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
IPM motordrive casestudy
Simulationresults
Experimentalresults
Introduction
Introduction
SLED–PRECEDE 2013 Sensorless control using High Frequency injection signals 3
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Introduction
DTC newformulation
IPM motordrive casestudy
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
IPM motordrive casestudy
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
DTC new formulation
DTC new formulation
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Introduction
DTC newformulation
IPM motordrive casestudy
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
DTC new formulation
Mathematical 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
IPM motordrive casestudy
Simulationresults
Experimentalresults
DTC new formulation
Mathematical 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
IPM motordrive casestudy
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
IPM motordrive casestudy
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
IPM motordrive casestudy
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
IPM motordrive casestudy
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
IPM motordrive casestudy
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
IPM motor drive case study
IPM motor drive case study
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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
IPM motor drive case study
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
IPM motor drive case study
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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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
IPM motor drive case study
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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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
IPM motor drive case study
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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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
IPM motor drive case study
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
IPM motor drive case study
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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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
IPM motor drive case study
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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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
IPM motor drive case study
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.
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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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Simulation results
Simulation results
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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Simulation results
Simulation scheme
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Introduction
DTC newformulation
IPM motordrive casestudy
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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Introduction
DTC newformulation
IPM motordrive casestudy
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
IPM motordrive casestudy
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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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
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
d-axis current, id(A)
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
IPM motordrive casestudy
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Experimental resultsTest bench
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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
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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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
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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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Experimental resultsTests with 1–step and 2–steps
prediction
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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
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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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Tests with 1 − step and 2 − steps prediction
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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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Tests with 1 − step and 2 − steps prediction
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Experimental resultsTests with PWM vectors graph and
control performance
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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Experimental resultsSpeed control with 2–step prediction
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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
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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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Speed control with 2 − steps prediction
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Speed control with 2 − steps prediction
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
MTPAMTPVConstant torque lociIso flux curve
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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Speed control with 2 − steps prediction
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
MTPAMTPVConstant torque lociIso flux curve
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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
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
d-axis current, id(A)
q-ax
is c
urre
nt, i
q(A)
currentiso fluxiso fluxiso torque
1
0
1
0
0.028
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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Control trouble due to the double stable points
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
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Control trouble due to the double stable points
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
d-axis current, id(A)
q-ax
is c
urre
nt, i
q(A)
currentiso fluxiso torque
0.088
0.02
0
1
SLED–PRECEDE 2013 Sensorless control using High Frequency injection signals 51
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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Control trouble due to the double stable points
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
d-axis current, id(A)
q-ax
iscu
rren
t,i q(A
)
currentiso fluxiso torque
0.088
0.02
0
1
SLED–PRECEDE 2013 Sensorless control using High Frequency injection signals 52
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Introduction
DTC newformulation
IPM motordrive casestudy
Simulationresults
Experimentalresults
Experimental results
Thank youfor your attention
SLED–PRECEDE 2013 Sensorless control using High Frequency injection signals 53
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Appendix index
EP
SLED–PRECEDE 2013 Sensorless control using High Frequency injection signals 54
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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
SLED–PRECEDE 2013 Sensorless control using High Frequency injection signals 55
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