Abstract:
A control system for an electric motor. Field Oriented Control, FOC, uses a digital control system, and attempts to maintain the rotating stator magnetic field ninety degrees ahead of the rotor field, in order to maximize torque. However, if a three-phase motor is used, which is very common, large amounts of computation are required, to convert from a three-phase coordinate system to an orthogonal coordinate system. The invention replaces the three-phase motor with a two-phase motor, which has been found to impose certain additional costs, but also provides benefits.

Description:
This application is related to an application entitled “FAULT-HANDLING SYSTEM FOR ELECTRIC POWER-STEERING SYSTEM,” which is concurrently filed herewith on Oct. 31, 2003, and which is hereby incorporated by reference. 
   The invention relates to electric power-steering systems in vehicles, and simplified control systems used therein. 
   BACKGROUND OF THE INVENTION 
   This discussion will first explain how a simple synchronous motor operates, and then explain one type of prior-art speed control used with such a motor. 
   Synchronous Motor 
     FIG. 1  illustrates schematically three stator coils  3 ,  6 , and  9 , which are contained in a three-phase synchronous motor (not shown).  FIG. 2  shows the coils, but with connecting wires W of  FIG. 1  omitted, to avoid clutter. In  FIG. 2 , currents  13 ,  16 , and  19  are generated in the respective coils. Each current produces a magnetic field B 3 , B 6 , and B 9 , as indicated. 
   The coils  3 ,  6 , and  9  are physically positioned to be 120 degrees apart, as shown, so that the fields B 3 , B 6 , and B 9  are also positioned 120 spatial degrees apart. This arrangement allows creation of a magnetic field which rotates in space at a constant speed, if proper currents are generated in the coils, as will now be explained. 
     FIG. 3  illustrates three-phase currents. The vertical axis on the coordinates runs from negative unity to positive unity for simplicity. In practice, one would multiply the values of unity by the actual peak-to-peak values of the currents being used. 
   Currents in the form of sine waves SIN 3 , SIN 6 , and SIN 9  are created respectively in coils  3 ,  6 , and  9 , as indicated. Coil  3  resides at zero physical degrees. SIN 3  begins at zero electrical degrees, as indicated on the plot. 
   Similarly, coil  6  stands at 120 degrees from coil  3 . SIN 6  begins at 120 degrees, as indicated on the plot. Similarly, coil  9  stands at 240 degrees from coil  3 . Correspondingly, SIN 9  begins at 240 degrees, as indicated on the plot. 
   Each coil  3 ,  6 , and  9  produces a magnetic field, as indicated. Those three magnetic fields add vectorially to produce a single magnetic field. Two examples will illustrate this vectorial addition. 
   In the first example, time T 1  is chosen in  FIG. 3 , which corresponds to 255 electrical degrees. T 1  is also indicated in  FIG. 4 . At this time T 1 , the values of the currents I 3 , I 6 , and I 9  are indicated. Those currents exist in coils  3 ,  6 , and  9  in  FIGS. 2 and 3 . Those currents produce magnetic fields which are roughly proportional to the currents. 
   Since the coils  3 ,  6 , and  9  are physically positioned at angles of zero, 120, and 240 degrees, the magnetic fields are also positioned at those angles. The magnetic fields are indicated as B 3 , B 6 , and B 9  in  FIG. 4 . 
   It is noted that field B 3  is positioned at 180 degrees, rather than zero degrees. This occurs because current  13  is negative, thus producing a magnetic field B 3  which is 180 degrees from the magnetic field which would be produced by a positive current. 
   Field-vectors B 3 , B 6 , and B 9  are re-positioned within circle C 1 , to show vector addition. They sum to the resultant vector R 1 . Resultant R 1  represents the vector sum of the three magnetic fields, and is an actual magnetic field vector located in space. Resultant R 1  is the magnetic field produced by the three coils, and is termed the stator field. 
   In the second example, time T 2  in  FIG. 3  is chosen, which corresponds to 330 electrical degrees. T 2  is also indicated in  FIG. 5 . At this time T 2 , the particular values of currents I 3 , I 6 , and I 9  are indicated. 
   It is noted that I 3  and I 6  are superimposed over each other: the same arrow represents both. It should be observed that these two identical currents produce two magnetic fields of the same size at this time. However, because the two currents I 3  and I 6  are applied to coils which are physically 120 degrees apart, the magnetic fields are oriented differently in space. 
   The magnetic field vectors produced are indicated as B 3 , B 6 , and B 9  in  FIG. 5 . 
   It is noted that fields B 3  and B 6  are positioned at 180 and 300 degrees, respectively, rather than at zero and 120 degrees. As before, this occurs because currents I 3  and I 6  are negative, producing magnetic fields B 3  and B 6  which are 180 degrees rotated from the magnetic field which would be produced by positive voltages. 
   Field-vectors B 3 , B 6 , and B 9  are re-positioned within circle C 2 , to show vector addition. They sum to the resultant vector R 2 . Resultant R 2  represents the vector sum of the three magnetic fields, and is an actual magnetic field vector located in space. It is the stator field. 
   If these two examples are repeated for every angle from zero to 360 in  FIG. 3 , it will be found that a resultant R in  FIG. 6  is produced at each angle, and that all resultants R are identical in length. It will also be found that, as one computes resultant R for sequential angles, that resultant R rotates at a uniform speed around circle C. 
   The arrangement just described produces a constant magnetic field which rotates at a constant speed. This rotating field can be used as shown in  FIG. 7 . 
     FIG. 7  illustrates the coils of  FIG. 2 . A rotor ROT is added, which contains a rotor magnetic field RF, produced by a magnetic field source FS, which may be a permanent magnet or electrical coil. Because of the laws of physics, the rotor field RF will attempt to follow the rotating resultant R. Consequently, the rotating resultant R induces rotation in the rotor ROT, producing motor-action. 
   Control System 
   A prior-art approach to controlling speed of the motor just described will be given. In one approach, the basic idea is to maintain the resultant stator field R in  FIG. 7  at 90 degrees ahead of the rotor field RF. ( FIG. 7  shows the resultant R at zero degrees with respect to RF.) 
   The particular approach to be explained is sometimes termed “Field Oriented Control,” FOC. In FOC, the stator field is transformed, or superimposed, onto a rotating coordinate system, and is then compared with the rotor field, within the rotating coordinate system. Under this approach, two fields (stator and rotor) are, ideally, not changing with respect to each other and, when they do change, they change slowly, with respect to each other. FOC reduces bandwidth requirements, especially in Proportional Integral controllers, used to control the error between the two fields. 
   Perhaps an analogy can explain the bandwidth reduction. Assume two race horses traveling on a circular track. Each, in essence, can be represented by a hand on a clock. In one approach, a stationary observer can, say, every second, compute position of each horse, compare the positions, and deduce a difference between positions. In essence, the observer computes an angle for each hand of the clock, and continually compares those changing angles. However, even if the horses are running nose-to-nose, the observer still must compute an angle for each horse every second, and each angle changes, second-to-second. 
   In the FOC approach, the observer, in essence, rides along with the horses. If the horses are nose-to-nose, the observer computes a steady zero difference. When one horse passes the other, the observer computes a slowly changing difference. 
   The FOC approach reduces the number of a certain type of computation which must be done, thereby reducing bandwidth requirements. 
   In explaining FOC, a current in a coil will sometimes be treated interchangeably with the magnetic field which the coil produces. One reason is that the two parameters are approximately proportional to each other, unless the coil is saturated. Thus, a current and the field it produces differ only by a constant of proportionality. 
     FIG. 8  is a schematic of the connection of the three coils C 3 , C 6 , and C 9  in one type of synchronous motor (not shown). They are connected in a WYE configuration, with point PN representing neutral. 
   A significant feature of the WYE configuration is that the currents in the coils are not mutually independent. Instead, by virtue of Kirchoffs Current Law, the three currents must sum to zero at point PN. Thus, only two independent currents are present, because once they are specified, the third is thereby determined. One significance of this feature will be explained later, in connection with the present invention. 
   A CONTROLLER  50  measures and controls the currents I 3 , I 6 , and I 9 , in a manner to be described. It is again emphasized that each current I 3 , I 6 , and I 9  produces a respective magnetic field B 3 , B 6 , and B 9  which are separated in space by 120 degrees, as indicated. (B 3 , B 6 , and B 9  in  FIG. 8  only show the different directions in space, but not different magnitudes.) 
   The CONTROLLER  50  undertakes the processes which will be explained with reference to  FIGS. 9–14 . Block  55  in  FIG. 9  indicates that the CONTROLLER  50  in  FIG. 8 , or an associated device, measures each current I 3 , I 6 , and I 9 . In block  57  in  FIG. 9 , a data point, or vector, for each current is computed, giving the magnitude and direction of the magnetic field produced by each current. For example, if the measurement occurred at time T 1  in  FIG. 4 , then vectors B 3 , B 6 , and B 9  would be computed. Those vectors are shown adjacent block  57  in  FIG. 9 . 
   In block  60 , two orthogonal vectors are computed which produce the equivalent magnetic field to the resultant of the vectors computed in block  57 . The two graphs adjacent block  60  illustrate the concept. STATOR FIELD is the vector sum of the three vectors B 3 , B 6 , and B 9  which were previously computed in block  57 . Two orthogonal vectors a and b are now computed, which are equivalent to that vector sum, namely, the STATOR FIELD. Parameter a is the length of a vector parallel with the x-axis. Parameter b is the length of a vector parallel with the y-axis. 
     FIG. 10  illustrates how this computation is performed, and is presented to illustrate one complexity in the prior art which the present invention eliminates, or reduces.  FIG. 10  illustrates three generalized vectors I 1 , I 2 , and I 3 , which are illustrated across the top of  FIG. 10 . The overall procedure is to (1) compute the x- and y-coordinates for each vector, (2) add the x-coordinates together, and (3) add the y-coordinates together. The result is two orthogonal vectors. 
   As to the x-coordinates, as indicated at the top center of  FIG. 10 , the x-coordinate of I 2  is I 2 (COS 120). As indicated at the top right, the x-coordinate of I 3  is I 3 (COS 240). As indicated at the top left, the x-coordinate of I 1  is I 1 (COS 180). These three x-coordinates are added at the lower left, producing a vector Ia. 
   As to the y-coordinates, as indicated at the top center of  FIG. 10 , the y-coordinate of I 2  is I 2 (SIN 120). As indicated at the top right, the y-coordinate of I 3  is I 3 (SIN 240). There is no y-coordinate for I 1 , because it always stands at either zero or 180 degrees. These y-coordinates are added at the lower right, producing a vector Ib. 
     FIG. 11  shows the two vectors Ia and Ib. Their vector sum is the STATOR FIELD, as indicated. These two vectors Ia and Ib correspond to the two vectors computed in block  60  in  FIG. 9 . 
   In block  70  in  FIG. 12 , rotor angle, theta, is measured. A shaft encoder (not shown) is commonly used for this task. Rotor angle is an angle which indicates the rotor field vector, either directly or through computation. 
   In block  80 , the two vectors computed in block  60  in  FIG. 9  are transformed into a coordinate system which rotates with the rotor. (The angle theta is continually changing.) The graphs adjacent block  80  illustrate the concept. The STATOR FIELD, as computed in block  60  in  FIG. 9 , is on the left, and has x-y coordinates of (a,b). Block  80  transforms the coordinates to a 1  and b 1 , shown on the right, which are the coordinates for the same STATOR FIELD, but now in a rotating u-v coordinate system. 
     FIG. 13  illustrates how this transformation may be accomplished. Plot  100  illustrates a generalized point P, representing a generalized stator field vector, having coordinates (a, b) in an x-y coordinate system. Plot  105  illustrates how the u-coordinate, of value a 1 , can be computed for a rotated u-v coordinate system. Plot  110  illustrates how the v-coordinate, of value b 1 , can be computed for the rotated u-v coordinate system. Equations  115  summarize the results. 
   Parameters a 1  and b 1  are the variables computed by block  80  in  FIG. 12 . It is noted that coordinate b 1  corresponds to a vector which is parallel to the LEADING ORTHOGONAL in the graph adjacent block  90 . The significance of this will become clear shortly. 
   Block  90  in  FIG. 12  computes the error, if any, between the STATOR FIELD (shown adjacent block  80 ) and the LEADING ORTHOGONAL in the plot adjacent block  90 . The LEADING ORTHOGONAL is a vector which is perpendicular to the ROTOR FIELD, and leads the ROTOR field. In order to maximize torque, the stator field is controlled so that it continually remains aligned parallel with the LEADING ORTHOGONAL, also called the quadrature vector to the ROTOR FIELD. (In generator action, as opposed to motor action, the quadrature vector lags the ROTOR FIELD.) This evaluation is done in the rotating coordinate system u-v, as block  90  indicates. 
   Block  130  in  FIG. 14  indicates that the vector coordinates of the required stator field are computed, but in the rotating coordinate system. The graph adjacent block  130  illustrates the concept. The NEEDED FIELD is that which is orthogonal with the ROTOR FIELD. In the graph, the STATOR FIELD illustrated is not orthogonal, and corrective action must be taken. 
   The coordinates computed in block  130  for the required stator field lie in the rotating u-v coordinate system. Block  135  transforms those coordinates into the stationary x-y coordinate system, using inverses of the operations shown in  FIG. 13 . The inverse operations are
 
 x=u  COS (theta)− v  SIN (theta)
 
 y=u  SIN (theta)+ v  COS (theta)
 
   Block  140  in  FIG. 14  then computes the required voltages needed for the coils to attain the required stator field. In concept, block  135  specifies a vector analogous to resultant R 1  in  FIG. 4 . Block  140  in  FIG. 14  computes the voltages analogous to V 3 , V 6 , and V 9  required to produce that vector R 1 . 
   The computation of block  140  is of the same type as that shown in  FIG. 10 . In the latter, two orthogonal vectors are derived which are equivalent to three vectors. In block  140 , three vectors are derived from two orthogonal vectors. 
   Then the processes of  FIGS. 9–14  are continually repeated during operation of the motor. 
   The preceding was a simplification. In practice, various prior art control strategies are used in the process of converging the stator field to the required stator field, that is, in reducing the error of block  90  to zero, by adjusting the currents in the coils. These control strategies were not discussed. 
   The Inventors have developed a less expensive approach to controlling a synchronous motor. 
   SUMMARY OF THE INVENTION 
   In one form of the invention, a two-phase motor is used to provide power assist for steering system in a vehicle. A control system of the FOC type is used, but one which requires no conversion of magnetic field vectors to equivalent orthogonal vectors. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIGS. 1–14  and  20  are labeled “Prior Art.”  FIG. 1  illustrates three coils, as used in a three-phase motor. 
       FIG. 2  illustrates magnetic fields generated in the coils of  FIG. 1 . 
       FIG. 3  illustrates three-phase waveforms. 
       FIGS. 4 and 5  illustrate two resultant vectors, at two different points in time, created by the waveforms of  FIG. 3 . 
       FIG. 6  illustrates that the resultant vector under discussion rotates about in a circle. 
       FIG. 7  is a schematic view of a synchronous motor. 
       FIG. 8  illustrates a schematic of a controller  50  used to control a prior-art three-phase synchronous motor. 
       FIGS. 9 ,  12 , and  14  illustrate a flow chart explaining operation of the controller  50  of  FIG. 8 . 
       FIGS. 10 and 11  illustrate graphically the transform undertaken by block  60  in  FIG. 9 . 
       FIG. 13  illustrates graphically the transform undertaken by block  80  in  FIG. 12 . 
       FIG. 15  illustrates one form of the invention. 
       FIG. 16  is a schematic of the stator of a two-phase motor. 
       FIGS. 17 and 18  are flow charts describing processes undertaken by one form of the invention. 
       FIG. 19  illustrates a three-phase waveform. 
       FIG. 20  illustrates a WYE-connected coil set, and equations illustrating computations of voltage and power therein. 
       FIG. 21  illustrates directions of currents in the two-phase motor of  FIG. 16 , for each of the four Cartesian quadrants. 
       FIG. 22  illustrates how currents are initialized in the coils of the stator of  FIG. 16 , for each of the four Cartesian quadrants. 
       FIGS. 23–25  illustrate how a particular waveform can be generated. 
       FIG. 26  illustrates a sequence of the waveforms of  FIG. 25 . 
       FIGS. 27–29  illustrate how different average voltages can be generated by adjusting the duty cycle of the switch SW in  FIGS. 23–25 . 
       FIG. 30  illustrates how the average voltages described in connection with  FIGS. 27–29  can be caused to generate a sequence which describes a sine wave. 
       FIG. 31  illustrates soft switching, used by the invention. 
       FIG. 32  illustrates how alteration of the duty cycle of switch SW in  FIGS. 23–25  can create a different average voltage. 
       FIG. 33  illustrates three pulse trains applied to a WYE-connected coil set. 
       FIG. 34  illustrates a shift of a waveform in a three-phase set. 
       FIG. 35  illustrates various possible combinations of control systems and motors, for the main purpose of classifying the present invention. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
     FIG. 15  illustrates one form of the invention. A two-phase synchronous motor  200  is controlled by a controller  205 . In one embodiment, the two-phase motor  200  is contained within a vehicle  210 , and provides a steering assist, of the type commonly called “power steering.” The controller  205  detects that the steering wheel  215  is calling for a turn of the forward wheels  220 . The controller  205  orders the motor  200  to produce power as long as the change in direction of wheels  220  is being made. Motor  200  turns the wheels  220  through a linkage  225 . 
   A two-phase synchronous motor has two stator coils CX and CY, as in  FIG. 16 , which are physically located at 90 degrees to each other. The two coils CX and CY thus produce magnetic fields BX and BY which are also 90 degrees from each other. 
   If proper currents IX and IY are applied to the coils  210  and  215 , they produce magnetic field vectors which add vectorially to a single rotating magnetic field vector, of constant magnitude and constant angular velocity. For example, if the current IX is described by the expression COS T and the current IY described by SIN T, then their sum is a resultant RSUM. The preceding sentence described the equation RSUM=COS T+SIN T, defines a circle, in parametric terms, the parameter being T. 
   Unlike the three-phase case of  FIG. 8 , the two-phase motor  200  of  FIG. 15  contains coils which are unconnected with each other. That is, the current IX in  FIG. 16  can be controlled independently of IY, and those currents are the only stator currents in the motor which drive the rotor. Restated, all currents in the coils are controllable independently. 
   The controller  205  in  FIG. 15  undertakes the processes illustrated in  FIGS. 17–19 . In block  230  in  FIG. 17 , the currents IX and IY in  FIG. 16  are measured.  FIG. 17  uses the conventional symbology of Ia and Ib. 
   These currents produce magnetic fields Ba and Bb which are 90 spatial degrees apart, as indicated. Thus, measurement of the currents Ia and Ib directly indicates the STATOR FIELD VECTOR, because the magnetic fields Ba and Bb equal the currents multiplied by a constant (outside saturation). 
   That is, any need for the transformation of block  60  in  FIG. 9 , and shown in  FIGS. 10 and 11 , is eliminated. Under the invention, two vectors, analogous to Ia and Ib in  FIG. 11 , are read directly by block  230  in  FIG. 16 . Restated, two orthogonal currents, and by implication two orthogonal magnetic fields, which sum vectorially to the STATOR FIELD VECTOR adjacent block  230  in  FIG. 17 , are read directly by block  230 . These two currents reside in the stationary x-y coordinate system. 
   In block  235  in  FIG. 17 , rotor angle, theta, is measured. In block  240 , Ia and Ib are transformed into a rotating u-v coordinate system positioned at rotor angle theta, as indicated.  FIG. 13  illustrates the type of transformation. 
   In block  245  in  FIG. 18 , the error between (1) the stator angle, in rotating u-v coordinates, and (2) the LEADING ORTHOGONAL, also in rotating u-v coordinates, is computed. This computation seeks the difference in angle between b 1  and q, both adjacent block  245 . 
   Block  250  in  FIG. 18  computes the needed stator angle, i.e., the required stator magnetic field vector, in rotating u-v coordinates. 
   Block  260  in  FIG. 18  transforms the required stator angle from rotating u-v coordinates to stationary x-y coordinates, using a known transform, such as the inverse discussed in the Background of the Invention, in connection with  FIG. 13 . 
   Block  270  computes the required currents. At this point, the currents required in the two stator coils CX and CY in  FIG. 16  are known directly. There is no requirement for a transformation of the type indicated in block  140  in  FIG. 14 . That type of transformation, wherein two orthogonal vectors in x-y coordinates are converted to an equivalent set of three vectors in x-y coordinates, is, as stated, not necessary. 
   Block  280  in  FIG. 18  indicates that currents are generated in the coils, and the particular instantaneous currents generated depend on the control strategy used. Thus, block  280  is closely related to block  270 . An example will illustrate this. 
   Consider a single cycle through the processes described by the flow chart of  FIGS. 17 and 18 . During that cycle, assume that the quadrature vector q adjacent block  245  in  FIG. 18  is computed at 120 degrees. Assume that the STATOR FIELD VECTOR adjacent block  230  in  FIG. 17  is computed at 110 degrees, or ten degrees away from its desired position. 
   The question then arises on how to reduce this error of ten degrees. Should it be gradually and linearly done over the next, say, ten seconds? Or should the error be reduced as rapidly as possibly? Or should the error be reduced extremely rapidly at first until it reaches, say, one degree, and then reduced gradually? 
   Each of the three approaches has advantages and disadvantages, in terms of stability, overshoot, behavior under other conditions, and other factors. Thus, the particular actions taken in blocks  270  and  280 , once the error is computed, depend on the particular control strategy used. As later discussed, in one embodiment, a Proportional Integral, PI, strategy is preferred. 
   Additional Considerations 
   1. One advantage of the invention is that it produces more power, for a given supply voltage, compared with the prior art. Assume that the three-phase voltage synthesized in a vehicle having a 12-volt battery corresponds to that in  FIG. 19 . Zero volts is only taken as a reference. It is a fact of engineering that, in the WYE-connected system of  FIG. 20 , the voltage across any phase, Vp, equals the line voltage divided by the square root of 3, assuming balanced conditions wherein zero current flows in the neutral line. (The square root of 3 will be taken as 1.7 herein.) The line voltage is that between any two lines, such as Va and Vb. 
   In general, for a motor vehicle application, the maximum line voltage available in  FIG. 19  will be the vehicle system voltage of 12 volts. (Of course, separate power supplies could be provided which deliver a different voltage. Nevertheless, some system voltage exists which will equal the line voltage.) Thus, the maximum voltage which can be applied to any coil in the stator represented in  FIG. 20  is about 12/1.7, or 6.9 volts. In contrast, the maximum voltage which can be applied by the invention to each coil CX and CY in  FIG. 16  is the full line voltage of 12 volts. 
   This is significant because, at any given stage of technology in a society, the wiring available to construct the coils is the same in both cases of  FIG. 16  and  FIG. 20 . Thus, for a given physical coil, the invention provides (1) a voltage across the coil which is 1.7 times larger, (2) a current which is 1.7 times larger, and (3) thus a magnetic field which is 1.7 times larger, compared with  FIG. 20 . 
   If the system of  FIG. 20  is to provide the same magnetic field per coil as the invention, then larger diameter wire must be used in the coils. Or wire of lower resistance, and thus higher cost, must be used. Or a higher voltage must be used. All to overcome the factor of 1.7 just discussed. 
     FIG. 20  also illustrates another fact of engineering, namely, that the power delivered in a WYE-connected system equals Vline times Iphase, multiplied by the square root of 3. (If the phases were connected in parallel, then the total power delivered would be three times that produced by an individual phase.) In contrast, the invention, using two coils in  FIG. 16 , delivers total power equaling twice the produced by each phase. 
   2. A second characteristic of the invention relates to the soft switching techniques utilized. First, a generalized explanation of creation of sinusoidal voltages through switching techniques will be explained. 
     FIG. 21 , adjacent  FIG. 16 , illustrates the direction of the currents passing through coils CX and CY in each of the four Cartesian quadrants, I–IV. The dashed vector in each quadrant represents the generalized sums of the currents, and is used to illustrate direction only. 
     FIG. 22  illustrates the H-bridges used to generate the currents in each quadrant I–IV, and is considered self-explanatory. These H-bridges generate sinusoidal currents in the coils CX and CY of  FIG. 16 , as will now be explained. 
     FIG. 23  illustrates an RL circuit, with a switch SW connected to ground. If, in  FIG. 24 , the switch SW is connected to the supply voltage of 12 volts, the output current will rise exponentially, as shown in  FIG. 24 . The parameter T is the time constant, which depends on the values of R and L. The graph follows the form I=12−12×EXP(−t*T), wherein t is time and T is the time constant. 
   If the switch SW is grounded at 5 T, as in  FIG. 25 , then I decays exponentially to zero, as indicated. 
   If the switch SW alternates between the two positions every 5 T, the waveform of  FIG. 26  is generated. 
   Assume that the switch SW initially was in the position of  FIG. 23 , and then is moved to the position of  FIG. 24 . As shown in  FIG. 27 , current I rises, and follows path  300 . Assume now that, at time T (one time constant later), switch SW is grounded, as in  FIG. 25 . Now I starts to decay, along path  305  in  FIG. 27 . Assume that switch SW is re-connected to 12 volts at 1.5 T. I now rises again. 
   As indicated in  FIG. 27 , inductor L is alternately charged (when switch SW is connected to 12 volts) and discharged (when switch SW is connected to ground.) However, inductor L is not allowed to completely charge, or completely discharge. 
   If this alternation of position of switch SW, every 0.5T, is maintained, the sawtooth waveform of  FIG. 27  will be created. That waveform has an average AVG. 
   Now assume that, in  FIG. 28 , switch SW is not grounded at time 4T. I will attempt to rise exponentially to Imax, and attempt to follow path  307 . Assume that, at 5T, the oscillation of switch SW is resumed, but now every 0.25T. The sawtooth waveform of  FIG. 29  will be obtained, having a different average voltage AVG. 
   In the general case, by adjusting the duty cycle at which switch SW opens and closed, one can adjust the average current produced. If the timing is chosen properly, one can generate a sinusoidal waveform, such as that shown in  FIG. 30 . That is, one chooses the proper duty cycles, and arranges them in the proper sequence, to produce the sequence represented by AVG 1 , AVG 2 , and so on in  FIG. 30 . 
   This can be explained from another perspective.  FIG. 32  is another representation of the switching events of switch SW in  FIG. 23 , but with the switch SW absent in  FIG. 32 . Instead, the voltage created at point Pin by the switch SW is plotted. 
   In plot  350 , Pin is held at 12 volts, except for brief intervals such as  355 , wherein Pin is driven to zero volts. This is a large duty cycle. Thus, current I can be thought as being held at Imax, which equals the input voltage divided by R, except for the brief intervals  355 , at which it attempts to exponentially fall to zero. However, before current I falls very far, Pin is again raised to 12 volts. Thus, lout is kept at the relatively high average voltage  360 , which is slightly below Imax. 
   Conversely, in plot  375 , Pin is held at zero volts, except for brief intervals such as  380 , wherein Pin is driven to 12 volts. This is a low duty cycle. Thus, current I can be thought as being held at zero, except for the brief intervals  380 , at which it attempts to exponentially rise to Imax. However, before current I rises very far, Pin is again dropped to zero volts. Thus, current I is kept at the relatively low average voltage  390 , which is slightly above zero. 
   In a similar manner, plot  400  causes current I to remain at average voltage  410 , between the two extremes just described. In the general case, the average voltage of current I depends on the relative duration of intervals  355  and  380 , or the duty cycle. 
   Under the invention, the type of switching just described is undertaken using the switches of  FIG. 22 , to create sinusoidal currents in the coils of the 2-phase motor. Further, “soft switching” is used. 
   In the opposite, namely, “hard switching,” switch  450  in  FIG. 22  would be repeatedly opened and closed, analogous to switch SW in  FIG. 23 , in order to develop an average waveform of the current through coil CX of the proper value. 
   Whenever switch  450  is opened, the voltage at point PH tends to jump to a high value. To accommodate this jump, a diode D is provided, connected between PH and the 12 volt line. The jumping voltage now generates a current which is fed back to the power supply. A diode is provided for each switch in  FIG. 22 , the switches taking the form of transistors. 
   In soft switching, current through the coil, for example coil CX in quadrant I in  FIG. 31 , is handled by first charging coil CX through a closed switch  500 . A rising current is generated, which is supplied by a DC bus capacitor (not shown) analogous to rising current  300  in  FIG. 27 . A similar rising current is generated in coil CY, and supplied by the same bus capacitor (not shown). As explained later, the clocks for coils CX and CY are not simultaneous, although possibly of identical frequency. This lack of simultaneity eliminates any requirement that the bus capacitor supply two rising currents together, to both coils CX and CY at once. The bus capacitor supplies CX and then CY. 
   Then, switch  500  in  FIG. 31  is opened and switch  505  is simultaneously closed, allowing the existing current to discharge, analogous to the falling current  305  in  FIG. 27 . 
   Repetition of the process shown in  FIG. 31 , indicated by arrow  515 , generates a current having an average value in coil CX, analogous to the situations of  FIGS. 27–32 . 
   From another point of view, a voltage is first applied to coil CX, which generates a rising current. Then, the voltage is removed, and a resistance is placed in parallel with CX, to absorb the current present, which then decays. That resistance includes the resistance of switch  505 ,  510 , and components between them external to the H-bridge. Further, resistance within coil CX dissipates some energy. 
   Similar soft-switching occurs in the other quadrants of  FIG. 31 , as indicated by the horizontal double-ended arrows. 
   3. A significant feature of the invention is that the voltages applied to coils CX and CY in  FIG. 16  need not be synchronized. That is, each coil CX and CY carries its own current, and generates its own field vector B, without regard to the other. Of course, in one mode of operation, the currents are arranged to work together to generate a rotating field vector. 
   Nevertheless, the currents in each coil are in many respects independent. For example, the exact frequency at which the switches are cycled in  FIG. 31  are independent for coil X, compared with coil Y. For example, the switching of coil X may be 20 KHz, and coil CY may be 21 KHz. Further, those frequencies need not be related, nor synchronous. This will be explained in the context of the prior-art 3-phase case. 
   In  FIG. 33 , three voltages are applied to three coils in a WYE-connected system. However, the rising edges of the voltages must be synchronized, as indicated by dashed line  550 . That is, if pulse train  400 B were shifted by time T 1 , that would have the effect of shifting the phase of SIN 6  to SIN 6 A in  FIG. 34 . That phase shift cannot be allowed, because that would alter the voltages applied to the other coils in  FIG. 33 . Restated, that shift would create unbalanced operation in the WYE-connected system. 
   4.  FIG. 35  illustrates several columns of options available to a designer of a motor system. The designer would select the appropriate elements from each column, to design a system. Of course, some elements are incompatible with others. For example, speed control through control of the rotor field, by adjusting voltage, is not, in general, applicable to a stepper motor, which can be classified as a switched-reluctance motor. 
   The present invention utilizes a specific combination of elements in  FIG. 35 , namely, a two-phase motor having a synchronous type stator, as indicated. Also, a buried permanent magnet rotor is possible. The control system implements Field-Oriented Control, FOC, to maintain the stator field in quadrature with the rotor field. 
   5. The invention reduces computation required, by eliminating the transformations from a three-phase reference frame, to a two-phase frame, and the converse transformations. One quantitative measure of the saving in computation is the following. 
   One or more of the Inventors, or their designees, wrote computer code for FOC of a three-phase system of the type shown in  FIG. 2 . The code ran on a 16-bit Digital Signal Processor, DSP, running at a given clock speed. This processor is of the type which executes one program instruction per clock cycle. 
   When compiled, the code occupied about N lines, or N instructions. 
   Functionally equivalent code was written for FOC for the 2-phase system of  FIGS. 15  et seq., for the same DSP at the same clock speed. The code, when compiled, occupied about 0.69 N lines, or 0.69 N instructions. The reduction was about 30 percent. A reduction greater than 25 percent is accurately descriptive. 
   6. One characterization of the invention can be based on the preceding point 5. In a system wherein energy from a vehicle battery is converted into mechanical power in a power steering system, a three-phase FOC requires a program, running on a controller, and containing N instructions. For the same energy conversion context, the invention, by substituting a 2-phase motor and associated FOC, reduces the number of instructions to 0.69N. 
   7. Some comparisons of the 3-phase system, compared with the 2-phase system, will be given. For a given DSP, the latter requires a program containing 30 percent fewer instructions. As explained in the co-pending application, the 3-phase system requires two transistors per coil, or six transistors total. In the invention, four transistors per coil are required, for a total of eight transistors for two coils. 
   In the 3-phase case, three wires enter the motor, the neutral being contained within the motor. Under the invention, four wires enter the motor. 
   In the 3-phase case, a relay is required, as explained in the co-pending application. The invention eliminates the relay. 
   In the 3-phase case, maximum line voltage is not delivered to each coil. Under the invention, maximum line voltage is available to each coil. Also, in the two-phase case, the peak current is radical 3-over-2, or 0.866, of the peak current in the three-phase case. This reduction in peak current reduces Joule heating losses in the switching transistors, and other resistive elements in the power converter. Further, the lower current allows the use of switching transistors of lower current rating, which are less expensive. 
   As explained above, the invention generates a sine wave for each coil in the two-phase motor. One advantage of the invention is that the duty cycle needed for the PWM can be computed directly from the sine and cosine of rotor angle, as opposed to the three-phase case which requires a much more cumbersome process. For example, if the rotor angle is 45 degrees, and the controller specifies a phase angle of Beta, then the duty cycle for one coil will be sine(45+Beta) and the duty cycle for the other coil will be cosine(45+Beta) at that instant. 
   Numerous substitutions and modifications can be undertaken without departing from the true spirit and scope of the invention. What is desired to be secured by Letters Patent is the invention as defined in the following claims.