A new three-phase varying-band hysteresiscurrent controller for voltage-source inverters

Vinciane Chereau, Francois Auger, Luc LoronInstitut de Recherche en Electrotechnique et Electronique de Nantes Atlantique (IREENA)

CRTT, 37 Bd de l’Universite, BP 406, F-44602 Saint Nazaire cedex, France.

Abstract— Three independent hysteresis controllers areoften used to control the currents of a neutral isolated well-balanced load through a three-phase voltage-fed inverter,because of their simplicity and robustness. But this controlstructure also has some drawbacks, such as the destructionof the inverter if its highest switching frequency is not limitedand the interdependence of the load phases. This articleshows how to limit this frequency in a single-phase case andhow to extend this result to the three-phase case, througha transformation avoiding the complexity of most solutionsproposed until now.

Index Terms—adaptive hysteresis band current control,decoupling three-phase into two single-phase

I. INTRODUCTION

Pulse Width Modulation (PWM) for three-phasevoltage-fed inverters supplying three-phase isolated neu-tral loads has lead to a great number of current controlstrategies [2]. One of the simpliest is hysteresis cur-rent control. This kind of control is robust and allowsto achieve a good accuracy. Nevertheless, this methodpresents some drawbacks compared to other strategies.Indeed, it does not take into account the interaction be-tween the phases when used on each phase independently,and the switching frequency of the inverter can not becontained, so that it can lead to its destruction [1], [2],[3], [4]. This explains why other methods are usuallypreferred.

To solve the switching frequency limitation problem,some techniques are used such as using a combinatorialalgorithm [5], using a quasi-sinusoidal hysteresis band [6]or a sinusoidal one [7], using feed-forward and feedbacktechniques [8], calculating the hysteresis band with theload parameters and the neutral potential measure [9],using phase locked loop control [10], [11] or varying theband with the system parameters (load and supply) [12].

To solve the phase dependency problem, some strate-gies have been developped such as using a switchingtable between the controllers and the inverter [13], usingmulti-level hysteresis controllers and a switching table[14], switching between two different control strategies(hysteresis and optimum voltage vector calculation) [15],using state-vector modulation with a region detection tochoose in a switching table how to drive the inverter [16].

All these methods have their advantages and disadvan-tages, but a common drawback is that they complexify the

1This work was supported by Electronavale Technologies 332 bdMarcel Paul, ZIL CP0604, 44806 Saint Herblain Cedex, France

R,L

S

SVdc2

Vdc2

[a]

∆I

T

iref

t0 t1 t2

[b]

Fig. 1. [a] single-phase case, [b] Hysteresis band and switching period.

controller. This paper shows how to simply calculate thehysteresis band in the single-phase case to limit the in-verter switching frequency (section II). Then it shows howto simply transform a three-phase case into a two single-phases case so that well operating single-phase strategiessuch as hysteresis control can be extended to three-phaseapplications and so that the previous calculation can beextended to the three-phase case (section III). Finally,experimental results are given (section IV).

II. SWITCHING FREQUENCY LIMITATION

In this section, we will first see how the switchingfrequency can be limited theoretically and then, some sim-ulations made with Matlab/Simulink are provided to showthis limitation and to compare it with two methods usinga fixed-band and a sinusoidal band hysteresis controller.

A. Theoretical calculation

In this paragraph, we will design a simple methodin the single-phase case to limit the inverter switchingfrequency, as it is one of the main drawbacks of hysteresiscontrol. The load consists of a resistance R and an induc-tance L and is suplied by a voltage source ±Vdc/2 throughan inverter leg (see Fig. 1.a). The current reference irefis considered as constant during each switching period.Looking at Fig. 1.b (where ∆I is the hysteresis band andT the switching period), we can define the current at eachswitching time by :

I0 = i(t0) = iref − ∆I/2I1 = i(t1) = iref + ∆I/2 (1)

I2 = i(t2) = iref − ∆I/2

Between t0 and t1, the current is the solution of

Ldi(t)dt

+ R i(t) = Vdc/2

and equals

i(t) = I0 e−RL (t−t0) +

Vdc

2R

(1 − e−

RL (t−t0)

)(2)

Between t1 and t2, the current is the solution of

Ldi(t)dt

+ R i(t) = −Vdc/2

and equals

i(t) = I1 e−RL (t−t1) − Vdc

2R

(1 − e−

RL (t−t1)

)(3)

Equations (2) and (3) clearly show that |i(t)| ≤ Vdc2 R (the

maximum value is obtained when t goes to +∞). Besides,Fig. 1.b, with this current limit, assumes that the followingconditions are satisfied:

iref +∆I

2≤ Vdc

2R(4)

iref − ∆I

2≥ −Vdc

2R(5)

The equations (2) to (5) also allow to compute theswitching period:

t1 − t0 =L

Rln

(Vdc2 R + ∆I

2 − irefVdc2 R − ∆I

2 − iref

)(6)

t2 − t1 =L

Rln

(Vdc2 R + ∆I

2 + irefVdc2 R − ∆I

2 + iref

)(7)

T = t2 − t0 =L

Rln

((Vdc2 R + ∆I

2

)2 − i2ref(Vdc2 R − ∆I

2

)2 − i2ref

)(8)

To obtain the desired switching frequency 1T , we deduce

an expression of the hysteresis band ∆I:

∆I =Vdc

R

1 + δ

1 − δ± 2

√(Vdc

R

)2δ

(1 − δ)2+ i2ref (9)

with δ = e−RL T . Provided that |iref | < Vdc

2 R , the two valuesof ∆I are strictly positive, but, the only one that satisfiesconditions (4) and (5) is

∆I =Vdc

R

1 + δ

1 − δ− 2

√(Vdc

R

)2δ

(1 − δ)2+ i2ref (10)

It should be underlined that the hysteresis band does notdepend on the reference sign and is lower- and upper-bounded:

0 < ∆I ≤ Vdc

R

1 −√δ

1 +√

δ

One can also notice that when R goes to 0 Ω, ∆I goesto Vdc T

4 L and becomes independent of iref .

B. Simulation results

In this paragraph, some simulation results are providedto point out the effectiveness of the previous calculation(called hereafter method 3). In order to do that, we willshow, first, that the switching frequency is limited andthen, we will compare these results to those of a fixedband hysteresis controller (method 1) and of a sinusoidalband one [7], [6] (method 2). In our simulations, thesampling frequency is 1 MHz to avoid distorsions dueto this frequency, those distorsions will be shown at theend of the paragraph. The load is a single-phase one

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0.14

0.252

0

0.1

0.2

∆I (

A)

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0.10.14

0.2

0.252

∆I (

A)

40 42 44 46 48 50 52 54 56 58 60

0.14

0.252

0

0.1

0.2

time (ms)

∆I (

A)

Fig. 2. Hysteresis band during one period of the reference: method 1(top), method 2 (middle), method 3 (bottom).

with R = 6.6 Ω and L = 4 mH, the DC-bus voltageis Vdc = 20 V and the desired switching frequency is1T = 5 kHz. The reference current is a sinusoid and hasan amplitude of Iref = 1 A and a frequency of fref = 50Hz. It should be noticed that the reference iref is notconstant in our case but is a regularly sampled version ofiref(t) = Iref sin(2π fref t).For method 3, the band is given by equation (10):

∆I(t) =Vdc

R

1 + δ

1 − δ− 2

√(Vdc

R

)2δ

(1 − δ)2+ i2ref(t)

As this band should limit the switching frequency, we willuse this expression for method 1 and 2. Method 1 has afixed band and as the highest switching frequency occurswhen the reference is the smallest (in our case, iref = 0A), the band is

∆I(t) = ∆I1 =Vdc

R

1 −√δ

1 +√

δ

Method 2 has a sinusoidal band that follows the referenceand, for the same reason as method 1, we choose

∆I(t) = ∆I1 |sin(2π fref t)|Fig. 2 shows how the bands evolve for each method.

On Fig. 3, the current waveforms and the spectrumsare presented. Fig. 3.a shows that the switching frequency,for method 1, varies with the reference. When the refer-ence value is the biggest (|iref | = 1 A), the switchingfrequency falls down and when this amplitude decreasesto |iref | = 0 A, the switching frequency increases. Formethod 2, we have the same but as the band goes tozero, the switching frequency increases more. In fact,for this method, the switching frequency is limited bythe sampling frequency (fswitching max = fsampling/2)or by adding a lockout circuit [7] but the frequency ofthis circuit has to be higher than the desired frequencyas a lockout frequency close to the desired one leadsto the same results as method 1. Method 3 shows adifferent result as the band varies inversely with thereference amplitude, so the switching frequency staysnearly constant. Indeed, the band calculation was made

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curr

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curr

ent (

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time (ms)

curr

ent (

A)

[a]

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0

spec

trum

(dB

)

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102

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0

spec

trum

(dB

)

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102

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0

frequency (kHz)

spec

trum

(dB

)

100

101

−60−50−40−30

100

101

−60−50−40−30

100

101

−60−50−40−30

[b]

Fig. 3. Current waveform (load current (solid blue) and reference(dashed red)) [a] and load current spectrum [b]: method 1 (top), method2 (middle), method 3 (bottom).

under the hypothesis that the currents at the beginningand at the end of the switching period are the same. Sincethe reference varies, this hypothesis is not satisfied andso the switching frequency is not strictly constant.

Fig. 3.b confirms these results. We can see that formethod 1, the spectrum (except the fondamental) is under−30 dB and is located between 2.5 kHz and 5 kHz(see the zoom between 1 and 10 kHz). Therefore, theswitching frequency is limited under the chosen frequency(5 kHz) and varies. For method 2, the spectrum is under−32 dB and is not well located, there are harmonics andsub-harmonics higher than the desired frequency and thiscan lead to the inverter destruction. For method 3, thespectrum is under −30 dB and is much more located thanfor the other methods, as it varies between 4 kHz and 5kHz only.

Table I shows the total harmonic distorsion (THD) foreach method. For a sampling frequency of 1 MHz, ourmethod is between method 1 and 2. As we have not addeda lockout circuit, the THD yield by method 2 is better thanif we would have added it and the switching frequency isonly limited by the sampling frequency. For a samplingfrequency of 50 kHz, we have the same results but theTHD is worse as sampling distorts the currents.

To conclude, we can say that our method gives better

TABLE ISINGLE-PHASE SIMULATION THD

Method

Sampling frequency 1 2 3

1 MHz 10.30 % 7.31 % 8.22 %

50 kHz 12.97 % 9.54 % 10.45 %

R,LR,L

R,LA

BNC

SA

SA

SB SC

SB SC

iA

iB

iC

Vdc2

Vdc2

Fig. 4. Three phase load with DC-AC inverter.

results than a fixed band as it limits the upper boundof the switching frequency and its variations, and thecurrent is less distorted. The sinusoidal band seems togive better results than our method as the THD and thespectrum amplitude are less important but the switchingfrequency cannot be limited below the desired frequencywith this method. Besides, this method only works witha sinusoidal reference, whereas our method does not careof the reference shape. Therefore, our method seems tobe more powerfull.

III. 3-TO-2 PHASE TRANSFORMATION

In this section, we will first show how to transforma three-phase case into a two single-phase case to takethe dependency between the load phases into account andto extend the previous calculation to limit the inverterswitching frequency. Some simulation results are alsoprovided.

A. Theoretical calculation

In this paragraph, we will see how to decouple thethree phases to obtain two single-phase systems so thatthe previous calculation can easily be extended.The load is a well-balanced three-phase R-L load withisolated neutral (see Fig. 4) supplied by a DC voltagesource ±Vdc/2 through a DC-AC inverter (with switchesSA, SB and SC). The phase currents can be written as

LdiAdt

+ R iA = VA − VN

LdiBdt

+ R iB = VB − VN (11)

LdiCdt

+ R iC = VC − VN ,

where VX is the potential at point X. VA, VB and VC

can only be equal to Vdc/2 or −Vdc/2 (depending on theinverter switches) and as the phase current sum equalszero, the neutral potential is:

VN =VA + VB + VC

3(12)

TABLE II2-PHASE CURRENTS

case SA SB SC iα(t) iβ(t)

1 0 0 0 iα0 (t) iβ0 (t)

2 0 0 1 iα0 (t) − Vdc3

1−γ(t)R

iβ0 (t) − Vdc√3

1−γ(t)R

3 0 1 0 iα0 (t) − Vdc3

1−γ(t)R

iβ0 (t) +Vdc√

3

1−γ(t)R

4 0 1 1 iα0 (t) − 2 Vdc3

1−γ(t)R

iβ0 (t)

5 1 0 0 iα0 (t) +2 Vdc

31−γ(t)

Riβ0 (t)

6 1 0 1 iα0 (t) +Vdc3

1−γ(t)R

iβ0 (t) − Vdc√3

1−γ(t)R

7 1 1 0 iα0 (t) +Vdc3

1−γ(t)R

iβ0 (t) +Vdc√

3

1−γ(t)R

8 1 1 1 iα0 (t) iβ0 (t)

R,LR,L

α β

Sα Sβ

Sα Sβ

Vdc3

Vdc√3

Vdc3

Vdc√3

Fig. 5. Equivalent two single-phases.

Substituting (12) in (11) leads to:

LdiAdt

+ R iA =2VA − VB − VC

3

LdiBdt

+ R iB =2VB − VA − VC

3(13)

LdiCdt

+ R iC =2VC − VA − VB

3Equations (13) can be solved with (t0, IA0 , IB0 , IC0) asinitial conditions:

iA(t) = IA0 γ(t) +2VA − VB − VC

3R(1 − γ(t))

iB(t) = IB0 γ(t) +2VB − VA − VC

3R(1 − γ(t)) (14)

iC(t) = IC0 γ(t) +2VC − VA − VB

3R(1 − γ(t))

with γ(t) = e−RL (t−t0). To go from 3-phase (A,B,C) to

2-phase (α, β), the Clarke transform [16] is used:

iα(t) =23

(iA(t) − iB(t) + iC(t)

2

)= iA(t)(15)

iβ(t) =√

33

(iB(t) − iC(t)) (16)

As the potential at each point A, B and C can onlytake two different values, this leads to eight possiblecombinations (see Table II, where, for the switches SA,SB and SC , 1 stands for on and 0 for off and where iαand iβ are achieved by substituting (14) in (15) and (16),by replacing the potentials by their values and by settingiα0(t) = IA0 γ(t) and iβ0(t) = IB0−IC0√

3γ(t)).

If we extract from Table II the 4 cases (2, 3, 6, 7), it canbe noticed, in Table III, that iα and iβ have expressions

TABLE III2 SINGLE-PHASE CURRENTS

case Sα Sβ iα(t) iβ(t)

2 0 0 iα0 (t) − Vdc3

1−γ(t)R

iβ0 (t) − Vdc√3

1−γ(t)R

3 0 1 iα0 (t) − Vdc3

1−γ(t)R

iβ0 (t) +Vdc√

3

1−γ(t)R

6 1 0 iα0 (t) +Vdc3

1−γ(t)R

iβ0 (t) − Vdc√3

1−γ(t)R

7 1 1 iα0 (t) +Vdc3

1−γ(t)R

iβ0 (t) +Vdc√

3

1−γ(t)R

TABLE IVEQUIVALENCE 1-PHASE AND 2 VIRTUAL 1-PHASE

changing element 1-phase α β

DC voltage Vdc2

Vdc3

Vdc√3

reference phase 0 0 −π2

that are equivalent to single-phase current ones (they looklike equations (2) and (3)). As a consequence, we cancontrol a three-phase load as two single-phase loads (seeFig. 4 and 5). Furthermore, there is no need for switchingtables, as it is easy to see in Table II and III that thevirtual switch Sα of Fig.5 is directly the switch SA andthe virtual switch Sβ of this figure is the switch SB andthe complementary of the switch SC . It should be noticedthat this transform can be extented to applications withback-emf such as AC motors.

Since we have shown that it is possible to control athree-phase load as two single-phase loads, we can extendthe calculation previously performed for a simple single-phase load to control the currents iα and iβ .In order to do that, we have built table IV which gives thechanging elements between the single-phase model (Fig.1.a) and the two equivalent single-phase models in α andβ (Fig. 5). The current references are chosen as sampledversions of:

irefA(t) = Iref sin(2π fref t) (17)

irefB(t) = Iref sin(2π fref t − 2π

3) (18)

irefC(t) = Iref sin(2π fref t +2π

3) (19)

and so, with Clarke transform:

irefα(t) = Iref sin(2π fref t) (20)

irefβ (t) = Iref sin(2π fref t − π

2) (21)

We can achieve the hysteresis band for the α-phaseand for the β-phase by substituting in equation (10) theelements of the second column of Table IV by those of thethird column for ∆Iα, and by those of the fourth columnfor ∆Iβ (with δ = e−

RL T ):

∆Iα(t) =2Vdc

3R

1 + δ

1 − δ

−2

√(2Vdc

3R

)2δ

(1 − δ)2+ i2refα

(t) (22)

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ent (

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)

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spec

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(dB

)

100

101

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100

101

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100

101

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[b]

Fig. 6. Phase A current waveform (load current (solid blue) andreference (dashed red)) [a] and load current spectrum [b]: method 1(top), method 2 (middle), method 3 (bottom).

∆Iβ(t) =2Vdc√

3 R

1 + δ

1 − δ

−2

√(2Vdc√

3R

)2δ

(1 − δ)2+ i2refβ

(t) (23)

This choice for ∆Iα and ∆Iβ allows to limit the switchingfrequency on the virtual α and β phases. Since Sα =SA and Sβ = SB = SC (tables II and III), limiting theswitching frequency of Sα and Sβ is equivalent to limitingthe frequency of SA, SB , SC .

B. Simulation results

In this paragraph, some simulations are presented toillustrate the previous theoretical results. We will compareour method with two others, as for the single-phase case:

• method 1: an hysteresis controller with a fixed bandon each three phase,

• method 2: an hysteresis controller with a sinusoidalband on each three phase [7] without lockout circuit,

• method 3: an hysteresis controller on each twovirtual phase (α, β) with a variable band as definedby equations (22) and (23).

The sampling period is 1 MHz, as in the single-phasecase, to remove the current distorsion due to this fre-quency. We will also give the THD computed for each

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time (ms)

curr

ent (

A)

[a]

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−40

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spec

trum

(dB

)

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spec

trum

(dB

)

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10−1

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101

102

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0

frequency (kHz)

spec

trum

(dB

)

100

101

−50−40−30−20

100

101

−50−40−30−20

100

101

−50−40−30−20

[b]

Fig. 7. Phase B current waveform (load current (solid blue) andreference (dashed red)) [a] and load current spectrum [b]: method 1(top), method 2 (middle), method 3 (bottom).

method with a sampling frequency of 50 kHz to com-pare with experimental results. The load parameters areR = 2.2 Ω and L = 4 mH, the DC-bus voltage isVdc = 20 V and the desired switching frequency is 1

T = 5kHz. The reference current has an amplitude of Iref = 1A and a frequency of fref = 50 Hz.

The hysteresis bands are computed with the samereasoning as with a single-phase load. For method 1, wechoose the phase bands as:

∆IA(t) = ∆IB(t) = ∆IC(t) = ∆I1

with ∆I1 defined in section II-B.For method 2, the phase bands are:

∆IA(t) = ∆I1 | sin(ωt)|∆IB(t) = ∆I1 | sin(ωt − 2π

3)|

∆IC(t) = ∆I1 | sin(ωt +2π

3)|

For method 3, ∆Iα(t) and ∆Iβ(t) are calculated with eq.(22) and (23).

Fig. 6.a and 7.a show the current waveforms for phaseA and B. The currents for methods 1 and 2 have flatparts due to the phase interferences. These phenomenonscannot be found on the currents of the third method as

TABLE V3-PHASE SIMULATION THD

Method

Sampling frequency Phase 1 2 3

1 MHz A 10.69 % 6.66 % 6.51 %

1 MHz B 11.39 % 7.92 % 11.34 %

50 kHz A 12.43 % 8.08 % 8.02 %

50 kHz B 12.92 % 9.18 % 13.87 %

the phases are decoupled thanks to our transform. Formethod 3, the current on phase B is distorted comparedto the current on phase A. This is due to the fact thatthe proposed transform implies that phase A is phase αand so is directly controlled as a single-phase, whereas,phase B is controlled with phase C through phase β. Formethod 1 and 2, the switching frequency on each phaseis varying, whereas, for method 3, it seems to stay nearlyfixed.

Fig. 6.b and 7.b are the current spectrums of phaseA and B for each method. First, method 1 and 2 hasa phase A spectrum close to phase B one. It is not thecase for method 3 for the reason explained previously.Then, we can see that the spectrum is well located formethod 3 whereas it is not for the other methods. Indeed,for phase A, method 1 spectrum is greater than −40 dBuntil 4.5 kHz, method 2 spectrum is under this value butdoes not show clear limits for the switching frequency.Method 3 has its spectrum located between 4.2 kHz and5.2 kHz, its value is higher than for the other methodsbut the switching frequency is nearly constant. For phaseB, method 1 and 2 have the same results whereas method3 shows a highest value of spectrum for the switchingfrequency than for phase A but also a well locatedspectrum.

The THD has been computed for each method (seetable V). For a sampling frequency of 1 MHz, the THDobtained with method 3 is better on phase A. For phaseB, method 3 is close to method 1. This is due to thedistorsion implied by the chosen transform. It should benoticed that, as for the single-phase simulation, method2 has no lockout circuit, so the THD is better than itshould be. For a sampling frequency of 50 kHz, the THDrates are worse as a lower sampling frequency leads tomore distorsions on the current. Phase A still shows betterresults for method 3 but phase B has its worst THD forthis method. So, the sampling frequency has to be highenough to perform correctly.

To conclude, we have shown that our transform allowsto decouple the phases easily and that the proposed bandcalculation can be extended to a three-phase load andleads to a limited switching frequency. The importanceof a sufficiently high sampling frequency has been under-lined to reduce the current distorsions for the proposedmethod.The reduced computational cost of our algorithm should

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curr

ent (

A)

[a]

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(dB

)

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101

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100

101

−60−50−40−30

100

101

−60−50−40−30

[b]

Fig. 8. Current waveform (load current (solid blue) and reference(dashed red)) [a] and load current spectrum [b]: method 1 (top), method2 (middle), method 3 (bottom).

allow its implementation on current microcontrollers.

IV. EXPERIMENTAL RESULTS

In this section, we will compare the experimentalresults of the three hysteresis control strategies, with thesimulation results presented in paragraphs II-B and III-B.These methods are implemented on a DS1103 dSpaceenvironment with a sampling frequency of 50 kHz. Thesystem’s parameters are still L = 4 mH, Vdc = 20 V,1T = 5 kHz, Iref = 1 A and fref = 50 Hz. The loadresistance is R = 6.6 Ω for the single-phase load andR = 2.2 Ω for the three-phase one. The experimentaldata have been recorded with a Yokogawa ScopeCorderDL750 with a sampling rate of 500 kHz.

A. Single-phase results

Fig. 8.a shows the current waveforms for each method.We can notice that, as in simulation, the switching fre-quency seems to stay mostly constant for method 3 thanfor the others. Method 1 has its greater switching fre-quency when the reference amplitude decreases. Method2 has the same behaviour but its switching frequencyincreases more.

Fig. 8.b shows the current spectrum of each method.All the methods give a spectrum under −40 dB butmethod 3 has the most located spectrum. Indeed, it is

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Fig. 9. Phase A current waveform (load current (solid blue) andreference (dashed red)) [a] and load current spectrum [b]: method 1(top), method 2 (middle), method 3 (bottom).

TABLE VISINGLE-PHASE THD

Method

Sampling frequency 1 2 3

50 kHz 13.85 % 11.03 % 12.19 %

located between 3.5 kHz and 5.2 kHz whereas method 1spectrum goes from 2 kHz to 5 kHz and method 2 onegoes from 3 kHz to nearly 6 kHz.

The THD has been computed for each method (tableVI). As in simulation, method 3 is between methods 1and 2 but method 2 has no lockout circuit, so, its THDshould be worse.

These results confirm that the proposed method limitsthe switching frequency and yields a correct THD.

B. Three-phase results

Fig. 9.a and 10.a shows the current waveforms obtainedfor a three-phase load for phase A and B. As in simula-tion, we can see that method 3 gives more regular currentsthan the two other methods, without interferences betweenphases. Phase B current is more distorted than phase Abecause of our transform.

Fig. 9.b and 10.b are the current spectrums for each

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Fig. 10. Phase B current waveform (load current (solid blue) andreference (dashed red)) [a] and load current spectrum [b]: method 1(top), method 2 (middle), method 3 (bottom).

TABLE VII3-PHASE THD

Method

Sampling frequency Phase 1 2 3

50 kHz A 12.41 % 9.26 % 8.81 %

50 kHz B 11.87 % 8.49 % 13.17 %

method. Method 3 has a well located spectrum on phaseA (from 4 kHz to 5 kHz) and on phase B (from 3.5 kHzto 4.5 kHz) whereas methods 1 and 2 have spectrumswith no distinct limits. The harmonics and sub-harmonicshave values under −40 dB for method 1, under −42 dBfor phase A and −45 dB for phase B for method 2 andunder −40 dB for phase A and −35 dB for phase Bfor method 3. Therefore, there is not a great differencebetween the three methods.

The THD has been computed for each method (seetable VII). The results are similar to those obtained insimulation for phase A. Method 3 has better results thanthe other methods whereas, for phase B, it has worseresults than these methods. This is due to the switchingfrequency and to the fact that method 2 has no lockoutcircuit.

To conclude, we have shown that the experimentalresults obtained leads to the same conclusion as thesimulation ones: the switching frequency is limited withthe proposed method and the phase currents are welldecoupled thanks to the proposed transform.

V. CONCLUSION

In this article, we have first shown how to achieve theinverter switching frequency limitation through a simplehysteresis band calculation in a single-phase case. To thisaim, we assume that the current reference is constantduring each switching period. As it is not generally thecase, the switching frequency slightly varies instead ofremaining constant. However, it can be considered as anadvantage, as a small variation on the switching frequencyavoids uncomfortable noises.

Then, we have transformed, in a very simple manner,a three-phase load application with isolated neutral ina two single-phases application, so that we can easilyapply single-phase control strategies to three-phase cases.Besides, it allows to extend the previous single-phasecalculation to a three-phase application.For this, we used the Clarke transform (α β plane). Bychoosing four of the eight possible configurations of theinverter switches, we succeeded to decouple the phases.Furthermore, these four configurations allow to controldirectly the switching frequency on phase A through thephase α, and to control the switching frequencies onphases B and C through the phase β. Therefore, thereis no need of a switching table.

The simulation and experimental results obtained withour hysteresis band calculation and our transform haveshown their interest compared to two other control strate-gies: a fixed-band hysteresis comparator and a sinusoidal-band hysteresis one [7] for a single-phase load and fixed-band hysteresis controllers and sinusoidal-band hysteresiscontrollers [7] on each phase for a three-phase load.In the single-phase case, we have shown that the switchingfrequency can be controlled easily with our band calcula-tion and in the three-phase case, the phases are decoupled,thanks to our transform, so that we can control the loadcurrent without phase interferences and, as with a single-phase load, the switching frequency is nearly constant.It should also be noticed that the system parameters andthe sampling frequency have a great importance on theresults obtained. If the system parameters have values thatleads to a very small variation of the band, our method isequivalent to a fixed-band one. If the sampling frequencyis not high enough, the current is distorted and therefore,the THD becomes worse.

Finally, the proposed method leads to a located spec-trum below the desired switching frequency. Hence, ap-plications that need to filter the load current to extract thefundamental can use this method as the filter design willbe easier.

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