Abstract:
A method and system for estimating an angular position and an angular velocity of a rotor in a dynamoelectric machine measures an AC current and a potential for each of a plurality of windings coupled to a stator of the dynamoelectric machine, transforms the measured currents and potentials to a stationary frame to produce transformed currents and transformed potentials, and processes the transformed currents and transformed potentials to produce a first intermediate signal and a second intermediate signal. The first intermediate signal and the second intermediate signal are cross-coupled by being processed to obtain a first extended rotor flux value and a second extended rotor flux value that are each functions of the first intermediate signal and the second intermediate signal. The first extended rotor flux value and the second extended rotor flux value are applied to a phase lock loop to derive an estimated rotor angular position and an estimated rotor angular velocity for the dynamoelectric machine.

Description:
BACKGROUND OF THE INVENTION 
     This invention relates to rotor angular position and velocity sensing systems for mechanical shaft sensorless control of dynamoelectric machines, and more particularly to an improved system for resolving the position and velocity of a rotor for a dynamoelectric machine using an estimate of extended rotor flux. 
     In some vehicles, including some aircraft, a motor may be utilized both as a motor and as a generator. Because of this dual function, the motor may be called a dynamoelectric machine. A typical motor comprises a stationary stator, and a rotating rotor. In some motors, it is necessary to detect a position of a rotor in order to sustain operation of the motor. Determining a rotor position typically requires a shaft position sensor. It is desirable to eliminate a mechanical shaft sensor to reduce cost and improve reliability. 
     Some methods of sensorless rotor position detection include the back EMF method, which determines rotor position based on voltage, the signal injection method, which injects high frequencies into a system, and the method discussed in U.S. Pat. No. 7,072,790 which uses flux to determine rotor position. It is desirable to improve the method U.S. Pat. No. 7,072,790 for applications operating at low speeds or at a standstill. 
     SUMMARY OF THE INVENTION 
     A method and system for estimating an angular position and an angular velocity of a rotor in a dynamoelectric machine measures an AC current and a potential for each of a plurality of windings coupled to a stator of the dynamoelectric machine, transforms the measured currents and potentials to a stationary frame to produce transformed currents and transformed potentials, and processes the transformed currents and transformed potentials to produce a first intermediate signal and a second intermediate signal. The first intermediate signal and the second intermediate signal are cross-coupled by being processed to obtain a first extended rotor flux value and a second extended rotor flux value that are each functions of the first intermediate signal and the second intermediate signal. The first extended rotor flux value and the second extended rotor flux value are applied to a phase lock loop to derive an estimated rotor angular position and an estimated rotor angular velocity for the dynamoelectric machine. 
     These and other features of the present invention can be best understood from the following specification and drawings, the following of which is a brief description. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates a block diagram of a mechanical sensorless rotor angular position and velocity sensing system. 
         FIG. 2  illustrates a phasor diagram of electrical parameters related to extended rotor flux. 
         FIG. 3  illustrates a block diagram of operations performed within the system of  FIG. 1  to calculate a first extended rotor flux value and a second extended rotor flux value. 
         FIG. 4  illustrates a block diagram of how a microprocessor of  FIG. 1  uses a phase lock loop (PLL) to obtain an estimated rotor angular position and an estimated rotor angular velocity. 
         FIG. 5  illustrates how a stationary frame of the system of  FIG. 1  aligns with multiple phases of AC. 
         FIG. 6  illustrates an initial stator voltage and an initial rotor position as a function of time. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     As shown in  FIG. 1 , the rotor angular position and velocity sensing system  10  comprises a motor  12  that is able to operate as a starter to start an engine  14 , or as a generator to power a load (not shown). Because of this dual function, the motor  12  may be called a dynamoelectric machine. In one example, the motor  12  is a brushless motor that requires a controller to know a position of its rotor to operate. 
     To start the motor  12 , an AC power supply  16  provides an AC voltage along supply lines  18  to a rotating exciter  19 . In the example of  FIG. 1 , the AC power supply  16  provides three phases of AC power, however it is understood that other quantities of phases of AC power could be provided. The rotating exciter is connected to a shaft  21  that is also connected to the motor  12  and the engine  14 . 
     The AC voltage from the supply lines  18  induces an AC voltage along motor terminals  20   a ,  20   b , and  20   c . The induced voltage causes a current to flow through an output filter  22 . A microprocessor  24  measures a voltage  26  and a current  28  from each of the terminals  20   a ,  20   b , and  20   c . A position and speed estimator  30  uses the voltage and current measurements to estimate a flux of the motor  12  and to estimate a rotor position  32  and a rotor angular velocity  34 . 
     Once the estimated rotor position  32  and estimated rotor angular velocity  34  have been calculated, an inverter  38  is turned ON. The microprocessor  24  processes the estimated rotor position  32  and estimated rotor angular velocity  34  to control a pulse width modulated (PWM) generator  36 . An inverter  38  is coupled to the PWM generator  36  and converts a DC voltage from DC voltage supply lines  40  to AC. This voltage enables AC to flow through the output filter  22 , which improves power quality by filtering out harmonics and reducing electromagnetic interference (EMI). The AC from the terminals  20   a ,  20   b , and  20   c  then flows to a stator of the motor  12  to sustain operation of the motor  12 . 
       FIG. 1  illustrates how the microprocessor  24  processes the estimated rotor position  32  and estimated rotor angular velocity  34  to control the inverter  38 . An abc to d-q frame transformer  42  uses the estimated rotor position  32  to transform the current measurements  28  to a rotating d-q frame to obtain current values I d  and I q . A torque current profile generator  44  uses the estimated rotor angular velocity  34  to lookup reference current values I d * and I q *. Comparators  45  and  46  compare the transformed I d  and I q  values to reference current values I d * and I q * to determine differences ΔI d  and ΔI q  between the transformed values and the reference values. 
     Proportional and integral (PI) regulators  47  and  48  process the differences ΔI d  and ΔI q  using proportional and integral gains, and transmit an output signal to d-q to alpha-beta frame transformer  50 , which converts the output into a stationary α-β frame to produce V alpha * and V beta * signals which are transmitted to the PWM generator  36 . The PWM generator then controls the inverter  38  accordingly to produce a desired AC voltage. 
     The output filter  22  comprises an inductor and a capacitor (not shown) in each phase. An input current I invt  flows from the inverter  38  along the windings  23   a ,  23   b , and  23   c  to the output filter  22 , and an output current I s  flows from the output filter  22  along the terminals  20   a ,  20   b , and  20   c  to the motor  12 . The current flowing through the capacitor can be calculated by the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       ⋒ 
                     
                     c 
                   
                   = 
                   
                     C 
                     ⁢ 
                     
                       
                         ⅆ 
                         
                           V 
                           s 
                         
                       
                       
                         ⅆ 
                         t 
                       
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   #1 
                 
               
             
           
         
       
     
     where Î c  is an estimated capacitor current; and
         V s  is one of the voltage measurements  26 .       

     A motor current can then be calculated using the following equation:
 
 I   s   =I   invt   −Î   c   equation #2
 
     where I s  is the calculated motor current; and
         I invt  is the inverter output current.       

     Equations 1 and 2 apply to all three phases A, B, and C corresponding to the three windings  20   a ,  20   b , and  20   c.    
     The voltage measurements  26  and current measurements  28  are measured from each of the three terminals ( 20   a ,  20   b ,  20   c ) and each of the three windings ( 23   a ,  23   b ,  23   c ) in an a-b-c frame. The current measurement  28  is a measurement of the inverter output current I invt . A flux estimation is implemented in an alpha-beta (α-β) stationary frame. The relationship between the α-β frame and the a-b-c frame is described in the following equation: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             f 
                             α 
                           
                         
                       
                       
                         
                           
                             f 
                             β 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         2 
                         3 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 
                                   3 
                                 
                                 2 
                               
                             
                             
                               
                                 - 
                                 
                                   
                                     3 
                                   
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               f 
                               a 
                             
                           
                         
                         
                           
                             
                               f 
                               b 
                             
                           
                         
                         
                           
                             
                               f 
                               c 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   #3 
                 
               
             
           
         
       
     
     where f can be replaced with voltage, current, or flux;
         a, b, and c represent the phases of current on the terminals  20   a ,  20   b , and  20   c  in the a-b-c frame; and   α and β represent axes of the α-β frame.       

     The stationary α-β frame is a two phase frame and is a necessary step in calculating flux. Equation #3 is used to determine an α-axis voltage V α , a β-axis voltage V β , an α-axis current I α , and a β-axis current I β . 
     The following equation can then be used to determine an extended rotor flux in the α-β stationary frame: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             λ 
                             ext_α 
                           
                         
                       
                       
                         
                           
                             λ 
                             ext_β 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         1 
                         s 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     V 
                                     α 
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     β 
                                   
                                 
                               
                             
                             ] 
                           
                           - 
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       R 
                                       s 
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                       R 
                                       s 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       I 
                                       α 
                                     
                                   
                                 
                                 
                                   
                                     
                                       I 
                                       β 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         [ 
                         
                           
                             
                               
                                 L 
                                 q 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 L 
                                 q 
                               
                             
                           
                         
                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 I 
                                 α 
                               
                             
                           
                           
                             
                               
                                 I 
                                 β 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   #4 
                 
               
             
           
         
       
     
     where λ ext     —     α  is an alpha extended rotor flux;
         λ ext     —     β  is a beta extended rotor flux;   R s  is a stator resistance;   Lq is a q-axis inductance; and   1/s is an integrator.       

     Equation #4 can be used to determine flux in both salience and non-salience motors. As shown in equation #4, an integrator 1/s is required to calculate extended rotor flux. The integrator 1/s is an operator, not a variable. 
     One problem that may arise when using a pure integrator, such as “1/s”, is a DC drift problem, in which a small DC component in an AC signal can cause a substantial error in a flux determination. To avoid the DC drift problem associated with a pure integrator, lag functions, such as 
                 1     s   +     ω   i         ⁢           ⁢   and   ⁢           ⁢       ω   i       s   +     ω   i           ,         
may be used, as shown in the following equation:
 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             λ 
                             ext_α 
                           
                         
                       
                       
                         
                           
                             λ 
                             ext_β 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               
                                 
                                   ω 
                                   i 
                                 
                                 
                                   s 
                                   + 
                                   
                                     ω 
                                     i 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   
                                     ω 
                                     i 
                                   
                                   
                                     s 
                                     + 
                                     
                                       ω 
                                       i 
                                     
                                   
                                 
                               
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         1 
                         
                           s 
                           + 
                           
                             ω 
                             i 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     V 
                                     α 
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     β 
                                   
                                 
                               
                             
                             ] 
                           
                           - 
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                         
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                         
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       I 
                                       α 
                                     
                                   
                                 
                                 
                                   
                                     
                                       I 
                                       β 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         [ 
                         
                           
                             
                               
                                 L 
                                 q 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 L 
                                 q 
                               
                             
                           
                         
                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 I 
                                 α 
                               
                             
                           
                           
                             
                               
                                 I 
                                 β 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   #5 
                 
               
             
           
         
       
     
     where ω i  is a selected corner frequency. 
       FIG. 2  illustrates a phasor diagram in the rotating d-q frame.  FIG. 2  illustrates the relationship between extended rotor flux and back EMF. 
     A flux λ s  in the stator of the motor  12  is represented by a phasor  60 . A stator current I s  is represented by a phasor  62 . A stator potential V s  is represented by a phasor  64 . A phasor  66  represents I s *L q  where L q  is a q-axis rotor inductance. A vector sum of the phasor  60 , representing λ s , and the phasor  66 , representing I s *L q , is an extended rotor flux λ ext , which aligns with the d-axis of the d-q frame, and is represented by a phasor  67 . 
     A back electromotive force (EMF) E s  is represented by a phasor  68 . As shown in  FIG. 2 , the back EMF E s  is perpendicular to the stator flux λ s . The back EMF E s , represented by the phasor  68 , is a vector sum of the stator potential V s  represented by phasor  64  and stator resistance potential drop I s *R s  represented by a phasor  70 , where R s  is the stator resistance. 
     An extended back electromotive force (EEMF), E ext , in the stator is represented by a phasor  72 , and aligns with the q-axis of the d-q frame. I s *X q , where X q  is a q-axis stator reactance, is represented by a phasor  74 . The extended back EMF represented by phasor  72  is a vector sum of E s  represented by phasor  68  and I s *X q  represented by a phasor  74 . 
       FIG. 3  illustrates a block diagram of the flux estimation algorithm shown in equation #5. As shown in  FIG. 3 , a transformed measured current I α  for the α-axis on a signal path  80  is multiplied by the stator resistance R s    82  to produce I α *R s  on a signal path  84 . A summer  86  subtracts I α *R s  on the signal path  84  from the transformed potential V α  on a signal path  88  to produce V α −(I α *R s ) on a signal path  90 . V α −(I α *R s ) on the signal path  90  is multiplied by a 
             1     s   +     ω   i             
first lag function  92  to produce
 
               1     s   +     ω   i         ⁢     (       V   α     -     (       I   α     *     R   s       )       )           
on a signal path  93 .
 
               1     s   +     ω   i         ⁢     (       V   α     -     (       I   α     *     R   s       )       )           
on the signal path  93  is multiplied by a
 
               ω   i       s   +     ω   i             
second lag function  94  to produce
 
                 ω   i       s   +     ω   i         ⁢     (       V   α     -     (       I   α     *     R   s       )       )           
on a signal path  95 .
 
     Additionally, a transformed measured current I β  for the β-axis on a signal path  96  is multiplied by the stator resistance R s    98  to produce I β *R s  on a signal path  100 . A summer  102  subtracts I β *R s  on the signal path  100  from the transformed potential V β  on a signal path  104  to produce V β −(I β *R s ) on a signal path  106 . V β −(I β *R s ) on the signal path  106  is multiplied by the 
             1     s   +     ω   i             
first lag function  108  to produce
 
               1     s   +     ω   i         ⁢     (       V   β     -     (       I   β     *     R   s       )       )           
on a signal path  110 .
 
               1     s   +     ω   i         ⁢     (       V   β     -     (       I   β     *     R   s       )       )           
on the signal path  110  is multiplied by the
 
               ω   i       s   +     ω   i             
second lag function  112  to produce
 
                 ω   i         (     s   +     ω   i       )     2       ⁢     (       V   β     -     (       I   β     *     R   s       )       )           
on a signal path  114 .
 
     The transformed measured current I α  for the α-axis on the signal path  80  is also multiplied by a q-axis inductance L q    116  to produce I α *L q  on the signal path  118 . A summer  120  subtracts I α *L q  on the signal path  118  from 
                 ω   i         (     s   +     ω   i       )     2       ⁢     (       V   α     -     (       I   α     *     R   s       )       )           
on the signal path  93  and adds
 
                 ω   i         (     s   +     ω   i       )     2       ⁢     (       V   β     -     (       I   β     *     R   s       )       )           
from the signal path  114  to produce
 
     
       
         
           
             
               
                 1 
                 
                   s 
                   + 
                   
                     ω 
                     i 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     V 
                     α 
                   
                   - 
                   
                     ( 
                     
                       
                         I 
                         α 
                       
                       * 
                       
                         R 
                         s 
                       
                     
                     ) 
                   
                 
                 ) 
               
               ⁢ 
               
                 
                   ω 
                   i 
                 
                 
                   
                     ( 
                     
                       s 
                       + 
                       
                         ω 
                         i 
                       
                     
                     ) 
                   
                   2 
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     V 
                     β 
                   
                   - 
                   
                     ( 
                     
                       
                         I 
                         β 
                       
                       * 
                       
                         R 
                         s 
                       
                     
                     ) 
                   
                 
                 ) 
               
             
             - 
             
               
                 I 
                 α 
               
               * 
               
                 L 
                 q 
               
             
           
         
       
     
     which corresponds to the extended rotor flux on the α-axis {circumflex over (λ)} ext     —     α  on the signal path  122 . The “^”notation indicates that the extended rotor flux is an estimate based on measured values. 
     Additionally, the transformed measured current I β  for the β-axis on the signal path  96  is also multiplied by a q-axis inductance L q    124  to produce I β *L q  on the signal path  126 . A summer  128  subtracts I β *L q  on the signal path  126  from 
               1     s   +     ω   i         ⁢     (       V   β     -     (       I   β     *     R   s       )       )           
on the signal path  110  and subtracts
 
                 ω   i         (     s   +     ω   i       )     2       ⁢     (       V   α     -     (       I   α     *     R   s       )       )           
on the signal path  95  from
 
               1     s   +     ω   i         ⁢     (       V   β     -     (       I   β     *     R   s       )       )           
on the signal path  110  to produce
 
     
       
         
           
             
               
                 1 
                 
                   s 
                   + 
                   
                     ω 
                     i 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     V 
                     β 
                   
                   - 
                   
                     ( 
                     
                       
                         I 
                         β 
                       
                       * 
                       
                         R 
                         s 
                       
                     
                     ) 
                   
                 
                 ) 
               
             
             - 
             
               
                 
                   ω 
                   i 
                 
                 
                   
                     ( 
                     
                       s 
                       + 
                       
                         ω 
                         i 
                       
                     
                     ) 
                   
                   2 
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     V 
                     α 
                   
                   - 
                   
                     ( 
                     
                       
                         I 
                         α 
                       
                       * 
                       
                         R 
                         s 
                       
                     
                     ) 
                   
                 
                 ) 
               
             
             - 
             
               
                 I 
                 β 
               
               * 
               
                 L 
                 q 
               
             
           
         
       
     
     which corresponds to the extended rotor flux on the β-axis {circumflex over (λ)} ext     —     β  on the signal path  130 . Once again, the “^” notation indicates that the extended rotor flux is an estimate based on measured values. 
     As shown in  FIG. 3 , the signal paths  95  and  114  cross-couple the signal paths  93  and  110 . 
     The following equation can be used to describe the relationship between the extended rotor flux and the rotor position: 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               λ 
                               ext_α 
                             
                           
                         
                         
                           
                             
                               λ 
                               ext_β 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       | 
                       λ 
                       | 
                       
                         [ 
                         
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   ; 
                 
               
               
                 
                   equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   #6 
                 
               
             
           
         
       
     
     where θ is the rotor position; and
         λ is a flux amplitude.       

     Using equation #6, it would be possible to use an arctangent function to calculate a rotor position. Another option is to used a phase-locked loop (PLL) to derive position and angular velocity information. 
       FIG. 4  is a block diagram illustrating how the microprocessor  24  uses a phase lock loop (PLL) to improve an estimate of rotor angular position and rotor angular velocity. The estimated α-axis extended rotor flux {circumflex over (λ)} ext     —     α  and the estimated β-axis extended rotor flux {circumflex over (λ)} ext     —     β  are applied to the signal paths  122  and  130 . A multiplier  132  multiplies the estimated α-axis extended rotor flux {circumflex over (λ)} ext     —     α  with a feedback signal on a signal path  134  from a sine function  136  to produce an α-axis multiplier output signal on a signal path  138 . Likewise, a multiplier  140  multiplies the estimated β-axis extended rotor flux {circumflex over (λ)} ext     —     β  with a feedback signal on a signal path  142  from a cosine function  144  to produce a β-axis multiplier output signal on a signal path  146 . 
     A summer  148  subtracts the α-axis multiplier output signal on the signal path  138  from the β-axis multiplier output signal on the signal path  146  to produce a difference signal on a signal path  150 . A proportional and integral (PI) regulator function  152  multiplies the difference signal on the signal path  150  by the function 
               K   p     +       K   i     s           
to produce a PI output signal on a signal path  154 . K i  is an integral gain of the PI function  152 , and K p  is a proportional gain of the PI function  152 . Both K i  and K p  are constants based on a design of the system  10  as shown in  FIG. 1 .
 
     An integral function  156  multiplies the PI output signal on the signal path  154  by the function 1/s to produce an integration output signal on a signal path  158 . The integration output signal on the signal path  158  is also fed into the inputs of the sine function  136  and the cosine function  144  to provide the PLL. 
     A low pass filter (LPF) function  160  multiplies the PI output signal on the signal path  154  by a third lag function 
               ω   c       s   +     ω   c             
to produce an estimated rotor angular velocity {circumflex over (ω)} on a signal line  162 , where ω c  is a corner or cutoff frequency of the LPF function  160 . A low pass filter associated with the LPF function  160  is used to smooth out the signal on the signal line  154 .
 
     The integration output signal on the signal path  158  is compensated by an offset Δθ to obtain a final estimated rotor angular position {circumflex over (θ)}. The offset Δθ can be a lump-sum error of miscellaneous delays, including delays introduced by the lag functions  92 ,  108  of the  FIG. 3 , digital sampling delays introduced in measured voltage and current signals, and computation delays in the microprocessor  24  as shown in  FIG. 1 . A lookup table  164  may be used to compensate for this phase delay Δθ. The lookup table  164  generates a suitable phase delay Δθ on a signal path  166 , and a summer  168  subtracts the phase delay Δθ from the integration output signal on the signal path  158  to produce the estimated rotor angular position {circumflex over (θ)} on a signal path  170 . 
       FIG. 5  illustrates how the α-β frame  173  comprises an α-axis  174  and a β-axis  175  that are perpendicular to each other. The α-β frame  173  aligns with a first phase  178 , a second phase  180  and a third phase  182  of the system  10 . A rotor  172  rotates, and its displacement from the α-axis is shown by the angle θ  184 , which is the rotor angular position to be estimated. 
     When the motor  12  is at a standstill, as magnetic flux in the motor  12  changes in magnitude, a voltage is induced on the motor terminals  20   a ,  20   b , and  20   c , which can be sensed by the microprocessor  24 . The induced stator voltages in the α-β frame can be described by the following equation: 
     
       
         
           
             
               V 
               α 
             
             = 
             
               
                 
                   ⅆ 
                   
                     λ 
                     s 
                   
                 
                 
                   ⅆ 
                   t 
                 
               
               ⁢ 
               
                 cos 
                 ⁡ 
                 
                   ( 
                   
                     θ 
                     0 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               V 
               β 
             
             = 
             
               
                 
                   ⅆ 
                   
                     λ 
                     s 
                   
                 
                 
                   ⅆ 
                   t 
                 
               
               ⁢ 
               
                 sin 
                 ⁡ 
                 
                   ( 
                   
                     θ 
                     0 
                   
                   ) 
                 
               
             
           
         
       
     
     where λ s  is a magnitude of stator flux; and
         θ 0  is an initial rotor position angle at standstill.       

     The measured voltage  26  can be transformed to an alpha-beta frame. The transformed measured voltages V α  and V β  may be fed into the PLL as shown in  FIG. 4  to obtain the initial position angle θ 0 . In that case, the voltage V α  replaces the flux {circumflex over (λ)} ext     —     α  on the signal path  122 , and the voltage V β  replaces the flux {circumflex over (λ)} ext     —     β  on the signal path  130  in  FIG. 4 . 
       FIG. 6  illustrates an initial stator voltage and an initial rotor position as a function of time. During startup the rotating exciter  19  is powered on by the ac power supply  16  in  FIG. 1 . Graph  186  illustrates a line-to-line voltage for each phase of the motor  12  as a function of time, and graph  188  illustrates an estimated rotor position as a function of time in the motor  12 . The inverter  38  is OFF in the time periods shown in graphs  186  and  188 . A voltage  190  corresponds to a V BC  line-to-line voltage, a voltage  192  corresponds to a V CA  line-to-line voltage, and a voltage  194  corresponds to a V AB  line-to-line voltage. An estimated rotor position θ  200  corresponds to an angle of the rotor  172 . 
     During an initial time period  196 , the AC power supply  16  is OFF, and the three voltages  190 ,  192 , and  194  voltage close to zero and the estimated rotor position θ 0    200  cannot be used to determine actual rotor position. At time  198 , the AC power supply  16  turns ON and current flows to the rotating exciter  19  through the supply line  18  in the system  10 . During this period, an excitation magnetic field of the motor  12  is arising. The rising magnetic flux induces voltage at the terminals  20   a ,  20   b , and  20   c  of the system  10 . The magnitude of voltages  190 ,  192  and  194  is sufficient for the microprocessor  24  to be able to estimate rotor position θ 0    200 . In graph  188 , from time  198  to approximately time  202  the value of θ 0  remains stable, and after time  202  the value starts to fluctuate due to a decaying voltage signal as shown in graph  186 . This stable period demonstrates that a rotor position can be estimated from the voltages  190 ,  192 , and  194  during the stable time period. 
     Although a preferred embodiment of this invention has been disclosed, a worker of ordinary skill in this art would recognize that certain modifications would come within the scope of this invention. For that reason, the following claims should be studied to determine the true scope and content of this invention.