Abstract:
The invention includes a motor controller and technique for controlling a permanent magnet motor. In accordance with one aspect of the present technique, a permanent magnet motor is controlled by receiving a torque command, determining a physical torque limit based on a stator frequency, determining a theoretical torque limit based on a maximum available voltage and motor inductance ratio, and limiting the torque command to the smaller of the physical torque limit and the theoretical torque limit. Receiving the torque command may include normalizing the torque command to obtain a normalized torque command, determining the physical torque limit may include determining a normalized physical torque limit, determining a theoretical torque limit may include determining a normalized theoretical torque limit, and limiting the torque command may include limiting the normalized torque command to the smaller of the normalized physical torque limit and the normalized theoretical torque limit.

Description:
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH &amp; DEVELOPMENT 
     This invention was made with Government support under contract number NREL-ZCL-3-32060-03; W(A)-03-011, CH-1137 awarded by Department of Energy. The Government has certain rights in the invention. 
    
    
     BACKGROUND 
     The invention relates generally to torque control of a permanent magnet motor. More particularly, the invention relates to a technique for torque control of a permanent magnet motor operating above base speed. 
     Three phase interior permanent magnet synchronous motors (IPMSM) receive three phases of electrical voltage, which enter three stator windings of the motor to produce a rotating magnetic stator field. The rotating magnetic stator field interacts with a magnetic field associated with the permanent magnets of the motor. The rotor rotates based on interactions between the magnetic stator field and the permanent magnetic field of the rotor. 
     To more precisely control the output torque and speed of the motor, a synchronous reference frame may be employed, which is represented by a quadrature axis (q) and a direct axis (d) defined by the relative location of the rotor to the stator windings. In the synchronous reference frame, voltages for obtaining a particular torque and speed may be more easily determined. Once direct-axis and quadrature-axis voltages are obtained for the motor in the synchronous reference frame, a mathematical transformation may be used to produce the equivalent three-phase voltage in a stationary reference frame, in which (a), (b), and (c) axes are defined by the location of the stator windings of the motor. The three-phase voltage in the stationary reference frame may subsequently be used to drive the motor. 
     When operating below base speed (generally considered to be a speed at which a voltage limit has been reached and additional speed may be achieved primarily by weakening the magnetic fields of permanent magnets in the rotor), the amount of torque output by the motor may generally be adjusted by changing the magnitude and frequency of the driving voltages in a relatively straightforward manner. To operate above base speed, the stator current in the direct axis of the permanent magnet motor operates to weaken the magnetic field of the permanent magnets, and thus a motor operating above base speed may be referred to be operating in a field-weakening region. However, as the magnetic field of the permanent magnets is reduced, control becomes much more complex. Though techniques for generating a maximum torque per amperes with a permanent magnet motor have been developed, the techniques remain limited to very specific applications and may vary considerably from one motor to another. 
     BRIEF DESCRIPTION 
     The invention includes a motor controller and technique for controlling a permanent magnet motor. In accordance with one aspect of the present technique, a permanent magnet motor is controlled by receiving a torque command, determining a physical torque limit based on a stator frequency, determining a theoretical torque limit based on a maximum available voltage and motor saliency, and limiting the torque command to the smaller of the physical torque limit and the theoretical torque limit. The physical torque limit may be determined by multiplying a ratio of a rated stator frequency to the stator frequency with a rated torque and an overload torque ratio. For universality, receiving the torque command may include normalizing the torque command to obtain a normalized torque command, determining the physical torque limit may include determining a normalized physical torque limit, determining a theoretical torque limit may include determining a normalized theoretical torque limit, and limiting the torque command may include limiting the normalized torque command to the smaller of the normalized physical torque limit and the normalized theoretical torque limit. In each case, the normalized torque command, the normalized physical torque limit, and the normalized theoretical torque limit may be normalized to a characteristic current of the motor. Determining the theoretical torque limit may include determining the maximum available voltage based on the stator frequency, magnetic flux of permanent magnets in the motor, and a rated voltage of the motor, and determining the maximum available voltage may include determining a ratio of the rated voltage to a multiplication of the stator frequency and magnetic flux of the permanent magnets in the motor. 
     In accordance with another aspect of the present invention, a motor controller is provided that may include an inverter configured to supply a three-phase voltage to a permanent magnet motor, driver circuitry configured to cause the inverter to supply the three-phase voltage to the permanent magnet motor based on a voltage command, and control circuitry configured to receive a torque command and generate the voltage command based on the torque command. The control circuitry may be configured to limit the torque command to a theoretical torque limit based at least in part on a saliency of the permanent magnet motor. Additionally, the control circuitry may be configured to limit the torque command to the theoretical torque limit based at least in part on the saliency of the permanent magnet motor, a maximum available voltage, and a maximum available flux-producing current, and the control circuitry may be configured to limit the torque command to a lesser of the theoretical torque limit or a physical torque limit based on a stator frequency. 
    
    
     
       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 simplified diagram of an interior permanent magnet synchronous motor; 
         FIG. 2  is a dynamic block diagram of the interior permanent magnet synchronous motor of  FIG. 1  in a synchronous reference frame; 
         FIG. 3  is a vector diagram representing the phasors of operating parameters of the interior permanent magnet synchronous motor of  FIG. 1 ; 
         FIG. 4  is a block diagram of a torque limit control of the interior permanent magnet synchronous motor of  FIG. 1  for above base speed operation in accordance with an aspect of the invention; 
         FIG. 5  is a block diagram of an optimum torque control of the interior permanent magnet synchronous motor of  FIG. 1  for above base speed operation employing a three-dimensional table in accordance with an aspect of the invention; 
         FIG. 6  represents an exemplary two-dimensional component of a three-dimensional table for application in the optimum torque control block diagram depicted in  FIG. 5 ; and 
         FIG. 7  is a block diagram of an optimum torque control of the interior permanent magnet synchronous motor of  FIG. 1  for above base speed operation employing a closed-loop solver in accordance with an aspect of the invention. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a simplified diagram of an interior permanent magnet synchronous motor  10 . The motor  10  includes conductive material  12  (typically ferromagnetic) and permanent magnets  14 . Three stator windings  16 ,  18 , and  20  receive a three-phase current to produce a stator magnetic field. The stator magnetic field interacts with a magnetic field caused by permanent magnets  14  within a rotor  22 , causing the rotor  22  to rotate accordingly. 
     Three-phase voltage may generally be supplied to the stator windings  16 ,  18 , and  20  by way of an inverter module (not shown), which may receive power from a DC voltage supply. Driver circuitry may direct the inverter module to output the three-phase power at a desired frequency, based upon control signals received by the driver circuitry from control circuitry. The control circuitry may generally determine the appropriate control signals to send to the driver circuitry based upon a torque signal received from an operator or remote controller, as well as feedback from the motor  10 , the inverter module, the driver circuitry, and from calculations carried out within the control circuitry. To perform motor control operations, the control circuitry may include an appropriate processor, such as a microprocessor or field programmable gate array, and may perform a variety of motor control calculations, including those techniques described herein. The control circuitry may include a memory device or a machine-readable medium such as Flash memory, EEPROM, ROM, CD-ROM or other optical data storage media, or any other appropriate storage medium which may store data or instructions for carrying out the foregoing techniques. 
     To simplify the analysis of the motor  10 , it may be assumed that material  12  has a permeability equal to infinity (i.e., there is no saturation), that the stator windings are assumed to be sinusoidal distributed (i.e., magneto motive force (mmf) space harmonics and slot harmonics may be neglected), and that the stator winding fields are assumed to be sinusoidally distributed, (i.e., only the first harmonic is shown). Additionally, the stator windings may be assumed to be symmetric, and thus winding turns, resistance, and inductances may be assumed to be equal. Further, a lumped-parameter circuit model may also be assumed. 
     In a stationary reference frame, the voltage and torque associated with stator windings  16 ,  18 , and  20  of motor  10  may be represented by the following equations, where x represents the phases a, b, or c of motor  10 : 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       x 
                     
                     = 
                     
                       
                         R 
                         · 
                         
                           i 
                           x 
                         
                       
                       + 
                       
                         
                           ⅆ 
                           
                             ψ 
                             x 
                           
                         
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
             
               
                 
                   
                     Σ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     T 
                   
                   = 
                   
                     J 
                     · 
                     
                       
                         
                           ⅆ 
                           ω 
                         
                         
                           ⅆ 
                           t 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     In equation (1), V x  represents the instantaneous phase voltage [V], R represents the stator phase resistance [Ω], i x  represents the instantaneous stator phase current [A], and Ψ x  represents the instantaneous stator flux linkage [Wb]. In equation (2), ΣT represents a sum of all torque [Nm], including load torque, of the motor  10 , J represents the moment of inertia [kg m 2 ], and ω represents rotor speed [rad/sec]. 
     Flux linkage ψ a , ψ b , and ψ c  for stator windings  16 ,  18 , and  20  of motor  10  may be described according to the following equations: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               ψ 
                               a 
                             
                             = 
                             
                               
                                 
                                   L 
                                   a 
                                 
                                 · 
                                 
                                   i 
                                   a 
                                 
                               
                               + 
                               
                                 
                                   L 
                                   ab 
                                 
                                 · 
                                 
                                   i 
                                   b 
                                 
                               
                               + 
                               
                                 
                                   L 
                                   
                                     a 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     c 
                                   
                                 
                                 · 
                                 
                                   i 
                                   c 
                                 
                               
                               + 
                               
                                 ψ 
                                 am 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ψ 
                               b 
                             
                             = 
                             
                               
                                 
                                   L 
                                   ba 
                                 
                                 · 
                                 
                                   i 
                                   a 
                                 
                               
                               + 
                               
                                 
                                   L 
                                   b 
                                 
                                 · 
                                 
                                   i 
                                   b 
                                 
                               
                               + 
                               
                                 
                                   L 
                                   bc 
                                 
                                 · 
                                 
                                   i 
                                   c 
                                 
                               
                               + 
                               
                                 ψ 
                                 bm 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ψ 
                               c 
                             
                             = 
                             
                               
                                 
                                   L 
                                   ca 
                                 
                                 · 
                                 
                                   i 
                                   a 
                                 
                               
                               + 
                               
                                 
                                   L 
                                   cb 
                                 
                                 · 
                                 
                                   i 
                                   b 
                                 
                               
                               + 
                               
                                 
                                   L 
                                   c 
                                 
                                 · 
                                 
                                   i 
                                   c 
                                 
                               
                               + 
                               
                                 ψ 
                                 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   m 
                                 
                               
                             
                           
                         
                       
                     
                     } 
                   
                   . 
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     In equation (3) above, L a , L b , and L c  represent self-inductance [H], L ab , L ba , L ac , L ca , L bc , and L cb  represent mutual inductances [H], and ψ am , ψ bm , and ψ cm  represent flux linkage [Wb] from the permanent magnets  14 . The values of self inductance and mutual inductance are a function of the rotor  22  position, which vary as the rotor  22  rotates relative to the stator windings a  16 , b  18 , and c  20 . Mutual-inductances L ab , L ba , L ac , L ca , L bc , and L cb  also conform to the following equations: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               L 
                               ab 
                             
                             = 
                             
                               L 
                               ba 
                             
                           
                         
                       
                       
                         
                           
                             
                               L 
                               
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 c 
                               
                             
                             = 
                             
                               L 
                               ca 
                             
                           
                         
                       
                       
                         
                           
                             
                               L 
                               
                                 b 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 c 
                               
                             
                             = 
                             
                               L 
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 b 
                               
                             
                           
                         
                       
                     
                     } 
                   
                   . 
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Continuing to view  FIG. 1 , motor  10  may be understood to operate in a synchronous reference frame as well as a stationary frame. The synchronous reference frame includes a quadrature axis (q)  24  and a direct axis (d)  26 . The quadrature axis (q)  24  is defined by the relative location of the permanent magnets of the rotor  22 , and direct axis (d)  26  is defined relative to an angular position γ e    28  from stator windings a  16 , b  18 , and c  20 . The equations for the motor  10  in the synchronous reference frame may be written in the following form: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       d 
                     
                     = 
                     
                       
                         R 
                         · 
                         
                           I 
                           d 
                         
                       
                       + 
                       
                         
                           L 
                           d 
                         
                         · 
                         
                           
                             ⅆ 
                             
                               I 
                               d 
                             
                           
                           
                             ⅆ 
                             t 
                           
                         
                       
                       - 
                       
                         
                           L 
                           q 
                         
                         · 
                         
                           I 
                           q 
                         
                         · 
                         
                           
                             ⅆ 
                             
                               γ 
                               e 
                             
                           
                           
                             ⅆ 
                             t 
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       V 
                       q 
                     
                     = 
                     
                       
                         R 
                         · 
                         
                           I 
                           q 
                         
                       
                       + 
                       
                         
                           L 
                           q 
                         
                         · 
                         
                           
                             ⅆ 
                             
                               I 
                               q 
                             
                           
                           
                             ⅆ 
                             t 
                           
                         
                       
                       + 
                       
                         
                           ( 
                           
                             
                               
                                 L 
                                 d 
                               
                               · 
                               
                                 I 
                                 d 
                               
                             
                             + 
                             
                               ψ 
                               
                                 m 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                             
                           
                           ) 
                         
                         · 
                         
                           
                             ⅆ 
                             
                               γ 
                               e 
                             
                           
                           
                             ⅆ 
                             t 
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     J 
                     · 
                     
                       
                         
                           ⅆ 
                           2 
                         
                         ⁢ 
                         
                           γ 
                           e 
                         
                       
                       
                         ⅆ 
                         
                           t 
                           2 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         3 
                         2 
                       
                       · 
                       
                         p 
                         n 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               Ψ 
                               
                                 m 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                             
                             · 
                             
                               I 
                               q 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   L 
                                   d 
                                 
                                 - 
                                 
                                   L 
                                   q 
                                 
                               
                               ) 
                             
                             · 
                             
                               I 
                               d 
                             
                             · 
                             
                               I 
                               q 
                             
                           
                         
                         ] 
                       
                     
                     - 
                     
                       
                         T 
                         load 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     The equations above describe motor  10  in the synchronous reference frame (d, q). As such, V d  represents a direct-axis voltage [V] and V q  represents a quadrature-axis voltage [V] applied to the motor  10 , R represents the stator resistance [Ω], I d  represents a flux-producing current [A] and I q  represents a torque-producing current [A], L d  and L q  represent direct-axis and quadrature-axis inductances [H], respectively, γ e  represents angular distance γ e    28  [rad/sec], Ψ m0  represents the magnetic flux [Wb] of the pole pairs of the rotor  22 , p n  represents the number of permanent magnets of the rotor  22 , and T load  represents the torque [Nm] exerted against motor  10  by the load. Self inductance L d  and L q  may be further represented according to equations (8) and (9) below. It should be noted that usually for a surface permanent magnet synchronous motor, L d  is equal to L q . 
     
       
         
           
             
               
                 
                   
                     
                       L 
                       d 
                     
                     = 
                     
                       
                         
                           3 
                           2 
                         
                         · 
                         
                           ( 
                           
                             
                               L 
                               0 
                             
                             + 
                             
                               L 
                               m 
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           3 
                           2 
                         
                         · 
                         
                           L 
                           
                             
                               d 
                               . 
                               p 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             h 
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     L 
                     q 
                   
                   = 
                   
                     
                       
                         3 
                         2 
                       
                       · 
                       
                         ( 
                         
                           
                             L 
                             0 
                           
                           - 
                           
                             L 
                             m 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         3 
                         2 
                       
                       · 
                       
                         
                           L 
                           
                             
                               q 
                               . 
                               p 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             h 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Turning to  FIG. 2 , a dynamic block diagram  30  represents the interior permanent magnet synchronous motor  10  in a synchronous reference frame (d, q). With known values of the three-phase voltage waveforms  32  and the relative angular position γ e    34 , a coordinate transformation  36  may be performed, producing direct-axis voltage V d    38 , and quadrature-axis voltage V q    40 . In accordance with equation (5) above, a summer  42  adds direct-axis voltage V d    38 , subtracts direct-axis current I d    44  multiplied by stator resistance R  46 , and adds voltage E d    48 , outputting direct-axis change in flux 
               ⅆ     Ψ   d         ⅆ   t             50 , which is equal to
 
               L   d     ·         ⅆ     I   d         ⅆ   t       .           
The origin of voltage E d    48  will be discussed further below.
 
     Because direct-axis change in flux 
               ⅆ     Ψ   d         ⅆ   t             50  is equal to
 
                 L   d     ·       ⅆ     I   d         ⅆ   t         ,         
multiplying the inverse of direct-axis inductance
 
             1     L   d             52  by direct-axis change in flux
 
                 ⅆ     Ψ   d         ⅆ   t       ⁢           ⁢   50         
produces direct-axis change in current
 
                 ⅆ     I   d         ⅆ   t       ⁢           ⁢   54.         
When direct-axis change in current
 
                 ⅆ     I   d         ⅆ   t       ⁢           ⁢   54         
is integrated by the Laplace operator
 
                 1   s     ⁢           ⁢   56     ,         
direct-axis current I d    44  results.
 
     By multiplying direct-axis current I d    44  with direct-axis inductance L d    58 , direct-axis flux ψ d    60  is produced. Direct-axis flux ψ d    60  may subsequently enter multiplier  62  with stator frequency ω e    64 , producing voltage L d I d ω e    66 . Permanent magnet flux Ψ m0    68  and stator frequency ω e    70  multiplied in multiplier  72  produce voltage E 0    74 . When voltage L d I d ω e    66  is added to voltage E 0    74  in summer  76 , voltage E q    78  results. 
     As apparent from equation (6), when summer  80  subtracts voltage E q    78  from quadrature-axis voltage V q    40  and quadrature-axis current I q    82  multiplied by stator resistance R  84 , the result is quadrature-axis change in flux 
                   ⅆ     Ψ   q         ⅆ   t       ⁢           ⁢   86     ,         
which is equal to
 
               L   q     ·         ⅆ     I   q         ⅆ   t       .           
Multiplying by the inverse of quadrature-axis inductance
 
               1     L   q       ⁢           ⁢   88         
thus produces quadrature-axis change in current
 
                   ⅆ     I   q         ⅆ   t       ⁢           ⁢   90     ,         
which may be integrated via the Laplace operator
 
               1   s     ⁢           ⁢   92         
to produce the quadrature-axis current I q    82 .
 
     When quadrature-axis current I q    82  is multiplied by quadrature-axis inductance L q    94 , the result is quadrature-axis flux ψ q    96 . Multiplying quadrature-axis flux ψ q    96  and stator frequency ω e    64  in multiplier  98  produces voltage E d    48 , which enters summer  42 , as discussed above. 
     Quadrature-axis current I q    82  enters multiplier  100  where it is multiplied by direct-axis current I d    44 . The result is multiplied by block  102 , which represents a value of the direct-axis inductance L d  less the quadrature-axis inductance L q , and subsequently enters summer  104 . Meanwhile, permanent magnet flux Ψ m0    68  is multiplied by quadrature-axis current I q    82  in multiplier  106  which enters summer  104 . The output of summer  104  is subsequently multiplied by block  108 , which represents the value 
                 3   2     ·     p   n       ,         
to produce a motor torque T mot    110  representing the torque output by motor  10 .
 
     The load torque T load    112  may be subtracted from motor torque T mot    110  in summer  114 , producing an excess torque 
             J   ⁢         ⅆ   2     ⁢     γ   e         ⅆ     t   2         ⁢           ⁢   116.         
In block  118 ,
 
               1     J   ·   s       ,         
the moment of inertia J is divided from excess torque
 
             J   ⁢         ⅆ   2     ⁢     γ   e         ⅆ     t   2         ⁢           ⁢   116         
and excess torque
 
             J   ⁢         ⅆ   2     ⁢     γ   e         ⅆ     t   2         ⁢           ⁢   116         
is integrated, resulting in motor frequency ω  120  of motor  10 .
 
     By multiplying motor frequency ω  120  by permanent magnet pole pairs p n , stator frequency ω e    64  and  70  may be obtained. Motor frequency ω  120  may also be integrated in Laplace integral 
               1   s     ⁢           ⁢   122         
to produce a motor angular position γ  124 . The motor angular position γ  124  may be subsequently multiplied by permanent magnet pole pairs p n    126  to obtain angular position γ e    34 .
 
     As illustrated by dynamic block diagram  30 , equations (5) and (6) may be rewritten for a steady state condition, according to the following equations:
 
 V   d   =R·I   d   −L   q   ·I   q ·ω e    (10);
 
 V   q   =R·I   q +ω e   ·L   d   ·I   d +ω e ·ψ m0    (11).
 
     Equations (10) and (11) may alternatively be expressed in the following form:
 
 V   d   =R·I   d   −E   d    (12);
 
 V   q   =R·I   q   +E   q    (13).
 
     In equations (12) and (13), E d  and E q  may be defined according to the following equations:
 
 E   d   =L   q   ·I   q ·ω e    (14);
 
 E   q   =L   d   ·I   d ·ω e +Ψ m0 ·ω e   =L   d   ·I   d ·ω e   +E   0    (15);
 
 E   Σ =√{square root over ( E   d   2   +E   q   2 )}=ω e ·√{square root over (( L   q   ·I   q ) 2 +( L   d   ·I   d +Ψ m0 ) 2 )}{square root over (( L   q   ·I   q ) 2 +( L   d   ·I   d +Ψ m0 ) 2 )}  (16).
 
       FIG. 3  represents a basic vector diagram  128  of the motor  10 , based upon equations (10)-(16) above. A stator current I st    130  may be broken into direct-axis and quadrature-axis components, direct-axis current I d    132  and quadrature-axis current I q    134 , based upon on the magnitude of stator current I st    130  and an angle β  136 . A stator voltage V a    138  may be broken into direct-axis and quadrature-axis components direct-axis voltage V d    140  and quadrature-axis voltage V q    142 . 
     Direct-axis voltage V d    140  may be further broken into a voltage E d    144  component less a voltage drop I d ·R  146 , representing the voltage drop across stator resistance R caused by direct-axis current I d . 
     Quadrature-axis voltage V q    142  may also be broken into additional components. Voltage E q    148  is equal to voltage E 0    150 , which represents a value of stator frequency ω e  multiplied by permanent magnet flux Ψ m0 , less a voltage ω e ·L d ·I d    152 . Quadrature-axis voltage V q    142  may be obtained by adding voltage drop I q ·R  154 , representing a voltage drop across stator resistance R caused by quadrature-axis current I q , to voltage E q    148 . 
     Flux  156  may be broken into direct-axis and quadrature-axis components, direct-axis flux Ψ d    158  and quadrature-axis flux Ψ q    160 . To obtain flux  156 , vectors representing permanent magnet flux Ψ m0    162 , quadrature-axis flux Ψ q    160 , and flux L d ·I d    164  may be summed. 
     To obtain an optimum torque control algorithm for above base speed operation, a torque equation should be considered. A general equation representing motor torque T mot  in the synchronous reference frame may be written as follows: 
     
       
         
           
             
               
                 
                   
                     T 
                     mot 
                   
                   = 
                   
                     
                       3 
                       2 
                     
                     · 
                     
                       p 
                       n 
                     
                     · 
                     
                       I 
                       q 
                     
                     · 
                     
                       
                         [ 
                         
                           
                             Ψ 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   L 
                                   q 
                                 
                                 - 
                                 
                                   L 
                                   d 
                                 
                               
                               ) 
                             
                             · 
                             
                               I 
                               d 
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     By rewriting equations (10) and (11) with the assumption that the voltage drop across stator resistance R is negligible above base speed, the following equations may be obtained:
 
 V   d =−ω e   ·L   q   ·I   q    (18);
 
 V   q =ω e   ·L   d   ·I   d +ω e ·ψ m0    (19).
 
     When motor  10  operates above base speed, motor voltage remains constant according to the following equation:
 
 V   d.rtd   2   +V   q.rtd   2   =V   rtd   2    (20).
 
     To achieve above base speed operation, direct-axis current I d  may cause permanent magnets  14  of motor  10  to become temporarily weakened or demagnetized. A direct-axis current I d  that fully demagnetizes the permanent magnets  14  may be referred to as the “characteristic current.” The characteristic current I df  may be represented by the following equation: 
     
       
         
           
             
               
                 
                   
                     I 
                     df 
                   
                   = 
                   
                     - 
                     
                       
                         
                           Ψ 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                         
                         
                           L 
                           d 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     The torque equation may be normalized to the characteristic current I df . Accordingly, a normalized torque is defined according to the following equations: 
     
       
         
           
             
               
                 
                   
                     T 
                     ^ 
                   
                   = 
                   
                     
                       
                         T 
                         mot 
                       
                       
                         T 
                         base 
                       
                     
                     = 
                     
                       
                         
                           T 
                           mot 
                         
                         
                           
                             3 
                             2 
                           
                           · 
                           
                             p 
                             n 
                           
                           · 
                           
                             Ψ 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                           
                           · 
                           
                             I 
                             df 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     In the above equation (22), normalized base torque T base  is defined according to the following equation: 
     
       
         
           
             
               
                 
                   
                     T 
                     base 
                   
                   = 
                   
                     
                       3 
                       2 
                     
                     · 
                     
                       p 
                       n 
                     
                     · 
                     
                       Ψ 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                     
                     · 
                     
                       
                         I 
                         df 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     Substituting equation (17) into equation (22) produces the following equation: 
     
       
         
           
             
               
                 
                   
                     T 
                     ^ 
                   
                   = 
                   
                     
                       
                         
                           I 
                           q 
                         
                         
                           I 
                           df 
                         
                       
                       · 
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               
                                 L 
                                 d 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     
                                       L 
                                       q 
                                     
                                     
                                       L 
                                       d 
                                     
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                               · 
                               
                                 I 
                                 d 
                               
                             
                             
                               Ψ 
                               
                                 m 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         
                           
                             I 
                             q 
                           
                           
                             I 
                             df 
                           
                         
                         · 
                         
                           [ 
                           
                             1 
                             - 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         L 
                                         q 
                                       
                                       
                                         L 
                                         d 
                                       
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 · 
                                 
                                   I 
                                   d 
                                 
                               
                               
                                 
                                   Ψ 
                                   
                                     m 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     0 
                                   
                                 
                                 
                                   L 
                                   d 
                                 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         
                           
                             I 
                             q 
                           
                           
                             I 
                             df 
                           
                         
                         · 
                         
                           
                             [ 
                             
                               1 
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         L 
                                         q 
                                       
                                       
                                         L 
                                         d 
                                       
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 · 
                                 
                                   
                                     I 
                                     d 
                                   
                                   
                                     I 
                                     df 
                                   
                                 
                               
                             
                             ] 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Thus, the following normalized torque equation may be rewritten as the following equation:
 
 {circumflex over (T)}=Î   q ·(1 −K·Î   d )   (25).
 
     Equations (18) and (19) may also be rewritten in a normalized, per-unit form, according to the following equations: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             V 
                             d 
                           
                           
                             
                               ω 
                               
                                 e 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                             
                             · 
                             
                               Ψ 
                               
                                 m 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                             
                           
                         
                         = 
                         
                           
                             V 
                             ^ 
                           
                           d 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               
                                 ω 
                                 e 
                               
                             
                             · 
                             
                               L 
                               q 
                             
                             · 
                             
                               I 
                               q 
                             
                             · 
                             
                               L 
                               d 
                             
                           
                           
                             
                               ω 
                               
                                 e 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                             
                             · 
                             
                               Ψ 
                               
                                 m 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                             
                             · 
                             
                               L 
                               d 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               
                                 
                                   ω 
                                   ^ 
                                 
                                 e 
                               
                             
                             · 
                             
                               ( 
                               
                                 K 
                                 + 
                                 1 
                               
                               ) 
                             
                             · 
                             
                               
                                 I 
                                 ^ 
                               
                               q 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           or 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 V 
                                 ^ 
                               
                               d 
                             
                             
                               
                                 ω 
                                 ^ 
                               
                               e 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                               ( 
                               
                                 K 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           · 
                           
                             
                               
                                 I 
                                 ^ 
                               
                               q 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             V 
                             q 
                           
                           
                             
                               ω 
                               
                                 e 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                             
                             · 
                             
                               Ψ 
                               
                                 m 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                             
                           
                         
                         = 
                         
                           
                             V 
                             ^ 
                           
                           q 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 ω 
                                 e 
                               
                               · 
                               
                                 L 
                                 d 
                               
                               · 
                               
                                 I 
                                 d 
                               
                             
                             
                               
                                 ω 
                                 
                                   e 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                               · 
                               
                                 Ψ 
                                 
                                   m 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                             
                           
                           + 
                           
                             
                               
                                 ω 
                                 e 
                               
                               · 
                               
                                 Ψ 
                                 
                                   m 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                             
                             
                               
                                 ω 
                                 
                                   e 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                               · 
                               
                                 Ψ 
                                 
                                   m 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 ω 
                                 ^ 
                               
                               e 
                             
                             · 
                             
                               ( 
                               
                                 
                                   
                                     I 
                                     ^ 
                                   
                                   d 
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           or 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 V 
                                 ^ 
                               
                               q 
                             
                             
                               
                                 ω 
                                 ^ 
                               
                               e 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   I 
                                   ^ 
                                 
                                 d 
                               
                               + 
                               1 
                             
                             ) 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     The motor voltage in a normalized form may thus be written as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 V 
                                 ^ 
                               
                               
                                 d 
                                 . 
                                 rtd 
                               
                               2 
                             
                             + 
                             
                               
                                 V 
                                 ^ 
                               
                               
                                 q 
                                 . 
                                 rtd 
                               
                               2 
                             
                           
                           
                             
                               ω 
                               ^ 
                             
                             e 
                             2 
                           
                         
                         = 
                         
                           
                             
                               V 
                               ^ 
                             
                             rtd 
                             2 
                           
                           
                             
                               ω 
                               ^ 
                             
                             e 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             V 
                             ^ 
                           
                           max 
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 I 
                                 ^ 
                               
                               q 
                               2 
                             
                             · 
                             
                               
                                 ( 
                                 
                                   K 
                                   + 
                                   1 
                                 
                                 ) 
                               
                               2 
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       I 
                                       ^ 
                                     
                                     d 
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                               2 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     In equation (28) above, normalized quadrature-axis current Î q , normalized direct-axis current Î d , normalized stator frequency {circumflex over (ω)} e , normalized maximum available voltage {circumflex over (V)} max , and motor inductance ratio K may be described according to the following equations: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   I 
                                   ^ 
                                 
                                 q 
                               
                               = 
                               
                                 
                                   I 
                                   q 
                                 
                                 
                                   I 
                                   df 
                                 
                               
                             
                             ; 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   I 
                                   ^ 
                                 
                                 d 
                               
                               = 
                               
                                 
                                   I 
                                   d 
                                 
                                 
                                   I 
                                   df 
                                 
                               
                             
                             ; 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   ω 
                                   ^ 
                                 
                                 e 
                               
                               = 
                               
                                 
                                   ω 
                                   e 
                                 
                                 
                                   ω 
                                   
                                     e 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     0 
                                   
                                 
                               
                             
                             ; 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   V 
                                   ^ 
                                 
                                 max 
                               
                               = 
                               
                                 
                                   
                                     
                                       V 
                                       ^ 
                                     
                                     rtd 
                                   
                                   
                                     
                                       ω 
                                       ^ 
                                     
                                     e 
                                   
                                 
                                 = 
                                 
                                   
                                     
                                       
                                         V 
                                         rtd 
                                       
                                       · 
                                       
                                         ω 
                                         
                                           e 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           0 
                                         
                                       
                                     
                                     
                                       
                                         ω 
                                         
                                           e 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           0 
                                         
                                       
                                       · 
                                       
                                         Ψ 
                                         
                                           m 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           0 
                                         
                                       
                                       · 
                                       
                                         ω 
                                         e 
                                       
                                     
                                   
                                   = 
                                   
                                     
                                       V 
                                       rtd 
                                     
                                     
                                       
                                         Ψ 
                                         
                                           m 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           0 
                                         
                                       
                                       · 
                                       
                                         ω 
                                         e 
                                       
                                     
                                   
                                 
                               
                             
                             ; 
                           
                         
                       
                       
                         
                           
                             K 
                             = 
                             
                               ( 
                               
                                 
                                   
                                     L 
                                     q 
                                   
                                   
                                     L 
                                     d 
                                   
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                     } 
                   
                   . 
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     From the equations above, two normalized equations sharing two unknown variables normalized direct-axis current Î d  and normalized quadrature-axis current Î q , an above base speed operation may be determined. Normalized torque {circumflex over (T)} and normalized maximum available voltage {circumflex over (V)} max  may be obtained by way of the following equations:
 
 {circumflex over (T)}=Î   q ·(1 −K·Î   d )   (30);
 
 {circumflex over (V)}   max   2   =Î   q   2 ·( K +1) 2 +( Î   d +1) 2    (31).
 
     From equations (30) and (31), normalized torque {circumflex over (T)} may be reduced to a function of normalized direct-axis current Î d , normalized maximum available voltage {circumflex over (V)} max , and motor inductance ratio K, in accordance with the following equation:
 
 {circumflex over (T)}   2 ·( K +1) 2 =(1 −k·Î   d ) 2   ·└{circumflex over (V)}   max   2 −(1 +Î   d ) 2 ┘  (32).
 
     Equation (32) may be rewritten according to the following equation: 
     
       
         
           
             
               
                 
                   
                     T 
                     ^ 
                   
                   = 
                   
                     
                       
                         1 
                         - 
                         
                           K 
                           · 
                           
                             
                               I 
                               ^ 
                             
                             d 
                           
                         
                       
                       
                         K 
                         + 
                         1 
                       
                     
                     · 
                     
                       
                         
                           
                             
                               V 
                               ^ 
                             
                             max 
                             2 
                           
                           - 
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     I 
                                     ^ 
                                   
                                   d 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     From equations (32) or (33), a theoretical maximum torque value may be determined as a function of normalized direct-axis current Î d . A local maximum of normalized torque {circumflex over (T)} may be obtained with a derivative according to the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       ⅆ 
                       
                         
                           T 
                           ⁢ 
                           
                               
                           
                         
                         ^ 
                       
                     
                     
                       ⅆ 
                       
                         
                           I 
                           ^ 
                         
                         d 
                       
                     
                   
                   = 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           - 
                           
                             K 
                             
                               K 
                               + 
                               1 
                             
                           
                         
                         · 
                         
                           
                             
                               
                                 V 
                                 ^ 
                               
                               max 
                               2 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       I 
                                       ^ 
                                     
                                     d 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       - 
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               
                                 K 
                                 · 
                                 
                                   
                                     I 
                                     ^ 
                                   
                                   d 
                                 
                               
                             
                             ) 
                           
                           · 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   I 
                                   ^ 
                                 
                                 d 
                               
                             
                             ) 
                           
                         
                         
                           
                             ( 
                             
                               K 
                               + 
                               1 
                             
                             ) 
                           
                           · 
                           
                             
                               
                                 
                                   V 
                                   ^ 
                                 
                                 max 
                                 2 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       
                                         I 
                                         ^ 
                                       
                                       d 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                     = 
                     0. 
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     Equation (34) may alternatively be rewritten as the following equation:
 
− K·[{circumflex over (V)}   max   2 −(1 +Î   d     —     max ) 2 ]−(1 −K·Î   d     —     max )·(1 +Î   d     —     max )=0   (35).
 
     By manipulating equation (35), the following equation may be derived: 
     
       
         
           
             
               
                 
                   
                     
                       
                         I 
                         ^ 
                       
                       d_max 
                       2 
                     
                     + 
                     
                       
                         
                           
                             3 
                             · 
                             K 
                           
                           - 
                           1 
                         
                         
                           2 
                           · 
                           K 
                         
                       
                       · 
                       
                         I 
                         d_max 
                       
                     
                     - 
                     
                       
                         1 
                         2 
                       
                       · 
                       
                         ( 
                         
                           
                             1 
                             K 
                           
                           + 
                           
                             V 
                             max 
                             2 
                           
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                   = 
                   0. 
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     It will be apparent that equation (36) is a quadratic equation in normalized direct-axis current Î d . When motor inductance ratio K is greater than zero, meaning that the permanent magnets  14  of motor  10  are located beneath the surface of the rotor  22 , the solution of the equation is equal to the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       ^ 
                     
                     
                       d_max 
                       ⁢ 
                       
                         ( 
                         
                           K 
                           &gt; 
                           0 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         1 
                         - 
                         
                           3 
                           · 
                           K 
                         
                         - 
                         
                           
                             
                               
                                 K 
                                 2 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     8 
                                     · 
                                     
                                       
                                         V 
                                         ^ 
                                       
                                       max 
                                       2 
                                     
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             + 
                             
                               2 
                               · 
                               K 
                             
                             + 
                             1 
                           
                         
                       
                       
                         4 
                         · 
                         K 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     When motor inductance ratio K is equal to zero, meaning that the permanent magnets  14  of motor  10  are located on the surface of the rotor  22 , the following equations may be obtained: 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       ^ 
                     
                     = 
                     
                       
                         
                           
                             V 
                             ^ 
                           
                           max 
                           2 
                         
                         - 
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   I 
                                   ^ 
                                 
                                 d 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ⅆ 
                       
                         T 
                         ^ 
                       
                     
                     
                       ⅆ 
                       
                         
                           I 
                           ^ 
                         
                         
                           d 
                           ⁡ 
                           
                             ( 
                             
                               K 
                               = 
                               0 
                             
                             ) 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           ( 
                           
                             I 
                             + 
                             
                               
                                 I 
                                 ^ 
                               
                               d 
                             
                           
                           ) 
                         
                         
                           
                             
                               
                                 V 
                                 ^ 
                               
                               max 
                               2 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       I 
                                       ^ 
                                     
                                     d 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                     = 
                     0. 
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     As a result, when motor inductance ratio K is equal to zero, the normalized maximum direct-axis current Î d     —     max(K=0)  is equal to negative 1, as illustrated in the following equation:
 
 Î   d     —     max(K=0) =−1   (40).
 
     A normalized maximum torque may be found by substituting the maximum direct access current Î dmax  into equation (34). As a result, the normalized maximum torque {circumflex over (T)} max(K&gt;0)  when motor inductance ratio K is greater than zero will be equal to the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       ^ 
                     
                     
                       max 
                       ⁡ 
                       
                         ( 
                         
                           K 
                           &gt; 
                           0 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             K 
                             · 
                             
                               
                                 I 
                                 ^ 
                               
                               d_max 
                             
                           
                         
                         ) 
                       
                       
                         K 
                         + 
                         1 
                       
                     
                     · 
                     
                       
                         
                           
                             
                               V 
                               ^ 
                             
                             max 
                             2 
                           
                           - 
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     I 
                                     ^ 
                                   
                                   d_max 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     Accordingly, the normalized maximum torque {circumflex over (T)} max(K=0)  when motor inductance ratio K is equal to zero may be described according to the equation (40) and (41) below:
 
 {circumflex over (T)}   max(K=0)   ={circumflex over (V)}   max    (42).
 
     Turning to  FIG. 4 , a torque limit control block diagram  166  for above base speed operation of motor  10  illustrates a manner of appropriately limiting a reference torque command T ref    168  to produce a normalized torque command {circumflex over (T)} command     —     pu    170 . The normalized torque command {circumflex over (T)} command     —     pu    170  is limited to prevent motor control circuitry and driver circuitry from attempting to exact a torque from motor  10  that would be impossible or that could result in a loss of control of motor  10 . 
     To obtain the normalized torque command {circumflex over (T)} command     —     pu    170 , the reference torque command T ref    168  may first be normalized by multiplying the reference torque command T ref    168  by the contents of block  172 , producing a normalized reference torque {circumflex over (T)} ref     —     pu    174 . The normalized reference torque {circumflex over (T)} ref     —     pu    174  subsequently enters a torque limiter  176 . If the normalized referenced torque {circumflex over (T)} ref     —     pu    174  exceeds a normalized torque limit {circumflex over (T)} Lim    178 , to be described in greater detail below, the normalized torque command {circumflex over (T)} command     —     pu    170  is limited to, or made to equate, the normalized torque limit {circumflex over (T)} Lim    178 . 
     The normalized torque limit {circumflex over (T)} Lim    178  represents a choice of the smallest  180  of either a normalized general torque limit {circumflex over (T)} Lim     —     general     —     pu    182  or a normalized theoretical torque limit {circumflex over (T)} Lim     —     theoretical     —     pu    184 . To obtain a normalized general torque limit {circumflex over (T)} Lim     —     general     —     pu    182 , a general torque limit T Lim     —     general    186  is normalized through multiplication by the contents of block  188 . 
     The torque limit control block diagram  166  may be broken down into sub-diagrams  190  and  192 . Sub-diagram  190  represents a portion of the torque limit control block diagram  166  that outputs the general torque limit T Lim     —     general    186 , which represents a physical torque limit above which the motor  10  may not physically produce additional torque. 
     To obtain the general torque limit T Lim     —     general    186 , a rated stator frequency ω e.rtd    194  is first divided in division block  196  over a value representing an amount of stator frequency ω e    198  which, in absolute value  200  terms, exceeds rated stator frequency ω e.rtd  by way of processing in block  202 . A frequency ratio ω ratio    204  results, which may be understood to represent a ratio of the rated stator frequency ω e.rtd    194  to the amount of stator frequency ω e    198  above base speed. The general torque limit T Lim     —     general    186  is produced by multiplying a torque overload ratio T ovrload     —     ratio    206 , a rated torque T rated    208 , and the frequency ratio ω ratio    204  in multiplier  210 . 
     Continuing to view  FIG. 4 , sub-diagram  192  represents the portion of torque limit control block diagram  166  that outputs the normalized theoretical torque limit {circumflex over (T)} Lim     —     theoretical     —     pu    184 , represents a theoretical torque limit above which control circuitry may lose some control of the motor  10 . To obtain the normalized theoretical torque limit {circumflex over (T)} Lim     —     theoretical     —     pu    184 , permanent magnet flux Ψ m0    212  first enters a multiplier  214  to be multiplied against a value representing the amount of stator frequency ω e    198  which, in absolute value  200  terms, exceeds the rated stator frequency ω e.rtd  by way of processing in block  202 . A voltage Ψ m0 ·ω e    216  results, which may also be represented as voltage E 0 . 
     Dividing a voltage V dc    218  in division block  220  over √{square root over (3)}  222  produces a numerator N  224  with a value equal to a rated voltage V rtd . Numerator N  224  is subsequently divided over voltage Ψ m0 ·ω e    216  in division block  226 , which in turn produces a normalized maximum available voltage {circumflex over (V)} max    228 . 
     If the value of block  230 , which represents motor inductance ratio K  232 , is approximately greater than zero, as illustrated in block  234 , then a normalized maximum direct-axis current Î d     —     max    236  should be calculated via equation block  238 . Regardless as to the value of motor inductance ratio K  232 , normalized theoretical torque limit {circumflex over (T)} Lim     —     theoretical     —     pu    184  is obtained by way of equation block  240 , which accepts as inputs the normalized maximum direct-axis current Î d     —     max    236 , motor inductance ratio K  232 , and normalized maximum available voltage {circumflex over (V)} max    228 . 
     As discussed above, the smallest  180  value of either the normalized general torque limit {circumflex over (T)} Lim     —     general     —     pu    182  or the normalized theoretical torque limit {circumflex over (T)} Lim     —     theoretical     —     pu    184  subsequently represents the normalized torque limit {circumflex over (T)} Lim    178 . After the reference torque command T ref    168  is normalized in block  172 , resulting in the normalized reference torque {circumflex over (T)} ref     —     pu    174 , the normalized reference torque {circumflex over (T)} ref     —     pu    174  is limited to a maximum of the normalized torque limit {circumflex over (T)} Lim    178  through torque limiter  176 . The limited torque is ultimately output as the normalized torque command {circumflex over (T)} command     —     pu    170 . 
     A universal adaptive torque control algorithm which may provide maximum torque per amperes control to a permanent magnet motor with any value of inductance ratio K may also be obtained. To obtain a universal torque control algorithm, normalized direct-axis current Î d  may be obtained as a function of normalized torque {circumflex over (T)}, normalized maximum available voltage {circumflex over (V)} max , and motor inductance ratio K. From equations (30) and (31), the following fourth order equation may be determined:
 
 {circumflex over (T)}   2 ·( K +1) 2   ={circumflex over (V)}   max   2 ·(1 −K·Î   d ) 2 −(1 −K·Î   d ) 2 ·(1 +Î   d ) 2    (43).
 
     Equation (43) may not be easily solved analytically. Thus, a practical implementation may involve solving the above equations numerically or with a closed loop solver.  FIGS. 5-7  illustrate algorithms which may provide optimum torque control according to equation (43). 
       FIG. 5  illustrates an optimum torque control block diagram  242  employing a three-dimensional table to obtain a numerical solution of equation (43). In the three-dimensional table, values for normalized torque {circumflex over (T)}, normalized maximum available voltage {circumflex over (V)} max , and motor inductance ratio K serve as inputs, and normalized direct-axis current Î d  is output. Since the three-dimensional table employs calculations based on per-unit, or normalized, variables, the table may be said to be universal. Being universal, once the three-dimensional table is derived, the table may apply to any permanent magnet motor, as the table accounts for characteristic current I df  and motor inductance ratio K. 
     The optimum torque control block diagram  242  begins when a reference torque command T ref    244  enters a torque limiter  246 , which limits the reference torque command T ref    244  to a torque limit T Lim    248 . It should be noted, however, the torque limiter  246  may also limit the reference torque command T ref    244  using the method illustrated by the torque limit control block diagram  166  of  FIG. 4 . 
     To obtain torque limit T Lim    248 , a rated stator frequency ω e.rtd    250  is divided in division block  252  over a value representing an amount of stator frequency ω e    254  which, in terms of absolute value  256 , exceeds rated stator frequency ω e.rtd  by way of processing in block  258 . A frequency ratio ω ratio    260  results, which may be understood to represent a ratio of the rated stator frequency ω e.rtd    250  to the amount of stator frequency ω e    254  above base speed. The torque limit T Lim    248  is obtained by multiplying a maximum torque limit T Lim.Max    262  in multiplier  264  by frequency ratio ω ratio    260 . 
     By multiplying the output of torque limiter  246  with the contents of block  266 , a normalized reference torque {circumflex over (T)} ref    268  is obtained. In another location on the optimum adaptive torque control block diagram  242 , permanent magnet flux Ψ m0    270  is multiplied in multiplier  272  with a value representing an amount of stator frequency ω e    254  which, in terms of absolute value  256 , exceeds rated stator frequency ω e.rtd  by way of processing in block  258 . As a result, multiplier  272  outputs a voltage Ψ m0 ·ω e    274 . Rated voltage V rtd    276  may be divided by voltage Ψ m0 ·ω e    274  in division block  278  to produce the normalized maximum available voltage {circumflex over (V)} max    280 . In another location on the optimum torque control block diagram  242 , block  282  represents an equation that outputs motor inductance ratio K  284 . 
     Normalized reference torque {circumflex over (T)} ref    268 , normalized maximum available voltage {circumflex over (V)} max    280 , and motor inductance ratio K  284  enter a three-dimensional table  286 , which represents a universal numerical solution to equation (43). For the given normalized reference torque {circumflex over (T)} ref    268 , normalized maximum available voltage {circumflex over (V)} max    280 , and motor inductance ratio K  284 , the three-dimensional table  286  may provide an optimum normalized direct-axis current Î d.table    288 . 
     The optimum normalized direct-axis current Î d.table    288  may be subsequently limited by a current limiter  290 , which may limit the optimum normalized direct-axis current Î d.table    288  to a maximum normalized stator current Î st.Max . The resulting current is represented by normalized direct-axis command current Î d.com    292 . Multiplying motor inductance ratio K  284  with the normalized command current Î d.com    292  in multiplier  294  produces an interim current value K·Î d    296 . Interim current value K·Î d    296  and normalized reference torque {circumflex over (T)} ref    268  are employed by equation block  298  to determine a normalized quadrature-axis current Î q . 
     The normalized quadrature-axis current Î q  output by equation block  298  and the normalized direct-axis command current Î d.com    292  may enter a current limiter  300 , which subsequently may limit the normalized quadrature-axis current Î q  to a value √{square root over (Î st.Max   2 −Î d.com   2 )}, which produces a normalized quadrature-axis command current Î q.com    302 . By multiplying the normalized direct-axis command current Î d.com    292  and normalized quadrature-axis command current  302  by the contents of block 
             304   ,       Ψ     m   ⁢           ⁢   0         L   d       ,         
otherwise known as the characteristic current I df , a direct-axis command current in amperes I d.com    306  and a quadrature-axis command current in amperes I q.com    308  may be obtained. The direct-axis command current I d.com    306  and the quadrature-axis command current I q.com    308  may subsequently be input in blocks  310  and  312 , respectively, which represent the direct-axis and quadrature-axis current loops, to obtain direct-axis command voltage V d.com    314  and quadrature-axis command voltage V d.com    316 .
 
       FIG. 6  depicts an exemplary two-dimensional component  318  of the three-dimensional table  286 . The two-dimensional component  318  represents a table of numerical solutions for flux current  320 , otherwise known as the normalized direct-axis current Î d  or the flux-producing current, at a single value D  322 . Value D  322  is equal to the normalized maximum available voltage {circumflex over (V)} max . 
     For varying values of torque per-unit  324 , otherwise known as the normalized reference torque {circumflex over (T)} ref , and values of motor inductance ratio K  326  ranging from zero to any appropriate value at any appropriate intervals, various numerical solutions to equation (43) for flux current  320  are displayed. It should be appreciated that any appropriate level of detail may be calculated and that the exemplary two-dimensional component of the three-dimensional table does not display all the values that may be desired for the three-dimensional table  286 . 
     Rather than implement a numerical solution, as described in  FIGS. 5 and 6 , a closed loop solver solution may be derived based on equation (43). The closed-loop solver equation may be described as follows: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       ^ 
                     
                     max 
                     2 
                   
                   = 
                   
                     
                       
                         
                           
                             T 
                             ^ 
                           
                           2 
                         
                         · 
                         
                           
                             ( 
                             
                               K 
                               + 
                               1 
                             
                             ) 
                           
                           2 
                         
                       
                       
                         
                           ( 
                           
                             1 
                             - 
                             
                               K 
                               · 
                               
                                 
                                   I 
                                   ^ 
                                 
                                 d 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 I 
                                 ^ 
                               
                               d 
                             
                           
                           ) 
                         
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
       FIG. 7  depicts an optimum torque control block diagram  328  based on the closed loop solver equation (44) above. In a manner similar to that of the optimum torque control block diagram  242  of  FIG. 5 , the optimum torque control block diagram  328  begins as a reference torque command T ref    330  enters a torque limiter  332 , which limits the reference torque command T ref    330  to a torque limit T Lim    334 . It should be noted, however, the torque limiter  332  may also limit the reference torque command T ref    330  using the method illustrated by the torque limit control block diagram  166  of  FIG. 4 . 
     To obtain torque limit T Lim    334 , a rated stator frequency ω e.rtd    336  is divided in division block  338  over a value representing an amount of stator frequency ω e    340  which, in terms of absolute value  342 , exceeds rated stator frequency ω e.rtd  by way of processing in block  344 . A frequency ratio ω ratio    346  results, which may be understood to represent a ratio of the rated stator frequency ω e.rtd    336  to the amount of stator frequency ω e    340  above base speed. The torque limit T Lim    334  is obtained by multiplying a maximum torque limit T Lim.Max    348  in multiplier  350  by frequency ratio ω ratio    346 . 
     By multiplying the output of torque limiter  332  with the contents of block  352 , a normalized reference torque {circumflex over (T)} ref    354  may be obtained. The normalized reference torque {circumflex over (T)} ref    354  will be employed elsewhere in the optimum torque control block diagram  328 . 
     In another location on the optimum torque control block diagram  328 , permanent magnet flux Ψ m0    356  is multiplied in multiplier  358  with a value representing stator frequency ω e    360  which, in terms of absolute value  342 , exceeds rated stator frequency ω e.rtd  by way of processing in block  344 . As a result, multiplier  358  outputs a voltage Ψ m0 ·ω e    362 . Rated voltage V rtd    364  may be divided by voltage Ψ m0 ·ω e    362  in division block  366 , producing normalized maximum available voltage {circumflex over (V)} max . 
     When the output of division block  366 , equal to normalized maximum available voltage {circumflex over (V)} max , is multiplied in multiplier  368  against itself, the output is {circumflex over (V)} max.ref   2    370 . The value of {circumflex over (V)} max.ref   2    370  enters a summer  372 , from which feedback {circumflex over (V)} max.fbk   2    374  is subtracted. The result enters a current controller  376 , which outputs an optimum normalized direct-axis reference current Î d.ref    378 . The optimum normalized direct-axis reference current Î d.ref    378  enters a summer  380  to which the contents of block  382 , a value of 1, are added. The output of summer  380  is multiplied against itself in multiplier  384 , producing (1+Î d ) 2    386 . 
     At another location on the optimum torque control block diagram  328 , motor inductance ratio K  388  enters a summer  390  with the contents of block  382 , a value of 1. The output of summer  390  is multiplied against itself in multiplier  392 , the result of which is subsequently multiplied in multiplier  394  with the a square of the normalized reference torque {circumflex over (T)} ref    354 , which results when the normalized reference torque {circumflex over (T)} ref    354  is multiplied against itself in multiplier  396 . The output of the multiplier  394  is {circumflex over (T)} 2 ·(k+1) 2    398 . 
     Motor inductance ratio K  388  also enters multiplier  399  with the optimum normalized direct-axis reference current Î d.ref    378 , the output of which is subtracted from the contents of block  400 , a value of 1, in summer  402 . The output of summer  402  is subsequently multiplied against itself in multiplier  404  to produce (1−K·Î d ) 2    406 . The value {circumflex over (T)} 2 ·(k+1) 2    398  is divided by (1−K·Î d ) 2    406  in division block  408 , the result of which subsequently enters summer  410  with (1+Î d ) 2    386 . Summer  410  ultimately outputs feedback {circumflex over (V)} max.fbk   2    374 . 
     The optimum normalized direct-axis reference current Î d.ref    378  also enters a current limiter  412 , which may limit the optimum normalized direct-axis reference current Î d.ref    378  to a maximum normalized stator current Î st.Max . The resulting current is represented by a normalized direct-axis command current Î d.com    414 . Multiplying motor inductance ratio K  388  with the normalized command current Î d.com    414  in multiplier  416  produces an interim current value K·Î d    418 . Interim current value K·Î d    418  and normalized reference torque {circumflex over (T)} ref    354  are subsequently employed by equation block  420  to determine a normalized quadrature-axis current Î q.ref . 
     The normalized quadrature-axis current Î q.ref  output by equation block  420  and the normalized direct-axis command current Î d.com    414  may enter a current limiter  422 , which subsequently may limit the normalized quadrature-axis current Î q.ref  to a value √{square root over (Î st.Max   2 −Î d.com   2 )}, which produces a normalized quadrature-axis command current Î q.com    424 . 
     By multiplying the normalized direct-axis command current Î d.com    414  by the contents of block  426 , 
                 Ψ     m   ⁢           ⁢   0         L   d       ,         
otherwise known as the characteristic current I df  a direct-axis command current in amperes I d.com  may be obtained. The direct-axis command current in amperes I d.com    428  may subsequently be input in block  430 , which represents the direct-axis current loop, to obtain an optimum direct-axis command voltage V d.com    432 . Similarly, by multiplying the normalized quadrature-axis command current Î q.com    424  by the contents of block  426 ,
 
                 Ψ     m   ⁢           ⁢   0         L   d       ,         
otherwise known as the characteristic current I df , a quadrature-axis command current in amperes I q.com    434  may be obtained. The quadrature-axis command current I q.com    434  may subsequently be input in block  436 , which represents the quadrature-axis current loop, to obtain an optimum quadrature-axis command voltage V q.com    438 .
 
     It should be noted that the torque limit algorithm described in  FIG. 4  may be employed by the universal adaptive torque control approaches of  FIGS. 5-7 . Additionally or alternatively, the torque limit algorithm may be employed with any appropriate algorithm for controlling a motor for either above base speed or below base speed. Such algorithms for controlling a motor may include, for example, a voltage loop algorithm, a minimum flux algorithm, or an algorithm for maintaining control of a motor operating below base speed after a decrease in line voltage. 
     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.