Abstract:
Provided is a motor controller and method of controlling a motor with an improved maximum junction temperature rating. In accordance with one aspect of the present invention, a motor is controlled by varying a common mode voltage duty ratio for a plurality of solid state switching devices in a power inverter, generating drive signals for the solid state switching devices based at least in part upon the common mode voltage duty ratio, and changing states of the solid state switching devices based upon the drive signals.

Description:
BACKGROUND 
     The present invention relates generally to the field of electrical power converters and inverters. More particularly, the invention relates to techniques for controlling such devices so as to reduce their thermal load. 
     As the field of power electronics continues a trend toward greater packaging density, higher switching speed, and rapid power dissipation, the systems which employ power semiconductors increasingly operate at higher temperatures. Higher operating temperatures, however, tend to increase system failure rates and reduce reliability. A manufacturer may improve reliability by limiting the maximum operating temperature, choosing power semiconductor modules with appropriately low maximum junction temperature ratings. Often, however, lower maximum junction temperature ratings are achieved with a cost of decreased performance. It is thus advantageous to reduce the maximum junction temperature of a power semiconductor module while maintaining the same output power. 
     A three-phase inverter module is one such power semiconductor which may benefit from a reduction of maximum junction temperature rating. Commonly found in variable frequency motor drives, a three phase inverter module takes in DC power and outputs a three-phase sine wave at a desired output frequency. The three phase inverter module achieves the output waveform by rapidly switching insulated gate bipolar transistors (IGBTs) located on three parallel legs connecting a DC power source and a motor. By varying the fraction of the switching time each IGBT remains open on each of the three legs, a three phase waveform is produced. 
     The maximum junction temperature of a three-phase inverter module increases as the output frequency decreases, reaching a zenith when the output frequency nears 0 Hz, known as DC condition. Two primary reasons may explain the relationship. First, when a three-phase inverter module operates at low-frequency AC condition or at DC condition, power losses arise among the IGBTs unevenly. These effects may become magnified under a DC condition, when at certain output phase angles the IGBTs of only two of the three legs may sustain substantially all power losses. Secondly, under a low-frequency AC condition, the junction-to-case thermal time constants of the inverter module may grow comparatively small relative to the output period. Heat may thus dissipate from IGBTs unevenly. As a result, a much higher variation in temperature among the IGBTs of an inverter module occurs when operating at a low-frequency AC condition or at a DC condition. 
     While attempts have been made to mitigate the effects of a high maximum junction temperature, such efforts either fail to reduce the maximum junction temperature or tend to introduce additional problems. For example, using a thermal manager to predict temperature and avoid excessive temperatures may protect an inverter module from failure, but does not reduce the module&#39;s maximum junction temperature. As a result, the overall rating of the inverter cannot be maximized. Similarly, discontinuous pulse width modulation (DPWM) may be used to reduce junction temperature when an inverter module operates at low-frequency AC conditions or at DC conditions, but doing so increases the harmonic content and ripple current of the system. 
     BRIEF DESCRIPTION 
     The invention provides a motor controller and method of controlling a motor to address such needs. In accordance with one aspect of the present invention, a motor is controlled by varying a common mode voltage duty ratio for a plurality of solid state switching devices in a power inverter, generating drive signals for the solid state switching devices based at least in part upon the common mode voltage duty ratio, and changing states of the solid state switching devices based upon the drive signals. 
     In accordance with another aspect of the present invention, the common mode voltage duty ratio is varied by choosing an optimal common mode voltage duty ratio dependent on a phase angle of an output current of the inverter. The optimal common mode voltage duty ratio is chosen by determining a first possible optimal common mode voltage duty ratio which reduces the temperature differences among a plurality of solid state switching devices and associated diodes on a leg carrying maximum current in a power inverter. A second possible optimal common mode voltage duty ratio is determined which causes the temperature of a plurality of solid state switching devices on a leg not carrying maximum current to equate to the temperature of the plurality of solid state switching devices on the leg carrying maximum current in the power inverter. A third possible optimal common mode voltage duty ratio is also determined which causes the temperature of a plurality of diodes on a leg not carrying maximum current to equate to the temperature of the plurality of solid state switching devices on the leg carrying maximum current in the power inverter. An optimal common mode voltage duty ratio is then chosen from among the first possible optimal common mode voltage duty ratio, second possible common mode voltage duty ratio, and third possible common mode voltage duty ratio. 
     In accordance with yet another aspect of the present invention, an amplitude of the optimal common mode voltage duty ratio is stored in memory, and the optimal common mode voltage duty ratio is estimated in real-time as a periodic function with an amplitude equivalent to the amplitude previously stored in memory. 
    
    
     
       DRAWINGS 
       These and other features, aspects, and advantages of the present invention will become better understood when the following detailed description is read with reference to the accompanying drawings in which like characters represent like parts throughout the drawings, wherein: 
         FIG. 1  is a diagrammatical model of an exemplary motor drive system employing control circuitry for reducing the maximum junction temperature of the inverter module in accordance with one embodiment of the present invention; 
         FIG. 2  is a diagrammatical representation of exemplary control circuitry for a motor drive system in accordance with one embodiment of the present invention; 
         FIG. 3  represents a simplified model of one leg of an inverter module employed by a motor drive system in accordance with one embodiment of the present invention; 
         FIG. 4  is a graph illustrating the relationship between duty ratio and the positive and negative gating signals of each switch in accordance with one embodiment of the present invention; 
         FIG. 5  is a flowchart illustrating a method of determining an optimal common mode voltage duty ratio in accordance with one embodiment of the present invention; 
         FIG. 6  represents a plot of an optimized common mode voltage duty ratio over one output period at a DC or very low speed AC condition when the switching frequency is 4 kHz, in accordance with one embodiment of the present invention; 
         FIG. 7  represents a plot of an optimized common mode voltage duty ratio over one output period at a DC or very low speed AC condition when the switching frequency is 8 kHz, in accordance with one embodiment of the present invention; 
         FIG. 8  illustrates two plots of real-time maximum junction temperature over one output period under low speed AC condition when the switching frequency is 4 kHz, the first plot calculated using a traditional common mode voltage duty ratio and the second calculated using an optimized common mode voltage duty ratio in accordance with one embodiment of the present invention; 
         FIG. 9  illustrates two plots of real-time maximum junction temperature over one output period under low speed AC condition when the switching frequency is 8 kHz, the first calculated using a traditional common mode voltage duty ratio and the second calculated using an optimized common mode voltage duty ratio in accordance with one embodiment of the present invention; 
         FIG. 10  illustrates the difference between traditional and optimized maximum junction temperatures over one output period under low speed AC condition when the switching frequency is 4 kHz, in accordance with one embodiment of the present invention; and 
         FIG. 11  illustrates the difference between traditional and optimized maximum junction temperatures over one output period under low speed AC condition when the switching frequency is 8 kHz, in accordance with one embodiment of the present invention; 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a diagrammatical representation of an exemplary motor drive system employing control circuitry for reducing the maximum junction temperature of the inverter module. A three phase power source  2 , represented by three legs comprising an AC voltage source  4 , a source resistance  6 , and a source inductance  8 , provides three phase AC power of constant frequency to rectifier circuitry  10 . In the rectifier circuitry  10 , six diodes  12  provide full wave rectification of the three phase voltage waveform. Each leg of the three phase power source  2  entering rectifier circuitry  10  is coupled between two diodes  12  arranged in series, anode to cathode, which span from a negative DC bus line to a positive DC bus line. Line inductance  14  resides on both the positive and negative lines of the rectifier circuitry  10 , but two capacitors  16  are configured to minimize their effect. The two capacitors  16  link the positive DC bus line and negative DC bus line to neutral  18 , largely removing the AC component of the rectified voltage waveform. The resulting voltage difference between the positive DC bus line and the negative DC bus line remains essentially a DC voltage difference (labeled V dc ). 
     An inverter module  20  accepts the positive and negative lines of DC voltage from the rectifier circuitry  10  as inputs, outputting a discretized three phase waveform at a desired frequency, independent of the frequency of three phase power source  2 . At the positive end of each leg a, b, and c, an insulated gate bipolar transistor (IGBT) switch  22  (labeled S ap , S bp , or S cp ) paired with a diode  24  (labeled D ap , D bp  or D cp ) interconnects the positive DC bus line to the leg a, b, or c output. Similarly, at the negative end of each leg a, b, or c, an insulated gate bipolar transistor (IGBT) switch  22  (labeled S an , S bn , or S cn ), paired with a diode  24  (labeled D an , D bn  or D cn ), interconnects the negative DC bus line to the leg a, b, or c output. Driver circuitry  26  signals the IGBT switches  22  to rapidly close and open, resulting in a three phase waveform output across legs a, b, and c. Phase A is output on leg a, phase B is the output on leg b, and phase C is output on leg c. 
     The three legs a, b, and c provide three phase power to an external load, such as a motor  28 . Sensors  30  gather information about the power supplied to the motor  28 , which serves as feedback  32  for the control circuitry  34 . The control circuitry  34  takes commands  36  and feedback  32  to formulate the signals necessary to induce a three phase waveform in the inverter module through driver circuitry  26 . A networked remote control monitor  38  may be employed to supply commands  36  to the control circuitry  34 . 
       FIG. 2  diagrammatically represents exemplary control circuitry  34  in greater detail. To determine the necessary switching signals to send to driver circuitry  26 , a processor  40  obtains feedback data through a feedback interface  42 , commands through a command interface  44 , and other data, such as standard preset values in a lookup table, from memory circuitry  46 . The processor  40  then rapidly calculates the signals required to produce a three phase waveform by switching each IGBT switch  22  of inverter module  20 . For a given switching frequency f s  (often 4 kHz or 8 kHz) at each given voltage command phase angle θ v , the signals from processor  40  represent the precise fractions of each switching period that a switch remains closed or open. Because the voltage command phase angle θ v  changes continually, the switching signals similarly change continually to produce the three phase output waveform. 
     According to a method of the present invention, the processor  40  may first determine a common mode voltage duty ratio (d cm ) with d cm  determination circuitry  48 . The common mode voltage duty ratio d cm  represents one component of the total duty ratio for each leg a, b, and c relative to the negative DC bus (termed d an , d bn , and d cn ). Having determined the common mode voltage duty ratio d cm , the processor  40  may then calculate the total duty ratio for each phase A, B, and C with d n  determination circuitry  50 . After determining the three duty ratios d an , d bn , and d cn , the processor may next determine the necessary IGBT switch  22  control signals using command circuitry  52 . Command circuitry  52  may apply pulse width modulation (PWM) using the three duty ratios to obtain the precise amount of time each switch must remain open and closed. 
     The d cm  determination circuitry  48 , d n  determination circuitry  50 , and command circuitry  52  operate to minimize the maximum junction temperature according to one method of the present technique. To simplify further discussion of the present technique, seven approximations or assumptions are stipulated. First, the motor drive operates at steady state with very low speed AC or DC conditions. Second, the command circuitry  52  utilizes sine wave/triangle wave comparison pulse width modulation (PWM), though a similar improvement may be obtained using space vector pulse width modulation (SVPWM). Third, because the stator resistance value for a motor drive operating at steady state may be generally 3% to 5% of the rated resistance, the maximum amplitude of the output voltage need only reach 3% to 5% of maximum positive and negative voltage to supply all necessary current to the motor; the maximum output voltage amplitude ratio (k) is thus selected to be 0.05. Fourth, since the motor is operating at a very low frequency, the output power factor of the inverter is treated as being equal to 1. Fifth, because the junction-to-heat-sink thermal time constants of the IGBT switches remain far smaller than the period of the output waveform, any corresponding thermal capacitance is negligible. Sixth, the bottom of the heat-sink beneath the IGBT switches is treated as isothermal with a uniform temperature. Finally, power losses are calculated under rated conditions for a Rockwell 480V/50 hp drive with a DC bus voltage of 650V, 4 kHz switching frequency, and 77 A rms  inverter current; as such, the junction-to-heat-sink thermal impedances for an IGBT switch (R i0 ) and for a diode (R d0 ) are estimated as 0.42 K/W and 0.66 K/W, respectively. 
     The duty ratios for phases A, B, and C (d a , d b , and d c ), as a function of the common mode voltage duty ratio d cm  and phase angle of voltage command of the inverter θ v , may be represented as: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             d 
                             a 
                           
                           = 
                           
                             0.5 
                             + 
                             
                               d 
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 m 
                               
                             
                             + 
                             
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     θ 
                                     v 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             d 
                             b 
                           
                           = 
                           
                             0.5 
                             + 
                             
                               d 
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 m 
                               
                             
                             + 
                             
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       v 
                                     
                                     - 
                                     
                                       
                                         2 
                                         ⁢ 
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             d 
                             c 
                           
                           = 
                           
                             0.5 
                             + 
                             
                               d 
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 m 
                               
                             
                             + 
                             
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                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         v 
                                       
                                       + 
                                       
                                         
                                           2 
                                           ⁢ 
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                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     In equation (1), above, θ v  represents the angle of the voltage command of the inverter and, as previously discussed, k represents the output voltage amplitude ratio, approximated at 0.05: 
     
       
         
           
             
               
                 
                   
                     θ 
                     v 
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
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                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       k 
                     
                     = 
                     
                       
                         
                           V 
                           m 
                         
                         
                           
                             V 
                             dc 
                           
                           / 
                           2 
                         
                       
                       ≈ 
                       
                         0.05 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     With the above approximations or assumptions, the output voltages for phases A, B, and C relative to the neutral DC bus voltage (V an , V bn , and V cn ) may be represented as: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             V 
                             an 
                           
                           = 
                           
                             
                               ( 
                               
                                 
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                                   a 
                                 
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                                   d 
                                   
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                                     ⁢ 
                                     
                                         
                                     
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                             · 
                             
                               
                                 V 
                                 dc 
                               
                               / 
                               2 
                             
                           
                         
                       
                     
                     
                       
                         
                           
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                             bn 
                           
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                                   d 
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                                 dc 
                               
                               / 
                               2 
                             
                           
                         
                       
                     
                     
                       
                         
                           
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                           = 
                           
                             
                               ( 
                               
                                 
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                                 dc 
                               
                               / 
                               2. 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Similarly, when I m  represents the maximum amplitude of the output current, the output current for each phase A, B, and C may be represented as: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               i 
                               a 
                             
                             ≈ 
                             
                               
                                 I 
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                                   ( 
                                   
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                                 ( 
                                 
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                                 ⁡ 
                                 
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                                     + 
                                     
                                       
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                                       3 
                                     
                                   
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                           = 
                           
                             
                               I 
                               m 
                             
                             ⁢ 
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       v 
                                     
                                     + 
                                     
                                       
                                         2 
                                         ⁢ 
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
       FIG. 3  represents a model of a single phase leg of the three phase inverter module, which may provide, in combination with equations (1)-(4) above, a manner of calculating the conduction losses and switching losses of a complete three phase inverter module. Representing the voltage difference between the positive and negative DC bus lines of a three phase inverter, DC voltage supply  54  provides a constant supply of DC voltage (labeled V dc ). An IGBT switch  22  (S p ), paired with a diode  24  (D p ), interconnects the positive DC voltage node to the output. Similarly, an IGBT switch  22  (S n ), paired with a diode  24  (D n ), interconnects the negative DC voltage node to the output. The S n  IGBT switch  22  and the S p  IGBT switch  22  each receive a distinct gating signal, switching each IGBT switch closed when the respective gating signal is high and open when the signal is low. Output current  56  (labeled I c ) connects the output to the negative voltage node. 
       FIG. 4  depicts a graph relating the gating signals of the two IGBT switches  22  of  FIG. 3  and the duty ratio (labeled d). The abscissa of the graph represents time across one switching period  58 . Because the switching frequency f s  is typically very fast, e.g. 4 kHz or 8 kHz, one switching period  58  is correspondingly short, necessarily repeating thousands of times each second. For each switching period  58 , a prescribed duty ratio  60 , overlaid against a triangle pulse of one switching period with a peak value of 1, determines the S p  IGBT switch gating signal  62  and the S n  IGBT switch gating signal  64 . The point in time at which the duty ratio value first intersects the triangle pulse represents the point in time at which the S p  IGBT switch gating signal  62  rises from low to high and the S n  IGBT switch gating signal  64  drops from high to low. The second intersection of the duty ratio value and the triangle pulse represents the point in time at which the S p  IGBT switch gating signal  62  returns to low from high and the S n  IGBT switch gating signal  64  returns to high from low. Since the duty ratio varies as a function of the voltage output phase angle, as apparent in equation (2), above, the S p  IGBT switch gating signal  62  and the S n  IGBT switch gating signal  64  also depend on the voltage output phase angle. 
     Under the model of the single phase leg represented in  FIGS. 3 and 4 , the forward voltages of the S p  and S n  IGBT switches  22  and D p  and D n  diodes  24  may be further represented as a voltage source plus a small resistance during conduction. The forward voltage drop of an IGBT switch  22 , V cc (t), and the forward voltage drop of a diode  24 , V f (t), may be represented as:
 
 V   ce ( t )= V   ce0   +R   I   I   c ( t )
 
 V   f ( t )= V   f0   +R   d   I   c ( t )  (5).
 
In the above equation, V ce0  and V f0  represent the forward voltage when current is almost zero, and R I  and R d  represent the IGBT switch or diode slope resistance. Values for V ce0 , V f0 , R I  and R d  may be obtained from the IGBT datasheet directly.
 
     Total power losses for the single phase leg may be calculated as both conduction losses and switching losses. Conduction occurs for the S p  IGBT switch  22  and the D n  diode  24  when the S p  IGBT switch gating signal  62  is high and the S n  IGBT switch gating signal  64  is low. Similarly, conduction occurs for the S n  IGBT switch  22  and the D p  diode  24  when the S n  IGBT switch gating signal  64  is high and the S p  IGBT switch gating signal  62  is low. Accordingly, conduction losses may be calculated as: 
     
       
         
           
             
               
                 
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                   6 
                   ) 
                 
               
             
           
         
       
     
     Switching losses for the single phase leg may be calculated from the motor drive datasheet. The average switching losses of an IGBT switch  22  and a diode  24 , P sw.IGBT (t) and P sw.diode (t), may be calculated as: 
                         P     sw   ·   IGBT       ⁡     (   t   )       =           E   onoff     ·       I   c     ⁡     (   t   )         ⁢         f   s     ⁢     V   dc           V   nom     ⁢     I   nom           =       k   s     ⁢     f   s     ⁢     V   dc     ⁢            I   c     ⁡     (   t   )                  ⁢     
     ⁢           P     sw   ·   Diode       ⁡     (   t   )       =           E   rec     ·       I   c     ⁡     (   t   )         ⁢         f   s     ⁢     V   dc           V   nom     ⁢     I   nom           =       k   d     ⁢     f   s     ⁢     V   dc     ⁢            I   c     ⁡     (   t   )                  ,             (   7   )               
where E onoff  and E rec  represent energy spent during switching, and V nom  and I nom  represent the rated DC bus voltage and current. The symbols k s  and k d  are employed to simplify the single phase leg switching loss equation (7).
 
     The total power losses for a three-phase inverter may be obtained in light of the power loss equations for a single phase leg (6) and (7). By combining equations (1)-(6), the conduction losses for any phase x (where x represents a, b, or c) of the inverter may be represented as: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       P 
                                       
                                         cond 
                                         , 
                                         sxp 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   = 
                                   
                                     
                                       d 
                                       x 
                                     
                                     · 
                                     
                                       
                                         V 
                                         ce 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     · 
                                     
                                       
                                         i 
                                         x 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       P 
                                       
                                         cond 
                                         , 
                                         sxn 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   = 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       P 
                                       
                                         cond 
                                         , 
                                         dxp 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   = 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       P 
                                       
                                         cond 
                                         , 
                                         dxn 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   = 
                                   
                                     
                                       ( 
                                       
                                         1 
                                         - 
                                         
                                           d 
                                           x 
                                         
                                       
                                       ) 
                                     
                                     · 
                                     
                                       
                                         V 
                                         f 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     · 
                                     
                                       
                                         i 
                                         x 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 for 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     i 
                                     x 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                               ≥ 
                               0 
                             
                             , 
                             and 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       
                                         P 
                                         
                                           cond 
                                           , 
                                           sxp 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     = 
                                     0 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         P 
                                         
                                           cond 
                                           , 
                                           sxn 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     = 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           
                                             d 
                                             x 
                                           
                                         
                                         ) 
                                       
                                       · 
                                       
                                         
                                           V 
                                           ce 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                       · 
                                       
                                          
                                         
                                           
                                             i 
                                             x 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             t 
                                             ) 
                                           
                                         
                                          
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         P 
                                         
                                           cond 
                                           , 
                                           dxp 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     = 
                                     
                                       
                                         d 
                                         x 
                                       
                                       · 
                                       
                                         
                                           V 
                                           f 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                       · 
                                       
                                          
                                         
                                           
                                             i 
                                             x 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             t 
                                             ) 
                                           
                                         
                                          
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         P 
                                         
                                           cond 
                                           , 
                                           dxn 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     = 
                                     0 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               for 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   i 
                                   x 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             &lt; 
                             0. 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Equation (8), above, illustrates that the conduction loss calculation for leg of phase x (where x represents a, b, or c) of the three phase inverter is influenced by the common mode voltage duty ratio d cm . More particularly, when i x (t) is greater than 0, an increase in the common mode voltage duty ratio d cm  causes the conduction losses of the S xp  IGBT switch  22  to increase and the conduction losses of the D xn  diode  24  to decrease. On the other hand, when i x (t) is less than 0, an increase in the common mode voltage duty ratio d cm  causes the conduction losses of the S xn  IGBT switch  22  to decrease and the conduction losses of the D xp  diode  24  to increase. From equation (8), above, the total conduction losses of the S xp  and S xn  IGBT switches  22  and the D xp  and D xn  diodes  24  for each leg of phase x (where x represents a, b, or c) may be described as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               P 
                               
                                 cond 
                                 , 
                                 sx 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           = 
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       P 
                                       
                                         cond 
                                         , 
                                         sxp 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         i 
                                         x 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     ≥ 
                                     0 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       P 
                                       
                                         cond 
                                         , 
                                         sxn 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         i 
                                         x 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     &lt; 
                                     0 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         and 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         P 
                         
                           cond 
                           , 
                           dx 
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   P 
                                   
                                     cond 
                                     , 
                                     dxn 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   
                                     i 
                                     x 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 ≥ 
                                 0 
                               
                             
                           
                           
                             
                               
                                 
                                   P 
                                   
                                     cond 
                                     , 
                                     dxp 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   
                                     i 
                                     x 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 &lt; 
                                 0 
                               
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Similarly, the switching losses for each leg of phase x (where x represents a, b, or c) do not depend on the common mode voltage duty ratio d cm . From equation (7), above, the following equations may be derived for each leg of phase x: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       P 
                                       
                                         sw 
                                         , 
                                         sxp 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   = 
                                   
                                     
                                       
                                         k 
                                         s 
                                       
                                       · 
                                       
                                         V 
                                         
                                           d 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           c 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         
                                           i 
                                           x 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                       · 
                                       
                                         f 
                                         s 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       P 
                                       
                                         sw 
                                         , 
                                         sxn 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   = 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       P 
                                       
                                         sw 
                                         , 
                                         dxp 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   = 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       P 
                                       
                                         sw 
                                         , 
                                         dxn 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   = 
                                   
                                     
                                       
                                         k 
                                         d 
                                       
                                       · 
                                       
                                         V 
                                         
                                           d 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           c 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         
                                           i 
                                           x 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                       · 
                                       
                                         f 
                                         s 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 for 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     i 
                                     x 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                               ≥ 
                               0 
                             
                             , 
                             and 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       
                                         P 
                                         
                                           sw 
                                           , 
                                           sxp 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     = 
                                     0 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         P 
                                         
                                           sw 
                                           , 
                                           sxn 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     = 
                                     
                                       
                                         k 
                                         s 
                                       
                                       · 
                                       
                                         V 
                                         
                                           d 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           c 
                                         
                                       
                                       · 
                                       
                                          
                                         
                                           
                                             i 
                                             x 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             t 
                                             ) 
                                           
                                         
                                          
                                       
                                       · 
                                       
                                         f 
                                         s 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         P 
                                         
                                           sw 
                                           , 
                                           dxp 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     = 
                                     
                                       
                                         k 
                                         d 
                                       
                                       · 
                                       
                                         V 
                                         
                                           d 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           c 
                                         
                                       
                                       · 
                                       
                                          
                                         
                                           
                                             i 
                                             x 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             t 
                                             ) 
                                           
                                         
                                          
                                       
                                       · 
                                       
                                         f 
                                         s 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         P 
                                         
                                           sw 
                                           , 
                                           dxn 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     = 
                                     0 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               for 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   i 
                                   x 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             &lt; 
                             0. 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     From equation (10), above, the total switching losses of the S xp  and S xn  IGBT switches  22  and the D xp  and D xn  diodes  24  for each leg of phase x (where x represents a, b, or c) may also be described as follows: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 P 
                                 
                                   sw 
                                   , 
                                   sx 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             = 
                             
                               
                                 
                                   k 
                                   s 
                                 
                                 · 
                                 
                                   V 
                                   
                                     d 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     c 
                                   
                                 
                               
                               ⁢ 
                               
                                  
                                 
                                   
                                     i 
                                     x 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                  
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 P 
                                 
                                   sw 
                                   , 
                                   dx 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             = 
                             
                               
                                 
                                   k 
                                   d 
                                 
                                 · 
                                 
                                   V 
                                   
                                     d 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     c 
                                   
                                 
                               
                               ⁢ 
                               
                                  
                                 
                                   
                                     i 
                                     x 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                  
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Because the conduction losses and switching losses represent the total losses for each leg of phase x (where x represents a, b, or c), total losses may be represented as: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 P 
                                 sx 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             = 
                             
                               
                                 
                                   P 
                                   
                                     cond 
                                     , 
                                     sx 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   P 
                                   
                                     sw 
                                     , 
                                     sx 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 P 
                                 dx 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             = 
                             
                               
                                 
                                   P 
                                   
                                     cond 
                                     , 
                                     dx 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   P 
                                   
                                     sw 
                                     , 
                                     dx 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     The maximum junction temperatures of the IGBT switches  22  and diodes  24  of the three phase inverter module may be estimated using the equations describing the power losses of the three phase inverter module, above, keeping in mind three stipulations or assumptions. First, the bottom of the heat sink is treated as isothermal. Second, thermal coupling between the two chips is weak and may be neglected. Third, operating at a very low output frequency, because the output period is very large relative to the junction-to-heat-sink thermal time constants of the IGBT switches  22  and diodes  24 , any corresponding thermal capacitance is negligible. Accordingly, the maximum junction temperature for the IGBT switches  22  and diodes  24  of each leg of the three phase inverter, T jsx,max  and T jds,max , respectively, may be represented as follows (where x represents leg a, b, or c): 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               T 
                               
                                 jsx 
                                 , 
                                 max 
                               
                             
                             ≈ 
                             
                               
                                 
                                   P 
                                   sx 
                                 
                                 · 
                                 
                                   R 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     0 
                                   
                                 
                               
                               + 
                               
                                 T 
                                 h 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               T 
                               
                                 jdx 
                                 , 
                                 max 
                               
                             
                             ≈ 
                             
                               
                                 
                                   P 
                                   dx 
                                 
                                 · 
                                 
                                   R 
                                   
                                     d 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     0 
                                   
                                 
                               
                               + 
                               
                                 T 
                                 h 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     In equation (13), above, T h  represents the temperature at the bottom of the heat sink, which is determined largely by the ambient temperature and air flow through the heat sink. To simplify further analysis, T h  may be treated as very nearly fixed, such that only the maximum temperature difference between the junction temperature and the heat sink temperature T h  may be obtained. The maximum junction temperature relative to the heat sink temperature T h , for the IGBT switches  22  and diodes  24  of each leg of the three phase inverter, T jhsx,max  and T jhdx,max , respectively, may thus be represented as follows (where x represents leg a, b, or c): 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             T 
                             
                               jhsx 
                               , 
                               max 
                             
                           
                           = 
                           
                             
                               
                                 T 
                                 
                                   jsx 
                                   , 
                                   max 
                                 
                               
                               - 
                               
                                 T 
                                 h 
                               
                             
                             ≈ 
                             
                               
                                 P 
                                 sx 
                               
                               · 
                               
                                 R 
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             T 
                             
                               jhdx 
                               , 
                               max 
                             
                           
                           = 
                           
                             
                               
                                 T 
                                 
                                   jdx 
                                   , 
                                   max 
                                 
                               
                               - 
                               
                                 T 
                                 h 
                               
                             
                             ≈ 
                             
                               
                                 P 
                                 dx 
                               
                               · 
                               
                                 R 
                                 
                                   d 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     An optimal maximum junction temperature for the entire three phase inverter may occur when total power losses are spread as evenly as possible across the components of the inverter. To simplify the analysis, only the output current phase angle θ i  between (−π/6) and (π/6) is initially examined, when phase A reaches its maximum phase current and phases B and C are both negative. The following equation thus represents a configuration when the maximum junction temperature relative to T h  of the IGBT switches  22  of the leg is the same as that of the diodes  24  of the leg:
 
 P   sa ( t )· R   i0   =P   da ( t )· R   d0   (15).
 
     By combining equation (15) with equations (8)-(11), an optimal common mode voltage duty ratio for phase A while i a ≧0, d cm1 , may be calculated: 
     
       
         
           
             
               
                 
                   
                     d 
                     
                       c 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       d 
                                       a 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   V 
                                   f 
                                 
                                 ⁢ 
                                 
                                   R 
                                   
                                     d 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     0 
                                   
                                 
                               
                               - 
                               
                                 
                                   d 
                                   a 
                                 
                                 ⁢ 
                                 
                                   V 
                                   ce 
                                 
                                 ⁢ 
                                 
                                   R 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     0 
                                   
                                 
                               
                             
                             ] 
                           
                           · 
                           
                             i 
                             a 
                           
                         
                         + 
                         
                           P 
                           
                             sw 
                             , 
                             da 
                           
                         
                         - 
                         
                           P 
                           
                             sw 
                             , 
                             sa 
                           
                         
                       
                       
                         
                           ( 
                           
                             
                               
                                 V 
                                 ce 
                               
                               ⁢ 
                               
                                 R 
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                             
                             + 
                             
                               
                                 V 
                                 f 
                               
                               ⁢ 
                               
                                 R 
                                 
                                   d 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                             
                           
                           ) 
                         
                         · 
                         
                           i 
                           a 
                         
                       
                     
                     - 
                     
                       0.5 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Recalling equation (8), above, when i a (t)≧0, a reduction of the common mode voltage duty ratio d cm  lowers the conduction losses of the S ap  IGBT switch  22  and raises the conduction losses of the D an  diode  24 . If the S ap  IGBT switch  22  is producing greater conduction losses than D an  diode  24 , reducing the common mode voltage duty ratio d cm  would allow for greater parity in conduction losses between the S ap  IGBT switch  22  and D an  diode  24 . However, because i b (t)&lt;0 and i c (t)&lt;0 when the output current phase angle θ i  is between (−π/6) and (π/6), any reduction in d cm  would simultaneously raise the conduction losses of the S bn  and S cn  IGBT switches  22 . Reducing d cm  to lower conduction losses in the S ap  IGBT switch  22  by too great an amount could therefore result in excessive heat losses in the S bn  and S cn  IGBT switches  22 . To avoid reducing d cm  excessively, the following condition should also be met:
 
 P   sa ( t )≧ P   sb ( t ) and  P   sa ( t )≧ P   sc ( t )  (17).
 
     A lower bound of the optimized common mode voltage duty ratio d cm2  for phases B and C may thus be determined by combining equations (17) and (8)-(12). Representing the lowest possible optimal common mode voltage duty ratio d cm  when the output current phase angle θ i  is between (−π/6) and (π/6), d cm2,y  may be represented as follows (where y represents phase b or c; V ce,y  represents the forward voltage drop of the particular phase IGBT switch  22 ): 
     
       
         
           
             
               
                 
                   
                     d 
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                       , 
                       y 
                     
                   
                   = 
                   
                     
                       
                         
                           P 
                           
                             sw 
                             , 
                             sy 
                           
                         
                         - 
                         
                           P 
                           
                             sw 
                             , 
                             sa 
                           
                         
                         + 
                         
                           
                             V 
                             
                               ce 
                               , 
                               y 
                             
                           
                           · 
                           
                              
                             
                               i 
                               y 
                             
                              
                           
                           · 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 d 
                                 y 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             V 
                             
                               ce 
                               , 
                               a 
                             
                           
                           · 
                           
                             i 
                             a 
                           
                           · 
                           
                             d 
                             a 
                           
                         
                       
                       
                         
                           
                             V 
                             
                               ce 
                               , 
                               a 
                             
                           
                           · 
                           
                             i 
                             a 
                           
                         
                         + 
                         
                           
                             V 
                             
                               ce 
                               , 
                               y 
                             
                           
                           · 
                           
                              
                             
                               i 
                               y 
                             
                              
                           
                         
                       
                     
                     - 
                     
                       0.5 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     Again recalling equation (8), above, when i a (t)≧0, an increase of the common mode voltage duty ratio d cm  lowers the conduction losses of the D an  diode  24  and raises the conduction losses of the S ap  IGBT switch  22 . If the D an  diode  24  is producing greater conduction losses than the S ap  IGBT switch  22 , increasing the common mode voltage duty ratio d cm  would allow for greater parity in conduction losses between the S ap  IGBT switch  22  and D an  diode  24 . However, because i b (t)&lt;0 and i c (t)&lt;0 when the output current phase angle θ i  is between (−π/6) and (π/6), any increase in d cm  would simultaneously raise the conduction losses of the D bn  and D cn  diodes  24 . Increasing d cm  to lower conduction losses in the D an  diode  24  by too great an amount could therefore result in excessive heat losses in the D bn  and D cn  diodes  24 . To avoid increasing d cm  excessively, the following condition should also be met:
 
 P   da ( t )≧ P   db ( t ) and  P   da ( t )≧ P   dc ( t )  (19)
 
     An upper bound of the optimized common mode voltage duty ratio d cm3  for phases B and C may thus be determined by combining equations (17) and (8)-(12). Representing the highest possible optimal common mode voltage duty ratio d cm  when the output current phase angle θ i  is between (−π/6) and (π/6), d cm3,y  may be represented as follows (where y represents phase b or c; V f,y  represents the forward voltage drop of the particular phase IGBT switch  22 ): 
     
       
         
           
             
               
                 
                   
                     d 
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         3 
                       
                       , 
                       y 
                     
                   
                   = 
                   
                     
                       
                         
                           P 
                           
                             sw 
                             , 
                             da 
                           
                         
                         - 
                         
                           P 
                           
                             sw 
                             , 
                             dy 
                           
                         
                         + 
                         
                           
                             V 
                             
                               f 
                               , 
                               a 
                             
                           
                           · 
                           
                             i 
                             a 
                           
                           · 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 d 
                                 a 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             V 
                             
                               f 
                               , 
                               y 
                             
                           
                           · 
                           
                              
                             
                               i 
                               y 
                             
                              
                           
                           · 
                           
                             d 
                             y 
                           
                         
                       
                       
                         
                           
                             V 
                             
                               f 
                               , 
                               a 
                             
                           
                           · 
                           
                             i 
                             a 
                           
                         
                         + 
                         
                           
                             V 
                             
                               f 
                               , 
                               y 
                             
                           
                           · 
                           
                              
                             
                               i 
                               y 
                             
                              
                           
                         
                       
                     
                     - 
                     
                       0.5 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     For any instantaneous output current phase angle θ i  between (−π/6) and (π/6), an optimal common mode voltage duty ratio d cm  may be obtained from equations (16), (18), and (20). On the leg carrying phase A, optimal common mode voltage duty ratio d cm1  represents a value for d cm  which most evenly distributes the conduction losses between the S ap  IGBT switch  22  and D an  diode  24 . If the value of d cm1  remains above the lower bound optimal common mode voltage duty ratio d cm2  (where d cm2  is the higher of d cm2,b  and d cm2,c ) and below the upper bound common mode voltage duty ratio d cm3  (where d cm3  is the lower of d cm3,b  and d cm3,c ), then d cm1  also represents the optimized common mode voltage d cm . However, if d cm1  is lower than the lower bound optimal common mode voltage duty ratio d cm2 , then d cm2  represents the optimized common mode voltage duty ratio d cm . Similarly, if d cm1  is higher than the upper bound optimal common mode voltage duty ratio d cm3 , then d cm3  represents the optimized common mode voltage duty ratio d cm . When the control circuitry  34  of a motor control system employs the above common mode voltage duty ratio d cm , the resulting control signals may provide the inverter module  20  a higher maximum junction temperature rating. 
     Though the above analysis stipulates an output current phase angle θ i  between (−π/6) and (π/6),  FIG. 5  illustrates a more general method of obtaining an optimal common mode voltage duty ratio d cm  for any given output current phase angle θ i . The d cm  determination flowchart  66  begins with step  68 , to locate the phase with maximum current. If the output current phase angle θ i  is between (−π/6) and (π/6), as well as any multiple of (2π) in addition to or subtracted from between (−π/6) and (π/6), the maximum current occurs in phase A, while the currents of phases B and C are negative. If the output current phase angle θ i  is between (π/6) and (5π/6), as well as any multiple of (2π) in addition to or subtracted from between (π/6) and (5π/6), the maximum current occurs in phase B, while the currents of phases A and C are negative. Finally, if the output current phase angle θ i  is between (5π/6) and (3π/2), as well as all multiples of (2π) in addition to or subtracted from between (5π/6) and (3π/2), the maximum current occurs in phase C, while the currents of phases A and B are negative. 
     After locating the phase with the maximum current per step  68 , step  70  next prescribes calculating the value of the maximum phase current optimal common mode voltage duty ratio d cm1 . As discussed above, d cm1  represents a value for d cm  which most evenly distributes the conduction losses between the S xp  IGBT switch  22  and D xn  diode  24  (where x represents the phase with maximum current located in step  68 ). If d cm1  is a value less than zero, the reduction in d cm  would cause fewer conduction losses in the S xp  IGBT switch  22  and greater conduction losses in the D xn  diode  24 . On the other hand, if d cm1  is a value greater than zero, the increase in d cm  would cause fewer conduction losses in the D an  diode  24  and greater conduction losses in the S ap  IGBT switch  22 . The value of d cm1  may be expressed in the general equation below (where x represents the phase with maximum current located in step  68 ): 
     
       
         
           
             
               
                 
                   
                     d 
                     
                       c 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       d 
                                       x 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   V 
                                   f 
                                 
                                 ⁢ 
                                 
                                   R 
                                   
                                     d 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     0 
                                   
                                 
                               
                               - 
                               
                                 
                                   d 
                                   x 
                                 
                                 ⁢ 
                                 
                                   V 
                                   ce 
                                 
                                 ⁢ 
                                 
                                   R 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     0 
                                   
                                 
                               
                             
                             ] 
                           
                           · 
                           
                             i 
                             x 
                           
                         
                         + 
                         
                           P 
                           
                             sw 
                             , 
                             dx 
                           
                         
                         - 
                         
                           P 
                           
                             sw 
                             , 
                             sx 
                           
                         
                       
                       
                         
                           ( 
                           
                             
                               
                                 V 
                                 ce 
                               
                               ⁢ 
                               
                                 R 
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                             
                             + 
                             
                               
                                 V 
                                 f 
                               
                               ⁢ 
                               
                                 R 
                                 
                                   d 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                             
                           
                           ) 
                         
                         · 
                         
                           i 
                           x 
                         
                       
                     
                     - 
                     
                       0.5 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Step  72  next prescribes calculating the value of the lower bound optimal common mode voltage duty ratio d cm2 . To obtain d cm2 , the value d cm2,y  may be first calculated for each phase y (where y represents the phases other than phase x, the phase with maximum current located in step  68 ): 
     
       
         
           
             
               
                 
                   
                     d 
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                       , 
                       y 
                     
                   
                   = 
                   
                     
                       
                         
                           P 
                           
                             sw 
                             , 
                             sy 
                           
                         
                         - 
                         
                           P 
                           
                             sw 
                             , 
                             sx 
                           
                         
                         + 
                         
                           
                             V 
                             
                               ce 
                               , 
                               y 
                             
                           
                           · 
                           
                              
                             
                               i 
                               y 
                             
                              
                           
                           · 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 d 
                                 y 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             V 
                             
                               ce 
                               , 
                               x 
                             
                           
                           · 
                           
                             i 
                             x 
                           
                           · 
                           
                             d 
                             x 
                           
                         
                       
                       
                         
                           
                             V 
                             
                               ce 
                               , 
                               x 
                             
                           
                           · 
                           
                             i 
                             x 
                           
                         
                         + 
                         
                           
                             V 
                             
                               ce 
                               , 
                               y 
                             
                           
                           · 
                           
                              
                             
                               i 
                               y 
                             
                              
                           
                         
                       
                     
                     - 
                     
                       0.5 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     The value of d cm2,y  for each phase y represents the lower bound optimal common mode voltage duty ratio of that phase, or the point at which d cm  could not be further reduced without creating excessive conduction losses in the S yn  IGBT switches  22  of each phase y (where y represents the phases other than phase x, the phase with maximum current located in step  68 ). Accordingly, the higher of the two values of d cm2,y  represents the lower bound optimal common mode voltage duty ratio d cm2  of the entire inverter module. Step  72  thus completes when the higher value of d cm2,y  is chosen as the lower bound optimal common mode voltage duty ratio d cm2 . 
     Proximate step  74  prescribes calculating the value of the upper bound optimal common mode voltage duty ratio d cm3 . To obtain d cm3 , the value d cm3,y  may be first calculated for each phase y (where y represents the phases other than phase x, the phase with maximum current located in step  68 ): 
     
       
         
           
             
               
                 
                   
                     d 
                     
                       
                         c 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         3 
                       
                       , 
                       y 
                     
                   
                   = 
                   
                     
                       
                         
                           P 
                           
                             sw 
                             , 
                             dx 
                           
                         
                         - 
                         
                           P 
                           
                             sw 
                             , 
                             dy 
                           
                         
                         + 
                         
                           
                             V 
                             
                               f 
                               , 
                               x 
                             
                           
                           · 
                           
                             i 
                             x 
                           
                           · 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 d 
                                 x 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             V 
                             
                               f 
                               , 
                               y 
                             
                           
                           · 
                           
                              
                             
                               i 
                               y 
                             
                              
                           
                           · 
                           
                             d 
                             y 
                           
                         
                       
                       
                         
                           
                             V 
                             
                               f 
                               , 
                               x 
                             
                           
                           · 
                           
                             i 
                             x 
                           
                         
                         + 
                         
                           
                             V 
                             
                               f 
                               , 
                               y 
                             
                           
                           · 
                           
                              
                             
                               i 
                               y 
                             
                              
                           
                         
                       
                     
                     - 
                     
                       0.5 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     Since the value of d cm3,y  for each phase y represents the upper bound optimal common mode voltage duty ratio of that phase, or the point at which d cm  could not be further increased without creating excessive conduction losses in the D yp  diodes  24  of each phase y (where y represents the phases other than phase x, the phase with maximum current located in step  68 ). Accordingly, the lower of the two values of d cm3,y  represents the lower bound optimal common mode voltage duty ratio d cm3  of the entire inverter module. Step  74  terminates when the lower value of d cm3,y  is chosen as the upper bound optimal common mode voltage duty ratio d cm2 . 
     Continuing to follow the d cm  determination flowchart  66  of  FIG. 5 , and having calculated d cm1 , d cm2 , and d cm3 , the remaining decision blocks and steps describe how to choose the optimal common mode voltage duty ratio d cm  from among d cm1 , d cm2 , and d cm3 . Decision block  76  compares the values of the maximum phase current optimal common mode voltage duty ratio d cm1  with the lower bound optimal common mode voltage duty ratio d cm2 . As discussed above, d cm2  represents the lowest possible reduction of d cm  available before the S yn  IGBT switches  22  (where y represents the two phases not carrying the maximum current) would produce excessive conduction losses. If d cm1  is not greater than the lower bound d cm2 , decision block  76  directs the process flow to step  78 , illustrating that the maximum inverter module junction temperature reduction will occur when the common mode voltage duty ratio d cm  is equal to d cm2 . If decision block  76  determines d cm1  is greater than or equal to the lower bound d cm2 , the process instead flows to decision block  80 . 
     Decision block  80  compares the values of the maximum phase current optimal common mode voltage duty ratio d cm1  with the upper bound optimal common mode voltage duty ratio d cm3 . As discussed above, d cm3  represents the greatest possible increase in d cm  available before the D yp  diodes  24  (where y represents the two phases not carrying the maximum current) would produce excessive conduction losses. If d cm1  is less than the upper bound d cm3 , decision block  80  directs the process flow to step  82 , as the maximum inverter module junction temperature reduction will occur when the common mode voltage duty ratio d cm  is equal to d cm1 . If d cm1  is not less than or equal to the upper bound d cm3 , however, decision block  80  instead directs the process flow to step  84 , as the maximum inverter module junction temperature reduction will occur when the common mode voltage duty ratio d cm  is equal to d cm3 . Upon reaching any of steps  78 ,  82 , or  84 , the value of d cm  for a given output current phase angle θ i  is determined and the d cm  determination flowchart  66  terminates. 
     It should be noted that in the approach outlined in  FIG. 5 , the maximum current is positive. In a more general case, the algorithm may determine whether the maximum current is positive or negative, and, if it is negative, the values for the phase currents are inverted in step  68  (i.e., i a =−i a , i b =−i b , i c =−i c ). The calculations and comparisons summarized above may then proceed based upon these inverted values. However, in the case of a negative maximum current, the resulting value of d cm  is also inverted in step  48  (i.e., d cm =−d cm ). That is, the value of d cm  is negative when the maximum current is negative. 
       FIGS. 6 and 7  illustrate the real-time determinations of d cm (θ i ) for a Rockwell 480V/50 hp drive with a DC bus voltage of 650V and 77 A rms  inverter current across one output period, operating with a switching frequency f s  of 4 kHz and 8 kHz, respectively. Turning to  FIG. 6 , with an ordinate reflecting values of d cm  and an abscissa reflecting the inverter module output current phase angle θ i  from 0 to 6.28 (2π) radians, plot  86  shows the real-time relationship between d cm  and θ i . When θ i  is 0 radians, real-time d cm (θ i ) value  88  begins near −0.07 and remains flat until around 0.35 (π/9) radians, when it rises essentially linearly to near 0.07 by 0.70 (2π/9) radians. The real-time d cm (θ i ) value  88  remains near 0.07 from 0.70 (2π/9) radians to around 1.40 (4π/9) radians, before returning again to −0.07 by 1.75 (5π/9) radians. From 1.75 (5π/9) radians to 2.45 (7π/9) radians, the real-time d cm (θ i ) value  88  remains near −0.07, at which point the real-time d cm (θ i ) value  88  then rises again to 0.07 by 2.80 (8π/9) radians. Plot  86  makes apparent that the real-time d cm (θ i ) value  88  varies with a period of (2π/3) radians, reaching a trough from (−π/9) to (π/9) radians and a crest from (2π/9) to (4π/9) radians. 
     Turning now to  FIG. 7 , with an ordinate reflecting values of d cm (θ i ) and an abscissa reflecting the inverter module output current phase angle θ i  from 0 to 6.28 (2π) radians, plot  90  similarly illustrates the real-time relationship between d cm  and θ i  when the switching frequency is 8 kHz. Plot  90  makes apparent that the real-time d cm (θ i ) value  92  varies with a period of (2π/3) radians, reaching a trough of around −0.17 from (−π/9) to (π/9) radians and a crest of around 0.17 from (2π/9) to (4π/9) radians. Although the real-time d cm (θ i ) value  92  briefly retreats by a small amount after reaching the minimum and minimum values of d cm (θ i ), plot  90  of  FIG. 7  only significantly varies from plot  86  in the magnitude of the real-time d cm (θ i ). 
     Though a motor drive system may calculate the optimal common mode voltage duty ratio d cm (θ i ) in real-time, two alternative methods of closely estimating d cm (θ i ) may prove more efficient. Under both methods of closely estimating optimal common mode voltage d cm (θ i ), the amplitude of d cm (θ i ), termed d m , is first determined according to the equation below:
 
 d   m   =d   cm (0)  (24).
 
     The amplitude d m  may be calculated by the control circuitry  34  of a motor drive if supplied necessary data, or may be calculated prior to manufacture and stored in a lookup table in memory. If calculated prior to manufacture, a range of possible amplitudes d m  may be calculated for a variety potential motor drive components. 
     A first method of estimating d cm (θ i ) appears in the equation below, where d cm (θ i ) is estimated as a cosine wave with amplitude d m :
 
 d   cm ′(θ i )= d   m  cos(3·Θ i )  (25).
 
     A second method of estimating d cm (θ i ) involves the discrete equation below: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             d 
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               m 
                             
                             ″ 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 θ 
                                 i 
                               
                               + 
                               
                                 k 
                                 · 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                   
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           { 
                           
                             
                               
                                 
                                   d 
                                   m 
                                 
                               
                               
                                 
                                   - 
                                   
                                     ⁣ 
                                     
                                       
                                         π 
                                         9 
                                       
                                       &lt; 
                                       
                                         θ 
                                         i 
                                       
                                       &lt; 
                                       
                                         π 
                                         9 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   
                                     π 
                                     9 
                                   
                                   &lt; 
                                   
                                     θ 
                                     i 
                                   
                                   &lt; 
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                     9 
                                   
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     d 
                                     m 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                     9 
                                   
                                   &lt; 
                                   
                                     θ 
                                     i 
                                   
                                   &lt; 
                                   
                                     
                                       4 
                                       ⁢ 
                                       π 
                                     
                                     9 
                                   
                                 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   
                                     
                                       4 
                                       ⁢ 
                                       π 
                                     
                                     9 
                                   
                                   &lt; 
                                   
                                     θ 
                                     i 
                                   
                                   &lt; 
                                   
                                     
                                       5 
                                       ⁢ 
                                       π 
                                     
                                     9 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           k 
                           = 
                           0 
                         
                         , 
                         1 
                         , 
                         
                           … 
                           ⁢ 
                           
                               
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     The second method of equation (26), above, provides a slightly closer approximation of d cm (θ i ) than the first method, offering an inverter module maximum junction temperature only very slightly higher than a real-time calculation of d cm (θ i ). To implement the second method of estimating d cm (θ i ), the control circuitry  34  of a motor drive may employ a lookup table based on the value of output current phase angle θ i . Through a lookup table or similar mechanism, the second method of estimating d cm (θ i ) could provide an efficient method of reducing the maximum junction temperature of an inverter module operating at very low frequency or a DC condition. 
       FIGS. 8-11  illustrate the potential improvement in maximum junction temperature rating for a motor drive operating at very low frequency or a DC condition under the present methods.  FIGS. 8 and 9  illustrate the maximum junction temperature under traditional conditions, T jh,max,trad , and maximum junction temperature when the common mode voltage duty ratio d cm  is optimized, T jh,max opt , for an inverter module operating with a switching frequency f s  of 4 kHz and 8 kHz, respectively.  FIGS. 10 and 11  illustrate the maximum junction temperature difference ΔT j,max  between the maximum junction temperature under traditional conditions and maximum junction temperature when the common mode voltage duty ratio d cm  is optimized, for an inverter module operating with a switching frequency of 4 kHz and 8 kHz, respectively. 
     Plot  94  of  FIG. 8  has an ordinate representing the maximum junction temperature in ° C. and an abscissa in radians over one period of the output current phase angle θ i , describing the maximum junction temperature of an inverter operating at a switching frequency f s  of 4 kHz. The real-time maximum junction temperature under traditional conditions T jh,max,trad    96  reaches a peak of almost 86° C. at 0 radians and drops to 72° C. by 0.52 (π/6) radians, repeating every 1.05 (π/3) radians. In contrast, the optimized real-time maximum junction temperature T jh,max  opt  98  reaches a peak of around 80° C. at 0 radians and drops to 72° C. by 0.52 (π/6) radians, also repeating every 1.05 (π/3) radians. 
     Plot  100  of  FIG. 9  has an ordinate representing the maximum junction temperature in ° C. and an abscissa in radians over one period of the output current phase angle θ i , describing the maximum junction temperature of an inverter operating at a switching frequency f s  of 8 kHz. The real-time maximum junction temperature under traditional conditions T jh,max,trad    102  reaches a peak of 129° C. at 0 radians and drops to 109° C. by 0.52 (π/6) radians, repeating every 1.05 (π/3) radians. In contrast, the optimized real-time maximum junction temperature T jh,max,opt  reaches a peak of almost 117° C. at 0 radians and drops to 109° C. by 0.52 (π/6) radians, also repeating every 1.05 (π/3) radians. 
       FIG. 10  features a plot  106  of the maximum junction temperature difference, ΔT j,max , between the maximum junction temperature under traditional conditions and the optimized maximum junction temperature of an inverter module operating with a switching frequency f s  of 4 kHz. With an ordinate illustrating the difference in maximum temperature in ° C. and an abscissa in radians over one period of the output current phase angle θ i , plot  106  illustrates a real-time maximum junction temperature difference ΔT j,max    108  of up to nearly 5.5° C. when phase angle θ i  is 0 and at multiples of 1.05 (π/3) radians. 
       FIG. 11  provides a plot  110  of the maximum junction temperature difference, ΔT j,max , between the maximum junction temperature under traditional conditions and the optimized maximum junction temperature of an inverter module operating with a switching frequency f s  of 8 kHz. With an ordinate illustrating the difference in maximum temperature in ° C. and an abscissa in radians over one period of the output current phase angle θ i , plot  110  illustrates a real-time maximum junction temperature difference ΔT j,max    112  of up to nearly 13° C. when phase angle θ i  is 0 and at multiples of 1.05 (π/3) radians. 
     As a result of higher maximum junction temperature ratings made available from an optimized common mode voltage duty ratio d cm , the inverter module current rating may increase from 5% to 12.5%, depending on the switching frequency f s . The table below describes an improvement possible for a Rockwell 480V/50 hp drive with a DC bus voltage of 650V: 
     
       
         
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 Switching Frequency f s   
               
             
          
           
               
                   
                 4 kHz 
                 8 kHz 
                 12 kHz 
                 16 kHz 
               
               
                   
                   
               
             
          
           
               
                 I smax,trad  (traditional 
                 77.0 
                 53.7 
                 40.5 
                 32.3 
               
               
                 method) (A rms ) 
               
               
                 I smax,opt  (optimal 
                 81.1 
                 59.1 
                 45.6 
                 36.1 
               
               
                 method) (A rms ) 
               
               
                 Current Rating 
                 5.3% 
                 10.0% 
                 12.5% 
                 11.6% 
               
               
                 Increase 
               
               
                   
               
             
          
         
       
     
     While only certain features of the invention have been illustrated and described herein, many modifications and changes will occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the invention.