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. da Silva Departamento de Engenharia ElCtrica, Universidade Federal da Paraiba

58109-970 Campina Grande, PB, Brasil, Caixa Postal 10105 Fax: + +55-83-3 10 10 15 Email: j acobina@dee.ufp b. 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 combinations 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 modulator. Simu- lation and experimental results are presented to corroborate the analytical developments.

I. INTRODUCTION

The conventional structure of a three-phase voltage in- verter comprises three legs, six power switches (SSTPI), a pair of complementary switches for each phase. The four-switch three-phase inverter (FSTPI) employs only two legs, that 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. The method is based on the so called space vector modulation but the paper also presents its scalar version. The 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. The influence of different switching patterns on the output voltage symmetry, current waveform and switching frequency are examined. The paper also dis- cusses how the use of the wye and delta connections of the machine windings affects the implementation of pulse width modulator. The utilization of an induction ma- chine with its windings connected in delta is studied here

R e C t I f e r I

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 FSTPI to supply a three-phase induction motor.

11. SPACE VECTOR ANALYSIS

With respect to the circuit of Fig. la let us assume that 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 - 41 and 94 = 1 - 92.

The voltages VAO, VBO and VCO, depend 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 0 1999 IEEE 94 1

TABLE 1. Available vectors in the a@ plane for the wye c

41 42 v = vup + jv, 0 0 v1 = [E/&)e-j2"I3

:onnection

v c o = 0 . ( 3 ) The space vector modulation and the problem of select-

ing the appropriate switching sequence are better under- stood if the three-phase quantities are transformed into (YP quantities. The transformed (YO variables are given by

vupp = AV123 (4)

with VI23 = [VI v2 v3IT , vap = [wa vplT and the transformation matrix being

A . Wye connection ,

Fig. l b shows a three-phase induction machine with the windings connected in wye. In this case the line-to- neutral voltages are v1 = VAO - v ~ o , 212 = 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. The a@ voltage components are given by:

(7)

The combinations of the states of the switches originate four different vectors in the (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 (YP plane 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 4 times bigger than the amplitude of the pair v 1 and v 3 . Also, the vectors v1 and v 3 are opposite in direction (VI = - v 3 ) .

v42 A

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 Table 2. These vectors are also 7r/2 away from each other but their amplitude is fi times bigger than the vectors of the wye connection (see Fig. ab).

In the following sections the analytical formulation of B. Delta connection

Fig. IC also shows a three-phase induction machine but in this case the windings are connected in delta. In this case vi = VAO - VBO, 212 = OBO - vco and vg = vco - VAO

the space vector modulation will be~derived for the case of wye connected load. Further, in section VI it will be demonstrated how to map these results for the case of a delta connected load.

111. SPACE VECTOR PWM and 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. vup = - 42)E (*)

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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 , t 2 , t 3 and t 4 restricted to

T = t i + t 2 + t 3 + t 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: + j v z , v, = VI =

Replacing the vectors v, and ve into (10) results in -VQ = U,, + jv,p and Ve = ~2 = - v 4 = Ve, + j vep .

v * T = ~ , t 1 3 + vet24 (12)

with t13 = t l - t 3 and t24 = t 2 - t 4 .

Rewriting (12) in terms of the (YP components gives

v:T = Voatl3 + Wed24

v ~ T = vopt l3 + vept24.

(13)

(14)

Considering the wye connection voa = -&+, vop = -d$, vea = &f and vep = -A$, then from (13) one find that t13 and t24 are given by

As it can be seen from the above equations, the compu- tation of the time weights is an under-determined problem i.e., there 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. The present paper proposes an elegant way to 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~ = - It131 - lt241. (17) Now introduce an apportioning 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 et al. [8]. Also, the switching patterns given in Tables 4 and 3 have already

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TABLE 5. Grouped sectors around the voltage vector

Vectors Condition Sector U

41A

I

0

0

I

I

I

t

I

I - I C , A I;,- . . . , , . . , * I , I ,

, . . , . . , . , < , , . . 71 I

, . . , . I . . . . , . . . ,

4;‘ I C 2 - j : j - - I 3 - I2-1,*14*1~*1;-1~- f ; * I

< T T 1-

Fig. 3. Sectors A, B , C and D in 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.

The above analysis has demonstrated how the selection of a specific switching sequence is decided by a single vari- able, the apportioning factor p.

IV. SCALAR PWM

The use of the space vector approach provides simple analytical way to explain the functioning of the FSTPI. However, to implement the modulator 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 strategy defined for a SSTPI 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 o sup- plied to the modulator that controls the FSTPI must obey specific phase shift relationships. From the Fig. l a and lb , the voltages vi0 vgo are given by vT0 = v i o = U; +‘UNO, v ; ~ = vh0 = v; +VNO, vzo = v;o = v z + v ~ o = 0, and then VNO = -vB . This implies that vro = vT -U;, vfo = VI - v; . Consequently, if the line-to-neutral reference voltages v; ,

4 1 L 1.1 *+ I;,-

v; and v: are written in terms of v: and v;, the reference voltages may be given by

vfo = 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 FSTPI, are determined by

T T 2 E

7 2 = - + f i - v ; .

It can be noted that for both cases represented in Fig. 4, 71 = t 2 + t 3 and 7 2 = t 3 + t4 . These relationships demon- strate that both the space vector and the scalar techniques give equivalent results.

The generation of the command of the switches of q1

and q2 is done by using programmable 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:’) and 75 and 7’2’

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(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. IMPLEMENTING THE 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 , t 2 , t3 and t4 based on equations (18) to

iv) If t24 < 0 program the timers to count as follows: q1 timer is first loaded with t,l = t3 and after with tLl = tl +t4; q 2 timer is first loaded with tc2 = t3 +t4 and after with tb2 = tl f t z .

v) If t24 >_ 0 program the timers to counts as follows: q1 timer is first loaded with t,l = tz + t3 and after with tLl = tl + t4; 42 timer is first loaded with tc2 = t3 and after with tL2 = t 2 + tl.

The time intervals tell tLl, tc2 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. DELTA CONNECTION PWM

For the delta connection voa = 0, v,,p = -E/&, Vea = m E and vep = 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 ; ~ ) in terms of v z A and v i 4 (crp voltages for the delta connection). Using matrix A it can be shown that v : ~ and v i A 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 al p=O-l

I 0 I

'0 0.2 0.4 0.6 0.8 rn

p=o.5 o j ~ = 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 RESULTS

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 x = 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 Thdp as follows

TH D = 4 Tldu + 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

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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 THD when four vectors are used. To maintain the same switching frequency, the period T used for the THD 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 . EXPERIMENTAL RESULTS

The proposed modulation scheme was implemented in a microcomputer-based FSTPI drive system. The FSTPI employs IGBTs that switches at 5kHz and supplies a 1/3hp three-phase squirrel induction motor. The 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 stator 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 the delta 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 method 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. The paper also presented PWM strategies suitable to applied with delta connections of the machine windings. That 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. In Conf. Rec. EPE, 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 modulation 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.

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