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A Modified Sychronous Current Regulator for Brushless Motor Control

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A Modified Sychronous Current Regulator for Brushless Motor Control Powered By Docstoc
					A Modified Sychronous Current Regulator
for Brushless Motor Control


Shane Colton <scolton@mit.edu>
Graduate Student, Department of Mechanical Engineering
Massachusetts Institute of Technology

Rev0 - Doctoral Qualifying Examination, January 26, 2011

                                                           1
Overview
•   This work details a torque controller for brushless Permanent Magnet
    Synchronous Motors (PMSM).

•   Methods of controlling PMSM:
     • Brushless DC Control
     • Field-Oriented Control (FOC): Synchronous Current Regulator (SCR)

•   The author’s contribution is a modified SCR that:
     • uses Hall effect sensors (instead of an encoder).
     • is more computational efficient (low-cost processing).
     • has the potential for improved transient response.

•   The design of the controller and an experimental application to low-cost
    personal transportation will be detailed.




                                                                               2
Outline
Theoretical Analysis
• Permanent Manget Synchronous Motor Model
• Field Oriented Control Principles
• Synchronous Current Regulator (SCR)
• Modified Synchronous Current Regulator (mSCR)

Applied Analysis
• Plant Information
• Controller Hardware
• Controller Design
• Controller Simulations: SCR and mSCR
• Experimental Testing and Data

•   Future Work
•   Questions / Feedback

•   Motor Control Overview
•   Current Sensing
•   Simplified Plant Closed-Loop Transfer Function and Root-Locus
•   A more fair transient response comparison.
•   High-Speed Operation
•   Error Handling and Failsafes
•   Connection to Adaptive Feed-Forward Cancellation (AFC)          3
PMSM Model
Three-phase permanent magnet synchronous motor (PMSM) electromechanical
model:
                     PMSM
                Ia              R
       ~                              ~
                         L
       Va       Ib                    Ea
                                R
       ~                              ~
                         L
       Vb       Ic                    Eb                              τ, Ω
                                R
       ~                              ~
                         L
       Vc                             Ec



                                     I a  Ea  I b  Eb  I c  Ec
            Power Conversion:   
                                                   
                                                                             4
PMSM Model
•    To control torque, both the phase and the magnitude of current must be
     controlled.

•    One option: high-bandwidth current controllers on each phase of the brushless
     motor. The closed-loop bandwidth must be significantly faster than the
     commutation of the motor (the AC frequency):


                                 Iar +                Va                  Ia
    AC References:                           Gc (s)        Z (s)
                                                                 1

                                         -
      I xr  I sin(t   x ),
                                 Ibr +                Vb                  Ib
       x  {a, b, c}                         Gc (s)        Z (s)   1

                                         -

                                 Icr +                Vc                  Ic
                                             Gc (s)        Z (s)   1

                                         -


                                                                               5
Field-Oriented Control Principles
By exploiting symmetry of the three-phase variables and transforming to the
reference frame of the rotor, the controller can act on quantities which are DC
in steady-state operation.

(Similar to adaptive feed-forward cancellation with sinusoidal input.)




Field-Oriented Current control works without the need for high-bandwidth
control loops.


 •Easier to implement on fixed-point, low-
 cost microcontrollers.
 •Better high-speed performance.


                                                                                  6
Field-Oriented Control Principles
Vector Motor Quantities, D/Q Axes

•   Controller operates in a two-dimensional coordinate system that is
    attached to the rotor: rotor/synchronous reference frame.

                                       •       Direct (D) Axis: Aligned with a North
                                               magnet pole.
                   ×           ·       •       Quadrature (Q) Axis: Exactly between
                   ×
                       A       ·
                   ×           ·               two magnet poles.
                   ×           ·
                           Q           •       In a two-pole motor, they are
                                               physically perpendicular.



                                   D
          ·




                                           ×
         ·




                                            ×
      ·




                                             ×
     ·




                                              ×




                                           B                           South-Face Magnet
          C
                                                                       North-Face Magnet
                                   ·
             ×




                                       ·
            ×




                                       ·
           ×




                                                                       Steel
                                           ·
          ×




                                                                       Copper Winding
                                                                                       7
Field-Oriented Control Principles
Vector Motor Quantities, D/Q Axes

•   Controller operates in a two-dimensional coordinate system that is
    attached to the rotor: rotor reference frame.

                                    •       Direct (D) Axis: Aligned with a North
                                            magnet pole.
                   ×       ·        •       Quadrature (Q) Axis: Exactly between
                   ×
                       A   ·
                   ×       ·                two magnet poles.
                   ×       ·
                                    •       The axes are attached to the rotor. Q
                   Q                        always leads D in the direction of
             Ω                              rotation.
                                D
          ·




                                        ×
         ·




                                         ×
      ·




                                          ×
     ·




                                           ×




                                        B                           South-Face Magnet
          C
                                                                    North-Face Magnet
                                 ·
             ×




                                    ·
            ×




                                     ·
           ×




                                                                    Steel
                                        ·
          ×




                                                                    Copper Winding
                                                                                    8
Field-Oriented Control Principles
Vector Motor Quantities, D/Q Axes

•   Controller operates in a two-dimensional coordinate system that is
    attached to the rotor: rotor reference frame.

                                    •       Direct (D) Axis: Aligned with a North
                                            magnet pole.
                   ×       ·        •       Quadrature (Q) Axis: Exactly between
                   ×
                       A   ·
                   ×       ·                two magnet poles.
                   ×       ·
                                    •       In a four-pole motor, they are
                                            separated by 45º mechanical. They
                               Q            are always separated by 90º electrical.


                                D
          ·




                                        ×
         ·




                                         ×
      ·




                                          ×
     ·




                                           ×




                                        B                           South-Face Magnet
          C
                                                                    North-Face Magnet
                                   ·
             ×




                                    ·
            ×




                                       ·
           ×




                                                                    Steel
                                        ·
          ×




                                                                    Copper Winding
                                                                                      9
 Field-Oriented Control Principles
 Vector Motor Quantities, D/Q Axes

  •   All motor quantities that have “direction” can be projected onto the d/q
      axes as vectors:
                              Stator Current / Flux: Vector sum of coil
                          I
                              current/flux defined by right hand rule.
Back EMF:
Always on the q-      ×        ·
                      ×
                          A    ·
axis.                 ×        ·
                      ×        ·         λ
      d     E                                  Rotor Flux Linkage: Always on the d-axis
  E                                            for a permanent magnet motor.
      dt             Q
            Ω
                                    D                           N
            ·




                                            ×
           ·




                                             ×
        ·




                                              ×
       ·




                                               ×




                                         B                           South-Face Magnet
            C
                                                                     North-Face Magnet
                                     ·
               ×




                                        ·
              ×




                                         ·
             ×




                                                                     Steel
                       I E
                                            ·
            ×




                                                                   Copper Winding
                                                                                    10
Field-Oriented Control Principles
Unrealistic Zero-Inductance Motor


                        Q                •   Voltage applied in-phase
                                             with back-EMF.
                                         •   Current also in-phase with
                                             back-EMF.
                        V
                   IR                    •   Torque per amp is optimal.
                        I                •   Reasonable approximation if
                    E                        inductance or speed is low:
                                                    L  R
                                    λr   D
                                                I               R



                                                ~   V           E    ~


                                                                          11
Field-Oriented Control Principles
Motor with Inductance


                        Q                  •   Voltage applied in-phase
                                               with back-EMF.
                                           •   Current lags due to the
                                               motor inductance.
                            IωL
                    V                      •   Torque per amp is no longer
                        IR                     optimal. Current and back
                                  I
                    E                          EMF are not in phase:

                                                       IE  0
                                      λr   D
                                                  I               R
                                                          L

                                                  ~   V           E      ~


                                                                             12
Field-Oriented Control Principles
Phase Advance to Correct for Inductance Lag


                         Q                    •    Voltage applied ahead of
                                                   back EMF.
                                              •    Current lags due to the
                                                   motor inductance such that
                   IωL
                                                   it is in phase with back EMF.
                         IR
               V                              •    Torque per amp is optimal.
                     E   I
                     ϕ
                                                    f (V , I , , K t , R, L,...)
                                 λr           D
                                                       I                R
                                                                L

                                                       ~   V             E    ~


                                                                                  13
Field-Oriented Control Principles
Field Weakening for High-Speed Operation


                       Q                   •   Voltage and current both
                                               lead back EMF.
                                           •   Stator flux counteracts rotor
                                               flux: “field weakening”
                      IR                   •   Torque per amp is not
                IωL
                                               optimal but…
            V          E
                  I                        •   Maximum achievable speed
                                               per volt is higher.
                                λr         D
                                                  I                R
                                                           L

                                                  ~   V            E    ~


                                                                            14
Field-Oriented Control Principles
Park Transform / Inverse Park Transform

 •   Tranforms used to convert from/to stator frame {a,b,c} quantities to/from
     rotor frame {d,q} quantities.
 •   Require rotor position, θ, as an input.


      xd     xa              cos         cos  23      cos  23  
                                                                            

     x   T x             2                                              
      q      b         T   sin          sin   23 
                                                           
                                                                  sin   23 
                              3
      x0 
             xc 
                               1
                                 2
                                                     1
                                                     2
                                                                        1
                                                                        2
                                                                                
                                                                                


      xa         xd            cos             sin       1
      x   T 1  x                                           
      b          q     T 1  cos  23   sin   23  1
                                             

      xc 
                 x0 
                                cos  23   sin   23  1
                                  
                                                             
                                                                  


                                                                                    15
Synchronous Current Regulator
                                                                              θ

                  +          d-axis     Vd                PWMa
0 or Idr                   controller        dq
                      -                                   PWMb
          +
                                        Vq                PWMc
                                                                                         M   Encoder
    Iqr                      q-axis               abc
              -
                           controller
                                             Inverse
                                         Park Transform
                                                                             Ia Ib
                                         Park Transform
                          Id
                                             dq
                          Iq                                             -
                                                  abc
                                                           Ic = -Ia-Ib        -
                                                                                     θ


•    Park and inverse Park transform convert into and out of rotor reference frame.
•    Two “independent” controllers for the d- and q-axis.
•    Requires rotor position, typically from an encoder or resolver.
                                                                                                       16
Synchronous Current Regulator
                                                                              θ

                  +          d-axis     Vd                PWMa
0 or Idr                   controller        dq
                      -                                   PWMb
          +
                                        Vq                PWMc
                                                                                         M   Encoder
    Iqr                      q-axis               abc
              -
                           controller
                                             Inverse
                                         Park Transform
                                                                             Ia Ib
                                         Park Transform
                          Id     1
                               s  1        dq
                          Iq                                             -
                                 1                abc
                                                           Ic = -Ia-Ib        -
                               s  1                                                θ
                          Current Filters

•    Because the controllers run in the rotor frame, where values are “DC” in steady
     state, the controllers may operate at low bandwidth, below commutation
     frequency, and long time-constant current filtering can be implemented.
                                                                                                       17
Modified Synchronous Current Regulator
Initial Motivation
•   For sufficient resolution of rotor position, an encoder or resolver is typically
    required for field oriented control. (Sensorless techniques also exist.)
•   However, less expensive motors use three Hall effect sensors to derive rotor
    position with 60º electrical resolution:


                     ×           ·                  A
                     ×
                         A       ·                  B
                     ×           ·                  C
                     ×           ·
                             Q                                                time
       C                                      A


                                                                   Hall Effect Sensor
                                     D
           ·




                                             ×




                                                                   South-Face Magnet
         ·




                                              ×
      ·




                                               ×
     ·




                                                ×




           C                             B                         North-Face Magnet
                                                                   Steel
                                     ·
              ×




                                         ·
             ×




                                         ·
            ×




                                             ·
           ×




                                 B                                 Copper Winding
                                                                                       18
Modified Synchronous Current Regulator
Initial Motivation
In sensored brushless DC control, the six Hall effect sensor states directly map to
phase voltage outputs.
                                                      State    Va       Vb       Vc
                                                          1    PWM        0V    High-Z
    A
    B                                                     2   High-Z      0V     PWM
    C                                                     3      0V    High-Z    PWM
                                                          4      0V     PWM     High-Z
           1    2    3   4   5    6                       5   High-Z    PWM        0V
                                                          6    PWM     High-Z      0V


•       Pros: very simple algorithm (state table), can run on low-cost processor.
•       Cons: fixed timing, torque ripple, audible noise

Initial Motivation: Can the Synchronous Current Regulator be modified to work with
Hall effect sensor inputs, with interpolation?


                                                                                         19
Modified Synchronous Current Regulator
  Slow Loop (100-1,000Hz)                   Fast Loop (10kHz)
                                                                      Hall Effect
                                                                      Interpolator         3
                                                   
                                                                                     Hall Effect
                                      ϕ                                                 Sensors
              -         d-axis                                    PWMa
  0 or Idr            controller
                  +                                               PWMb
         +
                                      |V|                         PWMc
                                                                                               M
   Iqr                  q-axis
          -
                      controller
                                             Sine Wave              Synchronous
                                              Generator             Measurement
                                                                               Ia        Ib
                                            Park Transform
                           Id        1
                                   s  1      dq
                           Iq                                                   -
                                     1               abc
                                                                 Ic = -Ia-Ib          -
                                   s  1


                                                                                                   20
Modified Synchronous Current Regulator
There are several practical differences:
     •   The controller is explicity split into fast and slow loops; only PWM
         generation and rotor position estimatation need be in the fast loop.
     •   PWM generation is done by a sine table look-up, which is faster to
         compute than an inverse Park transform.
     •   The rotor position is estimated by interpolating between Hall effect
         sensor absolute states using the last known speed.
     •   As long as rotor position and phase currents are sampled
         synchronously by the slow loop, the slow loop bandwidth can be
         arbitrarily low.



•   The modified synchronous current regulator can be run on fixed-point
    processors to control low-cost motors with Hall effect sensors.
•   It can achieve AC servo motor-like control with brushless DC motors.

                                                                                21
Modified Synchronous Current Regulator
The primary theoretical difference is the controller outputs:

                                                  Standard SCR
               +             d-axis      Vd       • Vd and Vq fully-define a voltage vector.
      0                    controller             • D-axis gain: [V/A]
                   -
      +                                  Vq       • Q-axis gain: [V/A]
Iqr                          q-axis
          -
                           controller
                                                  •   Simulate with:

              Iq Id

                                                  Modified SCR
                                         ϕ = V   • |V| and V fully-define a voltage vector.
                   -          d-axis
      0                     controller            • D-axis gain: [rad/A]
                       +
          +
                                          |V|     • Q-axis gain: [V/A]
Iqr                           q-axis
           -
                            controller
                                                  •   Simulate with:

               Iq Id                                                                           22
Modified Synchronous Current Regulator
Consider a step increase in torque command via Iqr:


                          Q                       SCR:
        ΔVmSCR                                                         Kd   Vd
                 ΔVSCR                                0       +
                                                                   -
                                                                        s
                  IωL                                                       Vq
                                                      +                Kq
            V2            IR                              -
                  V                                                    s
                      E   I
                      ϕ
                                                              Iq Id
                                  λr       D
                                                 mSCR:
                                                               -
                                                                       Kd   ϕ
                                                      0                 s
                                                                   +
                                                      +
                                                                            |V|
                                                                       Kq
                                                       -               s




                                                              Iq Id               23
Applied Analysis




                   24
Plant Information
Overview

   •   The controller presented here has been tested on several plants.
   •   The example used for this presentation is a 500W electric kick scooter.

                                    •   Custom-designed and built hub motor.
                                    •   Rear wheel direct drive, 1:1.
                                    •   33V, 4.4Ah LiFePO4 battery.
                                    •   Torque command by hand throttle.




                                                                                 25
Plant Information
Important Specifications


  Symbol                    Description                Value           Units
     2p      Number of poles.                                   14        -
     Ra      Per-phase motor resistance.                    0.084        Ω
     Ls      Synchronous inductance.                    0.2 × 10-3       H
     Kt      Per-phase torque/back EMF constant.               0.10   V/(rad/s)
     V       Nominal DC voltage.                               33.0      V
      J      Plant inertia, reflected to rotational.           0.40    kg·m²




                                                                               26
Controller Hardware
Overview

 •   Custom 48V/40A three-phase inverter drive
 •   Hall effect-based current sensing (phase and DC).
 •   v1,2: Texas Instruments MSP430F2274 (16-bit, no hardware multiplier)
     v3: STMicroelectronics STM32F103 (32-bit, w/ hardware multiplier)
 •   2.4GHz wireless link for data acquisition.




                                                                            27
Controller Hardware
Important Specifications


  Symbol                   Description                  Value          Units
     Rds     On-resistance of each phase leg.               7.5×10-3    Ω
     fsw     PWM switching frequency.                        15,625     Hz
     ffast   Fast-loop frequency. Handles position   MSP430: 14,500     Hz
             estimate, sine wave generation.         STM32: 10,000
    fslow    Slow-loop frequency. Handles current      MSP430: 122      Hz
             sampling, control computation.           STM32: 1,000
     ftx     Data transmit frequency. For data                   20     Hz
             display and logging.




                                                                             28
Controller Design
Overview
                              Synchronous Current Regulator:
   (Idr – Id)                  Vd
                  D-Axis
                 Controller          dq
   (Iqr – Iq)                  Vq                                        M
                  Q-Axis                   abc
                 Controller                       V{abc}

                Controllers    Inverse Park Transform      Amplifier   Motor


                         Modified Synchronous Current Regulator:

   (Idr – Id)                 V
                  D-Axis
                 Controller
   (Iqr – Iq)                  |V|                                       M
                  Q-Axis
                 Controller                       V{abc}

                Controllers     Sine Wave Generator        Amplifier   Motor   29
Controller Design
Simplified Plant: Q-Axis Only, Stalled

 •   At stall, both the d-axis and the q-axis look like resistors.
 •   Modeling the q-axis (torque-producing) controller and plant:


                  +                Kq                 1
          Iqr                                                           Iq
                            Iqe     s         Vq      Ra
                      -
                                   Gc(s)              Gp(s)


                                               1
                             Iqf            d s 1
                                             H(s)

 •   Closed-loop poles can be placed anywhere in the left half-plane,
     bandwidth set by filter frequency and damping ratio set by Kq.

                                                                             30
    Controller Design
    Simplified Plant: Q-Axis Only, Stalled

                                        L(s)
                      40
                                  System: L
                      20          Frequency (rad/sec): 14.5
                                                                                             1
     Magnitude (dB)




                       0
                                  Magnitude (dB): -1.97
                                                                           L( s ) 
                                                                                    ( Ra s )( d s  1)
                      -20
                      -40
                      -60                                                         To leave 75º
                      -80                                                         phase margin:
                        0
                                                                                              V
                      -45                                                        K q  1. 2
Phase (deg)




                                  System: L
                                  Frequency (rad/sec): 14.5                                   A s
                      -90         Phase (deg): -105



                -135
                                                                                    m  75
                -180 0
                   10       10
                              1
                                               10
                                                              2
                                                                  10
                                                                       3              0.75
                             Frequency (rad/sec)

                                                                                                     31
Controller Design
Simplified Plant: Q-Axis Only, Stalled

                                       Simplified Plant Closed-Loop Step Response
                                 1.4
                                                                          Kq = 1.2 V/A/s
                                 1.2                                      Kq = 1.6 V/A/s
                                                                          Kq = 2.5 V/A/s

                                  1
     Normalized Iq
                     Amplitude




                                 0.8

                                 0.6


                                 0.4

                                 0.2

                                  0
                                   0   0.05      0.1      0.15      0.2      0.25      0.3
                                                       Time (sec)
                                                                                             32
Controller Simulations
Synchronous Current Regulator

•   Full motor simulation with vector quantities and complex impedance using
    measured motor parameters (Ra, Ls, Kt).
•   Current filtering as described above.
•   Speed fixed at 500rpm. (Load dynamics not considered.)
•   Idr = 0, Iqr steps from 15A to 30A.



                       +
                               Kd   Vd
                0               s           Kd = {1.2, 1.6, 2.5} V/A/s
                           -
               +
                                    Vq
                               Kq
                   -                        Kq = {1.2, 1.6, 2.5} V/A/s
                               s




                       Iq Id


                                                                               33
Controller Simulations
Synchronous Current Regulator

                             SCR: Vector Step Response, Voltage
                 9
                                                           Kq = Kd = 1.2 V/A/s
                                                           Kq = Kd = 1.6 V/A/s
                8.5
                                                           Kq = Kd = 2.5 V/A/s

                 8
       Vq [V]




                7.5


                 7


                6.5


                 6
                 -3   -2.5         -2       -1.5      -1          -0.5           0
                                           Vd [V]


                                                                                     34
Controller Simulations
Synchronous Current Regulator             9
                                                      SCR: Vector Step Response, Voltage

                                                                                    Kq = Kd = 1.2 V/A/s
                                                                                    K = K = 1.6 V/A/s
                                         8.5                                         q    d
                                                                                    K = K = 2.5 V/A/s
                                                                                     q    d


                         Q                8

        ΔVmSCR




                                Vq [V]
                 ΔVSCR                   7.5
                                                                                    ΔVSCR
                                          7

            V2
                  V                      6.5


                                          6
                                          -3   -2.5         -2       -1.5      -1             -0.5        0
                                                                    V [V]
                                                                     d




                                               D



                                                                 What am I looking at?




                                                                                                              35
Controller Simulations
Synchronous Current Regulator

                             SCR: Vector Step Response, Voltage
                 9
                                                           Kq = Kd = 1.2 V/A/s
                                                           Kq = Kd = 1.6 V/A/s
                8.5
                                                           Kq = Kd = 2.5 V/A/s

                 8
       Vq [V]




                7.5


                 7


                6.5


                 6
                 -3   -2.5         -2       -1.5      -1          -0.5           0
                                           Vd [V]


                                                                                     36
Controller Simulations
Synchronous Current Regulator

                                 SCR: Vector Step Response, Current
                 34

                 32

                 30

                 28

                 26
       I q [A]




                 24

                 22

                 20
                             Kq = Kd = 1.2 V/A/s
                 18          Kq = Kd = 1.6 V/A/s
                 16          Kq = Kd = 2.5 V/A/s

                 14
                  -10   -8     -6    -4     -2         0     2   4    6   8   10
                                                   I d [A]


                                                                                   37
Controller Simulations
Synchronous Current Regulator

                                      SCR: Step Response, Torque
                      5.5
                       5

                      4.5

                       4
                      3.5
        Torque [Nm]




                       3
                      2.5

                       2
                      1.5                                  K = K = 1.2 V/A/s
                                                             q     d
                       1                                   K = K = 1.6 V/A/s
                                                             q     d
                      0.5                                  Kq = Kd = 2.5 V/A/s

                       0
                       15   15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9         16
                                               Time [s]
                                                                                      38
Controller Simulations
Modified Synchronous Current Regulator

•   Full motor simulation with vector quantities and complex impedance using
    measured motor parameters (Ra, Ls, Kt).
•   Current filtering as described above.
•   Speed fixed at 500rpm. (Load dynamics not considered.)
•   Idr = 0, Iqr steps from 15A to 30A.



                      -
                              Kd   ϕ
               0               s            Kd = 1.0 rad/A/s       (Is this fair?)
                          +
                +
                                   |V|
                              Kq
                 -                          Kq = {1.2, 1.6, 2.5} V/A/s
                              s




                     Iq Id


                                                                                     39
Controller Simulations
Modified Synchronous Current Regulator

                         mSCR: Vector Step Response, Voltage
                 9


                8.5


                 8
       Vq [V]




                7.5


                 7
                       K = 1.2 V/A/s
                         q
                       K = 1.6 V/A/s
                6.5      q
                       K = 2.5 V/A/s
                         q

                 6
                 -3   -2.5      -2      -1.5      -1       -0.5   0
                                       V [V]
                                         d


                                                                      40
Controller Simulations
Synchronous Current Regulator             9
                                                     mSCR: Vector Step Response, Voltage



                                         8.5



                         Q                8

        ΔVmSCR                                                               ΔVmSCR




                                Vq [V]
                 ΔVSCR                   7.5


                                          7
                                                   Kq = 1.2 V/A/s
            V2
                  V                      6.5
                                                   Kq = 1.6 V/A/s
                                                   Kq = 2.5 V/A/s

                                          6
                                          -3   -2.5          -2       -1.5     -1      -0.5   0
                                                                     V [V]
                                                                      d




                                               D



                                                                  What am I looking at?




                                                                                                  41
Controller Simulations
Modified Synchronous Current Regulator

                         mSCR: Vector Step Response, Voltage
                 9


                8.5


                 8
       Vq [V]




                7.5


                 7
                       K = 1.2 V/A/s
                         q
                       K = 1.6 V/A/s
                6.5      q
                       K = 2.5 V/A/s
                         q

                 6
                 -3   -2.5      -2      -1.5      -1       -0.5   0
                                       V [V]
                                         d


                                                                      42
Controller Simulations
Modified Synchronous Current Regulator

                                   mSCR: Vector Step Response, Current
                  34

                  32

                  30

                  28

                  26
        I q [A]




                  24

                  22

                  20
                              K = 1.2 V/A/s
                               q
                  18          K = 1.6 V/A/s
                               q
                  16          Kq = 2.5 V/A/s

                  14
                   -10   -8    -6      -4      -2     0     2   4    6   8   10
                                                    I [A]
                                                    d


                                                                                  43
Controller Simulations
Modified Synchronous Current Regulator

                                     mSCR: Step Response, Torque
                      5.5
                       5

                      4.5

                       4
                      3.5
        Torque [Nm]




                       3

                      2.5

                       2
                      1.5
                                                               Kq = 1.2 V/A/s
                       1                                       Kq = 1.6 V/A/s
                      0.5                                      Kq = 2.5 V/A/s
                       0
                       15   15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9        16
                                               Time [s]
                                                                                     44
         Controller Simulations
         Comparison

                            Voltage                                                                              Current                                                                  Torque
                      SCR: Vector Step Response, Voltage                                                SCR: Vector Step Response, Current                                              SCR: Step Response, Torque
          9                                                                             34                                                                              5.5
                                                    Kq = Kd = 1.2 V/A/s
                                                                                        32                                                                               5
                                                    Kq = Kd = 1.6 V/A/s
         8.5
                                                                                        30                                                                              4.5
                                                    Kq = Kd = 2.5 V/A/s
                                                                                        28                                                                               4
          8
                                                                                                                                                                        3.5
                                                                                        26




                                                                                                                                                          Torque [Nm]
Vq [V]




                                                                                                                                                                         3




                                                                              I q [A]
         7.5                                                                            24
                                                                                                                                                                        2.5
                                                                                        22
          7                                                                                                                                                              2
                                                                                        20
                                                                                                    Kq = Kd = 1.2 V/A/s                                                 1.5                                  Kq = Kd = 1.2 V/A/s
                                                                                        18          Kq = Kd = 1.6 V/A/s
         6.5                                                                                                                                                             1                                   Kq = Kd = 1.6 V/A/s
                                                                                        16          Kq = Kd = 2.5 V/A/s                                                                                      Kq = Kd = 2.5 V/A/s
                                                                                                                                                                        0.5
          6                                                                             14                                                                               0
          -3   -2.5         -2       -1.5      -1          -0.5           0              -10   -8     -6    -4       -2       0     2   4    6   8   10                  15   15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9            16
                                    Vd [V]                                                                                I d [A]                                                                Time [s]

                  mSCR: Vector Step Response, Voltage                                                  mSCR: Vector Step Response, Current                                             mSCR: Step Response, Torque
          9                                                                             34                                                                              5.5

                                                                                        32                                                                               5
         8.5                                                                                                                                                            4.5
                                                                                        30

                                                                                        28                                                                               4
          8
                                                                                                                                                                        3.5
                                                                                        26




                                                                                                                                                          Torque [Nm]
                                                                                                                                                                         3
Vq [V]




                                                                              I q [A]




         7.5                                                                            24
                                                                                                                                                                        2.5
                                                                                        22
                                                                                                                                                                         2
          7
                Kq = 1.2 V/A/s                                                          20
                                                                                                    Kq = 1.2 V/A/s                                                      1.5
                                                                                                                                                                                                                     Kq = 1.2 V/A/s
                Kq = 1.6 V/A/s                                                          18          Kq = 1.6 V/A/s
         6.5                                                                                                                                                             1                                           Kq = 1.6 V/A/s
                Kq = 2.5 V/A/s                                                          16          Kq = 2.5 V/A/s                                                      0.5                                          Kq = 2.5 V/A/s
          6                                                                             14                                                                               0
          -3   -2.5         -2       -1.5      -1          -0.5           0              -10   -8     -6    -4       -2       0     2   4    6   8   10                  15   15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9            16
                                    Vd [V]                                                                                I d [A]                                                                Time [s]



                                                                                                                                                                                                                         45
        Experimental Testing and Data
        Baseline: Q-axis Control Only


                                    Q-axis Control Only (Fixed Phase)           •   Q-axis (torque producing)
                800
                700                                                                 current controlled.
                600
Speed [rpm]




                500                                                             •   D-axis current increases
                400                                                                 with speed.
                300
                200
                100
                  0
                      40           60        80              100    120   140

                 40
                 30
                 20
  Current [A]




                 10
                  0
                -10
                -20        I
                               q
                -30        I
                               d
                -40
                      40           60        80              100    120   140
                                                  Time [s]

                                                                                                        46
 Experimental Testing and Data
 Baseline: Q-axis Control Only


                   Q-Axis Control Only (Fixed Phase)             •   Q-axis (torque producing)
     40
                                                                     current controlled.
     30                                                          •   D-axis current increases
                                                                     with speed.
     20

     10

       0
Iq




     -10

     -20

     -30

     -40
       -40   -30   -20   -10       0      10     20    30   40
                                  I
                                   d


                                                                                         47
        Experimental Testing and Data
        Full mSCR
                                       Full mSCR
                800
                700
                600                                              •   D-axis current controlled to
Speed [rpm]




                500                                                  be zero.
                400
                300                                              •   Phase advanced as speed
                200
                100
                                                                     increases.
                  0
                   0              50   100           150   200

                    40
                    30
                    20
  Current [A]




                    10
                     0
                   -10
                   -20
                          Iq
                   -30
                   -40    I
                      0       d   50   100           150   200
     Phase [deg]




                    30
                                        Time [s]
                    20
                    10
                     0
                      0           50   100           150   200
                                          Time [s]
                                                                                          48
    Experimental Testing and Data
    Full mSCR


                                    Full mSCR                       •   In the postive torque
          40
                                                                        quadrant, Id is effectively
          30                                                            regulated.
                                                                    •   Negative torque still needs
          20
                                                                        work, but it’s better than
          10                                                            open-loop.
I q [A]




            0

          -10

          -20

          -30

          -40
            -40   -30   -20   -10       0       10   20   30   40
                                      I [A]
                                       d


                                                                                              49
Future Work
•   Range testing (or directly measure energy consumption) with SCR vs.
    mSCR in real-world use.

•   Controlled dynamometer experiment of SCR vs. mSCR transient torque
    response, to verify simulations. (Requires high-speed data acquisition.)

•   Sensorless control using a state observer for rotor position.

•   Fault detection and recovery to increase controller robustness, possibly
    using sensorless control as a “back-up” in the event of sensor failure.

•   More high-speed testing.

•   Larger-scaled motor and controllers.




                                                                               50
Questions / Feedback




                       51
References
[1] J.R. Mevey. Sensorless Field Oriented Control of Brushless Permanent
Magnet Motors. M.S. Thesis. Kansas State University, Manhattan, 2009.
[2] J.L Kirtley. Permanent Magnet ‘Brushless DC’ Motors. Chapter 7 of Course
Notes for 6.685 - Electric Machines. Massachusetts Institute of Technology,
Cambridge, 2005.
[3] A. Hughes. Electric Motors and Drives: Fundamentals, Types, and
Applications. Third Edition. Newness, TK, 2005.
[4] T.M. Rowan, R.J. Kerkman. “A new synchronous current regulator and an
analysis of current-regulated PWM inverters,” IEEE Transactions on Industry
Applications, vol. IA-22, no. 4, pp. 678-690, Jul./Aug. 1986.
[5] F. Briz, M.W. Degner, R.D. Lorenz. “Analysis and Design of Current
Regulators Using Complex Vectors.” IEEE Transactions on Industry
Applications, vol. 36, no. 3, pp. 817-825, May/Jun. 2000.




                                                                               52
Motor Control Overview
•   Electric motors convert electrical power (voltage, current) to mechanical power
    (torque, speed), with some power lost as heat in the motor.



            I
                                                I
                               L       R              Kt I
       +                                    +
      V                                 E
        -                                   -       E  Kt 
                                                                      τ, Ω


                               Brushed DC Motor Model


•   The torque constant (Kt) and back EMF constant are identical due to power
    conservation. The conversion from current and back EMF to torque and speed
    is lossless; all losses are accounted for externally.

                                                                               53
Motor Control Overview
•   A brushed DC motor can be modeled as a SISO system (voltage to speed)
    with an internal feedback loop of back EMF:




     V +            1        I             τ                    Ω
                                    Kt             Gload (s )
        -         Ls  R
                             E
                                    Kt




                                                                            54
Motor Control Overview
•   A current control loop provides the ability to command torque. Current is
    directly proportional to torque, and easy to measure.
•   Depending on the load, an integral controller may be sufficient to track the
    reference current with zero steady-state error.




Ir +   -     Ki     V
             s          +       1        I                  τ
            Gc(s)                                  Kt              Gload (s )
                    -         Ls  R
                                             E                       Ω
                                                   Kt

                                                 Plant, Gp(s)

                                                                                   55
Current Sensing
Overview
                                                      1kHz Sampling
                           Digital                                    Analog

                         Rotor Position
                           Estimator
                                 θ
                                             Trigger
                          Latch Value
                                 θlatch

   Id                                                                          Ia
                           dq                     -

                                                  -
   Iq                            abc                                           Ic

           Digital LPF   Park Transform   I a  Ib  Ic  0      Analog LPF



                                                                                    56
Current Sensing
Analog Filtering: Second-Order Low Pass

    1. Buffered output filter on ACS714 Hall effect current sensor.
    2. Local 2:1 voltage divider and RC filter at ADC pin.


      ACS714 Current Sensor                                           STM32F103


      I               1.7k                   R2
           Signal                                                     ADC
           Cond.                                                       In
                                                  R2    C2


                                 CF



                    1  1                    1  1.7k C F 
           F (s)  
                     s  1   s  1 
                                      
                    1        2              2  1 R2C2
                                                      2
                                                                                  57
Current Sensing
Analog Filtering: Second-Order Low Pass

    •   The goal is to do as little filtering of the AC current signal as
        possible, so as not to distort the phase of the current. (Less than 5º
        phase lag desireable.)
    •   The PWM frequency (15,625Hz) is an obvious target for filtering.
         1. Actual current ripple will be at this frequency.
         2. Power transient-induced noise will be here, too.
    •   The filtering after the Park Transform can be much more aggressive,
        so noise in the AC current signal is acceptable.
    •   Component Selection:
          C F  10nF
           R2  10k                                         1  1 
                                                              s  1   s  1 
                                                    F (s)                    
          C2  10nF
                                                             1        2       
           1  1.7k 10nF   17 s
            2  1 (10k)(10nF )  50 s
                 2
                                                                                     58
Current Sensing
Analog Filtering: Second-Order Low Pass

                                         Current Sensor AC Filter, F(s)
                         0

                       -20
     Magnitude (dB)




                       -40

                       -60                                                 PWM Frequency
                                                                            -20dB Filtering
                       -80

                      -100
                         0

                       -45
     Phase (deg)




                       -90                 Maximum Commutation Frequency
                                                   4º Phase Lag
                      -135

                      -180
                            2        3             4              5              6                 7
                         10     10             10          10               10                10
                                             Frequency (rad/sec)                                       59
Current Sensing
Digital Filtering: First-Order Low Pass

    •   The digital filter acts on Id and Iq, the outputs of the Park transform.
    •   At steady-state, these are DC quantities. The filter time constant can
        be much slower than the commutation frequency.
    •   The bandwidth lower limit is driven by the target performance of the
        current (torque) controller.
    •   The bandwidth upper limit is driven by the sampling frequency. The
        filter time constant should be much longer than the sampling
        interval.
    •   Where Δt is the sampling interval, a first-order digital low pass filter
        on Id and Iq can be implemented with the following difference
        equations:
                                                 Equivalent continuous time constant:
                                         
         I q  a  I q 1  (1  a )  I q
           n         n
                                                                a
                                                         d       t
         I d  a  I d 1  (1  a )  I d
           n         n
                                                               1 a
                                                                                   60
Current Sensing
Digital Filtering: First-Order Low Pass

    •   Parameter Selection:

              t  1ms             a         0.95     
                             d        t       1ms   19ms
              a  0.95             1  a   0.05       
    •   The filter time constant is significantly longer than the sampling
        interval, so a “continuous time” analysis is appropriate:

                                             1
                                 H (s) 
                                          d s 1

    •   The bandwidth is 1/τd, 52.6rad/s, or 8.38Hz.




                                                                             61
Simplified Plant
Closed-Loop Transfer Function and Root Locus


                          +                      Kq                 1
               Iqr                                                                    Iq
                                          Iqe     s         Vq      Ra
                              -
                                                 Gc(s)              Gp(s)


                                                             1
                                           Iqf            d s 1
                                                           H(s)
                                                                                 jω


                                  Kq                                   ζ=0.707
L( s )  Gc G p H 
                          Ra s d s  1                                                  σ
                 Gc G p                   K q d s  K q
Gcl ( s )                        
              1  Gc G p H            Ra d s 2  Ra s  K q                                   62
Controller Simulations
A more fair transient response comparison.

                -
                        Kd   ϕ
          0              s             Kd = 1.0 rad/A/s       (Is this fair?)
                    +
          +
                             |V|
                        Kq
           -                           Kq = {1.2, 1.6, 2.5} V/A/s
                         s




               Iq Id

One possible way to make a more fair comparison is by using the initial voltage
vector to normalize the new d-axis gain:

               Kq

                                    Kd 
                                             1.2   1.6 2.2 rad
                                                    V0       A s
                        Kd
        V0
                                                                                63
  Controller Simulations
  A more fair transient response comparison.




                      SCR: Vector Step Response, Voltage                                          mSCR Vector Step Response, Voltage
          9                                                                           9
                                                    K = K = 1.2 V/A/s                                                             K = 1.2 V/A/s
                                                     q   d                                                                         q
                                                    K = K = 1.6 V/A/s                                                             K = 1.6 V/A/s
         8.5                                         q   d                           8.5                                           q
                                                    K = K = 2.5 V/A/s                                                             K = 2.5 V/A/s
                                                     q   d                                                                         q

          8                                                                           8
Vq [V]




                                                                            Vq [V]
         7.5                                                                         7.5


          7                                                                           7


         6.5                                                                         6.5


          6                                                                           6
          -3   -2.5         -2       -1.5      -1            -0.5       0             -3   -2.5         -2       -1.5      -1          -0.5        0
                                    V [V]                                                                       V [V]
                                     d                                                                           d




                                                                                                                                              64
 Controller Simulations
 A more fair transient response comparison.




                           SCR: Vector Step Response, Current                                           mSCR Vector Step Response, Current
          34                                                                           34

          32                                                                           32

          30                                                                           30

          28                                                                           28

          26                                                                           26




                                                                             I q [A]
I q [A]




          24                                                                           24

          22                                                                           22

          20                                                                           20
                      K = K = 1.2 V/A/s                                                        K = 1.2 V/A/s
                       q     d
                                                                                       18          q
          18          K = K = 1.6 V/A/s
                       q     d                                                                 K = 1.6 V/A/s
                                                                                                   q
          16          K = K = 2.5 V/A/s                                                16
                       q     d                                                                 K = 2.5 V/A/s
                                                                                                   q
          14                                                                           14
           -10   -8     -6       -4   -2     0     2   4        6   8   10              -10   -8       -6   -4    -2     0     2   4     6   8        10
                                           I [A]                                                                       I [A]
                                           d                                                                           d




                                                                                                                                                 65
  Controller Simulations
  A more fair transient response comparison.




                              SCR: Step Response, Torque                                                   mSCR Step Response, Torque
              5.5                                                                           5.5
               5                                                                             5
              4.5                                                                           4.5
               4                                                                             4
              3.5                                                                           3.5




                                                                              Torque (Nm)
Torque [Nm]




               3                                                                             3
              2.5                                                                           2.5
               2                                                                             2
              1.5                                                                           1.5
                                                   Kq = Kd = 1.2 V/A/s                                                                  Kq = 1.2 V/A/s
               1                                   K = K = 1.6 V/A/s                         1
                                                     q     d                                                                            Kq = 1.6 V/A/s
              0.5                                  Kq = Kd = 2.5 V/A/s                      0.5                                         Kq = 2.5 V/A/s
               0                                                                             0
               15   15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9         16                  15   15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9           16
                                       Time [s]                                                                      Time [s]




                                                                                                                                                  66
High Speed Operation
•   Sensing and control becomes more difficult as speed increases:
     •   ωL ≈ R, large phase angle.
     •   Significant lag due to current sensing / AC-side filtering.
     •   Analysis of digital effects (sampling, fitlering) becomes important.


                                           •   Poles: 2

                                           •   Max Speed: 35,000RPM
                                               (without field weakening)
                                               ω = 3,665rad/s, f = 583Hz

                                           •   Current sensor phase lag with
                                               components specified: ~20º!




                                                                                67
High Speed Operation
                                      4
                                   x 10
                              4

                              3
                Speed [rpm]


                              2

                              1

                              0
                              205         210    215     220     225         230         235     240    245    250   255

                              60
        Phase Angle (deg)




                              45

                              30

                              15

                              0
                              200         205   210    215     220     225         230     235    240    245   250   255

                              40
                                                                                                                     Iq
     Current [A]




                              20                                                                                     Id

                              0

                            -20
                              200         205   210    215     220     225    230          235    240    245   250   255
                                                                        Time [s]




                                                                                                                           68
Error Handling and Failsafes
•    Hall effect sensor failure presents a significant risk to the controller.

Failure Mode                         Effect                              Countermeasure
The entire sensor cable becomes      Comlete loss of ability to          Pull-down resistors take the
unplugged.                           commutate the motor.                sensor state to {0,0,0}, which is
                                                                         invalid. The output driver shuts
                                                                         down. Motor coasts.
Transient sensor glitch.             An unexpected state transition,     If new state is not as expected,
< 1/6 cycle (single sensor glitch)   resulting in large current/torque   trust rotor speed interpolation for
                                     transient when voltage vector is    the next 60º segment.
                                     applied at the wrong angle.
Permanent sensor failure.            Repeated loss of two states per     Follow same rules as above, but
> 1/6 cycle                          cycle.                              with a counter that talleys
                                                                         unexpected state transitions per
                                                                         unit time. If larger than some
                                                                         threshold, shut down.
                                                   …
 •   Sensorless or hybrid techniques will significantly change the FMEA.
 •   Future work: Ability to switch to sensorless control if a Hall effect sensor
     fault is detected.
                                                                                                       69
Connection to Adaptive Feed-Forward Cancellation
•   The SCR and mSCR are applications of adaptive feed-forward cancellation
    (AFC) to three-phase variables.
•   In one implementation of AFC, a feed-forward path allows for zero-error
    tracking of a sinusoidal input at a specific frequency:




            Reference:
            Cattell, Joseph H. Adaptive Feedforward Cancellation Viewied from an
            Oscillator Amplitude Control Perspective. S.M. Thesis, Massachusetts
            Institute of Technology, 2003.


                                                                                   70
Connection to Adaptive Feed-Forward Cancellation
•   By manipulating the block diagram of a the SCR, focusing on the amplitude of
    a single phase of current, the SCR can be related to single-oscillator AFC (not
    proven here).

•   The modified SCR is related to single-oscillator AFC with a phase advance
    offset, which has been proven to improve transient response.


                                                        •   In both cases, the Park
                                                            Transform provides the
                                                            sinuosoidal multiplier for the
                                                            input and output.

                                                        •   In AFC with phase advance, ϕi
                                                            is set as the plant phase angle
                                                            (initial voltage vector angle).

    Reference:
    Cattell, Joseph H. Adaptive Feedforward Cancellation Viewied from an
    Oscillator Amplitude Control Perspective. S.M. Thesis, Massachusetts
    Institute of Technology, 2003.
                                                                                             71

				
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