effective formulation of the dtc strategy for convergence...

Effective Formulation of the DTC Strategyfor Convergence and Stability Analysis

The IPM Motor Drive Case Study

Adriano Faggion Silverio Bolognani

Electric Drives LaboratoryDepartment of Industrial Engineering

University of Padova - Italy

IEEE - SLED–PRECEDE 20132nd Symposium on Predictive Control of Electrical Drives and

Power ElectronicsMunich, 17-19 October 2013, Germany

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Introduction

DTC newformulation

IPM motordrive casestudy

Simulationresults

Experimentalresults

Outline

1 Introduction

2 DTC new formulation

3 IPM motor drive case study

4 Simulation results

5 Experimental results

SLED–PRECEDE 2013 Sensorless control using High Frequency injection signals 2

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Introduction

DTC newformulation


Simulationresults

Experimentalresults

Introduction

Introduction


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

Introduction

Direct Torque Control technique

The Direct Torque Control DTC is the control technique thatdefines the three–phase voltages source inverter state onthe basis of the torque and flux errors, without current controlloops.

DTC has two variants:1 Finite Control Set, if the selection of the voltage vector

is performed among the six active inverter spatialvectors (U1 · · · U6) and the two null spatial vectors (U0and U7)

2 Continuous Control Set, if voltage vectors of any phaseangle are available (thanks to a PWM voltage control).


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

Introduction

Direct Torque Control technique

DTC FormulationA novel formulation of the DTC base principle is presentedthat can be an effective tool for understanding andcomparing implementation variants but also for studyingconvergence and stability issues.

IPM applicationAn IPM synchronous motor drive controlled by a FiniteControl Set DTC is assumed as case study for exemplifyingthe proposes approach of analysis.


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

DTC new formulation

DTC new formulation


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

DTC new formulation

Mathematical formulation

New complex variableThe DTC is a technique based on the control of:

torque mmodule of flux vector |λ|

A new complex variable z can be introduced:

z =m

MN+ j

|λ|ΛN

where MN and ΛN are the torque and flux nominal values.


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

DTC new formulation


ErrorFor a given reference torque m∗ and reference flux |λ|∗, it ispossible to define the error ε as:

ε = z∗ − z =m∗ − m

MN+ j

|λ|∗ − |λ|ΛN

= εm + jε|λ|

|ε| =√

ε2m + ε2

|λ|


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

DTC new formulation


Target of the control

The target of the control is to maintain the actual vector zvery close to the reference z∗.

|ε| ≤ Emax

where Emax is a prefixed value.In the case in which the inequality is satisfied the controlcontinues applying the same voltage vector.


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

DTC new formulation

Adopted strategy

Right case

When ε ≥ Emax an action has to be taken to reduce |ε|choosing the new vector voltage in order to obtain ε in thenext step.

In this case the applied voltage is correctly selected.


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

DTC new formulation

Adopted strategy

Wrong case

On the contrary, in this case the voltage selection is wrong.


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

DTC new formulation

Adopted strategy

Wrong case

Control convergence condition is that the selected U has to

meetd |ε|dt

≤ 0.

In the case of Finite Control Set the possible voltage arechosen among the six inverter spatial vectors (U1 . . . U6)and the two null spatial vectors (U0 and U7).


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

DTC new formulation

Convergence condition

d |ε|dt

=d√

ε2m + ε2

|λ|

dt=

εmεm + ε|λ|ε|λ|√ε2

m + ε2|λ|

Then equivalent convergence conditions are:

∆ = εmεm + ε|λ|ε|λ| ≤ 0

∆ = εm

(− m

MN

)+ ε|λ|

(−

˙|λ|ΛN

)≤ 0


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

DTC new formulation

Convergence condition

Prediction nature of DTC

∆ = εm

(− m

MN

)+ ε|λ|

(−

˙|λ|ΛN

)≤ 0

Prediction nature of DTC is evident from the last equationsas the control is decided by the future (of course predicted)error (or feedback) derivatives.


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

IPM motor drive case study



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Introduction

DTC newformulation


Simulationresults

Experimentalresults


Control features

Equations are expressed in discrete form being thecontrol implemented in discrete time.Last sampling time index is kThe chosen U(k) will be imposed at instant k + 1Then a 1 − step prediction of currents and otherquantities must be done.The time–line is moved at the instant k + 1 because ofthe inverter delay.


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Introduction

DTC newformulation


Simulationresults

Experimentalresults


Control algorithm

If the error is inside the limit, then no action is taken and thevoltage vector of the previously step is maintained.Otherwise, if the error exceeds the limit, the new voltagevector is chosen in order to bring back the error inside thelimit.


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DTC newformulation


Simulationresults

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Torque and flux equation

The two controlled variable are the torque and the stator fluxmodule that can be expressed as:

m(k + 1) =32

pΛmg iq(k + 1) +32

p(Ld − Lq)id (k + 1)iq(k + 1)

|λ(k + 1)| =√

λd (k + 1)2 + λq(k + 1)2

λd (k + 1) = Ld id (k + 1) + Λmg

λq(k + 1) = Lq iq(k + 1)

Torque and flux can be calculated at any sampling time,starting from the measured or predicted current (id and iq).


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DTC newformulation


Simulationresults

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Control algorithm-Choosing of voltage vector

The new voltage vector are chosen in order to minimize ∆j ,i.e. to determine its higher negative value.

∆j = εU j

m

(− mj

MN

)+ εU j

|λ|

(−

˙|λ|j

ΛN

)≤ 0


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DTC newformulation


Simulationresults

Experimentalresults


Control algorithm

Prediction of currentsThe currents are predicted considering the voltage balanceequations:

i Uj

d (k + 2) = Tsdiddt

(k + 1) + id (k + 1)

= Ts

(uj

d (k + 1)

Ld−

RLd

id (k + 1) + ωmeLq

Ldiq(k + 1)

)+ id (k + 1)

i Uj

q (k + 2) = Tsdiqdt

(k + 1) + iq(k + 1)

= Ts

(uj

q(k + 1)

Lq−

RLq

iq(k + 1) −ωme

Lq(Ld id (k + 1) + Λmg)

)+ iq(k + 1)

mj (k + 2) and |λ|j (k + 1) are predicted from i jd (k + 2) and i jq(k + 1).


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Introduction

DTC newformulation


Simulationresults

Experimentalresults


Control algorithm

Key points of the control:No switching table is required by this approach.To reduce the switching frequency it is possible toimplement a switch state graph with the law that onlyone inverter leg can be switched.Electrical machine model must be well known.


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d– and q–axis current plane

• The machine workingpont is defined by achosen m∗ and |λ|∗.• Fixed the referencevalues, there are twopossible Working Points(WP) given by the inter-sections between theconstant torque andconstant flux curves.

d-axis current, id(A)

q-ax

iscu

rren

t,i q(A

)

-20 -15 -10 -5 00

5

10

15

20

Costant Torque lociMTPACurrent limitMTPVIsoflux curves


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DTC newformulation


Simulationresults

Experimentalresults


d– and q–axis current plane

• The presence of twoWP is a drawback of thecontrol strategies.• The machine could beoperated in one or theother point indiscrimi-nately.• WP1 is along theMTPA trajectory. Coor-dination between torqueand flux references isneeded, in order to workon MTPA locus.


q-ax

iscu

rren

t,i q(A

)

-20 -15 -10 -5 00

5

10

15

20

Costant Torque lociMTPACurrent limitMTPVIsoflux curves


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

Simulation results

Simulation results


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

Simulation results

Simulation scheme


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DTC newformulation


Simulationresults

Experimentalresults

Simulation results

Simulation featuresThe motor has been simulated taking into account itsnon–linear magnetic characteristics.The actual torque–flux relationships extracted byexperimental results are used.The DTC is designed assuming linear magneticcharacteristics.


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DTC newformulation


Simulationresults

Experimentalresults

Simulation results

Speed reference step

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

time, t(s)

flux

mod

ule,

|λ|(

Vs)

actualcalculatedreference

0 0.05 0.1 0.15 0.2 0.25-5

0

5

10

15

torq

ue, m

(Nm

)

actualcalculatedreference

0 0 .05 0 .1 0 .15 0 .2 0 .25-0.2

0

0.2

0.4

0.6

0.8

1

spee

d, n

(krp

m)

referenceactual


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

Simulation results

Speed reference stepThe speed presents an oscillation around the referencevalue of 0.3 rpm.The actual torque and flux presents an error, respect tothe reference value, due to the linear model used in theDTC control.Flux reference value is chosen from torque reference inorder to control the machine along the MTPA trajectory.


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DTC newformulation


Simulationresults

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Simulation results

Speed reference step

-2

-2

-1-1

0 0

11

2

23

3

4

4

5

5

6

6

7

7

8

8

9

9

10

11

12

13


q-ax

is c

urre

nt,

i q(A

)

-15 -10 -5 0-2

0

2

4

6

8

10

12

14

16 Constant torque lociid-iq

trajectory

MTPA

Linear MTPA

Current limit

Linear flux ellipse


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

Simulation results

Two stable points

The error |ε| is plotted for a given m∗ and |λ|∗, in the planeid–iq.

−20 −15 −10 −5 00

10

20

0

0.5

1

1.5

2

10

d−axis current, id(A)

10

10

(A)

q−axis current, iq

Normalized

Error,

id−iqtrajectory

MTPAMTPVConstant torque lociIso flux curve

The function presents two minimum points that representsthe two possible working–points of the machine.


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Introduction

DTC newformulation


Simulationresults

Experimentalresults

Experimental results

Experimental resultsTest bench


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DTC newformulation


Simulationresults

Experimentalresults


Test bench

IPM machineX Some tests have been carried out in order to

confirm the mathematical analysis.X The test bench is equipped with a master

motor that can be speed or torque controlledby an industrial inverter.

X The machine under test is an IPM motor with12–slot and 10–pole.

X The IPM motor is controlled by means of alaboratory inverter coupled to a dSpace 1104control board.


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DTC newformulation


Simulationresults

Experimentalresults


Delay on the duty–cycles updating

• Firmware is imple-mented in the dSpace1104.• New duty–cyclevalue are calculatedinside the ISR on themaster PPC.• ISR is trigged by thePWM interrupt.• Master transfersthe new values to theslave, that store themein a global variables.

ISRDSP

ISRDSP

ISRDSP

PWM signal

Slave (DSP)

Master (PPC)

Timer slave DSP

t

t

t

t

Timer=0,duty cycle update

Globalvariable

Compare register

DSPinterrupt

DSPinterrupt

DSPinterrupt

ST1PWM ST1PWM ST1PWM

TPWM

ISR-PPC ISR-PPC ISR-PPC

Write dutycycle


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Introduction

DTC newformulation


Simulationresults

Experimentalresults


Delay on the duty–cycles updating

• At TPWM/2 slavecopied the values ofglobal variables intothe PWM compareregister unit.• Duty cycle is updatedfor the next PWMperiod if the new val-ues are stored beforeTPWM/2.• Otherwise, the dutycycles are updated inthe second next PWMperiod.

ISRDSP

ISRDSP

ISRDSP

PWM signal

Slave (DSP)

Master (PPC)

Timer slave DSP

t

t

t

t

Timer=0,duty cycle update

Globalvariable

Compare register

DSPinterrupt

DSPinterrupt

DSPinterrupt

ST1PWM ST1PWM ST1PWM

TPWM

ISR-PPC ISR-PPC ISR-PPC

Write dutycycle


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DTC newformulation


Simulationresults

Experimentalresults


Delay on the duty–cycle updating

X The DTC control code required a lot of computation timeand then the new values of duty–cycles are updated aftertwo PWM periods.X The control must be modified.X The currents must be predicted, with a prediction of twosteps.X From them all the other quantities can be predicted.X Finally the error ε is evaluated.X From it, when necessary, the new duty–cycle values areevaluated in according to the algorithm previouslydescribed.


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Introduction

DTC newformulation


Simulationresults

Experimentalresults


Experimental resultsTests with 1–step and 2–steps

prediction


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Introduction

DTC newformulation


Simulationresults

Experimentalresults


Tests with 1–step and 2–steps prediction

Effects of error prediction have been investigated atfirst.The machine is dragged at constant speed of about400 rpm.The reference torque is equal to 5 Nm and thereference flux is equal to 0.1337 Vs.Results with and without currents prediction will beshown.


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DTC newformulation


Simulationresults

Experimentalresults


Tests with 1 − step and 2 − steps prediction

Tests with 1 − step prediction

• At instant k error ε ishigher than Emax .• New voltage vector,able to cause a nega-tive error derivative, ischosen.• Such vector is ap-plied at instant k + 2.• The error starts to de-crease only at the in-stant k + 2.

0

0.05

0.1

0.150.11

0.12

0.13

0.14

0.15

0.16

4.5

5

5.5

6

01234567


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Introduction

DTC newformulation


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Tests with 2 − steps prediction

• Currents id (k + 2)and iq(k + 2) arepredicted.• Blue curves are forpredicted quantitieswhile black is for theactual ones.• One can realize thatpredictions anticipatethe actual variables bytwo steps with a goodaccuracy.• Discrepancies aremainly due to param-eter mismatch andmodel inaccuracy.

0

0.05

0.1

0.15

4.6

4.8

5

5.2

5.4

0.11

0.12

0.13

0.14

0.15

0

1

2

3

4

5

6

7


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DTC newformulation


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Tests with 2 − steps prediction

• At instant k the predictederror exceeds the limit Emaxwhile the actual error is wellinside the limit.• Therefore a new voltagevector is calculated that willbe applied at time k + 2.• At instant k +2 actual errorhas just overcame the limitand is forced to come back.• Error exceeds the limit onlyfor a single sampling time,provided that prediction isperformed accurately.

0

0.05

0.1

0.15

4.6

4.8

5

5.2

5.4

0.11

0.12

0.13

0.14

0.15

0

1

2

3

4

5

6

7


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Introduction

DTC newformulation


Simulationresults

Experimentalresults


Experimental resultsTests with PWM vectors graph and

control performance


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Introduction

DTC newformulation


Simulationresults

Experimentalresults


Tests with PWM vectors graph and control performance

A switch state graph can be adopted for limiting number ofinverter switch commutation.


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Introduction

DTC newformulation


Simulationresults

Experimentalresults


Tests with PWM vectors graph and control performance

Performance comparisonComparison of control performance with and withoutprediction, with and without switch state graph areperformed.Error and number of phase–a commutation in 1 s atsteady–state operation are taken into account.

With pred. With pred. Without pred. Without pred.without graph with graph without graph with graph

Average Error 0.0568 0.0605 0.0878 0.091] commutation

phase a 3093 2323 2200 2007


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Introduction

DTC newformulation


Simulationresults

Experimentalresults


Experimental resultsSpeed control with 2–step prediction


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Experimentalresults


Speed control with 2 − steps prediction

Features of controlReference speed step from 400 rpm to −400 rpmNominal torque equal to 7 NmNominal flux equal to about 0.17 Vs.


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Simulationresults

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time, t(s)0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

Error

4 5 60

0.05

0.1

0.15

0 1 2 3 4 5 6 7 8 9 10-10

-5

0

5

10

Tor

que,

m(N

m)

0 1 2 3 4 5 6 7 8 9 100.05

0.1

0.15

0.2

Flu

x, |Λ

|(V

s)

0 1 2 3 4 5 6 7 8 9 10

0

100

200

300

400

speed,ω(rpm)


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Introduction

DTC newformulation


Simulationresults

Experimentalresults



DTC performance in response to torque step

• Current trajectoryin the id–iq plane.• |ε| surface calcu-lated for given steptorque and flux refer-ence.

−20−10

0

0

10

200

0.5

1

1.5

2

0

0

d−axis current, i d(A)

0

5

5

q−axis current, iq (A)

Nor

mal

ized

err

or,

id−iq

trajectory



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Introduction

DTC newformulation


Simulationresults

Experimentalresults



DTC performance in response to torque step• Initially the trajec-tory current remainsaround the planeorigin.• When the referencestep occurs, the op-erating point moveto the error surfaceand slides towards thenearest hollow.• This is a stableoperating point thatguaranties a null erroron the MTPA locus.

−20−10

0

0

10

200

0.5

1

1.5

2

0

0

d−axis current, i d(A)

0

5

5

q−axis current, iq (A)

Nor

mal

ized

err

or,

id−iq

trajectory



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Introduction

DTC newformulation


Simulationresults

Experimentalresults


Control trouble due to the double stable points

id–iq trajectory

• By reducing theflux level with a giventorque, the two in-tersection pointsapproach each–other.• The operation pointcould jump casuallyfrom one of the twohollows to the other.

-12 -10 -8 -6 -4 -2-5

-4

-3

-2

-1

0

1

2

3

4

5


q-ax

is c

urre

nt, i

q(A)

currentiso fluxiso fluxiso torque

1

0

1

0

0.028


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Simulationresults

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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-12

-10

-8

-6

-4

-2

d-ax

is c

urre

nt, i

d(A)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

1

1.5

2

2.5

time, t(s)

q-ax

is c

urre

nt, i

q(A)

One can note that the currents id and iq assume twodifferent values casually.


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Introduction

DTC newformulation


Simulationresults

Experimentalresults



Wright operating point

• Increasing the fluxlevel again, the twominima of the er-ror surface distancethemselves and theoperating point re-mains trapped in oneof the two.• In this case the op-erating point movestowards lower cur-rents on the MTPAlocus.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2-6

-4

-2

0

2

4

6


q-ax

is c

urre

nt, i

q(A)

currentiso fluxiso torque

0.088

0.02

0

1


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Introduction

DTC newformulation


Simulationresults

Experimentalresults



Bad operating point

• On the contrary, in thiscase the operating pointmoves towards higher cur-rents on the right side ofMTPV locus.• This cause the interventionof the current protection ofthe drive and the currents goto zero.

-16 -14 -12 -10 -8 -6 -4 -2 0-8

-6

-4

-2

0

2

4

6

8


q-ax

iscu

rren

t,i q(A

)

currentiso fluxiso torque

0.088

0.02

0

1


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Introduction

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Simulationresults

Experimentalresults


Thank youfor your attention


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Appendix index

EP


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Electrical Parameter

Parameter Symbol Value UnitPair–pole p 5

Slot number sl 12Phase resistance R 0.063 Ωd–axis inductance Ld 0.012 Hq–axis inductance Lq 0.020 H

Residual flux linkage Λmg 0.088 VsNominal voltage UN 80 V


effective formulation of the dtc strategy for convergence...

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