Abstract:
A multi-phase transformer system is provided having a main transformer fed by an N-phase voltage and a separate auxiliary transformer fed by the N-phase voltage. Windings in the main transformer are connected to secondary windings in the auxiliary transformer to provide pairs of connected windings. Each pair of connected windings has one of the windings of the main transformer and one of the secondary windings of the auxiliary transformer. The windings in such connected pair are arranged to produce voltages having different phases with each pair of windings producing an output voltage equal to the vector sum of the voltages produced by the such connected pair of windings. With such an arrangement, by having two separate transformers, i.e., the main transformer and the auxiliary transformer, fabrication of a multi-phase transformer system is simplified. Further, leakage inductance in the auxiliary transformer may be readily adjusted and increased to thereby suppress higher harmonic distortion without the need for an additional filter. In one embodiment, secondary windings of the main transformer are connected to the secondary windings of the auxiliary transformer. In such embodiment, each pair of connected windings includes one of the secondary windings of the main transformer and one of the secondary windings of the auxiliary transformer. In a second embodiment, the N-phase voltage is connected directly to the primary winding of the auxiliary transformer and indirectly to the primary windings of the main transformer through the secondary windings of the auxiliary transformer. In such embodiment, each pair of windings includes one of the secondary winding of the auxiliary transformer and one of the primary windings of the main transformer.

Description:
This invention relates generally to multi-phase transformer systems and more particularly to multi-phase power transformer systems having improved phasor balance and reduced total harmonic distortion (THD). 
     As is known in the art, many electrical systems require direct current power. Such direct current (DC) is typically produced by rectifying three-phase alternating current (AC) voltage. The rectifiers, however, induce harmonic distortion in the input line. Such effect is described in U.S. Pat. No. 4,779,181 entitled “Multiphase Low Harmonic Distortion Transformer”, inventors Traver et al., issued Oct. 18, 1988. The total harmonic distortion (THD) generated by rectification can be improved by increasing the number of AC phases fed to the rectifiers. Some of these multi-phase transformer systems are described in the U.S. Pat. Nos.: 4,779,181, 4,255,784, 5,148,357, 4,532,581, and 4,488,211. The line harmonics for these systems are inversely proportional to the number of phases according to the following equation: 
     
       
         K H =2*m*(n+/−1), 
       
     
     where K H  is the harmonic order 
     m is the number of phases 
     n =0, 1,2, . . . 
     For example, the harmonics of a 12-phase system are:  23 ,  25 ,  47 ,  49 ,  71 ,  73 . 
     A schematic diagram for a conventional 18-phase, single transformer is shown in FIGS. 1A and 1B. Thus, the transformer  10  has a three-phase primary winding  12  magnetically coupled to a secondary winding section  16  through a core  14 . The secondary winding section  16  has a set of six main Y-configured, three-phase secondary windings  16   a - 16   f . The voltage produced in the three secondary windings of set  16   a  are {overscore (A)}, {overscore (B)} and {overscore (C)} where {overscore (A)}, {overscore (B)} and {overscore (C)} have equal magnitudes and 120 degrees of relative phase shift with respect to each other. The voltages produced in the three secondary windings of set  16   b  are K 1  {overscore (A)}, K 1 {overscore (B)} and K 1 {overscore (C)} where K 1  is a less than one. Thus, the number of turns in each of the three windings in set  16   a  are equal to each other and the number of turns set  16   b  are equal to each other the number of turns in the three sets of windings in set  16   b  are a fraction of the number of turns in the three windings in set  16   a . Thus, the voltages K 1 {overscore (A)}, K 1 {overscore (B)} and K 1 {overscore (C)} have equal magnitudes, here 1/K 1  th the voltage in each of the windings set  16 , and 120 degrees of relative phase shift with respect to each other. That is, the voltages K 1 {overscore (A)}, K 1 {overscore (B)} and K 1 {overscore (C)} are in-phase with the voltages {overscore (A)}, {overscore (B)}, and {overscore (C)} in set  16   a . In like manner, the voltages in sets  16   c  through  16   f  are: K 2 {overscore (A)}, K 2 {overscore (B)} and K 2 {overscore (C)}; K 3 {overscore (A)}, K 3 {overscore (B)} and K 3 {overscore (C)}; K 4 {overscore (A)}, K 4 {overscore (B)} and K 4 {overscore (C)}; and K 5 {overscore (A)}, K 5 {overscore (B)} and K 5 {overscore (C)}, respectively, where K 3 =1, K 1 =K 4 , K 2 =K 5 , K 2 &lt;K 1  and such relationship is determined by the relative number of turns in the windings. 
     The secondary section  16  also includes six sets  16   g - 16   l  of auxiliary windings magnetically coupled to the primary  12  though core  14 . Each set has three windings. Each one of the windings in the set produces a voltage in-phase with a corresponding one of the three voltage {overscore (A)}, {overscore (B)}, and {overscore (C)}. The magnitudes of the voltages in sets  16   g - 16   l  are scaled relatively to the magnitudes of the voltages {overscore (A)}, {overscore (B)}, and {overscore (C)} by factors of: 1/K 6  through 1/K 11 , respectively. It is noted that the windings in sets  16   a  through  16   f  are connected to the windings in sets  16   g  through  16   l  selectively as shown to thereby produce voltages V O1  through V O18  which may be represented as: {overscore (C)}+K 6 {overscore (A)}; 
     {overscore (A)}+K 6 {overscore (B)}; 
     {overscore (B)}+K 6 {overscore (C)}; 
     K 1 {overscore (C)}+K 7 {overscore (A)}; 
     K 1 {overscore (A)}+K 7 {overscore (B)}; 
     K 1 {overscore (B)}+K 7 {overscore (C)}; 
     K 2 {overscore (C)}+K 8 {overscore (A)}; 
     K 2 {overscore (A)}+K 8 {overscore (B)}; 
     K 2 {overscore (B)}+K 8 {overscore (C)}; 
     K 3 {overscore (C)}+K 9 {overscore (A)}; 
     K 3 {overscore (A)}+K 9 {overscore (B)}; 
     K 3 {overscore (B)}+K 9 {overscore (C)}; 
     K 4 {overscore (C)}+K 10 {overscore (A)}; 
     K 4 {overscore (A)}+K 10 {overscore (B)}; 
     K 4 {overscore (B)}+K 10 {overscore (C)}; 
     K 5 {overscore (C)}+K 11 {overscore (A)}; 
     K 5 {overscore (A)}+K 11 {overscore (B)}; 
     K 5 {overscore (B)}+K 11 {overscore (C)}, respectively. 
     These voltages VO 1  through VO 11  are fed to a rectification system, as shown. The rectified voltages are combined in combiner  20  to produce the here 18-phase combined output voltage, VOUT. 
     SUMMARY 
     In accordance with the present invention, a multi-phase transformer system is provided having a main transformer fed by an N-phase voltage and a separate auxiliary transformer fed by the N-phase voltage. Windings in the main transformer are connected to secondary windings in the auxiliary transformer to provide pairs of connected windings. Each pair of connected windings has one of the windings of the main transformer and one of the secondary windings of the auxiliary transformer. The windings in such connected pair are arranged to produce voltages having different phases with each pair of windings producing an output voltage equal to the vector sum of the voltages produced by the such connected pair of windings. 
     With such an arrangement, by having two separate transformers, i.e., the main transformer and the auxiliary transformer, fabrication of a multi-phase transformer system is simplified. Further, leakage inductance in the auxiliary transformer may be readily adjusted and increased to thereby suppress higher harmonic distortion without the need for an additional filter. The increased leakage inductance of the auxiliary transformer does not cause higher harmonic distortion in the low frequency part of the spectrum that occurs if the leakage inductance of the main transformer is increased. 
     In one embodiment, secondary windings of the main transformer are connected to the secondary windings of the auxiliary transformer. In such embodiment, each pair of connected windings includes one of the secondary windings of the main transformer and one of the secondary windings of the auxiliary transformer. 
     In a second embodiment, the N-phase voltage is connected directly to the primary winding of the auxiliary transformer and indirectly to the primary windings of the main transformer through the secondary windings of the auxiliary transformer. In such embodiment, each pair of windings includes one of the secondary windings of the auxiliary transformer and one of the primary windings of the main transformer. 
     In accordance with still another aspect of the invention, a multi-phase transformer system is provided having a main transformer and a separate auxiliary transformer. The main transformer includes a main secondary winding section magnetically coupled to a main primary winding section. One of the winding sections of the main transformer includes a plurality of M sets of main windings, where M is an integer greater than one. Each one of the M sets has a plurality of N main windings for producing N voltages having the same amplitudes and a predetermined phase relationship. The amplitudes of the voltages produced by one of the sets are different from the amplitude of the voltages produced by another one of the sets. The auxiliary transformer includes an auxiliary primary winding section having inputs connected to the main transformer. The auxiliary transformer includes a plurality of M auxiliary secondary winding sets magnetically coupled to an auxiliary primary winding section. Each one of the M sets of auxiliary secondary winding sets is connected to a corresponding one of the M sets of main windings. Each one of the auxiliary secondary windings in each one of the M sets thereof produces N voltages having the predetermined phase relationship. The amplitudes of the voltage produced in each one of the M sets of auxiliary secondary windings are equal. The amplitudes of the voltages produced in one of the M sets of auxiliary secondary windings are different from the amplitudes of the voltages produced in another one of the sets M sets of auxiliary secondary windings. Each one the windings in each one of the sets M sets of auxiliary secondary windings is connected to a corresponding one of the windings in the one of the M sets of main windings to form a pair of connected windings. The windings in the connected pair produce voltages having different amplitudes and phases. Each one of the connected pair of windings produces an output voltage equal to the vector sum of the voltages produced by the connected pair of windings. 
     In one embodiment, the M sets of main windings are secondary windings of the main transformer. In such embodiment, each pair of connected windings includes one of the secondary windings of the main transformer and one of the secondary windings of the auxiliary transformer. 
     In a second embodiment, the M sets of main windings are primary windings of the main transformer. In such embodiment, each pair of windings includes one of the secondary winding of the auxiliary transformer and one of the primary windings of the main transformer. 
     The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 
    
    
     DESCRIPTION OF DRAWINGS 
     FIGS. 1A and 1B is a schematic diagram of a multiphase transformer according to the PRIOR ART; and 
     FIG. 2 is a schematic diagram of a multiphase transformer according to the invention. 
     FIG. 3 is a schematic diagram of a multiphase transformer according to another embodiment of the invention. 
     FIG. 4 is a phasor diagram for an 18-phase voltage. 
    
    
     Like reference symbols in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     Referring now to FIG. 2, a multi-phase transformer system  100  is provided here arranged to provide an eighteen-phase output voltage. The transformer system  100  includes a pair of main transformers  102 ,  104  and an auxiliary transformer  106 . The pair of main transformers  102 ,  104  are identical in construction except as described below, and each includes a primary winding section  108 ,  110 , respectively, here having three main primary windings arranged in a delta-configuration, as shown, and connected to the same, three-phase, AC voltages V A , V B  and V C . Each one of the main transformers includes a magnetic core  112 ,  114 , respectively, having a secondary winding section  116 ,  118  magnetically coupled to the primary winding sections  108 ,  110 , respectively, through the cores  112 ,  114 , respectively, as shown. The main secondary winding sections  116 ,  118  each has a plurality of M sets of main secondary windings  120   a ,  120   b ,  120   c  for main transformer  102 ; and sets  120   d ,  120   e ,  120   f  for main transformer  104 , respectively, as shown, where M is an integer greater than one. Here, M is three. Each one of the M sets  120   a ,  120   b ,  120   c  of main transformer  102  has a plurality of N main secondary windings, where here N is three. The windings in each set produces N voltages having the same amplitudes and a predetermined phase relationship, here 360 degrees/N or 120 degrees. The amplitudes of the voltages produced by one of the sets are different from the amplitudes of the voltages produced by another one of the sets. Thus, considering one of the two main transformers  102 ,  104  here main transformer  102 , the voltages produced by set  120   a  are {overscore (A)}, {overscore (B)} and {overscore (C)} where {overscore (A)}, {overscore (B)} and {overscore (C)} have equal magnitudes and 120 degrees of relative phase shift with respect to each other. The voltages produced in the three secondary windings of set  120   b  are K 1 {overscore (A)}, K 1 {overscore (B)} and K 1 {overscore (C)} where K 1  is a less than one. Thus, the number of turns in each of the three windings in set  102   a  are equal to each other and the number of turns set  102   b  are equal to each other the number of turns in the three sets of windings in set  102   b  are a fraction of the number of turns in the three windings in set  102   a . Thus, the voltages K 1 {overscore (A)}, K 1 {overscore (B)} and K 1 {overscore (C)} in set  102   b  have equal magnitudes, here 1/K 1  th the voltage in each of the windings in set  102   a , and 120 degrees of relative phase shift with respect to each other. That is, the voltages K 1 {overscore (A)}, K 1 {overscore (B)} and K 1 {overscore (C)} are in-phase with the voltages {overscore (A)}, {overscore (B)}, and {overscore (C)} in set  102   a . In like manner, the voltages in set  102   c  are: K 2 {overscore (A)}, K 2 {overscore (B)} and K 2 {overscore (C)}. Considering the second main transformer  104 , the voltages in the first set  120   d  of secondary windings are: K 3 {overscore (A)}, K 3 {overscore (B)} and K 3 {overscore (C)}. In like manner, the voltages in the second set  102   e  of secondary windings are K 4 {overscore (A)}, K 4 {overscore (B)} and K 4 {overscore (C)} and the voltages in the third set  102   f  of windings are: K 5 {overscore (A)}, K 5 {overscore (B)} and K 5 {overscore (C)} where angles α1-α4 are as shown in FIG. 4, and coefficients K 1 -K 5  are given by expressions below:          k   1     =       cos                   α   2     *     (     1   -       tan                   α   2         tan                 60      °         )         cos                   α   1     *     (     1   -       tan                   α   1         tan                 60      °         )                   k   2     =       cos                   α   3     *     (     1   -       tan                   α   3         tan                 60      °         )         cos                   α   2     *     (     1   -       tan                   α   2         tan                 60      °         )                     K   3     =   1     ,                  K   4     =     K   1       ,                  K   5     =     K   2       ,                  and                   K   1       &gt;       K   2     .                              
     The auxiliary transformer  106  has an auxiliary primary winding section  140 . The auxiliary primary winding section  140  has three primary windings here arranged in a delta-configuration, as shown, connected to the AC voltage V A , V B , and V C , which is fed to the primary windings  108 ,  110  of the main transformers  102 ,  104 . The auxiliary transformer  106  has a secondary winding section  142  magnetically coupled to the primary winding section  140  through core  144 . The secondary winding section  142  includes a plurality of M auxiliary winding sets,  120   g  through  120   l , magnetically coupled to the auxiliary primary winding section  140  through core  144  of the auxiliary transformer  106 . Each one of such M sets  120   g  through  120   l , is connected to a corresponding one of the M sets  120   a  through  120   f , of main secondary windings, respectively, as shown. Each one of the auxiliary windings in each one of the M sets  120   g  through  120   l  thereof produces N voltages having the predetermined phase relationship, here the 120 degree phase relationship. The amplitudes of the voltage produced in each one of the sets  16 ′ g  through  16 ′ l  thereof are equal to each other. The amplitudes of the voltages produced in one of the sets  120 ′ g  through  120 ′ l  are different from the amplitudes of the voltages produced in another one of the sets  120 ′ g  through  120 ′ l  thereof. Thus, the voltages produced in set  120   g  are: K 6 {overscore (A)}; K 6 {overscore (B)}; and K 6 {overscore (C)}, where K 6  is an integer less than K 5 . In like manner, the voltages produced in set  120   h  through  120   l  are: K 7 {overscore (A)}, K 7 {overscore (B)}, and K 7 {overscore (C)}; K 8 {overscore (A)}, K 8 {overscore (B)}, and K 8 {overscore (C)}; K 9 {overscore (A)}, K 9 {overscore (B)}, and K 9 {overscore (C)}; K 10 {overscore (A)}, K 10 {overscore (B)}, and K 10 {overscore (C)}; and, K 11 {overscore (A)}, K 11 {overscore (B)}, and K 11 {overscore (C)}, respectively. Coefficients K 6 -K 11  are given by expressions below:          k   6     =       tan                   α   1         sin                 60      °   *     (     1   -       tan                   α   1         tan                 60      °         )                   k   7     =       tan                   α   2         sin                 60      °   *     (     1   -       tan                   α   2         tan                 60      °         )                   k   8     =       tan                   α   3         sin                 60      °   *     (     1   -       tan                   α   3         tan                 60      °         )                                
     K 9 =K 6 , 
     K 10 =K 7    
     K 11 =K 8    
     Each one of the windings in each one of the sets  120   g  through  120   l  is connected to a corresponding one of the windings in the one of the sets  120   a  through  120   f  of two main secondary windings corresponding thereto to form a pair of connected windings. Thus windings in the connected pair produce voltages that have different amplitudes and phases. The auxiliary secondary winding in such connected pair produce an output voltage equal to the vector sum of the voltages produced by the main secondary windings and the auxiliary secondary winding in such connected pair of windings. 
     More particularly, the voltage produced by the auxiliary secondary windings in sets  120   g  through  120   l  i.e., the output voltages V O1  through V O18 , as indicated, such output voltages being represented as: 
     {overscore (C)}+K 6 {overscore (A)}; 
     {overscore (A)}+K 6 {overscore (B)}; 
     {overscore (B)}+K 6 {overscore (C)}; 
     K 1 {overscore (C)}+K 7 {overscore (A)}; 
     K 1 {overscore (A)}+K 7 {overscore (B)}; 
     K 1 {overscore (B)}+K 7 {overscore (C)}; 
     K 2 {overscore (C)}+K 8 {overscore (A)}; 
     K 2 {overscore (A)}+K 8 {overscore (B)}; 
     K 2 {overscore (B)}+K 8 {overscore (C)}; 
     K 3 {overscore (C)}+K 9 {overscore (B)}; 
     K 3 {overscore (A)}+K 9 {overscore (C)}; 
     K 3 {overscore (B)}+K 9 {overscore (A)}; 
     K 4 {overscore (C)}+K 10 {overscore (B)}; 
     K 4 {overscore (A)}+K 10 {overscore (C)}; 
     K 4 {overscore (B)}+K 10 {overscore (A)}; 
     K 5 {overscore (C)}+K 11 {overscore (B)}; 
     K 5 {overscore (A)}+K 11 {overscore (C)}; 
     K 5 {overscore (B)}+K 11 {overscore (A)}; respectively. 
     Here, in this example, there are 45 turns on the secondary windings  120   a  of each one of the windings thereof. Likewise, there are 45 turns on the secondary windings  120   d  of each one of the windings thereof. Further, here K 1 =39/45; K 2 =32/45; K 3 =1; K 4 =K 1 ; K 5 =K 2 ; K 6 =−5/45; K 7 =−14/45; K 8 =−23/45; K 9 =K 6 ; K 10 =K 7 ; and, K 11 =K 8 ; where the negative sign (−) indicates an opposite sense in the direction of the winding. 
     These voltages V O1  through V O18  are fed to a rectification system  140 , as shown. The rectified voltages are combined in combiner  142  to produce the here 18-phase combined output voltage, VOUT. 
     It is noted that the leakage inductance in the secondary winding section of the auxiliary transformer may be increased compared with the leakage inductance of the secondary windings sections of the two main transformers by, here for example, increasing the separation between the secondary windings in the auxiliary transformers from the primary windings thereof compared to the separations in the two main transformers 
     Thus, with the multi-phase transformer described above in connection with FIG. 2, the construction, cooling, and internal connections of such transformer are greatly simplified because all transformers are three-phase units and their phase-to-phase connections are external. Conventional off the shelf transformers may be used if the form factor and cooling requirements are satisfied. Further, because the power transfer and phase-shifting functions are separated among the main and auxiliary transformers, the transformer leakage inductances of both the main and auxiliary transformers can be independently adjusted to compensate for voltage imbalance caused by the non-ideal turns ratio. Finally, the leakage inductance of either the auxiliary or main transformers can be increased to perform the function of an external reactor. It can be done without affecting resultant secondary voltages at the rectifier inputs because neither unit has leg-to-leg cross coupling of leakage flux. For example, the phase-shifting transformer leakage inductance can be made arbitrarily large while the main transformers would remain tightly coupled. As indicated above, leakage inductance can be adjusted to balance the resultant secondary voltages. Because the auxiliary transformer provides line reactance to the rectifiers, its coupling factor is low (Ka=0.999). As noted above, it is possible to adjust the leakage inductances of each phase thereby correcting the phasor imbalance caused by the transformer turns ratios. 
     Referring now to FIG. 3, a multi-phase transformer system  100 ′ is provided here arranged to provide an eighteen-phase output voltage. The transformer system  100 ′ includes six main transformers  102   a ′,  102   b ′,  102   c ′,  104   a ′,  104   b ′ and  104   c ′ and an auxiliary transformer  106 ′. The main transformers  102   a ′,  102   b ′,  102   c ′,  104   a ′,  104   b ′ and  104   c ′ are identical in construction except as described below, and each includes a primary winding section  108   a ′.  108   b ′,  108   c ′,  110   a ′,  110   b ; and  110   c ; respectively as indicated, each one being and connected to the same, three-phase, AC voltages V A , V B  and V C  through secondary windings of the auxiliary transformer  106 ′, as shown. Each one of the main transformers includes a magnetic core  112   a ′,  112   b ′,  112   c ;  114   a ′,  114   b ′ and  114   c ; respectively, each having a delta configured secondary winding set  116   a ′,  116   b ′,  116   c ;  118   a ′,  118   b ′,  118   c ; respectively, as indicated magnetically coupled to the primary winding set  108   a ′,  108   b ′,  108   c ′,  110   a ′,  110   b ; and  110   c ; respectively as indicated, respectively, through the cores  112   a ′,  112   b ′,  112   c ;  114   a ′,  114   b ′,  114   c ; respectively, as shown. Each one of the secondary winding sets  116   a ′,  116   b ′,  116   c ;  118   a ′,  118   b ′,  118   c  has a plurality of M main secondary windings, where M is an integer greater than one. Here, M is three. Each one of the primary winding sets  108   a ′,  108   b ′,  108   c ′,  110   a ′,  110   b ; and  110   c  has a plurality of N main primary windings, where here N is three. The windings in each set produce N voltages having the same amplitudes and a predetermined phase relationship, here 120 degrees. The amplitudes of the voltages produced by one of the sets are different from the amplitudes of the voltages produced by another one of the sets. Thus, considering one of the six main transformers  102   a ′,  102   b ′,  102   c ′,  104   a ′,  104   b ′ and  104   c ′ here main transformer  102   a ′, the voltages produced by set  108   a ′ are {overscore (A)}, {overscore (B)} and {overscore (C)} where {overscore (A)}, {overscore (B)} and {overscore (C)} have equal magnitudes and 120 degrees of relative phase shift with respect to each other. The voltages produced in the three primary windings of set  108   b ′ are K 1 {overscore (A)}, K 1 {overscore (B)} and K 1 {overscore (C)} where K 1  is a less than one. Thus, the number of turns in each of the three windings in set  108   a ′ is equal to each other. The number of turns in set  108   b ′ is equal to each other. The number of turns in the three windings in set  108   b ′ is a fraction of the number of turns in the three windings in set  108   a ′. Thus, the voltages K 1 {overscore (A)}, K 1 {overscore (B)} and K 1 {overscore (C)} in set  108   b ′ have equal magnitudes, here 1/K 1  th the voltage in each of the windings in set  108   a ′, and 120 degrees of relative phase shift with respect to each other. That is, the voltages K 1 {overscore (A)}, K 1 {overscore (B)} and K 1 {overscore (C)} are in-phase with the voltages {overscore (A)}, {overscore (B)} and {overscore (C)} in set  108   a . In like manner, the voltages in set  108   c ′ are: K 2 {overscore (A)}, K 2 {overscore (B)} and K 2 {overscore (C)}. Considering the fourth main transformer  104   a ′, the primary voltages in the set  110   a ′ are K 3 {overscore (A)}, K 3 {overscore (B)} and K 3 {overscore (C)}. In like manner, the voltages in the fifth set  110   b ′ are K 4 {overscore (A)}, K 4 {overscore (B)} and K 4 {overscore (C)} and the voltages in the sixth set  110   c ′ of windings are: K 5 {overscore (A)}, K 5 {overscore (B)} and K 5 {overscore (C)} where K 3 =1, K 1 =K 4 , K 2 =K 5 , K 2 &lt;K 1  and such relationship is determined by the relative number of turns in the windings. 
     The auxiliary transformer  106  has an auxiliary primary winding section  140 . The auxiliary primary winding section  140  has three primary windings here arranged in a star-configuration, as shown, connected to the AC voltage V A , V B , and V C . The auxiliary transformer  106  has a secondary winding section  142  magnetically coupled to the primary winding section  140  through core  144 . The secondary winding section  142  includes a plurality of M auxiliary winding sets,  120   g  through  120   l , magnetically coupled to the auxiliary primary winding section  140  through core  144  of the auxiliary transformer  106 . Each one of such M sets  120   g ′ through  120   l ′, is connected to a corresponding one of the M sets  108   a ′,  108   b ′  108   c ′,  110   a ′,  110   b ′,  110   c ′, of main primary windings, respectively, as shown. Each one of the auxiliary windings in each one of the M sets  120   g  through  120   l  thereof produces N voltages having the predetermined phase relationship, here the 120 degree phase relationship. The amplitudes of the voltage produced in each one of the sets  120   g ′ through  120   l ′ thereof are equal to each other. The amplitudes of the voltages produced in one of the sets  120   g ′ through  120   l ′ are different from the amplitudes of the voltages produced in another one of the sets  120   g ′ through  120   l ′ thereof. Thus, the voltages produced in set  120   g ′ are: K 6 {overscore (A)}; K 6 {overscore (B)}; and K 6 {overscore (C)}, where K 6  is an integer less than K 5 . In like manner, the voltages produced in set  120   h ′ through  120   l ′ are: K 7 {overscore (A)}, K 7 {overscore (B)}, and K 7 {overscore (C)}; K 8 {overscore (A)}, K 8 {overscore (B)}, and K 8 {overscore (C)}; K 9 {overscore (A)}, K 9 {overscore (B)}, and K 9 {overscore (C)}; K 10 {overscore (A)}, K 10 {overscore (B)}, and K 10 {overscore (C)}; and, K 11 {overscore (A)}, K 11 {overscore (B)}, and K 11 {overscore (C)}, respectively. 
     Each one of the windings in each one of the sets  120   g ′ through  120   l ′ is connected to a corresponding one of the windings in the one of the sets  108   a ′,  108   b ′,  108   c ′,  110   a ′,  110   b ′,  110   c ′ of six main secondary winding sets corresponding thereto to form a pair of connected windings. Thus windings in the connected pair producing voltages have different amplitudes and phases. The resultant output voltage equals to the vector sum of the voltages produced by the main primary winding and the auxiliary secondary winding in such connected pair of windings. 
     More particularly, the voltage produced by the auxiliary secondary windings in sets  120   g ′ through  120 ′ l  i.e., the output voltages V O1  through V O18 , as indicated, such output voltages being represented as: 
     {overscore (C)}+K 6 {overscore (A)}; 
     {overscore (A)}+K 6 {overscore (B)}; 
     {overscore (B)}+K 6 {overscore (C)}; 
     K 1 {overscore (C)}+K 7 {overscore (A)}; 
     K 1 {overscore (A)}+K 7 {overscore (B)}; 
     K 1 {overscore (B)}+K 7 {overscore (C)}; 
     K 2 {overscore (C)}+K 8 {overscore (A)}; 
     K 2 {overscore (A)}+K 8 {overscore (B)}; 
     K 2 {overscore (B)}+K 8 {overscore (C)}; 
     K 3 {overscore (C)}+K 9 {overscore (B)}; 
     K 3 {overscore (A)}+K 9 {overscore (C)}; 
     K 3 {overscore (B)}+K 9 {overscore (A)}; 
     K 4 {overscore (C)}+K 10 {overscore (B)}; 
     K 4 {overscore (A)}+K 10 {overscore (C)}; 
     K 4 {overscore (B)}+K 10 {overscore (A)}; 
     K 5 {overscore (C)}+K 11 {overscore (B)}; 
     K 5 {overscore (A)}+K 11 {overscore (C)}; 
     K 5 {overscore (B)}+K 11 {overscore (A)}; respectively. 
     A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. Accordingly, other embodiments are within the scope of the following claims.