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

Fig. 220 Electrical system of a 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 in turn is expressed as a fourier series of the electrical rotor angle \(\theta_\mathrm{e}\):

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

\[k_\mathrm{e,a}(\theta_\mathrm{e}) = \sum_n K_{\mathrm{c},n}\cos(n\theta_\mathrm{e}) + K_{\mathrm{s},n}\sin(n\theta_\mathrm{e})\]

\[k_\mathrm{e,b}(\theta_\mathrm{e}) = \sum_n K_{\mathrm{c},n}\cos(n\theta_\mathrm{e}-\frac{2\pi n}{3}) + K_{\mathrm{s},n}\sin(n\theta_\mathrm{e}-\frac{2\pi n}{3})\]

\[k_\mathrm{e,c}(\theta_\mathrm{e}) = \sum_n K_{\mathrm{c},n}\cos(n\theta_\mathrm{e}+\frac{2\pi n}{3}) + K_{\mathrm{s},n}\sin(n\theta_\mathrm{e}+\frac{2\pi n}{3})\]

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:

\[L_\mathrm{a}(\theta_\mathrm{e}) = L_0 - M + \sum_n L_{\mathrm{c},n}\cos(n\theta_\mathrm{e}) + L_{\mathrm{s},n}\sin(n\theta_\mathrm{e})\]

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:

\[T_\mathrm{e} \;= \sum_{x = \mathrm{a,b,c}} k_{\mathrm{e},x}i_x + \frac{p}{2}\frac{dL_x}{d\theta_\mathrm{e}}i_x^2\]

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

\[T_\mathrm{cog}(\theta_\mathrm{e})\; =\; \sum_n T_{\mathrm{c},n}\cos(n\theta_\mathrm{e}) + T_{\mathrm{s},n}\sin(n\theta_\mathrm{e})\]

Mechanical System

Mechanical rotor speed:

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

Mechanical and electrical rotor angle:

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

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

Parameters

Back EMF shape coefficients

Fourier coefficients \(K_{\mathrm{c},n}\) and \(K_{\mathrm{s},n}\) of the back EMF shape function \(k_\mathrm{e, a}(\theta_\mathrm{e})\) in \((\mathrm{Vs})\).

Stator resistance

The stator resistance \(R\) in ohms \((\Omega)\).

Stator inductance

The constant inductance \(L_0 - M\) and the fourier coefficients \(L_{\mathrm{c},n}\), \(L_{\mathrm{s},n}\) of the phase a inductance \(L_\mathrm{a}(\theta_\mathrm{e})\) in henries \((\mathrm{H})\).

Cogging torque coefficients

Fourier coefficients \(T_{\mathrm{c},n}\), \(T_{\mathrm{s},n}\) of the cogging torque \(T_\mathrm{cog}(\theta_\mathrm{e})\) in \((\mathrm{Nm})\).

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 \((\mathrm{V})\).

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 \((\mathrm{Nm})\).

Cogging torque

The cogging torque \(T_\mathrm{cog}\) 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.