Elimination of Common Mode Voltage in Three-To-Six-Phase Matrix Converter

: The matrix converter (MC) is the n-phase input and m-phase output power electronic system. To synthesis the controllable sinusoidal output voltage and input current with controllable input displacement angle, the pulse width modulation method (PWM) is used in the MC. During the modulation process a problem of the common mode voltage (CMV) exists. The elimination of the CMV in three-to-six-phase MC by usage of only rotating voltage space vectors is analyzed in the paper. The carrier based implementation of the space vector modulation (SVM) with Venturini modulation functions is applied to the control of the three-to-six-phase MC. Entire elimination of the CMV in three-to-six-phase MC is presented in the paper. The simulation and experiment results conﬁrm utility of the proposed modulation method.


Introduction
Matrix converter (MC) is a direct AC/AC converter that is used to convert available AC supply into desirable AC outputs with variable voltage and variable frequency. Advantages commonly associated with the MC are: Sinusoidal input current and output voltage, input power factor control, regeneration capability, and compact circuitry. The synthesis of referenced sinusoidal output voltage and sinusoidal input current with controllable input displacement angle is the main goal in the modulation method used in controlling MCs. The control method of m-phase-to-n-phase MC, realizing requested output voltage and demanded input current, was worked out by inventors of the MC and authors of first papers published in eighties of previous century [1,2]. The authors of References [1,2] propose two types of modulation signals used in the carrier based pulse width modulation (PWM) control method. The next, many modulation methods in controlling of the MC, were developed. These methods were analyzed for the three-phase-to-three-phase MC and may be grouped as follows: Space vector modulation methods, scalar modulation methods, predictive control method, and carrier based modulation method.
In the last decade the multiphase topologies of the MC are investigated. The interest of the multiphase structure of the MC was dictated by development of multiphase drives, supplied by back-to-back converters. The MC, compared with the back-to-back converter, offers some advantages, like smaller size, bidirectional power flow and controllable input displacement angle. The multiphase (more than three) motor drives have some inherent advantages over the traditional three phase motor drives. To these advantages belong, among others: Increasing the frequency and reducing the amplitude of torque pulsations, reducing the rotor harmonic currents losses, decreasing acoustic noise, and lowering current per phase. Additionally, the multiphase motor drives improve the system reliability owing to their redundant structure. If one phase of the multiphase machine becomes open

Topology of Multiphase Matrix Converter
The multiphase drive is addressed into high power load. The most frequently considered multiphase machine for high power applications is the asymmetrical six-phase induction machine, that the stator has two sets of three-phase windings, spatially shifted by 30 electrical degrees [4], while the rotor winding is of squirrel cage type and is the same as for a three-phase machine. The six-phase induction machine needs six-phase input voltage source, which would be realized by application of the three-to-six-phase matrix converter. The power circuit topology of the MC (Figure 1) consists of six legs with each leg having three bidirectional power switches, so the total number of the switches is 18. Switching function of bidirectional switch is defined as (1). Taking the number of the switches into account, the total number of switching states combinations is 2 18 . For the MC to operate safely, the following constraints must be adhered to, i.e., the input phases should never be shorted, and the output phases should never be opened at any instant. It means that only one of three switches connecting given output phase with three input phases must be switched on (2). Considering these constraints for the three-to-six-phase MC, the number of valid switching states is 3 6 , i.e., there are 729 different switching combinations for connecting the six output phases to the three input phases.  The relation of the output voltages u o to the input voltages u i , and the input currents i i to output currents i o may be defined in terms of switching states as follows (3,4).
The output voltage in the three-to-six-phase MC can reach up to 75% of the input voltage, which is the physical limit of this MC [9,16]. The modulation method, presented in the paper, assumes the 50% value of voltage transfer ration. This low value of voltage transfer ratio results from the fact, that the presented in the paper modulation method is focused on obtaining entire elimination of the CMV, what demands to apply only rotating voltage space vectors, and it is connected with limitation of the maximum obtainable output voltage in MC [22,24,27].

Output Voltage Space Vectors in Three-to-Six-Phase MC
The output voltages of the three-to-six-phase MC could be presented using space vectors, defined by (5). Each of instantaneous output voltages: u 1 , u 2 , u 3 , u 4 , u 5 and u 6 in the relation (5) is equal, appropriately, one of the input phase voltage: u a , u b or u c . The label xxxxxx in the name → V xxxxxx of the space vector means the name of input phase chosen into output phase 1, 2, 3, 4, 5, and 6, appropriately.
where: a = e j 2π 6 . The switching combinations for connecting the six output phases to the three input phases in the three-to-six-phase MC can be analyzed in seven groups. The switching combinations are named as {x,y,z}, where x, y, and z represent the number of output phases connected to input phase A, phase B, and phase C, respectively.  {2,2,2} To every one input phases the two output phases are connected. One sub-group may have (6!/(2! × 4!)) × (4!/(2! × 2!)) × (2!/(2! × 0!)) = 90 combinations. As such, there is one sub-group and thus the total possible combination will be 1 × 90 = 90 switching combinations. In this group the active voltage vectors exist in the number of 36. Their maximum module is U i /

Output Voltage Space Vectors Reducing CMV in the Three-to-Six-Phase MC
Amongst all 118 rotating space vectors a special remark, in context of the CMV elimination, should be returned on 12 rotating voltage space vectors (Table 1) and 6 zero vectors (Table 2) in the seventh group described in Section 2.2. Only these 12 rotating voltage space vectors and 6 zero vectors (Table 2) could be used in control, realizing the elimination of the CMV in the three-to-six-phase MC. This conclusion is drawn from the analysis of all 729 space vectors, determined with assumption that the three-to-six-phase MC is supplied by balanced input voltage (6). The lay-out of the chosen rotating voltage space vectors in a complex plane is shown in the Figure 2. In Figure 2 the initial position of rotating space vectors is shown.   Table 2. The names and relations defining the zero space vectors resulting in elimination of the CMV.

Six-Phase Load Six-Phase Load with Two Sets of Three-phase Windings Open-End Three Phase Load
Figure2. Lay-out on complex plane of the voltage rotating space vectors, which implementation assure the CMV elimination.
In the presented paper the application of three-to-six-phase MC to supplying six-phase load is investigated. The CMV for the six-phase output voltage of MC is defined in (7). The application of the rotating voltage space vectors (Table1) and zero vectors (  (8) and (9). The modules of the In the presented paper the application of three-to-six-phase MC to supplying six-phase load is investigated. The CMV for the six-phase output voltage of MC is defined in (7). The application of the rotating voltage space vectors (Table 1) and zero vectors (  (8) and (9). The modules of the zero voltage space vectors are equal zero. Nonetheless, these vectors are not typical zero vectors where all output phases are connected to the same input phase like in switching combinations mentioned in group one. Here all input phases participate in shaping of the voltage in each output phases.
Apart from application of the three-to-six-phase MC to the typical six-phase load, this MC may be used to supply the six-phase load which phases are connected in two star points like in the asymmetrical six-phase induction machine, where the stator has two sets of three-phase windings, spatially shifted by 30 electrical degrees [4]. The respective sets of three-phase winding should be supplied by voltages of the three MC output phases with odd numbers and even numbers, appropriately. Taking into account the switching combination of the MC, corresponding to the distinguished rotating voltage space vectors, and the zero voltage space vectors, the space vector → V 135 and → V 246 representing the voltages supplying the load connected in two sets with separated star points, could be calculated as per (10) ( Table 1, Table 2).
The Tables 1 and 2 (11). All these space vectors are rotating vectors providing elimination of the CMV.
The performed analysis of all allowable switching combinations of the three-to-six-phase MC and corresponding voltage space vectors allow to choose the vectors synthesizing sinusoidal output voltage, sinusoidal input current and eliminating the CMV. As a result, the proposed modulation method is based on application solely the switching combinations belonging to seventh group, that correspond to the rotating voltage space vectors with magnitudes equal √ 3U i /2. The rotating space vectors are not usually used in modulation strategies, because they lay in different position, so it is difficult to create a repetitive pattern. However, in References [22,27] one can find that implementation of Venturini modulation function for determination of switch duty cycles in the conventional MC and in the Multilevel Matrix Converter could provide the control with use of only rotating voltage space vectors with elimination of the CMV. By analogy, the Venturini modulation functions are used in proposed strategy for modulation duty cycles in the three-to-six-phase MC.

Proposed Modulation Method
Synthesis of the output voltage with use of rotating voltage space vectors could be realized by applying carrier-based implementation of the SVM. To set out the switch duty cycles S nl the Venturini modulation functions are used in proposed method. In the case of the three-to-six-phase MC with symmetric sinusoidal input voltages (6) and sinusoidal output voltage with output amplitude and angular frequency indicated as U o , ω o , the Venturini modulation functions are denoted in relationships (12,13,14).
It is easy to see that if the MC synthesizes the output and input waveforms with the input displacement angle ϕ i = ϕ o (parameter α 2 = 0) there are six groups consisting of three identical modulation functions (15). The similar situation is when the parameter α 2 = 0, then the synthesized input current has the displacement angle opposite than output phase shift: ϕ i = −ϕ o , and some of the modulation functions also are the same as another (16). The application of the carrier based pulse width modulation allows for some arbitrariness in switch duty cycles order during the switching period T s . The same waveform of some modulation functions make possible such arrangement of the switch duty cycles during the switching period, that ensures usage the rotating voltage space vectors only.
The exemplary arrangement of the switch duty cycles during the period T s is shown in the Figure 3. While the modulation functions is defined by (15) the output voltages are represented by rotating space vectors named as CCW type, and lagging input displacement angle. The application of the modulation function (16) gives CW output voltage rotating space vectors and leading input displacement angle [28]. Both of them, i.e., modulation function (15) depending on difference: ω o − ω i of output and input frequency and modulation function (16) depending on sum: ω o + ω i result in output voltage amplitude equal to half of input voltage amplitude, at most.

Simulation
Simulation tests were realized in an EMTP-ATP program. The matrix of the bidirectional switches was modelled as a matrix of the ideal bidirectional switches, controlled using signals generated in TACS subroutine. The supply grid is represented by ideal sinusoidal voltage sources, with an RMS value of 220 V and a frequency of 50 Hz, while the load consists in star-connected resistance and inductance elements, with values of 2 Ω and 10 mH per phase. The six-phase load emulates six phase machine stator. Three different cases of the application of the three-to-six-phase MC to supplying induction machine are analysed. One of them is when the MC supplies the six-

Simulation
Simulation tests were realized in an EMTP-ATP program. The matrix of the bidirectional switches was modelled as a matrix of the ideal bidirectional switches, controlled using signals generated in TACS subroutine. The supply grid is represented by ideal sinusoidal voltage sources, with an RMS value of 220 V and a frequency of 50 Hz, while the load consists in star-connected resistance and inductance elements, with values of 2 Ω and 10 mH per phase. The six-phase load emulates six phase machine stator. Three different cases of the application of the three-to-six-phase MC to supplying induction  For all analysed cases the same trait is characteristic -the common mode voltage is entirely eliminated (Figures 5a, 6a, 7a, 9a,b, and 10a). The circular trajectories of output voltage space vector (Figures 5b, 6b, 7b, 8a,b, 9a and 10b), adequately to sinusoidal input voltage, indicates that the synthesis of output voltage is realized with use of rotating space vectors. The module of the rotating voltage space vectors, representing six-phase output voltages (Figures 5b, 6b and 7b) is equal 2 3 i U , while for three-phase output (Figure 8a,b) , 6b and 7b) the placement of zero vectors could be observed. These zero vectors, depicted in Table 2, are characterized by zero value of the module for six-phase voltage. For the same switching configuration the module of the vectors representing three of six phases is different than zero, what could be observed for case when the MC is used in supplying two sets of the six phase machine with two isolated star points (Figure 8). In both cases the zero vectors can be categorized as the rotating voltage space vectors, and they do not have negative impact on elimination of the CMV. The carrier frequency is f carr = 5 kHz. Two PWM control methods are tested, one with use of modulation functions (15)   For the case when the three-to-six-phase MC is applied to supply the open-end load, in the Figure 9, and Figure 10 the waveforms of the MC phase output voltages ( Figure 9) and load phase voltage ( Figure  10a) are depicted. In the same figure it could be seen the zero value of the CMV as well as for voltages of three MC output phases (Figure9) and also for three load phases (Figure 10a). In the trajectory of the load voltages (Figure 10b) could be observed the zero vectors, like for voltages of six output phases, and that three MC output phases (Figure9) and also for three load phases (Figure 10a). In the trajectory of the load voltages (Figure 10b) could be observed the zero vectors, like for voltages of six output phases, and that the magnitude of rotating voltage space vectors is equal i U 3 ( Table 1).
The analysis of load voltages allows to note, that the amplitude of basic component of this voltage is the same as amplitude of supplying voltage (Figure10), while the referenced voltage transfer ratio used in control of the MC is kU = 0.5.

Experiment
To further verify the proposed control method, measurement tests were performed. The experimental parameters are shown in Table 3. The laboratory station (Figure 12) is the universal research station provided for testing different configurations of the MC with use of different control methods, applying different modulation methods. The station consists of two three-to-three MC

Experiment
To further verify the proposed control method, measurement tests were performed. The experimental parameters are shown in Table 3. The laboratory station (Figure 12) is the universal research station provided for testing different configurations of the MC with use of different control methods, applying different modulation methods. The station consists of two three-to-three MC For all analysed cases the same trait is characteristic -the common mode voltage is entirely eliminated (Figure 5a, Figure 6a, Figure 7a, Figure 9a,b, and Figure 10a). The circular trajectories of output voltage space vector (Figure 5b, Figure 6b, Figure 7b, Figure 8a,b, Figures 9a and 10b), adequately to sinusoidal input voltage, indicates that the synthesis of output voltage is realized with use of rotating space vectors. The module of the rotating voltage space vectors, representing six-phase output voltages (Figures 5b, 6b and 7b) is equal √ 3U i /2, while for three-phase output (Figure 8a,b) is equal U i /2 (Table 1). For six-phase voltage (Figures 5b, 6b and 7b) the placement of zero vectors could be observed. These zero vectors, depicted in Table 2, are characterized by zero value of the module for six-phase voltage. For the same switching configuration the module of the vectors representing three of six phases is different than zero, what could be observed for case when the MC is used in supplying two sets of the six phase machine with two isolated star points (Figure 8). In both cases the zero vectors can be categorized as the rotating voltage space vectors, and they do not have negative impact on elimination of the CMV.
For the case when the three-to-six-phase MC is applied to supply the open-end load, in the Figure 9, and Figure 10 the waveforms of the MC phase output voltages ( Figure 9) and load phase voltage (Figure 10a) are depicted. In the same figure it could be seen the zero value of the CMV as well as for voltages of three MC output phases ( Figure 9) and also for three load phases (Figure 10a). In the trajectory of the load voltages ( Figure 10b) could be observed the zero vectors, like for voltages of six output phases, and that the magnitude of rotating voltage space vectors is equal Table 1). The analysis of load voltages allows to note, that the amplitude of basic component of this voltage is the same as amplitude of supplying voltage (Figure 10), while the referenced voltage transfer ratio used in control of the MC is k U = 0.5.

Experiment
To further verify the proposed control method, measurement tests were performed. The experimental parameters are shown in Table 3. The laboratory station ( Figure 12) is the universal research station provided for testing different configurations of the MC with use of different control methods, applying different modulation methods. The station consists of two three-to-three MC modules, what allows for testing the classic scheme of three-to-three-phase MC, the three-to-six-phase MC, the six-to-three-phase MC, the two-modules MC supplying open-end load and the multilevel MC with clamp capacitors. The research station works under control of computer, where the modulation methods, applied in control system is encoded. The control system is based on digital signal processor TMS320F2812, determining the duty cycles of bidirectional switches, and programmable logic device. The CPLD provides the generation of controlling signals with taking four-step commutation into account, proper succession the switches duty cycles during switching period, and protection system.   Presented in Figures 14 and 15, waveforms represent the results of measurements concerned with the two method of modulation. In first of the method (Figure 14), the PWM modulation is realised with use of modulation functions (15) what is concerned with referenced input displacement angle the same as load phase angle ( o i    ). In the second one (Figure 15), the modulation functions (16) are used, and referenced input displacement angle fulfil the condition: . The waveform Figure 13. Scheme of three-to-six-phase MC tested in experiment.  (16) are used, and referenced input displacement angle fulfil the condition: . The waveform of output voltage for three phases, supplying the load at the one side (Figures 14a and 15a), and the load current (Figure 14b and 15b) confirm correctness the proposed modulation method-the presented waveforms of current are sinusoidal. Additionally, the elimination of the CMV was obtained (Figures 14a and 15a). The maximum value of the CMV peak voltage did not exceed 5% of input voltage amplitude.

Conclusions
The control method of the three-to-six-phase MC, applied to supply the six-phase induction machine, with common star point, and to the six-phase machine with two isolated star points, and to the three-phase open-end machine, was investigated. All 729 admissible switching combinations for connecting the six output phases to the three input phases of MC were analysed, to find the voltage space vectors eliminating the CMV. There was found that 18 configurations fulfill the posited condition. The output MC voltages synthesized in 18 found configurations are represented by the six CCW, and six CW rotating voltage space vectors, and six zero space vectors while the MC supplies the six-phase machine with common star point. In case of the application of the MC to supply the sixphase induction machine with two isolated star points, half of the 18 configurations are represented by the CCW rotating voltage space vectors and half of them correspond to the CW rotating voltage space vectors. These vectors represent the output voltages of MC. In case of the open-end three-phase machine, the voltage space vectors represent the load phase voltage.
The carrier-based implementation of space vector modulation (SVM) was used to control of analyzed MC. As a modulation functions the modified Venturini modulation functions were chosen. The carrier-based SVM, using Venturini modulation functions, is valuable for control of the three-tosix-phase MC. The advantage of proposed strategy is a significant simplification of the MC control. The carrier based implementation of SVM is an alternative way to achieve SVM, that does not involve the knowledge of space vectors when it is derived. Additionally, it avoids any trigonometric and division operations that could be needed to implement the SVM using the general space vector approach. Implementation of Ventutini modulation functions, proposed by the authors of the paper, results in synthesis of output voltage with use of only rotating voltage space vectors, what is Presented in Figures 14 and 15, waveforms represent the results of measurements concerned with the two method of modulation. In first of the method (Figure 14), the PWM modulation is realised with use of modulation functions (15) what is concerned with referenced input displacement angle the same as load phase angle (ϕ i = ϕ o ). In the second one (Figure 15), the modulation functions (16) are used, and referenced input displacement angle fulfil the condition: ϕ i = −ϕ o . The waveform of output voltage for three phases, supplying the load at the one side (Figures 14a and 15a), and the load current (Figures 14b and 15b) confirm correctness the proposed modulation method-the presented waveforms of current are sinusoidal. Additionally, the elimination of the CMV was obtained (Figures 14a and 15a). The maximum value of the CMV peak voltage did not exceed 5% of input voltage amplitude.

Conclusions
The control method of the three-to-six-phase MC, applied to supply the six-phase induction machine, with common star point, and to the six-phase machine with two isolated star points, and to the three-phase open-end machine, was investigated. All 729 admissible switching combinations for connecting the six output phases to the three input phases of MC were analysed, to find the voltage space vectors eliminating the CMV. There was found that 18 configurations fulfill the posited condition. The output MC voltages synthesized in 18 found configurations are represented by the six CCW, and six CW rotating voltage space vectors, and six zero space vectors while the MC supplies the six-phase machine with common star point. In case of the application of the MC to supply the six-phase induction machine with two isolated star points, half of the 18 configurations are represented by the CCW rotating voltage space vectors and half of them correspond to the CW rotating voltage space vectors. These vectors represent the output voltages of MC. In case of the open-end three-phase machine, the voltage space vectors represent the load phase voltage.
The carrier-based implementation of space vector modulation (SVM) was used to control of analyzed MC. As a modulation functions the modified Venturini modulation functions were chosen. The carrier-based SVM, using Venturini modulation functions, is valuable for control of the three-to-six-phase MC. The advantage of proposed strategy is a significant simplification of the MC control. The carrier based implementation of SVM is an alternative way to achieve SVM, that does not involve the knowledge of space vectors when it is derived. Additionally, it avoids any trigonometric and division operations that could be needed to implement the SVM using the general space vector approach. Implementation of Ventutini modulation functions, proposed by the authors of the paper, results in synthesis of output voltage with use of only rotating voltage space vectors, what is precondition to entire elimination of CMV.
The entire elimination of the CMV is an important achievement, obtained by implementation of the proposed modulation method. The elimination of CMV was confirmed by the results of simulation and measurement tests. In the experiment, the peak value of the CMV is less than 5% of the amplitude of the supplying voltage and the peak value of the output voltage. The three-to-six-phase MC controlled with entire elimination of the CMV could be applied in high power six-phase drive.
The drawback of the proposed controlling method is the assumption that the control of the MC is realised with fixed value of input displacement angle-with the lagging or leading one. It could be overcome by applying the modulation method with use of the both modulation functions proposed here for switching time divided proportionally to the value of the referenced input displacement angle. This modulation method will be investigated by the authors soon.