Abstract:
An apparatus and method for driving a three-phase motor from an AC energy source includes a converter connected between the energy source and an inverter. The inverter includes two inverter circuits controlled by a logic circuit. The outputs of the inverter circuits have a phase angle therebetween of  60 ° and are connected directly to two inputs of the motor. The remaining motor input can be connected to either the first or second line of the energy source. Both the frequency and the magnitude of the voltage waveforms can be varied to control the speed of the motor.

Description:
This application is a continuation of U.S. patent application Ser. No. 08/419,298, filed Apr. 10, 1995 and entitled “Single Inverter Drive for a Three-Phase Motor”, which is a continuation-in-part of Ser. No. 08/072,511 filed Jun. 4, 1993, now U.S. patent No. 5,406,185, issued Apr. 11, 1995 and entitled “Two-Phase Inverter Drive for a Three-Phase Motor.” 
    
    
     BACKGROUND OF THE INVENTION 
     This invention relates to inverting or transmitting electrical power to a motor. 
     DESCRIPTION OF PRIOR ART 
     EMBODIMENT #1 
     The common and long-standing method to drive a three-phase AC motor is to apply three 120-degree-spaced AC voltages. This common method for a Variable-Voltage-Variable-Frequency (VVVF) drive from a DC energy source is illustrated in FIG. 1 for the case of a system with a battery  10 , a three-phase DC-to-AC inverter  20 , and a three-phase delta-connected induction motor  40 . 
     The battery  10  is a DC energy source that supplies two output voltages on lines  12  and  14 . The output voltage on line  12  is positive X volts and the output voltage on line  14  is negative X volts, where X is a variable that can be any practical value. The battery also has a ground  16  and internal cells  18  that actually produce the voltage potential. The operation of batteries is well defined in the literature and therefore is not discussed in detail here. 
     The inverter  20  takes the DC voltages on lines  12  and  14  from battery  10  and produces three phases of AC voltage. The three phases are called V1, V2, and V3 and are made available on terminals  24 ,  25 , and  26 , respectively. The three phases are produced by the inverter circuits  21 ,  22 , and  23 , which are controlled by the logic circuit  30 . The voltage waveforms V1, V2, and V3 approximate sine waves with amplitude X and with phase shifts of  30 ,  150 , and  270  degrees, respectively, for forward motor rotation, as detailed in Eqs. 1a through 3a. For reverse motor  40  rotation, the logic circuit  30  changes the sign of the phase angles for V1, V2, and V3 as detailed in Eqs. 1b through 3b. The accuracy to which these waveforms approximate the exact shape of a sine wave depends on the design of the inverter circuits  21 ,  22 , and  23 , as well as the logic circuit  30 . For the purpose of discussion, we will assume that V1, V2, and V3 each has the exact shape as a sine wave. However, the accuracy of their wave shape is not critical to the present invention. 
     The following equations describe inverter output voltages for forward rotation, where w is the period of the sine wave in radians and t is the time variable: 
     
       
         V1=X sin(wt+30)  (1a) 
       
     
     
       
         V2=X sin(wt+150)  (2a) 
       
     
     
       
         V3=X sin(wt+270)  (3a) 
       
     
     The following equations describe inverter output voltages for reverse rotation: 
     
       
         V1=X sin(wt−30)  (1b) 
       
     
     
       
         V2=X sin(wt−150)  (2b) 
       
     
     
       
         V3=X sin(wt−270)  (3b) 
       
     
     The design and operation of the inverter circuits  21 ,  22 , and  23  and of the logic circuit  30  are well documented in the prior art and therefore will not be described in detail here. For reference purposes, a typical inverter circuit is shown in FIG.  2 . Its power-conducting elements are transistor Q 1   50 , transistor Q 2   54 , diode D 1   52 , and diode D 2   56 . A set of silicon-controlled rectifiers (SCRs) could be used instead of the transistors. There are other peripheral circuit components which, for simplicity, are not shown. The logic circuit  30  could be composed of a microprocessor and/or other digital and analog circuit components. It typically produces a pulse-width-modulated (PWM) signal that controls when the transistors Q 1   50  and Q 2   54  are turned on. It is a simple matter for the logic circuit  30  to control the phase angle of each phase, as described in the prior literature. The design details of the inverter circuits  21 ,  22 , and  23  as well as the design details of the control circuit  30  are not pertinent to the present invention. 
     The motor  40  produces a useful mechanical torque from the balanced three-phase voltages Va, Vb, and Vc across the stator coils  44 ,  45 , and  46 , respectively. The three inverter  20  outputs V1, V2, and V3, on terminals  24 ,  25 , and  26 , respectively, are connected to the three motor input terminals  41 ,  42 , and  43 , respectively. The voltages on terminals  41 ,  42 , and  43  then cause voltage waveforms Va, Vb, and Vc to be applied to coils  44 ,  45 , and  46 , respectively, which make up the stator winding of the motor. Voltage waveforms Va, Vb, and Vc are defined in Eqs. 4a through 6a for forward motor rotation and in Eqs. 4b through 6b for reverse motor rotation. These voltages then cause currents to flow, which in turn produce magnetic flux, which in turn produces torque. The design and operation of an induction motor is well defined in the prior art and thus will not be discussed in detail here. The only critical point for our discussion is to note that, in Eqs. 4, 5, and 6, the voltage waveforms Va, Vb, and Vc are balanced. In other words, they have a 120-degree phase spacing between them and have the same amplitude. This is true for both forward and reverse cases and is a critical requirement for a three-phase motor to work properly. 
     The following equations describe motor coil voltages for forward rotation: 
     
       
         Va=V1−V2=1.73X sin(wt+0)  (4a) 
       
     
      Vb=V2−V3=1.73X sin(wt+120)  (5a) 
     
       
         Vc=V3−V1=1.73X sin(wt+240)  (6a) 
       
     
     where V1, V2, and V3 are given by Eqs. 1a, 2a, and 3a, respectively. 
     The following equations describe motor coil voltages for reverse rotation: 
     
       
         Va=V1−V2=1.73X sin(wt+0)  (4b) 
       
     
     
       
         Vb=V2−V3=1.73X sin(wt−120)  (5b) 
       
     
     
       
         Vc=V3−V1=1.73X sin(wt−240)  (6b) 
       
     
     where V1, V2, and V3 are given by Eqs. 1b, 2b, and 3b, respectively. 
     DESCRIPTION OF PRIOR ART—EMBODIMENT #2 
     The case of a VVVF drive for a three-phase motor with an single-phase AC energy source is shown in FIG.  3 . The system is comprised of an AC energy source  75 , a AC-to-DC converter  71 , a three-phase inverter  20 , and a three-phase delta connected motor  40 . 
     The AC energy source  75  is typical of a 240 volt 60 Hz single-phase source common in residences. Said energy source  75  is composed of two 120 volt 60 Hz AC energy sources  70 , supplies an AC voltage between lines  72  and  73 , and has a ground  16 . The shape of the voltage on line  72  is sinusoidal with respect to the voltage on line  73 , with any practical value for magnitude and any practical value for frequency. For illustration purposes, the magnitude of the voltage on line  72  with respect to the voltage on line  73  is x volts. 
     The AC-to-DC converter  71  produces a positive DC voltage on line  12  and a negative DC voltage on line  14 . The input to the converter  71  is the voltage between lines  72  and  73 . The input terminal  74  is considered the reference input to the converter  71  since the DC voltage on line  12  is positive with respect to terminal  74  and the DC voltage on line  14  is negative with respect to terminal  74 . For reference purposes, a typical AC-to-DC rectification circuit is shown in FIG.  4 . The values of capacitor  61 , capacitor  62 , inductor  64  and inductor  65  will need to be sufficient in order to supply near-constant DC voltages on lines  12  and  14  for three-hundred-sixty degrees so that adequate voltage is available for inverter  20  to produce output voltages V1, V2, and V3. There are many configurations for AC-to-DC converter circuits that are well defined in the literature and the design details of the converter  71  are not pertinent to the present invention and therefore will not be discussed in detail. 
     The inverter  20  is exactly the same as discussed previously in prior art example #1 so will not be discussed here. 
     The motor  40  is exactly the same as discussed previously in prior art example #1 so will not be discussed here. 
     The disadvantage of the common method illustrated for a VVVF drive is that it takes three inverter circuits, which are relatively expensive, to produce the balanced-voltage waveforms across the three stator coils of the motor. 
     DESCRIPTION OF PRIOR ART—EMBODIMENT #3 
     A method using conventional techniques to implement a Fixed-Voltage-Fixed-Frequency (FVFF) drive for a three-phase motor from a single phase source is illustrated in FIG. 5. A FVFF drive is a lower function drive than a VVVF drive since it can not control the speed of the motor, but is lower cost since only two inverter circuits are required. 
     The AC energy source  75  and the converter  71  are identical as previously discussed. 
     The inverter  20 ″ is comprised of inverter circuit  21  and inverter circuit  22  which supply voltage outputs V1and V2, respectively, to the motor  40  terminals  41  and  42 , respectively. The third input to the motor  40 , V3 on terminal  43 , is supplied directly from the voltage on line  72 . The logic circuit  30 ″ controls inverter  21  so that V1is the same magnitude as V3 and is 120 degrees ahead of V3 for forward rotation. Similarly, logic circuit  30 ″ will control inverter  22  so that V2is the same magnitude as V3 and is  120  degrees behind of V3 for forward rotation. 
     The disadvantage of the conventional method illustrated for a FVFF drive for a three-phase motor being powered from a single-phase AC source is that it takes two inverter circuits and requires relatively large values for the capacitors and inductors in the converter circuit. 
     SUMMARY OF THE INVENTION 
     Normal operation of a three-phase motor is achieved when the 1st input terminal and the 2nd input terminal have voltages applied to them that are approximately sinusoidal waveforms as measured from the 3rd input terminal and the amplitude and phase angle of the voltage applied to the 2nd input is maintained by some electronic or electrical means at pre-determined values relative to the amplitude and phase-angle of the voltage applied to the 1st input. The amplitude and phase-angle values are determined by the characteristics of the motor&#39;s windings. For the common case of a three-phase motor with windings spaced at 120 degrees, equivalent impedances, and equivalent torque constants, the amplitude of the voltage applied to the 2nd input as measured from the 3rd input terminal should be equivalent to the amplitude of the voltage applied to the 1st input as measured from the 3rd input terminal. And the phase angle of the voltage applied to the 2nd input should be 60 degrees from the phase angle of the voltage applied to the 1st input. The 3rd input terminal would be connected to ground in a system with a bipolar DC energy source. The 3rd input would be connected to the neutral terminal of the converter in a system with an AC energy source, a converter, and an inverter. The voltage for the 1st input could be produced by an inverter or taken directly from an AC line. The voltage for the 2nd input will be produced by some electrical or electronic means. 
     Conventional methods require that both the 2nd and 3rd voltage inputs be controlled relative to the 1st voltage input in order for normal operation of the motor to be achieved. For a common motor with windings spaced at 120 degrees, equivalent impedances, and equivalent torque constants, the conventional method for forward rotation requires that the 2nd voltage input have the same amplitude as the 1st voltage input and have a phase angle that is 120 degrees greater than the phase angle of the 1st voltage input and that the 3rd voltage input have the same amplitude as the 1st voltage input and have a phase angle that is 120 degrees less than the 1st voltage input&#39;s phase angle. For reverse rotation, the phase angles of the 2nd voltage input and the 3rd voltage input are switched, but the requirement of having to control both the 2nd and 3rd inputs relative to the 1st is still present. 
     The novel concept in the disclosed invention is that normal operation of a three-phase motor is achieved when only the 2nd voltage input is controlled relative to the 1st voltage input, when the amplitude of the 1st voltage input and the amplitude of the 2nd voltage input is measured from the 3rd input terminal. Theoretically, the 3rd input terminal can have any constant or any time-varying voltage applied to it. However, in practice the 3rd input will simply be connected to an available ground, neutral, or AC line so that the cost to control the 3rd input is avoided. The connection could be direct so that the 3rd input is always connected to a terminal of the power source, or it could be through an on/off switch which would always be in the on state when the motor is operational. 
     Accordingly, the reader will see that the two-phase inverter drive does indeed provide for balanced voltages of equal amplitude and phase spacing of 120 degrees across the respective stator coils of a three-phase motor, resulting in normal operation of the motor. Furthermore, the two-phase inverter drive has additional advantages in that: 
     (1) it is low cost since it only requires two inverter circuits for Variable-Voltage-Variable-Frequency control of a three-phase motor, instead of the three required by a conventional three-phase inverter, 
     (2) it is low cost since it only requires one inverter circuit for a Fixed-Voltage-Fixed-Frequency control of a three-phase motor, instead of the two inverters required by a conventional inverter, 
     (3) it allows the use of passive components to compose a Fixed-Voltage-Fixed-Frequency drive for a three-phase motor from a single-phase AC source, which is not feasible using conventional techniques, and 
     (4) it is low cost since it requires relatively small energy storage components in the AC-to-DC converter for a Variable-Voltage-Fixed-Frequency drive for a three-phase motor from an AC energy source. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     FIG. 1 shows a conventional VVVF three-phase inverter drive where the energy source is DC, with three inverter circuits. 
     FIG. 2 shows a typical inverter circuit that performs the DC-to-AC inversion for one phase. 
     FIG. 3 shows a conventional VVVF three-phase inverter drive where the energy source is AC, with three inverter circuits and a AC to DC converter. 
     FIG. 4 shows a typical circuit that performs the AC to DC conversion that is necessary when the energy source is AC. 
     FIG. 5 shows a FVFF two-phase inverter drive for a three-phase motor using conventional techniques. 
     FIG. 6 shows the disclosed VVVF two-phase inverter drive for a three-phase motor, where the energy source is DC. 
     FIG. 7 shows the disclosed VVVF two-phase inverter drive for a three-phase motor, where the energy source is AC, with a AC to DC converter. 
     FIG. 8 shows the disclosed FVFF single-phase inverter drive for a three-phase motor, where the energy source is AC, with a AC to DC converter. 
     FIG. 9 shows the disclosed FVFF passive drive for a three-phase motor powered from a single-phase AC source. 
     In all the drawings, the solid lines indicate circuit elements and conductors, and the dashed lines indicate the system components of energy source, converter, inverter, and motor. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Description of Embodiment #1 
     A first typical embodiment of the closure of the present invention, a two-phase VVVF inverter for a three-phase motor powered from a DC energy source, is illustrated in FIG.  6 . The features that are different from the previously discussed, common three-phase VVVF inverter for a three-phase motor are as follows: 
     (a) There are only two inverter circuits  21  and  22 . 
     (b) The inverter&#39;s  20 ′ output names are V1′ and V2′. 
     (c) The battery  10 ′ output voltages on lines  12 ′ and  14 ′ are +1.73X volts and −1.73X volts, respectively. 
     (d) The phase angle for the second phase is either +60 or −60 degrees relative to the first phase, depending on desired rotation direction. 
     (e) The third phase is grounded. 
     (f) The logic circuit  30 ′ controls only two inverter circuits. 
     The battery  10 ′ is a DC energy source that supplies two output voltages, one each on lines  12 ′ and  14 ′. The output voltage on line  12 ′ is +1.73X volts and the output voltage on line  14 ′ is −1.73X volts. These voltages were increased by the multiplier factor 1.73 over battery  10  output voltages so that the resulting voltages Va, Vb, and Vc across each of coils  44 ,  45 , and  46  would be identical for either the two-phase inverter  20 ′ or the three-phase inverter  20 . The battery also has a ground  16  and internal cells  18 ′ that actually produce the voltage potential. 
     The inverter  20 ′ takes the DC voltages on lines  12 ′ and  14 ′ from battery  10 ′ and produces two phases of AC voltage, which are called V1′ and V2′ and which are made available on terminals  24  and  25 , respectively. The two phases are produced by the inverter circuits  21  and  22 , which are controlled by the logic circuit  30 ′. The voltage waveforms V1′ and V2′ approximate sine waves with amplitude 1.73X volts and with phase shifts of 0 and −60 degrees, respectively, for forward motor rotation, as detailed in Eqs. 7a through 8a. For reverse motor  40  rotation, the logic circuit  30 ′ changes the phase angle for V2′ as detailed in Eqs. 7b through 8b. The accuracy to which V1′ and V2′ approximate the exact shape of a sine wave is not critical and depends on the design of the inverter circuits  21  and  22 , as well as of the logic circuit  30 ′. For the purpose of discussion, we will assume that V1′ and V2′ each has the exact shape as a sine wave. 
     The following equations describe the two-phase inverter output voltages for forward rotation: 
     
       
         V1′=1.73X sin(wt+60)  (7a) 
       
     
     
       
         V2′=1.73X sin(wt+0)  (8a) 
       
     
     The following equations describe the two-phase inverter output voltages for reverse rotation: 
     
       
         V1′=1.73V sin(wt−60)  (7b) 
       
     
     
       
         V2′=1.73V sin(wt+0)  (8b) 
       
     
     The design and operation of the inverter circuits  21  and  22  and of the logic circuit  30 ′ are well documented in the prior art and thus will not be described in detail here. The typical inverter circuit shown in FIG. 2 is still valid for the two-phase inverter. 
     The motor  40  produces a useful mechanical torque from the balanced three-phase voltages Va, Vb, and Vc across the stator coils  44 ,  45 , and  46 , respectively. The two-phase inverter  20 ′ outputs V1′ and V2′, on terminals  24  and  25 , respectively, are connected to the motor input terminals  41  and  42 , respectively. The third motor  40  terminal  43  is simply connected to ground. The voltages on terminals  41 ,  42 , and  43  then cause voltage waveforms Va, Vb, and Vc to be applied to coils  44 ,  45 , and  46 , respectively, which make up the stator winding of the motor. Voltages Va, Vb, and Vc are defined in Eqs. 10a, 11a, and 12a for forward motor rotation and in Eqs. 10b, 11b, and 12b for reverse motor rotation. The voltages Va, Vb, and Vc then cause currents to flow, which in turn produce magnetic flux, which in turn produces torque. The design and operation of a three-phase motor is well defined in the prior art and therefore will not be discussed in detail here. The only critical point for our discussion is to note that, in Eqs. 10, 11, and 12, the voltage waveforms Va, Vb, and Vc are balanced. In other words, they have a 120-degree phase spacing between them and have the same amplitude. This is true for both forward and reverse cases and is a critical requirement for a three-phase induction motor to work properly. In fact, a comparison of Va, Vb, and Vc in Eqs. 10, 11, and 12 for the two-phase inverter  20 ′ with their counterparts in Eqs. 4, 5, and 6 for the three-phase inverter  20  reveals that the coil voltages Va, Vb, and Vc are exactly the same! 
     The two-phase inverter coil equations for forward rotation are: 
     
       
         Va=V1′−V2′=1.73X sin(wt+0)  (10a) 
       
     
     
       
         Vb=V2′−V3′=1.73X sin(wt+120)  (11a) 
       
     
      Vc=V3′−V1′=1.73X sin(wt+240)  (12a) 
     where V1′ and V2′ are given in Eqs. 7a and 8a, respectively, and V3′ is zero. The two-phase inverter coil equations for reverse rotation are: 
     
       
         Va=V1′−V2′=1.73X sin(wt+0)  (10b) 
       
     
     
       
         Vb=V2′−V3′=1.73X sin(wt−120)  (11b) 
       
     
     
       
         Vc=V3′−V1′=1.73X sin(wt−240)  (12b) 
       
     
     Description of Embodiment #2 
     A second typical embodiment of the closure of the present invention for a VVVF drive for a three-phase motor from an AC energy source is illustrated in FIG.  7 . The features that are different from the previously discussed, common three-phase VVVF drive from an AC energy source shown in FIG. 3 are as follows: 
     (a) There are only two inverter circuits  21  and  22 . 
     (b) The inverter&#39;s  20 ′ output names are V1′ and V2′. 
     (c) The AC energy source&#39;s  75 ′ output voltage on line  72 ′ is 1.73x volts with respect to line  73 ′. 
     (d) The phase angle for the second phase is either +60 or −60 degrees relative to the first phase, depending on desired rotation direction. 
     (e) The third phase is connected to the reference input  74  to the converter  71 . 
     (f) The logic circuit  30 ′ controls only two inverter circuits. 
     The AC energy source  75 ′ supplies a voltage between lines  72 ′ and  73 ′. Line  73 ′ is connected to the input reference terminal  74  of the converter  71 . The magnitude of the voltage on line  72 ′ with respect to line  73 ′ is increased by a factor of 1.73 so that the resulting voltages across the windings will be the same magnitude as in the conventional case to make the comparison easier. 
     The inverter  20 ′ behaves exactly as previously described. The motor  40  operates as previously described except that the third terminal  43  is simply connected to the reference terminal  74  of the converter  71 . 
     Description of Embodiment #3 
     A third embodiment of the closure of the present invention is a Variable-Voltage-Fixed-Frequency (VVFF) drive for a three-phase motor driven from a single-phase AC source. This embodiment has the same circuit topology as illustrated in FIG. 7 for a VVVF drive. However, lower cost is achieved by requiring much smaller values for the capacitors and inductors in the converter circuit. This is achieved by the logic circuit controlling V1′ and V2′ so that they are the same frequency as the voltage on line  72 ′ and are close in phase with the voltage on line  72 ′. Therefore the majority of the energy comes directly from line  72 ′ and only a minority is stored in the capacitors and inductors for use through out the 360 degree cycle. By contrast, a typical converter has to store the majority of the energy that is available on line  72 ′ during limited portions of the 360 degree cycle for use at other times during the cycle. 
     Description of Embodiment #4 
     A fourth typical embodiment of the closure of the present invention for a FVFF drive for a three-phase motor from an AC energy source is illustrated in FIG.  8 . The features that are different from the previously discussed three-phase FVFF drive from an AC energy source shown in FIG. 5 are as follows: 
     (a) There is only one inverter circuit  21 . 
     (b) The inverter&#39;s  20 ′ output name is V1′. 
     (c) The phase angle for the second phase is either +60 or −60 degrees relative to the first phase, depending on desired rotation direction. 
     (e) The third phase is connected to the reference input  74  to the converter  20 ′″. 
     (f) The logic circuit controls only two inverter circuits. 
     The AC energy source  75  supplies a voltage between lines  72  and  73 . Line  73  is connected to the input reference  74  of the converter  71 . 
     The inverter  20 ′″ behaves exactly as previously described, except that it now only has one output, V1 on terminal  42 . The motor  40  operates as previously described except that the third terminal  43  is simply connected to the reference terminal  74  of the converter  71 . 
     Description of Embodiment #5 
     A fifth embodiment of the closure of the present invention is a Fixed-Voltage-Fixed-Frequency (FVFF) drive for a three-phase motor powered from a single-phase AC energy source and is shown in FIG.  9 . Passive circuit elements such as capacitors, inductors, and resistors can be used for this class of drive instead of active circuit elements such as the transistors used in an inverter. The system is comprised of an AC energy source  75 , a capacitor  80 , an inductor  81 , and a three-phase delta connected motor  40 . 
     The AC energy source  75  and the motor  40  have been previously discussed. The capacitor  80  is in series with the AC source  70  and motor terminal  41 . The inductor  81  is connected in parallel with motor terminal  41  and motor terminal  43 . 
     The values of capacitor  80  (C) and inductor  81  (L) should be chosen so that Equ. 13 is satisfied so that V2′ will be the same magnitude as V1′ and phase shifted 60 degrees from it. Z is the motor  40  winding impedance vector and w is the angular velocity of the AC source  75 . 
     
       
         1.73/Z=(0.5−j0.866)×wC+(0.5+j0.866)/wL  (13) 
       
     
     Summary of Embodiments 
     The novel idea of this invention that is revealed in the embodiments shown is that only two active phases are required to drive a three-phase motor if the two active phases have appropriately chosen phase angles and magnitudes. The importance of this idea is that a lower cost drive for a three-phase motor can be achieved. For the case of a VVVF drive, only two inverter circuits are required instead of the conventional three. For the case of a FVFF drive, only one inverter circuit is required instead of the conventional two. And the case of passive components being used to drive a three-phase motor from a single-phase source is now feasible. 
     That this idea is not obvious is clear, since it allows for a significant cost reduction yet it is neither used in practice nor mentioned in the prior art. 
     Although the preceding description contains many specificities, these should not be construed as limiting the scope of the invention but as merely providing illustrations of some of the currently preferred embodiments of this invention. For example, the motor being controlled could be a WYE connected three-phase motor or a brushless DC motor. Thus the scope of the invention should be determined by the appended claims and their legal equivalents, rather than by the examples given. 
     Theory of Operation for Present Invention 
     It is not obvious that two active phases and a grounded phase applied to a three-phase motor could result in balanced voltages across the motor&#39;s stator coils. However, thoughtful consideration of Eqs. 14, 15, and 16 will lead to the conditions that the two active phases must meet in order to accomplish balanced voltages across the three stator coils. Eq. 14 is a general equation that defines the resultant waveform Va when the second active phase V2′ is subtracted from the first active phase V1′, according to the polarity convention defined for Va in FIG.  6 . Eqs. 15 and 16 are general equations that define the waveforms Vb and Vc, respectively. The correctness of Eqs. 14, 15, and 16 can be verified by examining FIG.  6 . 
     The variables in Eq. 14 are PA1, the phase angle for the first phase; PA2, the phase angle for the second phase; PAa, the resulting phase angle for the voltage Va; and Vp, the resulting peak amplitude for Va. The peak amplitudes for V1′ and V2′ are set to unity (1.0) to simplify the analysis.                    Va   =       V1   ′     -     V2   ′                   =       sin   (     wt   +   PA1     )     -     sin   (     wt   +   PA2     )                   =     Vp                   sin   (     wt   +   PAa     )                     (   14   )                     Vb   =     V2   ′                 =     sin   (     wt   +   PA2     )                   (   15   )                     Vc   =     -     V1   ′                   =     -     sin   (     wt   +   PA1     )                   =     sin   (     wt   +   PA1   +   180     )                   (   16   )                                
     To further simplify the analysis, let PA1 be equal to zero. The four conditions for a two-phase inverter to properly drive a three-phase motor are as follows: 
     (1) The phase angle of the second phase, PA2, must be 60 degrees from the phase angle of the first phase, PA1. This is true since the sum of 180 degrees and PA1 must be 120 degrees from PA2, as dictated by Eqs. 15 and 16. Thus, in the case where PA1 is zero,  23  PA2 must be either+60 degrees or −60 degrees. 
     (2) When the second phase, V2′, is subtracted from the first phase, V1′, the resulting waveform, Va, must have the same amplitude as V1′ and V2′. Therefore, Vp in Eq. 14 must be unity since the amplitudes for V1′ and V2′ are unity. 
     (3) When the second phase, V2′, is subtracted from the first phase, V1′, the resulting waveform, Va, must have a phase angle that is 120 degrees from V2′. Therefore, PAa must be 120 degrees from PA2, as dictated by Eqs. 14 and 15. 
     (4) There must exist two values of PA2 which satisfy the above three conditions and these two values must be of equal magnitude and of opposite sign. 
     Table 1 lists the resulting values of Vp, PAa, and PA2 minus PAa over the range of possible values for PA2. 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Vp and PA2 − PAa vs PA2, with PA1 = 0 
               
             
          
           
               
                   
                 PA2 
                 Vp 
                 PAa 
                 PA2 − PAa 
               
               
                   
                 (Degrees) 
                 (Degrees) 
                 (Degrees) 
                 (Degrees) 
               
               
                   
                   
               
             
          
           
               
                   
                 0 
                 0 
                 0 
                 0 
               
               
                   
                 30 
                 0.517 
                 −75 
                 105 
               
               
                   
                 60 
                 1.000 
                 −60 
                 120 
               
               
                   
                 90 
                 1.414 
                 −45 
                 135 
               
               
                   
                 120 
                 1.732 
                 −30 
                 150 
               
               
                   
                 150 
                 1.932 
                 −15 
                 165 
               
               
                   
                 180 
                 2.000 
                 0 
                 180 
               
               
                   
                 −150 
                 1.932 
                 15 
                 −165 
               
               
                   
                 −120 
                 1.732 
                 30 
                 −150 
               
               
                   
                 −90 
                 1.414 
                 45 
                 −135 
               
               
                   
                 −60 
                 1.000 
                 60 
                 −120 
               
               
                   
                 −30 
                 0.517 
                 75 
                 −105 
               
               
                   
                   
               
             
          
         
       
     
     Examining Table 1 reveals that the first three conditions are met when PA2 is +60 degrees and when PA2 is −60 degrees! And the fourth condition is met since there are two values of PA2 of opposite sign and equal magnitude that satisfies the first three conditions. Therefore, if the phase angle difference between PA1 and PA2 is +60 degrees, a two-phase inverter can drive a three-phase motor in one direction. And when the difference between PA1 and PA2 is −60 degrees, a two-phase inverter can drive a three-phase motor in the other direction! 
     To make the comparison between the three-phase inverter and the two-phase inverter easier in the “Description and Operation of Present Invention”, PA1 was chosen to be 60 degrees and PA2 to be 0 degrees for the forward rotation case, as shown in Eqs. 7a and 8a. Any other values that are 60 degrees apart could have been used.