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.1 Relationship of Torque Constant K1 and Back-EMF Ke to Current Ipk and Irms Profiles
TABLE .19 Y Commutation Sequence-Three-Phase Four-Pole
ILL. 95 Y configuration: ( a) excitation pattern, and ( b) commutation pattern.
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 as sinusoidal drives. Ill. 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 12 electrical to achieve a 6 electri-cal torque commutation zone, as shown in Ill. 97. The sinusoidally driven BLDC motor's current and torque waveforms are shown in Ill. 98. A series of sinusoidally generated current waveforms create sinusoidally shaped torque wave-forms.
In reviewing the available information concerning back-emf K e, torque constant K t, 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 K e, K t, and RLL values from each motor drive when one has computed the individual phase Kt and Rph values for a sinusoidal or trapezoidal BLDC motor.
ILL. 96 Torque versus position waveform for the Y line configuration: (top) winding ED TAC, (middle) winding EF TAB, and (bottom) winding DF TCB.
ILL. 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.
ILL. 98 Sinusoidal torque and current waveforms: ( a) principles of the electrical circuit, and ( b) principles of torque generation.
TABLE .20 Ratios of Torque and Voltage Constants for Brushless Motors in Three-Phase Sine-Wave Systems, Motor Speed in krpm
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 and 54 mm (2.126 in) OD for the 36-slot.The magnet arc widths were varied from 70 to 8 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 .21 Ratios of Torque and Voltage Constants for Brushless Motors in Three-Phase Sine-Wave Systems, Motor Speed in rad/s. 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.
ILL. 99 Static torque function: 12 slots, 8 arc, no skew.
ILL. 100 Static torque function: 12 slots, 8 arc, 3.4 -slot skew.
ILL. 101 Static torque function: 12 slots, 7 arc, no skew.
ILL. 102 Static torque function: 12 slots, 7 arc, 3.4 -slot skew. What is more astounding is the shape of the motor T versus waveform. The commutation interval (angle) is 6 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 6 commutation interval. Most of the torque drop in Ill. 100 was caused by the motor's cogging torque. The 3.4 -slot skewed version ( Ill. 100) reduces the torque drop to 7 percent of total (peak) value over the 6 electrical interval. The overall torque versus position waveform shape for both skewed and unskewed stators (with an 8 arc width) is trapezoidal in shape.
Figures 10.101 and 10.102 show the torque versus position plot for a 12-slot stator with a 7 magnet arc width. Ill. 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 6 commutation interval in this setup. Skewing the stator stack by 3.4 slot changes the torque profile into a quasi-sinusoidal torque profile with lower instantaneous torque variation (see Table 10.22).
TABLE .22 Subjective Evaluation of Expected Torque Ripple Ill. 103 illustrates the torque profile of the 24-slot BLDC stator, no skew, with an 8 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 ( Ill. 104) smoothes the torque profile, and the waveform over the 6 electrical interval is closer to a trapezoidal waveform. Figures 10.105 and 10.106 show the effects on waveform of a 7 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. Ill. 107 displays the 8 rotor magnet arc width with no stator skew. The overall torque profile is decidedly sinusoidal. There is a 13 percent drop-off in torque magnitude over the 6 electrical commutation interval, and the torque waveform shows the cogging torque "bumps" quite prominently.
The 1-slot skew version of the 8 rotor magnet and 36-slot lamination displays a quasi-trapezoidal waveform with a 10 percent torque drop-off over the 6 electrical commutation interval.
The final torque-position waveform shows the last torque profile, which describes a 36-slot lamination with a 7 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 ( Ill. 109) and 20 percent ( Ill. 110). Figures 10.111 and 10.112 display copies of the actual 12-slot and 24-slot laminations used by Persson.
ILL. 103 Static torque function: 24 slots, 8 arc, no skew.
ILL. 104 Static torque function: 24 slots, 8 arc, 1-slot 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.
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.
Increasing the number of stator slots per phase per pole (n > 3) will increase the tendency toward sinusoidal torque waveforms.
.2 Phase Resistance
This section develops a basic comparison chart for the theoretically anticipated values for line-to-line resistance for the four major line-winding configurations. Using the phase resistance X values, the line-to-line resistance for a given configuration is as shown in Table 10.23.
ILL. 105 Static torque function: 24 slots, 7 arc, no skew.
ILL. 106 Static torque function: 24 slots, 7 arc, 1-slot skew.
ILL. 107 Static torque function: 36 slots, 8 arc, no skew.
ILL. 108 Static torque function: 36 slots, 8 arc, 1-slot skew.
ILL. 109 Static torque function: 36 slots, 7 arc, noskew.
.3 Establish Speed-Torque Curve-No-Load, Load, and Peak Torque Values
ILL. 110 Static torque function: 36 slots, 7 arc, 1-slot skew.
The brushless dc motor has one major difference from its brush dc counterpart. All testing including the "simple" back-emf Ke measurement requires that the brushless dc motor be tested with its electronic drive package. The brushless dc motor that uses rare earth magnets can usually outperform its companion drive in the region of maximum torque capability, and there are commutation limits based on motor speed times the number of commutation switching events per revolution. These two limits shape the torque-speed profile. Ill. 113 shows a representative torque-speed limit curve for a sinusoidal drive showing these limitations.
The calculations for establishing the various torque-speed performance points does proceed in the same manner used for a PM brush dc motor, except that the torque and speed limits established for the specific motor-drive combination must be used to limit the calculated values at both ends of the motor torque-speed curve.
The BLDC motor's torque and speed performance is linear over a major portion of the torque-speed curve as described.
ILL. 111 12-slot lamination.
ILL. 112 24-slot lamination.
Once the torque constant Kt and the phase resistance Rph have been established for a representative design, one can use simple algebraic formulas to compute the motor's basic torque versus speed performance. The basic performance parameters from the design example are shown in the Ill. 114 computer simulation. The computed values for Kt 21.046 (oz in)/A or 0.149 (N m)/A and RLL 0.275 for a full-wave Y-connected BLDC motor are used to establish the load torque Tload, peak torque Tpk, and other operating parameters.
Tload Iload Kt (10.10) where Iload is the value established by the motor's thermal constants
Tpk Kt (10.11) where
ET terminal voltage (12)
Ipk (13) The load torque can be computed by using Eq. (10.10) if the load current has been established. The inner line on the Ill. 113 representative torque-speed curve is the motor's safe operating ambient conditions (SOAC) curve. All load points (torque and speed) to the left of this boundary line are rated and continuous values.
The Iload is located somewhere along this curve. The actual motor terminal voltage would create a straight-line torque-speed curve based on constant voltage which would intersect the SOAC curve at a specific point.
For purposes of illustration, let us assume that this BLDC motor can dissipate 74 W at a specific point on the SOAC line. Using the estimated hot resistance value for
RLL which is assumed to be at 100C above ambient, the actual load current can be computed using the formula in Eq. (10.14):
ILD
RLL, Pdiss hot
13.8 A (10.14) where R H, hot (0.275)(1.4) 0.385
Therefore, using Eqs. (10.10), (10.11), and (10.12), the load torque Tload is computed at 290 oz in. The Tpk is calculated as (160/0.385) (21.046) 8746 oz in or 61.94 N m. The peak current is calculated as 415.8 A. The peak values are theoretical, because one can expect the drive limits to engage well before these calculated peak values would be reached. The establishment of the theoretical no-load speed NNL,th can be done by using the Ke value computed in equation 15.
NNL,th
ET Ke
(10.15) where: ET motor terminal voltage
Ke motor's Y-configuration back-emf
NNL,th 160 15.562 10.28 krpm where NNL,th is the motor's no-load speed.
Therefore, 10,280 rpm is the peak theoretical no-load speed. One can estimate or calculate the actual no-load speed if no load current is available. In any case, since the BLDC no-load rotating losses (windage, bearing system, friction, etc.) are very low, conservatively in the region of 2 to 3 percent, an adjusted no-load speed value of 9972 rpm (97 percent of theoretical value) can be established. Load speed NL can be computed as displayed in Eq. (10.16).
NL NNL (1 ) (10.16)
ET
RLL
Et
RLL where is the ratio of Tload/TPK,th
NL (9972) 1 290 8746 (9972)(0.967) 9641 rpm Any speed point along the constant-voltage derived torque-speed curve can be computed in a similar manner until the boundary for peak performance (intermit tent duty limit line in Ill. 113) is reached.
The values used in the Ill. 114 simulation are the original cold (20C) value for Ru 0.275 , so the calculated values in the simulation in Ill. 114 are 40 per cent higher than the ones computed in this section. Still, the procedures are the same.
.4 Optimizing Motor Constant Km
The thousands of applications that use a BLDC motor make it impossible to deter mine a simple criteria in selecting the best BLDC motor for a given application. For motor designers, there is a single Ill. of merit that can determine if the motor design engineer has optimized the design in terms of magnetics and stator winding conditions. The Ill. of merit (motor constant K m), along with the motor's dimensions does provide a relatively simple but effective method of evaluating a BLDC motor design. This also assumes that the proper magnet material has been selected, based on cost as well as dimensional requirements.
Km T
Win
Kt
RLL
(10.17) where T any torque value
Win input power at the chosen torque value
Kt motor torque constant
RLL motor line-to-line resistance This constant, when maximized for a given motor volume, ensures the design engineer of the best possible design. Many motor mechanical, magnetic, and electrical parameters are integrated into motor constant K m. Maximizing Km is maximizing the motor's copper to iron ratio for that specific motor size.
If one reviews the design process, the Kt value is primarily tied to magnet flux, number of turns, pole count, unit path reluctance, and so forth. RLL is a function of available slot space, winding pattern, pole count, and so forth. Optimizing both performance parameters optimizes the motor design in terms of the motor's magnetic circuit and winding selection.
Remember that other user requirements, such as the lowest possible winding inductance, would bias the motor constant Km to very low numbers of turns (conductors). Minimizing rotor inertia for certain incremental motion applications would also bias the Km optimization process as described.
The Km value listed in Ill. 114 is 40.17 (oz in)/W. One could significantly increase Km by the following means:
Increasing the number of poles Increasing the rotor OD up to a certain OD dimension; then the Km value would decrease beyond this dimensional limit Changing to high-energy magnets Increasing motor OD or AL Usually the motor's available input voltage and current levels will act as a limit to optimizing Km beyond a certain limit. The challenge for the motor design engineer is to maximize Km within specified electric inputs or friction and inertia requirements which will yield the best overall magnetic design.
TABLE 10.23 Basic Comparison Chart: Phase resistance; Hookup configuration, Line-to-line resistance X Y (half) X X Y (full) 2X X Delta 2.3 X X Independent X
ILL. 113 Typical BLDC motor torque-speed limit curve.
ILL. 114 Example simulation program results.
5 Power, Losses, Efficiency, and Others
There are two groups of motor performance parameters, the conventional and the special performance parameters. A list of the conventional motor performance parameters includes those parameters or figures of merit that relate to motor power.
For example, Eq. (10.18) determines dissipated power I 2 R, Eq. (19) computes input power, Eq. (10.20) calculates output power, and Eq. (10.21) establishes the power efficiency at that specific load point PLD. These parameters are designated as the conventional performance parameters and are computed as follows:
Pdiss ILD,RMS RLL (10.18)
Pin ET ILD,RMS (10.19)
Pout K
TLDNLD (10.20) where K 1352 for oz in and rpm values
K 1 for N m and rad/s values Efficiency
Pin Pout (.21)
The measuring of motor voltage and currents has become much more difficult with the inverter and the power amplifier unable to be separated from the motor.
The VRMS and IRMS current values are the most important ones to measure and require special equipment.
In most design computations, a 100C temperature rise value of RLL is used. Computations of these values are shown in Ill. 114, where a more optimistic room ambient RLL value is employed.
The more specialized figures of merit include motor characteristics such as theoretical acceleration T/ J, peak power rate (PPR), rated power rate (RPR), peak power density (PPD), and rated power density (RPD). The first three figures of merit are used in incremental motion applications. The other two are more often used in automotive, aerospace, and portable instrument applications. Theoretical acceleration can be determined by Eq. (22).
... where acceleration, rad/s 2 Tacc Tpk
Jm the motor rotor inertia This equation can be modified in the inertia term (J m) to include the load inertia
JL for load matching conditions.
Power rate is defined as the rate of change of power with respect to time and is important in incremental motion applications. Equations (10.23) and (10.24) detail the method of computation of peak and rated power rates (PPR and RPR).
... where m is the machine's mechanical time constant, RPR T
Power rate can be expressed in kilowatts per second. The BLDC motor with the highest power rate will produce the shortest time to move from one point to another.
Power density or power per unit volume has become popular in recent years with the advent of battery operated motors and actuators. These figures of merit relate output power to the unit volume and, indirectly, unit weight. Equations (25) and (.26) show the simple formulas for computing these figures of merit.
PPD Rated output power/unit volume(10.25) RPD Peak output power/unit volume(10.26) The units are watts per cubic inch or watts per cubic meter. The higher the value of power density, the more power per unit volume available.
