# Patent

## Title
Pulse width modulation control of a matrix converter

## Abstract

A matrix converter includes a plurality of switching elements and is adapted to receive a multi-phase alternating current (AC) input signal having an input frequency and to generate a multi-phase AC output signal having an output frequency. The phases of the input signal are sorted as a function of their instantaneous voltage amplitude (60). A reference signal is generated from output reference voltages that correspond to each phase of the output signal (56). Duty cycles are calculated for each phase of the output signal based on the sorted input signal phases and the reference signal (62). Switching functions, which each control one of the switching elements, are then generated based on the duty cycles for each phase of the output signal (64, 66).

## Background
BACKGROUND
The present invention relates to power systems. More specifically, the present invention relates to a pulse width modulation control method for a matrix converter or direct frequency changer.
A matrix converter is an electronic device that converts AC voltage of one frequency at its input to AC voltage of a different frequency at its output. The matrix converter may also change the amplitude and the number of phases between the input signal and the output signal. The matrix converter includes a plurality of switching devices that are controlled by pulse width modulation (PWM) to provide voltages in a single phase or multiple phases at the output of the matrix converter. The number of switching devices in the matrix converter is a function of the number of phases in the input and output lines. PWM changes the connections of the switches between the input and the output of the matrix converter such that the locally averaged output voltages follow reference voltages.
One application for a matrix converter is controlling the speed and torque of an AC motor. In this application, the matrix converter receives an AC input signal (for example, a three-phase signal from an electrical utility) and converts the input signal to a single phase or multi-phase output signal having a frequency and amplitude that is compatible with the AC motor. However, many control algorithms for converting the input signal to the appropriate output signal are very complicated and consume a large amount of processor resources. In addition, the incorporation of a matrix converter into a motor control system often necessitates complex control hardware, which increases the cost of the system.
SUMMARY
The present invention relates to control of a matrix converter including a plurality of switching elements. The matrix converter is adapted to receive a multi-phase alternating current (AC) input signal having an input frequency and to generate a multi-phase AC output signal having an output frequency. The phases of the input signal are sorted as a function of their instantaneous voltage amplitude. A reference signal is generated from output reference voltages that correspond to each phase of the output signal. Duty cycles are calculated for each phase of the output signal based on the sorted input signal phases and the reference signal. Switching functions, which each control one of the switching elements, are then generated based on the duty cycles for each phase of the output signal.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a schematic of a power system including a matrix converter with a plurality of switching elements and a controller for operating the switching elements.
FIG. 2A-2C are schematics of switching elements suitable for use in the matrix converter shown in FIG. 1.
FIG. 3 is a block diagram of the controller for generating switching functions for the plurality of switching elements from the input signals.
FIG. 4 is a graph showing the input signals sorted according to their instantaneous values.
FIG. 5A is a graph of a triangle comparison signal for generating pulse width modulation functions.
FIG. 5B is a graph of intermediate switching functions generated by a triangle comparison method.
FIG. 5C is a graph of switching functions derived from the intermediate switching functions for controlling the switching elements of the matrix controller.
FIG. 5D is a graph of an output voltage waveform for a phase of the matrix converter output.



DETAILED DESCRIPTION
FIG. 1 is a schematic view of power system 10, which receives multi-phase alternating current (AC) power from power supply 12 at an input frequency and provides multi-phase AC power to a load at an output frequency. Power system 10 includes LC filter 20, matrix converter 22, and matrix converter (MxC) controller 24. In the embodiment shown, power supply 12 is a three-phase power supply (including input phases R, S, and T) that provides input voltages v1, v2, and v3 and supplies input currents i1, i2, and i3 at the inputs of matrix converter 22. Also in the embodiment shown, matrix converter 22 provides three-phase power (including phases U, V, and W) to induction motor 14, including output voltages v1o, v2o, and v3o and output currents i1o, i2o, and i3o.
LC filter 20 includes inductors 26a, 26b, and 26c, and capacitors 28a, 28b, and 28c. Inductor 26a is connected in series with input phase R, inductor 26b is connected in series with input phase S, and inductor 26c is connected in series with input phase T. Capacitor 28a is connected across input phases R and S, capacitor 28b is connected across input phases S and T, and capacitor 28c is connected across input phases R and T. LC filter 20 controls the current levels and prevents voltage transients from power supply 12.
Matrix converter 22 includes switching elements s11, s21 s31, s12, s22, S32, s13, s23, and s33 (collectively referred to as switches sjk). Switching elements sjk are bi-directional switches that are connected to power supply 12 via LC filter 20 such that input voltage v1 is received on the input node of switching elements s11, s12, and s13, input voltage v2 is received on the input node of switching elements s21, s22, and s23, and input voltage v3 is received on the input node of switching elements s31, s32, and s33. The output node of switching elements s11, s21, and s31 is connected to provide output voltage v1o to motor 14, the output node of switching elements s12, s22, and s32 is connected to provide output voltage v2o to motor 14, and the output node of switching elements s13, s23, and s33 is connected to provide output voltage v3o to motor 14. White matrix converter 30 is shown receiving three-phase power at its input and providing three-phase power at its output, it will be appreciated that matrix converter 22 can be adapted to receive power from power supply 12 having any number of phases and to provide power to motor 14 with any number of phases.
MxC controller 24 is connected to each of switching elements sjk to provide switching functions Sjk that operate switches sjk to provide output signals to motor 14 at an output frequency. In other words, MxC controller 24 operates switching elements sjk to convert the frequency of the input signal from power supply 12 to an output frequency that is compatible with motor 14. MxC controller 24 receives input voltages v1, v2, and v3 as inputs and generates the switching function Sjk based on these inputs. The algorithm employed to develop the switching functions will be described in more detail below.
FIG. 2A-2C are schematics of devices suitable for switching elements sjk in matrix converter 22. Each of the devices receives input voltage vj on its input node and provides output voltage vko on its output node. Each of the devices is controlled by a switching function Sjk and its complement Sjk, which are provided by MxC controller 24. FIG. 2A shows device 40 including transistors 42 connected in an anti-parallel configuration (emitter to collector), with reverse blocking capability. FIG. 2B shows device 44 including transistors 42 connected in a common collector configuration. Each transistor 42 in device 44 is connected to a diode 46 in an anti-parallel configuration to provide opposite conductivity between each transistor 42 and diode 46. FIG. 2C shows device 48 including transistors 42 connected in a common emitter configuration. Each transistor 42 in device 48 is connected to a diode 46 in an anti-parallel configuration to provide opposite conductivity between each transistor 42 and diode 46. In some embodiments, transistors 42 in FIGS. 2A, 2B, and 2C are insulated gate bipolar transistors (IGBTs). It should be noted that devices 40, 44, and 48 are merely exemplary, and any device capable of controllable to provide bi-directional switching between two nodes may be employed for switching elements sjk.
Transistors 42 in devices 40, 44, and 48 may be controlled by pulse width modulation (PWM) signals, which provide pulses to the gates of transistors 42 to control current flow through them. The gating pulses may be modeled by switching functions Sjk, which assume values of “1” when switching element sjk is closed (i.e., conducting) and “0” when switching element sjk is open. If an inductive load is provided at the output of matrix converter 22 (such as inductive motor 14), one of switching elements sjk needs to be in a conduction state at any given time. In addition, to avoid a short circuit between input phases R, S, and T, no two switching elements sjk may be conducting at the same time. These constraints may be expressed as:




  
    
      
        
          
            
              
                ∑
                
                  j
                  =
                  1
                
                n
              
              ⁢
              
                S
                jk
              
            
            =
            1
          
          ;
          
            k
            =
            1
          
        
        ,
        
          m
          .
        
      
    
    
      
        (
        1
        )
      
    
  




From Equation 1, it follows that for a given k, there are only n−1 independent switching functions Sjk. Thus, the number of switching functions Sjk may be reduced from n×m to (n−1)×m switching functions.

As is shown in FIG. 1, the output signal for each output phase U, V, and W is generated by controlling three switching elements s1k, s2k, and s3k, corresponding to the three-phase input power from power supply 12. Thus, the 3×3 matrix converter 22 shown in FIG. 1 can be viewed as three converters each including three input phases and a single output phase having a signal based on control of switching elements s1, s2, and s3. The output voltage vo from a three-phase input and single phase output matrix converter is:




  
    
      
        
          
            v
            o
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          [
          
            
              
                
                  
                    
                      S
                      1
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                
                  
                    
                      S
                      2
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                
                  
                    
                      
                        
                          S
                          3
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                v
                                1
                              
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                          
                        
                        
                          
                            
                              
                                v
                                2
                              
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                          
                        
                        
                          
                            
                              
                                v
                                3
                              
                              ⁡
                              
                                (
                                t
                                )
                              
                            
                          
                        
                      
                      ]
                    
                  
                
              
            
            .
          
        
      
    
    
      
        (
        2
        )
      
    
  




Using local averaging over a short sampling interval Ts and assuming that input voltages v1, v2, and v3 are constant over sampling interval Ts, Equation 2 can be written as:

vo=d1v1+d2v2+d3v3  (3)

where d1, d2, and d3 are duty cycle functions defined as d1,2,3=T1,2,3/Ts. Ts is the sum of time intervals T1, T2, and T3, which correspond to the times that switching elements s1, s2, and s3, respectively, are conducting, and vo is the locally averaged output voltage. Thus, Equation 1 may expressed in terms of duty cycles as:

d1+d2+d3=1,  (4)

where 0≦d1, d2, d3≦1. Equation 4 shows that output voltage vo is a function of two of the duty cycle functions, since the third duty cycle function can be calculated from two known duty cycle functions.

Duty cycle functions d1, d2, and d3 may be employed not only to control output voltage vo, but also to provide additional criteria related to the distribution of output current io over particular input phases in one sampling interval. In particular, input currents i1, i2, and i3 are related to output current io in that

d1io=i1; d2io=i2; d3io=i3  (5)

The ratio of two locally averaged contributions from output current io to input currents i1, i2, and i3, may be selected to follow the desired ratio of phase shifted input voltages v1, v2, and v3, to control the displacement factor. This can be accomplished by introducing a current distribution factor a to the duty cycle functions, wherein current distribution factor a may be defined as:





  
    
      
        
          a
          =
          
            
              
                i
                2
              
              
                i
                3
              
            
            =
            
              
                
                  d
                  2
                
                
                  d
                  3
                
              
              =
              
                
                  v
                  2
                  *
                
                
                  v
                  3
                  *
                
              
            
          
        
        ,
      
    
    
      
        (
        6
        )
      
    
  




where voltages v*2 and v*3 are phase angle reference voltages. Voltages v*2 and v*3 may be generated by a phase-locked loop (PLL) system such that they are in phase with input voltages v2 and v3, respectively.

To reduce the number of unknown duty cycles from three to two, Equation 4 may be expressed as d1=1−(d2+d3), and substituted into Equation 3:

vo−v1=d2(v2−v1)+d3(v3−v1).  (7)

Furthermore, Equation 6 may be expressed as d2=ad3 and substituted into Equation 7 and rewritten to provide an expression for d3:





  
    
      
        
          d
          3
        
        =
        
          
            
              
                v
                o
              
              -
              
                v
                1
              
            
            
              
                (
                
                  
                    v
                    3
                  
                  -
                  
                    v
                    1
                  
                
                )
              
              +
              
                a
                ⁡
                
                  (
                  
                    
                      v
                      2
                    
                    -
                    
                      v
                      1
                    
                  
                  )
                
              
            
          
          .
        
      
    
    
      
        (
        8
        )
      
    
  




With d3 calculated to meet output voltage and input power factor demands, the remaining duty cycle functions d1 and d2 may be calculated backward from Equations 4 and 7.

FIG. 3 is a block diagram of a portion of MxC controller 24 (referred to as MxC controller portion 24a) for generating switching functions S1, S2, and S3 for switching elements s1, s2, and s3. MxC controller portion 24a is an embodiment of a system that generates switching functions pursuant to the constraints outlined above. MxC controller portion 24a includes phase-locked loop (PLL) module 50, linearity extender module 52, signal polarity module 54, level shifter module 56, sorting module 60, duty cycle module 62, pulse width modulation (PWM) module 64, and de-multiplexing module 66. Each of the modules of MxC controller portion 24a may be implemented in hardware, software, firmware, or combinations thereof. In order to provide output signals for all three output phases U, V, and W of MxC controller 24, three MxC controller portions 24a may be connected in parallel to the input phases from power supply 12.
PLL module 50 receives input voltages v1, v2, and v3 at its inputs and provides output reference voltages v1o*, v2o*, and v3o* to linearity extender 52. Linearity extender module 52 provides signals 70 to signal polarity module 54 based on output reference voltages v1o, v2o* and v3o and zero sequence signal vzs. Signal polarity module 54 provides signals 72 to level shifter module 56 based on the signal from linearity extender module 52 and polarity signal pol from duty cycle module 62. Level shifter module 56 generates modified output reference voltages Δv1o**, Δv2o**, and Δv3o**, and one of these modified output reference voltages are provided as an input to duty cycle module 62.
Sorting module 60 also receives input voltages v1, v2, and v3 at its inputs and generates sorted voltage signals vmin, vmid, and vmax at its output and provides a decoding signal to de-multiplexing module 66. Duty cycle module 62 generates duty cycle signals dmin, dmid, and dmax from sorted voltage signals vmin, vmid, and vmax. PWM module 64 generates switching functions Smin, Smid, and Smax from duty cycle signals dmin, dmid, and dmax, and de-multiplexing module 66 provides output switching functions S1, S2, and S3 based on switching functions Smin, Smid, and Smax and the decoding signal from sorting module 60.
Sorting module 60 receives input voltages v1, v2, and v3 and sorts them as a function of their instantaneous voltage amplitudes. Input voltages v1, v2, and v3 are sorted such that vmax is the input phase with the highest amplitude, vmin is the phase with lowest amplitude, and vmid is the phase with an amplitude intermediate vmax and vmin. Signals vmax, vmid, and vmin are provided at the outputs of sorting module 60 and the inputs of duty cycle module 62. Sorting module 60 also provides a decoding signal to de-multiplexing module 66 that associates sorted input voltages vmin, vmid, and vmax with their originating input voltages v1, v2, and v3.
PLL module 50 also receives input voltages v1, v2, and v3 at its inputs and generates output reference voltages v1o*, v2o*, and v3o* at its outputs. Output reference voltages v1o*, v2o*, and v3o* are phase-locked with input voltages v1, v2, and v3, respectively. Output reference voltages v1o*, v2o*, and v3o* are provided to linearity extender module 52 which extends the linearity of output reference voltages v1o*, v2o*, and v3o*. The linearity of output reference voltages v1o*, v2o*, and v3o* may be extended by adding a zero sequence signal vzs having a specific waveform and amplitude to reduce the peaks of output reference voltages v1o*, v2o*, and v3o*. In some embodiments, the zero sequence signal vzs is the third harmonic of one of output reference voltages v1o*, v2o*, and v3o*. With proper selection of zero sequence signal vzs, the linearity of output reference voltages v1o*, v2o*, and v3o* can be extended by a factor of 2/√{square root over (3)}, or up to 15.4%.
After zero sequence signal vzs has been added to output reference voltages v1o*, v2o*, and v3o*, signals 70 are provided to polarity module 54. Polarity module 54 receives a polarity signal pol from duty cycle module 62 that has a value of “1” when sorted input voltage vmid is zero or positive, and a value of “−1” when sorted input voltage vmid is negative. The signals from linearity extender 52 are multiplied by polarity signal pol, which assures that the criterion 0≦d1,d2,d3≦1 set forth above is satisfied.
FIG. 4 is a graph that illustrates the relationship between sorted input voltages vmin, vmid, and vmax and polarity signal pol. Input voltages v1, v2, and v3 are plotted versus time, and input voltages v1, v2, and v3 are phase shifted by about 120° with respect to each other. Line poi shows that the value of polarity signal pal changes as the polarity of the input voltage with the intermediate amplitude changes. For example, at instantaneous time Tinst sorting module 60 sorts input voltages v1, v2, and v3 such that {v1, v2, v3}={vmin, vmid, vmax}, and polarity signal pol has a value pol=1 because vmid≧0.
Referring back to FIG. 3, polarity module 54 provides the polarity adjusted output reference voltages v1o*, v2o*, and v3o* (i.e., signals 72) to level shifter module 56. Level shifter module 56 scales and shifts the signals to provide modified output voltages Δv1o**, Δv2o**, and Δv3o** having an amplitude of up to about 86.6% of input voltages v1, v2, and v3, respectively. One of modified output reference voltages Δv1o**, Δv2o**, and Δv1o** is provided to duty cycle module 62 for use in calculating duty cycle functions d1, d2, and d3.
Duty cycle module 62 receives sorted input voltages vmin, vmid, and vmax and one of modified output reference voltages Δv1o**, Δv2o**, and Δv3o** and generates duty cycles dmin, dmid, and dmax. The duty cycles for a sampling interval Ts are calculated pursuant to the following table. Signals vmin*, vmid*, and vmax* are phase-locked sorted input reference voltages vmin, vmid, and vmax, respectively, α is the current distribution factor described above, and ΔVo is the reference signal provided by level shifter module 56 to duty cycle module 62.










TABLE 1







If vmid ≧ 0, pol = 1
If vmid < 0, pol = −1









ΔV1 = vmid − vmin
ΔV1 = vmax − vmid



 







  a
  =
  
    
      v
      mid
      *
    
    
      v
      max
      *
    
  








  a
  =
  
    
      v
      mid
      *
    
    
      v
      min
      *
    
  







 



ΔVo = vo − vmin
ΔVo = −(vo − vmax)



ΔVh = vmax − vmin
ΔVh = vmax − vmin



ΔV = ΔVh + aΔV1
ΔV = ΔVh + aΔV1



 







  
    
      d
      h
    
    =
    
      
        Δ
        ⁢
        
          
        
        ⁢
        
          V
          o
        
      
      
        Δ
        ⁢
        
          
        
        ⁢
        V
      
    
  
  ;
  
    
      d
      1
    
    =
    
      ad
      h
    
  
  ;
  
    
      d
      mid
    
    =
    
      d
      1
    
  








  
    
      d
      h
    
    =
    
      
        Δ
        ⁢
        
          
        
        ⁢
        
          V
          o
        
      
      
        Δ
        ⁢
        
          
        
        ⁢
        V
      
    
  
  ;
  
    
      d
      1
    
    =
    
      ad
      h
    
  
  ;
  
    
      d
      mid
    
    =
    
      d
      1
    
  







 



dmin = 1 − (d1 + dh)
dmin = dh



dmax = dh
dmax = 1 − (d1 + dh)










After duty cycle functions dmin, dmid, and dmax are calculated, PWM module 64 generates modulation functions uhm and ulm that are a function of duty cycle functions dmin, dmid, and dmax. In some embodiments, ulm=dmid+dmax=(1+a)dmax and uhm=dmax. PWM module 64 compares modulation functions uhm and ulm to a triangular carrier signal of known frequency to generate the switching functions for switching elements s1, s2, and s3. FIG. 5A is a graph showing triangle carrier signal vtri with duty cycle functions dmin, dmid, and dmax and modulation functions uhm and ulm plotted on the graph.
The comparison of modulation functions uhm and ulm to triangle carrier signal vtri generates intermediate switching functions Smaxtc and Smidtc. FIG. 5B is a graph showing the waveforms for intermediate switching functions Smaxtc and Smidtc. Intermediate switching function Smaxtc has a logic “1” value when triangle carrier signal vtri is less than uhm and a logic “0” value at all other times. Intermediate switching function Smidtc has a logic “1” value when triangle carrier signal vtri is less than ulm and a logic “0” value at all other times.
FIG. 5C is a graph of switching functions Smin, Smid, and Smax, which are derived from the intermediate switching functions Smaxtc and Smidtc. Switching functions Smin, Smid, and Smax are derived as follows:

Smax=Smaxtc 

Smid =XOR(Smaxtc,Smidtc)

Smin =NOT(Smidtc)

Logic gates may be connected to conventional triangle comparison hardware to generate switching functions Smin, Smid, and Smax from intermediate switching functions Smaxtc and Smidtc.
Switching functions Smin, Smid, and Smax are then provided to de-multiplexing module 66, which associates switching functions Smin, Smid, and Smax with switching elements s1, s2, and s3 based on the decoding signal provided by sorting block 60. Thus, switching function S1 is provided to switching element s1, switching function S2 is provided to switching element s2, and switching function S3 is provided to switching element s3. FIG. 5D is a graph of the waveform generated switching functions S1, S2, and S3 control switching elements s1, s2, and s3, respectively, when vmid≧0. The output voltage vo for MxC controller portion 24a is the locally averaged contributions of vmin, vmid, and vmax.
In summary, the present invention relates to control of a matrix converter including a plurality of switching elements. The matrix converter is adapted to receive a multi-phase alternating current (AC) input signal having an input frequency and to generate a multi-phase AC output signal having an output frequency. The phases of the input signal are sorted as a function of their instantaneous voltage amplitude. A reference signal is generated from output reference voltages that correspond to each phase of the output signal. Duty cycles are calculated for each phase of the output signal based on the sorted input signal phases and the reference signal. Switching functions, which each control one of the switching elements, are then generated based on the duty cycles for each phase of the output signal.
Although the present invention has been described with reference to preferred embodiments, workers skilled in the art will recognize that changes may be made in form and detail without departing from the spirit and scope of the invention.

## Claims

1. A method for controlling a matrix converter including a plurality of switching elements, the matrix converter adapted to receive a multi-phase alternating current (AC) input signal having an input frequency and to generate a multi-phase AC output signal having an output frequency, the method comprising:
sorting the phases of the input signal as a function of instantaneous voltage amplitude of the input signal phases;
generating a reference signal from output reference voltages that correspond to each phase of the output signal;
calculating a current distribution factor based upon polarity of an intermediate input phase voltage and phase-locked input reference voltages;
calculating duty cycles functions for each phase of the output signal based on the sorted input signal phases, the current distribution factor, and the reference signal; and
generating switching functions based on the duty cycle function for each phase of the output signal, wherein each switching function controls one of the switching elements.

2. The method of claim 1, wherein sorting the phases of the input signal comprises associating each of the sorted input signal phases with the input signal phase from which it originated.

3. The method of claim 2, and further comprising:
associating each of the switching functions with a switching element based on the associations of the sorted input signal phases with the originating input signal phases.

4. The method of claim 1, wherein generating the reference signal comprises:
adding a zero sequence signal to each of the output reference voltages to provide modified output reference voltages; and
selecting one of the modified output reference voltages as the reference signal.

5. The method of claim 4, wherein adding the zero sequence signal comprises selecting the zero sequence signal to extend a linearity of the switching functions by about 15.4%.

6. The method of claim 4, wherein generating the reference signal further comprises scaling and shifting the output reference voltage signals after adding the zero sequence signal to provide output signal voltages having amplitudes of about 86.6% of corresponding input signal voltages.

7. The method of claim 1, wherein generating switching functions comprises:
calculating modulation functions based on the duty cycles for each phase of the output signal;
comparing the modulation functions to a trianglular comparison signal; and
generating the switching functions based on the comparison between the modulation functions and the triangular signal.

8. A method for controlling a matrix converter including a plurality of switching elements, the matrix converter adapted to receive an alternating current (AC) input signal having an input frequency with phase voltages v1, v2, and v3 and to generate a multi-phase AC output signal having an output frequency, the method comprising:
sorting input phase voltages v1, v2, and v3 as a function of their instantaneous amplitudes such that vmax is the input phase voltage with the highest amplitude, vmin is the input phase voltage with lowest amplitude, and vmid is the input phase voltage with an amplitude intermediate vmax and vmin;
generating a reference signal from output reference voltages that correspond to each phase of the output signal;
calculating a current distribution factor based upon polarity of vmid and phase-locked input reference voltages;
calculating duty cycle functions dmin, dmid, and dmax based on input phase voltages vmin, vmid, and vmax, respectively, the current distribution factor, and the reference signal;
calculating modulation functions based on duty cycle functions dmin, mid, and dmax for each output signal phase;
comparing the modulation functions to a triangular comparison signal; and
generating switching functions smin, smid, and smax based the comparison between the modulation functions and the triangular comparison signal, wherein each of switching functions smin, smid, and smax controls one of the switching elements associated with an output phase.

9. The method of claim 8, wherein sorting phases v1, v2, and v3 comprises associating each of vmax, vmid, and vmin with one of the input phase voltages v1, v2 and v3 from which it originated.

10. The method of claim 9, and further comprising:
associating each of switching functions smin, smid, and smax with a switching element based on the associations of sorted phase voltages vmax, vmid, and vmin with input phase voltages v1, v2, and v3.

11. The method of claim 8, wherein generating a reference signal comprises reversing a polarity of the output reference voltage associated with an output phase is reversed if vmid is negative on the input phase corresponding to the output phase.

12. The method of claim 8, wherein generating the reference signal comprises:
adding a zero sequence signal to each of the output reference voltages to provide modified output reference voltages; and
selecting one of the modified output reference voltages as the reference signal.

13. The method of claim 12, wherein adding the zero sequence signal comprises selecting the zero sequence signal to extend a linearity of the switching functions by about 15.4%.

14. A system for controlling a matrix converter including a plurality of switching elements adapted to receive a multi-phase alternating current (AC) input signal having an input frequency and to generate a multi-phase AC output signal having an output frequency, the system comprising:
a sorting module configured to sort the phases of the input signal as a function of instantaneous voltage amplitude of the input signal phases;
a phase-locked loop for generating phase-locked input reference voltages based upon the input signal;
a polarity module for determining polarity of an intermediate input phase voltage;
a reference signal module configured to generate a reference signal from output reference voltages that correspond to each phase of the output signal;
a duty cycle module to calculate duty cycle functions for each phase of the output signal based on the sorted input signal phases and the reference signal, and a current distribution factor based upon the polarity determined by the polarity module and the phase-locked input reference voltages; and
a switching function module configured to generate switching functions based on the duty cycle functions for each phase of the output signal, wherein each switching function controls one of the switching elements.

15. The system of claim 14, wherein the sorting module is configured to associate each of the sorted input signal phases with the input signal from which it originated.

16. The system of claim 15, wherein the switching function module includes a demultiplexing module configured to sort the switching functions based on the associations of the sorted input signal phases from the sorting module.

17. The system of claim 14, wherein the switching function module is configured to calculate modulation functions based on the duty cycles and to comparing the modulation functions to a triangular comparison signal to generate the switching functions.

18. The system of claim 14, wherein the switching function module is configured to add a zero sequence signal to each of the output reference voltages to provide modified output reference voltages and to select one of the modified output reference voltages as the reference signal.

19. The system of claim 18, wherein the zero sequence signal extends a linearity of the switching functions by about 15.4%.

20. A method for controlling a matrix converter including a plurality of switching elements, the matrix converter adapted to receive a multi-phase alternating current (AC) input signal having an input frequency and to generate a multi-phase AC output signal having an output frequency, the method comprising:
sorting the phases of the input signal as a function of instantaneous voltage amplitude of the input signal phases;
generating a reference signal from output reference voltages that correspond to each phase of the output signal; wherein generating the reference signal comprises:
adding a zero sequence signal to each of the output reference voltages to provide modified output reference voltages; and
selecting one of the modified output reference voltages as the reference signal;

calculating duty cycles for each phase of the output signal based on the sorted input signal phases and the reference signal; and
generating switching functions based on the duty cycles for each phase of the output signal, wherein each switching function controls one of the switching elements.

21. The method of claim 20, wherein adding the zero sequence signal comprises selecting the zero sequence signal to extend a linearity of the switching functions by about 15.4%.

22. The method of claim 20, wherein generating the reference signal further comprises scaling and shifting the output reference voltage signals after adding the zero sequence signal to provide output signal voltages having amplitudes of about 86.6% of corresponding input signal voltages.

23. A method for controlling a matrix converter including a plurality of switching elements, the matrix converter adapted to receive an alternating current (AC) input signal having an input frequency with phase voltages v1, v2, and v3 and to generate a multi-phase AC output signal having an output frequency, the method comprising:
sorting input phase voltages v1, v2, and v3 as a function of their instantaneous amplitudes such that vmax is the input phase voltage with the highest amplitude, vmin is the input phase voltage with lowest amplitude, and vmid is the input phase voltage with an amplitude intermediate vmax and vmin;
generating a reference signal from output reference voltages that correspond to each phase of the output signal; generating the reference signal comprises:
adding a zero sequence signal to each of the output reference voltages to provide modified output reference voltages; and
selecting one of the modified output reference voltages as the reference signal;

calculating duty cycles dmin, dmid, and dmax based on input phase voltages vmin, vmid, and vmax, respectively, and the reference signal;
calculating modulation functions based on duty cycles dmin, dmid, and dmax for each output signal phase;
comparing the modulation functions to a triangular comparison signal; and
generating switching functions smin, smid, and smax based the comparison between the modulation functions and the triangular comparison signal, wherein each of switching functions smin, dmid, and smax controls one of the switching elements associated with an output phase.

24. The method of claim 23, wherein adding the zero sequence signal comprises selecting the zero sequence signal to extend a linearity of the switching functions by about 15.4%.

25. A system for controlling a matrix converter including a plurality of switching elements adapted to receive a multi-phase alternating current (AC) input signal having an input frequency and to generate a multi-phase AC output signal having an output frequency, the system comprising:
a sorting module configured to sort the phases of the input signal as a function of instantaneous voltage amplitude of the input signal phases;
a reference signal module configured to generate a reference signal from output reference voltages that correspond to each phase of the output signal; wherein the switching function module is configured to add a zero sequence signal to each of the output reference voltages to provide modified output reference voltages and to select one of the modified output reference voltages as the reference signal;
a duty cycle module to calculate duty cycles for each phase of the output signal based on the sorted input signal phases and the reference signal; and
a switching function module configured to generate switching functions based on the duty cycles for each phase of the output signal, wherein each switching function controls one of the switching elements.

26. The system of claim 25, wherein the zero sequence signal extends a linearity of the switching functions by about 15.4%.