Digital Control Applications in Power Electronics Lez5

8/13/2019 Digital Control Applications in Power Electronics Lez5

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November 1999 1Simone Buso - University of Padova - Lesson 5

Lesson 5Lesson 5

Digital Control of Three-Phase DC/ACDigital Control of Three-Phase DC/AC

Converters: Current Control TechniquesConverters: Current Control Techniques

DigitalDigitalHysteresisHysteresisControlControl


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DigitalDigitalHysteresis ControlHysteresis Control

Thank to itsThank to its inherent non-linearity,inherent non-linearity,thethehysteresishysteresiscurrent control is capable of providingcurrent control is capable of providing the fastestthe fastest

possible dynamic response.possible dynamic response.

Using this technique, it is possible to achieve theUsing this technique, it is possible to achieve the

maximum exploitation of the power converter.maximum exploitation of the power converter.TheThe limitlimit to current regulation capability, in fact,to current regulation capability, in fact,

is only given by the poweris only given by the power converter design.converter design.

TheThehysteresishysteresiscurrent controlcurrent controlis inherentlyis inherently stablestable

and robustand robustto load variations or any other type ofto load variations or any other type ofdynamic perturbations.dynamic perturbations.


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The generation ofThe generation of non-dcnon-dccurrent waveformscurrent waveforms

implies theimplies the variation of the instantaneousvariation of the instantaneous

switching frequency.switching frequency.

This isThis is not desirablenot desirablebecause:because:

thethe design of converters output filtersdesign of converters output filtersisis

more complicated and less effective;more complicated and less effective;

in case ofin case of motor drive applications,motor drive applications,acousticacoustic

noise due tonoise due to mechanicmechanicresonancesresonancesmay bemay be

generated.generated.


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Single-phase schematicSingle-phase schematicof theof thehysteresishysteresiscurrentcurrent

control. Thecontrol. ThehysteresishysteresisbandsbandsBBpp,,BBnnare supposedare supposedto beto be fixed.fixed.

LL RRi*i*LL

++--

11

00

iiLL

ii

ee++

--VVdcdc/2/2

++VVdcdc/2/2-B-B nn

BBpp iiLL

uuu

!!!


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edt

diLiRu LL ++++== edtdiLiRu LL ++++== Load equationLoad equationLoad equation

*LLi ii ==*LLi ii == Current errorCurrent errorCurrent error

edt

diLiRu

*L*

L* ++++== e

dtdi

LiRu*L*

L* ++++== Reference voltageReference voltageReference voltage

dtdLRuuii* ++== dtdLRuu ii* ++== Current errordynamic equationCurrent errorCurrent errordynamic equationdynamic equation


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dtdLRuu ii

* ++==dt

dLRuu ii* ++==

NormallyNormallythethe resistive termresistive termcan becan be neglectedneglectedandand

thethe reference voltage ureference voltage u**can be consideredcan be consideredconstantconstantduring a modulation period.during a modulation period.

Therefore,Therefore, the current errorthe current errorhas ahas a triangulartriangular

shapeshape(right-hand side is constant) and the(right-hand side is constant) and the

average of uaverage of uover a modulation period isover a modulation period is equalequalto u*to u*(total current variation is 0).(total current variation is 0).


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Current errorand hysteresis

band.

CurrentCurrent errorerrorandandhysteresishysteresis

band.band.

Inverter voltage

pulses.

InverterInverter voltagevoltage

pulses.pulses.

Synchronizationclock.SynchronizationSynchronizationclock.clock.

iii

-Vdc/2--VVdcdc/2/2

Vdc/2VVdcdc/2/2

TTT

uuu


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==2

Vuu dc*

n

==2

Vuu dc*

n

It is possible to derive the followingIt is possible to derive the following periodperiodequation:equation:

(( ))2ndc u1V

L4T

==

(( ))2ndc u1V

L4T

==

where:where:


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TheThe period equationperiod equationshows that:shows that:

ifif isisconstantconstantandanduunnvariesvariesthe periodthe period T alsoT alsovaries;varies;

toto

get a constant switching frequency (1/Tget a constant switching frequency (1/T

dd))

thethehysteresishysteresisbandband has to behas to be dynamicallydynamicallymodified,modified, according to this equation:according to this equation:

(( )2nddc u1L4 TV == (( 2nddc u1L4TV

==


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The application of theThe application of thehysteresishysteresiscurrentcurrentcontrol tocontrol tothree phase systemsthree phase systemswith insulatedwith insulated

neutralneutralis made a little more complicated byis made a little more complicated by

the unavoidablethe unavoidable interference among the phaseinterference among the phase

currents.currents. It is anywayIt is anyway possible to strongly reducepossible to strongly reduce thethe

interference by suitablyinterference by suitably manipulating themanipulating the

current error.current error.

This only requires aThis only requires a simplesimpleadditional analogadditional analogcircuit.circuit.


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In aIn a three-phase system with insulated neutralthree-phase system with insulated neutral

the load equation must be modified to takethe load equation must be modified to take

into accountinto account the load midpoint voltage (neutralthe load midpoint voltage (neutral

voltage).voltage).


1uedt

idLiRu oLL ++++++== 1uedt

idLiRu oLL ++++++==

wherewhereuuoois the load midpoint voltageis the load midpoint voltage(neutral(neutral

voltage) andvoltage) and 11is theis the unity vector.unity vector.

(( ))

3

uuuu 321o

++++==

(( ))

3

uuuu 321o

++++==


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WithWith the same definitionsthe same definitionsand procedure of theand procedure of the

single phase case, the followingsingle phase case, the following dynamicdynamic

equationequationcan be derived for thecan be derived for the current error:current error:

dt

dLR1uuu i

io

* ++==dt

dLR1uuu i

io

* ++==

Because of theBecause of the presence ofpresence ofuuoo,,the currentthe current

error in a switching perioderror in a switching periodis not triangular andis not triangular and

its slopeits slope(on each phase)(on each phase) depends on the statedepends on the state

of all phasesof all phases(through(throughuuoo). This phenomenon is). This phenomenon is

known asknown as phase-interference.phase-interference.


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ToTo eliminate the phase interference,eliminate the phase interference,whichwhich

negatively affectsnegatively affectsthethehysteresishysteresiscontrolcontrol

behavior, abehavior, adecouplingdecouplingtermtermiican be definedcan be definedas follows:as follows:

1udtdLR oii ==

++ 1udtdLR oii ==

++

which can bewhich can be easily implementedeasily implementedbyby filteringfilteringthethe

instantaneous voltageinstantaneous voltageuuoowith an analogwith an analog lowlow

pass filterpass filterhaving thehaving the same time constantsame time constantof theof theloadload (L/R).(L/R).Tuning is normally required.Tuning is normally required.


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Re-writingRe-writingthe error dynamic equation as athe error dynamic equation as a

function of thefunction of thedecoupleddecoupledcurrent errorcurrent errorii::


iii == iii ==

it is possible to derive theit is possible to derive thedecoupleddecouplederrorerroriidynamic equation:dynamic equation:

dt

dLRuu ii

* ++==dt

dLRuu ii

* ++==

whichwhich no longer dependsno longer dependson the load midpointon the load midpoint

voltagevoltageuuoo..


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TheThe structurestructure of theof thedecoupleddecouplederror dynamicerror dynamic

equation isequation is exactly the sameexactly the sameof theof the single-phasesingle-phasecase.case.

Therefore, onceTherefore, once a suitablea suitabledecouplingdecouplingcircuit iscircuit is

implemented, theimplemented, the three-phase systemthree-phase systembehavesbehaves

exactly asexactly as three single phase ones,three single phase ones,withwithtriangulartriangulardecoupleddecoupledcurrent errors,current errors,whose slopewhose slope

depends onlydepends onlyon the state of theon the state of the correspondingcorresponding

inverter phase.inverter phase.

From now on,From now on, the explanationthe explanationof the digitalof the digitalcontrol system for thecontrol system for thehysteresishysteresiscontrollercontroller willwill

refer to the single phase case.refer to the single phase case.


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TheThe

controller maintains its analog structure,controller maintains its analog structure,

but abut a bandwidth digital controlbandwidth digital controlis added whichis added which

ensuresensures constant switching frequency.constant switching frequency.

LL RRi*i*LL

++-- 11

00

iiLL

ii

ee++

--VVdcdc/2/2

++VVdcdc/2/2

-B-B nn

BBpp iiLL

uu

DigitalDigital

ControlControl BBpp,,BBnn


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Bandwidth Control AlgorithmBandwidth Control Algorithm

T(k)T(k)T(k) T(k+1)T(k+1)T(k+1)

(k)(k)(k) (k+1)(k+1)(k+1)S+(k+1)SS++(k+1)(k+1)

S-(k+1)SS--(k+1)(k+1)

S+(k)SS++(k)(k)

S-(k)SS--(k)(k)

k:k:k:S+Ton= S-Toff= SS++TTonon= S= S--TToffoff==

Ton+ Toff= TTTonon++TToffoff= T= T


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From theFrom the previous equationsprevious equationsit is possible toit is possible to

derive:derive:

)k(S

)k(S)k(S)k(

)k(S

)k(

)k(S

)k()k(T

p

++

++

++==

++

==

)k(S

)k(S)k(S)k(

)k(S

)k(

)k(S

)k()k(T

p

++

++

++==

++

==

and assuming:and assuming:

)k(

)k(T)1k()1k(T

++==++

)k(

)k(T)1k()1k(T

++==++

)k(S)1k(S ++++ ==++ )k(S)1k(S ++++ ==++ )k(S)1k(S ==++ )k(S)1k(S ==++

we can get:we can get:


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From the previous equations, it is possible toFrom the previous equations, it is possible to

derive thederive the control equation:control equation:

)k(T

T)k()1k( d==++

)k(T

T)k()1k( d==++

where Twhere Tddis theis the desired switching period.desired switching period.

It is worth noting thatIt is worth noting that this reasoning leads tothis reasoning leads to

an algorithman algorithmwhich is equivalent to awhich is equivalent to a firstfirst

order dead-beat controlorder dead-beat controlof the switchingof the switchingperiod.period.


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The control algorithm is very simple. It onlyThe control algorithm is very simple. It only

requiresrequires time measurements,time measurements,which can bewhich can be

easily implemented byeasily implemented by using the captureusing the capture

function of any micro-controller.function of any micro-controller.

The algorithm isThe algorithm is auto-tuning,auto-tuning,does not requiredoes not require

anyany knowledge of the load parameters.knowledge of the load parameters.They areThey are

implicitlyimplicitly estimated by measuring the switchingestimated by measuring the switching

period with a knownperiod with a known ..

TheThe calculationcalculation of the newof the new requires arequires adivision.division.This implies a certainThis implies a certain computationcomputationtime.time.


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The control algorithm is able to guarantee aThe control algorithm is able to guarantee a

good frequency regulation.good frequency regulation.

Unfortunately, theUnfortunately, the switching pulsesswitching pulsesfor thefor the

three phases are not phase-controlled. Thisthree phases are not phase-controlled. This

means thatmeans that the allocation inside the modulationthe allocation inside the modulation

period of the switching pulses is random.period of the switching pulses is random.

ThisThis implies a slightly increased current rippleimplies a slightly increased current ripple

with respect with the optimum pulse allocationwith respect with the optimum pulse allocation

(centered pulses).(centered pulses).

Methods to improve the algorithm,Methods to improve the algorithm, includingincluding

pulses phase controlpulses phase controlare available.are available.


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A feasible way toA feasible way to

regulate the phaseregulate the phase

shiftshift

between the pulsesbetween the pulsesis tois to lock them to alock them to a

synchronizing clock.synchronizing clock.

TheThe time differencetime difference between thebetween thesynchronizingsynchronizing

clock and the inverter pulsesclock and the inverter pulsescan be measuredcan be measured

and used to modify the bandwidth.and used to modify the bandwidth.

AA proper regulator must be designedproper regulator must be designedto stabilizeto stabilize

the system.the system.

ThisThis solutionsolution is equivalent tois equivalent tothe implementationthe implementationofofaadigital Phase Locked Loop (PLL).digital Phase Locked Loop (PLL).


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Hysteresis

Comparator

HysteresisHysteresis

ComparatorComparator Switching PulsesSwitching PulsesSwitching Pulses

BandwidthCalculatorBandwidthBandwidthCalculatorCalculator

BandwidthCorrectorBandwidthBandwidthCorrectorCorrector ClockClock

Clock

iLiiLL

i*Lii**LL iii


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The designThe designof the bandwidth corrector is normallyof the bandwidth corrector is normally

complicated by the open loop gain variability.complicated by the open loop gain variability.

Small signal analysis shows that theSmall signal analysis shows that thehysteresishysteresis

comparatorcomparatorgaingainis given by:is given by:

(( ))

==

==

== d2n2dc fu1

L4

V

d

dfHC (( ))

==

==

== d2n2dc fu1

L4

VddfHC

This gain, together with theThis gain, together with the phase-detectorphase-detector

(integrator)(integrator)and theand the regulatorregulator (which is normally a(which is normally aPI)PI)gainsgainsgives thegives the open loop gain.open loop gain.


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[dB][dB][dB]

log10floglog1010ff

GGG

un= 0uunn= 0= 0

un= 0.8uunn= 0.8= 0.8

fzffzz

z2

z2n

2r

dc

p

Ts

sT1

u1

f

V

Lk8G

++

==

z2

z2n

2r

dc

p

Ts

sT1

u1

f

V

Lk8G

++

==

z

zp

sT

sT1k

d

dPI

++==

==

z

zp

sT

sT1k

d

dPI

++==

==

s2

dfdPhD ==== s

2dfdPhD ====


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To avoidTo avoid instabilityinstabilitywhen thewhen the modulation indexmodulation index

is maximum,is maximum,the bandwidth of the regulationthe bandwidth of the regulation

must be reduced.must be reduced.

Therefore,Therefore, the quality of thethe quality of theswitching pulsesswitching pulses

phase regulationphase regulationisis notnot very high.very high.

In case ofIn case of transients,transients,the regulator of the pulsethe regulator of the pulse

phase inducesphase induces oscillations,oscillations,which furtherwhich further

decrease the effectiveness of the control.decrease the effectiveness of the control.

AlternativeAlternativemethods to achieve the pulsesmethods to achieve the pulsesphase lockphase lock can be identified.can be identified.


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kk k+1k+1 k+2k+2

T*T*

ClockClock

tk+3k+3k-1k-1

BBpp(k)(k)

BB**

TTspsp(k)(k)

ttee(k)(k)

BBpp(k+2)(k+2)

TTclockclock

iiiiii

--BBnn(k)(k)

-B-B**

ttee(k+1)(k+1)

TTspsp(k+1)(k+1)


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2

T)1k(T)k(t)1k(t

*

spee ++++==++2

T)1k(T)k(t)1k(t

*

spee ++++==++


The algorithmThe algorithm modifies the twomodifies the twohysteresishysteresis

thresholdsthresholdsindependentlyindependentlyfrom each other.from each other.

TheThe input is the timing errorinput is the timing errorttee(k)(k)between thebetween the

current errorcurrent error zero crossingzero crossingand the externaland the external

synchronization clock.synchronization clock.

At any zero-crossingAt any zero-crossingttee(k) is measured(k) is measuredand theand the

followingfollowingttee(k+1) is estimated(k+1) is estimatedaccording to theaccording to the

following equation, wherefollowing equation, where T* is the desiredT* is the desired

period (= 2Tperiod (= 2Tclkclk):):


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TheThe half periodhalf periodTTspsp(k)(k)also needs to bealso needs to be

estimatedestimatedaccording to the following equation:according to the following equation:

)k(T)k(B

)k(B)1k(T sp

p

nsp ==++ )k(T

)k(B

)k(B)1k(T sp

p

nsp ==++

Note thatNote that at instant kat instant kthethe thresholdsthresholdsBBpp(k) and(k) and

BBnn(k) are known(k) are knownto the algorithm. The algorithmto the algorithm. The algorithm

modifies the threshold the current error is notmodifies the threshold the current error is not

going to.going to.In this caseIn this caseBBppwill be modifiedwill be modified to getto getthe timing error to zero.the timing error to zero.


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First,First, the correct threshold amplitudethe correct threshold amplitudemust bemust be

identified, according to the following equation:identified, according to the following equation:

)k(T

)k(B

2

T)k(B

sp

p*

* ==)k(T

)k(B

2

T)k(B

sp

p*

* ==

Then, the new value ofThen, the new value of thresholdthresholdBBpp(k+2),(k+2),whichwhicheliminates the timing error,eliminates the timing error,can be computed:can be computed:

2

T

)1k(t2

T

)k(B

)2k(B

*

e

*

*

p++

==

++

2

T

)1k(t2

T

)k(B

)2k(B

*

e

*

*

p++

==

++


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)k(T

)k(B)k(B)k(T)k(tT

)k(B)2k(Bsp

p

nspe

*

pp

==++)k(T

)k(B)k(B)k(T)k(tT

)k(B)2k(Bsp

p

nspe

*

pp

==++

Substituting all theSubstituting all the known quantitiesknown quantitiesin thein the

previous equationprevious equation the final equation,the final equation,basedbasedonon measured data,measured data,whichwhich updates theupdates the

thresholdthresholdand,and, theoretically,theoretically,eliminates theeliminates the

error, is obtained:error, is obtained:

A totallyA totally symmetrical equationsymmetrical equationcan be derivedcan be derived

forforBBnn..


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Based on this algorithm aBased on this algorithm a second order dead-second order dead-

beat control of the pulses timing errorbeat control of the pulses timing errorisisachieved.achieved.

In principle,In principle, the control eliminates any errorthe control eliminates any errorinin

frequency and phase with a modulation periodfrequency and phase with a modulation period

delaydelay (two clock cycles).(two clock cycles). BeingBeing the current zero-crossings synchronizedthe current zero-crossings synchronized

with the modulation period,with the modulation period,the pulses arethe pulses are

automatically centered.automatically centered.

Dead-times compensationDead-times compensationcan also be includedcan also be includedin the algorithm.in the algorithm.


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0us 100us 200us 300us 400us 500us

1.5

1.0

0.5

0.3

-0.3

0


Controls dynamic

response to a

bandwidth error.

Controls dynamicControls dynamic

response to aresponse to a

bandwidth error.bandwidth error.

Solid: normalized

switching frequency.

Dashed: instantaneousphase error.

Solid:Solid:normalizednormalized

switching frequency.switching frequency.

Dashed:Dashed:instantaneousinstantaneousphase error.phase error.

i

B*

Bp

-Bn

0us 100us 200us 300us 400us 500us

2.0

1.0

0.0

-1.0

-2.0

-B*


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0.6ms 0.8ms 1.0ms 1.2ms 1.4ms 1.6ms 1.8ms 2.0ms

1.0

0.5

0.0

-0.5

-1.0

Bp

-Bn

i

0.6ms 0.8ms 1.0ms 1.2ms 1.4ms 1.6ms 1.8ms 2.0ms

1.5

1.0

0.5


Control operation in

saturation mode.

Control operation inControl operation in

saturation mode.saturation mode.

Normalized switching

frequency.

Normalized switchingNormalized switching

frequency.frequency.


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Bp

n-B

i

e


Experimentalmeasurement.

Control operation in

saturation mode.

ExperimentalExperimentalmeasurement.measurement.

Control operation inControl operation in

saturation mode.saturation mode.

Converter ParametersConverter Parameters

DC Link VoltageDC Link Voltage 300 V300 V

Output InductorOutput Inductor 1.81.8mHmHSwitching FrequencySwitching Frequency 20 kHz20 kHz

Nominal Output PowerNominal Output Power 5 kW5 kW


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Experimental measurementsExperimental measurementsExperimental measurements

Phase current errorsPhasePhase current errorscurrent errors Phase voltage pulsesPhasePhase voltage pulsesvoltage pulses


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00

2020

4040

6060

8080

100100

120120

140140

5050 100100 200200 300300 400400 500500 700700 10001000

Frequency [Hz]Frequency [Hz]

Phase

Error[

Phase

Error[deg

deg]]

experimentalexperimental

simulation 1simulation 1

simulation 2simulation 2

Phase error in degrees between clock and

phase voltage pulses. All plotted data refer

to the generation of a sinusoidal current (6 A

peak value) with a 0.8 modulation index.

Phase error in degreesPhase error in degrees between clock andbetween clock and

phase voltage pulses. All plotted data referphase voltage pulses. All plotted data refer

to the generation of ato the generation of a sinusoidal current (6 Asinusoidal current (6 A

peak value) with a 0.8 modulation index.peak value) with a 0.8 modulation index.


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ReferencesReferences

[1][1] L.L.MalesaniMalesaniand P.and P.TentiTenti: "A novel: "A novelhysteresishysteresiscontrol methodcontrol methodforfor

current controlled VSI PWM inverterscurrent controlled VSI PWM inverters with constant modulationwith constant modulation

frequency"frequency"IEEE Trans. on Industry Applications,IEEE Trans. on Industry Applications,VolVol.26, No.1,.26, No.1,

Jan./Feb. 1990,Jan./Feb. 1990,pppp.88-92..88-92.

[2][2] Q.Q.YaoYao and D.G.Holmes: "A simple, novel method forand D.G.Holmes: "A simple, novel method for variable-variable-

hysteresishysteresis-band current control-band current control of a three phase inverter withof a three phase inverter with

constant switching frequency". IEEE IAS'93 Ann. Meet.constant switching frequency". IEEE IAS'93 Ann. Meet.ConfConf..RecRec.,.,

Toronto, Oct. 1993,Toronto, Oct. 1993,pppp. 1122-1129.. 1122-1129.

[3][3] L.L.MalesaniMalesani, L., L.RossettoRossetto, P., P.TomasinTomasin, A., A.ZuccatoZuccato: ": "Digital AdaptiveDigital Adaptive

HysteresisHysteresisCurrent ControlCurrent Controlwith Clocked Commutations and Widewith Clocked Commutations and Wide

Operating Range", IEEE Trans. on Industry Applications,Operating Range", IEEE Trans. on Industry Applications, VolVol. 32,. 32,

No. 2, March/April 1996,No. 2, March/April 1996,pppp. 316-325.. 316-325.

[4][4] L.L.MalesaniMalesani, P., P.MattavelliMattavelli, S., S.FasoloFasolo, S., S.BusoBuso:: AADeadDead--Beat AdaptiveBeat Adaptive

Hysteresis Current ControlHysteresis Current Control,, IEEE IAS99IEEE IAS99 AnnAnn.. MeetMeet.. ConfConf.. RecRec.,.,

PhoenixPhoenix,,OctOct. 1999,. 1999,pppp. 72-79.. 72-79.

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Digital Control Applications in Power Electronics Lez5