lesson 5 digital control of three-phase dc/ac converters ...antenor/pdffiles/lez5.pdfdigital...

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

Lesson 5Lesson 5

Digital Control of Three-Phase DC/ACDigital Control of Three-Phase DC/ACConverters: Current Control TechniquesConverters: Current Control Techniques

DigitalDigital Hysteresis Hysteresis Control Control


DigitalDigital Hysteresis Control Hysteresis Control

•• Thank to its Thank to its inherent non-linearity,inherent non-linearity, the the hysteresis hysteresiscurrent control is capable of providing current control is capable of providing the fastestthe fastestpossible dynamic response.possible dynamic response.

•• Using this technique, it is possible to achieve theUsing this technique, it is possible to achieve themaximum exploitation of the power converter.maximum exploitation of the power converter.The The limit limit to current regulation capability, in fact,to current regulation capability, in fact,is only given by the power is only given by the power converter design.converter design.

•• TheThe hysteresishysteresis current control current control is inherently is inherently stablestableand robustand robust to load variations or any other type of to load variations or any other type ofdynamic perturbations.dynamic perturbations.



•• The generation of The generation of non-dcnon-dc current waveforms current waveformsimplies the implies the variation of the instantaneousvariation of the instantaneousswitching frequency.switching frequency.

•• This is This is not desirablenot desirable because: because:

•• the the design of converter’s output filtersdesign of converter’s output filters is ismore complicated and less effective;more complicated and less effective;

•• in case of in case of motor drive applications,motor drive applications, acoustic acousticnoise due to noise due to mechanicmechanic resonances resonances may be may begenerated.generated.



Single-phase schematicSingle-phase schematic of the of the hysteresis hysteresis current currentcontrol. Thecontrol. The hysteresishysteresis bands bands B Bpp,, B Bnn are supposed are supposedto be to be fixed.fixed.

LL RRi*i*LL

++-- 11

00

iiLL

εεεεii

ee++

--VVdcdc/2/2

++VVdcdc/2/2

-B-B nn

BBpp iiLL

uuu

!!!



edtdiLiRu L

L ++++⋅⋅== edtdiLiRu L

L ++++⋅⋅== Load equationLoad equationLoad equation

*LLi ii −−==εε *LLi ii −−==εε Current errorCurrent errorCurrent error

edtdiLiRu

*L*

L* ++++⋅⋅== edt

diLiRu*L*

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

dtdLRuu i

i* εε++εε⋅⋅==−− dt

dLRuu ii

* εε++εε⋅⋅==−− Current errordynamic equationCurrent errorCurrent errordynamic equationdynamic equation



dtdLRuu i

i* εε++εε⋅⋅==−− dt

dLRuu ii

* εε++εε⋅⋅==−−

•• NormallyNormally the the resistive termresistive term can be can be neglectedneglected and andthe the reference voltage ureference voltage u** can be considered can be consideredconstantconstant during a modulation period. during a modulation period.

•• Therefore, Therefore, the current errorthe current error has a has a triangulartriangularshapeshape (right-hand side is constant) and the (right-hand side is constant) and theaverage of uaverage of u over a modulation period is over a modulation period is equalequalto u*to u* (total current variation is 0). (total current variation is 0).



Current error and hysteresisband.

Current Current errorerror andand hysteresishysteresisband.band.

Inverter voltagepulses.Inverter Inverter voltagevoltagepulses.pulses.

Synchronizationclock.SynchronizationSynchronizationclock.clock.

ββββββ εεiεεεεii

-Vdc/2--VVdcdc/2/2

Vdc/2VVdcdc/2/2

TTTφφφφφφ

uuu



== 2Vuu dc*

n

== 2Vuu dc*

n

•• It is possible to derive the following It is possible to derive the following ‘period’‘period’equation:equation:

(( ))2ndc u1V

L4T

−−⋅⋅

ββ==

(( ))2ndc u1V

L4T

−−⋅⋅

ββ==

where:where:



The The ‘period’ equation‘period’ equation shows that: shows that:

•• if if ββββ is is constantconstant and and uunn varies varies the period the period T alsoT alsovaries;varies;

•• to to get a constant switching frequency (1/Tget a constant switching frequency (1/Tdd))thethe hysteresis hysteresis band band ββββ has to be has to be dynamicallydynamicallymodified, modified, according to this equation:according to this equation:

(( ))2n

ddc u1L4TV

−−⋅⋅⋅⋅⋅⋅

==ββ (( ))2n

ddc u1L4TV

−−⋅⋅⋅⋅⋅⋅

==ββ



•• The application of theThe application of the hysteresis hysteresis current currentcontrol tocontrol to three phase systems three phase systems with insulatedwith insulatedneutralneutral is made a little more complicated by is made a little more complicated bythe unavoidable the unavoidable interference among the phaseinterference among the phasecurrents.currents.

•• It is anyway It is anyway possible to strongly reduce possible to strongly reduce thetheinterference by suitably interference by suitably manipulating themanipulating thecurrent error.current error.

•• This only requires a This only requires a simplesimple additional analogadditional analogcircuit.circuit.


•• In a In a three-phase system with insulated neutralthree-phase system with insulated neutralthe load equation must be modified to takethe load equation must be modified to takeinto account into account the load midpoint voltage (neutralthe load midpoint voltage (neutralvoltage).voltage).


1uedtid

LiRu oL

L ⋅⋅++++++⋅⋅== 1uedtid

LiRu oL

L ⋅⋅++++++⋅⋅==

wherewhere uuoo is the load midpoint voltage is the load midpoint voltage (neutral (neutralvoltage) and voltage) and 11 is the is the unity vector.unity vector.

(( ))3

uuuu 321

o++++

==(( ))

3

uuuu 321

o++++

==



•• With With the same definitionsthe same definitions and procedure of the and procedure of thesingle phase case, the following single phase case, the following dynamicdynamicequationequation can be derived for the can be derived for the current error:current error:

dtd

LR1uuu iio

* εε++εε⋅⋅==⋅⋅−−−− dt

dLR1uuu i

io* εε

++εε⋅⋅==⋅⋅−−−−

•• Because of the Because of the presence ofpresence of u uoo,, the current the currenterror in a switching perioderror in a switching period is not triangular and is not triangular andits slopeits slope (on each phase) (on each phase) depends on the statedepends on the stateof all phasesof all phases (through (through u uoo). This phenomenon is). This phenomenon isknown as known as phase-interference.phase-interference.



•• To To eliminate the phase interference,eliminate the phase interference, which whichnegatively affectsnegatively affects the the hysteresis hysteresis control controlbehavior, abehavior, a decouplingdecoupling term term εεεε’’’’

ii can be defined can be definedas follows:as follows:

1udtd

LR oi

i ⋅⋅−−==εε ′′′′

++εε ′′′′⋅⋅ 1udtd

LR oi

i ⋅⋅−−==εε ′′′′

++εε ′′′′⋅⋅

which can be which can be easily implementedeasily implemented by by filteringfiltering the theinstantaneous voltageinstantaneous voltage uuoo with an analog with an analog lowlowpass filterpass filter having the having the same time constantsame time constant of the of theload load (L/R).(L/R). Tuning is normally required. Tuning is normally required.


•• Re-writingRe-writing the error dynamic equation as a the error dynamic equation as afunction of thefunction of the decoupleddecoupled current error current error εεεε’’

ii::


iii εε ′′′′−−εε==εε′′ iii εε ′′′′−−εε==εε′′

it is possible to derive theit is possible to derive the decoupleddecoupled error error εεεε’’ii

dynamic equation:dynamic equation:

dtd

LRuu ii

* εε′′++εε′′⋅⋅==−− dt

dLRuu i

i* εε′′

++εε′′⋅⋅==−−

which which no longer dependsno longer depends on the load midpoint on the load midpointvoltagevoltage uuoo..


DigitalDigital Hysteresis Control Hysteresis Control•• The The structure structure of theof the decoupled decoupled error dynamic error dynamic

equation is equation is exactly the sameexactly the same of the of the single-phasesingle-phasecase.case.

•• Therefore, once Therefore, once a suitablea suitable decoupling decoupling circuit is circuit isimplemented, the implemented, the three-phase systemthree-phase system behaves behavesexactly as exactly as three single phase ones,three single phase ones, with withtriangulartriangular decoupled decoupled current errors, current errors, whose slope whose slopedepends onlydepends only on the state of the on the state of the correspondingcorrespondinginverter phase.inverter phase.

•• From now on, From now on, the explanationthe explanation of the digital of the digitalcontrol system for thecontrol system for the hysteresis hysteresis controller controller willwillrefer to the single phase case.refer to the single phase case.



The The controller maintains its analog structure,controller maintains its analog structure,but a but a bandwidth digital controlbandwidth digital control is added which is added whichensures ensures constant switching frequency.constant switching frequency.

LL RRi*i*LL

++-- 11

00

iiLL

εεεεii

ee++

--VVdcdc/2/2

++VVdcdc/2/2

-B-B nn

BBpp iiLL

uu

DigitalDigitalControlControl BBpp,, B Bnn


Bandwidth Control AlgorithmBandwidth Control Algorithm

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

ββ(k)ββββ(k)(k) ββ(k+1)ββββ(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 = ββSS++·T·Tonon = S = S--··TToffoff = = ββββ

Ton + Toff = TTTonon + + T Toffoff = T = T



•• From the From the previous equationsprevious equations it is possible to it is possible toderive:derive:

)k(S

)k(S)k(S)k(

)k(S

)k(

)k(S

)k()k(T

p

−−++

−−++

++⋅⋅ββ==

ββ++

ββ==

)k(S

)k(S)k(S)k(

)k(S

)k(

)k(S

)k()k(T

p

−−++

−−++

++⋅⋅ββ==

ββ++

ββ==

and assuming:and assuming:

)k(

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

ββ⋅⋅++ββ==++

)k(

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

ββ⋅⋅++ββ==++

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

we can get:we can get:



•• From the previous equations, it is possible toFrom the previous equations, it is possible toderive the derive the control equation:control equation:

)k(TT

)k()1k( d⋅⋅ββ==++ββ)k(T

T)k()1k( d⋅⋅ββ==++ββ

where Twhere Tdd is the is the desired switching period.desired switching period.

•• It is worth noting that It is worth noting that this reasoning leads tothis reasoning leads toan algorithman algorithm which is equivalent to a which is equivalent to a firstfirstorder dead-beat controlorder dead-beat control of the switching of the switchingperiod.period.



•• The control algorithm is very simple. It onlyThe control algorithm is very simple. It onlyrequires requires time measurements,time measurements, which can be which can beeasily implemented by easily implemented by using the captureusing the capturefunction of any micro-controller.function of any micro-controller.

•• The algorithm is The algorithm is auto-tuning,auto-tuning, does not require does not requireany any knowledge of the load parameters.knowledge of the load parameters. They are They areimplicitly implicitly estimated by measuring the switchingestimated by measuring the switchingperiod with a known period with a known ββββ..

•• The The calculation calculation of the ‘new’ of the ‘new’ ββββ requires arequires adivision.division. This implies a certain This implies a certain computationcomputationtime.time.



•• The control algorithm is able to guarantee aThe control algorithm is able to guarantee agood frequency regulation.good frequency regulation.

•• Unfortunately, the Unfortunately, the switching pulsesswitching pulses for the for thethree phases are not phase-controlled. Thisthree phases are not phase-controlled. Thismeans that means that the allocation inside the modulationthe allocation inside the modulationperiod of the switching pulses is random.period of the switching pulses is random.

•• This This implies a slightly increased current rippleimplies a slightly increased current ripplewith 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, includingincludingpulses phase controlpulses phase control are available. are available.



•• A feasible way to A feasible way to regulate the phaseregulate the phase shiftshiftbetween the pulsesbetween the pulses is to is to lock them to alock them to asynchronizing clock.synchronizing clock.

•• The The time difference time difference between thebetween the synchronizing synchronizingclock and the inverter pulsesclock and the inverter pulses can be measured can be measuredand used to modify the bandwidth.and used to modify the bandwidth.

•• A A proper regulator must be designedproper regulator must be designed to stabilize to stabilizethe system.the system.

•• This This solution solution is equivalent tois equivalent to the implementationthe implementationofof aa digital Phase Locked Loop (PLL). digital Phase Locked Loop (PLL).



Hysteresis ComparatorHysteresisHysteresis

Comparator Comparator Switching PulsesSwitching PulsesSwitching Pulses

BandwidthCalculatorBandwidthBandwidthCalculatorCalculator

BandwidthCorrectorBandwidthBandwidthCorrectorCorrector ClockClockClock

ββββββ

∆β∆β∆β∆β∆β∆β

ββ’ββββ’’

iLiiLL

i*Lii**LL εεiεεεεii



•• The designThe design of the bandwidth corrector is normally of the bandwidth corrector is normallycomplicated by the open loop gain variability.complicated by the open loop gain variability.

•• Small signal analysis shows that theSmall signal analysis shows that the hysteresishysteresiscomparatorcomparator gain gain is given by: is given by:

(( ))ββ

−−==−−⋅⋅ββ⋅⋅⋅⋅

−−==ββ

== d2n2

dc fu1

L4

V

d

dfHC (( ))

ββ−−==−−⋅⋅

ββ⋅⋅⋅⋅−−==

ββ== d2

n2dc f

u1L4

V

d

dfHC

•• This gain, together with the This gain, together with the phase-detectorphase-detector(integrator)(integrator) and the and the regulator regulator (which is normally a(which is normally aPI)PI) gainsgains gives the gives the open loop gain.open loop gain.



[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

++⋅⋅

−−⋅⋅

ππ−−==

z2

z2n

2r

dc

p

Ts

sT1

u1

f

V

Lk8G

++⋅⋅

−−⋅⋅

ππ−−==

z

zp sT

sT1k

d

dPI

++⋅⋅==

φφββ∆∆

==z

zp sT

sT1k

d

dPI

++⋅⋅==

φφββ∆∆

==

s2

dfd

PhDππ

==φφ

==s

2dfd

PhDππ

==φφ

==



•• To avoid To avoid instabilityinstability when the when the modulation indexmodulation indexis maximum,is maximum, the bandwidth of the regulation the bandwidth of the regulationmust be reduced.must be reduced.

•• Therefore, Therefore, the quality of thethe quality of the switching pulses switching pulsesphase regulationphase regulation is is not not very high.very high.

•• In case of In case of transients,transients, the regulator of the pulse the regulator of the pulsephase induces phase induces oscillations,oscillations, which further which furtherdecrease the effectiveness of the control.decrease the effectiveness of the control.

•• AlternativeAlternative methods to achieve the pulses methods to achieve the pulsesphase lock phase lock can be identified.can be identified.



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

εεiεεεεiiεε∗∗

iεεεε∗∗∗∗ii

--BBnn(k)(k)-B-B**

ttee(k+1)(k+1)

TTspsp(k+1)(k+1)


2

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

*

spee −−++++==++2

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

*

spee −−++++==++


•• The algorithm The algorithm modifies the twomodifies the two hysteresis hysteresisthresholdsthresholds independentlyindependently from each other. from each other.

•• The The input is the timing errorinput is the timing error t tee(k)(k) between the between thecurrent error current error zero crossingzero crossing and the external and the externalsynchronization clock.synchronization clock.

•• At any zero-crossingAt any zero-crossing ttee(k) is measured(k) is measured and the and thefollowingfollowing ttee(k+1) is estimated(k+1) is estimated according to the according to thefollowing equation, where following equation, where T* is the desiredT* is the desiredperiod (= 2Tperiod (= 2Tclkclk):):



•• The The half periodhalf period T Tspsp(k)(k) also needs to be also needs to beestimatedestimated according 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 that Note that at instant kat instant k the the thresholdsthresholds B Bpp(k) and(k) andBBnn(k) are known(k) are known to the algorithm. The algorithm to the algorithm. The algorithmmodifies the threshold the current error is notmodifies the threshold the current error is notgoing to.going to. In this case In this case BBpp will be modified will be modified to getto getthe timing error to zero.the timing error to zero.



•• First, First, the correct threshold amplitudethe correct threshold amplitude must be must beidentified, 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 of Then, the new value of thresholdthreshold B Bpp(k+2),(k+2), which whicheliminates 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

++−−==

++



)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 the Substituting all the known quantitiesknown quantities in the in theprevious equation previous equation the final equation,the final equation, based basedon on measured data,measured data, which which updates theupdates thethresholdthreshold and, and, theoretically,theoretically, eliminates the eliminates theerror, is obtained:error, is obtained:

•• A totally A totally symmetrical equationsymmetrical equation can be derived can be derivedforfor B Bnn..



•• Based on this algorithm a Based on this algorithm a second order dead-second order dead-beat control of the pulses timing errorbeat control of the pulses timing error is isachieved.achieved.

•• In principle, In principle, the control eliminates any errorthe control eliminates any error ininfrequency and phase with a modulation periodfrequency and phase with a modulation perioddelay delay (two clock cycles).(two clock cycles).

•• Being Being the current zero-crossings synchronizedthe current zero-crossings synchronizedwith the modulation period,with the modulation period, the pulses are the pulses areautomatically centered.automatically centered.

•• Dead-times compensationDead-times compensation can also be included can also be includedin the algorithm.in the algorithm.


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

1.5

1.0

0.5

0.3

-0.3

0


Control’s dynamicresponse to abandwidth error.

Control’s dynamicControl’s dynamicresponse to aresponse to abandwidth error.bandwidth error.

Solid: normalizedswitching frequency.Dashed: instantaneousphase error.

Solid:Solid: normalized normalizedswitching frequency.switching frequency.Dashed:Dashed: instantaneous instantaneousphase error.phase error.

εi

B*

Bp

-Bn

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

2.0

1.0

0.0

-1.0

-2.0

-B*


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

ε 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 insaturation mode.Control operation inControl operation insaturation mode.saturation mode.

Normalized switchingfrequency.Normalized switchingNormalized switchingfrequency.frequency.


Bp

n-B

ε i

e


Experimentalmeasurement.Control operation insaturation mode.

ExperimentalExperimentalmeasurement.measurement.Control operation inControl operation insaturation mode.saturation mode.

Converter ParametersConverter Parameters

DC Link VoltageDC Link Voltage 300 V300 V

Output InductorOutput Inductor 1.81.8 mHmH

Switching FrequencySwitching Frequency 20 kHz20 kHz

Nominal Output PowerNominal Output Power 5 kW5 kW


Bandwidth Control AlgorithmBandwidth Control AlgorithmExperimental measurementsExperimental measurementsExperimental measurements

Phase current errorsPhase Phase current errorscurrent errors Phase voltage pulsesPhase Phase voltage pulsesvoltage pulses



00

2020

4040

6060

8080

100100

120120

140140

5050 100100 200200 300300 400400 500500 700700 10001000

Frequency [Hz]Frequency [Hz]

Ph

ase

Err

or

[P

has

e E

rro

r [d

egd

eg]]

experimentalexperimental

simulation 1simulation 1

simulation 2simulation 2

Phase error in degrees between clock andphase voltage pulses. All plotted data referto the generation of a sinusoidal current (6 Apeak value) with a 0.8 modulation index.

Phase error in degreesPhase error in degrees between clock and between clock andphase voltage pulses. All plotted data referphase voltage pulses. All plotted data referto the generation of a to the generation of a sinusoidal current (6 Asinusoidal current (6 Apeak value) with a 0.8 modulation index.peak value) with a 0.8 modulation index.


ReferencesReferences

[1][1] L.L.MalesaniMalesani and P. and P.TentiTenti: "A novel: "A novel hysteresishysteresis control method control method for forcurrent controlled VSI PWM inverters current controlled VSI PWM inverters with constant modulationwith constant modulation

frequency"frequency" IEEE Trans. on Industry Applications, IEEE Trans. on Industry Applications, Vol Vol.26, No.1,.26, No.1,Jan./Feb. 1990,Jan./Feb. 1990, pp pp.88-92..88-92.

[2][2] Q.Q.YaoYao and D.G.Holmes: "A simple, novel method for and D.G.Holmes: "A simple, novel method for variable-variable-hysteresishysteresis-band current control-band current control of a three phase inverter with of a three phase inverter withconstant switching frequency". IEEE IAS'93 Ann. Meet.constant switching frequency". IEEE IAS'93 Ann. Meet. Conf Conf.. Rec Rec.,.,Toronto, Oct. 1993,Toronto, Oct. 1993, pp pp. 1122-1129.. 1122-1129.

[3][3] L.L. Malesani Malesani, L., L. Rossetto Rossetto, P., P. Tomasin Tomasin, A., A. Zuccato Zuccato: ": "Digital AdaptiveDigital AdaptiveHysteresisHysteresis Current Control Current Control with Clocked Commutations and Wide with Clocked Commutations and WideOperating Range", IEEE Trans. on Industry Applications,Operating Range", IEEE Trans. on Industry Applications, Vol Vol. 32,. 32,No. 2, March/April 1996,No. 2, March/April 1996, pp pp. 316-325.. 316-325.

[4][4] L.L.MalesaniMalesani, P., P.MattavelliMattavelli, S., S.FasoloFasolo, S., S.BusoBuso: : “A“A Dead Dead--Beat AdaptiveBeat AdaptiveHysteresis Current ControlHysteresis Current Control”,”, IEEE IAS’99 IEEE IAS’99 Ann Ann.. Meet Meet.. Conf Conf.. Rec Rec.,.,PhoenixPhoenix,, Oct Oct. 1999,. 1999, pp pp. 72-79.. 72-79.

lesson 5 digital control of three-phase dc/ac converters ...antenor/pdffiles/lez5.pdfdigital...

Documents