Lesson 4 Digital Control of Three-Phase DC/AC Converters ...antenor/pdffiles/lez4.pdfOctober 1999 Simone Buso - Universitá di Padova Lesson 4 6 Dead-Beat Current Control • The control

October 1999 1Simone Buso - Universitá di Padova Lesson 4

LessonLesson 4 4

DigitalDigital Control of Three Control of Three--PhasePhase DC/AC DC/ACConvertersConverters:: Current Control Techniques Current Control Techniques

DeadDead--Beat Current ControlBeat Current Control



•• The The deaddead--beat current control techniquebeat current control technique is isinherently the inherently the fastestfastest linear linear control strategy control strategythat can be adoptedthat can be adopted..

•• With deadWith dead--beat controlbeat control,, in principle in principle,, the currentthe currentlooploop replicates exactly the current referencereplicates exactly the current referencewithwith a a twotwo--cycle delaycycle delay..

•• The The control is based on an internal modelcontrol is based on an internal model of ofthethe converter converter load load,, which is used which is used toto predict the predict thesystem’s dynamic behaviorsystem’s dynamic behavior..

•• The The controller iscontroller is therefore inherently therefore inherently sensitivesensitivetoto model and parameter model and parameter mismatchesmismatches..



Typical control setTypical control set--upup

LL FF

abc

αβαβ

i* i*ββαα

SVM

EE

+

-+

-

PowerPowerConverterConverter

V*αα

V*ββ

iiLLFF

ZZLL VVLL

LoadLoad

Digital ControlDigital Control

DeadBeatControl

uuSSuuavav


Dead-Beat Current Control EquationDead-Beat Current Control Equation

current referencecurrent reference

inverter currentinverter current

average inverteraverage invertervoltagevoltage

load voltageload voltage

ii**LLFF

iiLLFF

uuss

[[ ]] )k(u)k(u2)k(i)k(iT

L)1k(u avsL

*L

sw

Fmav FF

−−++−−==++ [[ ]] )k(u)k(u2)k(i)k(iT

L)1k(u avsL

*L

sw

Fmav FF

−−++−−==++

uav


kTkT (k+1)T(k+1)T (k+2)T(k+2)T

ii**LLFF

iiLLFF

uu

TTswsw



•• The The control equationcontrol equation assumes a assumes a very simplevery simple load loadmodelmodel::

+LLFmFm

uuSS

The The parameter parameter LLFmFm and andvoltagevoltage uuSS must must bebeknownknown to compute the to compute therequired voltage required voltage uuavav(k+1) .(k+1) .

•• While While measuring measuring LLFmFm is quite is quite easy,easy, some problems some problemsmay arise with may arise with voltagevoltage u uSS..

•• Note Note also thatalso that the equation the equation isis independent independent of of E.E.



•• The controlThe control equation equation assumesassumes that the that the voltagevoltage at atthe load connection point the load connection point uuSS is known is known.. This Thisvoltage can either be voltage can either be measuredmeasured or or estimatedestimated..

•• InIn any any case, for case, for the control system the control system to to work workproperlyproperly,, the load must exhibit voltage sourcethe load must exhibit voltage sourcecharacteristicscharacteristics,, with with lowlow series series impedance within impedance withinthe current controller bandwidththe current controller bandwidth..

•• If this is not theIf this is not the case, a case, a model mismatchmodel mismatch is ispresent and present and significant deviationssignificant deviations fromfrom the theexpected expected closed loopclosed loop performance performance can take place can take place((instabilities or lightly damped oscillationsinstabilities or lightly damped oscillations).).



[[ ]])k(i)1k(iT

L)1k(u)1k(e

FF LLsw

Fmavs −−−−++−−==−− [[ ]])k(i)1k(i

T

L)1k(u)1k(e

FF LLsw

Fmavs −−−−++−−==−−

•• The controlThe control equation can be easily equation can be easily rearranged rearranged totocreate ancreate an estimation algorithm estimation algorithm forfor the load the loadvoltage voltage uuSS,, as as the following the following::

where where eeSS stands stands for for the the ““estimationestimation”” of of uuSS..

•• ActuallyActually,, the value of uthe value of uSS(k)(k) is needed in the is needed in thecontrol equationcontrol equation;; therefore therefore some some interpolation isinterpolation isneededneeded toto generate generate e eSS(k)(k) from from eeSS(k-1).(k-1).

,,,



•• The simplest The simplest wayway to to generate e generate eSS(k)(k) from from eeSS(k-1)(k-1) isisby means ofby means of linear linear interpolation interpolation.. Applying thisApplying this to toeeSS(k-1)(k-1) we get we get to: to:

10),2k(e)1k(e)1()k(es ≤≤αα≤≤−−⋅⋅αα−−−−⋅⋅αα++== 10),2k(e)1k(e)1()k(es ≤≤αα≤≤−−⋅⋅αα−−−−⋅⋅αα++==

•• By varyingBy varying αααα,, it is possibleit is possible to to vary the dynamics of vary the dynamics ofthe estimatorthe estimator.. ThisThis affects the system stability affects the system stability ininthe presencethe presence ofof mismatches mismatches. . InIn the the ideal case, ideal case,therethere are are no no effects effects on the stabilityon the stability if if αααα is changed is changed..

•• It isIt is important important toto notice notice thatthat control and estimation control and estimationequationequation are are dynamically not independent dynamically not independent..


DeadDead--Beat Control BehaviorBeat Control Behavior

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-50

-40

-30

-20

-10

0

10

20

Current reference tracking (no PWM)Current reference tracking (no PWM)

The controller is ableto replicate thecurrent referencewith a two-cycledelay. Note theabsence of anytransient at the exitof saturated mode ofoperation.

The controller is ableThe controller is ableto to replicate thereplicate thecurrent referencecurrent referencewith a with a two-cycletwo-cycledelay.delay. Note the Note theabsence of absence of anyanytransienttransient at the exit at the exitof of saturated modesaturated mode of ofoperation.operation.


3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

x 10-3

7

7.5

8

8.5

9

9.5

10

2Tsw2T2Tswsw


Current reference tracking (no PWM)Current reference tracking (no PWM)

This generates aThis generates atracking error.tracking error.

The The reference valuereference valueis is correctlycorrectlyreproducedreproduced only onlyafter after two controltwo controlperiods.periods.



Load voltage estimation in ideal conditionsLoad voltage estimation in ideal conditions

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

In ideal conditions,i.e. when noparameter or modelmismatches arepresent, the error inthe estimation isvery small. In thiscase, we have αα = 1.

In In ideal conditions,ideal conditions,i.e. when i.e. when nonoparameter or modelparameter or modelmismatchesmismatches are arepresent, present, the errorthe error in inthe estimation the estimation isisvery small. very small. In thisIn thiscase, we havecase, we have αααα = 1. = 1.



Load voltage estimation in ideal conditionsLoad voltage estimation in ideal conditions

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Estimation error as afunction of time andparameter αα:

αα = 0.0αα = 0.5αα = 0.75αα = 1.0

A minimum is almostreached for αα = 0.75.

Estimation errorEstimation error as a as afunction of function of timetime and andparameter parameter αααα::

αααα = 0.0 = 0.0αααα = 0.5 = 0.5αααα = 0.75 = 0.75αααα = 1.0 = 1.0

A A minimumminimum is almost is almostreached for reached for αααα = 0.75. = 0.75.


DeadDead--Beat Control StabilityBeat Control Stability

•• InIn principle principle,, if there if there are are no no parameter and parameter and / / or ormodel errorsmodel errors,, the closed loop current control the closed loop current control isisdynamicallydynamically equivalent equivalent to a to a double unity double unity--delaydelay,,i.e. :i.e. :

iiLLFF(k) = i(k) = i**

LLFF (k-2)(k-2)

•• FromFrom a a stability analysisstability analysis standpoint this means standpoint this meansthat the that the closed loop systemclosed loop system exhibits exhibits two polestwo poleslocated located in the origin of the complexin the origin of the complex Z-plane Z-plane and andnono zeroes zeroes.. In In other words other words,, it is equivalent it is equivalent to a to asecond order allsecond order all--passpass FIR FIR filter filter..



•• UnfortunatelyUnfortunately,, parameter mismatchesparameter mismatches are arealways encountered in practicealways encountered in practice.. The The sensitivitysensitivityof the control’s stabilityof the control’s stability to to errors errors,, at least in the at least in theidentification of identification of parameterparameter L LFF,, needs needs to to be beinvestigatedinvestigated..

•• In general,In general, it is possible it is possible to to calculate the calculate the closedclosedloop polesloop poles of the system of the system as a as a function of thefunction of therelative identificationrelative identification error error ∆∆∆∆LL%%,, defined defined as asfollowsfollows::

∆∆∆∆LL%%=1 - =1 - LLFmFm/L/LFF


%2,1 Lp ∆∆±±== %2,1 Lp ∆∆±±==


•• It is possible to It is possible to verifyverify that, in the case of that, in the case of loadloadvoltage measurement,voltage measurement, the closed loop systems the closed loop systemshas has two simple polestwo simple poles given by: given by:

•• As can be seen, As can be seen, in the ideal casein the ideal case the poles are the poles arelocated in the origin,located in the origin, but but move away move away from it asfrom it assoon assoon as there isthere is an identification error. an identification error.



•• Anyway, Anyway, the stability robustnessthe stability robustness is very is very good.good. In Inprinciple, the relative error can be as high aprinciple, the relative error can be as high a100%,100%, before before instability occurs.instability occurs.

•• Practically, as soon as the error goes over Practically, as soon as the error goes over 60%60%the controller behavior the controller behavior becomes unacceptable,becomes unacceptable,i.e. lightly damped oscillations i.e. lightly damped oscillations appear.appear.

•• The situation is The situation is totally differenttotally different if, instead of if, instead ofmeasuring the load voltage, an measuring the load voltage, an estimationestimationalgorithmalgorithm is adopted. is adopted.

•• This modifies the This modifies the system dynamics,system dynamics, negatively negativelyaffecting the affecting the stability robustness.stability robustness.


0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-10

0

10

20

30

40

50


Example of control instability.Example of control instability.

The system showspersistentoscillations, at a halfof the samplingfrequency (Nyquistfrequency). Note thatthe simulation doesnot include PWM, sothe high frequency isnot current ripple.

The system showsThe system showspersistentpersistentoscillations,oscillations, at a half at a halfof the samplingof the samplingfrequency frequency ((NyquistNyquistfrequency).frequency). Note that Note thatthe simulation the simulation doesdoesnot include PWM, not include PWM, sosothethe high frequency is high frequency isnot not current current ripple.ripple.



•• To To analyze the control stabilityanalyze the control stability when voltage when voltageestimation estimation is employed,is employed, it is necessary to it is necessary toinclude in the system’s dynamic equations alsoinclude in the system’s dynamic equations alsothe the estimator equation.estimator equation.

•• The The system order is increased by one,system order is increased by one, but the but theanalytic calculationanalytic calculation of the closed loop poles is of the closed loop poles isstill possible.still possible. It can be found that: It can be found that:

%%3 L2zL3z)z( ∆∆++∆∆−−==λλ %%3 L2zL3z)z( ∆∆++∆∆−−==λλ

is is the system’s characteristic polynomial,the system’s characteristic polynomial, whose whoseroots are the system’s roots are the system’s eigenvalueseigenvalues..


1

-1

1

(a)

(b)(c)

(c)(b)

(c) (b)

Re(z)

Im(z)


System System eigenvalueseigenvalues plot in Z-plane plot in Z-plane

Real and imaginaryReal and imaginarycomponents of thecomponents of theclosed loop polesclosed loop poles(solutions of (solutions of λλλλ(z)=0)(z)=0)for different valuesfor different valuesof of ∆∆∆∆LL%% parameter: parameter:

a) a) ∆∆∆∆LL% % = 0%,= 0%,

b) b) ∆∆∆∆LL% % = 20%,= 20%,

c) c) ∆∆∆∆LL% % = 30%.= 30%.Instability!Instability!Instability!


0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-0.5

0

0.5

1

1.5

2

Example of control instability (voltage estimation)Example of control instability (voltage estimation)


Estimation errorEstimation errorEstimation errorEstimated voltageEstimated voltageEstimated voltage

Load voltageLoad voltageLoad voltage

As can be seen, withonly a 25% error onthe value ofparameter LF thesystem goes clearlyunstable.

As can be seen, withAs can be seen, withonly a only a 25%25% error on error onthe value ofthe value ofparameter Lparameter LFF the thesystem goes system goes clearlyclearlyunstable.unstable.



•• AA different type of instability different type of instability may affect themay affect thedead-beat control in case of a dead-beat control in case of a model mismatch.model mismatch.

•• A A model mismatchmodel mismatch arises any time arises any time the actualthe actualload of the power converter differs from theload of the power converter differs from theinternal modelinternal model the controller uses to compute the controller uses to computethe control variable.the control variable.

•• A A typical casetypical case is the presence of is the presence of capacitivecapacitivefiltersfilters at the converter’s output, at the converter’s output, needed inneeded incertain applications to reduce the current ripple.certain applications to reduce the current ripple.This is the case of This is the case of PWM rectifiers PWM rectifiers or or UPS’sUPS’s..

•• These cases need to be analyzed.These cases need to be analyzed.


uuSS

Not included in the control algorithmNot included in the control algorithm

PWM filterPWM filter

RR CC

CCuuCC

iiSS

++__

++

abcabc

PWMPWMRectifierRectifier

++

++--

--

αβαβαβαβ

Dead-BeatDead-BeatControlControl

UUαααα

UUββββ

Uαα

UββSpaceSpaceVectorVectorModulatorModulator

DigitalDigitalControllerController

iibb iicciiaaRRLL

ii αααα

LLFFLLSSCC LL


Typical control setTypical control set--upup for a for a PWMPWM rectifier rectifier application application

iiLLFF

ααααii**

ii**ββββ

iiββββ


−−−−−−

−−−−−−

==

==

⋅⋅

++⋅⋅

++⋅⋅==

0C1

C1

L1

LR

LR

L1

LR

LR

A,

u

i

i

x

u

0L10

u

0

0L1

xAdtxd

ss

c

s

c

FF

c

F

c

c

s

L

ss

av

F

Fr

rr

−−−−−−

−−−−−−

==

==

⋅⋅

++⋅⋅

++⋅⋅==

0C1

C1

L1

LR

LR

L1

LR

LR

A,

u

i

i

x

u

0L10

u

0

0L1

xAdtxd

ss

c

s

c

FF

c

F

c

c

s

L

ss

av

F

Fr

rr


The dynamicequations of theinput PWM filterare not included inthe converterinternal model.Stability analysiscan be performedby computing theclosed loopeigenvalues of thewhole system.

The The dynamicdynamicequations of theequations of theinput PWM filterinput PWM filterare are not included innot included inthe converterthe converterinternal model.internal model.Stability analysisStability analysiscan be performedcan be performedby by computing thecomputing theclosed loopclosed loopeigenvalueseigenvalues of theof thewhole system.whole system.



•• The control equation, together with a The control equation, together with a discretizeddiscretizedversion of the load equationsversion of the load equations can be used to can be used toderive a derive a complete state space representationcomplete state space representation of ofthe controlled system.the controlled system.

•• The The eigenvalues eigenvalues of the A matrixof the A matrix can then be can then becomputed.computed.

•• Assuming the Assuming the output voltage is measuredoutput voltage is measured and andthere is there is no identification error in parameter Lno identification error in parameter LFF it itis possible to evaluate the effect of the inputis possible to evaluate the effect of the inputcapacitive capacitive filter alone.filter alone.



Real and imaginarycomponents of theclosed loop polesfor different valuesof capacitor C:

a) C = 0.005 pu

b) C = 0.01 pu

c) C = 0.015 pu

d) C = 0.5 pu

Real and imaginaryReal and imaginarycomponents of thecomponents of theclosed loop polesclosed loop polesfor different valuesfor different valuesof of capacitor C:capacitor C:

a) C = 0.005 a) C = 0.005 pupu

b) C = 0.01 b) C = 0.01 pupu

c) C = 0.015 c) C = 0.015 pupu

d) C = 0.5 d) C = 0.5 pupu

1

(a)

(c)

(b)

Re(z)

Im(z)

(d)

(a)

(a) (b)

(c)

(d)

10

(a)

(d)

(d)

instability!instability!instability!




•• If the output voltage is If the output voltage is estimated,estimated, essentially the essentially thesame analysis proceduresame analysis procedure can be applied. can be applied.

•• The The estimator equationestimator equation must be must be added to theadded to thecontrol and load onescontrol and load ones (the latter in discrete-time (the latter in discrete-timeformat) to include its dynamics in the systemformat) to include its dynamics in the systemanalytical representation.analytical representation.

•• Again, matrix Again, matrix A of the complete system stateA of the complete system statespace representationspace representation can be computed and its can be computed and itseigenvalues eigenvalues evaluated, as a function of the evaluated, as a function of the sizesizeof the of the capacitive capacitive filter.filter.

•• The stability robustnessThe stability robustness is, in general, is, in general, reduced.reduced.


(a)

(c)

(b)

(d)

Re(z)

Im(z)

1

(d)

(d)(d)

(d)(a)

(a)

(a)

(b)(a)

(c)

Real and imaginaryReal and imaginarycomponents of thecomponents of theclosed loop polesclosed loop poles for fordifferent values ofdifferent values ofcapacitor C:capacitor C:

a) C = 0.005a) C = 0.005 pu pu,,

b) C = 0.01b) C = 0.01 pu pu,,

c) C = 0.015c) C = 0.015 pu pu,,

d) C = 0.5d) C = 0.5 pu pu..



instability!instability!instability!



•• The last analysis shows that The last analysis shows that when the outputwhen the outputvoltage is estimated the stability robustness isvoltage is estimated the stability robustness isworsened.worsened.

•• The The stability limitstability limit moves from moves from less than 0.01 less than 0.01 pupucapacitive capacitive filter filter to about 0.015 to about 0.015 pupu..

•• In fact, In fact, the critical capacitor valuesthe critical capacitor values are a little are a littlesmaller than the ones typically adoptedsmaller than the ones typically adopted to filter to filterthe current ripple.the current ripple.

•• As a consequence, As a consequence, in general only a littlein general only a littleoversizing oversizing of the of the capacitive capacitive filterfilter may be may benecessary to necessary to guarantee the system’s stability.guarantee the system’s stability.


•• With With a little increasea little increase in the algorithm complexity in the algorithm complexityit is anyway it is anyway possible to improve the stabilitypossible to improve the stabilityrobustness.robustness.

•• The oscillationsThe oscillations that take place in the controlled that take place in the controlledsystem when it goes unstable system when it goes unstable are due to theare due to theinteraction between the estimation algorithminteraction between the estimation algorithmand the current control.and the current control.

•• The The propagation of such oscillationspropagation of such oscillations can be can beavoided remembering that the avoided remembering that the estimated outputestimated outputvoltage waveform is known to be sinusoidal.voltage waveform is known to be sinusoidal.

DeadDead--Beat Control Stability ImprovementBeat Control Stability Improvement


•• The control algorithm robustness can be greatlyThe control algorithm robustness can be greatlyimproved with aimproved with a pass-band filter centered at thepass-band filter centered at thesupply frequencysupply frequency applied to the estimated voltage.applied to the estimated voltage.

LLFmFmTTswsw zz-1-1ReferenceReference

GenerationGeneration

SelectiveSelectiveFilterFilter

EstimatorEstimator

GGeqeq++--

+ +

+

+

-

iiLLFF(k)(k)ii** (k) (k)LLFF

-

u (k)u (k)avav

u (k)u (k)ssTTswsw

LLFF

11

z - 1z - 1..

221

2211

zmzcosm21

z)1m(z)m1(cos2)z(W

−−−−

−−−−−−

++λλ−−

−−++−−λλ==

221

2211

zmzcosm21

z)1m(z)m1(cos2)z(W

−−−−

−−−−−−

++λλ−−

−−++−−λλ==Pass-band filterPass-band filter



•• The The selective band pass filterselective band pass filter eliminates theeliminates theoscillationsoscillations from the estimated voltage from the estimated voltagewaveform and so waveform and so avoids their propagation inavoids their propagation inthe current loop.the current loop.

•• The idea is to The idea is to exploit the informationexploit the information about the about theinput voltage waveform, which is known to beinput voltage waveform, which is known to besinusoidal at the line frequency.sinusoidal at the line frequency.

•• The The stability analysis can now be repeatedstability analysis can now be repeatedtaking into account the taking into account the presence of thepresence of theselective filterselective filter in the control system. in the control system.



(a)

(a)

-1

1

(a)

(a)

(a)(c)

(c)

(c)

(c)

Re(z)

Im(z)

(b)

(b) (a)

(c)

0


•• Real and imaginaryReal and imaginarycomponents of thecomponents of theclosed loop polesclosed loop polesfor different valuesfor different valuesof of capacitor C, withcapacitor C, with∆∆∆∆LL%% = 20%. = 20%.

•• Note how the polesNote how the polesnow now remain insideremain insidethe unity circle.the unity circle.Filter dynamicsFilter dynamicsFilter dynamics



PWM rectifier behaviorwith the conventionalestimation techniqueand ∆∆L% = 20%.

Upper trace: linevoltage (uS).

Middle trace: converteroutput current (- iLF

).

Lower trace: estimatedvoltage (eS).

PWM rectifier behaviorPWM rectifier behaviorwith the with the conventionalconventionalestimation techniqueestimation techniqueandand ∆∆∆∆LL%% = 20%.= 20%.

Upper trace: Upper trace: linelinevoltage (voltage (uuSS).).

Middle trace: Middle trace: converterconverteroutput current (- output current (- iiLLFF

).).

Lower trace: Lower trace: estimatedestimatedvoltage (voltage (eeSS).).


Modified estimationModified estimationtechnique andtechnique and∆∆∆∆LL% % = 20%.= 20%.

Upper trace: Upper trace: linelinevoltage (voltage (uuSS).).

Middle trace: Middle trace: converterconverteroutput current (-output current (-iiLLFF

).).

Lower trace: Lower trace: estimatedestimatedvoltage (voltage (eeSS).).



•• ADSP 21020: ADSP 21020: floating point CPU (32-40 bits)floating point CPU (32-40 bits)4040 ns ns cycle cycle

•• ADMC 200: ADMC 200: motion control board includingmotion control board includingPWM modulator (25kHz max)PWM modulator (25kHz max)12 bit A/D converter12 bit A/D converter

DSP based control system featuring:DSP based control system featuring:

ExperimentalExperimental Tests Tests

Prototype Ratings:Prototype Ratings:

•• Line Voltage (RMS)Line Voltage (RMS) 85 V85 V•• Bridge Output PowerBridge Output Power 1 kW1 kW•• DC Link VoltageDC Link Voltage 300 V300 V•• Converter InductanceConverter Inductance 1.8 1.8 mHmH•• Switching FrequencySwitching Frequency 10 kHz10 kHz


Experimental ResultsExperimental ResultsPWM RectifierPWM Rectifier

Conventional estimation

∆∆∆∆LL ≈ ≈ ≈ ≈ 12%12%

Test conditions:Test conditions:no filterno filter on the estimated line on the estimated linevoltage;voltage;VVdcdc = 120V; = 120V;∆∆∆∆LL%% = 0.12; = 0.12;time-scale: 2ms/div;time-scale: 2ms/div;Upper trace:Upper trace: estimated line estimated linevoltage (200mV/div)voltage (200mV/div)Middle trace:Middle trace: line to neutral line to neutralmeasured voltage (40V/div)measured voltage (40V/div)Lower trace:Lower trace: phase current phase current(1A/div)(1A/div)


Experimental ResultsExperimental ResultsPWM RectifierPWM Rectifier

∆∆∆∆LL ≈ ≈ ≈ ≈ 30%30%

Modified estimation

Test conditions:Test conditions:no filterno filter on the estimated line on the estimated linevoltage;voltage;VVdcdc = 120V; = 120V;∆∆∆∆LL%% = 0.30; = 0.30;time-scale: 2ms/div;time-scale: 2ms/div;Upper trace:Upper trace: estimated line estimated linevoltage (200mV/div)voltage (200mV/div)Middle trace:Middle trace: line to neutral line to neutralmeasured voltage (40V/div)measured voltage (40V/div)Lower trace:Lower trace: phase current phase current(1A/div)(1A/div)


ReferencesReferences

[1][1] L.L. Malesani Malesani, P., P. Mattavelli Mattavelli, S., S. Buso Buso:: “Robust Dead-Beat Current “Robust Dead-Beat Current

Control for PWM Rectifiers and Active Filters”, IEEE TransactionsControl for PWM Rectifiers and Active Filters”, IEEE Transactionson Industry Applications,on Industry Applications, VolVol. 35, No. 3, May/June 1999, pp. 613-. 35, No. 3, May/June 1999, pp. 613-620.620.

[2] [2] L.L. Malesani Malesani, P., P. Mattavelli Mattavelli, S., S. Buso Buso: : "Dead-Beat Current Control for"Dead-Beat Current Control forActive Filters"Active Filters" 24th Annual Conference of the IEEE Industrial 24th Annual Conference of the IEEE Industrial

Electronics Society Electronics Society (IECON),(IECON), Aachen Aachen, Germany, August 31 -, Germany, August 31 -September 4, 1998, pp. 1859-1864.September 4, 1998, pp. 1859-1864.

[3][3] L.L. Malesani Malesani, P., P. Mattavelli Mattavelli, S., S. Buso Buso:: “On the Applications of Active “On the Applications of ActiveFilters to Generic Loads”, Filters to Generic Loads”, 88thth International Conference on International Conference onHarmonics and the Quality of Power Harmonics and the Quality of Power ICHQP ’98,ICHQP ’98, Athens, Greece, Athens, Greece,

October 14-16, 1998, pp. 310-319.October 14-16, 1998, pp. 310-319.

Documents

Lesson 4 Digital Control of Three-Phase DC/AC Converters ...antenor/pdffiles/lez4.pdfOctober 1999 Simone Buso - Universitá di Padova Lesson 4 6 Dead-Beat Current Control • The control