s)

Curr

ent (

A)

(b) Stator current



0

50

100

150

200Sp

eed

(rm

in)

4 45 5 55 6

t (s)


4 45 5 55 60

50

100

150

200

Spee

d (r

min

)

t (s)



4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)



3 4 5 6 70

255075

100125

Spee

d (r

min

)

t (s)


3 4 5 6 70255075

100125

Spee

d (r

min

)

t (s)



3 4 5 6 7minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


3 4 5 6 7t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)



3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)


3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2 3 4 5 60

50

100

150

200Sp

eed

(rm

in)

t (s)


2 3 4 5 60

50

100

150

200

Spee

d (r

min

)

t (s)



2 3 4 5 6t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)


2 3 4 5 6t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)



2 3 4 5 6 70

20406080

100

t (s)

Spee

d (r

min

)


2 3 4 5 6 70

20406080

100

Spee

d (r

min

)

t (s)



2 3 4 5 6 7t (s)

minus1

minus05

0

05

1

Curr

ent (

A)


2 3 4 5 6 7t (s)

minus1

minus05

0

05

1

Curr

ent (

A)



2 3 4 5 6 7 8 9 100

25

50

Spee

d (r

min

)

t (s)

(a) Motor speed

2 3 4 5 6 7 8 9 10minus03

minus02

minus01

0

01

02

03

t (s)

Curr

ent (

A)

(b) Stator current



0

50

100

150

200Sp

eed

(rm

in)

4 45 5 55 6

t (s)


4 45 5 55 60

50

100

150

200

Spee

d (r

min

)

t (s)



4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)



3 4 5 6 70

255075

100125

Spee

d (r

min

)

t (s)


3 4 5 6 70255075

100125

Spee

d (r

min

)

t (s)



3 4 5 6 7minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


3 4 5 6 7t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)



3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)


3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

50

100

150

200Sp

eed

(rm

in)

4 45 5 55 6

t (s)


4 45 5 55 60

50

100

150

200

Spee

d (r

min

)

t (s)



4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


4 45 5 55 6minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)



3 4 5 6 70

255075

100125

Spee

d (r

min

)

t (s)


3 4 5 6 70255075

100125

Spee

d (r

min

)

t (s)



3 4 5 6 7minus15

minus1

minus05

0

05

1

15

t (s)

Curr

ent (

A)


3 4 5 6 7t (s)

minus15

minus1

minus05

0

05

1

15

Curr

ent (

A)



3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)


3 4 5 6 7t (s)

minus01

0

01

02

03

04

Torq

ue (N

middotm)






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article stator flux observer for induction motor

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