Brushless DC Machine (Simple)

Purpose

Simple model of brushless DC machine excited by permanent magnets

Library

Electrical / Machines

Description

The simplified Brushless DC Machine is a model of a permanent magnet synchronous machine with sinusoidal or trapezoidal back EMF.

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

../../_images/bldcsimple_schema.svg — Fig. 221 Electrical system of a (simple model) brushless DC machine

The back EMF voltages are determined by a shape function \(k_\mathrm{e}\) and the mechanical rotor speed \(\omega_\mathrm{m}\). The shape function is a sinusoidal or an ideal trapezoidal function scaled with the back EMF constant \(K_\mathrm{E}\).

\[e_\mathrm{x}(\theta_\mathrm{e},\omega_\mathrm{m}) = k_{\mathrm{e},x}(\theta_\mathrm{e}) \cdot \omega_\mathrm{m}\]

../../_images/sinusoidal.svg — Fig. 222 Sinusoidal shape function

../../_images/trapezoidal.svg — Fig. 223 Ideal trapezoidal shape function

Electromechanical System

The electromagnetic torque is:

\[T_\mathrm{e} \;= \sum_{x = \mathrm{a,b,c}} k_{\mathrm{e},x}i_x\]

Mechanical System

Mechanical rotor speed:

\[\dot{\omega}_\mathrm{m}\; =\; \frac{1}{J} \left( T_\mathrm{e} - F\omega_\mathrm{m} - T_\mathrm{m} \right)\]

Mechanical and electrical rotor angle:

\[\dot{\theta}_\mathrm{m}\; =\; \omega_\mathrm{m}\]

\[\theta_\mathrm{e}\; =\; p \cdot \theta_\mathrm{m}\]

Parameters

Back EMF shape: Choose between sinusoidal and trapezoidal back EMF.
Back EMF constant: The back EMF constant \(K_\mathrm{E}\) in \((\mathrm{Vs})\).
Stator resistance: The stator resistance \(R\) in ohms \((\Omega)\).
Stator inductance: The stator inductance \(L - M\) in henries \((\mathrm{H})\).
Inertia: Combined rotor and load inertia \(J\) in \((\mathrm{Nms^2})\).
Friction coefficient: Viscous friction \(F\) in \((\mathrm{Nms})\).
Number of pole pairs: Number of pole pairs \(p\).
Initial rotor speed: Initial mechanical speed \(\omega_\mathrm{m,0}\) in radians per second \(\left( \frac{\mathrm{rad}}{\mathrm{s}} \right)\).
Initial rotor angle: Initial mechanical rotor angle \(\theta_\mathrm{m,0}\) in radians.
Initial stator currents: A two-element vector containing the initial stator currents \(i_\mathrm{a,0}\) and \(i_\mathrm{b,0}\) of phase a and b in amperes \((\mathrm{A})\).

Probe Signals

Stator phase currents: The three-phase stator winding currents \(i_\mathrm{a}\), \(i_\mathrm{b}\) and \(i_\mathrm{c}\), in amperes \((\mathrm{A})\). Currents flowing into the machine are considered positive.
Back EMF: The back EMF voltages \(e_\mathrm{a}\), \(e_\mathrm{b}\), \(e_\mathrm{c}\) in volts \((\mathrm{V})\).
Stator flux (dq): The stator flux linkages \(\Psi_\mathrm{d}\) and \(\Psi_\mathrm{q}\) in the stationary reference frame in \((\mathrm{Vs})\).
Rotational speed: The rotational speed \(\omega_\mathrm{m}\) of the rotor in radians per second \(\left( \frac{\mathrm{rad}}{\mathrm{s}} \right)\).
Rotor position: The mechanical rotor angle \(\theta_\mathrm{m}\) in radians.
Electrical torque: The electrical torque \(T_\mathrm{e}\) of the machine in \((\mathrm{Nm})\).

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.