Research Article Mathematical Modeling and Analysis of …downloads.hindawi.com/journals/jam/2014/623982.pdf · 2019-07-31 · Research Article Mathematical Modeling and Analysis

Research ArticleMathematical Modeling and Analysis of Different VectorControlled CSI Fed 3-Phase Induction Motor Drive

Arul Prasanna Mark Rajasekaran Vairamani and Gerald Christopher Raj Irudayaraj

Department of EEE PSNA College Engineering amp Technology Dindigul Tamilnadu 624622 India

Correspondence should be addressed to Arul Prasanna Mark arulprasannapsnaceteduin

Received 2 April 2014 Accepted 26 May 2014 Published 9 July 2014

Academic Editor Kerim Guney

Copyright copy 2014 Arul Prasanna Mark et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Themain objective of this paper is to build a simplemathematical competentmodel that describes the circuits and interconnectionsof a 3-phase squirrel cage induction motor used for industrial applications This paper presents the detailed analysis of theoreticalconcepts used in mathematical modeling simulationrsquo and hardware implementation The objective of this work is to compare thedynamic performances of the vector control methods for CSI fed IM drives Based on the results dynamic performances of theproposed drives are individually analysed using the sensitivity tests The tests that are chosen for the comparison are step changesin the reference speed and torque of the motor drive Here the IM is mathematically modeled in different reference frames for inputoutput linearization (IOL) control field oriented control (FOC) and direct torque control method (DTC) which are designedusing hardware equivalent mathematical equations The most important contributions in this paper are mathematical simulationstructure of IM model in rotor flux frame using current and speed that were developed and implemented in MATLAB-SimulinkThe operation and performance of the different vector control methods are verified by simulation usingMATLABSIMULINK andexperimental results

1 Introduction

The alternating current (AC) motor especially asynchronousthree-phase induction motor (IM) has been the motor ofchoice in industrial settings for about the past half century aspower electronics can be used to control its output behaviorBefore that the direct current (DC) motor especially theseparately excited one was widely used because of its easyspeed and torque controllabilityThe IM is a rugged structuremotor because it is brushless and has very fewer internal partsthat needmaintenance or replacementThis makes it cheaperin comparison to other motors such as the DC motor ThusIM and its drive system have been gaining market share inindustry and even in alternative applications such as hybridelectric vehicles For variable speed electric motor applica-tions in low to moderate power pulse width modulation(PWM) with voltage source inverter (VSI) is usually usedHowever the switched voltages produce high voltage slopesover the stator windings which stress the insulations andcause bearing current problems A possible solution is to use

PWM with current source inverter (CSI) Phillips [1] Lipoand Cornell [2] Palaniappan et al [3] Kaimoto et al [4]Krishnan et al [5] Kazmierkowski and Koepcke [6] andHombu et al [7] tried to develop the CSI and make itsuitable for ac motor (especially IM) variable speed drivessince both stator current and voltage waveforms are close tothe sinusoidal waveform the above problems are reducedFurther CSI can be used for high power applications Inthe recent years the high performance CSI fed variablespeed drives especially IM drives have been dominated byvarious control methods The important methods are vectorbased 119881119891 control input output linearization (IOL) controlindirect field oriented control (IFOC) and direct torquecontrol (DTC) as each method has its own unique features119881119891 is by far the simplest drive since it requires no parameterknowledge and is essentially an open loop drive IOL is onetype of nonlinear state feedback control which is completelyinput-output decoupled at all times even in transients IFOCoffers similar dynamic torque control performance to that ofDC motors giving fast near step changes in machine torque

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 623982 13 pageshttpdxdoiorg1011552014623982

2 Journal of Applied Mathematics

and DTC provides simpler control architecture with a similardynamic performance as that of IFOC

Similar studies of vector controlled inverter fed drives arediscussed mostly with the voltage source inverters and veryfew papers are available with the study of vector controlledcurrent source inverters Veerachary [8] Salo and Tuusa [9]and Babaei and Heydari [10] This paper presents a compar-ative analysis of different mathematical modeling of vectorcontrolled CSI fed inductionmotor drives and in particularlythe dynamic responses of the drive motor speed againstchange in speed as well as the torque references in a singlesimulation cycle and verified experimentally and to ascertaindifferent mathematical dynamic IM modeling for CSI feddrives as Holmes et al [11] and to apply control methodson them The MATLAB s-functions are described to designIM modeling and drives Finally the above mentioned drivestrategies are compared as fairly as possible by comparingtheir dynamic responses when there is step change in thereferences (usually stator currents motor speed and loadtorque)

2 Mathematical Modeling of3-Phase Induction Motor Drive

In this section Mathematical model of IM is shown inFigure 1 and described In general IM is used to transformelectrical energy to mechanical energy It consists of elec-tric circuitry electromagnetic circuitry and electromechaniccircuitry Krishnan [12] The main objective of the motormodeling is to build a simple but competent model thatdescribes these circuits and their interconnectionsThe mainpath in themodel is in relationwith the voltage or current andthe stator phases in input and magnetic flux inside the motorand electromagnetic torque in the output

As mentioned earlier the squirrel cage IM is preparedfor industry applications The IM can either be supplied by avoltage source or a current sourceThese sources are assumedto be ideal thismeans that they can supply any desired voltageor current without losses According to Krause and Thomas[13] the basic equations of the IM are the stator voltageequation (1) the rotor voltage equation (2) the equationsof the flux linkages of the stator and the rotor windingsystems (3) and (4) and the torque equation (5) Considerthe following

V119904119904= 119877119904119894119904

119904+ 119904

119904 (1)

V119903119903= 0 = 119877

119903119894119903

119903+ 119903

119903 (2)

120582119886

119904= 120582119886

119898+ 119871119897119904119894119886

119904 (3)

120582119886

119903= 120582119886

119898+ 119871119897119903119894119886

119903 (4)

119879119890= [119862(

120587

2) 120582119886

119898]

119879

119894119886

119904 (5)

The equations are given in vector format Bose [14] Everyvector variable has two components the first component isparallel to the reference frame axis or ldquo119889rdquo axis and the secondcomponent is perpendicular to the reference axis or ldquo119902rdquo axis

The symbols are Vmdashvoltage 119894mdashcurrent 120582119898mdashflux linkage

119877mdashresistance and 119871119897mdashleakage inductance subscript ldquo119904rdquo

indicates the stator winding system ldquo119903rdquo the rotor windingsystem superscript ldquo119904rdquo indicates the stator reference frame(stator coordinates) ldquo119903rdquo the rotor reference frame andldquo119886rdquo any arbitrary reference frame 119879

119890is the electromagnetic

torque 119862(1205872) is a rotation matrix over 1205872 radians the dot( ) indicates a time derivative The stator voltage equationis restricted to the stator reference frame because of thedifferential of the stator flux vector whereas the rotor voltageequation is restricted to the rotor reference frame becauseof the differentiation The flux equations and the torqueequation can be defined in any reference frame denoted bysuperscript ldquo119886rdquo

It is evident from (1) to (5) that themotor voltage flux andtorque can be controlled easily by controlling the current ofthe motor So the CSI is the right choice to control the motordirectly because the inputs of the current fed IM model arethe stator currents

Before these equations are completed to obtain the IMmodel the reference frame of the model in relation to theproblem being investigated and the type of computer (analogor digital) are chosen In order to evaluate the usefulness ofeach reference frame the above equations are simulated usingMatlab software to predict the transient performance of anIM

The mathematical modeling of 3-phase induction motordrive which can be run at any of the three rotating referenceframe models namely stator rotating reference frame rotorrotating reference frame and synchronous rotating referenceframe in MatlabSimulink is shown below in Figure 2

3 Transient Performance of DifferentMathematical Reference Frame Models

Figure 3 shows the transient performance of ldquo119886rdquo phase and119889-axis stator currents in each reference frame when thereis a step change in the reference torque from 0 to full loadat 14 seconds From Figure 3(a) it is evident that in statoror stationary reference frame stator 119889-axis current behavesexactly the same way as do the stator ldquo119886rdquo phase current ofthe motor If the stationary reference frame is used then thestator 119889-axis current is identical to the stator phase currentThis would be useful when interest is specifically confinedto stator variables for example variable speed stator-fedIM drives Similarly if the rotor reference frame is usedthen the rotor 119889-axis current will behave in exactly thesame manner as the rotor ldquo119886rdquo phase current This wouldbe useful when interest is confined to rotor variables onlyas for example variable speed rotor-fed IM drives FromFigure 3(b) it is evident that in rotor rotating reference framestator 119889-axis current does not behave as the stator phasecurrent FromFigure 3(c) it is apparent that in synchronouslyrotating reference frame stator 119889-axis current behaves likeDC quantity If the synchronously rotating reference frameis used then the stator and rotor current behave like DCquantity Thus the controller design is simple and is widelyused in AC drives

Journal of Applied Mathematics 3

Rotor

Stator

4

3

2

1

[0 0]

rot1

rot

1s

1s

-K-

-K-3

2

1

fr

fs

is

irfr = rotor flux

fs = stator flux

is = stator current

Te = electromagnetic torque

Te

ir = rotor current

Flux-current relationship+minus

minus

+minus

+minus

minus

+minus

+minus

times

times

r

Rr

Rs

wm

Gm1

Gm

Gs

Gr

wk

s

K lowast u

K lowast u

∙

(Lrl + Lm ) (Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)

Lm (Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)


(Lsl + Lm)(Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)

Figure 1 Induction motor modeling in arbitrary reference frame

dqreference

frame

Stator

Synchronous

Rotor

Speed

Angle

-K-

Torque

Speed

Frequency

dq

ab

dq2ab

Currents

Amplitude

ab

etakdq

ab2dq

ab2ABC

Step load pu

Mechanicalsystem

1s

1s

Inductionmotor

pu

-K--

f(u)

f(u)

f(u)

0

0

ABC2ab dqABC ab

etak

Variable-amplitude variable-frequency3-phase sinusoidal voltages

K lowast u

wm

wk

wmwm

wmwkws

ws

wo

sTe Te

Te

Te

Tl

Tl

i dq

i dq

i ab

i abi ABC

is

120579r

120579k

120579s

120579s

K lowast u

Figure 2 Mathematical modeling of 3-phase induction motor drive

Figure 4 shows the transient performance of 119902-axis statorcurrent and 119889-rotor flux in each reference frame when thereis a step change in the reference torque from 0 to full loadat 14 seconds From Figures 4(a) and 4(b) it is apparentthat there is a coupling effect between stator 119902-axis currentand rotor 119889-axis flux So independent control is difficultFigure 4(c) shows that the rotor 119889-axis flux and stator 119902-axis current (control variables) have linear relationship with

torque (speed) Thus it is useful in the rotor flux oriented(RFO) control methods

Figure 5 shows the transient performance of 119902-axis statorcurrent and 119889-stator flux in each reference frame when thereis a step change in the reference torque from 0 to full loadat 14 seconds Figures 5(a) and 5(b) state that there is acoupling effect between stator 119902-axis current and stator119889-axis flux and so independent control is difficult From


minus10

0

10

0 05 1 15 2Time (s)

I ac

andI d

c(P

U)

(a)

minus10

0

10

0 05 1 15 2Time (s)

I ac

andI d

c(P

U)

(b)

0 05 1 15 2minus10

0

10

Stator a phase currentStator b-axis current

Step change in load torque

Time (s)

I ac

andI d

c(P

U)

(c)

Figure 3 Simulated results of ldquo119886rdquo phase and 119889-axis stator currents in different rotating reference frames

minus10

0

10

0 05 1 15 2Time in seconds

I qs

and

FLU

Xdr

(a)

minus10

0

10


I qs

and

FLU

Xdr

(b)

0 05 1 15 2minus10

minus5

0

5


Stator q-axis currentRotor d-axis flux

Time (s)

I qs

and

FLU

Xdr

(PU

)

(c)

Figure 4 Simulated results of 119902-axis stator current and 119889-axis rotorflux in different rotating reference frames

minus10

0

10

0 05 1 15 2Time (s)

I qs

and

FLU

Xds

(a)

minus10

0

10

0 05 1 15 2Time (s)

I qs

and

FLU

Xds

(b)

0 05 1 15 2minus10

minus5

0

5


Time (s)

Stator q-axis currentStator d-axis flux

I qs

and

FLU

Xds

(c)

Figure 5 Simulated results of 119902-axis stator current and 119889-axis statorflux in different rotating reference frames


Figure 5(c) it is evident that the stator119889-axis flux and stator 119902-axis current (control variables) have linear relationship withtorque (speed) It is used in the stator flux oriented (SFO)control methods

The basic IM equations from Krause et al [15] aresimulated for three different reference frames usingMATLAB76 software and the results obtained are compared with eachreference frame From the comparison it is observed that ifthe stator rotating reference frame is used then stator 119889-axiscurrent is identical to that of the stator ldquo119886rdquo phase current asshown in Figure 3(a) which is used in stator fed IM drivesSimilarly in rotor reference frame rotor 119889-axis current isidentical to the rotor ldquo119886rdquo phase current and useful in rotor fedIM drives If the synchronous reference frame is used thenthe stator 119889-axis current behaves like DC quantity as shownin Figure 3(c) From Figures 4(c) and 5(c) it is evident thatsynchronous reference frame is useful in RFO and SFO drivecontrol methods In this paper synchronous reference frameis preferred to model the IM in 119881119891 IOL control methodsRFO control is used in IFOCmethod and SFO is used inDTCmethod

4 IOL Control for CSI Fed IM Drives

In this section a type of IOL control method for the CSIfed IM drive is described The controller is designed based

on pole placement technique of Wonham [16] to a 119889-119902 axisstate space linearized model of the drive which includes areduced order rotor current observer A control law has beendeveloped to improve the dynamic response of the drive byusing the concepts from Veerachary [8] which includes feedforward as well as feedback controls

41 Mathematical Modeling of IOL Control Method Theequivalent circuit for the CSI capacitor and IM in a 119889-119902 axis rotating reference frame is shown in Figure 6 Thecorresponding state equation (differential equations) withmotor flux linkages as variables is derived fromWu et al [17]

Veerachary [8] presented the linearized CSI-IM drivestate space model in synchronously rotating reference frameas follows

= 119860119909 + 119861119906 (6)

where

=119889119909

119889119905 (7)

119910 = 119862119909 (8)

where

119909 =

[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]

]

=119889

119889119905

[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]

]

119906 =

[[[[

[

Δ1198811015840

119889

Δ120596119890

120596119887

Δ119879119871

]]]]

]

119910 =[[

[

Δ119894119902119904

Δ120596119903

120596119887

]]

]

119860 =

[[[[[[[[[[[[[[

[

minus1199091015840

1199031199030120596119887

11990901199091015840119903minus 1199092119898

1199091198981198771015840

119903120596119887

11990901199091015840119903minus 1199092119898

minus12059611989001199091015840

119903119909119898(1 minus 119904)

11990901199091015840119903minus 1199092119898

minus1199091198981199091015840

1199031198941015840

1198891199030120596119887

11990901199091015840119903minus 1199092119898

1205961198871199091198981199030

11990901199091015840119903minus 1199092119898

minus11990901198771015840

119903120596119887

11990901199091015840119903minus 1199092119898

1205961198900(1199092

119898minus 11990411990901199091015840

119903)

11990901199091015840119903minus 1199092119898

11990901199091015840

1199031198941015840

1198891199030120596119887

11990901199091015840119903minus 1199092119898

1199041205961198900119909119898

1199091015840119903

1199041205961198900

minus1205961198871198771015840

119903

1199091015840119903

minus

120596119887(1199091198981198941199021199040+ 1199091015840

1199031198941015840

1199021199030)

1199091015840119903

1199091198981198941015840

1198891199030

21198670

1199091198981198941199021199040

21198670

]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[

[

1199091015840

119903120596119887

11990901199091015840119903minus 1199092119898

0 0

minus119909119898120596119887

11990901199091015840119903minus 1199092119898

minus1205961198871198941015840

11988911990300

0 minus

120596119887(1199091198981198941199021199040+ 1199091015840

1199031198941015840

1199021199030)

1199091015840119903

0

0 0 minus1

2119867

]]]]]]]]]]]]]]]

]

119862 = [1 0 0 0

0 0 0 1]

1198811015840

119889=120587119881dc

3radic3 119903

0= 119877119904+1205872

119877dc18

1199090= 119909119904+1205872

119909119889

18

(9)


Rs

Rs

Rr

Vin Idi = 0

VqsLm

Lm

Ll

Ll

p 120582ds

p 120582qs p 120582qr

p 120582dr

+ minus

+minus

c

c

120596r 120582qr

120596r 120582qr

IdsIdi

IqsIqi

Vds

CSI

Iqi = MdIdc Vqs

Figure 6 Equivalent circuit of CSI-IM section

With usual state feedback control there can be the trackingerror in steady state To eliminate this error Smith andDavidson [18] introduced an integral control which includesfeedback as well as feed forward control In this work anoptimal control law is derived by using the concepts of Smithand Davidson [18] and Veerachary [8]

42 Design of Optimal Control Law A new multivariableoptimal controller incorporating state feedback as well asfeed forward control for fast regulation and stability of a CSIfed IM drive is presented in this section The design of thestate feedback controller is based on the industrial regulatortheory together with pole placement technique applied ona linearized 119889-119902 axes state space model of the drive andincludes a reduced order observer to estimate the inaccessiblestates like 119889-119902 axes rotor currents A feed forward control interms of reference and disturbance inputs has been addedto the feedback controller-observer to obtain faster dynamicresponse either from (10) or (11) as follows (Figure 7)

119906 = 1198701119909 + 119870

2int

119905

0

(119910 minus 119910119903) 119889119905 + 119870

119865119865[119889

119910119903

] (10)

or

119906 = 1198701(2times4)

[[[[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]]]]

]

+ 1198702(2times2)

[[[[[

[

int

119905

0

(119894119902119904minus 119894lowast

119902119904) 119889119905

int

119905

0

(120596119903minus 120596lowast

119903) 119889119905

]]]]]

]

+ 119870119865119865(2times3)

[[[[

[

119879119871

119894lowast

119902119904

120596lowast

119903

]]]]

]

(11)

5 IFOC for CSI Fed IM Drives

In this section a high performance current fed indirect RFOcontrol method is described for IMs The IM is mathe-matically modeled in the rotor flux reference frame that is

CSIfed IM

Reduced order

observer

yy

x

r = [ iqslowast120596rlowast] y= [ iqs

120596r

]u

u

y

y

= [Vd

120596e

]

K1(2times4)

K2(2times2)

KFF(2times3)

+ +++

minus

d = TL

int

u998400

Figure 7 Combined feed forward and integral feedback controller

presented The rotor flux orientation is also obtained In thelow speed region the rotor flux components can be synthe-sized more easily with the help of speed and current signalsDesign of indirect vector controller Blaschke [19 20] androtor flux estimator is discussed Implementation of IFOCto CSI fed IM drive and transient response analysis of othersimulation results are also discussed

51 MathematicalModeling of IFOControlMethod The rotorequations of the IM containing flux linkages as variables aregiven by Krause et al [15] as follows

119877119903119894119890

119889119903+ 119901120582119890

119889119903minus (120596119890minus 120596119903) 120582119890

119902119903= 0

119877119903119894119890

119902119903+ 119901120582119890

119902119903+ (120596119890minus 120596119903) 120582119890

119889119903= 0

(12)

where

120596sl = 120596119890 minus 120596119903 (13)

then (12) becomes

119877119903119894119890

119889119903+ 119901120582119890

119889119903minus 120596sl120582

119890

119902119903= 0

119877119903119894119890

119902119903+ 119901120582119890

119902119903+ 120596sl120582

119890

119889119903= 0

(14)

The resultant rotor flux linkage 120582119903 also known as the rotor

flux linkages phasor is assumed to be on the direct axisto reduce the number of variables in the equations by oneMoreover it relates to reality that rotor flux linkages are asingle variable Hence aligning d axis with rotor flux phasoryields

120582119903= 120582119890

119889119903 (15)

120582119890

119902119903= 0 (16)

119901120582119890

119902119903= 0 (17)

Substituting (15) into (14) results in the new rotor equationsas follows

119877119903119894119890

119889119903+ 119901120582119903= 0

119877119903119894119890

119902119903+ 120596sl120582119903 = 0

(18)


The rotor currents in terms of the stator currents are derivedfrom (18) as

119894119890

119902119903= minus

119871119898

119871119903

119894119890

119902119904

119894119890

119889119903=120582119903

119871119903

minus119871119898

119871119903

119894119890

119889119904

(19)

119894119890

119889119904= 119894119891= [1 +

119871119903

119877119903

119901]120582119903

119871119898

119894119890

119889119904= 119894119891= [1 + 120591

119903119901]

120582119903

119871119898

(20)

120596sl =119877119903119871119898

119871119903

119894119890

119902119904

120582119903

120596sl =119877119903119871119898

119871119903

119894119879

120582119903

120596sl =119871119898

120591119903

119894119879

120582119903

(21)

The 119902- and 119889-axes currents are relabeled as torque (119894119879) and

flux producing (119894119891) components of the stator-current phasor

respectively 120591119903denotes rotor time constant Similarly by the

same substitution of the rotor currents from (19) into torqueexpression the electromagnetic torque is derived as

119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904minus 120582119890

119902119903119894119890

119889119904)

119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904)

119879119890= 119870119905119890120582119903119894119890

119902119904= 119870119905119890120582119903119894119879

(22)

where the torque constant119870119905119890is defined as

119870119905119890=3

2

119875

2

119871119898

119871119903

(23)

Note that the torque is proportional to the product of rotorflux linkage and the stator 119902-axis current This resemblesthe air gap torque expression of the DC motor which isproportional to the product of the field flux linkages and thearmature current If the rotor flux linkage is maintained asconstant the torque is proportional to the torque producingcomponent of the stator current This is in relation to theseparately excited DC motor with armature current controlwhere the torque is proportional to the armature currentwhen the field current is constant Further here too the timeconstant is measured in the order of a few milliseconds Therotor flux linkages and air gap torque given in (21) and (22)respectively complete the transformation of the IM into anequivalent separately-excited dc motor from a control pointof view

52 Design of IFO Controller The IFO controller was de-signed using the concepts of Vas [21] and Salo and Tuusa [9]

From that the stator-current phasor is the phasor sum of the119889- and 119902-axis stator currents in any frames and it is given as

119894119904= radic(119894119890

119902119904)2

+ (119894119890

119889119904)2

(24)

and the 119889119902 axes (2120601) to (3120601) abc phase current relationship isobtained from

[

[

119894119890

119902119904

119894119890

119889119904

]

]

=2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+4120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+4120587

3)

]]]

]

[

[

119894119886119904

119894119887119904

119894119888119904

]

]

(25)

And it can be written as

119894119902119889= [119879] [119894

119886119887119888] (26)

where

119894119902119889= [119894119890

119902119904119894119890

119889119904]119905

(27)

119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905

(28)

[119879] =2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+2120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+2120587

3)

]]]

]

(29)

where 119894119886119904 119894119887119904 and 119894

119888119904are the three phase stator currents Note

that the elements in the 119879 matrix are cosinusoidal functionsof electrical angle 120579

119891 The electrical field angle in this case is

that of the rotor flux-linkages phasor and is obtained as thesum of the rotor and slip angles as follows

120579119891= 120579119903+ 120579sl (30)

and the slip angle is obtained by integrating the slip speed andis given as

120579sl = int120596sl (31)

IFO controller developed from these derivations accepts thetorque and flux requests and generates the torque and fluxproducing components of the stator-current phasor and theslip-angle commandsThe command values are denoted withasterisks throughout this thesis From (20) (21) and (23) thecommand values of 119894

119879 119894119891 and 120596sl are obtained as follows

119894lowast

119879=

119879lowast

119890

119870119905119890120582lowast119903

= (2

3) (

2

119875)119879lowast

119890

120582lowast119903

119871119903

119871119898

(32)

119894lowast

119891= (1 + 119901

119871119903

119877119903

)120582lowast

119903

119871119898

(33)

120596lowast

sl =119877119903119871119898

119871119903

119894lowast

119879

120582lowast119903

(34)


The command slip angle 120579lowastsl is generated by integrating 120596lowastsl The torque angle command is obtained as the arctangentof 119894lowast119879and 119894lowast119891 The field angle is obtained by summing the

command slip angle and rotor angleWith the torque and fluxproducing components of the stator current commands androtor field angle the 119902119889 axes current commands (119886119887119888 phasecurrent commands) are obtained as follows The relevantsteps involved in the realization of the IFO controller are asfollows

[

[

119894lowast

119902119904

119894lowast

119889119904

]

]

= [

[

cos 120579119891

sin 120579119891

minus sin 120579119891

cos 120579119891

]

]

[

[

119894lowast

119879

119894lowast

119891

]

]

(35)

[[[[

[

119894lowast

119886119904

119894lowast

119887119904

119894lowast

119888119904

]]]]

]

= [119879minus1

]

[[[[

[

119894lowast

119902119904

119894lowast

119889119904

0

]]]]

]

(36)

where

119879minus1

=

[[[[[[[

[

1 0 1

minus1

2minusradic3

21

minus1

2

radic3

21

]]]]]]]

]

(37)

By using (35) to (37) the stator 119902- and 119889-axes and abccurrent commands are derived as

119894lowast

119902119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119889119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 cos 120579lowast

119904

119894lowast

119886119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119887119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904minus2120587

3)

119894lowast

119888119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904+2120587

3)

(38)

where

120579lowast

119904= 120579119891+ 120579lowast

119879= 120579119903+ 120579lowast

sl + 120579lowast

119879 (39)

The implementation of IFOC on a CSI fed IM is shownin Figures 8 and 9 The torque command 119894lowast

119879is generated

as a function of the speed error signal generally processedthrough a PI controller The flux command 119894lowast

119891can be given

directly or as a function of speed The rotor position 120579119903can

be measured with an encoder and converted into necessarydigital information for feedbackThere has been a substantialamount of research in the development of rotor flux observersfor field orientation that are compensated for variationsin parameters by their feedback corrections Digital imple-mentation of integrators for the estimation of rotor flux ofan IM from the stator voltages and stator currents posesproblems associated with the offset in the sensor amplifiers

CSI

IM

PWMswitching

logic

Rotor speed

regulator

Stator current

generator

DC link currentias

lowast

ibslowast

i cslowast

120579riTlowast is

iflowast

iflowast and iT

lowast120582rlowast

120596rlowast

120596sllowast

120596r

Telowast

(1 + pLr

Rr

) 120582rlowast

Lm

RrLm

Lr

iTlowast

120582rlowast

4

3P

Telowast

120582rlowast

Lr

Lm

+minus

Figure 8 Simplified diagram for IFOC method

islowast = radic(iTlowast)2 +(iflowast)2 ias

lowast = |islowast|sin120579s

lowast|islowast|

ibslowast = |is

lowast|sin120579slowast minus

2120587

3

icslowast = |is

lowast|sin120579slowast +

2120587

3

120579r 120579r

iTlowast

iflowast

120579Tlowast

120579slowast

120596sllowast 120579sl

lowastint

iaslowast

ibslowast

i cslowast

tanminus1 iTlowast

iflowast

+++

Figure 9 Stator current generator

Traditional low-pass filters can replace the integrator Soin this work rotor flux estimated using low pass filter isdescribed The instantaneous flux linkage can be computedusing the measured d-axis stator current using (40) which isreferred to as the current model as follows

120582119903=119871119898119894119889119904

1 + 120591119903119904 (40)

The theoretical concepts of IFOC method for CSI fed IMdrive discussed in Sections 51 and 52 are verified throughsimulation using the Simulink toolbox of MATLAB Thedetailed simulations results are presented in Section 71

6 DTC for CSI Fed IM Drives

Variable speed drives are used in all industries to controlprecisely the speed of electric motors driving loads rangingfrom pumps and fans to complex drives on paper machinesrolling mill cranes and similar drives The most modernmethod for these drives is direct torque and stator flux vectorcontrol method usually called DTC The VSI fed version hasbeen realized in an industrial way by ABB by using thetheoretical background proposed by Takahashi and Ohmori[22] and Depenbrock [23] This solution is based both onfield oriented control (FOC) as well as on the direct self-control theory The idea is that motor flux and torque areused as primary control variables which is contrary to thewayin which traditional AC drives control input frequency andvoltage but is in principle similar to what is done with a DCdrive where it is much more straightforward to achieve


Casadei et al [24] has described that by controllingmotor torque directly DTCprovides dynamic speed accuracyequivalent to closed loop AC and DC systems and torqueresponse times that are 10 times faster It is also claimedthat the DTC does not generate noise like that produced byconventional PWM AC drives This work describes a newmodular approach to implement DTC method on a CSIfed IM drive the simulation implementation procedure Thedynamic response of the drive is analyzed and presented

61 Mathematical Modeling of DTC Control Method TheDTC method is developed from Babaei and Heydari [10]From Krause et al [15] in the stationary reference frame thestator voltage and flux equation can be written as

V119904= 119877119904119894119904+119889120582119904

119889119905 (41)

120582119904= int (V

119904minus 119877119904119894119904) 119889119905 = 120582

120572119904+ 119895120582120573119904 (42)

10038161003816100381610038161205821199041003816100381610038161003816 = radic120582

2

120572119904+ 1205822

120573119904 (43)

ang120593119904= tanminus1 (

120582120573119904

120582120572119904

) (44)

where 119877119904is the stator resistance and 120582

119904is the stator flux

vector in the stationary reference frame |120582119904| and ang120593

119904are the

amplitude and position of stator flux vector respectivelyIn DTC of CSI fed IM it is desirable to determine the

stator current reference so that the torque and stator fluxfollow their reference values Babaei and Heydari [10] Theproposed DTC system is based on the stator flux oriented(SFO) reference frame In this rotating reference frame it iswritten as

120582119902119904= 0 120582

119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)

where 119889 and 119902 are the real and imaginary axes in the SFOreference frame Therefore the torque equation is rewrittenas

119879119890=3

2(119875

2)1

120596119887

(120582119889119904119894119902119904) (46)

where 119894119902119904is the imaginary part of the stator current vector

in the SFO reference frame Therefore if |120582lowast119904| and 119879lowast

119890are

stator flux magnitude and torque references respectively theimaginary part of the stator current reference vector whichleads to torque and stator flux magnitude references is asfollows

119894lowast

119902119904=2

3

1

119875

119879lowast

119890

1003816100381610038161003816120582lowast

119904

1003816100381610038161003816

(47)

Because of the existence of a capacitor in the controlsystem and its corresponding losses when (47) is used tocontrol the torque for all rotor speeds the control system willnot be accurate enough

Currentlimiter

Telowast

Te

|120582slowast|

2

3

1

P

PI

Telowast

|120582slowast |

+

+

+minus iqs

lowast

Figure 10 119902-axis reference current calculator

In order to have amore accurate torque control the blockdiagram shown in Figure 10 is used to calculate 119894lowast

119902119904This block

diagram is a combination of (47) and a simple PI controllerThis controller removes the steady-state error and leads to anaccurate control The coefficients of this controller must besmall enough so that it remains stable and the ripples in 119894lowast

119902119904

are limited because a ripple of 119894lowast119902119904results in a ripple of the

generated torque If only a PI is used 119894lowast119902119904

will have a largevalue of ripple and the torque response time will be verylargeTherefore in order to have a faster transient response oftorque it is necessary to use (47) In DTC method with VSIthe real part of voltage vector (V

119889) in the stator flux reference

frame leads to direct control of stator flux magnitudeTherefore using the real part of the stator current vector

in the stator flux reference frame it is possible to control theflux magnitude indirectly In order to obtain the real part ofthe stator current reference a simple PI controller is usedTheinput of this PI controller is the error that resulted from thecomparison between the reference stator flux magnitude andthe estimated flux value In order to control themotor currentin all cases especially in the starting mode the output ofthis PI controller must be limited between proper valuesTheoutput signal of the PI controller is the real part of referencestator current 119894lowast

119889119904

Therefore the stator current reference vector that leads tocontrol the IM stator flux and torque is as follows

119894lowast

119904(119889119902)= 119894lowast

119889119904+ 119895119894lowast

119902119904 (48)

1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 =radic119894lowast2

119889119904+ 119894lowast2119902119904 (49)

The obtained reference current vector is in the statorflux reference frame In order to generate it by SVM of theCSI it must be transferred to the stationary reference frameTo do that the stator flux position is used to transfer thisvector from the stator flux reference frame to the stationaryreference frame as follows

119894lowast

119904(120572120573)= 119894lowast

120572119904+ 119895119894lowast

120573119904= (119894lowast

119889119904+ 119895119894lowast

119902119904) sdot 119890119895120593119904

=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 ang (120593119904 + tanminus1 (119894lowast

119902119904

119894lowast

119889119904

))

(50)

This reference current vectormust be generated in theCSIfirst and then applied to the motor after removing its har-monics with filtering capacitorThis current has a nearly sinu-soidal waveform that leads to a nearly sinusoidal voltage inmotor terminals after feeding the motor and passing through


Stator flux torque

estimator

PI

CSI

SVM IM

Gate Signals

PI

120582slowast

120591elowast

120582s

120582s

120591e

ang120593s

120591e

iqlowast

idlowast

dq

to120572120573

iaib

vb

va

|islowast| and idc

lowast

idclowast

+minus

+minus i120572

lowast

i120573lowast

Figure 11 Block diagram schematic of the proposed DTC

motor impedances The speed and stator flux control of anIM with DTC method is shown in Figure 11 Separate PIcontrollers are used to control the speed and flux The inputto the speed controller is the error caused by comparingthe reference speed and the rotor speed The output of thiscontroller is the imaginary part of stator current vector 119894lowast

119902119904

which is limited to a proper value

7 Results and Discussion

71 Simulation Results The simulationmodels of IOL IFOCand DTC control methods for CSI fed IM drive systemsare developed and simulated using MATLAB simulationsoftware From the simulation results dynamic performancesof the developed drives have been individually analyzedmainly using the sensitivity tests Here the results of all thedrives are compared with one another The tests that arechosen for the comparison are step changes in the referencespeed and torque of the motor drive

The first test is step change in reference speed which isuseful when trying to change from one speed to anotherThe next test is step change in reference torque that wouldbe useful in identifying the controller with a high torqueresponse Change in speed reference is the first test that isdone to compare the motor drives

The results of this test can be seen in the Figures 12 13 and14The speed command is varied from0 to its rated valueThespeed response of IOL drive is faster and has the settling timeof 05 second range while DTC and IFOC drives performsimilarly but slower than IOL (settling time of less than 1second)

The second test as mentioned earlier is the torque steptestThe results of this test can be seen in Figures 12 to 14Thetorque reference is varied from 0 to its rated value The speedresponse of IFOC drive is faster and has the settling timeof 02 seconds while IOL and DTC drives perform similarly(settling time of less than 05 second)

During this test the motor torque response of DTCdrive is almost instantaneous within the 02 second rangewhile IFOC drive closely follows behind it IOL and vectorbased 119881119891 have slower torque response than all other VectorControl techniques

Table 1 Prototype motor parameters

1 Hp 3 phase star connected 4 pole 415V 18 A 50Hz

Stator resistance 0087Ω Stator amp rotorleakage reactance 08119890minus3H

Rotor resistance 0228Ω Magnetizingreactance 347119890minus3H

72 Experimental Results Brief experimental results arepresented for IOL method and DTC method usingTMS320F2812 DSP based hardware setup Figure 15 showsthe schematic diagram for the hardware setup of theproposed laboratory prototype CSI fed induction motordrive The induction motor ratings are given in Table 1 Theinverter consists of CM75TL-12NF The gate driving signalis developed by using Texas instruments DSP processorTMS320F2812 DSP The software used to develop programsfor TMS320F2812 DSP is by code composer studio andMatlabSimulinkThe processor contains the program that isdownloaded to the computer

The clock speed of the DSP is 150MHz and it is capableof 32-bit operations The on board available flash memoryis 2048Mb It was created specifically for motor controloperation and therefore Parkrsquos and Clarkrsquos transformationsare conveniently built in Another convenient feature is thatit has sixteen 12-bit ADC pins that allow for a high degree ofprecision while taking many possible measurements

The above experimental result of IOL method of vectorcontrol technique in Figure 16 shows the fast settling timewith feed forward control compared to feedback techniqueSimilarly the experimental result shown in Figure 17 depictsa greater torque response within 02 seconds comparedto other vector control techniques With this experimentalverification it is clear that IOL is adaptive for speed responseand DTC is more suitable for torque response controlleddrives

From Tables 2 and 3 it seems that the dynamic speed andtorque performance of all four drives using change in speedreference It is evident from the table that IOL drive hasthe best speed response with DTC and IFOC occupyingpositions next to it It is also evident that DTC drive has thebest instantaneous torque response with IFOC and IOL nextto it

8 Conclusion

This paper presents a new approach in mathematical mod-eling of a 3-phase induction motor drive Mathematics con-cepts are used in developing the simulation and implementa-tion of IOL IFOC and DTC methods for CSI fed IM drivesIn this paper a rotor flux based reference frame controlmethod for PWM CSI fed IM with simplified simulationcircuits has been developed in which the motor speed andtorque follow the references closely in all the three vectorcontrol methods However a CSI fed induction motor drivehas some drawbacks such as slower dynamic response andbulky overall size due to the requirement of large smoothing


15

minus05

0

05

1

Spee

d (P

U)

0 1 2 3 4 5Time (s)

Without FF controlWith FF control

(a)

0 1 2 3 4 5

minus2

0

2

Time (s)

Torq

ue (P

U)


(b)

Figure 12 Speed torque response of IOL control method

0

05

1

Spee

d (P

U)

0 1 2 3 4 5minus4

minus2

0

24

Torq

ue (P

U)

Time (s)

Reference speedStep increase in speed command

Step decrease in speed command

Step increase in torque command

Step decrease in torque command Reference torque

Motor torque

Motor speed

Response due to change intorque reference

Response due to change in

speed reference

Figure 13 Speed torque response of IFOC method

0

05

1

Spee

d (P

U)

0 1 2 3 4 5minus2minus1

012

Time (s)

Torq

ue (P

U)


Step increase in speed commandReference speed Motor speed


Response due to change inspeed reference

Motor torque

Step decrease in torque commandReference torque


Figure 14 Speed torque response of DTC method

inductor at its dc source end and also a CSI drive systemwithon-line control strategy is more complex compared to aVSI drive due to the CSI gating requirements But depend-ing on some particular applications its advantages can attimes outweigh its disadvantages hence rendering it as asuitable choice for certain applications From the resultsit is concluded that DTC drive is the top most performerfor wide range of torque control applications (load varying

TMS320F2812DSP

ADC

PWM

-EV

B

PWM

-EVA

Supplyfilter

3120601controlled

rectifier3120601 CSI inverter

3120601IM

3120601supply

Texasinstruments

DSP processor

Speed sensor output

(Nr) Rpm

CM75TL-12NFseries

IGBT (6-PAC)

Ldc

Is (Amps)

Figure 15 Schematic diagram of the hardware setup of CSI fed IM

Torque output with feedforward control

1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads

Torque output withoutfeed forward control

Speed output with feedforward control

Speed output withoutfeed forward control

Figure 16 Experimental result torque-speed response of IOLcontrol method

applications) with respect to change in speed as well as torquereferences IOL drive is the good performer for speed controlapplications with respect to change in speed reference andIFOC drive is the good performer for speed control applica-tions with respect to change in torque reference Hence forindustrial drive applications with the requirement of torque


Step increase in speed command

Reference torque


Step decrease in speedcommand

Actual motor torque

Torque reference

Speed reference

Actual speed

radsec Actual motor speed 120596m = 140

1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads

Figure 17 Experimental result of torque-speed response of DTCcontrol method

Table 2 Transient response results for step change in speedreference command

Controlmethods

Settling time(119905119904) in ms

Rising time 119905119903

in msMaximum peakovershoot (119872

119901)

Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139

Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149

Table 3 Transient Response results for Step change in Torquereference Command

Controlmethods




119901)



to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod

is efficient with change in speed reference and IFOC methodis efficient with change in torque reference

Symbols

120596119887 Base speed

119903

119903 Change in rotor flux wrt in rotor reference frame

119904

119904 Change in stator flux wrt in stator reference frame

119870119894 Current controller gain

120591119894 Current controller time constant120591119894119891 Current filter time constant

120582119889119903 119889-axis rotor flux

120582119890

119889119903 119889-axis rotor flux in synchronous reference frame

120582119889119904 119889-axis stator flux

119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux

120582119890


120582119902119904 119902-axis stator flux

120582119903 Rotor flux

119871119903 Rotor inductance

1199091015840

119903 Rotor reactance referred from stator

1198771015840

119903 Rotor resistance referred from stator

120596119903 Rotor speed

120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant

119870119905119890 Torque coefficient

119894120572119904 120572-axis stator current in stationary reference frame


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972

[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975

[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979

[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982

[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983


[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985

[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987

[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002

[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005

[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010

[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012

[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002

[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965

[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006

[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002

[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967

[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994

[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972

[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971

[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972

[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992

[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989

[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006

Submit your manuscripts athttpwwwhindawicom

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Algebra

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Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


and DTC provides simpler control architecture with a similardynamic performance as that of IFOC

Similar studies of vector controlled inverter fed drives arediscussed mostly with the voltage source inverters and veryfew papers are available with the study of vector controlledcurrent source inverters Veerachary [8] Salo and Tuusa [9]and Babaei and Heydari [10] This paper presents a compar-ative analysis of different mathematical modeling of vectorcontrolled CSI fed inductionmotor drives and in particularlythe dynamic responses of the drive motor speed againstchange in speed as well as the torque references in a singlesimulation cycle and verified experimentally and to ascertaindifferent mathematical dynamic IM modeling for CSI feddrives as Holmes et al [11] and to apply control methodson them The MATLAB s-functions are described to designIM modeling and drives Finally the above mentioned drivestrategies are compared as fairly as possible by comparingtheir dynamic responses when there is step change in thereferences (usually stator currents motor speed and loadtorque)

2 Mathematical Modeling of3-Phase Induction Motor Drive

In this section Mathematical model of IM is shown inFigure 1 and described In general IM is used to transformelectrical energy to mechanical energy It consists of elec-tric circuitry electromagnetic circuitry and electromechaniccircuitry Krishnan [12] The main objective of the motormodeling is to build a simple but competent model thatdescribes these circuits and their interconnectionsThe mainpath in themodel is in relationwith the voltage or current andthe stator phases in input and magnetic flux inside the motorand electromagnetic torque in the output

As mentioned earlier the squirrel cage IM is preparedfor industry applications The IM can either be supplied by avoltage source or a current sourceThese sources are assumedto be ideal thismeans that they can supply any desired voltageor current without losses According to Krause and Thomas[13] the basic equations of the IM are the stator voltageequation (1) the rotor voltage equation (2) the equationsof the flux linkages of the stator and the rotor windingsystems (3) and (4) and the torque equation (5) Considerthe following

V119904119904= 119877119904119894119904

119904+ 119904

119904 (1)

V119903119903= 0 = 119877

119903119894119903

119903+ 119903

119903 (2)

120582119886

119904= 120582119886

119898+ 119871119897119904119894119886

119904 (3)

120582119886

119903= 120582119886

119898+ 119871119897119903119894119886

119903 (4)

119879119890= [119862(

120587

2) 120582119886

119898]

119879

119894119886

119904 (5)

The equations are given in vector format Bose [14] Everyvector variable has two components the first component isparallel to the reference frame axis or ldquo119889rdquo axis and the secondcomponent is perpendicular to the reference axis or ldquo119902rdquo axis

The symbols are Vmdashvoltage 119894mdashcurrent 120582119898mdashflux linkage

119877mdashresistance and 119871119897mdashleakage inductance subscript ldquo119904rdquo

indicates the stator winding system ldquo119903rdquo the rotor windingsystem superscript ldquo119904rdquo indicates the stator reference frame(stator coordinates) ldquo119903rdquo the rotor reference frame andldquo119886rdquo any arbitrary reference frame 119879

119890is the electromagnetic

torque 119862(1205872) is a rotation matrix over 1205872 radians the dot( ) indicates a time derivative The stator voltage equationis restricted to the stator reference frame because of thedifferential of the stator flux vector whereas the rotor voltageequation is restricted to the rotor reference frame becauseof the differentiation The flux equations and the torqueequation can be defined in any reference frame denoted bysuperscript ldquo119886rdquo

It is evident from (1) to (5) that themotor voltage flux andtorque can be controlled easily by controlling the current ofthe motor So the CSI is the right choice to control the motordirectly because the inputs of the current fed IM model arethe stator currents

Before these equations are completed to obtain the IMmodel the reference frame of the model in relation to theproblem being investigated and the type of computer (analogor digital) are chosen In order to evaluate the usefulness ofeach reference frame the above equations are simulated usingMatlab software to predict the transient performance of anIM

The mathematical modeling of 3-phase induction motordrive which can be run at any of the three rotating referenceframe models namely stator rotating reference frame rotorrotating reference frame and synchronous rotating referenceframe in MatlabSimulink is shown below in Figure 2

3 Transient Performance of DifferentMathematical Reference Frame Models

Figure 3 shows the transient performance of ldquo119886rdquo phase and119889-axis stator currents in each reference frame when thereis a step change in the reference torque from 0 to full loadat 14 seconds From Figure 3(a) it is evident that in statoror stationary reference frame stator 119889-axis current behavesexactly the same way as do the stator ldquo119886rdquo phase current ofthe motor If the stationary reference frame is used then thestator 119889-axis current is identical to the stator phase currentThis would be useful when interest is specifically confinedto stator variables for example variable speed stator-fedIM drives Similarly if the rotor reference frame is usedthen the rotor 119889-axis current will behave in exactly thesame manner as the rotor ldquo119886rdquo phase current This wouldbe useful when interest is confined to rotor variables onlyas for example variable speed rotor-fed IM drives FromFigure 3(b) it is evident that in rotor rotating reference framestator 119889-axis current does not behave as the stator phasecurrent FromFigure 3(c) it is apparent that in synchronouslyrotating reference frame stator 119889-axis current behaves likeDC quantity If the synchronously rotating reference frameis used then the stator and rotor current behave like DCquantity Thus the controller design is simple and is widelyused in AC drives


Rotor

Stator

4

3

2

1

[0 0]

rot1

rot

1s

1s

-K-

-K-3

2

1

fr

fs

is

irfr = rotor flux

fs = stator flux

is = stator current


Te

ir = rotor current


minus

+minus

+minus

minus

+minus

+minus

times

times

r

Rr

Rs

wm

Gm1

Gm

Gs

Gr

wk

s

K lowast u

K lowast u

∙






dqreference

frame

Stator

Synchronous

Rotor

Speed

Angle

-K-

Torque

Speed

Frequency

dq

ab

dq2ab

Currents

Amplitude

ab

etakdq

ab2dq

ab2ABC

Step load pu

Mechanicalsystem

1s

1s

Inductionmotor

pu

-K--

f(u)

f(u)

f(u)

0

0

ABC2ab dqABC ab

etak


K lowast u

wm

wk

wmwm

wmwkws

ws

wo

sTe Te

Te

Te

Tl

Tl

i dq

i dq

i ab

i abi ABC

is

120579r

120579k

120579s

120579s

K lowast u






minus10

0

10

0 05 1 15 2Time (s)

I ac

andI d

c(P

U)

(a)

minus10

0

10

0 05 1 15 2Time (s)

I ac

andI d

c(P

U)

(b)

0 05 1 15 2minus10

0

10



Time (s)

I ac

andI d

c(P

U)

(c)


minus10

0

10


I qs

and

FLU

Xdr

(a)

minus10

0

10


I qs

and

FLU

Xdr

(b)

0 05 1 15 2minus10

minus5

0

5



Time (s)

I qs

and

FLU

Xdr

(PU

)

(c)


minus10

0

10

0 05 1 15 2Time (s)

I qs

and

FLU

Xds

(a)

minus10

0

10

0 05 1 15 2Time (s)

I qs

and

FLU

Xds

(b)

0 05 1 15 2minus10

minus5

0

5


Time (s)


I qs

and

FLU

Xds

(c)










= 119860119909 + 119861119906 (6)

where

=119889119909

119889119905 (7)

119910 = 119862119909 (8)

where

119909 =

[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]

]

=119889

119889119905

[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]

]

119906 =

[[[[

[

Δ1198811015840

119889

Δ120596119890

120596119887

Δ119879119871

]]]]

]

119910 =[[

[

Δ119894119902119904

Δ120596119903

120596119887

]]

]

119860 =

[[[[[[[[[[[[[[

[

minus1199091015840

1199031199030120596119887

11990901199091015840119903minus 1199092119898

1199091198981198771015840

119903120596119887

11990901199091015840119903minus 1199092119898

minus12059611989001199091015840

119903119909119898(1 minus 119904)

11990901199091015840119903minus 1199092119898

minus1199091198981199091015840

1199031198941015840

1198891199030120596119887

11990901199091015840119903minus 1199092119898

1205961198871199091198981199030

11990901199091015840119903minus 1199092119898

minus11990901198771015840

119903120596119887

11990901199091015840119903minus 1199092119898

1205961198900(1199092

119898minus 11990411990901199091015840

119903)

11990901199091015840119903minus 1199092119898

11990901199091015840

1199031198941015840

1198891199030120596119887

11990901199091015840119903minus 1199092119898

1199041205961198900119909119898

1199091015840119903

1199041205961198900

minus1205961198871198771015840

119903

1199091015840119903

minus

120596119887(1199091198981198941199021199040+ 1199091015840

1199031198941015840

1199021199030)

1199091015840119903

1199091198981198941015840

1198891199030

21198670

1199091198981198941199021199040

21198670

]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[

[

1199091015840

119903120596119887

11990901199091015840119903minus 1199092119898

0 0

minus119909119898120596119887

11990901199091015840119903minus 1199092119898

minus1205961198871198941015840

11988911990300

0 minus

120596119887(1199091198981198941199021199040+ 1199091015840

1199031198941015840

1199021199030)

1199091015840119903

0

0 0 minus1

2119867

]]]]]]]]]]]]]]]

]

119862 = [1 0 0 0

0 0 0 1]

1198811015840

119889=120587119881dc

3radic3 119903

0= 119877119904+1205872

119877dc18

1199090= 119909119904+1205872

119909119889

18

(9)


Rs

Rs

Rr

Vin Idi = 0

VqsLm

Lm

Ll

Ll

p 120582ds

p 120582qs p 120582qr

p 120582dr

+ minus

+minus

c

c

120596r 120582qr

120596r 120582qr

IdsIdi

IqsIqi

Vds

CSI

Iqi = MdIdc Vqs




119906 = 1198701119909 + 119870

2int

119905

0

(119910 minus 119910119903) 119889119905 + 119870

119865119865[119889

119910119903

] (10)

or

119906 = 1198701(2times4)

[[[[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]]]]

]

+ 1198702(2times2)

[[[[[

[

int

119905

0

(119894119902119904minus 119894lowast

119902119904) 119889119905

int

119905

0

(120596119903minus 120596lowast

119903) 119889119905

]]]]]

]

+ 119870119865119865(2times3)

[[[[

[

119879119871

119894lowast

119902119904

120596lowast

119903

]]]]

]

(11)



CSIfed IM

Reduced order

observer

yy

x


120596r

]u

u

y

y

= [Vd

120596e

]

K1(2times4)

K2(2times2)

KFF(2times3)

+ +++

minus

d = TL

int

u998400




119877119903119894119890

119889119903+ 119901120582119890

119889119903minus (120596119890minus 120596119903) 120582119890

119902119903= 0

119877119903119894119890

119902119903+ 119901120582119890

119902119903+ (120596119890minus 120596119903) 120582119890

119889119903= 0

(12)

where

120596sl = 120596119890 minus 120596119903 (13)

then (12) becomes

119877119903119894119890

119889119903+ 119901120582119890

119889119903minus 120596sl120582

119890

119902119903= 0

119877119903119894119890

119902119903+ 119901120582119890

119902119903+ 120596sl120582

119890

119889119903= 0

(14)



120582119903= 120582119890

119889119903 (15)

120582119890

119902119903= 0 (16)

119901120582119890

119902119903= 0 (17)


119877119903119894119890

119889119903+ 119901120582119903= 0

119877119903119894119890

119902119903+ 120596sl120582119903 = 0

(18)



119894119890

119902119903= minus

119871119898

119871119903

119894119890

119902119904

119894119890

119889119903=120582119903

119871119903

minus119871119898

119871119903

119894119890

119889119904

(19)

119894119890

119889119904= 119894119891= [1 +

119871119903

119877119903

119901]120582119903

119871119898

119894119890

119889119904= 119894119891= [1 + 120591

119903119901]

120582119903

119871119898

(20)

120596sl =119877119903119871119898

119871119903

119894119890

119902119904

120582119903

120596sl =119877119903119871119898

119871119903

119894119879

120582119903

120596sl =119871119898

120591119903

119894119879

120582119903

(21)





119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904minus 120582119890

119902119903119894119890

119889119904)

119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904)

119879119890= 119870119905119890120582119903119894119890

119902119904= 119870119905119890120582119903119894119879

(22)


119870119905119890=3

2

119875

2

119871119898

119871119903

(23)




119894119904= radic(119894119890

119902119904)2

+ (119894119890

119889119904)2

(24)


[

[

119894119890

119902119904

119894119890

119889119904

]

]

=2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+4120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+4120587

3)

]]]

]

[

[

119894119886119904

119894119887119904

119894119888119904

]

]

(25)


119894119902119889= [119879] [119894

119886119887119888] (26)

where

119894119902119889= [119894119890

119902119904119894119890

119889119904]119905

(27)

119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905

(28)

[119879] =2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+2120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+2120587

3)

]]]

]

(29)

where 119894119886119904 119894119887119904 and 119894





120579119891= 120579119903+ 120579sl (30)


120579sl = int120596sl (31)



119894lowast

119879=

119879lowast

119890

119870119905119890120582lowast119903

= (2

3) (

2

119875)119879lowast

119890

120582lowast119903

119871119903

119871119898

(32)

119894lowast

119891= (1 + 119901

119871119903

119877119903

)120582lowast

119903

119871119898

(33)

120596lowast

sl =119877119903119871119898

119871119903

119894lowast

119879

120582lowast119903

(34)




[

[

119894lowast

119902119904

119894lowast

119889119904

]

]

= [

[

cos 120579119891

sin 120579119891

minus sin 120579119891

cos 120579119891

]

]

[

[

119894lowast

119879

119894lowast

119891

]

]

(35)

[[[[

[

119894lowast

119886119904

119894lowast

119887119904

119894lowast

119888119904

]]]]

]

= [119879minus1

]

[[[[

[

119894lowast

119902119904

119894lowast

119889119904

0

]]]]

]

(36)

where

119879minus1

=

[[[[[[[

[

1 0 1

minus1

2minusradic3

21

minus1

2

radic3

21

]]]]]]]

]

(37)


119894lowast

119902119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119889119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 cos 120579lowast

119904

119894lowast

119886119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119887119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904minus2120587

3)

119894lowast

119888119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904+2120587

3)

(38)

where

120579lowast

119904= 120579119891+ 120579lowast

119879= 120579119903+ 120579lowast

sl + 120579lowast

119879 (39)


119879is generated


119891can be given



CSI

IM

PWMswitching

logic

Rotor speed

regulator

Stator current

generator

DC link currentias

lowast

ibslowast

i cslowast

120579riTlowast is

iflowast

iflowast and iT

lowast120582rlowast

120596rlowast

120596sllowast

120596r

Telowast

(1 + pLr

Rr

) 120582rlowast

Lm

RrLm

Lr

iTlowast

120582rlowast

4

3P

Telowast

120582rlowast

Lr

Lm

+minus




lowast|islowast|

ibslowast = |is


2120587

3

icslowast = |is


2120587

3

120579r 120579r

iTlowast

iflowast

120579Tlowast

120579slowast


lowastint

iaslowast

ibslowast

i cslowast

tanminus1 iTlowast

iflowast

+++



120582119903=119871119898119894119889119904

1 + 120591119903119904 (40)







V119904= 119877119904119894119904+119889120582119904

119889119905 (41)

120582119904= int (V

119904minus 119877119904119894119904) 119889119905 = 120582

120572119904+ 119895120582120573119904 (42)

10038161003816100381610038161205821199041003816100381610038161003816 = radic120582

2

120572119904+ 1205822

120573119904 (43)

ang120593119904= tanminus1 (

120582120573119904

120582120572119904

) (44)




119904are the



120582119902119904= 0 120582

119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)


119879119890=3

2(119875

2)1

120596119887

(120582119889119904119894119902119904) (46)



119890are


119894lowast

119902119904=2

3

1

119875

119879lowast

119890

1003816100381610038161003816120582lowast

119904

1003816100381610038161003816

(47)


Currentlimiter

Telowast

Te

|120582slowast|

2

3

1

P

PI

Telowast

|120582slowast |

+

+

+minus iqs

lowast





119902119904







119889119904


119894lowast

119904(119889119902)= 119894lowast

119889119904+ 119895119894lowast

119902119904 (48)

1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 =radic119894lowast2

119889119904+ 119894lowast2119902119904 (49)


119894lowast

119904(120572120573)= 119894lowast

120572119904+ 119895119894lowast

120573119904= (119894lowast

119889119904+ 119895119894lowast

119902119904) sdot 119890119895120593119904

=1003816100381610038161003816119894lowast

119904


119902119904

119894lowast

119889119904

))

(50)



Stator flux torque

estimator

PI

CSI

SVM IM

Gate Signals

PI

120582slowast

120591elowast

120582s

120582s

120591e

ang120593s

120591e

iqlowast

idlowast

dq

to120572120573

iaib

vb

va

|islowast| and idc

lowast

idclowast

+minus

+minus i120572

lowast

i120573lowast



119902119904
















8 Conclusion



15

minus05

0

05

1

Spee

d (P

U)

0 1 2 3 4 5Time (s)


(a)

0 1 2 3 4 5

minus2

0

2

Time (s)

Torq

ue (P

U)


(b)


0

05

1

Spee

d (P

U)

0 1 2 3 4 5minus4

minus2

0

24

Torq

ue (P

U)

Time (s)





Motor torque

Motor speed



speed reference


0

05

1

Spee

d (P

U)


012

Time (s)

Torq

ue (P

U)





Motor torque





TMS320F2812DSP

ADC

PWM

-EV

B

PWM

-EVA

Supplyfilter

3120601controlled


3120601IM

3120601supply

Texasinstruments

DSP processor

Speed sensor output

(Nr) Rpm

CM75TL-12NFseries

IGBT (6-PAC)

Ldc

Is (Amps)



1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads








Reference torque



Actual motor torque

Torque reference

Speed reference

Actual speed


1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads



Controlmethods




119901)




Controlmethods




119901)





Symbols

120596119887 Base speed

119903


119904




120582119889119903 119889-axis rotor flux

120582119890


120582119889119904 119889-axis stator flux


120582119890


120582119902119904 119902-axis stator flux

120582119903 Rotor flux


1199091015840


1198771015840









References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Rotor

Stator

4

3

2

1

[0 0]

rot1

rot

1s

1s

-K-

-K-3

2

1

fr

fs

is

irfr = rotor flux

fs = stator flux

is = stator current


Te

ir = rotor current


minus

+minus

+minus

minus

+minus

+minus

times

times

r

Rr

Rs

wm

Gm1

Gm

Gs

Gr

wk

s

K lowast u

K lowast u

∙






dqreference

frame

Stator

Synchronous

Rotor

Speed

Angle

-K-

Torque

Speed

Frequency

dq

ab

dq2ab

Currents

Amplitude

ab

etakdq

ab2dq

ab2ABC

Step load pu

Mechanicalsystem

1s

1s

Inductionmotor

pu

-K--

f(u)

f(u)

f(u)

0

0

ABC2ab dqABC ab

etak


K lowast u

wm

wk

wmwm

wmwkws

ws

wo

sTe Te

Te

Te

Tl

Tl

i dq

i dq

i ab

i abi ABC

is

120579r

120579k

120579s

120579s

K lowast u






minus10

0

10

0 05 1 15 2Time (s)

I ac

andI d

c(P

U)

(a)

minus10

0

10

0 05 1 15 2Time (s)

I ac

andI d

c(P

U)

(b)

0 05 1 15 2minus10

0

10



Time (s)

I ac

andI d

c(P

U)

(c)


minus10

0

10


I qs

and

FLU

Xdr

(a)

minus10

0

10


I qs

and

FLU

Xdr

(b)

0 05 1 15 2minus10

minus5

0

5



Time (s)

I qs

and

FLU

Xdr

(PU

)

(c)


minus10

0

10

0 05 1 15 2Time (s)

I qs

and

FLU

Xds

(a)

minus10

0

10

0 05 1 15 2Time (s)

I qs

and

FLU

Xds

(b)

0 05 1 15 2minus10

minus5

0

5


Time (s)


I qs

and

FLU

Xds

(c)










= 119860119909 + 119861119906 (6)

where

=119889119909

119889119905 (7)

119910 = 119862119909 (8)

where

119909 =

[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]

]

=119889

119889119905

[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]

]

119906 =

[[[[

[

Δ1198811015840

119889

Δ120596119890

120596119887

Δ119879119871

]]]]

]

119910 =[[

[

Δ119894119902119904

Δ120596119903

120596119887

]]

]

119860 =

[[[[[[[[[[[[[[

[

minus1199091015840

1199031199030120596119887

11990901199091015840119903minus 1199092119898

1199091198981198771015840

119903120596119887

11990901199091015840119903minus 1199092119898

minus12059611989001199091015840

119903119909119898(1 minus 119904)

11990901199091015840119903minus 1199092119898

minus1199091198981199091015840

1199031198941015840

1198891199030120596119887

11990901199091015840119903minus 1199092119898

1205961198871199091198981199030

11990901199091015840119903minus 1199092119898

minus11990901198771015840

119903120596119887

11990901199091015840119903minus 1199092119898

1205961198900(1199092

119898minus 11990411990901199091015840

119903)

11990901199091015840119903minus 1199092119898

11990901199091015840

1199031198941015840

1198891199030120596119887

11990901199091015840119903minus 1199092119898

1199041205961198900119909119898

1199091015840119903

1199041205961198900

minus1205961198871198771015840

119903

1199091015840119903

minus

120596119887(1199091198981198941199021199040+ 1199091015840

1199031198941015840

1199021199030)

1199091015840119903

1199091198981198941015840

1198891199030

21198670

1199091198981198941199021199040

21198670

]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[

[

1199091015840

119903120596119887

11990901199091015840119903minus 1199092119898

0 0

minus119909119898120596119887

11990901199091015840119903minus 1199092119898

minus1205961198871198941015840

11988911990300

0 minus

120596119887(1199091198981198941199021199040+ 1199091015840

1199031198941015840

1199021199030)

1199091015840119903

0

0 0 minus1

2119867

]]]]]]]]]]]]]]]

]

119862 = [1 0 0 0

0 0 0 1]

1198811015840

119889=120587119881dc

3radic3 119903

0= 119877119904+1205872

119877dc18

1199090= 119909119904+1205872

119909119889

18

(9)


Rs

Rs

Rr

Vin Idi = 0

VqsLm

Lm

Ll

Ll

p 120582ds

p 120582qs p 120582qr

p 120582dr

+ minus

+minus

c

c

120596r 120582qr

120596r 120582qr

IdsIdi

IqsIqi

Vds

CSI

Iqi = MdIdc Vqs




119906 = 1198701119909 + 119870

2int

119905

0

(119910 minus 119910119903) 119889119905 + 119870

119865119865[119889

119910119903

] (10)

or

119906 = 1198701(2times4)

[[[[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]]]]

]

+ 1198702(2times2)

[[[[[

[

int

119905

0

(119894119902119904minus 119894lowast

119902119904) 119889119905

int

119905

0

(120596119903minus 120596lowast

119903) 119889119905

]]]]]

]

+ 119870119865119865(2times3)

[[[[

[

119879119871

119894lowast

119902119904

120596lowast

119903

]]]]

]

(11)



CSIfed IM

Reduced order

observer

yy

x


120596r

]u

u

y

y

= [Vd

120596e

]

K1(2times4)

K2(2times2)

KFF(2times3)

+ +++

minus

d = TL

int

u998400




119877119903119894119890

119889119903+ 119901120582119890

119889119903minus (120596119890minus 120596119903) 120582119890

119902119903= 0

119877119903119894119890

119902119903+ 119901120582119890

119902119903+ (120596119890minus 120596119903) 120582119890

119889119903= 0

(12)

where

120596sl = 120596119890 minus 120596119903 (13)

then (12) becomes

119877119903119894119890

119889119903+ 119901120582119890

119889119903minus 120596sl120582

119890

119902119903= 0

119877119903119894119890

119902119903+ 119901120582119890

119902119903+ 120596sl120582

119890

119889119903= 0

(14)



120582119903= 120582119890

119889119903 (15)

120582119890

119902119903= 0 (16)

119901120582119890

119902119903= 0 (17)


119877119903119894119890

119889119903+ 119901120582119903= 0

119877119903119894119890

119902119903+ 120596sl120582119903 = 0

(18)



119894119890

119902119903= minus

119871119898

119871119903

119894119890

119902119904

119894119890

119889119903=120582119903

119871119903

minus119871119898

119871119903

119894119890

119889119904

(19)

119894119890

119889119904= 119894119891= [1 +

119871119903

119877119903

119901]120582119903

119871119898

119894119890

119889119904= 119894119891= [1 + 120591

119903119901]

120582119903

119871119898

(20)

120596sl =119877119903119871119898

119871119903

119894119890

119902119904

120582119903

120596sl =119877119903119871119898

119871119903

119894119879

120582119903

120596sl =119871119898

120591119903

119894119879

120582119903

(21)





119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904minus 120582119890

119902119903119894119890

119889119904)

119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904)

119879119890= 119870119905119890120582119903119894119890

119902119904= 119870119905119890120582119903119894119879

(22)


119870119905119890=3

2

119875

2

119871119898

119871119903

(23)




119894119904= radic(119894119890

119902119904)2

+ (119894119890

119889119904)2

(24)


[

[

119894119890

119902119904

119894119890

119889119904

]

]

=2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+4120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+4120587

3)

]]]

]

[

[

119894119886119904

119894119887119904

119894119888119904

]

]

(25)


119894119902119889= [119879] [119894

119886119887119888] (26)

where

119894119902119889= [119894119890

119902119904119894119890

119889119904]119905

(27)

119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905

(28)

[119879] =2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+2120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+2120587

3)

]]]

]

(29)

where 119894119886119904 119894119887119904 and 119894





120579119891= 120579119903+ 120579sl (30)


120579sl = int120596sl (31)



119894lowast

119879=

119879lowast

119890

119870119905119890120582lowast119903

= (2

3) (

2

119875)119879lowast

119890

120582lowast119903

119871119903

119871119898

(32)

119894lowast

119891= (1 + 119901

119871119903

119877119903

)120582lowast

119903

119871119898

(33)

120596lowast

sl =119877119903119871119898

119871119903

119894lowast

119879

120582lowast119903

(34)




[

[

119894lowast

119902119904

119894lowast

119889119904

]

]

= [

[

cos 120579119891

sin 120579119891

minus sin 120579119891

cos 120579119891

]

]

[

[

119894lowast

119879

119894lowast

119891

]

]

(35)

[[[[

[

119894lowast

119886119904

119894lowast

119887119904

119894lowast

119888119904

]]]]

]

= [119879minus1

]

[[[[

[

119894lowast

119902119904

119894lowast

119889119904

0

]]]]

]

(36)

where

119879minus1

=

[[[[[[[

[

1 0 1

minus1

2minusradic3

21

minus1

2

radic3

21

]]]]]]]

]

(37)


119894lowast

119902119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119889119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 cos 120579lowast

119904

119894lowast

119886119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119887119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904minus2120587

3)

119894lowast

119888119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904+2120587

3)

(38)

where

120579lowast

119904= 120579119891+ 120579lowast

119879= 120579119903+ 120579lowast

sl + 120579lowast

119879 (39)


119879is generated


119891can be given



CSI

IM

PWMswitching

logic

Rotor speed

regulator

Stator current

generator

DC link currentias

lowast

ibslowast

i cslowast

120579riTlowast is

iflowast

iflowast and iT

lowast120582rlowast

120596rlowast

120596sllowast

120596r

Telowast

(1 + pLr

Rr

) 120582rlowast

Lm

RrLm

Lr

iTlowast

120582rlowast

4

3P

Telowast

120582rlowast

Lr

Lm

+minus




lowast|islowast|

ibslowast = |is


2120587

3

icslowast = |is


2120587

3

120579r 120579r

iTlowast

iflowast

120579Tlowast

120579slowast


lowastint

iaslowast

ibslowast

i cslowast

tanminus1 iTlowast

iflowast

+++



120582119903=119871119898119894119889119904

1 + 120591119903119904 (40)







V119904= 119877119904119894119904+119889120582119904

119889119905 (41)

120582119904= int (V

119904minus 119877119904119894119904) 119889119905 = 120582

120572119904+ 119895120582120573119904 (42)

10038161003816100381610038161205821199041003816100381610038161003816 = radic120582

2

120572119904+ 1205822

120573119904 (43)

ang120593119904= tanminus1 (

120582120573119904

120582120572119904

) (44)




119904are the



120582119902119904= 0 120582

119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)


119879119890=3

2(119875

2)1

120596119887

(120582119889119904119894119902119904) (46)



119890are


119894lowast

119902119904=2

3

1

119875

119879lowast

119890

1003816100381610038161003816120582lowast

119904

1003816100381610038161003816

(47)


Currentlimiter

Telowast

Te

|120582slowast|

2

3

1

P

PI

Telowast

|120582slowast |

+

+

+minus iqs

lowast





119902119904







119889119904


119894lowast

119904(119889119902)= 119894lowast

119889119904+ 119895119894lowast

119902119904 (48)

1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 =radic119894lowast2

119889119904+ 119894lowast2119902119904 (49)


119894lowast

119904(120572120573)= 119894lowast

120572119904+ 119895119894lowast

120573119904= (119894lowast

119889119904+ 119895119894lowast

119902119904) sdot 119890119895120593119904

=1003816100381610038161003816119894lowast

119904


119902119904

119894lowast

119889119904

))

(50)



Stator flux torque

estimator

PI

CSI

SVM IM

Gate Signals

PI

120582slowast

120591elowast

120582s

120582s

120591e

ang120593s

120591e

iqlowast

idlowast

dq

to120572120573

iaib

vb

va

|islowast| and idc

lowast

idclowast

+minus

+minus i120572

lowast

i120573lowast



119902119904
















8 Conclusion



15

minus05

0

05

1

Spee

d (P

U)

0 1 2 3 4 5Time (s)


(a)

0 1 2 3 4 5

minus2

0

2

Time (s)

Torq

ue (P

U)


(b)


0

05

1

Spee

d (P

U)

0 1 2 3 4 5minus4

minus2

0

24

Torq

ue (P

U)

Time (s)





Motor torque

Motor speed



speed reference


0

05

1

Spee

d (P

U)


012

Time (s)

Torq

ue (P

U)





Motor torque





TMS320F2812DSP

ADC

PWM

-EV

B

PWM

-EVA

Supplyfilter

3120601controlled


3120601IM

3120601supply

Texasinstruments

DSP processor

Speed sensor output

(Nr) Rpm

CM75TL-12NFseries

IGBT (6-PAC)

Ldc

Is (Amps)



1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads








Reference torque



Actual motor torque

Torque reference

Speed reference

Actual speed


1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads



Controlmethods




119901)




Controlmethods




119901)





Symbols

120596119887 Base speed

119903


119904




120582119889119903 119889-axis rotor flux

120582119890


120582119889119904 119889-axis stator flux


120582119890


120582119902119904 119902-axis stator flux

120582119903 Rotor flux


1199091015840


1198771015840









References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus10

0

10

0 05 1 15 2Time (s)

I ac

andI d

c(P

U)

(a)

minus10

0

10

0 05 1 15 2Time (s)

I ac

andI d

c(P

U)

(b)

0 05 1 15 2minus10

0

10



Time (s)

I ac

andI d

c(P

U)

(c)


minus10

0

10


I qs

and

FLU

Xdr

(a)

minus10

0

10


I qs

and

FLU

Xdr

(b)

0 05 1 15 2minus10

minus5

0

5



Time (s)

I qs

and

FLU

Xdr

(PU

)

(c)


minus10

0

10

0 05 1 15 2Time (s)

I qs

and

FLU

Xds

(a)

minus10

0

10

0 05 1 15 2Time (s)

I qs

and

FLU

Xds

(b)

0 05 1 15 2minus10

minus5

0

5


Time (s)


I qs

and

FLU

Xds

(c)










= 119860119909 + 119861119906 (6)

where

=119889119909

119889119905 (7)

119910 = 119862119909 (8)

where

119909 =

[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]

]

=119889

119889119905

[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]

]

119906 =

[[[[

[

Δ1198811015840

119889

Δ120596119890

120596119887

Δ119879119871

]]]]

]

119910 =[[

[

Δ119894119902119904

Δ120596119903

120596119887

]]

]

119860 =

[[[[[[[[[[[[[[

[

minus1199091015840

1199031199030120596119887

11990901199091015840119903minus 1199092119898

1199091198981198771015840

119903120596119887

11990901199091015840119903minus 1199092119898

minus12059611989001199091015840

119903119909119898(1 minus 119904)

11990901199091015840119903minus 1199092119898

minus1199091198981199091015840

1199031198941015840

1198891199030120596119887

11990901199091015840119903minus 1199092119898

1205961198871199091198981199030

11990901199091015840119903minus 1199092119898

minus11990901198771015840

119903120596119887

11990901199091015840119903minus 1199092119898

1205961198900(1199092

119898minus 11990411990901199091015840

119903)

11990901199091015840119903minus 1199092119898

11990901199091015840

1199031198941015840

1198891199030120596119887

11990901199091015840119903minus 1199092119898

1199041205961198900119909119898

1199091015840119903

1199041205961198900

minus1205961198871198771015840

119903

1199091015840119903

minus

120596119887(1199091198981198941199021199040+ 1199091015840

1199031198941015840

1199021199030)

1199091015840119903

1199091198981198941015840

1198891199030

21198670

1199091198981198941199021199040

21198670

]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[

[

1199091015840

119903120596119887

11990901199091015840119903minus 1199092119898

0 0

minus119909119898120596119887

11990901199091015840119903minus 1199092119898

minus1205961198871198941015840

11988911990300

0 minus

120596119887(1199091198981198941199021199040+ 1199091015840

1199031198941015840

1199021199030)

1199091015840119903

0

0 0 minus1

2119867

]]]]]]]]]]]]]]]

]

119862 = [1 0 0 0

0 0 0 1]

1198811015840

119889=120587119881dc

3radic3 119903

0= 119877119904+1205872

119877dc18

1199090= 119909119904+1205872

119909119889

18

(9)


Rs

Rs

Rr

Vin Idi = 0

VqsLm

Lm

Ll

Ll

p 120582ds

p 120582qs p 120582qr

p 120582dr

+ minus

+minus

c

c

120596r 120582qr

120596r 120582qr

IdsIdi

IqsIqi

Vds

CSI

Iqi = MdIdc Vqs




119906 = 1198701119909 + 119870

2int

119905

0

(119910 minus 119910119903) 119889119905 + 119870

119865119865[119889

119910119903

] (10)

or

119906 = 1198701(2times4)

[[[[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]]]]

]

+ 1198702(2times2)

[[[[[

[

int

119905

0

(119894119902119904minus 119894lowast

119902119904) 119889119905

int

119905

0

(120596119903minus 120596lowast

119903) 119889119905

]]]]]

]

+ 119870119865119865(2times3)

[[[[

[

119879119871

119894lowast

119902119904

120596lowast

119903

]]]]

]

(11)



CSIfed IM

Reduced order

observer

yy

x


120596r

]u

u

y

y

= [Vd

120596e

]

K1(2times4)

K2(2times2)

KFF(2times3)

+ +++

minus

d = TL

int

u998400




119877119903119894119890

119889119903+ 119901120582119890

119889119903minus (120596119890minus 120596119903) 120582119890

119902119903= 0

119877119903119894119890

119902119903+ 119901120582119890

119902119903+ (120596119890minus 120596119903) 120582119890

119889119903= 0

(12)

where

120596sl = 120596119890 minus 120596119903 (13)

then (12) becomes

119877119903119894119890

119889119903+ 119901120582119890

119889119903minus 120596sl120582

119890

119902119903= 0

119877119903119894119890

119902119903+ 119901120582119890

119902119903+ 120596sl120582

119890

119889119903= 0

(14)



120582119903= 120582119890

119889119903 (15)

120582119890

119902119903= 0 (16)

119901120582119890

119902119903= 0 (17)


119877119903119894119890

119889119903+ 119901120582119903= 0

119877119903119894119890

119902119903+ 120596sl120582119903 = 0

(18)



119894119890

119902119903= minus

119871119898

119871119903

119894119890

119902119904

119894119890

119889119903=120582119903

119871119903

minus119871119898

119871119903

119894119890

119889119904

(19)

119894119890

119889119904= 119894119891= [1 +

119871119903

119877119903

119901]120582119903

119871119898

119894119890

119889119904= 119894119891= [1 + 120591

119903119901]

120582119903

119871119898

(20)

120596sl =119877119903119871119898

119871119903

119894119890

119902119904

120582119903

120596sl =119877119903119871119898

119871119903

119894119879

120582119903

120596sl =119871119898

120591119903

119894119879

120582119903

(21)





119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904minus 120582119890

119902119903119894119890

119889119904)

119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904)

119879119890= 119870119905119890120582119903119894119890

119902119904= 119870119905119890120582119903119894119879

(22)


119870119905119890=3

2

119875

2

119871119898

119871119903

(23)




119894119904= radic(119894119890

119902119904)2

+ (119894119890

119889119904)2

(24)


[

[

119894119890

119902119904

119894119890

119889119904

]

]

=2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+4120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+4120587

3)

]]]

]

[

[

119894119886119904

119894119887119904

119894119888119904

]

]

(25)


119894119902119889= [119879] [119894

119886119887119888] (26)

where

119894119902119889= [119894119890

119902119904119894119890

119889119904]119905

(27)

119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905

(28)

[119879] =2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+2120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+2120587

3)

]]]

]

(29)

where 119894119886119904 119894119887119904 and 119894





120579119891= 120579119903+ 120579sl (30)


120579sl = int120596sl (31)



119894lowast

119879=

119879lowast

119890

119870119905119890120582lowast119903

= (2

3) (

2

119875)119879lowast

119890

120582lowast119903

119871119903

119871119898

(32)

119894lowast

119891= (1 + 119901

119871119903

119877119903

)120582lowast

119903

119871119898

(33)

120596lowast

sl =119877119903119871119898

119871119903

119894lowast

119879

120582lowast119903

(34)




[

[

119894lowast

119902119904

119894lowast

119889119904

]

]

= [

[

cos 120579119891

sin 120579119891

minus sin 120579119891

cos 120579119891

]

]

[

[

119894lowast

119879

119894lowast

119891

]

]

(35)

[[[[

[

119894lowast

119886119904

119894lowast

119887119904

119894lowast

119888119904

]]]]

]

= [119879minus1

]

[[[[

[

119894lowast

119902119904

119894lowast

119889119904

0

]]]]

]

(36)

where

119879minus1

=

[[[[[[[

[

1 0 1

minus1

2minusradic3

21

minus1

2

radic3

21

]]]]]]]

]

(37)


119894lowast

119902119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119889119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 cos 120579lowast

119904

119894lowast

119886119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119887119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904minus2120587

3)

119894lowast

119888119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904+2120587

3)

(38)

where

120579lowast

119904= 120579119891+ 120579lowast

119879= 120579119903+ 120579lowast

sl + 120579lowast

119879 (39)


119879is generated


119891can be given



CSI

IM

PWMswitching

logic

Rotor speed

regulator

Stator current

generator

DC link currentias

lowast

ibslowast

i cslowast

120579riTlowast is

iflowast

iflowast and iT

lowast120582rlowast

120596rlowast

120596sllowast

120596r

Telowast

(1 + pLr

Rr

) 120582rlowast

Lm

RrLm

Lr

iTlowast

120582rlowast

4

3P

Telowast

120582rlowast

Lr

Lm

+minus




lowast|islowast|

ibslowast = |is


2120587

3

icslowast = |is


2120587

3

120579r 120579r

iTlowast

iflowast

120579Tlowast

120579slowast


lowastint

iaslowast

ibslowast

i cslowast

tanminus1 iTlowast

iflowast

+++



120582119903=119871119898119894119889119904

1 + 120591119903119904 (40)







V119904= 119877119904119894119904+119889120582119904

119889119905 (41)

120582119904= int (V

119904minus 119877119904119894119904) 119889119905 = 120582

120572119904+ 119895120582120573119904 (42)

10038161003816100381610038161205821199041003816100381610038161003816 = radic120582

2

120572119904+ 1205822

120573119904 (43)

ang120593119904= tanminus1 (

120582120573119904

120582120572119904

) (44)




119904are the



120582119902119904= 0 120582

119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)


119879119890=3

2(119875

2)1

120596119887

(120582119889119904119894119902119904) (46)



119890are


119894lowast

119902119904=2

3

1

119875

119879lowast

119890

1003816100381610038161003816120582lowast

119904

1003816100381610038161003816

(47)


Currentlimiter

Telowast

Te

|120582slowast|

2

3

1

P

PI

Telowast

|120582slowast |

+

+

+minus iqs

lowast





119902119904







119889119904


119894lowast

119904(119889119902)= 119894lowast

119889119904+ 119895119894lowast

119902119904 (48)

1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 =radic119894lowast2

119889119904+ 119894lowast2119902119904 (49)


119894lowast

119904(120572120573)= 119894lowast

120572119904+ 119895119894lowast

120573119904= (119894lowast

119889119904+ 119895119894lowast

119902119904) sdot 119890119895120593119904

=1003816100381610038161003816119894lowast

119904


119902119904

119894lowast

119889119904

))

(50)



Stator flux torque

estimator

PI

CSI

SVM IM

Gate Signals

PI

120582slowast

120591elowast

120582s

120582s

120591e

ang120593s

120591e

iqlowast

idlowast

dq

to120572120573

iaib

vb

va

|islowast| and idc

lowast

idclowast

+minus

+minus i120572

lowast

i120573lowast



119902119904
















8 Conclusion



15

minus05

0

05

1

Spee

d (P

U)

0 1 2 3 4 5Time (s)


(a)

0 1 2 3 4 5

minus2

0

2

Time (s)

Torq

ue (P

U)


(b)


0

05

1

Spee

d (P

U)

0 1 2 3 4 5minus4

minus2

0

24

Torq

ue (P

U)

Time (s)





Motor torque

Motor speed



speed reference


0

05

1

Spee

d (P

U)


012

Time (s)

Torq

ue (P

U)





Motor torque





TMS320F2812DSP

ADC

PWM

-EV

B

PWM

-EVA

Supplyfilter

3120601controlled


3120601IM

3120601supply

Texasinstruments

DSP processor

Speed sensor output

(Nr) Rpm

CM75TL-12NFseries

IGBT (6-PAC)

Ldc

Is (Amps)



1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads








Reference torque



Actual motor torque

Torque reference

Speed reference

Actual speed


1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads



Controlmethods




119901)




Controlmethods




119901)





Symbols

120596119887 Base speed

119903


119904




120582119889119903 119889-axis rotor flux

120582119890


120582119889119904 119889-axis stator flux


120582119890


120582119902119904 119902-axis stator flux

120582119903 Rotor flux


1199091015840


1198771015840









References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















= 119860119909 + 119861119906 (6)

where

=119889119909

119889119905 (7)

119910 = 119862119909 (8)

where

119909 =

[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]

]

=119889

119889119905

[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]

]

119906 =

[[[[

[

Δ1198811015840

119889

Δ120596119890

120596119887

Δ119879119871

]]]]

]

119910 =[[

[

Δ119894119902119904

Δ120596119903

120596119887

]]

]

119860 =

[[[[[[[[[[[[[[

[

minus1199091015840

1199031199030120596119887

11990901199091015840119903minus 1199092119898

1199091198981198771015840

119903120596119887

11990901199091015840119903minus 1199092119898

minus12059611989001199091015840

119903119909119898(1 minus 119904)

11990901199091015840119903minus 1199092119898

minus1199091198981199091015840

1199031198941015840

1198891199030120596119887

11990901199091015840119903minus 1199092119898

1205961198871199091198981199030

11990901199091015840119903minus 1199092119898

minus11990901198771015840

119903120596119887

11990901199091015840119903minus 1199092119898

1205961198900(1199092

119898minus 11990411990901199091015840

119903)

11990901199091015840119903minus 1199092119898

11990901199091015840

1199031198941015840

1198891199030120596119887

11990901199091015840119903minus 1199092119898

1199041205961198900119909119898

1199091015840119903

1199041205961198900

minus1205961198871198771015840

119903

1199091015840119903

minus

120596119887(1199091198981198941199021199040+ 1199091015840

1199031198941015840

1199021199030)

1199091015840119903

1199091198981198941015840

1198891199030

21198670

1199091198981198941199021199040

21198670

]]]]]]]]]]]]]]

]

119861 =

[[[[[[[[[[[[[[[

[

1199091015840

119903120596119887

11990901199091015840119903minus 1199092119898

0 0

minus119909119898120596119887

11990901199091015840119903minus 1199092119898

minus1205961198871198941015840

11988911990300

0 minus

120596119887(1199091198981198941199021199040+ 1199091015840

1199031198941015840

1199021199030)

1199091015840119903

0

0 0 minus1

2119867

]]]]]]]]]]]]]]]

]

119862 = [1 0 0 0

0 0 0 1]

1198811015840

119889=120587119881dc

3radic3 119903

0= 119877119904+1205872

119877dc18

1199090= 119909119904+1205872

119909119889

18

(9)


Rs

Rs

Rr

Vin Idi = 0

VqsLm

Lm

Ll

Ll

p 120582ds

p 120582qs p 120582qr

p 120582dr

+ minus

+minus

c

c

120596r 120582qr

120596r 120582qr

IdsIdi

IqsIqi

Vds

CSI

Iqi = MdIdc Vqs




119906 = 1198701119909 + 119870

2int

119905

0

(119910 minus 119910119903) 119889119905 + 119870

119865119865[119889

119910119903

] (10)

or

119906 = 1198701(2times4)

[[[[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]]]]

]

+ 1198702(2times2)

[[[[[

[

int

119905

0

(119894119902119904minus 119894lowast

119902119904) 119889119905

int

119905

0

(120596119903minus 120596lowast

119903) 119889119905

]]]]]

]

+ 119870119865119865(2times3)

[[[[

[

119879119871

119894lowast

119902119904

120596lowast

119903

]]]]

]

(11)



CSIfed IM

Reduced order

observer

yy

x


120596r

]u

u

y

y

= [Vd

120596e

]

K1(2times4)

K2(2times2)

KFF(2times3)

+ +++

minus

d = TL

int

u998400




119877119903119894119890

119889119903+ 119901120582119890

119889119903minus (120596119890minus 120596119903) 120582119890

119902119903= 0

119877119903119894119890

119902119903+ 119901120582119890

119902119903+ (120596119890minus 120596119903) 120582119890

119889119903= 0

(12)

where

120596sl = 120596119890 minus 120596119903 (13)

then (12) becomes

119877119903119894119890

119889119903+ 119901120582119890

119889119903minus 120596sl120582

119890

119902119903= 0

119877119903119894119890

119902119903+ 119901120582119890

119902119903+ 120596sl120582

119890

119889119903= 0

(14)



120582119903= 120582119890

119889119903 (15)

120582119890

119902119903= 0 (16)

119901120582119890

119902119903= 0 (17)


119877119903119894119890

119889119903+ 119901120582119903= 0

119877119903119894119890

119902119903+ 120596sl120582119903 = 0

(18)



119894119890

119902119903= minus

119871119898

119871119903

119894119890

119902119904

119894119890

119889119903=120582119903

119871119903

minus119871119898

119871119903

119894119890

119889119904

(19)

119894119890

119889119904= 119894119891= [1 +

119871119903

119877119903

119901]120582119903

119871119898

119894119890

119889119904= 119894119891= [1 + 120591

119903119901]

120582119903

119871119898

(20)

120596sl =119877119903119871119898

119871119903

119894119890

119902119904

120582119903

120596sl =119877119903119871119898

119871119903

119894119879

120582119903

120596sl =119871119898

120591119903

119894119879

120582119903

(21)





119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904minus 120582119890

119902119903119894119890

119889119904)

119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904)

119879119890= 119870119905119890120582119903119894119890

119902119904= 119870119905119890120582119903119894119879

(22)


119870119905119890=3

2

119875

2

119871119898

119871119903

(23)




119894119904= radic(119894119890

119902119904)2

+ (119894119890

119889119904)2

(24)


[

[

119894119890

119902119904

119894119890

119889119904

]

]

=2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+4120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+4120587

3)

]]]

]

[

[

119894119886119904

119894119887119904

119894119888119904

]

]

(25)


119894119902119889= [119879] [119894

119886119887119888] (26)

where

119894119902119889= [119894119890

119902119904119894119890

119889119904]119905

(27)

119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905

(28)

[119879] =2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+2120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+2120587

3)

]]]

]

(29)

where 119894119886119904 119894119887119904 and 119894





120579119891= 120579119903+ 120579sl (30)


120579sl = int120596sl (31)



119894lowast

119879=

119879lowast

119890

119870119905119890120582lowast119903

= (2

3) (

2

119875)119879lowast

119890

120582lowast119903

119871119903

119871119898

(32)

119894lowast

119891= (1 + 119901

119871119903

119877119903

)120582lowast

119903

119871119898

(33)

120596lowast

sl =119877119903119871119898

119871119903

119894lowast

119879

120582lowast119903

(34)




[

[

119894lowast

119902119904

119894lowast

119889119904

]

]

= [

[

cos 120579119891

sin 120579119891

minus sin 120579119891

cos 120579119891

]

]

[

[

119894lowast

119879

119894lowast

119891

]

]

(35)

[[[[

[

119894lowast

119886119904

119894lowast

119887119904

119894lowast

119888119904

]]]]

]

= [119879minus1

]

[[[[

[

119894lowast

119902119904

119894lowast

119889119904

0

]]]]

]

(36)

where

119879minus1

=

[[[[[[[

[

1 0 1

minus1

2minusradic3

21

minus1

2

radic3

21

]]]]]]]

]

(37)


119894lowast

119902119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119889119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 cos 120579lowast

119904

119894lowast

119886119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119887119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904minus2120587

3)

119894lowast

119888119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904+2120587

3)

(38)

where

120579lowast

119904= 120579119891+ 120579lowast

119879= 120579119903+ 120579lowast

sl + 120579lowast

119879 (39)


119879is generated


119891can be given



CSI

IM

PWMswitching

logic

Rotor speed

regulator

Stator current

generator

DC link currentias

lowast

ibslowast

i cslowast

120579riTlowast is

iflowast

iflowast and iT

lowast120582rlowast

120596rlowast

120596sllowast

120596r

Telowast

(1 + pLr

Rr

) 120582rlowast

Lm

RrLm

Lr

iTlowast

120582rlowast

4

3P

Telowast

120582rlowast

Lr

Lm

+minus




lowast|islowast|

ibslowast = |is


2120587

3

icslowast = |is


2120587

3

120579r 120579r

iTlowast

iflowast

120579Tlowast

120579slowast


lowastint

iaslowast

ibslowast

i cslowast

tanminus1 iTlowast

iflowast

+++



120582119903=119871119898119894119889119904

1 + 120591119903119904 (40)







V119904= 119877119904119894119904+119889120582119904

119889119905 (41)

120582119904= int (V

119904minus 119877119904119894119904) 119889119905 = 120582

120572119904+ 119895120582120573119904 (42)

10038161003816100381610038161205821199041003816100381610038161003816 = radic120582

2

120572119904+ 1205822

120573119904 (43)

ang120593119904= tanminus1 (

120582120573119904

120582120572119904

) (44)




119904are the



120582119902119904= 0 120582

119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)


119879119890=3

2(119875

2)1

120596119887

(120582119889119904119894119902119904) (46)



119890are


119894lowast

119902119904=2

3

1

119875

119879lowast

119890

1003816100381610038161003816120582lowast

119904

1003816100381610038161003816

(47)


Currentlimiter

Telowast

Te

|120582slowast|

2

3

1

P

PI

Telowast

|120582slowast |

+

+

+minus iqs

lowast





119902119904







119889119904


119894lowast

119904(119889119902)= 119894lowast

119889119904+ 119895119894lowast

119902119904 (48)

1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 =radic119894lowast2

119889119904+ 119894lowast2119902119904 (49)


119894lowast

119904(120572120573)= 119894lowast

120572119904+ 119895119894lowast

120573119904= (119894lowast

119889119904+ 119895119894lowast

119902119904) sdot 119890119895120593119904

=1003816100381610038161003816119894lowast

119904


119902119904

119894lowast

119889119904

))

(50)



Stator flux torque

estimator

PI

CSI

SVM IM

Gate Signals

PI

120582slowast

120591elowast

120582s

120582s

120591e

ang120593s

120591e

iqlowast

idlowast

dq

to120572120573

iaib

vb

va

|islowast| and idc

lowast

idclowast

+minus

+minus i120572

lowast

i120573lowast



119902119904
















8 Conclusion



15

minus05

0

05

1

Spee

d (P

U)

0 1 2 3 4 5Time (s)


(a)

0 1 2 3 4 5

minus2

0

2

Time (s)

Torq

ue (P

U)


(b)


0

05

1

Spee

d (P

U)

0 1 2 3 4 5minus4

minus2

0

24

Torq

ue (P

U)

Time (s)





Motor torque

Motor speed



speed reference


0

05

1

Spee

d (P

U)


012

Time (s)

Torq

ue (P

U)





Motor torque





TMS320F2812DSP

ADC

PWM

-EV

B

PWM

-EVA

Supplyfilter

3120601controlled


3120601IM

3120601supply

Texasinstruments

DSP processor

Speed sensor output

(Nr) Rpm

CM75TL-12NFseries

IGBT (6-PAC)

Ldc

Is (Amps)



1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads








Reference torque



Actual motor torque

Torque reference

Speed reference

Actual speed


1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads



Controlmethods




119901)




Controlmethods




119901)





Symbols

120596119887 Base speed

119903


119904




120582119889119903 119889-axis rotor flux

120582119890


120582119889119904 119889-axis stator flux


120582119890


120582119902119904 119902-axis stator flux

120582119903 Rotor flux


1199091015840


1198771015840









References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Rs

Rs

Rr

Vin Idi = 0

VqsLm

Lm

Ll

Ll

p 120582ds

p 120582qs p 120582qr

p 120582dr

+ minus

+minus

c

c

120596r 120582qr

120596r 120582qr

IdsIdi

IqsIqi

Vds

CSI

Iqi = MdIdc Vqs




119906 = 1198701119909 + 119870

2int

119905

0

(119910 minus 119910119903) 119889119905 + 119870

119865119865[119889

119910119903

] (10)

or

119906 = 1198701(2times4)

[[[[[[[[[[

[

Δ119894119902119904

Δ1198941015840

119902119903

Δ1198941015840

119889119903

Δ120596119903

120596119887

]]]]]]]]]]

]

+ 1198702(2times2)

[[[[[

[

int

119905

0

(119894119902119904minus 119894lowast

119902119904) 119889119905

int

119905

0

(120596119903minus 120596lowast

119903) 119889119905

]]]]]

]

+ 119870119865119865(2times3)

[[[[

[

119879119871

119894lowast

119902119904

120596lowast

119903

]]]]

]

(11)



CSIfed IM

Reduced order

observer

yy

x


120596r

]u

u

y

y

= [Vd

120596e

]

K1(2times4)

K2(2times2)

KFF(2times3)

+ +++

minus

d = TL

int

u998400




119877119903119894119890

119889119903+ 119901120582119890

119889119903minus (120596119890minus 120596119903) 120582119890

119902119903= 0

119877119903119894119890

119902119903+ 119901120582119890

119902119903+ (120596119890minus 120596119903) 120582119890

119889119903= 0

(12)

where

120596sl = 120596119890 minus 120596119903 (13)

then (12) becomes

119877119903119894119890

119889119903+ 119901120582119890

119889119903minus 120596sl120582

119890

119902119903= 0

119877119903119894119890

119902119903+ 119901120582119890

119902119903+ 120596sl120582

119890

119889119903= 0

(14)



120582119903= 120582119890

119889119903 (15)

120582119890

119902119903= 0 (16)

119901120582119890

119902119903= 0 (17)


119877119903119894119890

119889119903+ 119901120582119903= 0

119877119903119894119890

119902119903+ 120596sl120582119903 = 0

(18)



119894119890

119902119903= minus

119871119898

119871119903

119894119890

119902119904

119894119890

119889119903=120582119903

119871119903

minus119871119898

119871119903

119894119890

119889119904

(19)

119894119890

119889119904= 119894119891= [1 +

119871119903

119877119903

119901]120582119903

119871119898

119894119890

119889119904= 119894119891= [1 + 120591

119903119901]

120582119903

119871119898

(20)

120596sl =119877119903119871119898

119871119903

119894119890

119902119904

120582119903

120596sl =119877119903119871119898

119871119903

119894119879

120582119903

120596sl =119871119898

120591119903

119894119879

120582119903

(21)





119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904minus 120582119890

119902119903119894119890

119889119904)

119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904)

119879119890= 119870119905119890120582119903119894119890

119902119904= 119870119905119890120582119903119894119879

(22)


119870119905119890=3

2

119875

2

119871119898

119871119903

(23)




119894119904= radic(119894119890

119902119904)2

+ (119894119890

119889119904)2

(24)


[

[

119894119890

119902119904

119894119890

119889119904

]

]

=2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+4120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+4120587

3)

]]]

]

[

[

119894119886119904

119894119887119904

119894119888119904

]

]

(25)


119894119902119889= [119879] [119894

119886119887119888] (26)

where

119894119902119889= [119894119890

119902119904119894119890

119889119904]119905

(27)

119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905

(28)

[119879] =2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+2120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+2120587

3)

]]]

]

(29)

where 119894119886119904 119894119887119904 and 119894





120579119891= 120579119903+ 120579sl (30)


120579sl = int120596sl (31)



119894lowast

119879=

119879lowast

119890

119870119905119890120582lowast119903

= (2

3) (

2

119875)119879lowast

119890

120582lowast119903

119871119903

119871119898

(32)

119894lowast

119891= (1 + 119901

119871119903

119877119903

)120582lowast

119903

119871119898

(33)

120596lowast

sl =119877119903119871119898

119871119903

119894lowast

119879

120582lowast119903

(34)




[

[

119894lowast

119902119904

119894lowast

119889119904

]

]

= [

[

cos 120579119891

sin 120579119891

minus sin 120579119891

cos 120579119891

]

]

[

[

119894lowast

119879

119894lowast

119891

]

]

(35)

[[[[

[

119894lowast

119886119904

119894lowast

119887119904

119894lowast

119888119904

]]]]

]

= [119879minus1

]

[[[[

[

119894lowast

119902119904

119894lowast

119889119904

0

]]]]

]

(36)

where

119879minus1

=

[[[[[[[

[

1 0 1

minus1

2minusradic3

21

minus1

2

radic3

21

]]]]]]]

]

(37)


119894lowast

119902119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119889119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 cos 120579lowast

119904

119894lowast

119886119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119887119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904minus2120587

3)

119894lowast

119888119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904+2120587

3)

(38)

where

120579lowast

119904= 120579119891+ 120579lowast

119879= 120579119903+ 120579lowast

sl + 120579lowast

119879 (39)


119879is generated


119891can be given



CSI

IM

PWMswitching

logic

Rotor speed

regulator

Stator current

generator

DC link currentias

lowast

ibslowast

i cslowast

120579riTlowast is

iflowast

iflowast and iT

lowast120582rlowast

120596rlowast

120596sllowast

120596r

Telowast

(1 + pLr

Rr

) 120582rlowast

Lm

RrLm

Lr

iTlowast

120582rlowast

4

3P

Telowast

120582rlowast

Lr

Lm

+minus




lowast|islowast|

ibslowast = |is


2120587

3

icslowast = |is


2120587

3

120579r 120579r

iTlowast

iflowast

120579Tlowast

120579slowast


lowastint

iaslowast

ibslowast

i cslowast

tanminus1 iTlowast

iflowast

+++



120582119903=119871119898119894119889119904

1 + 120591119903119904 (40)







V119904= 119877119904119894119904+119889120582119904

119889119905 (41)

120582119904= int (V

119904minus 119877119904119894119904) 119889119905 = 120582

120572119904+ 119895120582120573119904 (42)

10038161003816100381610038161205821199041003816100381610038161003816 = radic120582

2

120572119904+ 1205822

120573119904 (43)

ang120593119904= tanminus1 (

120582120573119904

120582120572119904

) (44)




119904are the



120582119902119904= 0 120582

119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)


119879119890=3

2(119875

2)1

120596119887

(120582119889119904119894119902119904) (46)



119890are


119894lowast

119902119904=2

3

1

119875

119879lowast

119890

1003816100381610038161003816120582lowast

119904

1003816100381610038161003816

(47)


Currentlimiter

Telowast

Te

|120582slowast|

2

3

1

P

PI

Telowast

|120582slowast |

+

+

+minus iqs

lowast





119902119904







119889119904


119894lowast

119904(119889119902)= 119894lowast

119889119904+ 119895119894lowast

119902119904 (48)

1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 =radic119894lowast2

119889119904+ 119894lowast2119902119904 (49)


119894lowast

119904(120572120573)= 119894lowast

120572119904+ 119895119894lowast

120573119904= (119894lowast

119889119904+ 119895119894lowast

119902119904) sdot 119890119895120593119904

=1003816100381610038161003816119894lowast

119904


119902119904

119894lowast

119889119904

))

(50)



Stator flux torque

estimator

PI

CSI

SVM IM

Gate Signals

PI

120582slowast

120591elowast

120582s

120582s

120591e

ang120593s

120591e

iqlowast

idlowast

dq

to120572120573

iaib

vb

va

|islowast| and idc

lowast

idclowast

+minus

+minus i120572

lowast

i120573lowast



119902119904
















8 Conclusion



15

minus05

0

05

1

Spee

d (P

U)

0 1 2 3 4 5Time (s)


(a)

0 1 2 3 4 5

minus2

0

2

Time (s)

Torq

ue (P

U)


(b)


0

05

1

Spee

d (P

U)

0 1 2 3 4 5minus4

minus2

0

24

Torq

ue (P

U)

Time (s)





Motor torque

Motor speed



speed reference


0

05

1

Spee

d (P

U)


012

Time (s)

Torq

ue (P

U)





Motor torque





TMS320F2812DSP

ADC

PWM

-EV

B

PWM

-EVA

Supplyfilter

3120601controlled


3120601IM

3120601supply

Texasinstruments

DSP processor

Speed sensor output

(Nr) Rpm

CM75TL-12NFseries

IGBT (6-PAC)

Ldc

Is (Amps)



1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads








Reference torque



Actual motor torque

Torque reference

Speed reference

Actual speed


1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads



Controlmethods




119901)




Controlmethods




119901)





Symbols

120596119887 Base speed

119903


119904




120582119889119903 119889-axis rotor flux

120582119890


120582119889119904 119889-axis stator flux


120582119890


120582119902119904 119902-axis stator flux

120582119903 Rotor flux


1199091015840


1198771015840









References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894119890

119902119903= minus

119871119898

119871119903

119894119890

119902119904

119894119890

119889119903=120582119903

119871119903

minus119871119898

119871119903

119894119890

119889119904

(19)

119894119890

119889119904= 119894119891= [1 +

119871119903

119877119903

119901]120582119903

119871119898

119894119890

119889119904= 119894119891= [1 + 120591

119903119901]

120582119903

119871119898

(20)

120596sl =119877119903119871119898

119871119903

119894119890

119902119904

120582119903

120596sl =119877119903119871119898

119871119903

119894119879

120582119903

120596sl =119871119898

120591119903

119894119879

120582119903

(21)





119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904minus 120582119890

119902119903119894119890

119889119904)

119879119890=3

2

119875

2

119871119898

119871119903

(120582119890

119889119903119894119890

119902119904)

119879119890= 119870119905119890120582119903119894119890

119902119904= 119870119905119890120582119903119894119879

(22)


119870119905119890=3

2

119875

2

119871119898

119871119903

(23)




119894119904= radic(119894119890

119902119904)2

+ (119894119890

119889119904)2

(24)


[

[

119894119890

119902119904

119894119890

119889119904

]

]

=2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+4120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+4120587

3)

]]]

]

[

[

119894119886119904

119894119887119904

119894119888119904

]

]

(25)


119894119902119889= [119879] [119894

119886119887119888] (26)

where

119894119902119889= [119894119890

119902119904119894119890

119889119904]119905

(27)

119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905

(28)

[119879] =2

3

[[[

[

cos 120579119891

cos(120579119891minus2120587

3) cos(120579

119891+2120587

3)

sin 120579119891

sin(120579119891minus2120587

3) sin(120579

119891+2120587

3)

]]]

]

(29)

where 119894119886119904 119894119887119904 and 119894





120579119891= 120579119903+ 120579sl (30)


120579sl = int120596sl (31)



119894lowast

119879=

119879lowast

119890

119870119905119890120582lowast119903

= (2

3) (

2

119875)119879lowast

119890

120582lowast119903

119871119903

119871119898

(32)

119894lowast

119891= (1 + 119901

119871119903

119877119903

)120582lowast

119903

119871119898

(33)

120596lowast

sl =119877119903119871119898

119871119903

119894lowast

119879

120582lowast119903

(34)




[

[

119894lowast

119902119904

119894lowast

119889119904

]

]

= [

[

cos 120579119891

sin 120579119891

minus sin 120579119891

cos 120579119891

]

]

[

[

119894lowast

119879

119894lowast

119891

]

]

(35)

[[[[

[

119894lowast

119886119904

119894lowast

119887119904

119894lowast

119888119904

]]]]

]

= [119879minus1

]

[[[[

[

119894lowast

119902119904

119894lowast

119889119904

0

]]]]

]

(36)

where

119879minus1

=

[[[[[[[

[

1 0 1

minus1

2minusradic3

21

minus1

2

radic3

21

]]]]]]]

]

(37)


119894lowast

119902119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119889119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 cos 120579lowast

119904

119894lowast

119886119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119887119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904minus2120587

3)

119894lowast

119888119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904+2120587

3)

(38)

where

120579lowast

119904= 120579119891+ 120579lowast

119879= 120579119903+ 120579lowast

sl + 120579lowast

119879 (39)


119879is generated


119891can be given



CSI

IM

PWMswitching

logic

Rotor speed

regulator

Stator current

generator

DC link currentias

lowast

ibslowast

i cslowast

120579riTlowast is

iflowast

iflowast and iT

lowast120582rlowast

120596rlowast

120596sllowast

120596r

Telowast

(1 + pLr

Rr

) 120582rlowast

Lm

RrLm

Lr

iTlowast

120582rlowast

4

3P

Telowast

120582rlowast

Lr

Lm

+minus




lowast|islowast|

ibslowast = |is


2120587

3

icslowast = |is


2120587

3

120579r 120579r

iTlowast

iflowast

120579Tlowast

120579slowast


lowastint

iaslowast

ibslowast

i cslowast

tanminus1 iTlowast

iflowast

+++



120582119903=119871119898119894119889119904

1 + 120591119903119904 (40)







V119904= 119877119904119894119904+119889120582119904

119889119905 (41)

120582119904= int (V

119904minus 119877119904119894119904) 119889119905 = 120582

120572119904+ 119895120582120573119904 (42)

10038161003816100381610038161205821199041003816100381610038161003816 = radic120582

2

120572119904+ 1205822

120573119904 (43)

ang120593119904= tanminus1 (

120582120573119904

120582120572119904

) (44)




119904are the



120582119902119904= 0 120582

119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)


119879119890=3

2(119875

2)1

120596119887

(120582119889119904119894119902119904) (46)



119890are


119894lowast

119902119904=2

3

1

119875

119879lowast

119890

1003816100381610038161003816120582lowast

119904

1003816100381610038161003816

(47)


Currentlimiter

Telowast

Te

|120582slowast|

2

3

1

P

PI

Telowast

|120582slowast |

+

+

+minus iqs

lowast





119902119904







119889119904


119894lowast

119904(119889119902)= 119894lowast

119889119904+ 119895119894lowast

119902119904 (48)

1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 =radic119894lowast2

119889119904+ 119894lowast2119902119904 (49)


119894lowast

119904(120572120573)= 119894lowast

120572119904+ 119895119894lowast

120573119904= (119894lowast

119889119904+ 119895119894lowast

119902119904) sdot 119890119895120593119904

=1003816100381610038161003816119894lowast

119904


119902119904

119894lowast

119889119904

))

(50)



Stator flux torque

estimator

PI

CSI

SVM IM

Gate Signals

PI

120582slowast

120591elowast

120582s

120582s

120591e

ang120593s

120591e

iqlowast

idlowast

dq

to120572120573

iaib

vb

va

|islowast| and idc

lowast

idclowast

+minus

+minus i120572

lowast

i120573lowast



119902119904
















8 Conclusion



15

minus05

0

05

1

Spee

d (P

U)

0 1 2 3 4 5Time (s)


(a)

0 1 2 3 4 5

minus2

0

2

Time (s)

Torq

ue (P

U)


(b)


0

05

1

Spee

d (P

U)

0 1 2 3 4 5minus4

minus2

0

24

Torq

ue (P

U)

Time (s)





Motor torque

Motor speed



speed reference


0

05

1

Spee

d (P

U)


012

Time (s)

Torq

ue (P

U)





Motor torque





TMS320F2812DSP

ADC

PWM

-EV

B

PWM

-EVA

Supplyfilter

3120601controlled


3120601IM

3120601supply

Texasinstruments

DSP processor

Speed sensor output

(Nr) Rpm

CM75TL-12NFseries

IGBT (6-PAC)

Ldc

Is (Amps)



1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads








Reference torque



Actual motor torque

Torque reference

Speed reference

Actual speed


1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads



Controlmethods




119901)




Controlmethods




119901)





Symbols

120596119887 Base speed

119903


119904




120582119889119903 119889-axis rotor flux

120582119890


120582119889119904 119889-axis stator flux


120582119890


120582119902119904 119902-axis stator flux

120582119903 Rotor flux


1199091015840


1198771015840









References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












[

[

119894lowast

119902119904

119894lowast

119889119904

]

]

= [

[

cos 120579119891

sin 120579119891

minus sin 120579119891

cos 120579119891

]

]

[

[

119894lowast

119879

119894lowast

119891

]

]

(35)

[[[[

[

119894lowast

119886119904

119894lowast

119887119904

119894lowast

119888119904

]]]]

]

= [119879minus1

]

[[[[

[

119894lowast

119902119904

119894lowast

119889119904

0

]]]]

]

(36)

where

119879minus1

=

[[[[[[[

[

1 0 1

minus1

2minusradic3

21

minus1

2

radic3

21

]]]]]]]

]

(37)


119894lowast

119902119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119889119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 cos 120579lowast

119904

119894lowast

119886119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin 120579lowast

119904

119894lowast

119887119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904minus2120587

3)

119894lowast

119888119904=1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 sin(120579lowast

119904+2120587

3)

(38)

where

120579lowast

119904= 120579119891+ 120579lowast

119879= 120579119903+ 120579lowast

sl + 120579lowast

119879 (39)


119879is generated


119891can be given



CSI

IM

PWMswitching

logic

Rotor speed

regulator

Stator current

generator

DC link currentias

lowast

ibslowast

i cslowast

120579riTlowast is

iflowast

iflowast and iT

lowast120582rlowast

120596rlowast

120596sllowast

120596r

Telowast

(1 + pLr

Rr

) 120582rlowast

Lm

RrLm

Lr

iTlowast

120582rlowast

4

3P

Telowast

120582rlowast

Lr

Lm

+minus




lowast|islowast|

ibslowast = |is


2120587

3

icslowast = |is


2120587

3

120579r 120579r

iTlowast

iflowast

120579Tlowast

120579slowast


lowastint

iaslowast

ibslowast

i cslowast

tanminus1 iTlowast

iflowast

+++



120582119903=119871119898119894119889119904

1 + 120591119903119904 (40)







V119904= 119877119904119894119904+119889120582119904

119889119905 (41)

120582119904= int (V

119904minus 119877119904119894119904) 119889119905 = 120582

120572119904+ 119895120582120573119904 (42)

10038161003816100381610038161205821199041003816100381610038161003816 = radic120582

2

120572119904+ 1205822

120573119904 (43)

ang120593119904= tanminus1 (

120582120573119904

120582120572119904

) (44)




119904are the



120582119902119904= 0 120582

119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)


119879119890=3

2(119875

2)1

120596119887

(120582119889119904119894119902119904) (46)



119890are


119894lowast

119902119904=2

3

1

119875

119879lowast

119890

1003816100381610038161003816120582lowast

119904

1003816100381610038161003816

(47)


Currentlimiter

Telowast

Te

|120582slowast|

2

3

1

P

PI

Telowast

|120582slowast |

+

+

+minus iqs

lowast





119902119904







119889119904


119894lowast

119904(119889119902)= 119894lowast

119889119904+ 119895119894lowast

119902119904 (48)

1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 =radic119894lowast2

119889119904+ 119894lowast2119902119904 (49)


119894lowast

119904(120572120573)= 119894lowast

120572119904+ 119895119894lowast

120573119904= (119894lowast

119889119904+ 119895119894lowast

119902119904) sdot 119890119895120593119904

=1003816100381610038161003816119894lowast

119904


119902119904

119894lowast

119889119904

))

(50)



Stator flux torque

estimator

PI

CSI

SVM IM

Gate Signals

PI

120582slowast

120591elowast

120582s

120582s

120591e

ang120593s

120591e

iqlowast

idlowast

dq

to120572120573

iaib

vb

va

|islowast| and idc

lowast

idclowast

+minus

+minus i120572

lowast

i120573lowast



119902119904
















8 Conclusion



15

minus05

0

05

1

Spee

d (P

U)

0 1 2 3 4 5Time (s)


(a)

0 1 2 3 4 5

minus2

0

2

Time (s)

Torq

ue (P

U)


(b)


0

05

1

Spee

d (P

U)

0 1 2 3 4 5minus4

minus2

0

24

Torq

ue (P

U)

Time (s)





Motor torque

Motor speed



speed reference


0

05

1

Spee

d (P

U)


012

Time (s)

Torq

ue (P

U)





Motor torque





TMS320F2812DSP

ADC

PWM

-EV

B

PWM

-EVA

Supplyfilter

3120601controlled


3120601IM

3120601supply

Texasinstruments

DSP processor

Speed sensor output

(Nr) Rpm

CM75TL-12NFseries

IGBT (6-PAC)

Ldc

Is (Amps)



1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads








Reference torque



Actual motor torque

Torque reference

Speed reference

Actual speed


1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads



Controlmethods




119901)




Controlmethods




119901)





Symbols

120596119887 Base speed

119903


119904




120582119889119903 119889-axis rotor flux

120582119890


120582119889119904 119889-axis stator flux


120582119890


120582119902119904 119902-axis stator flux

120582119903 Rotor flux


1199091015840


1198771015840









References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












V119904= 119877119904119894119904+119889120582119904

119889119905 (41)

120582119904= int (V

119904minus 119877119904119894119904) 119889119905 = 120582

120572119904+ 119895120582120573119904 (42)

10038161003816100381610038161205821199041003816100381610038161003816 = radic120582

2

120572119904+ 1205822

120573119904 (43)

ang120593119904= tanminus1 (

120582120573119904

120582120572119904

) (44)




119904are the



120582119902119904= 0 120582

119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)


119879119890=3

2(119875

2)1

120596119887

(120582119889119904119894119902119904) (46)



119890are


119894lowast

119902119904=2

3

1

119875

119879lowast

119890

1003816100381610038161003816120582lowast

119904

1003816100381610038161003816

(47)


Currentlimiter

Telowast

Te

|120582slowast|

2

3

1

P

PI

Telowast

|120582slowast |

+

+

+minus iqs

lowast





119902119904







119889119904


119894lowast

119904(119889119902)= 119894lowast

119889119904+ 119895119894lowast

119902119904 (48)

1003816100381610038161003816119894lowast

119904

1003816100381610038161003816 =radic119894lowast2

119889119904+ 119894lowast2119902119904 (49)


119894lowast

119904(120572120573)= 119894lowast

120572119904+ 119895119894lowast

120573119904= (119894lowast

119889119904+ 119895119894lowast

119902119904) sdot 119890119895120593119904

=1003816100381610038161003816119894lowast

119904


119902119904

119894lowast

119889119904

))

(50)



Stator flux torque

estimator

PI

CSI

SVM IM

Gate Signals

PI

120582slowast

120591elowast

120582s

120582s

120591e

ang120593s

120591e

iqlowast

idlowast

dq

to120572120573

iaib

vb

va

|islowast| and idc

lowast

idclowast

+minus

+minus i120572

lowast

i120573lowast



119902119904
















8 Conclusion



15

minus05

0

05

1

Spee

d (P

U)

0 1 2 3 4 5Time (s)


(a)

0 1 2 3 4 5

minus2

0

2

Time (s)

Torq

ue (P

U)


(b)


0

05

1

Spee

d (P

U)

0 1 2 3 4 5minus4

minus2

0

24

Torq

ue (P

U)

Time (s)





Motor torque

Motor speed



speed reference


0

05

1

Spee

d (P

U)


012

Time (s)

Torq

ue (P

U)





Motor torque





TMS320F2812DSP

ADC

PWM

-EV

B

PWM

-EVA

Supplyfilter

3120601controlled


3120601IM

3120601supply

Texasinstruments

DSP processor

Speed sensor output

(Nr) Rpm

CM75TL-12NFseries

IGBT (6-PAC)

Ldc

Is (Amps)



1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads








Reference torque



Actual motor torque

Torque reference

Speed reference

Actual speed


1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads



Controlmethods




119901)




Controlmethods




119901)





Symbols

120596119887 Base speed

119903


119904




120582119889119903 119889-axis rotor flux

120582119890


120582119889119904 119889-axis stator flux


120582119890


120582119902119904 119902-axis stator flux

120582119903 Rotor flux


1199091015840


1198771015840









References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Stator flux torque

estimator

PI

CSI

SVM IM

Gate Signals

PI

120582slowast

120591elowast

120582s

120582s

120591e

ang120593s

120591e

iqlowast

idlowast

dq

to120572120573

iaib

vb

va

|islowast| and idc

lowast

idclowast

+minus

+minus i120572

lowast

i120573lowast



119902119904
















8 Conclusion



15

minus05

0

05

1

Spee

d (P

U)

0 1 2 3 4 5Time (s)


(a)

0 1 2 3 4 5

minus2

0

2

Time (s)

Torq

ue (P

U)


(b)


0

05

1

Spee

d (P

U)

0 1 2 3 4 5minus4

minus2

0

24

Torq

ue (P

U)

Time (s)





Motor torque

Motor speed



speed reference


0

05

1

Spee

d (P

U)


012

Time (s)

Torq

ue (P

U)





Motor torque





TMS320F2812DSP

ADC

PWM

-EV

B

PWM

-EVA

Supplyfilter

3120601controlled


3120601IM

3120601supply

Texasinstruments

DSP processor

Speed sensor output

(Nr) Rpm

CM75TL-12NFseries

IGBT (6-PAC)

Ldc

Is (Amps)



1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads








Reference torque



Actual motor torque

Torque reference

Speed reference

Actual speed


1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads



Controlmethods




119901)




Controlmethods




119901)





Symbols

120596119887 Base speed

119903


119904




120582119889119903 119889-axis rotor flux

120582119890


120582119889119904 119889-axis stator flux


120582119890


120582119902119904 119902-axis stator flux

120582119903 Rotor flux


1199091015840


1198771015840









References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










15

minus05

0

05

1

Spee

d (P

U)

0 1 2 3 4 5Time (s)


(a)

0 1 2 3 4 5

minus2

0

2

Time (s)

Torq

ue (P

U)


(b)


0

05

1

Spee

d (P

U)

0 1 2 3 4 5minus4

minus2

0

24

Torq

ue (P

U)

Time (s)





Motor torque

Motor speed



speed reference


0

05

1

Spee

d (P

U)


012

Time (s)

Torq

ue (P

U)





Motor torque





TMS320F2812DSP

ADC

PWM

-EV

B

PWM

-EVA

Supplyfilter

3120601controlled


3120601IM

3120601supply

Texasinstruments

DSP processor

Speed sensor output

(Nr) Rpm

CM75TL-12NFseries

IGBT (6-PAC)

Ldc

Is (Amps)



1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads








Reference torque



Actual motor torque

Torque reference

Speed reference

Actual speed


1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads



Controlmethods




119901)




Controlmethods




119901)





Symbols

120596119887 Base speed

119903


119904




120582119889119903 119889-axis rotor flux

120582119890


120582119889119904 119889-axis stator flux


120582119890


120582119902119904 119902-axis stator flux

120582119903 Rotor flux


1199091015840


1198771015840









References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Reference torque



Actual motor torque

Torque reference

Speed reference

Actual speed


1 Div = 50Nm

1 Div = 50Nm

1 Div = 50 rads

1 Div = 50 rads



Controlmethods




119901)




Controlmethods




119901)





Symbols

120596119887 Base speed

119903


119904




120582119889119903 119889-axis rotor flux

120582119890


120582119889119904 119889-axis stator flux


120582119890


120582119902119904 119902-axis stator flux

120582119903 Rotor flux


1199091015840


1198771015840









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









Documents

Research Article Mathematical Modeling and Analysis of …downloads.hindawi.com/journals/jam/2014/623982.pdf · 2019-07-31 · Research Article Mathematical Modeling and Analysis