Recent Developments of Electrical Drives - Part 16 potx

II-1. A GENERAL DESCRIPTION OF HIGH-FREQUENCY POSITION ESTIMATORS FOR INTERIOR PERMANENT-MAGNET SYNCHRONOUS MOTORS Frederik M.L.L. De Belie, Jan A.A. Melkebeek, Kristof R. Geldhof, Lieven Vandevelde and Ren´e K. Boel Electrical Energy Laboratory (EELAB), Department of Electrical Energy, Systems and Automation (EESA), Ghent University (UGent), Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium frederik.debelie@ugent.be, Jan.Melkebeek@ugent.be, Kristof.Geldhof@ugent.be Lieven.Vandevelde@ugent.be, Rene.boel@ugent.be Abstract. This paper discusses fundamental equations used in high-frequency signal based interior permanent-magnet synchronous motor (IPMSM) position estimators. For this purpose, an IPMSM model is presented that takes into account the nonlinear magnetic condition, the magnetic interaction between the two orthogonal magnetic axes and the multiple saliencies. Using the novel equations, some recently proposed motion-state estimators are described. Simulation results reveal the position estimation error caused by estimators that neglect the presence of multiple saliencies or that consider the magnetizing current in the d-axis only. Introduction Vector control of a high-dynamical, high-performance interior permanent-magnet syn- chronous motor (IPMSM) requires the stator flux linkage vector. For small stator currents, this flux is mainly generated by the high-grade permanent magnets, buried within the rotor. In a lot of drives, using field-oriented control, the rotor flux vector is considered instead of the stator flux linkage vector. Moreover, the rotor flux direction can be approximated by the rotor position, measured with a mechanical sensor. During the last 15 years, motion-state estimation methods have been developed with the intention to remove the expensive mechanical transducer, which, due to temperature variations and mechanical vibrations, produces measurements of low reliability. Modern sensorless drives try to estimate themotionstatesfrom measurements ofelectricalvariables. Filtering techniques and obser ving strategies are used to estimate the back-EMF vector and from that the rotor speed and angle. However, for a slow rotor motion, small signals have to be measured orcalculatedthatare disturbed strongly bynoise produced by normaloperation of the PWM and motor. As a result, the precision of such estimators in the low speed region is insufficient to control the motor in a stable and efficient way. S. Wiak, M. Dems, K. Kom ˛ eza (eds.), Recent Developments of Electrical Drives, 141–153. C  2006 Springer. 142 De Belie et al. To obtain accurate position estimations at low speed, in recently proposed estimation methods, a high-frequency voltage are supplied which generates a high-frequency variation of small amplitude in the stator flux linkage [1–8]. In an IPMSM, without damper effects e.g. dueto short-circuited windings or eddy currents, thisflux variation mainly occursin the main flux instead of the leakage flux path. If saturation occurs or an important reluctance variation along the air gap exists, it will be shown that the high-frequency current response will be modulated with the air-gap flux position additionally to the rotor angle. For an IPMSM, considerable reluctance variations, called magnetic saliencies, can be detected due to the buried placement of the magnets within the rotor. Furthermore, several stator teeth are saturated due to the presence of an important permanent magnetic flux. This paper discusses fundamental equationsusedinhigh-frequency signal based IPMSM positionestimators.Forthispurpose,thesmallsignaldynamicfluxmodel,presentedin[9],is used whichtakesintoaccount the nonlinear magnetic conditionand themagnetic interaction between the direct and the quadrature magnetic axis. An addition to the model is given to tackle the presence of multiple saliencies. By using the novel equations some recently proposed motion-state estimators are described. It is shown that the higher the inductance difference between the two orthogonal magnetic axes, the higher the position estimation resolution. Furthermore, simulation results reveal the position estimation error caused by estimatorsthatdisregard theexistenceofmultiplesaliencies or thatconsiderthemagnetizing current vector in the d-axis only. General description of a PMSM Small signal dynamic flux model To obtain accurate position estimations at low speed, in recently proposed estimation meth- ods a high-frequency voltage is supplied, which generates a high-frequency variation of small amplitude in the stator, flux linkage [1–8]. This implies that, to describe position esti- mators, a small signal dynamic flux model can be used. In an IPMSM, without the damper effects e.g. due to short-circuited windings or eddy currents, the flux variation generated by the high-frequency voltage mainly occurs in the main flux φ m instead of the leakage flux path. As a result, the small signal dynamic model of a saturated synchronous machine, presented in [9], can be used. This model is given by the flux equation  qd m (t) = ⎛ ⎜ ⎝ L qmt cos 2μ + L qmo sin 2 μ 1 2 (L qmt − L qmo ) sin (2μ) 1 2 (L dmt − L dmo ) sin ( 2μ ) L dmt sin 2μ + L dmo cos 2 μ ⎞ ⎟ ⎠ i qd m (t) (1) written in a reference frame (qd) fixed to the physical quadrature and direct axis and with i m the magnetizing current, with, see Fig. 1, L qmo , L dmo the chord-slope magnetizing inductances and L qmt , L dmt the tangent-slope magnetizing inductances in quadrature and direct magnetic axis respectively, μ the angle between the q-axis and the vector i m and  denoting the small variation of a vector. In a current controlled drive, the vector i m and the angle μ are regulated to a constant during steady state. By using the flux equation (1) the saturation level in both magnetic axes is assumed to be determined by i m and as a result the proposed small signal dynamic flux model includes cross saturation or magnetic II-1. High-Frequency Position Estimators 143 Figure 1. Magnetizing characteristicwith i mo and φ mo the average modulusof i m and φ m respectively. interaction between d- and q-axis.However, this model neglects possible statorleakage flux saturation. In some high-frequency signal based sensorless drives, a small high-frequency stator current is supplied instead of a voltage. Therefore, fundamental equations used in position estimators are given for current as well as voltage sources. Nevertheless, it will follow from the discussion that both methods can be described in a similar way. High-frequency current source Anestimationalgorithmusingahigh-frequencycurrentsourcemeasures the high-frequency flux response. For these estimators, the flux equation (1) is written in a complex notation, with the real axis parallel to the q-axis, as φ qd m (t) = l · i qd m (t) (2) with the complex inductance l given by l = L + L rel − L sat (3) where L = L q + L d 2 (4) L rel (α) = L q − L d 2 e −j2α(t) (5) L sat (α, μ) = L q − L d 2 e −j2μ + L q + L d 2 e j2(μ−α(t)) (6) L q = L qmo + L qmt 2 L d = L dmo + L dmt 2 (7) L q = L qmo − L qmt 2 L d = L dmo − L dmt 2 (8) 144 De Belie et al. with α(t) the angle between i m and the q-axis. Indicating the complex conjugate with the operator *, the relationship in (2) can alternatively be written as φ qd m (t) =  L q + L d 2 − L q − L d 2 e −j2μ  i qd m (t) +  L q − L d 2 − L q + L d 2 e j2μ  i ∗ qd m (t) (9) High-frequency voltage source A lot of position estimators use a high-frequency flux generated with a voltage source, and measure the current response. For these estimation methods, the flux equation (1) is written as i qd m (t) = r ·φ qd m (t) (10) with the complex reluctance r given by r = M(L + L rel + L sat ) (11) where M = 1 L q L d + (L q L d − L d L q ) cos(2μ) − L q L d (12) L = L q + L d 2 (13) L rel (β) =− L q − L d 2 e −j2β(t) (14) L sat (β, μ) =  − L q − L d 2 + L q + L d 2 e −j2β(t)  e j2μ (15) with β the angle between φ m and the q-axis. The relationship in (10) can alternatively be written as i qd m (t) = 1 (L q L d + (L q L d − L d L q ) cos(2μ) − L q L d ) ×  L q + L d 2 − L q − L d 2 e j2  φ qd m (t) −  L q − L d 2 − L q + L d 2 e j2μ  φ ∗ qd m (t)  (16) Discussion As most estimators are based on a current response to a high-frequency voltage variation, the following discussion will be restricted to such strategies. However, as the equations for an estimator, using a high-frequency current source, are similar to those in (10)–(16), the following discussion applies to both cases. From the reluctance r in (11) it can be seen that, in addition to a current change in phase with φ m due to L in (13), two important components in the current variation can be II-1. High-Frequency Position Estimators 145 distinguished. As follows from (14), a part of the current change is proportional to the inductance difference between q- and d-axis and is phase shifted from φ m over −2β. Another current variation, according to (15), is linked with the differences between the chord-slope inductance and the tangent-slope inductance in both q- and d-axis. If the saturation level is low, the chord-slope inductance equals almost the tangent-slope inductance. Consequently, the inductance in (15) becomes small and a phase shift between i m and φ m is the result of the inductance difference in (14) only. Clearly, the component in (14) reflects the reluctance variation along the air gap with extrema in both orthogonal magnetic axes. The reluctance r in pu, in the case of an unsaturated salient-pole synchronous machine, is shown in Fig. 2(a). It is given for various values of the inductance difference between the two magnetic axes. The trajectories of i m for a circular trajectory of φ m , shown in Fig. 2(b), are elliptical with axes of symmetry in q- and d-directions, corresponding to the point of minimum and maximum modulus of r respectively. Furthermore, for a given value of β, the higher the difference between the q- and d-inductance, the higher the angle between i m and φ m . Figure 2. Reluctance r in pu and current response to small flux variations in the case of magnetic saliency with β as parameter. 146 De Belie et al. Figure 3. Reluctance r in pu and current response to small flux variations in the case of saturation with β . as parameter. If the PMSM has a uniform air-gap permeance, most controllers disregard the reluctance variation along the air gap. Consequently, the direction of the q-axis fixed to the rotor can be chosen deliberately. Furthermore, the reciprocity property, mentioned in [10], implies that L dmo − L qmo = L dmt − L qmt = 0 (17) As a result, the difference in (14) becomes zero and the inductance (15) reduces to L sat (β, μ) = L q + L d 2 e −j2(β(t)−μ) (18) This means that a noticeable phase shift between i m and φ m is caused by (18) only. In the case of a saturated smooth air-gap synchronous machine the reluctance r in pu is presented in Fig. 3(a) for a given modulus of i m . This figure shows that the direction of i m influences the phase shift between i m and φ m for the same β. The trajectory of i m for a circular trajectory of φ m is shown in Fig. 3(b). The figure shows elliptical trajectories with a maximum i m in the direction of i m , corresponding the point in Fig 3(a) with the maximum modulus of r . Multiple saliencies Due to the construction of the rotor, the reluctance variation along the air gap can display global extrema in q- and d-axis as well as several local extrema. Such a reluctance variation II-1. High-Frequency Position Estimators 147 Figure 4. Reluctance r in pu of an IPMSM with and without multiple saliencies with β as parameter. is called a multiple saliency. Assuming sinusoidal reluctance variations, these multiple saliencies can be modeled in a similar way as the reluctance variation with extrema in q- and d-axis only. As a result, by using previous discussion, the equation in (14) can be replaced for modeling multiple saliencies by L rel =  i L q,i − L d,i 2 e −j ( iβ(t)+ϕ i ) , i ∈ IN 0 (19) with ϕ i a possible space phase shift. To illustrate the model with multiple saliencies, the trajectory of r for an unsaturated salient-pole machine with β as parameter is calculated by using (19) instead of (14). Two cases, with and without an extra sinusoidal reluctance variation having four extrema per pole pitch (i equals to 2 and 4 in (19)), are shown in Fig. 4. This reluctance trajectory is also observablein [7] and mentionedin [8]. In [7],by applying finite element simulations,almost the same trajectory as in Fig. 4 can be observed. In [8] the effect of multiple saliencies is measured asa variation in thestator current insteadof aninductance. However, theseresults are not modeled such as in (19). Recently proposed estimators Approximated small signal dynamic flux model In an IPMSM the magnetizing current is mainly generated by the permanent magnets. For this reason, in some recently proposed sensorless drives, such as [1–8], the angle of i m is approximated by π/2. For μ equal to π/2, the equation (10) results in i dq m (t) = 1 L qmo L dmt  L qmo + L dmt 2 − L qmo − L dmt 2 e −j2β(t)  φ qd m (t) (20) By defining L = L qmo − L qmt 2 ,L = L qmo − L dmt 2 (21) the relationship (20) can also be written as i dq m (t) =  L L 2 − L 2 − L L 2 − L 2 e −j2β(t)  φ qd m (t) (22) 148 De Belie et al. In some estimators a stationary reference frame (αβ) is used. Furthermore, equation (16) ratherthantheonein(10)isconsidered.Transformationofavariationx fromthereference frame (qd ) to the stationary reference frame (αβ ), with the real axis parallel to the α-axis, is given by x qd (t) = e jθ r  jθ r x αβ 0 + x αβ  (23) with θ r the rotor angle defined as the angle between the α-axis and the q-axis and with x 0 the mean value of the vectors x at the beginning and end of the variation. Transformation of (16) to the stationary reference frame, by using (23), results in jθ r i αβ mo + i αβ m = L L 2 − LL 2  jθ r φ αβ mo + φ αβ m  − L L 2 − L 2  jθ r φ αβ mo + φ αβ m  ∗ e j2θ r (24) for μ equal to π/2. Inverting (24) jθ r φ αβ mo + φ αβ m = L  jθ r i αβ mo + i αβ m  + Le j2θ r  jθ r i αβ mo + i αβ m  (25) results in the matrix notation φ qd m (t) =  L +L cos ( 2θ r ) L sin ( 2θ r ) L sin ( 2θ r ) L −L cos ( 2θ r )  · i αβ m (t) (26) This equation is well-known as it shows the sinusoidal variation of the magnetic reluctance along the air gap with the pole pitch as period. Stator voltage equation By supplying a high-frequency voltage to the motor terminals, a high-frequency stator flux linkage variation of small amplitude is generated. In an IPMSM, without damper effects e.g. dueto short-circuited windings or eddy currents, thisflux variation mainly occursin the main flux instead of the leakage flux path. For this reason and by disregarding the voltage drop across the stator resistance and leakage inductance, the motor voltage equation, in a two-dimensional stationary reference system (αβ ), can be approximated by υ αβ s (t) = φ αβ m (t) t (27) with υ s the complex stator voltage. Transformation of (27) to the synchronous reference frame (qd) results in υ qd s (t) = φ qd m (t) t − j θ r t φ qd mo (t) (28) High-frequency voltage pulse train In modern IPMSM drives, a pulse-width-modulated (PWM) inverter is used. This means that, at normal operation, a voltage pulse train at high frequency is supplied to the motor II-1. High-Frequency Position Estimators 149 terminals. From equation (28) it follows that the current variation is piecewise linear. As a result, according to the model in (10), a magnetizing current variation occurs, which depends on the direction of the main flux variation in the reference frame (qd) and on the magnetizing current. For a current controlled drive, the current i m can be approximated by the desired i m calculated within the controller. Moreover, in the synchronous reference frame, the variation of i m equals i s , with i s the complex stator current, as the equivalent magnetizing current due to the magnets is constant. The high-frequency flux variation, generated by using a PWM, can be used to estimate the rotor angle. Calculating themain flux variation with (28)and transforming the measured stator current to an estimated synchronous reference frame (  qd ), it follows from (10) that an estimation of the reluctance r can be obtained. Substituting i m with its desired value calculated within the controller, the angle μ and the inductances in (7)–(8) can be approxi- mated, which result, together with the estimated r , in an estimation of the angle β. As the angle of the main flux variation can also be calculated in the stationary reference frame by using (27), a new estimation of the q-axis is obtained. If the motion-induced voltage is known, φ m can be calculated by using (28). However, for a slow rotor motion, a back-EMF of small amplitude has to be measured or calculated which is strongly disturbed by noise produced by normal operation of the PWM and motor. As for most drives the mechanical time constant is higher than the electrical one, the rotor speed and the motion-induced voltage can be assumed to be constant during a sufficiently small time period. Therefore, by subtracting the stator voltage generated by two successive PWM pulses, back-EMF measurements are avoided. Together with (10), this results in the following system φ qd m,2 − φ dq m,1 t = υ qd s,2 − υ qd s,1 (29) i qd s,2 − i dq s,1 t = r (β 2 ) φ qd m,2 t −r (β 1 ) φ qd m,1 t (30) In the sensorless drive presented in [1], the reluctance r is estimated by using equation (30). This method is called indirect flux detection by online reactance measurement (INFORM) as introduced by Schr¨odl. However, in such an estimator μ is approximated by π/2, which means thatthe reluctance r is estimatedby usingequation (22) instead of (10).Furthermore, a β 2 value is used that is equal to β 1 + π. As a result, the system of (29) and (30) together with the relationship in (22), results in υ qd s,1 =−υ qd s,2 (31) i qd s,2 − i dq s,1 t =  L L 2 − L 2 − L L 2 − L 2 e −j2β 2  2υ qd s,2 (32) as the reluctance r in (22) varies periodically with 2β. Furthermore, inthe INFORM method the estimation of r is repeated in the two other stator phases. The average reluctance of the three phases approximately coincides with r (β) + r(β + 2π 3 ) +r(β − 2π 3 ) 3 = L L 2 − L 2 (33) 150 De Belie et al. As a consequence by subtracting (33) from an estimation of the reluctance r in (32), calcu- lating 2β is doneby usingthe inverse tangentfunction only. Clearly, this method requires no knowledge about the inductances.However, it requiressuccessivefluxvariationsin opposite directions, which can disturbpropermotor operation [2]. Notethatthe higher theinductance difference between the two orthogonal magnetic axes, the smaller the rotor variation that can be detected. Sinusoidal high-frequency voltage Instead of using the PWM generated pulse train, in some estimators, such as in [3–5,8], the current response is measured on a sinusoidal high-frequency voltage within the stator voltage. In most of these strategies, calculations are performed in a stationary reference frame by using the equations in (24) and (27). The stator voltage is given by υ αβ s (t) = V αβ s (t) +υ αβ s,i (ω i t) (34) with V s a complex voltage and with ω i the pulsation of the injected high-frequency voltage υ s,i . With the voltage in (34), the voltage equation in (27), at a high frequency, is written as υ αβ s,i (ω i t) = dφ αβ m,i (ω i t) dt (35) Furthermore, at a sufficiently high frequency, the rotor angle can be assumed to be constant. Consequently, equation (24), for a sufficiently high frequency, results in di αβ s,i ( θ r ,ω i t ) dt = L L 2 − L 2 υ αβ s,i ( ω i t ) − L L 2 − L 2 e j2θ r  υ αβ s,i ( ω i t )  ∗ (36) In some methods a voltage rotating at a high pulsation ω i is added to the stator voltage υ αβ s,i (ω i t) = V i e jω i t (37) The high-frequency current response, obtained by using (36), will include a positive and negative rotating component i αβ s,i (θ r ,ω i t) = I 0 e jω i t + I 1 e j(2θ r −ω i t) (38) with I 0 = L L 2 − L 2 · V i ω i I 1 = L L 2 − L 2 · V i ω i (39) Transformation of the current response to a reference frame that rotates at ω i results in i ω i s,i (θ r ,ω i t) = I 0 + I 1 e j2(θ r −ω i t) (40) In other estimation methods a high-frequency voltage in an estimated quadrature axis is added to the fundamental voltage υ αβ s,i ( ˆ θ r ,ω i t) = V i cos ( ω i t ) e j ˆ θ r (41) . II-1. A GENERAL DESCRIPTION OF HIGH-FREQUENCY POSITION ESTIMATORS FOR INTERIOR PERMANENT-MAGNET SYNCHRONOUS MOTORS Frederik M.L.L. De Belie, Jan A.A. Melkebeek, Kristof R. Geldhof, Lieven. and Ren´e K. Boel Electrical Energy Laboratory (EELAB), Department of Electrical Energy, Systems and Automation (EESA), Ghent University (UGent), Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium frederik.debelie@ugent.be,. that consider the magnetizing current in the d-axis only. Introduction Vector control of a high-dynamical, high-performance interior permanent-magnet syn- chronous motor (IPMSM) requires the stator