Output voltages of a multiswitch inverter

Prof. B.H. Smith, B.E., Ph.D., C.Eng., M.I.E.E., andW.H. Charlton, B.E., Ph.D., C. Eng., M.I.E.E.

Indexing terms: Power electronics, Switching theory

Abstract: The well known 3-phase inverter incorporates three pairs of power switching devices. When thenumber of phases exceeds three, there is the possibility of obtaining a phase-balanced system from less switchpairs than the number of phases. With particular reference to multiphase machines, this paper examines theeffect of switch-pair elimination on voltage balance and switch loading. Graphical methods of interpretationare given, together with examples.

1 Introduction

Static conversion from unidirectional sources to balancedpolyphase a.c. supplies is normally accomplished by theprovision of one changeover switch per phase. Such change-over switches comprise a pair of thyristors or transistors,together with associated flywheel diodes and control andprotection circuitry, and constitute the major cost of theconvertor.

For the general case of S such switches, there arepotentially

n =S\

(s-2)!2! 0)

line—line voltages available, possibly but not necessarilyoccurring in balanced subsets. For the particular case of thefamiliar 3-phase supply, S = 3 and n = 3; but for larger S,n>S and it becomes possible to obtain a phase-balanced£-phase supply from S switches, where Q > S.

2 Voltages available

To evaluate the possibilities of such a situation, it is helpfulto represent polyphase inverter operation by the diagram ofFig. 1. The S switches may be represented by contactspositioned around the circumference of the drum at thepoints P1 -+Ps- If the drum completes one revolution in atime T equal to the period of one complete cycle of theinverter, the potential at each contact will comprise asquare wave of amplitude V and period r. The relative time'phase' between any two such output voltages will be equalto the relative angular positions of these output points on

Fig. 1 Schematic model of polyphase inversion

Paper 586B, received 6th September 1979Prof. Smith and Dr. Charlton are with the Department of ElectricalEngineering, University of Wollongong, N.S.W. 2500, Australia

IEE PROCEEDINGS, Vol. 127, Pt. B, No. 2, MARCH 1980

the drum, and hence the instantaneous potential differencebetween any two switch points Pi and Pj will be V-xi asshown in Fig. 2, in which the angle 6 is equal to the anglesubtended at the centre of the drum by the chord PfPj.Furthermore, the phase of the fundamental component ofthe voltage Vy may be represented by the perpendicularbisector of the chord PjPj.

If the voltages Vy and Vjk derived from switching pointsPi, Pj and Pk are to form part of a phase-balanced (?-phasesystem, then from Fig. 1

= k 2TT/Q (2)

where k is an integer. Hence the phase displacement betweenthe two points Pt and Pk given by

oc = 2TT-2/3 = (Q-2k)2n/Q (3)

is also an integral multiple of 2ir/Q. It follows that, if aphase-balanced Q-phase system is to be developed using Sswitches, where S < Q, the set S must be a subset takenfrom Q switches having relative operating phases of 2n/Q.

Restricting Q to be an odd number, the line—line voltagepotentially available from an unmodulated inverter havingproducing outputs at two levels only, and having successiverelative operating phases of 2n/Q, will comprise (Q —1)/2separate (?-phase sets, denoted GlfG2,. . ., G\Q_ t ) /2 andbalanced with respect to both phase and amplitude of thefundamental frequency components. Amplitudes of thelatter are proportional to sin(Q — l)n/2Q, sin (Q — 3)•n/2Q, . . ., sin 2n/Q, respectively. Each individual voltagewill be of the form shown in Fig. 2.

Elimination of one switching point only removes twoline—line voltages from each of the (Q — l)/2 setsGl}. . . G(Q_ 1 ) / 2 . Elimination of subsequent switching

I9

•2V

-2V

2ti

Fig. 2 Waveform of typical line-line voltage

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points removes an additional one or two line—line voltagesfrom particular sets, depending on whether or not thesesubsequent switching points are dependent on orindependent of the previously eliminated points as far as aparticular set G is concerned. Thus, when S <Q, althoughphase balance may be retained, amplitude balance of thefundamental frequency components across the whole of theQ phases clearly is precluded, but is possible within subsets.

For some applications, such as inverter-fed a.c. motors,the small differences in fundamental frequency componentamplitudes may be compensated by adjusting the number ofturns in the corresponding winding.

3 Selection of voltages

Practical considerations suggest the following:(a) As many voltages as possible from the set Gi should

be used, to make best use of the available supply voltage.(b) The output should comprise the sum of two or more

balanced subsets, to minimise the effect of harmonics andto supply the machine winding configuration.

(c) Each switch should have equal loading, to simplifyboth switch and machine winding design.

Constraint (b) requires the elimination of switches in apattern having rotational symmetry; as a consequence, theyare independent of each other with respect of G\, so thatonly Q —2(Q -S) = 2S-Q of the set Gt will remain. Ifthese 2S —Q voltages are to form a balanced subset of Q,then

2S-Q =

or

S =_ (l +K

Q (4)

where A T 1 = 3 , 5 , 7 , . . . This condition cannot be satisfiedif Q is prime.

If constraint (c) is to be satisfied, Q —S circuits having2(Q — S) connections must be evenly distributed among Sswitches; thus

2(Q-S) = K2S

where K2 is an integer, or

S = 2Q/(K2 +2) (5)

Constraint eqns. 4 and 5 are satisfied simultaneously onlywhen S/Q = 2/3.

4 Typical examples

4.1 Nine phases

These can be derived from S = 2 x 9/3 = 6 switches forwhich « = 5 x 3 = 15 possible voltages. If the successiverelative switching delays are given by 1/9—2/9—1/9—2/9 . . . , or the switch points Pi -*- P6 are set out as shownin Fig. 3, a set of nine phase-balanced line voltages will beobtained as shown in Fig. 4. Machine winding connectionsare shown in Fig. 5. Note that, because the relative phasesof the fundamental components of the voltages are given bythe angles between the perpendicular bisectors of theappropriate chords, and hence by the relative anglesbetween the chords themselves, and because the funda-mental voltage magnitudes are proportional to sin (6^/2),all the relevant information concerning phase, voltage

magnitude, inverter timing and machine winding configura-tion is given by Fig. 3.

4.2 Fifteen phases

To satisfy constraint eqns. 4 and 5, ten switches (n = 45)are required. With switching intervals of 1/15—2/15—1/15—2/15 . .. etc., a set of 15 phase-balanced voltages isobtained which comprises a balanced 5-phasor set with6 = 147r/15, and two additional balanced 5-phasor sets forwhich 6 = 47r/5.

If condition 5 is relaxed, nine switches (n = 36) may beused to obtain a range of combination of subsets which canprovide a 15-phase phase-balanced set of voltages, e.g.

6 x (0 = 14TT/15)P1US 9 x (0 = 2TT/3)

or6 x (0 = 14TT/15) plus 3 x (0 = 4TT/5)

plus 6 x (0 = 2TT/3)

etc.

Air/9

Fig. 3 Graphical display of magnitude, phase and switch timingfor nine phase-balanced voltages

r26

Fig. 4 Phasor representation of fundamentals of voltages in Fig. 3

Two amplitude-balanced sets

118 IEE PROCEEDINGS, Vol. 127, Pt. B, No. 2, MARCH 1980

5 Harmonics

When a balanced (?-phase p-pole-pair double-layer windingis fed with a balanced set of 0-phase voltages of the formshown in Fig. 2, the contribution to the /ith harmoniccomponent of the circumferential current-density distribu-tion due to the qth phase band is given by.1

(6)exp {j{hcot — nd)}

The resultant n-pole-pair space-harmonic contribution dueto the /ith time harmonic when all phases are excited by abalanced supply becomes1

"

[forn = p(h±2QK)]

w h e r e K is an in teger 0 , 1 , 2 . . .

(7)

Fig. 5 Connections for 9-phase machine

Turns-balanced in groups (W14, W36, WS2) and (Wl5, W26, W31, W42,W)

One of the advantages of a polyphase supply arises fromthe fact that, as Q increases, the number of significant spaceharmonics decreases in accordance with eqn. 7. Further-more, the contribution that a particular time harmonicmakes to the losses tends to decrease as Q increases, whereasits contribution to useful torque tends to increase with Q.

When the Q-phase supply comprises two or more balancedsubsets, eqn. 6 still applies; but the /ith harmoniccomponent of the applied voltage waveform is

SVVh = — sin hd/2 (8)

so that the number of turns in each phase winding must beadjusted such that N a s m 0/2. The product ^N will thenremain constant and eqn. 7 then holds, just as fora balanced(2-phase supply, for the fundamental component only.

However, it is clear from eqn. 8 that the harmonicsQi =£ 1) are not necessarily balanced with respect tomagnitude. Consequently, residual harmonic componentscorresponding to a balanced supply of lower-phase-numbersystems will be present.

For example, the 9-phase supply using six switches willhave, in addition to the harmonics arising from a balancedg-phase supply with 6 = 27r/3, those harmonics which arisefrom a 3-phase supply having harmonic amplitudes whichare effectively the difference between those arising from a3-phase 8n/9 system and those of a 3-phase 27r/3 system,after correction for the effect of the modified turns ratio. Itis a simple matter to show that, for two groups 1 and 2,

IhlNx _ sinfl2/2 sin hOt/2

Ih2N2 SU10J2 sin hd 2/2(9)

By substituting the difference between 7/,!^! and/h2A^2 intoeqn. 7 with Q = 3 these residual harmonics can be found.

3 Reference

1 MCLEAN, G.W., NIX, G.F., and ALWASH, J.R.: 'Performanceand design of induction motors with square-wave excitation'Proc. IEE, 1969, 116, (8), pp. 1405-1411

IEE PROCEEDINGS, Vol. 127, Pt. B, No. 2, MARCH 1980 119