Three-Phase Motors (Electric Motors)

10.11.2
Three-phase motors overwhelmingly dominate all others. The exact reasons for this dominance are not known, but the historical dominance of three-phase induction and synchronous motors and the minimal number of power electronic devices required are likely contributing factors. The addition of a third phase provides an additional degree of freedom over the two-phase motor, which manifests itself in more drive schemes and terminology. For example, wye (Y) and delta (A) connections are possible.
In three-phase motors, the power balance equation leads to

FIGURE 10.79 Square-wave back-emf waveforms for a three-phase motor.
Three-Phase-On Operation. The most obvious drive scheme for the three-phase motor is to extend the two-phase-on operation of the two-phase motor, as shown in Fig. 10.80. Here each phase conducts current at all times and contributes equally to the torque at all times. At each commutation point, one phase current changes sign and the others remain unchanged. The important aspects previously listed for the two-phase-on two-phase motor apply here as well.
Despite the conceptual simplicity of this drive scheme, it is hardly ever implemented in practice because three H bridges as shown in Fig. 10.75 are required, one for each phase winding. The resulting 12 power electronic devices make the drive expensive compared with other drive schemes.
Y Connection. Just as the Y connection is a popular configuration in three-phase power systems, it is also the most common configuration in three-phase brushless PM motors. As shown in Fig. 10.81, the center or neutral of the Y is not brought out, each external terminal or line is connected to a half-bridge circuit, and the collection of three half bridges is called a three-phase bridge. In this way, an H bridge appears

FIGURE 10.80 Three-phase-on operation.

Torque production follows the idea that current should flow in only two of the three phases at a time, and that there should be no torque production near the back-emf sign crossings. Figure 10.82 shows the phase currents superimposed on the back-emfs. Each phase conducts currents over the central 2n/3 rad electrical of each half cycle. The resulting torque is shown at the bottom of the figure with the letter designating the current polarities contributing to the torque. At each commutation point, one switch remains closed, one opens, another closes, and the rest remain open. There are six commutations per electrical period, and thus this drive scheme is often called a six-step drive (Murphy and Turnbull, 1988). The six numbered arrows shown in Fig. 10.81 illustrate these steps, as do the respective circled step numbers in Fig. 10.82.
Because only two phases are conducting current and contributing to torque production at any one time, the amplitude of the current must be 50 percent larger here

FIGURE 10.81 Y-connected three-phase motor and drive circuitry.
than in the three-phase-on case, where all three phases contribute simultaneously. When two phases are called upon to produce the same torque that three phases do, the current in each phase must be 3/2 as large, since (3/2)(2 phases) = (1)(3 phases). As a result, if this drive scheme is implemented, the current equations must be modified to reflect the current waveforms shown in Fig. 10.82.
The RMS phase currents are required to produce a specified rated torque. Based on the preceding discussion, these currents must be increased in amplitude by a factor of 3/2. Moreover, the equations must reflect the RMS value of the phase currents,

for the six-step driven three-phase motor. Compared with the three-phase-on case, the RMS phase current is approximately 22 percent larger and the ohmic motor loss is 50 percent greater. Thus, while the Y connection minimizes the number of power electronic devices used, it does not minimize losses.
To summarize, important aspects of this drive scheme include the following:

FIGURE 10.82 Torque production in a Y-connected three-phase motor.
• Ideally, constant ripple-free torque is produced.
• Only six switches are required, which is a minimum number.
• Phases are not required to produce torque in regions where their associated back-emf is changing sign. Thus, the back-emf can be more trapezoidal than square.
• Each phase contributes an equal amount to the total torque produced. Thus, each phase experiences equal losses, and the drive electronics are identical for each phase.
• Copper utilization is 67 percent, since at any one time only two of the three phases are conducting current.
• For the same output, ohmic motor losses are 50 percent greater than those in the three-phase-on drive scheme.
• The amount of torque produced can be varied by changing the amplitude of the square-wave currents.
^ Impossible-to-produce 120°-wide square-wave currents are required. The inherent finite rise and fall time of the current creates torque ripple, commonly called commutation torque ripple.
• Independent control of phase currents is not possible.

monics that are multiples of three, that is, triple-n or triplen harmonics (Murphy and Turnbull, 1988; Kassakian, Schlecht, and Verghese, 1991).
• Because phase windings appear in series, the supply voltage must be greater than the vector sum of the back-emfs at rated speed.
Delta Connection. The delta connection shown in Fig. 10.83 is the dual of the Y connection.This connection is not that popular because it has a major weakness, that being the additional ohmic motor loss and torque ripple due to circulating currents flowing around the delta. Three-phase power-system utility generators are never delta-connected for this reason. It is relatively easy to show that if the back-emf waveforms of each phase do not have exactly the same shape, are not exactly 120° out of phase with one another, or contain any triplen harmonics, circulating currents will flow around the delta. Because of this weakness, connected motors appear only in lower-performance motors at low-output power levels (e.g., in the fractional horsepower range), where their higher losses can be offset with lower material costs
(Miller, 1989).

FIGURE 10.83 Delta-connected three-phase motor and drive circuitry.
Based on the preceding discussion, a motor having the ideal square-wave back-emf shape as shown in Fig. 10.79 cannot be connected in the delta connection because a square-wave back-emf motor has very high triplen harmonic content. Given the nature of dual circuits, it is not surprising that swapping the current and back-emf waveforms of the Y connection gives a workable solution for the delta connection, as shown in Fig. 10.84. Creating a motor with 120°-wide square-wave

FIGURE 10.84 Torque production in a delta-connected three-phase motor.
back-emf waveforms is not difficult. Simply making the magnet arc narrower works, which results in the use of less magnet material.
To ease the explanation of the delta connection, the rising edges of the back-emfs and currents are aligned in Fig. 10.84. As shown, the back-emf of one phase is zero at all times. Each takes a turn at being zero for 60°. Because of this zero back-emf, the line current splits approximately equally through the remaining two phases, which conduct current in opposite directions. As before, the torque produced is given by applying Eq. (10.6). The lowercase letters under the torque curve signify the line currents during the respective commutation intervals. The line not given in each commutation interval is left floating electrically and is associated with the phase having zero back-emf. A comparison of these states with those of the Y connection in Fig. 10.82 show that the three-phase bridge circuit switches identically for both configurations. It is for this reason that the commutation logic in commercial driver ICs for small brushless motors works with either Y- or delta-connected motors.
To summarize, important aspects of this drive scheme include the following:
• Ideally, constant ripple-free torque is produced.
• Only six switches are required, which is a minimum number.
• Each phase contributes an equal amount to the total torque produced. Thus, each phase experiences equal losses, and the drive electronics are identical for each phase.
^ Copper utilization remains 67 percent, even though all three phases conduct current simultaneously. At all times, one phase is conducting current and adding to the ohmic motor loss but is not producing torque, since the back-emf is zero in each phase one-third of the time.
• The amount of torque produced can be varied by changing the amplitude of the square-wave currents.
• Impossible-to-produce square-wave currents are required. The inherent finite rise and fall time of the current creates torque ripple.
^ With all else being equal, ohmic motor losses are 50 percent greater than those in the Y connection, but the motor requires only two-thirds of the magnetic material
(Miller, 1989).
^ Just as in the Y-connected case, the phase current amplitude must be increased by 50 percent to make up for the fact that only two phases are producing the required torque. Since the phase currents are square waves, the current equation becomes

^ Compared with the three-phase-on case, ohmic motor losses are 125 percent greater.
• Independent control of phase currents is not possible.

monics that are multiples of three, that is, triple-n or triplen harmonics (Kassakian,
Schlecht, and Verghese, 1991).
• Because phases appear in parallel, the supply voltage need only be greater than the peak phase back-emfs at rated speed.
• The delta connection is traditionally found in low-power, lower-performance motors.
The Sine-Wave Motor. The sine-wave back-emf motor completes the discussion of three-phase motors. A three-phase motor with sinusoidal back-emf can be Y or delta connected because there are by definition no triplen harmonics. Excitation of a sinusoidal motor with sinusoidal current gives constant ripple-free torque just as the two-phase sinusoidal motor does. In this case, the back-emfs and currents are all offset from each other by 120° electrical. Following the notion used earlier, the torque is found by substitution into Eq. (10.6) and is given by

The simple elegance of Eqs. (10.3) and (10.8) is due to the pure sinusoidal content of back-emf and phase currents. Because of this elegance, a greater deal of work goes into the design of some motors to minimize the higher harmonics in the back-emf so that a sinusoidal drive can be implemented. The sinusoidal motor commonly appears in high-performance applications where high accuracy and minimal torque ripple are required.
As shown by Eq. (10.8), each phase produces torque proportional to half the peak value of the current and back-emf as compared with a unity ratio for the square-wave back-emf motor driven three-phase-on. Therefore, in a sinusoidal motor driven by sinusoidal currents, correction of the phse current equation is necessary to establish the rms phase current required to produce a specified torque. The