Indirect 3D-Space Vector Modulation for a Matrix Converter

(1)

Indirect 3D-Space Vector Modulation for a Matrix Converter

Ahmed Abdelrehim Department of Electrical Engineering

Alexandria University Alexandria, Egypt [email protected]

Mohamed El-Habrouk Department of Electrical Engineering

Samir Deghedie Erfan Department of Electrical Engineering

Karim Hassan Youssef Department of Electrical Engineering

Abstract—This paper discusses the indirect space vector modulation for a four-leg matrix converter. The four-leg matrix converter has been proven to be a reliable, cost-effective, and compact power electronic interface to supply unbalanced or nonlinear loads. However, the added fourth leg has shifted the inverter side modulation from simple two-dimension SVM into complex three-dimension. This paper employs a new technique to implement indirect 3D SVM in digital controllers with further simplification in the modulation process. Moreover, Simulink simulation using repetitive controller has been performed to regulate the output voltage for 400 Hz Power supplies.

Keywords-repetitive controller; 3D SVM; four-leg matrix converter

I. INTRODUCTION

The matrix converter is a static and direct AC to AC converter with unique features of unity input power factor, high power to volume ratio, high reliability and MTBF factor which has gained interest in applications aiming to produce a realization of a compact three-phase drive for military, industrial and aerospace systems [1-3]. Moreover, researchers utilized the matrix converter as an electronic interface layer between all resources (wind, solar, storage, etc.) of the energy matrix model and as an electronic transformer competing with the traditional magnetic transformer. The added fourth leg is placed to provide a return path for the zero-sequence current during unbalancing and add the capability of supplying different connected single-phase loads. Four-leg converters have a superior ability to produce balanced output voltage waveform even under severely unbalanced load or non-linear load conditions [4]. A four-leg matrix converter topology is shown in Figure 1. This paper investigates the indirect space vector modulation which decouples the modulation into two stages (rectifier + inverter) without intermediate energy storage. This decoupling is efficient and allows separate control

to each stage, while as there is no intermediate energy storage, a proper synchronization between the two stages is mandatory to fulfill the power balance equation as instantaneous input power shall be equal to the instantaneous output power for the load [5]. At high-frequency applications with precise control requirements as for naval and aerospace applications where the 115V-400Hz system is commonly used, traditional controllers have failed to achieve a proper regulation, due to the limited bandwidth so the repetitive controller is introduced as an optimum solution for this control problem [6].

II. MATRIX CONVERTER MODEL

Matrix converter can be represented mathematically by matrix M and switches are identified as , where X is the output phase, Y is the input phase and S is the linking switch between input and output. Rectifier switches are numbered from to while inverter side switches are numbered from to . The rectifier is represented in Figure 1. Governing equations are:

Vout=M*Vin (1)

Iout=M^T*Iin (2)

Vout=Mi*VDC (3) VDC=MR*Vin (4)

Vout=Mi*MR*Vin (5)

where , , , , , , , and are output voltage, input voltage, output current, input current, DC intermediate voltage, inverter side switching matrix, rectifier side switching matrix, matrix converter switching matrix and transposed matrix converter switching matrix respectively.

and Mi are described by (6) and (7) respectively:

= 1 3 5

2 4 6 (6)

(2)

spl rec ach out ass A.

con con rec

Fig. 1.

M =

S7 S8

S9 S1 S11 S1 S13 S1

M equals to M

M =

S7 S8

S9 S10 S11 S12 S13 S14

Equation (8)

=

S1 ∗ S7 S2 S1 ∗ S9 S2 S1 ∗ S11 S2 S1 ∗ S13 S2

=

SaA SaB SbA SbB ScA ScB SnA SnB

In Figure 1 v

The mathema

Fig. 2.

III. INDIREC

Indirect spac litting the mo ctifying stage hieve decoupl

tput voltage sumption of vi Rectifying St The main fu ntrolon the d ntrol the amp ctifying stage,

Schematic of a f 108

12 14

Mi*MR:

8 02 4

* S1 S3 S5S2 S4 S6

can be further

∗ S8 S3 ∗ S7

∗ S10 S3 ∗ S9

∗ S12 S3 ∗ S11

∗ S14 S3 ∗ S13 SaCSbC

ScC SnC

voltages are: V

atical modelin

Mathematical

CT SPACE VECT

ce vector mod odulation into and the seco ling between t control.as per irtual DC link tage

unction of the displacement a plitude of the

, only two sw

four-leg direct ma

56

r synthesized i

7 S4 ∗ S8 S S4 ∗ S10 S 1 S4 ∗ S12 S5 3 S3 ∗ S14 S5

= V V V V

and V

ng is summariz

modeling of matr

TOR MODULAT

dulation can b o two stages, ond is called

the input curr r (5) and (9)

as shown in F

e rectifying sta angle of the in virtual DC l witches must b

atrix converter

(7)

(8)

into

S5 ∗ S7 S6 ∗ S8 5 ∗ S9 S6 ∗ S10 5 ∗ S11 S6 ∗ S1 5 ∗ S13 S6 ∗ S1

(9)

V = V

V V .

zed in Figure 2

rix converter

TION

be implement the first is inversion sta rent control an

and based o Figure 1.

age is to achi nput current a link voltage. I be closed at a

8 0 1214

2.

ted by called ge, to nd the on the

ieve a and to In the time,

just sim num comwil

B.

abcang con sim ther com mo

the ind is nwil Fig vec sele the the rect mat The be

t one upper s multaneously to mbered from o mbinations are l occur as show

Fig. 3. Ni

Rectifier Spac SVM for the c to αβ coord gle by which w nverter modul mulation that sw

re is no nee mmonly hap dulations as sh

Fig

In Figure 4, θ actual hexa dex=Iref/IDC , 0 no need to syn l be switched gure 5. Modula ctor during t

ections have b number of th range of on tifying stage trix converter e displacemen zero. To impl

switch and on o avoid short c one to six as sh e only allowed wn in Figure 3

ine switching com

ce Vector Mod rectifying sta dination and c

we can define lation requirem

witching one v ed to synthes ppens with

hown in Figur

g. 4. Synthesi

θc is the angle agon sector, 0<mc<1. For nthesize the re d during ever ation process the time inte been translated he current sec ne to six. To and one of t is to have uni nt angle betwee

lement this, th

ne lower swi circuit. The re hown in Figure

to guarantee t 3.

mbinations for the

dulation age works by

calculating th e the working

ment, it has vector per sect size two vect conventional re 4.

is of reference cu

e of the refere mc the cu matrix conver ference vector ry sector dura happens by as erval of eve d into a train ctor, numbers o achieve the the most inter ity power facto en input curren he reference s

itch can be c ectifier switche e 1. Nine switc that no short c

e rectifying stage

transforming he reference v

sector. For m been approve tor is sufficien tors per secto l rectifier

urrent

ence current w urrent modul rter, mc=1 and r as just one v ation as show ssigning one a ry sector. V of pulses car are represent e second targ resting featur or at the input nt and voltage signal given t

closed es are ching circuit

input vector matrix ed by nt and

or as SVM

within lation there vector wn in

active Vector rrying ted in get of res of t side.

shall to the

(3)

mo ins

mo con app fun out add the thr

app unb app neu rol leg

cooinv the inv com by tur (ne coo con be ana Pri vec A.

odulation proc stead of the inp

Fig. 5.

Inversion st odulation, and

nverter, we w proach for th nction of inve tput voltage ded fourth leg e zero-sequenc ree-phase syste

+ + And the tra plied as it ha balanced syst plied. Due to utral current p le of 3D SVM gs.

+ + Transformati ordination du version stage, e same time in verter with mbinations ar p or n for ea rned on (posi egative). If th

ordination, th nventional SV divided into alog to secto ism number m ctor angle on α

Prism Identif If

0<θ<60 60<θ<12 120<θ<1 180<θ<24

cess has been put phase curr

. Rectifying s

III. INV

tage is conce d as this pa will address

he four-leg ersion stage i under balanc g is responsibl ce component em follows (10

=0 ansformation f

as happened tem (10) is n the unbalanc passing throug M to provide

≠ 0 ion to statio ue to the zer

switches on th n order to avo eight switc re sixteen. Sw

ach leg, where tive) and n in he sixteen-ve he result is VM are turned six prisms (F ors representat

may be ident αβγ coordinati fication

the

0 the

80 the 40 the

taken from i rent.

stage SVM vector

VERSION STAGE

erned with in aper discusse the 3D SVM matrix conve is to provide ced and unba e for providin t during unbal

0).

from abc to with the rect no longer vali cing of the sy gh the system accurate swit

onary frame ro-sequence c he same leg ca oid short circu ches, the p itching combi e p indicates ndicates the l ectors are rep a 3D hexago into prisms. T Figures 6-7).

tion on conv tified by defi ion.

en P=1 en P=2 en P=3 en P=4

input phase v

rs selection

E

nverter side es four-leg m M as a modu

erter. The pr a control ove alanced loads ng a current pa lance. The bal

(10 αβ coordinati tifier stage. F id and (11) is ystem, there w m and it is the

tching pulses

(11 happens into component. I an’t be turned uit. For the fo possible swit inations repres the upper swi lower switch presented into on and secto The 3D hexago The prisms a entional 2D ining the refe

oltage

SVM matrix ulation rimary er the . The ath for

lanced

0) ion is For an s now will be basic to all

) o αβγ In the d on at our-leg tching sented itch is is on o αβα ors of on can are an

SVM.

erence 8. E

B.

and imp bes thre swi

240<θ<30 300<θ<36 Vectors are d Each prism con

Fig. 6. 3D re

Fig.

Switching Vec As the referen d tetrahedron plemented by side each tetr ee switching v itching vector

00 then 60 then distributed into ntains four act

epresentation of sw

7. Prisms ide

8. Vector dist

ctor Selection nce vector loca

number, sel choosing the n rahedron[8, 9]

vectors, to ca the reference

n P=5 n P=6, where θ o each prism a tive vectors.

witching vectors i

entification for 3D

tribution into each

ation is define ection of sw nearest referen ]. Each tetrah alculate the du e required volt

θ= tan as shown in F

in αβγ coordinatio

D SVM

h prism

ed by prism nu witching vecto nce vectors lo hedron consis uty cycle for tage is transfo

Figure

on

umber ors is ocated

sts of each ormed

(4)

firs sw

wh the and ma var tetr po SV the

nu act (re cha rep rep tw ma

val var suc var Th op sho 10

st to αβγ coor witching vector

V

V V

= d

The duty cyc

d d

d = ∗ S

= 1

here S is the d e reference vec d d represents atrix is depend rying values rahedron. S m ssible coincid VM code check

e current value

The rectifyin mbers from 1 tive rectifier ectifier matrix arge of genera presenting the presented on

o matrices as i atrix converter

Fig. 9. M

A Simulink lidate the effe rying types o ccessfully va rious scenario he input and

eration. The s own in Table

-26.

rdination and rs [9] as per (1

d d V

V V

cles can be furt

∗ V V V

duty cycle mat

ctor, (V1 V2 V3

s the duty cy dent on the sel depending matrix calcula

ence between ks the prism a es of S.

IV. MATR

ng stage is de 1 to 6, these n

switches and x). At the sam

ating another current active (inverter in (5) allows u r. The procedu

Matrix converter

V. SIMUL

k simulation ectiveness of m

f loads. The alidated as w os such as bala

output wavef simulation was I. The simula

then is divid 2).

ther calculated

trix [8-10], (V

3)^Tis the curren ycle for each s

lected switchin on each se ations have b prisms and te and tetrahedron

RIX CONVERTE

edicated to gen numbers repre d would be re me time the in series of num e inverter swit matrix) as we us to find all p ure is shown in

code generation f

LINK SIMULATI

was successf matrix conver proposed indi well. Simulati anced and unb forms ensured s carried out u ation results ar

ed by the ava

(12

d as shown in

(13

(14 Vα-ref Vβ-ref Vδ-r

nt switching v switching vec ng vectors, so elected prism been done for etrahedrons. Th

n number to e

ER

nerating a ser esenting the c epresented on nversion stage mbers from 1

tches that wou ell. Multiplyin possible switch

n Figure 9.

from Mi and MR

ION

fully develop rter operation irect 3D SVM ion worked balanced linear d proper con using the param

re shown in F ailable

2)

(13).

3)

4)

ref)^T is vector, ctor. S it has m and

r each he 3D extract

ries of current n e is in to 16, uld be ng the hes for

ed to under M was

under r load.

nverter meters igures

R

A.

Source sys Load syst Switching freq

Input filter Output filter Repetitive Contr

Tracking con Unbalanced Balanced l Nonlinear l

Non-Linear L

Fig. 12. FF

TABLE I.

tem em quency

(LC) r (LC) roller Q(z)

ntroller

d load 5Ω

load load

Load

Fig. 10. O

(a) P

(b) P

(c) P

(d) N Fig. 11. O

FT analysis for ou

SYSTEM PARAM 440V 115V- 1280 56Ω/0.8m 180uH+70

1.72377 1.663 1.868 Ω+5.5mH, 10Ω+6 5Ω +5 3phase diode

Output voltage

Phase A

Phase B

Phase C

Neutral Output current

utput phase voltag METERS V-60Hz

-400Hz 00Hz mH/10uF 0uH/100uF

2 1

76 0.75754 3 0.9914 8 0.8735 6.2mH, 20Ω+7.5m 5.5 mH

e bridge +50Ω

ge (THD=3.3%) mH

(5)

B.

Fig. 13.

Unbalanced

Fig. 17. Inp

FFT analysis fo

Fig. 14.

Load

Fig. 15.

(a) P

(b) P

(c) P

(d) Fig. 16. O

put current multip

or output current

Error signal

Output voltage

Phase A

Phase B

Phase C

Neutral Output Currents

plied by factor 50

(THD=1.8%)

and input voltagee

C.

Fig. 19.

Fig. 20. FF

Non Linear L

F

Fig. 22.

Fig. 18.

FFT analysis fo

FT analysis for ou

Load

Fig. 21. Output

FFT analysis fo

Error signal

or input current (T

utput phase voltag

t voltage 115V/40

or input current (T

THD=48%)

ge (THD=3.3%)

00H

THD=48%)

(6)

mo beecon lim (11 con eff rep vo

[1]

[2]

[3]

[4]

[5]

[6]

[7]

Fig

Fig. 25. Inp

This paper odulation for en implement ntroller was s mited bandw

15V/400Hz).

nditions (bal fectiveness o petitive contro

ltage in an exc

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Fig. 23.

. 24. Repetitive

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CONCLUSION

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gulate the out high-freque tion results lanced, nonlin sed modulati the ability to

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