7. CLOSED LOOP MOTION CONTROL 
                      IN ELECTRIC DRIVES
                          7.1. INTRODUCTION
          By motion control we mean torque, speed or position control. Motion 
      control  systems  are  characterized  by  precision,  response  quickness  and 
      immunity to parameter detuning, torque and inertia perturbations and energy 
      conversion rates. Motion control through electric motors and power electronic 
      converters (P.E.Cs) may be approached by the theory and practice of linear 
      and nonlinear, continuous or discrete control systems.
      . PM d.c. - brush motors are characterized by a low electrical time constant t  
                                                                              e
      = L / R of a few miliseconds or less. The armature (torque) current is fully 
      decoupled from the PM field because of the orthogonality of the armature and 
      PM fields, both at standstill and for any rotor speed.
                                     Electric Drives                       2
      As  shown  in  later  chapters,  vector  control  of  a.c.  motors  does  also 
    decoupled flux and torque control if the orientation of the flux linkage is kept 
    constant. Consequently vector controlled a.c. motors are similar to d.c. brush 
    motors and thus the application of various motion control systems to the d.c. 
    motor  holds  notable  generality  while  also  eliminates  the  necessity  of  a 
    separate chapter on closed loop control of brushless motors.
      7.2. THE CASCADED MOTION CONTROL
    The PM d.c. - brush motor equations are:
                          di
                        (7.1)
                   VRiL PMr
                          dt
             Figure 7.1. Typical cascaded motion control
                       Electric Drives          3
                                                                J dr T  T
                                                                                 e       L
                                                                    dt (7.2)
                                                                     dr
                                                                          (7.3)
                                                                             r
                                                                       dt
                                                                   T  I
                                                                          (7.4)
                                                                      e        PM
        7.2.1. The torque loop
                 For  constant  (or  zero)  load  torque  the  PM  d.c.  brush  motor  current  / 
          voltage transfer function, from (7.1) - (7.4), becomes:
                                                   H s  is                             sem
                                                       V                 (7.5)2
                                                                                                           
                                                                  Vs            s eme sem 1R
                                                                         JR
          where                                                      em          (7.6)
                                                                              2
                                                                                PM
                                                                    Electric Drives                                                         4
                   Figure 7.2. PI torque loop for a PM d.c. brush motor
           In  what  follows  we  are  using  the  critical  frequency  w   and  phase 
                                                                 c
       margin j  contraints for the open - loop transfer function A(s) of the system 
              c
       on figure 7.2.:                  
                        As Ksi 1ssi    KCKTKIsem
                                       (7.7)
                                  s R    s2  s    1
                                    si       em e    em
       The critical frequency wc should be high - up to 1...2 kHz - to provide fast 
       torque (current) control.
                                    Electric Drives                       5
        Let us have as a numerical example a PM d.c. brush motor with the data: Vn = 
        110V, P  = 2kW, n  = 1800rpm, R = 1W, L = 20mH, K  = 1.1Nm/A, t  = 0.1 
                    n               n                                                        T                      em
        sec., K  = 25V/V, K = 0.5V/A, critical frequency f  = 500Hz, and the phase 
                   C                    i                                               c
        margin j  = 47°.
                     c
        The phase margin j  of A(s) from (7.7), for the critical frequency w  = 2pf , is:
                                     c                     0                                             c         c
                                              180 ArgA j 
                                                c                              c
                                          0         1                    1                 
                                 180 tan    tan                              c  em      
                                                                  (7.8)
                                                          c si             1  2  
                                                                                    c   em e 
        Consequently:                                                           25000.1
                     tan1   1800 470 tan1                                                        460
                                 c si                             (7.9)               2
                                                                       1 2500 0.10.02
        And thus:                                        tan460
                                                 si                0.3075ms
                                                                (7.10)
                                                          2500
                                                            Electric Drives                                               6