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Brushless DC Machine

Purpose

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

Library

Electrical / Machines

Description

pict

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 ωm  . The shape function in turn is expressed as a fourier series of the electrical rotor angle θe  :

ex(θe,ωm) = ke,x(θe)⋅ωm

        ∑
ke,a(θe) =   Kc,n cos(nθe)+ Ks,nsin(nθe)
         n

        ∑               2πn                2πn
ke,b(θe) =   Kc,n cos(nθe --3--)+ Ks,nsin(nθe ---3-)
         n

        ∑               2πn-               2πn-
ke,c(θe) =   Kc,n cos(nθe + 3  )+ Ks,nsin(nθe + 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(θe) = L0 - M + ∑ Lc,n cos(nθe)+ Ls,n sin(nθe)
                  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 =   ∑   k  i + p dLxi2
 e  x=a,b,c e,x x  2 dθe x

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

           ∑
Tcog(θe) =     Tc,ncos(nθe) + Ts,nsin(nθe)
            n

Mechanical System

Mechanical rotor speed:

ωm =  1-(Te + Tcog(θe)- Fωm - Tm)
      J

Mechanical and electrical rotor angle:

θ   = ω
 m     m

θe = p⋅θm

Parameters

Back EMF shape coefficients
Fourier coefficients Kc,n   and Ks,n   of the back EMF shape function ke,a(θe)   in Vs.
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(θe)   in henries (H).
Cogging torque coefficients
Fourier coefficients Tc,n  , Ts,n    of the cogging torque Tcog(θe)   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 ωm,0   in radians per second (s- 1  ).
Initial rotor angle
Initial mechanical rotor angle θm,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 ωm    of the rotor in radians per second (s-1  ).
(2) Rotor position
The mechanical rotor angle θm   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).

References

D.  Hanselman, "Brushless permanent magnet motor design, 2nd ed.", The Writers' Collective, Mar. 2003.
P. Pillay, R. Krishnan, "Modeling, simulation, and analysis of permanent-magnet motor drives, Part II: The brushless DC motor drive", IEEE Trans. on Ind. App., Vol. 25, No. 2, Mar./Apr. 1989.