8/17/2019 00785624.pdf

1/6

A New Approach to Generate

PWM

Patterns for Four-Switch

Three-phase Inverters

M.B.R. Correa, C.B. Jacobina, A.M.N. Lima, E.R.C. d a Silva

Departamento de Engenharia ElCtrica, Universidade Federal da Paraiba

58109-970 Campina Grande, PB, Brasil, Caixa Postal 10105

Fax:

+

+55-83-3 101015 Email:

j

acobina@dee.ufpb. br

Abstract:

This paper presents a new method

to generate pulse width modulated signals to con-

trol four switches three phase inverters. The pro-

posed method provides a simple scheme to se-

lect three or four vectors to synthesize the de-

sired output voltage. The method

is

based on

the

so

called space vector modulation but the pa-

per

also

presents its scalar version. The paper

presents a comparative study where the different

vector com binations are investigated. The paper

also

discusses how the use of the wye and delta

connections of the machine windings affects the

implementation of pulse width mod ulator. Simu-

lation and experimental results are presented to

corroborate the analytical developments.

I. INTRODUCTION

Th e conventional s truc ture of a three-phase voltage in-

verter comprises three legs, six power switches (SSTPI),

a pair of complementary switches for each phase. Th e

four-switch three-phase inverter (FSTPI) employs only

two legs, th at is four switches as shown in Fig. la. Several

papers have investigated this structure [l-91. The FSTPI

structure allows one to generate four active vectors in the

CY@ plane instead of six as usual in the SSTPI structure.

This paper presents a new method to generate pulse

width modulated signals to control four-switch three-

phase inverters. Th e method is based on the so called

space vector modulation but the paper also presents its

scalar version. Th e proposed method provides

a

simple

way to select three

or

four vectors to synthesize the de-

sired output voltage during the switching period. In the

proposed approach the choice between the use of three

or four vectors is parameterized by

a

single variable and

this permits to simulate and implement 'all the altern-

atives making possible

a

fair comparison of the different

techniques. Th e influence of different switching patterns

on the output voltage symmetry, current waveform and

switching frequency are examined. Th e paper also dis-

cusses how the use of the wye and delta connections of

the machine windings affects the implementation of pulse

width modulator. Th e utilization of an induction ma-

chine with its windings connected in delta is studied here

R

e

C

t

f

e

r

A

Fig. 1.

Ac

drive system configurations.

as an alternative to operate the machine with same dc

link voltage used for the SSTPI. Simulation and experi-

mental results are used to illustrate the use of the FS TPI

to supply a three-phase induction motor.

11.

SPACE

ECTORANALYSIS

With respect to the circuit of Fig. la let us assume tha t

the conduction state of the power switches is associated

to the binary variables

41

to

44.

Therefore, from now on

the binary 1 will indicate

a

closed switch and the 0 an

open one. The pairs 41-43 and

42 44

are complementary

and, as

a

consequence,

43 =

1- 1 and 94

=

1

-

2 .

The voltages V A O , B O nd VCO, epend upon the states

of the power switches and may be expressed in terms of

the binary variables

91

and

4 2 ,

as

follows:

0-7803-5421-4/99/ 10.00 1999IEEE

94

1

8/17/2019 00785624.pdf

2/6

TABLE 1.

Available vectors in the a@ plane

for

the wye

41 42 v

=

vup +

jv

0

0

v1 =

[ E / & ) e - j 2 I 3

:onnection

v c o =

0 .

( 3 )

Th e space vector modulat ion and the problem of select-

ing the appropriate switching sequence are better under-

stood if the three-phase quantities are transformed into

YP

quantities. Th e transformed YOvariables are given by

vupp = AV123 4)

with VI23

=

[VI v2 v3IT , vap

=

[wa vplT and the

transformation matrix being

A . Wy e connection

,

Fig. l b shows a three-phase induction machine with

the windings connected in wye. In th is case the line-to-

neutral voltages are

v1 =

V A O

- v ~ o ,

12

=

vgo

-

v ~ o

and 213

=

- U N O with V N O being the voltage between the

neutral ( N ) and the dc bus midpoint 0),

as

indicated

in Fig. la . The induction machine is symmetric and the

neutral wire is disconnected. Th e a@ voltage components

are given by:

(7)

Th e combinations of t he s tates of the switches originate

four different vectors in t he

YP

plane

as

given in Table

1.

These vectors are

7r/2

away from each other. Using the

above vector definitions one may split the YPplane into

four sectors, i.e.

I

I I I I I and I V , as showed in Fig. 2a.

The vectors v2 and v 4 are opposite in directior? ( v 2

=

-vq and their amplitude is imes bigger than the

amplitude of the pair v 1 and v 3 . Also, the vectors v and

v 3 are oppos ite in direction (VI = - v 3 ) .

v42

Fig.

2.

Vectors in the a@ plane for the same

dc

bus voltage. (a)

wye connection and (b) delta connection.

TABLE

2.

Available vectors in the cr@ plane

for

the delta connection

1

P = -(41

+ 42 - 1)E.

fi

9)

The combinations of the states of the switches originate

four different vectors in the aP plane as given in Tabl e 2.

These vectors are also 7r/2 away from each other but their

amplitude is fi imes bigger than the vectors of the wye

connection (see Fig. ab .

In the following sections the analytical formulation of

B.

Delta connection

Fig. ICalso shows a three-phase induction machine but

in this case the windings are connected in delta. In this

case vi = VAO - VBO 12 = OBO- vco nd vg = vco

A O

the space vector modu lation will be ~der ived or the case

of wye connected load. Further, in section VI i t will be

demonstrated how to map these results for the case of

a

delta connected load.

111. SPACE

VECTORPWM

nd consequently the YP voltage components are given

by:

Let v* represent the reference voltage to be synthes-

ized by the FSTPI within a switching period of length T.

up = - 2 ) E

(*)

942

8/17/2019 00785624.pdf

3/6

According to the space vector technique this implies that:

v*T

=

vltl + ~ 2 t 2 ~ 3 t 3 ~ 4 t 4

10)

with the time weights t l , 2 , t 3 and t 4 restricted to

T

=

t i + 2 + 3 + 4 .

11)

The problem is now to find out the values of the time

weights given v * and T . In order to simplify the algebraic

manipulation let us introduce v*

= v: + v z ,

v,

=

VI =

Replacing th e vectors v, and

ve

into

10)

results in

-VQ = U,,

+

v p and Ve = ~2

=

- v 4 = Ve, + j vep .

v * T = ~ , t 1 3+ v e t 2 4

12)

with t13

= t l -

3 and t24 = t 2

-

4 .

Rewriting

12)

in terms of the

(YP

components gives

v:T = Voatl3 +Wed24

v ~ Tv o p t l 3 + vept24.

13)

14)

Considering the wye connection

v o a

= - +, vop

=

- d ,

ea

=

f

and

vep

=

-A ,

hen from

13)

one find that

t13

and

t24

are given by

As it can be seen from the above equations, the compu-

tat ion of the ti me weights is an under-determined problem

i.e., t here are four unknowns but only three different equa-

tions. By considering that the switching frequency must

be constant there are two possibilities to solve this prob-

lem. The first alternative is to use all of the four vectors

while the second one is to select only three among the four

available vectors. Th e present paper proposes an elegant

way t o pass from one alternative to another as

it is shown

in the following.

From 12) the resultant odd vectors are applied during

t13 and the resultant even vector are applied during t24 .

Under these conditions the remaining time is given by:

6~ =

- t131 - t241. 17)

Now int roduce an apport ioning factor p

0

5 p 5

l ) ,

p for vectors v1 and v 3 and

1 -Y

for vectors v 2 and v 4 .

The use of the apportioning factor depends on the signs

of

t13

and

t24 as

described below:

Sector

I :

t13 > 0 and t24 2 0

18)

Sector 11: t13 5 0 and t24

>

0 19)

Sector

111:

t13

<

0

and

t24

5

0

20)

Sector IV: t13

2

0 and t24 < 0

21)

Note that equation

11)

is always satisfied and the ap-

portioning factor

p

indicates how many vectors with its

respective weights are employed. If

,U =

0 only three vec-

tors are employed v 2 , v 4 and v 1

or

v3 (see Table

3) .

If

0 < p

<

1 all the four vectors are employed. If p = 1

only three vectors, V I, v 3 and v 2 or v 4 are employed (see

Table 4).

TABLE 3.

Two

large and one small vectors

Vectors Sector p

v 4 v 1 v Z

I

0

v 9 v n v 4 11 0

v 2 v 3 v 4 111

0

v q v 1 v z IV

0

TABLE 4. Two small and one large vectors

Vectors Sector U

~ 1 V 2 ~ 3 I

1

VlVZV.? I I 1

VQV4V1 111

1

v 3 v 4 v 1 IV

1

By changing p one may use the three vectors which

are as close as possible of

v * ,

i.e., avoiding the use of

the farthest vector for a given v * . Table 5 shows how

to select

p

in order to always use only the three closest

vectors for

a

given v . Fig. 3 illustrates how the value of

p is mapped into the voltage sectors A , B , C and D of

ap plane. The row labelled Condition in Table 5 indicates

when the reference vector v * enters in a given sector.

The use of the switching patterns given in Table

5 

has

already been proposed by Blaabjerg

e t

al.

[8].

Also, the

switching patterns given in Tables 4 and

3

have already

943

8/17/2019 00785624.pdf

4/6

TABLE .

Grouped sectors around the voltage vector

Vectors Condition Sector

41A

0

0

t

- C

;,-

. . . ,

, . .

, I

, I ,

, . .

, . .

, . ,

< , , . .

71 I

, . . , .

I . . . .

, . . . ,

4 I C 2

- j

-

-

3

-

2-1, 14 1~ 1;-1~- f ;

I

<

T

T

1-

Fig. 3. Sectors A, B , C and D n the a@ plane.

been proposed by Jacobina

et

al. [5]. However, the gen-

eric analytical development describing all these switching

patterns parameterized in terms of

a

single quantity has

not been presented previously.

Th e above analysis has demonstrated how the selection

of a specific switching sequence is decided by a single vari-

able, the apporti oning factor

p .

IV.

SCALAR

WM

The use of the space vector approach provides simple

analytical way to explain the functioning of the FSTPI.

However, to implement t he m odulator with a timer based

hardware it is more appropriate to define a scalar and

equivalent version of the proposed technique. Moreover,

this development provides a good insight about how the

pulse width modulator should be implemented in soft-

ware.

The basic modification to convert a scalar PWM

stra tegy defined for a SSTP I to be used to control a FSTPI

relies on the reference waveform generation. In this case

the line-to-neutral reference voltages

vT0,

vzo

and

w z

sup-

plied to th e modulator th at controls the F STPI must obey

specific phase shift relationships. From the Fig. l a and l b,

the voltages vi0 vgo are given by

vT0 =

v i o

= U;

‘UNO,

v ; ~ vh0 = v; + V N O , vzo = v;o = v z + v ~ o

= 0,

and then

VNO

=

-vB . This implies th at v ro =

vT -U;,

vfo = V - ;

.

Consequently, if the line-to-neutral reference voltages v;

4 1 L

1  +

I;,-

v; and v: are written in terms of v and v; the reference

voltages may be given by

v fo =

AV;.

23)

Fig. 4  shows the typical waveforms of the command

signals for the switches q1 and 42 when t24

< 0

(Fig. 3a

and t24

2 0

(Fig. 4b) both for

0 <

p < 1. From 22),

the time intervals

7 1

and 7 2 , during which the switches

q1 and q2 must be switched on in order to obtain the

desired reference voltage at the output of the FST PI, are

determined by

T T

2 E

7 2 = - + i - v ; .

It can be noted that for both cases represented in Fig. 4,

7 = t 2

+

3 and 7 2

=

3

+

4 . These relationships demon-

strat e th at both the space vector and t he scalar techniques

give equivalent results.

The generation of the command of the switches of

q1

and q2 is done by using programmabl e timers. Fig. 4

shows that for the geaeral case in which all the four vector

are employed, the pulse widths 7 1 and 7 2 can be split in

two parts: 7’1 and 7:’ (71 = 7’1

+

7:’ nd 75 and 7’2’

944

8/17/2019 00785624.pdf

5/6

(72 = 7;+

7; . Then, the timers are programmed twice

at each time period

T .

Note that if one chooses to use

only three vectors, the timers are programmed once at

each time period

T .

V . IMPLEMENTINGHE PWM/FSTPI

Based on the equations presented in the previous sec-

tion it is possible to derive an algorithm that can be in-

cluded in the ac drive software. This algorithm is de-

scribed by the following steps:

i) compute 213 and t24 using

15)

and

16)

ii) compute

ST

using 17)

iii) compute t l

2

t3 and t4 based on equations (18) to

iv) If t24 < 0 program the timers to count as follows:

q 1

timer is first loaded with

t l

=

t3

and after with tLl

=

tl t4; q 2 timer is first loaded with

t c 2

= t3 t4 and after

with tb2

=

tl f t z .

v) If t24

>_ 0

program the timers to counts as follows:

q 1

timer is first loaded with

t l =

tz + t3 and after with

tLl

=

t l + t4; 42 timer is first loaded with t c 2

=

t3 and

after with tL2

=

t 2

+

tl.

The time intervals

t e l l L l

c2 and

tL2

are indicated in

Fig. 4.  Observing Fig. 4a and Fig. 4b it can be noted that

the number of commutations of the FSTPI switches is not

equally distributed. The tests included in steps iv) and

v) have been added in order to obtain in the average the

same number of commutation for all the FSTPI switches.

Also, steps iv) and v) may be defined in terms of t13.

21)

VI. DELTACONNECTION

PWM

For the delta connection voa =

0,

v,,p = -E/ , Vea =

m E

nd

v e p

= 0, then from 13) one may find out

that

t13

and

t24

are:

27)

Given

26)

and 27) it is possible to use same procedure

presented in section V. However, it is also possible to

obtain

v

and v; for the wye connection (named from

now on

as

v:y and

v ; ~ )

n terms of v zA and v i 4 (crp

voltages for the delta connection). Using matrix A it can

be shown tha t v : ~nd v i are given by: *

Then the pulse widths

for

the delta connection can be

determined

by

using the same expressions presented in

0 p = o

30 +

p = l

i

l p = O l

I

0

0 2 0.4 0.6 0.8

rn

p=o.5

o

~ = O 6

p=0.8

Fig. 

5. THD of the output voltage.

sections I11 and IV. Note that Tyaap can also be used

when the delta connection is considered for the case of a

SSTPI.

VII. SIMULATIONS

ESULTS

Fig. 5 presents the total harmonic distortion (THD) of

the FSTPI as a function of the modulation index

m).

The THD presented in Fig. 5 has been calculated by

where x indicate the axis x = (Y or = p , Vrm,, is

total root mean square voltage of the axis

2

and Vrmso(l)

is root mean square voltage of the fundamental in axis

x The total harmonic distortion of the voltage vector is

calculated from

Thda

and T h d p as follows

TH

=

4 ldu

+

T .

(29)

Fig. 5a presents the

T H D

for the case where only three

vectors are used. In this figure the label

p

= 0-1 indicates

that

p

varies as indicted in Table

5.  

Fig. 5b presents

945

8/17/2019 00785624.pdf

6/6

5

-1.5

I

0 0.01

0.02 0.03 t s )

1.5

7

-1.5

I

0 0.01

0.02 0.03

t s )

Fig. 6. Line current for different configurations. ( a) wye connection ,

(b)

delta connection.

the TH D when four vectors are used. To maintain the

same switching frequency, the period T used for the TH D

in Fig. 5b is 1.5 times the period used in Fig. 5a. For

high m all the alternatives present similar THD , but for

medium and low values of m the alternative with p = 1

is sensibly the best.

VIII . EXPERIMENTALESULTS

The proposed modulation scheme was implemented in

a

microcomputer-based FSTP I drive system. The FSTP I

employs IGBTs that switches at 5kHz and supplies a

1/3hp three-phase squirrel induction motor. Th e motor

was started-up with a standard v/f law and when the

steady-state was reached the current of the phase 1 has

been recorded. Fig. 6 presents the st ator current obtained

with the FSTPI supplying a induction machine for p = 1

and

m

=

0.8.

Figs. 6a and 6b show the line current for the

wye connection and for t he del ta connection, respectively.

IX. CONCLUSION

This paper has presented a new method to gener-

ate pulse width modulated signals to control four-switch

three-phase inverters. With the proposed met hod it was

possible to study several PWM schemes, using three

or

four vectors to synthesize the desired output voltage dur-

ing the switching period. The scalar version of the pro-

posed modulation technique can be implemented by soft-

ware and may easily included in drive software with a

negligible increase in the computational load. This study

have shown that is preferable to use three vectors, where

two are the small vectors. Th e paper also presented PWM

strategies suitable to applied with delta connections of

the machine windings. T ha t type of connection permit to

supply the machine with the same phase voltage of the

standard FSTPI drive.

REFERENCES

[l] H. W. Van der Broeck and J . D. Van Wyk. A compar-

ative investigation of a three-phase induction machine

drive with a component minimized voltage-fed inverter

under different control options. IEEE Transactions

on Industry Applications 20(2):309-320, March/April

1984.

[2] W . McMurray. Modulation of the chopping frequency

in dc choppers and pwm inverters having current-

hysteresis controllers. IEEE Transactions on Industry

Applications 20(4):763-768, JulyIAugust 1984.

[3] F. Blaabjerg, S. Freysson, H. H. Hansen, and

S. Hansen. Comparasion of a space-vector modulation

strategy for

a

three phase standard and a component

minimized voltage source inverter. I n

Conf. Rec. EP E

pages 1806-1813, 1995.

[4] F. Blaabjerg, S. Freysson, H. H. Hansen, and

S. Hansen. A new optimized space vector modulation

strategy for a component minimized voltage source in-

verter. In

Conf. Rec. APEC

pages 577-585, 1995.

[5] C. B. Jacobina, E. R . C. da Silva,

A.

M . N. Lima, and

R.

L.

A. Ribeiro. Vector and scalar control of

a

four

switch three phase inverter. In

Proc. IAS Conf. Rec.

pages 2422-2429,1995.

[6] G. Kim and T. A. Lipo. Vsi-pwm rectifierlinverter

system with a reduced switch count . In

Proc. IAS

Conf. Rec.

pages 2327

-

2332, 1995.

[7] R. L. A.Ribeiro,

C.

B. Jacobina, E.R. C. da Silva,

and A. M.

N .

Lima. Ac/ac converter with four switch

three phase structures. In

Proc. PESC Conf. Rec.

pages 134-139, June 1996.

[8]

F. Blaabjerg,

S.

Freysson, H.-H. Hansen, and

S.

Hansen. A new optimizied space-vector modu lation

strategy for a component-minimized voltage source

inverter.

IEEE Transactions on Power Electronics

[9] D.T.W. Liang and

J.

Li. Flux vector modulation

strategy for a four-switch three-phase inverter

for

mo-

tor drive applications. In Proc. PESC Conf. Rec.

pages 612-617, June 1997.

12(4):704-714, July 1997.

946