Patent Publication Number: US-9419555-B2

Title: Synchronous machine control apparatus

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a synchronous machine control apparatus provided with an electric-power conversion unit for rotation-driving a synchronous machine and particularly to a synchronous machine control apparatus that corrects an error between the true rotor position of a synchronous machine and a rotor position detected by a position detection unit. 
     2. Description of the Related Art 
     When in controlling a synchronous machine, there exists an error between the true rotor position of the synchronous machine and a rotor position detected by a position detector (referred to as a rotor position error, hereinafter), the synchronous machine cannot be controlled in such away as to have desired torque. Therefore, this error needs to be detected and corrected. 
     As an example of synchronous machine control apparatus in which such a problem is solved, a method has been proposed in which under the condition that the load torque is constant, a current command that makes the armature current minimum is searched and the rotor position error is calculated from the searched current command (for example, refer to Patent Document 1). 
     In addition, a method has been proposed in which there is provided a comparator that compares the U-phase voltage and the W-phase voltage of respective rotor armature windings and makes an output clock signal fall when detecting a point where the U-phase voltage and the W-phase voltage cross each other (the U-phase voltage rises/the W-phase voltage falls) and in which the average value of a predetermined number of position-detector outputs to be obtained each time the output clock signal falls is calculated so that the rotor position error is calculated (for example, refer to Patent Document 2). 
     [Patent Document 1] Japanese Patent Application Laid-Open No. 2001-8486 
     [Patent Document 2] Japanese Patent Application Laid-Open No. 2008-278606 
     However, the synchronous machine control apparatus disclosed in Patent Document 1 has a problem that it takes a long time to search a current command, for calculating a rotor position error, that makes the armature current minimum under the condition that the load torque is constant. The synchronous machine control apparatus disclosed in Patent Document 2 has a problem that for the purpose of calculating a rotor position error, a dedicated circuit is required. 
     SUMMARY OF THE INVENTION 
     The present invention has been implemented in order to solve the problems in the foregoing conventional technologies; the objective thereof is to provide a synchronous machine control apparatus that can correct a rotor position error without searching a current command or without any dedicated circuit. 
     A synchronous machine control apparatus, according to the present invention, for correcting a rotor position error that is a difference between a rotor position of a synchronous machine and a rotor position detected by a position detection unit for detecting a rotor position of the synchronous machine is provided with a current control device for performing control in such a way that respective current command values and respective current detection values in a generation direction of an armature interlinked magnetic flux and in a direction perpendicular to the generation direction of the armature interlinked magnetic flux coincide with each other, a magnetic flux calculation device for calculating a phase of an estimated armature interlinked magnetic flux, based on an armature current detection value of the synchronous machine and an armature voltage command value therefor, and a rotor position error calculation device for calculating the rotor position error from a phase of an estimated armature interlinked magnetic flux calculated by the magnetic flux calculation device; a rotor position error that is a difference between a rotor position of the synchronous machine and a rotor position detected by the position detection unit is corrected by use of a rotor position error calculated by the rotor position error calculation device. 
     A synchronous machine control apparatus according to the present invention makes it possible to correct a rotor position error without taking a time for searching a current command value and without any dedicated circuit. 
     The foregoing and other objects, features, aspects and advantages of the present invention will become more apparent from the following detailed description of the present invention when taken in conjunction with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a configuration diagram illustrating an example of synchronous machine control system including a synchronous machine control apparatus, a power source, and a synchronous machine, according to Embodiment 1 of the present invention; 
         FIG. 2  is a diagram illustrating an example of configuration of a magnetic flux calculation device according to Embodiment 1 of the present invention; 
         FIG. 3  is a chart representing the relationship among the axes dq, d′q′, γδ, and γ′δ′ according to Embodiment 1 of the present invention; 
         FIG. 4  is a configuration diagram illustrating an example of synchronous machine control system including a synchronous machine control apparatus, a power source, and a synchronous machine, according to Embodiment 2 of the present invention; 
         FIG. 5  is a configuration diagram illustrating an example of synchronous machine control system including a synchronous machine control apparatus, a power source, and a synchronous machine, according to Embodiments 3 and 4 of the present invention; 
         FIG. 6  is a configuration diagram illustrating an example of synchronous machine control system including a synchronous machine control apparatus, a power source, and a synchronous machine, according to Embodiment 5 of the present invention; and 
         FIG. 7  is a diagram illustrating an example of configuration of a magnetic flux calculation device according to Embodiment 5 of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Hereinafter, preferred embodiments of a synchronous machine control apparatus according to the present invention will be explained with reference to the drawings. 
     Embodiment 1 
       FIG. 1  is a configuration diagram illustrating an example of synchronous machine control system including a synchronous machine control apparatus  1 , a power source  2 , and a synchronous machine  3 , according to Embodiment 1 of the present invention. Hereinafter, there will be explained the configuration of a synchronous machine control apparatus  1  according to Embodiment 1 and the functions of the constituent elements thereof. 
     The synchronous machine control apparatus  1  is provided with an electric-power conversion unit  11 , a rotor position error calculation device  12 , a γ′δ′ to uvw coordinate converter  13 , a current detection unit  14 , a position detection device  15  as a position detection unit, a speed calculation device  16 , a magnetic flux calculation device  17 , a uvw to γ′δ′ coordinate converter  18 , and a current control device  19 . 
     At first, the configuration of the synchronous machine control apparatus  1  will be explained in sequence from the output side of the electric-power conversion unit  11 ; then, there will be explained the flow of generation of voltage commands at the input side of the electric-power conversion unit  11 . After that, there will be explained a method in which the rotor position error calculation device  12  is utilized so as to correct a rotor position error. 
     In the configuration of the synchronous machine control apparatus  1 , according to Embodiment 1, for driving the synchronous machine  3 , the armature winding of the synchronous machine  3  is connected with the electric-power conversion unit  11  including devices, for example, an inverter having a function of converting electric power supplied by the power source  2  into multi-phase AC electric power. The electric-power conversion unit  11  applies voltages to the synchronous machine  3 , based on voltage commands vu*, vv* and vw* obtained by the γ′δ′ to uvw coordinate converter  13  having an after-mentioned configuration, and drives the synchronous machine  3 . As a result, an output current is produced in the armature winding of the synchronous machine  3 . The power source  2  is a power supply unit that outputs a DC voltage or is a battery; it may be the one that obtains a DC voltage, through a well-known converter, from a single-phase or three-phase AC power source. 
     The current in the armature winding (referred to as an armature current, hereinafter), which is the output current of the synchronous machine  3 , is detected by the current detection unit  14  such as a current sensor. In the case where the synchronous machine  3  is a three-phase electric rotating machine, the current detection unit  14  may be configured in such a way as to detect all of the armature currents of three phases; alternatively, the current detection unit  14  may be configured in such a way that the output currents of two phases are detected so that the output current iw of one phase (e.g., W phase) is obtained through the relationship [iw=−iu−iv], by use of the detected output currents of two phases iu and iv. Moreover, the current detection unit  14  may be formed by use of a well-known technology in which the output current is detected from a DC link current that flows from the power source  2  to the electric-power conversion unit  11 , instead of being formed through a method in which the current of each phase is directly detected. 
     The position detection device  15  detects the rotor position of the synchronous machine  3  by use of a well-known resolver, an encoder, or the like. Here, θ and θ′ denote the true rotor position of the synchronous machine  3  and the rotor position detected by the position detection device  15 , respectively. The position detection device  15  can reduce the value of θ′ by θ′ offset, by setting an origin correction value θ′ offset. The speed calculation device  16  applies a differential operation to θ′ so as to calculate an electric angular velocity ω of the synchronous machine  3 . 
     In Embodiment 1, in the case where the synchronous machine  3  is a permanent-magnet synchronous machine, the permanent-magnet magnetic flux is defined as magnetic field flux, and in the case where the synchronous machine  3  is a coil-magnetic-field synchronous machine, the magnetic flux generated when a current is applied to the magnetic-field winding is defined as magnetic field flux. The true rotor position θ of the synchronous machine  3  denotes the N-pole-direction angle, of the magnetic field flux, with respect to an axis that is set by regarding the armature winding of u-phase as a reference; the d axis of a rotating biaxial coordinate system (expressed as a d-q axis, hereinafter) that rotates at the rotation speed (electric angular frequency ω) of the synchronous machine  3  is set along the N-pole direction of the magnetic field flux, and the explanation below will follow this definition. The q axis is set in a direction that is 90° advanced from and perpendicular to the d axis. 
     In contrast, the direction of the rotor position detected by the position detection device  15  is defined as a d′ axis; a direction that is 90° advanced from and perpendicular to the d′ axis is defined as a Q′ axis; then, the d′ axis and the q′ axis are collectively expressed as a d′q′ axis. 
     A direction in which the armature interlinked magnetic flux is generated is defined as a γ axis; a direction that is 90° advanced from and perpendicular to the γ axis is defined as a δ axis; then, the γ axis and the δ axis are collectively expressed as a γδ axis. In contrast, the direction of the armature interlinked magnetic flux calculated by the after-mentioned magnetic flux calculation device  17  is defined as a γ′ axis; a direction that is 90° advanced from and perpendicular to the γ′ axis is defined as a δ′ axis; then, the γ′ axis and the δ′ axis are collectively expressed as a γ′δ′ axis. 
     The magnetic flux calculation device  17  calculates an estimation value of the armature interlinked magnetic flux (hereinafter, expressed as estimated armature interlinked magnetic flux), based on voltage commands vu*, vv*, and vw* of the phases u, v, and w obtained from the γ′δ′ to uvw coordinate converter  13 , described later, current detection values iu, iv, and iw of the phases u, v, and w obtained from the current detection unit  14 , θ′ obtained from the position detection device  15 , and the electric angular velocity ω obtained from the speed calculation device  16 ; specifically, the magnetic flux calculation device  17  estimates a phase ∠Φ′ of the estimated armature interlinked magnetic flux. The phase ∠Φ′ of the estimated armature interlinked magnetic flux denotes the angle of the direction (the γ′-axis direction) of the estimated armature interlinked magnetic flux with respect to the d′ axis. 
       FIG. 2  is an example of configuration diagram of the magnetic flux calculation device  17  according to Embodiment 1. In  FIG. 2 , through calculation according to the equation (1) below, a uvw to d′q′ coordinate converter  101   a  converts the voltage commands vu*, vv*, and vw* into d′q′-axis voltage command values vd′* and vq′*, with respect to θ′. 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             v 
                             
                               d 
                               ′ 
                             
                             * 
                           
                         
                       
                       
                         
                           
                             v 
                             
                               q 
                               ′ 
                             
                             * 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 ′ 
                               
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 ′ 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 ′ 
                               
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 ′ 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 2 
                                 3 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   6 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   6 
                                 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               v 
                               u 
                               * 
                             
                           
                         
                         
                           
                             
                               v 
                               v 
                               * 
                             
                           
                         
                         
                           
                             
                               v 
                               w 
                               * 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     Through calculation according to the equation (2) below, a uvw to d′q′ coordinate converter  101  converts the output currents iu, iv, and iw of the synchronous machine  3  into d′q′-axis current detection values id′ and iq′, with respect to θ′. 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             i 
                             
                               d 
                               ′ 
                             
                           
                         
                       
                       
                         
                           
                             i 
                             
                               q 
                               ′ 
                             
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 ′ 
                               
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 ′ 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 ′ 
                               
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 ′ 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 2 
                                 3 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   6 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   6 
                                 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               i 
                               u 
                             
                           
                         
                         
                           
                             
                               i 
                               v 
                             
                           
                         
                         
                           
                             
                               i 
                               w 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     A voltage-type magnetic flux calculation device  102  firstly calculates d′q′-axis estimated interlinked magnetic flux Φd′ and Φq′, through calculation according to the equation (3). 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             Φ 
                             
                               d 
                               ′ 
                             
                           
                           = 
                           
                             
                               1 
                               ω 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   v 
                                   
                                     q 
                                     ′ 
                                   
                                   * 
                                 
                                 - 
                                 
                                   Ri 
                                   
                                     q 
                                     ′ 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             Φ 
                             
                               q 
                               ′ 
                             
                           
                           = 
                           
                             
                               - 
                               
                                 1 
                                 ω 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   v 
                                   
                                     d 
                                     ′ 
                                   
                                   * 
                                 
                                 - 
                                 
                                   Ri 
                                   
                                     d 
                                     ′ 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Then, through calculation according to the equation (4), the voltage-type magnetic flux calculation device  102  calculates the phase ∠Φ′ of the estimated armature interlinked magnetic flux from the d′q′-axis estimated interlinked magnetic flux Φd′ and Φq′ and outputs it. 
     
       
         
           
             
               
                 
                   
                     ∠Φ 
                     ′ 
                   
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           Φ 
                           
                             q 
                             ′ 
                           
                         
                         
                           Φ 
                           
                             d 
                             ′ 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Heretofore, the operation of the output side of the electric-power conversion unit  11  has been explained. Next, there will be explained the flow of generation of the voltage commands at the input side of the electric-power conversion unit  11 . 
     Through calculation according to the equation (5) below, a uvw to γ′δ′ coordinate converter  18  converts the output currents iu, iv, and iw of the synchronous machine  3  into γ′δ′-axis current detection values iγ′ and iδ′. 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             i 
                             
                               γ 
                               ′ 
                             
                           
                         
                       
                       
                         
                           
                             i 
                             
                               δ 
                               ′ 
                             
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     θ 
                                     ′ 
                                   
                                   + 
                                   
                                     ∠Φ 
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     θ 
                                     ′ 
                                   
                                   + 
                                   
                                     ∠Φ 
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       ′ 
                                     
                                     + 
                                     
                                       ∠Φ 
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     θ 
                                     ′ 
                                   
                                   + 
                                   
                                     ∠Φ 
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 2 
                                 3 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   6 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   6 
                                 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               i 
                               u 
                             
                           
                         
                         
                           
                             
                               i 
                               v 
                             
                           
                         
                         
                           
                             
                               i 
                               w 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Based on the difference between γ′δ′-axis current command values iγ′* and iδ′* and the γ′δ′-axis current detection values iγ′ and iδ′, the current control device  19  performs proportional-integral control (PI control) expressed in the equation (6) below so as to calculate γ′δ′-axis voltage commands vγ′* and vδ′*. 
     
       
         
           
               
             
               
                 
                   
                     { 
                     
                       
                         
                           
                             
                               v 
                               
                                 γ 
                                 ′ 
                               
                               * 
                             
                             = 
                             
                               
                                 ( 
                                 
                                   
                                     K 
                                     
                                       p 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         γ 
                                         ′ 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       K 
                                       
                                         i 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           γ 
                                           ′ 
                                         
                                       
                                     
                                     s 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     i 
                                     
                                       γ 
                                       ′ 
                                     
                                     * 
                                   
                                   - 
                                   
                                     i 
                                     
                                       γ 
                                       ′ 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               v 
                               
                                 δ 
                                 ′ 
                               
                               * 
                             
                             = 
                             
                               
                                 ( 
                                 
                                   
                                     K 
                                     
                                       p 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         δ 
                                         ′ 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       K 
                                       
                                         i 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           δ 
                                           ′ 
                                         
                                       
                                     
                                     s 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     i 
                                     
                                       δ 
                                       ′ 
                                     
                                     * 
                                   
                                   - 
                                   
                                     i 
                                     
                                       δ 
                                       ′ 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
           
         
       
     
     Here, Kpγ′ is a current control γ′-axis proportional gain, Kiγ′ is a current control γ′-axis integral gain, Kpδ′ is a current control δ′-axis proportional gain, and Kiδ′ is a current control δ′-axis integral gain. 
     The γ′δ′ to uvw coordinate converter  13  converts the γ′δ′-axis voltage command values vγ′* and vδ′* into the voltage commands vu*, vv*, and vw* of the phased of u, v, and w, through calculation according to the equation (7) below. 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             v 
                             u 
                             * 
                           
                         
                       
                       
                         
                           
                             v 
                             v 
                             * 
                           
                         
                       
                       
                         
                           
                             v 
                             w 
                             * 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 2 
                                 3 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               - 
                               
                                 1 
                                 
                                   6 
                                 
                               
                             
                           
                           
                             
                               1 
                               
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 1 
                                 
                                   6 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     θ 
                                     ′ 
                                   
                                   + 
                                   
                                     ∠Φ 
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               - 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       ′ 
                                     
                                     + 
                                     
                                       ∠Φ 
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     θ 
                                     ′ 
                                   
                                   + 
                                   
                                     ∠Φ 
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     θ 
                                     ′ 
                                   
                                   + 
                                   
                                     ∠Φ 
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               v 
                               
                                 γ 
                                 ′ 
                               
                               * 
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 δ 
                                 ′ 
                               
                               * 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     In this regard, however, it may be allowed that in the equation (7), considering a control calculation delay time (waste time) until the control calculation based on the output currents iu, iv, and iw of the synchronous machine  3  detected by the current detection unit  14  is reflected on the voltages vu, vv, and vw of the three phased outputted from the electric-power conversion unit  11 , the coordinate conversion is performed with a phase obtained by correcting (θ+∠Φ) by a phase correction amount θd2 based on the control calculation delay time. 
     As described above, the electric-power conversion unit  11  applies the voltages vu, vv, vw to the synchronous machine  3 , based on the voltage commands vu*, vv* and vw*. 
     Heretofore, the operation of the input side of the electric-power conversion unit  11  has been explained. Next, the processing contents of the rotor position error calculation device  12  will be explained. 
     The rotor position error calculation device  12  calculates a rotor position error Δθ from the phase ∠Φ′ of the estimated armature interlinked magnetic flux. Based on the true rotor position θ of the synchronous machine  3  and the rotor position θ′ detected by the position detection device  15 , the rotor position error Δθ is given by the equation (8) below.
 
Δθ=θ′−θ  (8)
 
     Hereinafter, there will be explained a method through which the rotor position error calculation device  12  calculates the rotor position error Δθ. By substituting the equation (3) for the equation (4), the equation (9) is derived. 
     
       
         
           
             
               
                 
                   
                     ∠Φ 
                     ′ 
                   
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         - 
                         
                           
                             
                               v 
                               
                                 d 
                                 ′ 
                               
                               * 
                             
                             - 
                             
                               Ri 
                               
                                 d 
                                 ′ 
                               
                             
                           
                           
                             
                               v 
                               
                                 q 
                                 ′ 
                               
                               * 
                             
                             - 
                             
                               Ri 
                               
                                 q 
                                 ′ 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
       FIG. 3  represents the relationship among the dq-axis, the d′q′-axis, the γδ-axis, and the γ′δ′-axis. As can be seen in  FIG. 3 , the d′q′ axis is advance by Δθ from the dq axis; therefore, the d′q′-axis voltage command values vd′* and vq′* are given by the equation (10), by use of Δθ and dq-axis voltage command values vd* and vq*. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               v 
                               
                                 d 
                                 ′ 
                               
                               * 
                             
                             = 
                             
                               
                                 
                                   v 
                                   d 
                                   * 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                               + 
                               
                                 
                                   v 
                                   q 
                                   * 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                         
                       
                     
                     
                       
                         
                           
                             v 
                             
                               q 
                               ′ 
                             
                             * 
                           
                           = 
                           
                             
                               
                                 - 
                                 
                                   v 
                                   d 
                                   * 
                                 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ⁢ 
                             
                                 
                             
                             + 
                             
                               
                                 v 
                                 q 
                                 * 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Δθ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     The d′q′-axis current detection values id′ and iq′ are given by the equation (11), by use of Δθ and dq-axis current detection values id and iq. 
     
       
         
           
             
               
                 
                     
                   
                     { 
                     
                       
                         
                           
                             
                               i 
                               
                                 d 
                                 ′ 
                               
                               * 
                             
                             = 
                             
                               
                                 
                                   i 
                                   d 
                                   * 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                               ⁢ 
                               
                                   
                               
                               + 
                               
                                 
                                   i 
                                   q 
                                   * 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               i 
                               
                                 q 
                                 ′ 
                               
                               * 
                             
                             = 
                             
                               
                                 
                                   - 
                                   
                                     i 
                                     d 
                                     * 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                               + 
                               
                                 
                                   i 
                                   q 
                                   * 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     The true phase ∠Φ of the armature interlinked magnetic flux is given by the equation (12) below. 
     
       
         
           
             
               
                 
                   ∠Φ 
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           Φ 
                           q 
                         
                         
                           Φ 
                           d 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     In the equation (12), Φd and Φq are d-axis interlinked magnetic flux and q-axis interlinked magnetic flux, respectively, and are given by the equation (13) below. 
     
       
         
           
             
               
                 
                     
                   
                     { 
                     
                       
                         
                           
                             
                               Φ 
                               d 
                             
                             = 
                             
                               
                                 1 
                                 ω 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     v 
                                     q 
                                     * 
                                   
                                   - 
                                   
                                     Ri 
                                     q 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               Φ 
                               q 
                             
                             = 
                             
                               
                                 - 
                                 
                                   1 
                                   ω 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     v 
                                     d 
                                     * 
                                   
                                   - 
                                   
                                     Ri 
                                     d 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     By substituting the equations (10) and (11) for the equation (9), the equation (14) is derived. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ∠Φ 
                           ′ 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               - 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           v 
                                           d 
                                           * 
                                         
                                         - 
                                         
                                           Ri 
                                           d 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δθ 
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           v 
                                           q 
                                           * 
                                         
                                         - 
                                         
                                           Ri 
                                           q 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δθ 
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       
                                         ( 
                                         
                                           
                                             v 
                                             d 
                                             * 
                                           
                                           - 
                                           
                                             Ri 
                                             d 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δθ 
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           v 
                                           q 
                                           * 
                                         
                                         - 
                                         
                                           Ri 
                                           q 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   
                                     - 
                                     
                                       1 
                                       ω 
                                     
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         v 
                                         d 
                                         * 
                                       
                                       - 
                                       
                                         Ri 
                                         d 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 - 
                                 
                                   
                                     1 
                                     ω 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         v 
                                         q 
                                         * 
                                       
                                       - 
                                       
                                         Ri 
                                         q 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       1 
                                       ω 
                                     
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         v 
                                         d 
                                         * 
                                       
                                       - 
                                       
                                         Ri 
                                         d 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 + 
                                 
                                   
                                     1 
                                     ω 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         v 
                                         q 
                                         * 
                                       
                                       - 
                                       
                                         Ri 
                                         q 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     By substituting the equation (13) for the equation (14), the equation (15) is derived. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ∠Φ 
                           ′ 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     Φ 
                                     q 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 - 
                                 
                                   
                                     Φ 
                                     d 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                               
                                 
                                   
                                     Φ 
                                     q 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 + 
                                 
                                   
                                     Φ 
                                     d 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       Φ 
                                       q 
                                     
                                     
                                       Φ 
                                       d 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 - 
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                               
                                 
                                   
                                     
                                       Φ 
                                       q 
                                     
                                     
                                       Φ 
                                       d 
                                     
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 + 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     The equation (16) is derived from the equation (12). 
     
       
         
           
             
               
                 
                   
                     
                       Φ 
                       q 
                     
                     
                       Φ 
                       d 
                     
                   
                   = 
                   
                     tan 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ∠Φ 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     By substituting the equation (16) for the equation (15), the equation (17) is derived. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ∠Φ 
                           ′ 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   tan 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ∠Φcos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 - 
                                 
                                   
                                     Φ 
                                     d 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                               
                                 
                                   tan 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ∠Φ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 + 
                                 
                                   
                                     Φ 
                                     d 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ∠Φ 
                                     
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ∠Φ 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 - 
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                               
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ∠Φ 
                                     
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ∠Φ 
                                     
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 + 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ∠Φ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 - 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ∠Φ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ∠Φcos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 + 
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ∠Φ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     ∠Φ 
                                     - 
                                     Δθ 
                                   
                                   ) 
                                 
                               
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     ∠Φ 
                                     - 
                                     Δθ 
                                   
                                   ) 
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               tan 
                               ⁡ 
                               
                                 ( 
                                 
                                   ∠Φ 
                                   - 
                                   Δθ 
                                 
                                 ) 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           ∠Φ 
                           - 
                           Δθ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     The equation (18) is derived from the equation (17).
 
Δθ=∠Φ−∠Φ′  (18)
 
     In the equation (18), assuming that the true phase ∠Φ of the armature interlinked magnetic flux is “0”, the rotor position error Δθ is given by the equation (19) below.
 
Δθ=−∠Φ  (19)
 
     The rotor position error calculation device  12  calculates the rotor position error Δθ through calculation according to the equation (19). The rotor position error Δθ is added to the origin correction value θ′ offset of the position detection device  15 , so that the rotor position error is corrected. 
     As described above, unlike the synchronous machine control apparatus disclosed in Patent Document 1, the synchronous machine control apparatus  1  according to Embodiment 1 makes it possible to correct a rotor position error without taking a time for searching a current command value. Moreover, unlike the synchronous machine control apparatus disclosed in Patent Document 2, the synchronous machine control apparatus  1  according to Embodiment 1 does not require any dedicated circuit for correcting a rotor position error. 
     Embodiment 2 
     Next, a synchronous machine control apparatus according to Embodiment 2 will be explained. In the foregoing synchronous machine control apparatus  1  according to Embodiment 1, a rotor position error is corrected based on the assumption that based on the assumption that the true phase ∠Φ of the armature interlinked magnetic flux can be regarded as “0”. However, because it cannot necessarily be assumed that ∠Φ is “0”, the accuracy of correcting a rotor position error may be deteriorated. In contrast, when the current command values are set in such a way that the true phase ∠Φ of the armature interlinked magnetic flux can be regarded as “0”, the accuracy of correcting a rotor position error is raised. Considering the foregoing fact, the current command values are set in such a way that ∠Φ can be regarded as “0” in a synchronous machine control apparatus according to Embodiment 2. 
       FIG. 4  is a diagram illustrating a synchronous machine control system including a synchronous machine control apparatus  1   a  according to Embodiment 2, a power source  2 , and a synchronous machine  3 . The configuration of the synchronous machine control apparatus  1   a  according to Embodiment 2 is almost the same as that of the synchronous machine control apparatus  1  according to Embodiment 1; however, the configuration of the synchronous machine control apparatus  1   a  is different from that of the synchronous machine control apparatus  1  in that a numerical value storage device  20  is added and in that the numerical value storage device  20  fixes the input signal iδ′* of the current control device  19  to be zero. 
     The equation (20) is derived from the equation (17).
 
∠Φ=∠Φ′+Δθ  (20)
 
     As can be seen from  FIG. 3 , the γδ axis is advance by ∠Φ from the dq axis. In contrast, the γ′δ′ axis is advance by (Δθ+∠Φ′) from the dq axis. Therefore, when the equation (20) is established, the γδ axis and the γ′δ′ axis coincide with each other and the equation (21) below is established. 
     
       
         
           
             
               
                 
                     
                   
                     { 
                     
                       
                         
                           
                             
                               i 
                               
                                 γ 
                                 ′ 
                               
                             
                             = 
                             
                               i 
                               γ 
                             
                           
                         
                       
                       
                         
                           
                             
                               i 
                               
                                 δ 
                                 ′ 
                               
                             
                             = 
                             
                               i 
                               δ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Thus, in the case where a δ′-axis current command value IδT* is set to “0” and a δ′-axis current detection value Iδ′ is controlled in such a way as to become “0”, the δ-axis current detection value Iδ becomes “0” and hence ∠Φ can be regarded as “0”. 
     As described above, in the synchronous machine control apparatus  1   a  according to Embodiment 2, ∠Φ can be regarded as “0”; therefore, in comparison with the synchronous machine control apparatus  1  according to Embodiment 1, the accuracy of correcting a rotor position error is raised. 
     Embodiment 3 
     Next, a synchronous machine control apparatus according to Embodiment 3 will be explained. In the foregoing synchronous machine control apparatus  1   a  according to Embodiment 2, a rotor position error is corrected with the δ′-axis current command value Iδ * fixed to “0”. However, in the case where when a rotor position error has been corrected and the synchronous machine control apparatus  1   a  is being operated with iδ′* set to non-zero, the mounting position of the position detection device  15  is displaced and hence the rotor position error changes, it is not made possible to detect and correct the change in the rotor position error. In this regard, however, in the case where the true phase ∠Φ of the armature interlinked magnetic flux is calculated without setting the δ′-axis current command value iδ′* to “0” and then the rotor position error Δθ is calculated through the equation (18), the accuracy of correcting the rotor position error can be raised in comparison with the synchronous machine control apparatus  1  according to Embodiment 1. 
     Unlike the synchronous machine control apparatus  1   a  according to Embodiment 2, a rotor position error can be corrected even when 1δ′* is set to non-zero. Considering the foregoing fact, in a synchronous machine control apparatus according to Embodiment 3, the value of the true phase ∠Φ of the armature interlinked magnetic flux is calculated and then the rotor position error Δθ is calculated through the equation (18). 
       FIG. 5  is a diagram illustrating a synchronous machine control system including a synchronous machine control apparatus  1   b  according to Embodiment 3, a power source  2 , and a synchronous machine  3 . The configuration of the synchronous machine control apparatus  1   b  according to Embodiment 3 is almost the same as that of the synchronous machine control apparatus  1  according to Embodiment 1; however, the configuration of the synchronous machine control apparatus  1   b  is different from that of the synchronous machine control apparatus  1  in that an armature interlinked magnetic flux phase calculation device  21  is added and in that as the input signal of a rotor position error calculation device  12   a , the true phase ∠Φ of the armature interlinked magnetic flux is added. 
     The armature interlinked magnetic flux phase calculation device  21  calculates the true phase ∠Φ of the armature interlinked magnetic flux from γ′δ′ current command values iγ′* and iδ′*. As represented in the equations (9) and (10), the value of ∠Φ is expressed as a function of a plurality of parameters such as the dq-axis voltage command values vγ*, vδ* and the dq-axis current detection values id and iq; however, the armature interlinked magnetic flux phase calculation device  21  is realized in a formation of a table or an equation in such a way that instead of calculating the value of ∠Φ as a function of these parameters, ∠Φ is regarded as a value that changes depending on the γ′δ′-axis current command values iγ′* and iδ′* and when the γδ-axis current command values iγ* and iδ* are inputted, the value of ∠Φ is outputted. The table or the equation is preliminarily created based on an analysis and an actual measurement. As represented in the equation (20), the γδ axis and the γ′δ′ axis coincide with each other; therefore, when the armature interlinked magnetic flux phase calculation device  21  is realized in such a manner as described above, the true phase ∠Φ of the armature interlinked magnetic flux can be calculated from γ′δ′ current command values iγ′* and iδ′*. Then, through calculation according to the equation (22), the rotor position error calculation device  12  calculates the rotor position error Δθ from the true phase ∠Φ of the armature interlinked magnetic flux and the phase ∠Φ′ of the estimated interlinked magnetic flux.
 
Δθ=∠Φ−∠Φ′  (22)
 
     As described above, in the synchronous machine control apparatus  1   b  according to Embodiment 3, in comparison with the synchronous machine control apparatus  1  according to Embodiment 1, the accuracy of correcting a rotor position error can be raised. Unlike the synchronous machine control apparatus  1   a  according to Embodiment 2, the rotor position error Δθ can be calculated even when Iδ′ is set to non-zero; therefore, even in the case where when a rotor position error has been corrected and the synchronous machine control apparatus  1   b  is being operated with iδ′* set to non-zero, the mounting position of the position detection device  15  is displaced, it is not made possible to correct the rotor position error. 
     Embodiment 4 
     Next, a synchronous machine control apparatus according to Embodiment 4 will be explained. In the respective synchronous machine control apparatuses  1 ,  1   a , and  1   b  according to Embodiments 1 through 3, described above, the magnetic flux calculation device  17  calculates the phase ∠Φ′ of the estimated armature interlinked magnetic flux by use of the dq-axis voltage command values vd′* and vq′*, instead of the d′q′-axis actual voltages vd′ and vq′. The foregoing method is performed under the assumption that there exists no difference between the d′q′-axis voltage command values vd′* and vq′* and the d′ q′-axis actual voltages vd′ and vq′; however, in fact, the effect of an error in the voltage output (e.g., a voltage error caused by the dead time of the inverter) of the electric-power conversion unit  11  may occur in a region where the absolute value of the line-to-line voltage is small. 
     Considering the foregoing fact, the synchronous machine control apparatus according to Embodiment 4 implements the operation of calculating the rotor position error Δθ, performed by the rotor position error calculation device  12  or  12   a  described in each of Embodiments 1 through 3, only when the line-to-line voltage is the same as or larger than a predetermined value or only when the absolute value of the electric angular velocity ω of the synchronous machine  3  is the same as or larger than a predetermined value. 
     The configuration of the synchronous machine control apparatus according to Embodiment 4 is the same as that of the synchronous machine control apparatus  1   b  according to Embodiment 3, explained with reference to  FIG. 5 . In the synchronous machine control apparatus  1   b  according to Embodiment 4, the rotor position error calculation device  12   a  performs calculation processing of the rotor position error Δθ only when the line-to-line voltage satisfies the condition given by the equation (23) below.
 
 V   rms   ≧V   min   (23)
 
     The line-to-line voltage Vrms in the equation (23) is determined by the equation (24) below, by use of the d′q′-axis voltage command values vd′* and vq′*.
 
 V   rms =√{square root over ( v   d′   2   +v   q′   2 )}  (24)
 
     In the equation (23), Vmin denotes the voltage lower limit in the calculation of a rotor position error and is determined, for example, by the equation (25) below.
 
 V   min =α vmin   ·V   td   (25)
 
     In the equation (25), αvmin denotes the voltage lower limit in the calculation of a rotor position error. Vtd denotes a voltage error approximate value caused by the dead time of the inverter and is expressed by the equation (26) below.
 
 V   td   =t   d   ·f   c   ·V   pn   (26)
 
     In the equation (26), td, fc, and Vpn denote the dead time of the inverter, the carrier frequency of the PWM (pulse Width Modulation), and the DC voltage supplied by the power source  2 , respectively. When αv in the equation (25) is set to be larger than 1, the rotor position error Δθ can be calculated under the condition that the absolute value of vd′* or vq′* is larger than the voltage error caused by the dead time. 
     When being restricted by the absolute value of the electric angular velocity ω instead of the line-to-line voltage, the calculation of the rotor position error Δθ is performed only in the case where ω satisfies the equation (27) below.
 
|ω|≧ω min   (27)
 
     In the equation (27), ωmin denotes the electric angular velocity lower limit in the calculation of a rotor position error and is determined, for example, by the equation (28) below. 
     
       
         
           
             
               
                 
                   
                     ω 
                     min 
                   
                   = 
                   
                     
                       α 
                       wmin 
                     
                     · 
                     
                       
                         V 
                         td 
                       
                       
                         Φ 
                         m 
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     In the equation (28), αWmin denotes the electric angular velocity lower limit coefficient in the calculation of a rotor position error, and Φm denotes the permanent-magnet magnetic flux. When αw in the equation (28) is set to be larger than 1, the rotor position error Δθ can be calculated with the electric angular velocity ω at which the line-to-line voltage is larger than the voltage error caused by the dead time. 
     As described above, the synchronous machine control apparatus  1   b  according to Embodiment 4 implements the operation of calculating the rotor position error Δθ, only when the line-to-line voltage is the same as or larger than a predetermined value or only when the absolute value of the electric angular velocity ω of the synchronous machine  3  is the same as or larger than a predetermined value; therefore, the effect of a voltage output error in the electric-power conversion unit  11  can be reduced. Thus, in comparison with each of the respective synchronous machine control apparatuses  1 ,  1   a , and  1   b  according to Embodiments 1 through 3, the accuracy of correcting a rotor position error can be raised. 
     Embodiment 5 
     Next, a synchronous machine control apparatus according to Embodiment 5 will be explained. Because the d′q′-axis current detection values id′ and iq′ are affected by noise included in the current detection values detected by the current detection unit  14 , the d′q′-axis voltage command values vd′* and vq′* calculated by use of the current detection values are also affected by the noise. Accordingly, the phase ∠Φ′ of the estimated armature interlinked magnetic flux calculated from vd′*, id′, and iq′ is also affected by the noise. This noise may deteriorate the accuracy of correcting a rotor position error. However, in the case where the noise components can be eliminated by a low-pass filter, the accuracy of correcting a rotor position error can be raised. 
     Considering the foregoing fact, in the synchronous machine control apparatus according to Embodiment 5, a low-pass filter for eliminating noise included in the current detection unit  14  is added. 
       FIG. 6  is a diagram illustrating a synchronous machine control system including a synchronous machine control apparatus  1   c  according to Embodiment 5, a power source  2 , and a synchronous machine  3 . 
     The configuration of the synchronous machine control apparatus  1   c  according to Embodiment 5 is the same as that of the synchronous machine control apparatus  1   b  according to Embodiment 3; however, the configuration of a magnetic flux calculation device  17   a  is different from that of the magnetic flux calculation device  17 . 
       FIG. 7  is an example of configuration diagram of the magnetic flux calculation device  17   a  according to Embodiment 5. The configuration of the magnetic flux calculation device  17   a  in  FIG. 7  is almost the same as that of the magnetic flux calculation device  17  in  FIG. 2 ; however, the configuration of the magnetic flux calculation device  17   a  is different from that of the magnetic flux calculation device  17  in that low-pass filters  103 ,  103   a , and  103   b  are added therein. 
     The low-pass filter  102  applies low-pass filtering processing to the d′q′-axis current detection values id′ and iq′ so as to output filtered d′q′-axis current detection values id′f and iq′f. The low-pass filter  103   a  applies low-pass filtering processing to the d′q′-axis voltage command values vd′* and vq′* so as to output filtered d′q′-axis voltage command values vd′*f and vq′*f. The low-pass filter  103   b  applies low-pass filtering processing to the phase ∠Φ′ of the estimated armature interlinked magnetic flux so as to output filtered phase ∠Φ′f of the estimated armature interlinked magnetic flux. 
     As described above, in the synchronous machine control apparatus  1   c  according to Embodiment 5, noise included in the current detection values is eliminated by the low-pass filters  103 ,  103   a , and  103   b ; therefore, the phase ∠Φ′ of the estimated armature interlinked magnetic flux is not affected by the noise. Thus, in comparison with each of the respective synchronous machine control apparatuses  1 ,  1   a , and  1   b  according to Embodiments 1 through 4, the accuracy of correcting a rotor position error is raised. 
     Various modifications and alterations of this invention will be apparent to those skilled in the art without departing from the scope and spirit of this invention, and it should be understood that this is not limited to the illustrative embodiments set forth herein. 
     
       
         
           
             
               
                 
                   
                       
                   
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                     7 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
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                     8 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
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                       θ 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     9 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∠Φ 
                       ′ 
                     
                     = 
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           - 
                           
                             
                               
                                 v 
                                 
                                   d 
                                   ′ 
                                 
                                 * 
                               
                               - 
                               
                                 Ri 
                                 
                                   d 
                                   ′ 
                                 
                               
                             
                             
                               
                                 v 
                                 
                                   q 
                                   ′ 
                                 
                                 * 
                               
                               - 
                               
                                 Ri 
                                 
                                   q 
                                   ′ 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expresssion 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     10 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             
                               v 
                               
                                 d 
                                 ′ 
                               
                               * 
                             
                             = 
                             
                               
                                 
                                   v 
                                   d 
                                   * 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                               + 
                               
                                 
                                   v 
                                   q 
                                   * 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               v 
                               
                                 q 
                                 ′ 
                               
                               * 
                             
                             = 
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     d 
                                     * 
                                   
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                               + 
                               
                                 
                                   v 
                                   q 
                                   * 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     11 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             
                               i 
                               
                                 d 
                                 ′ 
                               
                               * 
                             
                             = 
                             
                               
                                 
                                   i 
                                   d 
                                   * 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                               + 
                               
                                 
                                   i 
                                   q 
                                   * 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               i 
                               
                                 q 
                                 ′ 
                               
                               * 
                             
                             = 
                             
                               
                                 
                                   - 
                                   
                                     i 
                                     d 
                                     * 
                                   
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                               + 
                               
                                 
                                   i 
                                   q 
                                   * 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δθ 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     12 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     ∠Φ 
                     = 
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             Φ 
                             q 
                           
                           
                             Φ 
                             d 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     13 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             
                               Φ 
                               d 
                             
                             = 
                             
                               
                                 1 
                                 ω 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     v 
                                     q 
                                     * 
                                   
                                   - 
                                   
                                     Ri 
                                     q 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               Φ 
                               q 
                             
                             = 
                             
                               
                                 - 
                                 
                                   1 
                                   ω 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     v 
                                     d 
                                     * 
                                   
                                   - 
                                   
                                     Ri 
                                     d 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     14 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                             
                         
                         ⁢ 
                         
                           
                             ∠Φ 
                             ′ 
                           
                           = 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               { 
                               
                                 - 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             v 
                                             d 
                                             * 
                                           
                                           - 
                                           
                                             Ri 
                                             d 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       Δθ 
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           
                                             v 
                                             q 
                                             * 
                                           
                                           - 
                                           
                                             Ri 
                                             q 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       Δθ 
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           ( 
                                           
                                             
                                               v 
                                               d 
                                               * 
                                             
                                             - 
                                             
                                               Ri 
                                               d 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       Δθ 
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           
                                             v 
                                             q 
                                             * 
                                           
                                           - 
                                           
                                             Ri 
                                             q 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       Δθ 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   
                                     - 
                                     
                                       1 
                                       ω 
                                     
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         v 
                                         d 
                                         * 
                                       
                                       - 
                                       
                                         Ri 
                                         d 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 - 
                                 
                                   
                                     1 
                                     ω 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         v 
                                         q 
                                         * 
                                       
                                       - 
                                       
                                         Ri 
                                         q 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       1 
                                       ω 
                                     
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         v 
                                         d 
                                         * 
                                       
                                       - 
                                       
                                         Ri 
                                         d 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 + 
                                 
                                   
                                     1 
                                     ω 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         v 
                                         q 
                                         * 
                                       
                                       - 
                                       
                                         Ri 
                                         q 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     15 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                             
                         
                         ⁢ 
                         
                           
                             ∠Φ 
                             ′ 
                           
                           = 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     
                                       Φ 
                                       q 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δθ 
                                   
                                   - 
                                   
                                     
                                       Φ 
                                       d 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δθ 
                                   
                                 
                                 
                                   
                                     
                                       Φ 
                                       q 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δθ 
                                   
                                   + 
                                   
                                     
                                       Φ 
                                       d 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δθ 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       Φ 
                                       q 
                                     
                                     
                                       Φ 
                                       d 
                                     
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 - 
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                               
                                 
                                   
                                     
                                       Φ 
                                       q 
                                     
                                     
                                       Φ 
                                       d 
                                     
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 + 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     16 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         Φ 
                         q 
                       
                       
                         Φ 
                         d 
                       
                     
                     = 
                     
                       tan 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ∠ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Φ 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     17 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                             
                         
                         ⁢ 
                         
                           
                             ∠Φ 
                             ′ 
                           
                           = 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     tan 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ∠ΦcosΔθ 
                                   
                                   - 
                                   
                                     
                                       Φ 
                                       d 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δθ 
                                   
                                 
                                 
                                   
                                     tan 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ∠ΦsinΔθ 
                                   
                                   + 
                                   
                                     
                                       Φ 
                                       d 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δθ 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ∠Φ 
                                     
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ∠Φ 
                                     
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 - 
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                               
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ∠Φ 
                                     
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ∠Φ 
                                     
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                                 + 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ∠ΦcosΔθ 
                                 
                                 - 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ∠Φsin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ∠ΦcosΔθ 
                                 
                                 + 
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ∠Φsin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δθ 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 sin 
                                 ( 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ∠Φ 
                                   - 
                                   Δθ 
                                 
                                 ) 
                               
                               
                                 cos 
                                 ( 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ∠Φ 
                                   - 
                                   Δθ 
                                 
                                 ) 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               tan 
                               ( 
                               
                                   
                               
                               ⁢ 
                               
                                 ∠Φ 
                                 - 
                                 Δθ 
                               
                               ) 
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           ∠Φ 
                           - 
                           Δθ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     18 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Δθ 
                     = 
                     
                       ∠Φ 
                       - 
                       
                         ∠Φ 
                         ′ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     19 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Δθ 
                     = 
                     
                       - 
                       
                         ∠Φ 
                         ′ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     20 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     ∠Φ 
                     = 
                     
                       
                         ∠Φ 
                         ′ 
                       
                       + 
                       Δθ 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     21 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             
                               i 
                               
                                 γ 
                                 ′ 
                               
                             
                             = 
                             
                               i 
                               γ 
                             
                           
                         
                       
                       
                         
                           
                             
                               i 
                               
                                 δ 
                                 ′ 
                               
                             
                             = 
                             
                               i 
                               δ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     22 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Δθ 
                     = 
                     
                       ∠Φ 
                       - 
                       
                         ∠Φ 
                         ′ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     23 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       rms 
                     
                     ≥ 
                     
                       V 
                       min 
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     24 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       rms 
                     
                     = 
                     
                       
                         
                           v 
                           
                             d 
                             ′ 
                           
                           2 
                         
                         + 
                         
                           v 
                           
                             q 
                             ′ 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     25 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       min 
                     
                     = 
                     
                       
                         α 
                         vmin 
                       
                       · 
                       
                         V 
                         td 
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     26 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       td 
                     
                     = 
                     
                       
                         t 
                         d 
                       
                       · 
                       
                         f 
                         c 
                       
                       · 
                       
                         V 
                         pn 
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     27 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                        
                       ω 
                        
                     
                     ≥ 
                     
                       ω 
                       min 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     28 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       ω 
                       min 
                     
                     = 
                     
                       
                         α 
                         wmin 
                       
                       · 
                       
                         
                           V 
                           td 
                         
                         
                           Φ 
                           m 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   )