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Brushless DC Machine

Purpose

Detailed model of a brushless DC machine excited by permanent magnets.

Library

Machines

Description

[Picture]

A brushless DC machine is a type of permanent magnet synchronous machine in which the back electromotive force (EMF) is not sinusoidal but has a more or less trapezoidal shape. Additionally, the variation of the stator inductance with the rotor position is not necessarily sinusoidal.

The machine operates as a motor or generator; the sign of the mechanical torque determines the mode of operation (positive for motoring, negative for generating). In the component icon, phase a of the stator winding is marked with a dot.

Electrical System

pict

The back EMF voltages are determined by a shape function ke and the mechanical rotor speed !m. The shape function in turn is expressed as a fourier series of the electrical rotor angle e:

ex(  e;!m) = ke;x( e)  !m

        X
ke;a(  e) =   Kc;n cos(n  e)+ Ks;nsin(n  e)
         n

        X               2  n                2  n
ke;b(  e) =  Kc;n cos(n  e   -3--)+ Ks;nsin(n  e   --3-)
         n

        X               2--n-               2---n
ke;c(  e) =  Kc;n cos(n  e + 3  )+ Ks;nsin(n  e +  3 )
         n

The stator self inductance is also expressed as a fourier series of the electrical rotor angle. The mutual inductance M between the stator phases is assumed to be constant. Since the stator windings are star connected, the mutual inductance can simply be subtracted from the self inductance:

La(  e) = L0  M  + X  Lc;n cos(n  e)+ Ls;n sin(n  e)
                  n

Electromechanical System

The electromagnetic torque is a superposition of the torque caused by the permanent magnet and a reluctance torque caused by the non-constant stator inductance:

T  =  X   k  i + p dLxi2
 e   x=a;b;c e;x x  2 d  e x

The cogging torque is again expressed as a fourier series of the electrical rotor angle:

           X
Tcog(  e) =    Tc;ncos(n  e) + Ts;nsin(n  e)
            n

Mechanical System

Mechanical rotor speed:

!m =  1-(Te + Tcog( e)   F!m    Tm)
      J

Mechanical and electrical rotor angle:

    = !
m     m

e = p    m

Parameters and Dialog Box

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Back EMF shape coefficients
Fourier coefficients Kc;n and Ks;n of the back EMF shape function ke;a(e) in volts per second (Vs1).
Stator resistance
The stator resistance R in ohms (_O_).
Stator inductance
The constant inductance L0 M and the fourier coefficients Lc;n, Ls;n of the phase a inductance La(e) in henries (H).
Cogging torque coefficients
Fourier coefficients Tc;n, Ts;n of the cogging torque Tcog(e) in Nm.
Inertia
Combined rotor and load inertia J in Nms2.
Friction coefficient
Viscous friction F in Nms.
Number of pole pairs
Number of pole pairs p.
Initial rotor speed
Initial mechanical speed !m;0 in radians per second (s1).
Initial rotor angle
Initial mechanical rotor angle m;0 in radians.
Initial stator currents
A two-element vector containing the initial stator currents ia;0 and ib;0 of phase a and b in amperes (A).

Inputs and Outputs

Mechanical torque
The input signal Tm represents the mechanical torque at the rotor shaft, in Nm.

The output vector "m" contains the following 7 signals:

(1) Rotor speed
The rotational speed !m of the rotor in radians per second (s1).
(2) Rotor position
The mechanical rotor angle m in radians.
(3) Electrical torque
The electrical torque Te of the machine in Nm.
(4) Cogging torque
The cogging torque Tcog of the machine in Nm.
(5-7) Back EMF voltages
The back EMF voltages ea, eb, ec in volts (V).