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The most fundamental parameters of a brushless permanent-magnet motor drive are the number of phases and the method of controlling the current. The drive circuit configuration depends primarily on the number of phases. Broadly speaking there is a division into two groups:

(1) Low-cost brushless DC drives with one or two phases, widely used in blowers and fans; and (2) Polyphase brushless DC drives with smooth torque and high efficiency.

Group 1 drives often use single-ended drive circuits, sometimes with bifilar windings. Group 2 drives use bridge circuits for high efficiency and maximum controllability. In some instances, full H-bridge circuits are used with 1-phase or 2-phase motors to achieve high efficiency. With 1 or 2 phases, it is generally not possible to achieve reliable starting in both directions from any position, unless auxiliary methods are adopted, including tapered airgaps and reluctance poles, [16].

The simplest type of “drive” is a plain AC voltage source with 2 or 3 phases. In the steady state, a motor operating from such a supply can be represented by its phasor diagram, provided that it is sinewound or approximately so. Even before the motor is connected to the supply, the open-circuit phasor diagram can be drawn as shown in Fig. 2.26. Since the motor is on open-circuit there is no current, and the phasor of the generated EMFE leads the phasor of the fundamental flux-linkage QQQQ_1Md by 90E. The fundamental flux-linkage Q_1Md is the product of the effective number of series turns/phase k_w1T_ph and the fundamental component of magnet flux in the airgap M1Md, where kw1 is the fundamental harmonic

winding factor and Tph is the number of series turns per phase.5 Thus if B1Md is the peak value of the

fundamental flux-density produced by the magnet on open-circuit,

and is centred on the d-axis. The subscript 1 refers to the fundamental space-harmonic component. The phasor relationship between E and QQQQ_1Md is

where the subscript 1 emphasizes the fundamental, q the q-axis, d the d-axis, and M the magnet. The 90E phase lead comes from Faraday’s law, when the phasors are expressed as complex numbers, and it is normal to consider the magnet flux-linkage phasor to be aligned with the d-axis while the generated EMF is aligned with the q-axis.6

7_{This definition of power factor is valid only if the voltage and current are sinusoidal.}

8_{Also known as “amortisseur” windings, these are usually in the form of a cast cage similar to the cage of an induction motor} rotor. See, for example, Jordan [1983], Miller [1984].

Fig. 2.28 Phasor diagram Fig. 2.27 Circuit diagram of one phase of nonsalient pole brushless PM motor, with AC voltage supply The connection of the motor to a balanced AC voltage source is shown in

Fig. 2.27. If the motor is nonsalient pole it can be represented electrically by its generated EMF E, its phase resistance R and its synchronous reactance X_s (which is equivalent to X_d in a salient-pole machine). The AC voltage source is assumed to have a constant voltage V and no internal impedance. The phasor diagram for the circuit in Fig. 2.27 is shown in Fig. 2.28. It shows that the supply voltage V comprises the series combination of the volt-drops RI, jXsI, and E, which must be added

“vectorially” because they are not in phase with one another. The volt- drop RI is in phase with the current I and its phasor is parallel to the current phasor, but the volt-drop jX_sI is at right-angles to the current and leads it in phase because Xs is an inductive impedance.

The angles shown in Fig. 2.28 are important. The angle N between V and I is the power-factor angle such that the power factor is cos N.7 _{The angle} ( between E and I is the “torque angle”, which is very important in drives which control the phase and magnitude of the current relative to the shaft position. The angle * between V and E is called the “load angle” and is also sometimes called the “torque angle” especially when the motor is operating from an AC voltage source (i.e., without current control).

Operation from an AC voltage source is useful for understanding the basic concepts of the phasor diagram, but in practice it is very rare. Brushless PM motors operated in this way are inherently unstable and tend to “hunt” or oscillate about the synchronous speed unless they are fitted with damper windings on the rotor.8 _{Moreover, without the damper winding the motor has no means of} starting. With damper windings, such motors are referred to as “line-start” motors. They are used in certain applications where several motors operate in synchronism from a single inverter, or direct from the mains supply where, for some reason, exact synchronous speed is required. Several attempts have been made to develop such machines to compete with induction motors on the basis of high efficiency, even with single-phase capacitor motors, and although some very good technical results have been reported, there has been little commercial development because of the cost and certain application problems.

9_{AC induction motors do not have obvious d,q axes because (i) both the stator and the rotor are cylindrical, i.e., rotationally} symmetric; and (ii) there is no distinct excitation winding. However, dq axis theory still applies to the induction motor. See, for example, Fitzgerald and Kingsley, [1961]. In descriptions of field-oriented control of induction motors, since about 1975, the labels d,q are applied extensively, but several different frames of reference are used and there is probably no single definition of d,q axes that would work equally well for all variants. In dealing with PM brushless machines it is natural to fix the dq axes to the rotor, as in the wound-field synchronous machine of Fig. 2.29, but this is not the only convention that can be adopted.

Fig. 2.29 Wound-field synchronous machine Fig. 2.30 Surface-magnet motor

Fig. 2.31 Interior-magnet motor