**10.13 **

**10.13.1 **

**Relationship of Torque Constant K1 and Back-EMF Ke to Current Ipk and Irms Profiles**

There are two major approaches to driving a BLDC motor: square-wave current drives, also known as trapezoidal drives, and sine wave current drives also known

**TABLE 10.19 Y Commutation Sequence—Three-Phase Four-Pole**

Stator-rotor | Drive | ||

mechanical degrees | windings | Positive | Negative |

0-30 | DF | C | B |

30-60 | EF | A | B |

60-90 | ED | A | C |

90-120 | FD | B | C |

120-150 | FE | B | A |

150-180 | DE | C | A |

180-210 | DF | C | B |

210-240 | EF | A | B |

240-270 | ED | A | C |

270-300 | FD | B | C |

300-330 | FD | B | A |

330-360 | DE | C | A |

** **** **

**FIGURE 10.95 Y configuration: (a) excitation pattern, and (b) commutation pattern.**

as sinusoidal drives. Figure 10.97 describes the torque and current profiles for the trapezoidally driven BLDC motor. The current winding excitation requires that the individual phase currents be excited for 120° electrical to achieve a 60° electrical torque commutation zone, as shown in Fig. 10.97. The sinusoidally driven BLDC motor’s current and torque waveforms are shown in Fig. 10.98. A series of sinusoidally generated current waveforms create sinusoidally shaped torque waveforms.

In reviewing the available information concerning back-emf Ke, torque constant Kt, and torque and phase currents for both sinusoidal and trapezoidal motor drive combinations, there appear to be conflicting and confusing data. The trapezoidal motor drive produces more peak torque for the same peak current inputs, while the sinusoidal drive produces slightly more peak torque for the same RMS current. The major difference between the two drive schemes is unit price, with the sinusoidal

drive being the more expensive. Table 10.20 details the performance comparisons of both drive systems undertaken by Tomasek (1986), Welsh (1994), Comstock (1986), and Miller (1992).Table 10.20 also details the Ke, Kt, and RLL values from each motor drive when one has computed the individual phase K, and Rph values for a sinusoidal or trapezoidal BLDC motor.

How does one design a sinusoidal BLDC motor? The answer to that question may have been provided by Erland Persson (1989). Persson focused on varying the PM arc width or length for three commercially available laminations, a 12-slot, a 24-slot, and a 36-slot lamination with a four-pole PM arc segment rotor structure. The stator lamination ODs were 49 mm (1.929 in) OD for the 12- and 24-slot laminations

**TABLE 10.20 Ratios of Torque and Voltage Constants for Brushless Motors in Three-Phase Sine-Wave Systems, Motor Speed in krpm**

**FIGURE 10.97 Trapezoidal torque and current waveforms: (a) typical 3/0 brushless motor inverter, (b) composite torque function for a 3/0 BLDC motor, and (c) square-wave current waveforms for a 3/0 BLDC motor.**

**FIGURE 10.98 Sinusoidal torque and current waveforms: (a) principles of the electrical circuit, and (b) principles of torque generation.**

TABLE 10.21 Ratios of Torque and Voltage Constants for Brushless Motors in Three-Phase Sine-Wave Systems, Motor Speed in rad/s.

and 54 mm (2.126 in) OD for the 36-slot.The magnet arc widths were varied from 70 to 85° mechanical, and the actual measured torque versus position waveforms were recorded by using a plotter and mounted torque transducer. The stator stack axial lengths (SALs) were all 3.0 in long. Persson also tested stators with no slot skew and with either a 3/4-slot skew or a 1-slot skew.

Table 10.21 shows Persson’s data summary. Persson was interested in the torque ripple measured for the 18 stator slot and skew combinations.The impact of skew on reducing torque ripple is also demonstrated. The impact of more teeth per pole in reducing torque ripple was not demonstrated by the experimental data. There was no attempt to alter tooth shapes, air gaps, or magnet material, just to supply the reader with a reference point in terms of solid experimental data. Magnet material used was a low grade of ferrite magnet material, so there was no magnetic saturation in any of the motor soft iron (steel) members.

What is more astounding is the shape of the motor T versus 6 waveform. The

**FIGURE 10.99 Static torque function: 12 slots, 80° arc, no skew.**

**FIGURE 10.100 Static torque function: 12 slots, 80° arc, 3/-slot skew.**

commutation interval (angle) is 60° electrical as in the previous theoretical examples. Figures 10.99 and 10.100 show the static torque function for the 12-slot stator in both unskewed and 3/4-skewed stator configurations, respectively. The overall shape is flat for trapezoidal drive conditions with a maximum drop of 9 percent in torque over the 60° commutation interval. Most of the torque drop in Fig. 10.100 was caused

**FIGURE 10.101 Static torque function: 12 slots, 70° arc, no skew.**

**FIGURE 10.102 Static torque function: 12 slots, 70° arc, 3/-slot skew.**

by the motor’s cogging torque. The 3/4-slot skewed version (Fig. 10.100) reduces the torque drop to 7 percent of total (peak) value over the 60° electrical interval. The overall torque versus position waveform shape for both skewed and unskewed sta-tors (with an 80° arc width) is trapezoidal in shape.

Figures 10.101 and 10.102 show the torque versus position plot for a 12-slot sta-tor with a 70° magnet arc width. Figure 10.101 displays the torque vs. position plot with no stator slot skew. The cogging torque signature is the primary cause of the 8 percent torque variation over the 60° commutation interval in this setup. Skewing the stator stack by % slot changes the torque profile into a quasi-sinusoidal torque profile with lower instantaneous torque variation (see Table 10.22).

Figure 10.103 illustrates the torque profile of the 24-slot BLDC stator, no skew, with an 85° rotor magnet arc width. There is a 9 percent torque drop-off, but the waveform has a near-trapezoidal torque profile. Adding a 1-slot skew (Fig. 10.104) smoothes the torque profile, and the waveform over the 60° electrical interval is closer to a trapezoidal waveform. Figures 10.105 and 10.106 show the effects on waveform of a 70° arc with a 24-slot stator, both skewed and unskewed, respectively. The waveform becomes more sinusoidal. The 36-slot lamination performance is shown in Figures 10.107 and 10.108. Figure 10.107 displays the 80° rotor magnet arc width with no stator skew. The overall torque profile is decidedly sinusoidal. There is

**TABLE 10.22 Subjective Evaluation of Expected Torque Ripple**

Magnet arc | 12 slot | 24 slot | 36 slot | |||

Straight | Slot skew | Straight | Slot skew | Straight | Slot skew | |

85° | 8 | 5 | 8 | 7 | 14 | 5 |

80° | 9 | 7 | 9 | 7 | 13 | 10 |

70° | 8 | 16 | 16 | 14 | 26 | 20 |

**FIGURE 10.103 Static torque function: 24 slots, 85° arc, no skew.**

a 13 percent drop-off in torque magnitude over the 60° electrical commutation interval, and the torque waveform shows the cogging torque “bumps” quite prominently. The 1-slot skew version of the 80° rotor magnet and 36-slot lamination displays a quasi-trapezoidal waveform with a 10 percent torque drop-off over the 60° electrical commutation interval.

The final torque-position waveform shows the last torque profile, which describes a 36-slot lamination with a 70° rotor magnet arc width. The overall shape is definitely sinusoidal in both unskewed and skewed examples. The torque drop-off has increased to 26 percent (Fig. 10.109) and 20 percent (Fig. 10.110). Figures 10.111 and 10.112 display copies of the actual 12-slot and 24-slot laminations used by Persson.

**FIGURE 10.104 Static torque function: 24 slots, 85° arc, 1-slot skew.**

**FIGURE 10.105 Static torque function: 24 slots, 70° arc, no skew.**

Summarizing Persson’s results:

• Decreasing the magnet arc width will lead to sinusoidal torque waveforms.

• Skewing the rotor will reduce cogging and increase the tendency toward quasi-sinusoidal torque waveforms.

^ Increasing the number of stator slots per phase per pole (n > 3) will increase the tendency toward sinusoidal torque waveforms.

**FIGURE 10.106 Static torque function: 24 slots, 70° arc, 1-slot skew.**

**FIGURE 10.107 Static torque function: 36 slots, 80° arc, no skew.**

^ Use a wide pole arc magnet and a minimum number of stator slots per phase per pole (n = 1) to achieve a trapezoidal torque profile.