Brushless DC Machine
Purpose
Detailed model of brushless DC machine excited by permanent magnets
Library
Electrical / Machines
Description
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; if the mechanical torque has the same sign as the rotational speed the machine is operating in motor mode, otherwise in generator mode. In the component icon, phase a of the stator winding is marked with a dot.
Electrical System
The back EMF voltages are determined by a shape function and the mechanical
rotor speed
. The shape function in turn is expressed as a fourier series of the
electrical rotor angle
:
The stator self inductance is also expressed as a fourier series of the
electrical rotor angle. The mutual inductance 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:
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:
The cogging torque is again expressed as a fourier series of the electrical rotor angle:
Mechanical System
Mechanical rotor speed:
Mechanical and electrical rotor angle:
Parameters
- Back EMF shape coefficients
- Fourier coefficients
and
of the back EMF shape function
in Vs.
- Stator resistance
- The stator resistance
in ohms (
).
- Stator inductance
- The constant inductance
and the fourier coefficients
,
of the phase a inductance
in henries (H).
- Cogging torque coefficients
- Fourier coefficients
,
of the cogging torque
in Nm.
- Inertia
- Combined rotor and load inertia
in
.
- Friction coefficient
- Viscous friction
in Nms.
- Number of pole pairs
- Number of pole pairs
.
- Initial rotor speed
- Initial mechanical speed
in radians per second (
).
- Initial rotor angle
- Initial mechanical rotor angle
in radians.
- Initial stator currents
- A two-element vector containing the initial stator currents
and
of phase a and b in amperes (A).
Inputs and Outputs
- Mechanical torque
- The input signal
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
of the rotor in radians per second (
).
- (2) Rotor position
- The mechanical rotor angle
in radians.
- (3) Electrical torque
- The electrical torque
of the machine in Nm.
- (4) Cogging torque
- The cogging torque
of the machine in Nm.
- (5-7) Back EMF voltages
- The back EMF voltages
,
,
in volts (V).
Probe Signals
- Stator phase currents
- The three-phase stator winding currents
,
and
, in A. Currents flowing into the machine are considered positive.
- Back EMF
- The back EMF voltages
,
,
in volts (V).
- Rotational speed
- The rotational speed
of the rotor in radians per second (
).
- Rotor position
- The mechanical rotor angle
in radians.
- Electrical torque
- The electrical torque
of the machine in Nm.
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.