Accurate Torque Ripple Measurement for PMSM - IEEE Xplore

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Accurate Torque Ripple Measurement for PMSM Greg Heins, Member, IEEE, Mark Thiele, and Travis Brown, Student Member, IEEE

Abstract—Torque ripple in permanent-magnet synchronous motors is generally undesirable. Significant work has been done to minimize this torque, either by modifying the mechanical motor design or by careful controller design. Surprisingly, however, little work has been published on the accuracy of torque ripple measurement. A successful measurement requires a mechanical design with readily modeled dynamics, sensors with suitable bandwidth and resolution, a method of applying a smooth load to the motor, and a method for calibrating the measurement. This paper presents a thorough approach to the accurate measurement of torque ripple. The proposed system has been validated by finite-element modeling, analytical calculations, and experimental analysis. Index Terms—Cogging torque, permanent-magnet machines, piezoelectric transducers, pulsating torque, resonance, torque measurement, torque ripple, vibration measurement.

I. I NTRODUCTION

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INIMIZING torque ripple is a common goal for the magnetic design or control system design of permanentmagnet synchronous motors (PMSMs). Low-torque ripple is essential for applications that require precise tracking. These applications include arc welding, laser cutting, numerical control machining, and antenna tracking [1], [2]. Despite considerable research of torque ripple, accurate measurement remains a problem. While the suggested methods of minimization predominantly affect the air-gap torque, measurement is usually done with a shaft torque sensor. This suggests that there is a known relationship between air-gap torque and shaft torque. It is suggested in [3], however, that “On occasion, shaft response will be looked at as an indication of air-gap torque. There are so many unknown factors which affect the shaft response that this method is of little value.” To overcome these unknown factors and ensure that the measured shaft torque is an adequate representation of the actual motor torque require the following: 1) known mechanical dynamics; 2) appropriate torque sensor choice; 3) appropriate load choice; and 4) a means of calibrating the system. The goal of this paper is to define the critical requirements of a torque ripple measurement system and propose and validate a specific design to meet these requirements. The test motor used is a 1-hp axial flux PMSM. Manuscript received December 19, 2010; revised March 4, 2011; accepted March 13, 2011. The Associate Editor coordinating the review process for this paper was Dr. Salvatore Baglio. The authors are with the School of Engineering and Information Technology, Charles Darwin University, Darwin, N.T. 0909, Australia (e-mail: [email protected]; [email protected]; travis.brown@cdu. edu.au). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2011.2138350

Section II reviews past work and specifies the critical requirements. The design process used to meet these requirements is explained in Section III. Results of design validation using impact testing and calibration are presented and discussed in Section IV. Section V concludes and discusses the significance of these results for future torque ripple research. II. S YSTEM R EQUIREMENTS A. Mechanical Dynamics To ensure accurate torque ripple measurement, the transfer function between motor torque and the measured torque should be time and frequency independent over the experimental operating range. The measured torque will not be a valid indication of motor torque if resonant frequencies affect the frequency range or if a torque sensor location is chosen, which measures inertial or bearing loads. The effect of resonance can be avoided by static or quasi-static measurements [4], [5]. However, if torque ripple components other than cogging torque are to be measured, this approach is not feasible. If resonances affect the frequency measurement range, the torque ripple created by the motor may be amplified before it is measured. To ensure that the measurement linearity error is less than 5%, all frequencies measured should be at least five times lower than the fundamental resonant frequency of the system [6, p. 166]. While torque sensor bandwidth is often considered, this more stringent requirement for the bandwidth of the total mechanical system is usually overlooked. If a measurement bandwidth of 1 kHz is desired, the fundamental resonant frequency of the system should be ≥ 5 kHz. In practice, unless the motor studied is very small, it is unlikely that the system can be designed with such a high fundamental resonant frequency. Instead, the system should be designed to have the highest practical fundamental resonant frequency and the operating range of the motor adjusted to ensure that all harmonics of interest fall within the resulting accurate measurement range. A light stiff design will have the highest fundamental resonant frequency, so flexible elements such as couplings should be avoided. The problems associated with flexible couplings are discussed in [7]. Another common source of flexibility is the torque sensor, so sensor stiffness is a high priority when selecting a suitable sensor (Section II-B). Torque sensor location is critical to avoid the measurement of inertial or bearing loads. Inertial loads are created by speed variation of the rotor. If an in-line torque sensor connected to the rotor is used, it will measure the inertial loads. Attempts have been made to minimize these effects by imposing a constant speed [8] or to compensate for them [9, eq. (14)]; however, they can be completely avoided by using a reaction torque

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sensor connected to the stator. In this configuration, there are no inertial loads, as the stator does not move [10]. Bearing loads (sometimes referred to as shunt loads [10]) can either be a viscous torque from the oil or grease or could be a higher frequency disturbance if imperfections exist in the bearing. The presence of shunt loads in the measured torque depends on the relative location of the sensor and bearings. B. Sensor Selection Section II-A explained the importance of a stiff sensor to achieve a high overall system bandwidth. This requirement is usually the most stringent specification, so in general, the stiffest sensor available should be chosen. In addition, the chosen torque sensor must have adequate resolution, a bandwidth above the total system bandwidth, and exhibit minimum crosstalk. Achieving adequate resolution can be a problem when measuring torque ripple. A torque sensor with a large-enough range to deal with the maximum load will not necessarily have the resolution to measure the torque ripple [9]. For a moderate level of torque ripple (approximately 5%–10% [11]), only a small portion of the total measurement range will include relevant information. To use the majority of the measurement range for the torque ripple (or ac component of the torque), it is desirable to use a measurement system that will reject the constant (or dc component) of the torque. Many torque sensors exhibit the limitation of being sensitive to the torques and forces applied in the direction perpendicular to their axis of measurement [12]. Good-quality sensors should minimize and specify this “crosstalk” so that the induced error can be quantified and the effect can be minimized. C. Load Application Operation of the test motor over its operating range usually requires a means of applying a load torque. This load must not create its own torque ripple. To apply a load, previous torque ripple researchers have either used dc motors, hysteresis brakes, eddy current brakes, or hydraulic motors. DC generators are a common method for applying a load to a test motor. They are advantageous, as the torque applied can be readily controlled by adjusting the current flowing out of the generator. Some issues can arise, however, when measuring torque ripple. In some experiments [13], the torque ripple from the dc generator interfered with the measurement of the torque ripple from the test motor. This can be minimized with a careful choice of load generator; however, commutations will always create some torque ripple. In a hysteresis brake, a disk made of magnetic material rotates between a series of magnetic poles. Poles previously induced in the disk interact with the stationary magnetic poles, and torque is transmitted until the disk begins to slip. Once slipping, the poles in the disk move as the material is magnetized and demagnetized. Energy is dissipated due to the hysteresis of the material. Load torque is adjusted by varying the strength of the applied magnetic field.

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Hysteresis brakes have the benefit that the torque applied is independent of speed [14], so torque can be applied down to zero speed. This means, however, that if speed is to be regulated, then a control system is required to do so. Sometimes, the controller from commercially available hysteresis brakes is not able to keep a constant low speed [7]. Another problem associated with hysteresis brakes is the potential for residual cogging torque to be created if the brake is incorrectly shut down [15, p. 53]. In an eddy current brake, a highly conducting disk (such as copper or aluminum) rotates in a magnetic field. The magnetic field induces eddy currents in the disk. These eddy currents, in turn, create a magnetic field that opposes the original magnetic field. As the current produced is a function of velocity, the torque produced is a function of motor speed. This proportionality of torque to speed limits the applicability of eddy current brakes, as they cannot be used for zero-speed testing (locked rotor). They do however have the benefit that the torque applied is completely smooth and the speed is self-regulating. Hydraulic motors can be used to apply a load in a similar way to a dc motor [16]. Instead of controlling the torque by the amount of current, the pressure is regulated. As with dc motors, if the motor is not chosen carefully, it can create torque ripple. Due to the peripheral equipment required, hydraulic motors are usually more complex and expensive than other options. D. Calibration Any measurement system is of limited value if it cannot be calibrated to validate its accuracy. In the case of torque ripple measurement, there should be a way of ensuring that the torque sensor output is an accurate representation of the motor torque. Limited efforts, however, have been made to calibrate torque ripple sensors. In [17], when developing a polymer-based piezoelectric torque sensor, the expected sixth harmonics from the motor torque were compared with the measured torque. This validated that the measurement system was linear for one harmonic; however, calibration should be done for the entire frequency range. One commonly used method of calibration is the use of an impact hammer. It is explained in [18], however, that due to the difficulty in controlling the direction of the force from the hammer, this technique has limited benefits for torque sensor calibration. III. S YSTEM D ESIGN Section II outlined the general requirements for an accurate experimental setup for measuring torque ripple in PMSM. This section will discuss the design of a specific setup used for measuring a 1-hp axial flux PMSM. A. Initial Mechanical Layout Section II-A explained that the measured torque will not be a valid indication of motor torque if the torque sensor and bearing arrangement lead to the measurement of inertial or bearing

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Fig. 1.

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Traditional layout for torque ripple measurement.

Fig. 3. Trifilar rig for stator inertia measurement. Fig. 2.

Proposed layout for torque ripple measurement.

(shunt) loads. With careful mechanical layout, the torque sensor can be located such that these loads are not measured. Fig. 1 shows a schematic of the traditional torque ripple measurement layout. Although an axial flux motor arrangement is shown, the concepts are equally applicable to a radial flux arrangement. To determine which torques are measured by the sensor requires consideration of the equation of motion containing the torque sensor term ksens (θR − θB ) + bbear θ˙R + Tmotor = JR θ¨R

(1)

where torque sensor stiffness; ksens rotor angular position; θR brake angular position; θB bearing viscous friction coefficient; bbear Tmotor motor torque; JR rotor angular moment of inertia. The information about the measured torque comes from the term with the sensor stiffness, so Tmeas = ksens (θR − θB ) = JR θ¨R − bbear θ˙R − Tmotor

(2)

Tmeas = ksens (θR − θB ) = Tmotor .

(5) (6)

Therefore, only motor torque is measured when using a reaction torque sensor with the rotor and stator mounted on separate shafts. B. Stator Inertia Section II-A explained that avoiding resonant frequencies in the measurement range usually requires ensuring the lowest torsional resonant frequency to be as high as possible. For a reaction torque sensor arrangement as shown in Fig. 2, the limit for the lowest resonant frequency is defined by the rotational moment of inertia of the test stator (JS ) and the torsional stiffness of the sensor (ksens ). Accurate theoretical determination of the stator moment of inertia was difficult because of its complex construction composed of laminated windings and copper. To ensure accuracy, inertia was determined using a trifilar rig [19], as shown in Fig. 3. This method gave JS = 2.14 × 10−3 kg · m2 .

(3) C. Sensor Selection

where Tmeas is the measured torque. From (2), it is clear that, in attempting to measure the motor torque, the inertial and bearing torques are also measured. To avoid this problem, the layout shown in Fig. 2 is proposed. The reaction torque sensor ensures that inertial loads are not measured and the separate rotor and stator shafts avoid the inclusion of bearing loads. Additional dynamics associated with flexible couplings are eliminated by mounting the rotor and load on the same shaft. Again, calculate the equation of motion that contains the torque sensor term ksens θS + bbear θ˙S + Tmotor = JS θ¨S

This time, the inertial terms are related to the stator which should not be moving θ˙S = θ¨S = 0, so

(4)

where θS is the stator angular position and JS is the stator angular moment of inertia.

The best torque sensor for measuring torque ripple will be a reaction torque sensor (Section III-A) with high stiffness (Section II-B). One of the most common styles of torque sensors used for torque ripple measurement is based on strain gauges which measure force by measuring the displacement of an element of known stiffness [20]. This requirement for displacement limits the potential stiffness of the sensor. Another style of torque sensor uses a torsion bar with two encoders [21], which also introduces flexibility. Usually, only sensors based on piezoelectric technology are stiff enough to ensure a suitably high system bandwidth. As an example, for our application with a measurement range of ±10 N · m, the two possible sensor options were Himmelstein (strain gauge) and Kistler (piezoelectric) sensors.



The stiffest strain gauge sensor in the Himmelstein range [10] has a stiffness of 11 130 N · m/rad, whereas a piezoelectric Kistler sensor (9339A) [12] has a stiffness of 96 000 N · m/rad, which is over eight times stiffer. If the system is simplified to a single degree of freedom system, the fundamental torsional resonant frequency is defined by ksens . (7) ωn = JS Therefore, a sensor that is eight times stiffer will have a band√ width 8 = 2.8 times higher. When used with the correct interfacing equipment, both of these sensors had adequate resolution and bandwidth, so for this research, the Kistler 9339A sensor was chosen for its superior stiffness. Combining the chosen sensor stiffness with the experimentally determined moment of inertia suggested that the upper limit for the fundamental torsional resonant frequency of the system was 1066 Hz. This limited the accurate measurement range to 213 Hz (five times lower as specified in Section II-A). Although the motor used for this research only required a sensor with a measurement range of ±10 N · m, the sensor chosen is part of a range of sensors that can measure up to ±1000 N · m (9389A). This allows the potential for a similar rig design catering for much larger motors. One potential issue with piezoelectric sensors is their inability to measure average or dc torque. However, as the focus of the research was torque ripple, this was not a substantial issue. In some ways, as discussed in Section II-B, only measuring ac torque is a benefit in that all of the measurement range could be used for torque ripple, without a large amount being taken up measuring the average (or dc) component. D. Load Application An eddy current brake was chosen, as it was the only one of the reviewed load application methods guaranteed not to generate torque ripple. The proportionality of torque to speed was an added advantage in that it allowed self speed regulation, which is a factor that Wu and Chapman [7] had difficulty with when using their hysteresis brake. The overall system bandwidth of 213 Hz limited the maximum motor speed during testing. For the test motor considered, 120 mechanical harmonics were of interest. This limited the maximum motor speed to 1.78 Hz. The eddy current brake was designed to provide a rated motor load at this speed. The eddy current brake design was done using the model presented in [22]. Due to space requirements, electromagnets were not possible. Instead, two permanentmagnet poles were located in a sliding arrangement, so they could be advanced or retracted to achieve the required torque. Optimizing the eddy current brake design affected the mechanical frequency response, so brake design was done in conjunction with the detailed mechanical design described in Section III-E. E. Detailed Mechanical System Design With the basic layout, torque sensor, and load selected, the detailed design could be completed to ensure appropriate

Fig. 4.

FEA boundary conditions.

mechanical frequency response. In Section III-C, the upper limit for the fundamental resonant frequency was calculated as 1066 Hz. In practice, this frequency was not achievable, as the flanges and shafts required to mount the stator to the torque sensor increased the moment of inertia and the flexibility. However, to ensure that the stator-torque sensor combination remained the limiting constraint, 1066 kHz was chosen as the design goal. An additional source of flexibility was the stator itself. At this stage, it was assumed that the wound stator itself did not have any resonant frequencies that would affect the accurate measurement range. To ensure that the system components were sufficiently light and stiff to meet this resonant frequency target, modal analysis was completed in ANSYS. These components included the following: 1) the stator mounted on bearings that left only one degree of freedom to be measured by the reaction torque sensor; 2) the rotor mounted on bearings; 3) the eddy current brake as selected in Section III-D; 4) an encoder. Fig. 4 shows the boundary conditions used to model the critical stator mounting. The curved frictionless support “B” was carefully designed to model the single bearing that would not support a moment. All components were modeled as structural steel apart from the stator (green on left) for which the density was modified to match the experimentally calculated moment of inertia and the torque sensor (yellow on right) for which the elastic modulus was modified to match the data sheet stiffness. The scale of this model can be approximated from the diameter of the stator mounting flange, which is 100 mm. Once additional hardware was added to mount the stator to the reaction torque sensor, the highest possible fundamental resonant frequency based on the ANSYS model was 498 Hz. This lowered the potential operating range of the motor to 0.8 Hz. This lower operating speed necessitated modifications to the eddy current brake design to ensure that the rated motor torque is still achievable. Section IV-B explains that these ANSYS results were later validated with closely matching analytical and experimental results. Fig. 5 shows the final design.


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Fig. 6. Stator impact testing.

Fig. 5.

Final design.

F. Alternate Analytical Method In conjunction with the finite-element analysis (FEA) method, an analytical method was used to determine the fundamental resonant frequency. While this method is potentially useful for researchers without access to modal analysis FEA software, for this research, it was used to validate the FEA results and as a parametric design tool. An iterative design using FEA software can be slow. Using a simple analytic model provided a quick method for initial appraisal of potential designs. As explained in Section III-C, the fundamental natural frequency for a single degree of freedom torsional system is a function of the stiffness and moment of inertia. For the complete torque ripple measurement system, the total stiffness (sensor and shaft) and the total inertia (stator and mounting flange) can be used. The total stiffness of two flexible elements in series is determined by the inverse sum 1 1 1 = + (8) ktotal ksens kshaft where ksens = 96 000 N · m/rad (Section III-C). Moreover, kshaft can be determined by kshaft =

Gπd4 32L

(9)

where G = 80 GPa (shear modulus of steel), d = 30 mm (diameter of shaft), and L = 100 mm (length of shaft). The total inertia is a standard sum Jtotal = JS + Jﬂange

(10)

where JS = 2.14 × 103 kg · m2 (from Section III-B) and Jﬂange = 1.89 × 103 kg · m2 (from a computer-aided design model). By substituting the values from (8) and (10) in (7), the fundamental natural frequency of the rig was calculated to be 490 Hz. This result validated the FEA result of 498 Hz. G. Calibration Process Design To calibrate the measurement setup, the transfer function between the motor torque and the measured torque was determined by defining a frequency response function (H(ω)) [23, eq. (7.10)] H(ω) =

Tmeas B(ω) = A(ω) Tmotor

(11)

where H(ω) frequency response function; A(ω) frequency input function; B(ω) frequency output function. With torque ripple measurement, full control of the input is possible, so a better estimate of the transfer function can be obtained by multiplying the numerator and denominator of H(ω) by the complex conjugate of A(ω) [23] H 1 (ω) =

GAB (ω) GAA (ω)

(12)

where H 1 (ω) alternate frequency response function; GAB (ω) cross spectrum; GAA (ω) input auto spectrum. The known input was created by injecting torque harmonics over the frequency range. These injected harmonics are created using a field-oriented torque controller implemented in a Texas Instruments TMS320F2812 DSP. The torque bandwidth of this controller is defined by the bandwidth of the current control loop, which was experimentally determined to be 900 Hz. To ensure that this bandwidth did not affect the results, the actual motor current was measured, and the resulting electromagnetic torque was reconstructed offline. Using this approach, any existing motor torque ripple will not be included in the input but will be measured in the output. To overcome this issue, those harmonics known to exhibit torque ripple were disregarded. If there are no mechanical resonances in the system, the plotted frequency response function (12) will have the same magnitude over the frequency range and zero phase lag. IV. R ESULTS AND D ISCUSSION A. Impact Testing In Section III-E, it was assumed that there were no resonant frequencies in the wound stator itself that would affect the measurement range. Validation of this assumption with FEA modal analysis was not practical due to the complex geometry and anisotropic material properties. Instead, experimental impact testing was conducted. Fig. 6 shows that the first resonant frequency (815 Hz) is above the first natural frequency predicted for the entire system (498 Hz). To ensure that the issues with impact testing raised in Section III-D were mitigated, impact tests were done from different directions, each time producing a similar result.



Fig. 7. Transfer function of measured torque to injected torque. (a) All injected frequencies. (b) Known torque ripple frequencies removed.

B. Calibration Using the method outlined in Section III-G, torque ripple harmonics were injected into the system. Fig. 7(a) shows the resulting transfer function. The low-frequency spikes are easily attributable to the standard expected torque ripple frequencies for the research motor used. For this motor, these harmonics are multiples of the electrical frequency and the number of magnets. Fig. 7(b) shows the transfer function with these harmonics removed. The peak response which corresponds with the fundamental torsional resonant frequency is at 518 Hz. This compares closely with the 498 Hz from the FEA analysis (3.9% error) and the 490 Hz from the analytical solution (5.4% error). This measured resonance peak leads to an accurate measurement range below 104 Hz. V. C ONCLUSION This paper has presented the essential background for accurate measurement of torque ripple in PMSM. The proposed design method ensured that the critical elements that were addressed have resulted in a system with a known accurate measurement range. Close correlation between FEA, analytical, and experimental results has validated the system and design process. To ensure accurate experimental results, torque ripple researchers should consider the issues raised and potentially follow a similar design and validation process for their measurement system. As the torque sensor used is part of a range of sensors that can cater for much larger torques, the proposed measurement system should be able to be scaled to cater for considerably larger motors than the one used in this research. R EFERENCES [1] F. Aghili, M. Buehler, and J. Hollerbach, “Torque ripple minimization in direct-drive systems,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 1998, vol. 2, pp. 794–799. [2] G. Ferretti, G. Magnani, and P. Rocco, “Force oscillations in contact motion of industrial robots: An experimental investigation,” IEEE/ASME Trans. Mechatronics, vol. 4, no. 1, pp. 86–91, Mar. 1999.

[3] E. L. Owen, H. D. Snively, and T. A. Lipo, “Torsional coordination of high speed synchronous motors—Part II,” IEEE Trans. Ind. Appl., vol. IA-17, no. 6, pp. 572–580, Nov. 1981. [4] Z. Q. Zhu, “A simple method for measuring cogging torque in permanent magnet machines,” in Proc. IEEE PES Gen. Meeting, 2009, pp. 1–4. [5] R. Gobbi, N. C. Sahoo, and R. Vejian, “Experimental investigations on computer-based methods for determination of static electromagnetic characteristics of switched reluctance motors,” IEEE Trans. Instrum. Meas., vol. 57, no. 10, pp. 2196–2211, Oct. 2008. [6] K. G. McConnell and P. S. Varoto, Vibration Testing: Theory and Practice. Hoboken, NJ: Wiley, 1995. [7] A. Wu and P. Chapman, “Simple expressions for optimal current waveforms for permanent-magnet synchronous machine drives,” IEEE Trans. Energy Convers., vol. 20, no. 1, pp. 151–157, Mar. 2005. [8] N. Bianchi and S. Bolognani, “Design techniques for reducing the cogging torque in surface-mounted PM motors,” IEEE Trans. Ind. Appl., vol. 38, no. 5, pp. 1259–1265, Sep./Oct. 2002. [9] L. Sun, H. Gao, Q. Song, and J. Nei, “Measurement of torque ripple in PM brushless motors,” in Conf. Rec. 37th IEEE IAS Annu. Meeting, 2002, vol. 4, pp. 2567–2571. [10] Bulletin 770d: Precision Reaction Torquemeters, Himmelstein, Hoffman Estates, IL, 2007. [Online]. Available: http://www.himmelstein.com/ PDF_Files/B770D.pdf. (Accessed Dec. 6, 2010). [11] T. Jahns and W. Soong, “Pulsating torque minimization techniques for permanent magnet AC motor drives—A review,” IEEE Trans. Ind. Electron., vol. 43, no. 2, pp. 321–330, Apr. 1996. [12] Data Sheet: Reaction Torque Sensors (Type 9329a, 9339a, 9349a, 9369a, 9389a), Kistler, Winterthur, Switzerland, 2009. [Online]. Available: http://www.kistler.com/mediaaccess/000-463e-02.09.pdf. (Accessed Dec. 6, 2010). [13] W. Qian, S. Panda, and J. Xu, “Speed ripple minimization in PM synchronous motor using iterative learning control,” IEEE Trans. Energy Convers., vol. 20, no. 1, pp. 53–61, Mar. 2005. [14] B. Guillemin and H. Smith, “Design and implementation of a rodent voluntary wheel-running exercise facility incorporating dynamically controllable torque load,” IEEE Trans. Instrum. Meas., vol. 55, no. 3, pp. 839– 845, Jun. 2006. [15] Hysteresis Dynamometer—User Manual, Magtrol, New York, 2004. [Online]. Available: http://www.magtrol.com/manuals/hdmanual.pdf. (Accessed Dec. 6, 2010). [16] F. Aghili, M. Buehler, and J. Hollerbach, “Experimental characterization and quadratic programming-based control of brushless-motors,” IEEE Trans. Control Syst. Technol., vol. 11, no. 1, pp. 139–146, Jan. 2003. [17] P. Beccue, J. Neely, S. Pekarek, and D. Stutts, “Measurement and control of torque ripple-induced frame torsional vibration in a surface mount permanent magnet machine,” IEEE Trans. Power Electron., vol. 20, no. 1, pp. 182–191, Jan. 2005. [18] K.-J. Xu, C. Li, and Z.-N. Zhu, “Dynamic modeling and compensation of robot six-axis wrist force/torque sensor,” IEEE Trans. Instrum. Meas., vol. 56, no. 5, pp. 2094–2100, Oct. 2007. [19] J. du Bois, N. Lieven, and S. Adhikari, “Error analysis in trifilar inertia measurements,” Exp. Mech., vol. 49, no. 4, pp. 533–540, Aug. 2009. [20] H. Dzapo, Z. Stare, and N. Bobanac, “Digital measuring system for monitoring motor shaft parameters on ships,” IEEE Trans. Instrum. Meas., vol. 58, no. 10, pp. 3702–3712, Oct. 2009. [21] J.-H. Kim and J.-B. Song, “Control logic for an electric power steering system using assist motor,” Mechatronics, vol. 12, no. 3, pp. 447–459, Apr. 2002. [22] D. Schieber, “Braking torque on rotating sheet in stationary magnetic field,” Proc. Inst. Elect. Eng., vol. 121, no. 2, pp. 117–122, Feb. 1974. [23] R. Randall, Frequency Analysis, 3rd ed. Glostrup, Denmark: Bruel and Kjaer, 1987.

Greg Heins (S’04–M’08) received the B.Eng. degree from the University of New South Wales, Sydney, Australia, in 2000 and the Ph.D. degree from Charles Darwin University, Darwin, Australia, in 2008. He was a Manufacturing Engineer with Robert Bosch Australia. He is currently a Senior Lecturer with Charles Darwin University. His research interests include the design and control of electric motors, vibration analysis, and system modeling and identification.


Mark Thiele received the B.Eng. degree from the University of Swinburne, Melbourne, Australia, in 1995 and the B.Sci. and B.Ed. degrees from Charles Darwin University, Darwin, Australia, in 2007, where he is currently working toward the M.Eng. degree in engineering. After he received his B.Eng. degree, he worked in the automotive industry for nine years. He was a Lecturer of mechanical engineering with Charles Darwin University. His research fields focus on cogging torque of axial-flux permanent-magnet machines.

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Travis Brown (S’10) received the B.Eng. degree in mechanical engineering from Charles Darwin University, Darwin, Australia, in 2010, where he is currently working toward the Ph.D. degree in the field of biomedical engineering. His research interests include permanent-magnet synchronous motor modeling and design, image processing, and auditory evoked potentials.