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Vibrationless alignment algorithm for incremental encoder based BLDC drives
Electric Power Systems Research 95 (2013) 225–231
Contents lists available at SciVerse ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Vibrationless alignment algorithm for incremental encoder based BLDC drives C. Concari, G. Franceschini, A. Toscani ∗ Department of Information Engineering, University of Parma, Viale G.P. Usberti, 181/A, 43124 Parma, Italy
a r t i c l e
i n f o
Article history: Received 7 October 2011 Received in revised form 31 July 2012 Accepted 21 September 2012 Available online 1 November 2012 Keywords: Brushless machines Alignment Digital control Modeling Starting Vibration control
a b s t r a c t Incremental encoder based BLDC drives are widely employed in industry whenever smooth and efficient operation is needed in the whole speed range. Their weak spot lies in the need for a phase of initial alignment, traditionally performed by forcing the motor through a series of fixed phase supply configurations in an open loop stepping mode. This strategy of alignment causes ample mechanical vibrations which can be intolerable for certain applications. This paper presents a novel algorithm for determining the initial rotor sector and aligning the encoder with the motor while minimizing vibrations and oscillations. © 2012 Elsevier B.V. All rights reserved.
1. Introduction DC brushless motors are widely employed in industry due to their high performances and ruggedness. Since such motors have no brushes, they need a solid state commutation circuit in order to supply the stator windings according to rotor position. Rotor position, therefore, must be determined, and this is typically performed using Hall sensors, incremental encoders or sensorless techniques. Hall sensors allow a great simplicity of control but they limit the operating temperature of the motor, need precise alignment at the factory and allow only coarse speed estimation at low speed. Sensorless techniques are widely used especially in applications which run mainly at medium to high speeds, such as fans and compressors; in such applications the reduction of vibrations during start-up is welcome but not mandatory [1–3]. Other techniques exploit inductance variation with rotor position [4,5], the d-axis current after applying voltage pulses [6,7] or signal injection [8] for initial position sensing without appreciably moving the rotor. Ref. [9] detects the initial rotor position by the time periods of discharge of stator windings, which are excited before discharge. Another initial position detection method combines an iterative sequence of voltage pulses with a fuzzy logic processing of the current response and phase currents derivation based on the
∗ Corresponding author. Tel.: +39 0521 906007; fax: +39 0521 905822. E-mail addresses: [email protected] (C. Concari), [email protected] (G. Franceschini), [email protected] (A. Toscani). 0378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.epsr.2012.09.010
DC-link current measurements [10]. Some algorithms are specific for avoiding reverse motion in unidirectional applications [11]. All sensorless techniques rely on the measurement of electrical variables such as currents and BEMFs, and they cannot be used at low speed due to signal/noise ratio degradation. Sensorless algorithms typically revert to step operation at low speeds, which leads to low efficiency and high vibration levels during start-up (even higher than with a traditional incremental encoder starting algorithm as described in Section 4). Incremental encoders allow optimal speed and position measurement over the full drive speed range, justifying the additional cost for applications in which the motor frequently accelerates from standstill or operates at low speed for extended periods of time [12,13]. Moreover, when used in clean environments such as food processing factories, they do not significantly reduce the reliability of the motor drive. A typical example of such an application is represented by weighing tension rollers used in the manufacturing and packaging industry, in which properly aligned encoder-based BLDC drives ensure smooth operation preventing objects with high center of gravity (e.g. bottles) from falling during acceleration and deceleration. The downside of this approach is that when the plant is powered on, e.g. at the beginning of the workday or after a power outage, incremental encoder based motor drives need initial alignment. While a smooth operation is important in every phase of the work cycle, alignment is typically performed by supplying motor windings with a predetermined sequence, which results in vibrations and oscillations with the risk of dropping the transported objects.
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Fig. 1. BLDC back-emf (top) and proper phase currents (bottom). Sectors, phase configurations and equilibrium positions are shown.
This paper presents a novel algorithm that allows determining the initial rotor sector and aligning the incremental encoder while minimizing vibrations and oscillations. In the following, Section 2 establishes notational conventions useful in later sections; Section 3 develops a simple dynamical analytical model of the BLDC motor; Section 4 introduces the proposed vibrationless alignment method; Section 5 reports experimental results; finally, Section 6 concludes the paper with the final remarks. 2. BLDC operation and definitions Before entering the details of the proposed alignment algorithm, it is useful to establish a set of definitions and conventions which describe the operation of BLDC motors and motor drives. Considering 2-phase-on operation, phase configurations will be referred to by stating the positive and negative supplied phases, e.g. phase configuration U V¯ means that phase U is supplied with positive current, phase V with negative current and phase W is floating. Rotor position can be said to lie in one of six angular sectors of 60 electrical degrees, labeled with numbers from 0 to 5. During properly aligned operation there exists a one-to-one correspondence between sectors and phase configurations; Fig. 1 depicts this correspondence. Supplying the motor phases with the configurations shown in Fig. 1 the motor generates clockwise (CW) torque, which conventionally pushes the rotor toward increasing sector numbers and increasing numbers of encoder quadrature counts (QCs). We will refer to the number of QCs in a sector of 60 electrical degrees as n; typical values range from tens to hundreds of QCs per angular sector. If the motor drive fails to commutate the phase configuration, the rotor will continue to move beyond the prescribed commutation point and will stop in the equilibrium position corresponding to the phase configuration the motor driver is stuck into. With 2-phase-on operation, the equilibrium positions are located 60 electrical degrees beyond the prescribed commutation positions (Fig. 2). Equilibrium positions will be labeled with lowercase letters “a” to “f”; for example, with reference to Fig. 1, the equilibrium position of configuration U V¯ is point “b”. These equilibrium positions have traditionally been exploited for the initial alignment: the BLDC motor is forced through a series of fixed phase supply configurations; after the rotor is stuck in a known equilibrium position, encoder pulse counting can begin and aligned operation can be achieved from then on. The traditional method of alignment described above is very simple, but it can generate vibrations and oscillations unacceptable for certain applications. The next section presents a dynamical analytical model that justifies this behavior.
Fig. 2. Rotor equilibrium position “b” corresponding to phase configuration U V¯ .
3. BLDC motor dynamic model BLDC motors operating in “2-phase-on” mode can be modeled with simple equations similar to those that describe permanent magnet DC motors. BLDC motors have a trapezoidal back-emf; nevertheless, to this paper aims, a first harmonic sinusoidal model is sufficient for describing the motor behavior. The electromagnetic torque can be written as: TEM = −kT · i · sin ϑ
(1)
where kT is the machine torque constant, i is the current flowing in the supplied phases, and ϑ is the electrical angle between the stator and rotor magnetic fields (torque angle). The minus sign accounts for the electromagnetic torque acting toward decreasing the torque angle. At the low speeds typical of initial alignment viscous and ventilation friction are negligible; the only friction torque is due to the rolling of the ball bearings, and the dynamical behavior of the motor can be described by a motion equation with Coulomb friction: TEM − TL = J
dω ± TF dt
(2)
In Eq. (2) TL is the load torque, J the combined moment of inertia of rotor and mechanical load, ω the mechanical speed of the rotor, and TF the Coulomb friction torque, with constant magnitude and direction opposite to speed. If no phase commutation occurs the stator magnetic field remains static, and a differential relationship can be written between the torque angle and the mechanical speed: ω=
1 dϑ , P dt
(3)
where P is the number of motor pole pairs. Assuming for simplicity null static load torque (TL = 0) and substituting (1) and (3) into (2) yields, in case of positive speed: J d2 ϑ − TF = −kT · i · sin ϑ P dt 2
(4)
Near the equilibrium points it is ϑ → 0 and sinϑ ∼ = ϑ: P · kT · i d2 ϑ ·ϑ =0 − TF + J dt 2
(5)
which leads to a damped oscillation with pulsation:
ωn =
P · kT · i J
(6)
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Fig. 4. Simulation results of the traditional BLDC starting sequence with Coulomb friction. Starting position is in sector 2. Fig. 3. Flowchart of the traditional alignment method.
Eq. (5) has no simple solution because the sign of TF depends on the direction of movement; every half cycle must be considered separately, using the final position attained during a half period as the starting position for the next one. A simplified method which allows a closed form solution is to replace Coulomb friction with an equivalent viscous friction, such that the energy dissipated in the first oscillation is the same for the two kinds of friction. The equivalent viscous friction coefficient kF can be calculated as [14]: kF =
4 · TF · ωn · ϑ
(7)
where ϑ is the angular amplitude of the first oscillation, which can be approximated with half the angle corresponding to an angular sixth. This approach is advantageous if we are mainly interested in the magnitude of the oscillations. The equation with viscous friction becomes: P · kT · i kF dϑ d2 ϑ + + · ·ϑ =0 J J dt dt 2
(8)
If kF2 − 4 · P · kT · i/J < 0 (typical in case of small friction coefficient) the differential Eq. (8) has a damped oscillating solution. Before stopping, the rotor oscillates around the equilibrium position; oscillations have the same natural pulsation ωn calculated by (6) and a damping coefficient given by: =
kF 2
1 . J · P · kT · i
that the motor is not stuck (it should move n QCs in the desired direction). Two sets of simulations were performed, one using Coulomb friction and the other with the equivalent viscous friction calculated by (7). Fig. 4 was obtained using Coulomb friction. The motor starts with the rotor initially in sector 2, phase configuration U V¯ is applied ¯ . The stroke for 0.4 s, then the phases switch to configuration U W of the second configuration (and all eventual subsequent ones) is always one angular sixth corresponding to n QCs, while the stroke resulting from the application of the first configuration can vary from zero to 3n in the worst case, and the amplitude of the resulting oscillations is proportional to the stroke. Coulomb friction leads to an arithmetic progression of the cycle amplitude until the rotor comes to a halt. Fig. 5 has been obtained in the same conditions as Fig. 4 except with viscous friction equivalent to the given Coulomb friction. In this case the amplitude of the oscillations follows a decreasing exponential progression with theoretically infinite duration. Fig. 6 compares the rotor position obtained with the two friction models. It can be seen that equivalent viscous friction can be used as a good approximation of Coulomb friction as far as the amplitude and frequency of the oscillations is concerned.
(9)
If the inertia and/or the supplied current is high and the friction coefficient is low, the resulting oscillations can be ample and persistent. 4. Traditional starting sequence simulation Simulations have been performed in the MATLAB/PLECS environment in order to verify the model presented in the previous section. The simulated machine is a 48 V, 500 W, 5 Nm, 6 pole pairs BLDC motor with J = 0.0015 kg m2 and TF = 0.025 Nm. The incremental encoder has 500 pulses per revolution, resulting in 2000 QCs per revolution and 55.6 QCs per angular sector. A typical traditional starting sequence was applied in which the motor is supplied with a prefixed sequence of two adjacent phase configurations, as shown in Fig. 3). The first configuration aligns the rotor in a known position, the second one can be used to verify
Fig. 5. Simulation results of the traditional BLDC starting sequence with equivalent viscous friction. Starting position is in sector 2.
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The procedure starts in iteration 1 by supplying a fixed phase configuration (U V¯ ) with a constant current whose value suffices for moving the motor even in case of maximum load. After the detection of a few coherent QCs in the same direction the algorithm enters iteration 2, in which the phase configuration is selected according to the direction of movement detected in iteration 1: the ¯ if the rotor moved counterclockwise, W V¯ next configuration is U W if it moved clockwise. The procedure is carried on with the same logic until one of the following events happens: • a reversal of direction is detected, or • no reversal is detected after applying three different phase configurations. Fig. 6. Rotor position as a result of simulations of the traditional BLDC starting sequence. Comparison between Coulomb friction (dark trace) and viscous friction (light trace).
5. Proposed alignment algorithm The proposed alignment method is an iterative algorithm and Fig. 7 shows its corresponding flow chart. During every iteration the motor is supplied with a suitable phase configuration. Each iteration ends as soon as the direction of rotation is univocally determined, i.e. when a few consecutive encoder steps are coherently measured in the same direction.
At this point the starting sector is univocally determined. The particular case in which the rotor is already in the equilibrium position corresponding to the supplied phase configuration is managed by a timeout condition (not represented in the flowchart of Fig. 7). If the rotor is not already in an equilibrium position, after determining the starting sector it is necessary to properly align the encoder to ensure that subsequent phase commutations occur at the correct points. This is done by supplying the motor with the phase configuration corresponding to the sector preceding the determined one and controlling the speed to prevent vibrations
Fig. 7. Flowchart of the proposed algorithm for determining the initial rotor position.
C. Concari et al. / Electric Power Systems Research 95 (2013) 225–231
Fig. 8. Typical rotor position profile during the proposed alignment procedure.
until the rotor stops in the corresponding equilibrium position after, at most, 60 electrical degrees. An absolute match between quadrature counts and rotor position is finally established, and normal operation can ensue. Fig. 8 shows the typical smooth rotor position profile obtained during the whole starting sector determination and alignment procedure. Two examples can help understanding the operation of the proposed algorithm. In the first example the initial position of the rotor is in sector 2, that is between equilibrium positions b and c (Fig. 9(a)). During iteration 1 of the algorithm, the drive supplies the motor phases with configuration U V¯ : the rotor moves counterclockwise (Fig. 9(b)). When the algorithm detects a few consistent QCs in the
229
CCW direction it enters iteration 2 switching the motor phases to ¯ . This time the rotor moves clockwise (Fig. 9(c)); configuration U W after a few QCs are detected, the initial position is univocally determined to lie in sector 2. In the second example the initial position of the rotor is in sector 4 (Fig. 10(a)). During iteration 1 of the algorithm, the motor drive supplies the motor phases with configuration U V¯ : the rotor moves counterclockwise (Fig. 10(b)). When the algorithm detects a few consistent QCs in the CCW direction, it enters iteration 2 switching ¯ . This time the rotor continthe motor phase to configuration U W ues moving counterclockwise (Fig. 10(c)). After detecting a few QCs the algorithm enters iteration 3 switching the motor phase to con¯ . The rotor still moves counterclockwise (Fig. 10(d)); figuration V W when a few QCs are detected, the initial position is univocally determined to lie in sector 4. This procedure allows the determination of the initial sector with minimum rotor movement. For example, if a threshold of five QCs is used to determine rotor direction, a maximum theoretical rotor angular displacement of 15 QCs is obtained, corresponding to a few mechanical degrees using standard encoders. This results in barely perceptible movement during motor drive initialization, even if inertia should cause a slightly larger rotor displacement as will be seen in the experimental results in Section 4.
6. Experimental results Experimental tests have been performed to confirm the validity of the proposed algorithm. The motor and encoder used for the tests have the same specifications as reported for the simulations in Section 4. Fig. 11 reports motor position in QCs as a function of time with the traditional alignment method. Motor windings are supplied
Fig. 9. Example of operation of the proposed algorithm with initial position in sector 2. Initial position (a); iteration 1 (b); iteration 2 (c).
Fig. 10. Example of operation of the proposed algorithm with initial position in sector 4. Initial position (a); iteration 1 (b); iteration 2 (c); iteration 3 (d).
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Fig. 11. Rotor position oscillations typical of traditional alignment procedures. Starting rotor position in sector 2 (a), starting rotor position in sector 4 (b).
Fig. 12. Rotor position profile obtained using the proposed algorithm with the rotor initially in sector 2 (a), enlargement of the starting sector determination and alignment procedures (b).
¯ , and each phase with prefixed phase configurations U V¯ and U W configuration lasts 400 ms. After the alignment, the normal operation ensues. The result in Fig. 11(a) is almost identical to the simulation result with Coulomb friction, and very similar to the result obtained in simulation with viscous friction. The behavior shown in Fig. 11 can be considered as the reference associated with the traditional starting strategy, upon which to improve using the proposed technique. This starting procedure causes large amounts of vibrations and resonances, especially if the rotor starts far from the equilibrium position of the first used phase configuration (Fig. 11(b)). Fig. 12(a) shows the results obtained applying the proposed algorithm when the rotor starts in sector 2, as in the first example
reported in Section 5. Fig. 12(b) is an enlargement of the starting sector determination and alignment procedures. Fig. 13 reports similar experimental results obtained when the initial rotor position lies in sector 4, as in the second example of Section 5. This is a worst case example in which the rotor displacement is maximum during the starting sector determination procedure. Figs. 12 and 13 were acquired using a threshold of 5 QCs for rotor movement detection. Due to rotor inertia the rotor displacement can result slightly larger than the theoretical maximum of 15 QCs (Fig. 12(b)), nevertheless a comparison of Figs. 12 and 13 with Fig. 11 shows that the proposed technique reduces the magnitude of the vibrations exhibited by the motor by almost an order of magnitude with respect to the traditional starting technique. The
Fig. 13. Rotor position profile obtained using the proposed algorithm with the rotor initially in sector 4 (a), enlargement of the starting sector determination and alignment procedures (b).
C. Concari et al. / Electric Power Systems Research 95 (2013) 225–231
proposed algorithm is also faster because it needs to wait a shorter time for vibrations to settle and for the rotor position to stabilize. 7. Conclusion Many industrial applications of BLDC motors require smooth and vibrationless operation during the whole work cycle, but traditional step-like techniques for the initial alignment of incremental encoders fail to deliver smooth motion during motor drive start-up. A new alignment algorithm for incremental encoder based BLDC drives has been presented in this paper with the aim of minimizing the amplitude of vibrations during startup. Experimental tests confirm the validity of the proposed algorithm and show a definite improvement in the smoothness of the starting procedure with respect to traditional alignment techniques. References [1] K.-W. Lee, D.-K. Kim, B.-T. Kim, B.-I. Kwon, A novel starting method of the surface permanent-magnet BLDC motors without position sensor for reciprocating compressor, IEEE Transactions on Industry Applications 44 (January–February (1)) (2008) 85–92. [2] D.-K. Kim, S.-H. Rhyu, K.-W. Lee, B.-T. Kim, D.-H. Chung, B.-I. Kwon, Comparison of starting method for position sensorless BLDC motor driven reciprocating compressor, in: IEEE-IAS 2008, Edmonton, Alberta, Canada, 5–9 October, 2008.
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[3] Z. Anping, W. Jian, Compare the starting methods of the position-sensorless BLDC motor, in: CCIE 2010, Wuhan, China, 5–6 June, 2010. [4] G.H. Jang, J.H. Park, J.H. Chang, Position detection and start-up algorithm of a rotor in a sensorless BLDC motor utilising inductance variation, IEE Proceedings on Electric Power Application 149 (March (2)) (2002) 137–142. [5] W.-J. Lee, S.-K. Sul, A new starting method of BLDC motors without position sensor, IEEE Transactions on Industry Application 42 (6) (2006) 1532–1538. [6] S. Nakashima, Y. Inagaki, I. Miki, Sensorless initial rotor position estimation of surface permanent-magnet synchronous motor, IEEE Transactions on Industry Application 36 (November–December) (2000) 1598–1603. [7] K. Tanaka, T. Yuzawa, R. Moriyama, I. Miki, Initial rotor position estimation for surface permanent magnet synchronous motor, IEEE IAC 4 (September–October) (2001) 2592–2597. [8] H. Hao, X. Guoqing, H. Bo, A new start method for sensorless brushless DC motor based on pulse injection, in: APPEEC 2009, Shenzhen, China, 27–31 March, 2009. [9] Y.-S. Lai, F.-S. Shyu, S.-S. Tseng, New initial position detection technique for three-phase brushless DC motor without position and current sensors, IEEE Transactions on Industry Application 39 (March–April) (2003) 485–491. [10] M. Tursini, R. Petrella, F. Parasiliti, Initial rotor position estimation method for PM motors, IEEE Transactions on Industry Application 39 (November–December) (2003) 1630–1640. [11] Y.-C. Chang, Y.-Y. Tzou, A new sensorless starting method for brushless DC motors without reversing rotation, in: IEEE-PESC 2007, Orlando, FL, USA, 17–21 June, 2007, pp. 619–624. [12] A. Murray, P. Kettle, F. Moynihan, Advances in brushless motor control, in: ACC’97, Albuquerque, NM, USA, 4–6 June, vol. 6, 1997, pp. 3685–3689. [13] G. Franceschini, S. Raboni, M. Pignedoli, C. Concari, P. Giorgioni, Single current sensing for brushless DC drives: a fully digital solution, in: EPE 2003, Toulouse, France, 2–4 September, 2003. [14] W. Weaver Jr., S.P. Timoshenko, D.H. Young, Vibration Problems in Engineering, 5th ed, Wiley-Interscience, Hoboken, NJ, USA, January 1990.
Contents lists available at SciVerse ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Vibrationless alignment algorithm for incremental encoder based BLDC drives C. Concari, G. Franceschini, A. Toscani ∗ Department of Information Engineering, University of Parma, Viale G.P. Usberti, 181/A, 43124 Parma, Italy
a r t i c l e
i n f o
Article history: Received 7 October 2011 Received in revised form 31 July 2012 Accepted 21 September 2012 Available online 1 November 2012 Keywords: Brushless machines Alignment Digital control Modeling Starting Vibration control
a b s t r a c t Incremental encoder based BLDC drives are widely employed in industry whenever smooth and efficient operation is needed in the whole speed range. Their weak spot lies in the need for a phase of initial alignment, traditionally performed by forcing the motor through a series of fixed phase supply configurations in an open loop stepping mode. This strategy of alignment causes ample mechanical vibrations which can be intolerable for certain applications. This paper presents a novel algorithm for determining the initial rotor sector and aligning the encoder with the motor while minimizing vibrations and oscillations. © 2012 Elsevier B.V. All rights reserved.
1. Introduction DC brushless motors are widely employed in industry due to their high performances and ruggedness. Since such motors have no brushes, they need a solid state commutation circuit in order to supply the stator windings according to rotor position. Rotor position, therefore, must be determined, and this is typically performed using Hall sensors, incremental encoders or sensorless techniques. Hall sensors allow a great simplicity of control but they limit the operating temperature of the motor, need precise alignment at the factory and allow only coarse speed estimation at low speed. Sensorless techniques are widely used especially in applications which run mainly at medium to high speeds, such as fans and compressors; in such applications the reduction of vibrations during start-up is welcome but not mandatory [1–3]. Other techniques exploit inductance variation with rotor position [4,5], the d-axis current after applying voltage pulses [6,7] or signal injection [8] for initial position sensing without appreciably moving the rotor. Ref. [9] detects the initial rotor position by the time periods of discharge of stator windings, which are excited before discharge. Another initial position detection method combines an iterative sequence of voltage pulses with a fuzzy logic processing of the current response and phase currents derivation based on the
∗ Corresponding author. Tel.: +39 0521 906007; fax: +39 0521 905822. E-mail addresses: [email protected] (C. Concari), [email protected] (G. Franceschini), [email protected] (A. Toscani). 0378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.epsr.2012.09.010
DC-link current measurements [10]. Some algorithms are specific for avoiding reverse motion in unidirectional applications [11]. All sensorless techniques rely on the measurement of electrical variables such as currents and BEMFs, and they cannot be used at low speed due to signal/noise ratio degradation. Sensorless algorithms typically revert to step operation at low speeds, which leads to low efficiency and high vibration levels during start-up (even higher than with a traditional incremental encoder starting algorithm as described in Section 4). Incremental encoders allow optimal speed and position measurement over the full drive speed range, justifying the additional cost for applications in which the motor frequently accelerates from standstill or operates at low speed for extended periods of time [12,13]. Moreover, when used in clean environments such as food processing factories, they do not significantly reduce the reliability of the motor drive. A typical example of such an application is represented by weighing tension rollers used in the manufacturing and packaging industry, in which properly aligned encoder-based BLDC drives ensure smooth operation preventing objects with high center of gravity (e.g. bottles) from falling during acceleration and deceleration. The downside of this approach is that when the plant is powered on, e.g. at the beginning of the workday or after a power outage, incremental encoder based motor drives need initial alignment. While a smooth operation is important in every phase of the work cycle, alignment is typically performed by supplying motor windings with a predetermined sequence, which results in vibrations and oscillations with the risk of dropping the transported objects.
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Fig. 1. BLDC back-emf (top) and proper phase currents (bottom). Sectors, phase configurations and equilibrium positions are shown.
This paper presents a novel algorithm that allows determining the initial rotor sector and aligning the incremental encoder while minimizing vibrations and oscillations. In the following, Section 2 establishes notational conventions useful in later sections; Section 3 develops a simple dynamical analytical model of the BLDC motor; Section 4 introduces the proposed vibrationless alignment method; Section 5 reports experimental results; finally, Section 6 concludes the paper with the final remarks. 2. BLDC operation and definitions Before entering the details of the proposed alignment algorithm, it is useful to establish a set of definitions and conventions which describe the operation of BLDC motors and motor drives. Considering 2-phase-on operation, phase configurations will be referred to by stating the positive and negative supplied phases, e.g. phase configuration U V¯ means that phase U is supplied with positive current, phase V with negative current and phase W is floating. Rotor position can be said to lie in one of six angular sectors of 60 electrical degrees, labeled with numbers from 0 to 5. During properly aligned operation there exists a one-to-one correspondence between sectors and phase configurations; Fig. 1 depicts this correspondence. Supplying the motor phases with the configurations shown in Fig. 1 the motor generates clockwise (CW) torque, which conventionally pushes the rotor toward increasing sector numbers and increasing numbers of encoder quadrature counts (QCs). We will refer to the number of QCs in a sector of 60 electrical degrees as n; typical values range from tens to hundreds of QCs per angular sector. If the motor drive fails to commutate the phase configuration, the rotor will continue to move beyond the prescribed commutation point and will stop in the equilibrium position corresponding to the phase configuration the motor driver is stuck into. With 2-phase-on operation, the equilibrium positions are located 60 electrical degrees beyond the prescribed commutation positions (Fig. 2). Equilibrium positions will be labeled with lowercase letters “a” to “f”; for example, with reference to Fig. 1, the equilibrium position of configuration U V¯ is point “b”. These equilibrium positions have traditionally been exploited for the initial alignment: the BLDC motor is forced through a series of fixed phase supply configurations; after the rotor is stuck in a known equilibrium position, encoder pulse counting can begin and aligned operation can be achieved from then on. The traditional method of alignment described above is very simple, but it can generate vibrations and oscillations unacceptable for certain applications. The next section presents a dynamical analytical model that justifies this behavior.
Fig. 2. Rotor equilibrium position “b” corresponding to phase configuration U V¯ .
3. BLDC motor dynamic model BLDC motors operating in “2-phase-on” mode can be modeled with simple equations similar to those that describe permanent magnet DC motors. BLDC motors have a trapezoidal back-emf; nevertheless, to this paper aims, a first harmonic sinusoidal model is sufficient for describing the motor behavior. The electromagnetic torque can be written as: TEM = −kT · i · sin ϑ
(1)
where kT is the machine torque constant, i is the current flowing in the supplied phases, and ϑ is the electrical angle between the stator and rotor magnetic fields (torque angle). The minus sign accounts for the electromagnetic torque acting toward decreasing the torque angle. At the low speeds typical of initial alignment viscous and ventilation friction are negligible; the only friction torque is due to the rolling of the ball bearings, and the dynamical behavior of the motor can be described by a motion equation with Coulomb friction: TEM − TL = J
dω ± TF dt
(2)
In Eq. (2) TL is the load torque, J the combined moment of inertia of rotor and mechanical load, ω the mechanical speed of the rotor, and TF the Coulomb friction torque, with constant magnitude and direction opposite to speed. If no phase commutation occurs the stator magnetic field remains static, and a differential relationship can be written between the torque angle and the mechanical speed: ω=
1 dϑ , P dt
(3)
where P is the number of motor pole pairs. Assuming for simplicity null static load torque (TL = 0) and substituting (1) and (3) into (2) yields, in case of positive speed: J d2 ϑ − TF = −kT · i · sin ϑ P dt 2
(4)
Near the equilibrium points it is ϑ → 0 and sinϑ ∼ = ϑ: P · kT · i d2 ϑ ·ϑ =0 − TF + J dt 2
(5)
which leads to a damped oscillation with pulsation:
ωn =
P · kT · i J
(6)
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Fig. 4. Simulation results of the traditional BLDC starting sequence with Coulomb friction. Starting position is in sector 2. Fig. 3. Flowchart of the traditional alignment method.
Eq. (5) has no simple solution because the sign of TF depends on the direction of movement; every half cycle must be considered separately, using the final position attained during a half period as the starting position for the next one. A simplified method which allows a closed form solution is to replace Coulomb friction with an equivalent viscous friction, such that the energy dissipated in the first oscillation is the same for the two kinds of friction. The equivalent viscous friction coefficient kF can be calculated as [14]: kF =
4 · TF · ωn · ϑ
(7)
where ϑ is the angular amplitude of the first oscillation, which can be approximated with half the angle corresponding to an angular sixth. This approach is advantageous if we are mainly interested in the magnitude of the oscillations. The equation with viscous friction becomes: P · kT · i kF dϑ d2 ϑ + + · ·ϑ =0 J J dt dt 2
(8)
If kF2 − 4 · P · kT · i/J < 0 (typical in case of small friction coefficient) the differential Eq. (8) has a damped oscillating solution. Before stopping, the rotor oscillates around the equilibrium position; oscillations have the same natural pulsation ωn calculated by (6) and a damping coefficient given by: =
kF 2
1 . J · P · kT · i
that the motor is not stuck (it should move n QCs in the desired direction). Two sets of simulations were performed, one using Coulomb friction and the other with the equivalent viscous friction calculated by (7). Fig. 4 was obtained using Coulomb friction. The motor starts with the rotor initially in sector 2, phase configuration U V¯ is applied ¯ . The stroke for 0.4 s, then the phases switch to configuration U W of the second configuration (and all eventual subsequent ones) is always one angular sixth corresponding to n QCs, while the stroke resulting from the application of the first configuration can vary from zero to 3n in the worst case, and the amplitude of the resulting oscillations is proportional to the stroke. Coulomb friction leads to an arithmetic progression of the cycle amplitude until the rotor comes to a halt. Fig. 5 has been obtained in the same conditions as Fig. 4 except with viscous friction equivalent to the given Coulomb friction. In this case the amplitude of the oscillations follows a decreasing exponential progression with theoretically infinite duration. Fig. 6 compares the rotor position obtained with the two friction models. It can be seen that equivalent viscous friction can be used as a good approximation of Coulomb friction as far as the amplitude and frequency of the oscillations is concerned.
(9)
If the inertia and/or the supplied current is high and the friction coefficient is low, the resulting oscillations can be ample and persistent. 4. Traditional starting sequence simulation Simulations have been performed in the MATLAB/PLECS environment in order to verify the model presented in the previous section. The simulated machine is a 48 V, 500 W, 5 Nm, 6 pole pairs BLDC motor with J = 0.0015 kg m2 and TF = 0.025 Nm. The incremental encoder has 500 pulses per revolution, resulting in 2000 QCs per revolution and 55.6 QCs per angular sector. A typical traditional starting sequence was applied in which the motor is supplied with a prefixed sequence of two adjacent phase configurations, as shown in Fig. 3). The first configuration aligns the rotor in a known position, the second one can be used to verify
Fig. 5. Simulation results of the traditional BLDC starting sequence with equivalent viscous friction. Starting position is in sector 2.
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The procedure starts in iteration 1 by supplying a fixed phase configuration (U V¯ ) with a constant current whose value suffices for moving the motor even in case of maximum load. After the detection of a few coherent QCs in the same direction the algorithm enters iteration 2, in which the phase configuration is selected according to the direction of movement detected in iteration 1: the ¯ if the rotor moved counterclockwise, W V¯ next configuration is U W if it moved clockwise. The procedure is carried on with the same logic until one of the following events happens: • a reversal of direction is detected, or • no reversal is detected after applying three different phase configurations. Fig. 6. Rotor position as a result of simulations of the traditional BLDC starting sequence. Comparison between Coulomb friction (dark trace) and viscous friction (light trace).
5. Proposed alignment algorithm The proposed alignment method is an iterative algorithm and Fig. 7 shows its corresponding flow chart. During every iteration the motor is supplied with a suitable phase configuration. Each iteration ends as soon as the direction of rotation is univocally determined, i.e. when a few consecutive encoder steps are coherently measured in the same direction.
At this point the starting sector is univocally determined. The particular case in which the rotor is already in the equilibrium position corresponding to the supplied phase configuration is managed by a timeout condition (not represented in the flowchart of Fig. 7). If the rotor is not already in an equilibrium position, after determining the starting sector it is necessary to properly align the encoder to ensure that subsequent phase commutations occur at the correct points. This is done by supplying the motor with the phase configuration corresponding to the sector preceding the determined one and controlling the speed to prevent vibrations
Fig. 7. Flowchart of the proposed algorithm for determining the initial rotor position.
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Fig. 8. Typical rotor position profile during the proposed alignment procedure.
until the rotor stops in the corresponding equilibrium position after, at most, 60 electrical degrees. An absolute match between quadrature counts and rotor position is finally established, and normal operation can ensue. Fig. 8 shows the typical smooth rotor position profile obtained during the whole starting sector determination and alignment procedure. Two examples can help understanding the operation of the proposed algorithm. In the first example the initial position of the rotor is in sector 2, that is between equilibrium positions b and c (Fig. 9(a)). During iteration 1 of the algorithm, the drive supplies the motor phases with configuration U V¯ : the rotor moves counterclockwise (Fig. 9(b)). When the algorithm detects a few consistent QCs in the
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CCW direction it enters iteration 2 switching the motor phases to ¯ . This time the rotor moves clockwise (Fig. 9(c)); configuration U W after a few QCs are detected, the initial position is univocally determined to lie in sector 2. In the second example the initial position of the rotor is in sector 4 (Fig. 10(a)). During iteration 1 of the algorithm, the motor drive supplies the motor phases with configuration U V¯ : the rotor moves counterclockwise (Fig. 10(b)). When the algorithm detects a few consistent QCs in the CCW direction, it enters iteration 2 switching ¯ . This time the rotor continthe motor phase to configuration U W ues moving counterclockwise (Fig. 10(c)). After detecting a few QCs the algorithm enters iteration 3 switching the motor phase to con¯ . The rotor still moves counterclockwise (Fig. 10(d)); figuration V W when a few QCs are detected, the initial position is univocally determined to lie in sector 4. This procedure allows the determination of the initial sector with minimum rotor movement. For example, if a threshold of five QCs is used to determine rotor direction, a maximum theoretical rotor angular displacement of 15 QCs is obtained, corresponding to a few mechanical degrees using standard encoders. This results in barely perceptible movement during motor drive initialization, even if inertia should cause a slightly larger rotor displacement as will be seen in the experimental results in Section 4.
6. Experimental results Experimental tests have been performed to confirm the validity of the proposed algorithm. The motor and encoder used for the tests have the same specifications as reported for the simulations in Section 4. Fig. 11 reports motor position in QCs as a function of time with the traditional alignment method. Motor windings are supplied
Fig. 9. Example of operation of the proposed algorithm with initial position in sector 2. Initial position (a); iteration 1 (b); iteration 2 (c).
Fig. 10. Example of operation of the proposed algorithm with initial position in sector 4. Initial position (a); iteration 1 (b); iteration 2 (c); iteration 3 (d).
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Fig. 11. Rotor position oscillations typical of traditional alignment procedures. Starting rotor position in sector 2 (a), starting rotor position in sector 4 (b).
Fig. 12. Rotor position profile obtained using the proposed algorithm with the rotor initially in sector 2 (a), enlargement of the starting sector determination and alignment procedures (b).
¯ , and each phase with prefixed phase configurations U V¯ and U W configuration lasts 400 ms. After the alignment, the normal operation ensues. The result in Fig. 11(a) is almost identical to the simulation result with Coulomb friction, and very similar to the result obtained in simulation with viscous friction. The behavior shown in Fig. 11 can be considered as the reference associated with the traditional starting strategy, upon which to improve using the proposed technique. This starting procedure causes large amounts of vibrations and resonances, especially if the rotor starts far from the equilibrium position of the first used phase configuration (Fig. 11(b)). Fig. 12(a) shows the results obtained applying the proposed algorithm when the rotor starts in sector 2, as in the first example
reported in Section 5. Fig. 12(b) is an enlargement of the starting sector determination and alignment procedures. Fig. 13 reports similar experimental results obtained when the initial rotor position lies in sector 4, as in the second example of Section 5. This is a worst case example in which the rotor displacement is maximum during the starting sector determination procedure. Figs. 12 and 13 were acquired using a threshold of 5 QCs for rotor movement detection. Due to rotor inertia the rotor displacement can result slightly larger than the theoretical maximum of 15 QCs (Fig. 12(b)), nevertheless a comparison of Figs. 12 and 13 with Fig. 11 shows that the proposed technique reduces the magnitude of the vibrations exhibited by the motor by almost an order of magnitude with respect to the traditional starting technique. The
Fig. 13. Rotor position profile obtained using the proposed algorithm with the rotor initially in sector 4 (a), enlargement of the starting sector determination and alignment procedures (b).
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proposed algorithm is also faster because it needs to wait a shorter time for vibrations to settle and for the rotor position to stabilize. 7. Conclusion Many industrial applications of BLDC motors require smooth and vibrationless operation during the whole work cycle, but traditional step-like techniques for the initial alignment of incremental encoders fail to deliver smooth motion during motor drive start-up. A new alignment algorithm for incremental encoder based BLDC drives has been presented in this paper with the aim of minimizing the amplitude of vibrations during startup. Experimental tests confirm the validity of the proposed algorithm and show a definite improvement in the smoothness of the starting procedure with respect to traditional alignment techniques. References [1] K.-W. Lee, D.-K. Kim, B.-T. Kim, B.-I. Kwon, A novel starting method of the surface permanent-magnet BLDC motors without position sensor for reciprocating compressor, IEEE Transactions on Industry Applications 44 (January–February (1)) (2008) 85–92. [2] D.-K. Kim, S.-H. Rhyu, K.-W. Lee, B.-T. Kim, D.-H. Chung, B.-I. Kwon, Comparison of starting method for position sensorless BLDC motor driven reciprocating compressor, in: IEEE-IAS 2008, Edmonton, Alberta, Canada, 5–9 October, 2008.
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