Compensating the influence of the stator resistor and inverter nonlinearities in signal-injection based sensorless strategies Fabien Gabriel∗‡ , Frederik De Belie† , Pascal Druyts∗ , Xavier Neyt∗ , Jan Melkebeek† and Marc Acheroy∗ ∗ Signal
and Image Centre (SIC), Royal Military Academy (RMA) avenue de la Renaissance 30, 1000 Brussels, Belgium † Dept. Electrical Energy, Systems & Automation (EESA), Ghent University (UGent) St-Pietersnieuwstraat 41, 9000 Gent, Belgium ‡ phone: +32 2 742 66 65, fax: +32 2 742 64 72, e-mail:
[email protected]
Abstract—Among the sensorless position estimation methods in electrical machines drives, the injection of voltage test pulses is a promising strategy. Several papers are studying and applying this strategy, in particular for permanent magnet synchronous machine (PMSM) at low speed or standstill. Recently, an adaptive test pulses sequence reducing the current distortions has been proposed. However, the test pulses can be influenced by the voltage drops such as across stator resistor and semiconductor switches. By neglecting these voltage drops, the distortion in the current samples can not be fully reduced. In this paper, we improve the adaptive test pulses strategy by estimating and compensating the resistive drops. We also discuss the impact of the inverter nonlinearities. Finally, we present experimental results on a PMSM. Index Terms—Sensorless Control, Signal-injection, Adaptive Test Pulses, Parameters identification, Inverter nonlinearities.
I. I NTRODUCTION Nowadays, controlling the movements of electrical machines without mechanical speed or position sensor is an important topic of research [1]. The advantages are reduction of the cost and improvement of the reliability in rough environments, using only measurement of stator currents and/or voltages at motor terminals to estimate the speed or position. Applications are found in vehicle propulsion [2] and railway traction [3]. Sensorless methods based on the back electromotive force [4] have been proposed. However, those methods loose their precision at low rotor speed since the electromotive force disappears. Methods using signals injection are then better suited. Signal-injection methods exploit the magnetic saliences [5], also named magnetic anisotropic properties [6], of some electric machines due to geometric variation (salient poles or any variation of the air-gap) or due to magnetic saturation effects [6], [7]. These effects are modelled through inductance variations depending on the angular position of the saliences. By design, permanent-magnet synchronous machines (PMSM) have important magnetic saliences effects from magnetic saturation, which is due to the permanent-magnet mounted on the rotor surface. Therefore the signal-injection methods are suited to that type of machines in order to estimate the rotor
position. By using a signal-injection method, the response to the injected signal is measured to identify the salience position, following the concept presented in [7]. Two different signal approaches exist: either oscillating voltage signals are injected [6]; or the signals are formed by regular sequences of voltage pulses [7], [8], [9]. In that second approach, the sequences are named test pulses sequences or simply test sequences. The development proposed in this paper is based on the test pulses, recently improved with the adaptive test pulses method [10], [11] that will be further described. The voltage of the test pulses causes a current ripple from which an estimation of the salience position can be derived. Only current sensors, those used by the current controller, are required. No other specific sensors are necessary, consequently reducing the cost and the complexity. The injected test pulses may disturb the current controller. To avoid this, the adaptive test pulses sequence is computed in order to minimize as much as possible the distortion in the current samples that are used by the digital current controller. However, the current behaviour during the test sequence can be influenced by the voltage drops across the stator resistor and by the nonlinearities of the inverter. As a result, the distortion of the current samples will not be fully reduced by the adapted test pulses and this distortion may still disturb the current controller. In this paper, a compensation method correcting the applied test voltage in order to reduce the current distortion is presented. This compensation method is swiftly computed and can easily be added to any test pulses method. Among the inverter nonlinearities, the voltage drop of the semiconductor devices can be modelled around an operating point as a threshold voltage and a differential resistance. The presented method calculates the compensation using an equivalent model resistor including the stator resistor and that differential resistance of the semiconductors. The value of the equivalent model resistor varies significantly with the operating conditions and the stator’s temperature. It is therefore needed to estimate it online. The method performs an estimation of the resistor after each test pulse sequence.
Other inverter nonlinearities effects, as the dead time effects but also the limits of the semiconductor model, will also be discussed. They are mainly disturbing the adaptive test pulse strategies around zero current values and cannot be compensated by the presented method. Solutions will be suggested. Experimental results are performed on a wheel motor developed for electrical automotive propulsion projects. That wheel motor is a very low inductance PMSM designed to be powered by a low voltage supply. For this motor, the voltage drops are not negligible compared to the supplied voltage. Therefore their impact is of importance and must be compensated. The results are presented to compare the current response for the adaptive test pulse sequence with and without compensation method. The results show that the current distortion is significantly reduced with the proposed method. The paper is organized as follows: Section II describes the model of the machine and gives the expression of the current response used in the injection-based sensorless strategies. In Section III, the injection based strategies and the adaptive test pulse strategy are presented and the voltage drops considerations are introduced. Section IV develops the compensation method and the online identification of the resistor. In Section V, the inverter nonlinearities, and the specific problems related to the zero crossing current, are presented. Section VI shows the experimental results and Section VII concludes this paper. II. M ODEL OF THE MACHINE A. Stator circuit model The injection-based sensorless strategies are designed starting from the stator circuit model of the electrical machine written with spatial phasors in a qd reference-frame [12] attached to the salience. The circuit quantities are governed by V =
d Ψ +R·I +E dt l Ψl = L · I
(1a) (1b)
where V is the voltage phasor applied at the stator terminals, I the stator current phasor, Ψl the stator flux phasor induced by I, and E is the back electromotive force phasor induced in the stator by the variation of the rotor flux linking the stator. L is the stator inductance factor and R the stator resistor factor. They are represented by complex quantities as will be discussed in the next section. B. Inductance model The complex form of the inductance L in phasor formalism results from the interaction between the different inductances of the phases due to the magnetic salience. In a surface PMSM, the salience is due to the variation of the magnetic saturation level as a function of the angle, due to the presence of permanent magnets mounted on the rotor [4], [5]. As a consequence the inductance phasor is linked to the rotor position. This is the basis of the rotor position identification strategies [7].
m Y
−2γ e
yq
yd
Fig. 1: Inverse inductance Y locus as a function of the γ angle
C. Resistor model The resistor values perceived by the different phases may be slightly different. This difference is due to physical differences between the stator circuit phases. This can generally be reduced by a good design of the machine. In addition the resistor appearing in the model includes the effect of the voltage drops in the semiconductor devices of the inverter. Those parameters depend on the operating point, defined by the value of the current. As a result, the resistor can be different for each phase. In the phasor formalism, the resistor differences yield a complex factor R. D. Current response To express the current response relation as a function of the applied voltage, the equations are simplified by defining an inductance voltage phasor V l . This voltage corresponds to the voltage applied on the stator inductance d Ψ (2) dt l The inductance voltage applied between two measurement times of period Δt creates a flux variation ΔΨl following V l dt = ΔΨl = Δ (L · I) (3) Vl =V −R·I −E =
Δt
Assuming that the variation of the salience orientation between the two measurements can be neglected, the variation of the flux can be written Δ (L · I) ≈ L · ΔI. Finally, the expression of the current response becomes ΔI = Y · V˜ l Δt
(4)
−1
where Y = (L) is the inverse inductance and where ˜ = X Xdt Δt denotes the mean value of a phasor X Δt on a period of interest Δt. The identification of the inverse inductance is based on expression (4). This will be discussed in Section III. E. Inverse Inductance expression The expression of Y has been described in [4], [5] and can be written (5) Y = Y + YΔ e−j2γ with Y = (yq + yd ) /2 and YΔ = (yq − yd ) /2, and where γ is the angle of the phasor V˜ l with respect to the q direction. The extrema values yq and yd are measured for V˜ l oriented along
d
V˜ l,test
d
ΔI
test period
ΔI test
V˜test,1
γtest
V˜ l,contr
γcontr
I contr q
V˜test,2
V˜contr
ΔI contr q
V˜l,test,1
Fig. 2: Induction voltage phasor V˜ l,test added to the command
induction voltage phasor from the current controller V˜ l,contr , and their current phasor responses ΔI l,test and ΔI l,contr
q and d respectively. When the angle γ varies, Y describes a circle as illustrated in Fig.1. Reminding that the qd reference-frame is attached to the salience, and thus to the rotor in a PMSM, the determination of the angle γ allows to find the rotor orientation. This angle can be found from inverse inductance estimation and, for example, from an identification of yd and yq .
A. Signal-Injected Strategy
ΔItest,2
Icontr Idist
t0
T 2
t1
T 2
t2
Fig. 3: Injected test pulses, induction voltages and current responses
B. Inverse Inductance identification
The current variation resulting from the current controller normal voltage is generally too small to allow accurate inverse inductance identification, in particular in steady-state situations. Therefore, to perform the identification, a test voltage V test is injected in addition to the normal voltage V contr computed by the current controller. In a situation without test voltage, the induction voltage V l,contr is related to V contr and I contr by means of (2). Assuming rotation speed and hence E is not affected by the test injection, the total induction voltage V l,test +V l,contr when V test + V contr is applied gives V l,contr + V l,test = (6)
where I test is the additional test current. Subtracting the contribution V l,contr from that total induction voltage gives V l,test = V test − R · I test . Notice that this equation is independent of the back electromotive force E. The voltage applied to the stator terminals by means of PWM signals is in general not measured. However, neglecting the inverter nonlinearities, the mean value of the voltage applied during a PWM period is equal to the command voltage. Inverter nonlinearities will be discussed in Section V. In terms of mean values, the test induction voltage is V˜ l,test = V˜ test − R · I test
ΔItest,1
(valid for real and imaginary parts of the phasors)
III. A DAPTIVE T EST P ULSE S TRATEGY
V contr + V test − R · (I contr + I test ) − E
V˜l,test,2
(7)
where V˜ test is the test command voltage, and where the mean stator resistor voltage drop is defined by R ·I = (R · I) dt Δt. A method to estimate this voltage drop Δt will be presented in Section IV.
In a situation without test, the current response ΔI contr is related to V˜ l,contr by means of (4). The additional test induction voltage V˜ l,test will produce an additional current variation ΔI test , as illustrated in Fig.2. Assuming the injected signal does not significantly affect the local saturation levels, the superposition principle of the flux authorizes to use (4) to relate the test current variation ΔI test to V˜ l,test . Therefore, from the applied test induction voltage and its response, the inverse inductance can be estimated as follows 1 ΔI test V˜ l,test (8) Y (γtest ) = Δt where γtest is the direction of V˜ l,test . In practice, ΔI = ΔI contr + ΔI test is measured and ΔI test remains to be estimated. This will be discussed in Section III-E. C. Adaptive Test Pulse Strategy The adaptive test pulses strategy aims at reducing the effect of the injected signal on the current controller [10], [11]. Therefore, a test sequence is formed by two consecutive opposite test pulses that generate a transient test current. It is illustrated in Fig.3 which presents ideal constant applied test voltages, the related induction voltages and the current responses (valid for real and imaginary part of the phasors). The two test pulses V˜ test,1 and V˜ test,1 applied between t0 and t1 , and t1 and t2 respectively, produce the current responses ΔI test,1 and ΔI test,2 respectively. The remaining current variation I dist produced by the test sequence is given by I dist = ΔI test,1 + ΔI test,2
(9)
The current measurements used by the current controller are sampled at t0 and t2 . Therefore I dist is indeed the disturbance
seen by the current controller. If I dist = 0, the current controller will not be disturbed by the test sequence. According (4), this require that Y (γtest,1 )· V˜ l,test,1 = Y (γtest,2 )· V˜ l,test,2 . It will be the case if
¯ between the same and to define an average resistor R i moments satisfying
V˜ l,test,2 = −V˜ l,test,1
¯ and R ¯ are different, but The authors are well aware that R 1 2 this difference is assumed to be small enough to be neglected. Looking at Fig.3 and inspecting the variations of the currents, the average currents calculated from (12) are I¯test,1 = ΔI test,1 2 (14a) ¯ I 2 (14b) = ΔI +I
(10)
Indeed, that solution corresponds to γtest,2 = γtest,1 + π and consequently to Y (γtest,1 ) = Y (γtest,2 ), as it can be seen in Fig.1. D. Voltage drops considerations Most of the existing signal-injected based strategies neglect voltage drops across resistors and semiconductor components. As a result the inductance voltage V˜ l,test = V˜ test (10), and the test sequence simply consists in two opposite test pulses of same amplitude V˜ test,2 = −V˜ test,1 . This approximation is accurate for most drives. However, in some cases where the resistive voltage drop is not negligible, the voltage drops can strongly affect the test current response. This is illustrated Fig.3 where I dist = 0. A method to modify the test pulses to get I dist = 0 even when the voltage drop is not negligible is presented next section. E. Current response measure The current I contr may variate during the test period. Therefore it is not obvious to extract ΔI test from the total measured current variation ΔI. However, assuming ΔI contr,1 = ΔI contr,2 and I dist = 0, this variation can be canceled by calculation [11] ΔI 1 − ΔI 2 = ΔI test,1 − ΔI test,2 = 2 ΔI test,1
(11)
IV. C OMPENSATION OF THE RESISTOR INFLUENCE A. Estimation of the voltage drop To compute (7), the mean resistive voltage drop R · I test must be estimated. In most of the driving situations, as the test periods are much smaller than the electrical time constants, it can be assumed that the current varies piecewise linearly in time during these test periods [11]. Consequently, the mean current over half a test period I˜test can be approached by the average current value between its initial and final times. However the linear behaviour of the current is not strictly satisfied in very low inductance machines for which the electrical time constants are small. Consequently, as the inductance lowers, the current behaviour becomes more exponential. In addition, due to inverter nonlinearities, the resistor R1 during the first haft test sequence may differ from the resistor R2 during the second half test sequence. Nevertheless, as illustrated in Fig.3, the difference between the real current behaviour and the linear approximation during the half first and second periods are approximately compensated. Therefore, linear behaviour can be assumed valid. By consequence, it is convenient to define an average current I¯test,i between ti−1 and ti as I¯test,i = (I test (ti ) + I test (ti−1 )) /2
(12)
¯ · I¯ R i test,i = R · I test
test,2
test,1
(13)
dist
Note that in practice, a PWM signal is applied and should be considered to evaluate the average voltage drop. However, the centered signal between two measurement instants of a symmetric PWM system should not affect significantly the validity of the average current approximation. B. Voltage drop compensation Assume the compensation to be efficient such that I dist = 0. ¯ ≈R ¯ and ΔI ¯ ≈R If R 1 2 test,1 are known, according to (7), (13) and (14), the test voltages V˜ test,1 and V˜ test,2 to apply are ¯ · ΔI V˜ test,1 = V˜ l,test,1 + R (15a) test,1 2 ˜ ˜ ¯ V 2 (15b) = −V + R · ΔI test,2
l,test,1
test,1
The test current variation ΔI test,1 cannot be measured before the test is applied. The solution is to calculate the response directly from (4). It follows T ¯ ˜ R · Y · V˜ l,test,1 V test,1 = 1+ (16a) 4 T ¯ V˜ test,2 = −1 + R · Y · V˜ l,test,1 (16b) 4 If the index k denotes the test sequence in progress, the inverse inductance and the resistors are approached by their previous ¯ (k−1) respectively. To be valid, it is estimation Y (k−1) and R assumed that the angle of the test pulse phasor in the qd (k) (k−1) reference-frame has not significantly moved γtest ≈ γtest (k) (k−1) ¯ ≈R ¯ and that R . The test pulses to inject is T ¯ (k−1) (k) (k) V˜ test,1 = R 1+ · Y (k−1) · V˜ l,test,1 (17a) 4 T ¯ (k−1) (k) (k) V˜ test,2 = −1 + R · Y (k−1) · V˜ l,test,1 (17b) 4 To use these equations, the mean resistor remains to be estimated. This is now discussed. C. Online estimation of the resistor Assuming that the remaining test current distortion I dist comes from an error in the resistor value, the resistor can be identified from a feedback correction loop based on the measure of the current distortion. The relation between resistor error and current distortion is obtained from (9), where the
Y (γtest,1 )
Equ.(8)
V˜ test,1
V˜ l,test,1
Equ.(17) V˜ test,2 ˆ R
switching device
freewheeling device one phase terminal
ΔI test,1 Inverter + Machine
I dist
Equ.(23)
Fig. 5: Inverter circuit of one phase
Fig. 4: Flowchart of the Adaptive Test Pulse Strategy with resistor estimation and voltage drop compensation r˜sc
current deviations ΔI test,1 and ΔI test,2 are expressed in terms of (4) T ˜ V l,test,1 + V˜ l,test,2 = I dist (18) Y 2 Using (7), (13) and (14) to develop the inductance voltages V˜ l,test,1 and V˜ l,test,2 in (18), the result is Y
T ˜ ¯ · ΔI V test,1 + V˜ test,2 − R test,1 2 ¯·Y T ·I = 1+R dist 4
HR =
Fig. 6: Characteristic of the semiconductor devices and model around an operating point
(19)
A. The PWM voltage-source inverter A 3-phase PWM voltage-source inverter is considered. This type of inverter is widely used in industrial power drive applications. Figure 5 shows the inverter circuit for one of the three phases. There are mainly two nonlinearities to consider: the voltage drop characteristic of the semiconductor devices (freewheeling and switching) and the switching dead time effect. B. Voltage drop of the semiconductor devices
ˆ−R ¯ I dist = H R · R with
i 0
V. C ONSIDERATIONS ON INVERTER NONLINEARITIES
The test pulses V˜ test,1 and V˜ test,2 are computed using (15), ˆ is used. Introducing ¯ written R, where the last estimation of R, this expression in (19) yields T ˆ ¯ ¯·Y T ·I R − R · ΔI test,1 = 1 + R (20) Y dist 2 4 which can be rewritten
operating point
u ˜sc
2 1 ¯ L+ R T 2
(21)
−1 · ΔI test,1
(22)
The estimator used is a discrete accumulator applied on I dist to give a new estimation of the resistor, following ˆ (k) = R ˆ (k−1) − K (k) I (k) R dist
(23)
where the index k denotes the last test sequence applied. The dynamic of the closed loop estimator is a first order (k) system. The time constant τ can be fixed by means of K (k) (k) following τ = −Ttest / ln 1 − K H R , where Ttest is the time between two test sequences. In practice, the current distortion is small and its signal to noise ratio is low, therefore the time constant should be high enough to filter that noise. (k) (k) Here H R is calculated from the measure ΔI test,1 and the ˆ (k−1) . Other solutions could be last estimations L(k−1) and R studied in the future. The resistive estimator is illustrated in Fig.4.
A typical voltage/current characteristic of a freewheeling device and a switching device is presented Fig.6. From this curve, one may define a threshold voltage usc and a differential resistance rsc [1], [13]. Those parameters are generally slightly different between the two devices, but in practice, only mean values are considered: u ˜sc and r˜sc . They also variate significantly with the temperature and the operating point, but they are assumed constant during a control period. Assuming u ˜sc is not affected by the test sequence, its effect will be compensated by the current controller and has no effect on I test . The mean differential resistance r˜sc is added to the circuit resistance and included in the resistor to be estimated. C. Switching dead time Due to inherent turn off time of the switching devices, the switching dead time in drive signals (also named lag time [14]) is indispensable to avoid switching devices of the two legs to be simultaneously ON (conducing), and so to prevent direct voltage link shortage. During this dead time, both phase switching devices are OFF (blocking) and the phase current i flows through one of the two freewheeling devices, depending on the current flow direction. Consequently, the phase terminal potential will be mainly determined by the
dead time ON OFF
switching device +
ON OFF
switching device −
torque production, and to skip the test during the time a phase current crosses zero. However, this solution affects the machine behaviour and should be analyzed and improved by further works. It is also important to avoid injecting a test sequence that causes a current deviation for which the current crosses zero. In that case it can be sufficient to choose correctly the angle of the test pulses.
i (ideal) 0
i (real)
t
VI. E XPERIMENTAL RESULTS A. Experimental setup
Fig. 7: Zero clamp phenomenon during dead time
current flow direction. The dead time effect may significantly change the applied voltage. This disturbance is even more important for higher switching frequencies as the dead time to PWM period ratio increases. The modelling as well as the compensation of the dead time effect on the inverter output has widely been studied for oscillating phase currents in open-loop control [14], [15], [16]. The papers [13], [17] include also the turn off and turn on time in their models. For each phase, the dead time effect during a PWM sequence is modeled by an equivalent constant voltage multiplied by the sign of the current. The main assumption of the modelling is to neglect zero crossing effects. The equivalent constant voltage is compensated by the current controller and has no effect on I test . D. Zero crossing current problem Strong nonlinearities occur when the current of one phase crosses zero [18]. Indeed, Fig.6 shows that the characteristic is strongly nonlinear for a current close to zero and the linear approximation is less accurate. Moreover, when the phase current crosses zero during the dead time, this current is clamped to zero for the rest of the dead time duration [19]. This is named zero clamp phenomenon which is shown in Fig.7. The modelling of those nonlinearities is complicated by the current ripples produced by the PWM sequence. Under a certain desired controlled current value, the ripples of the real current may cross the zero several times, making it difficult to estimate the voltage drop accurately. The solution presented in [19] uses the periodical zero crossing occurring in a rotating machine to estimate the voltage drop and to introduce a predictive correction. Obviously, this is not adapted to machines working at low speed. Solutions with additional detectors of the current zero crossing moment, like the one proposed by [14], can only compensate the dead time effect, but not the nonlinearities of the semiconductor characteristic curve. Moreover, the use of additional sensors reduce the reliability and increase the price, and therefore should be avoided in some applications. E. Discussion to avoid the zero crossing problems It should be possible to compensate the threshold voltage u ˜sc , but the problem of the zero clamp remains. The solution is to inject an current offset in the direct axis, avoiding
The motor used is brushless DC wheel motor. This is a 3kW three-phase PMSM with very low inductance (around 70μH) developed by Technicréa, France, and intended for the automotive research. The nominal starting current is 134A and the nominal current at the maximum speed of 500rpm is 50A. The machine is fed with an IGBT voltage inverter from SEMIKRON, which is supplied by a 50V voltage rectifier. The PWM generator works at 20kHz. At this switching frequency, the dead time effect is significant, especially when the current of one phase crosses zero. The current controller works at 10kHz and takes a measurement every two PWM periods. The experiments are performed at standstill. The implementation of the test pulse method with the voltage drop compensation is summarized in Fig.4. The test sequence covers two PWM periods (T =0.1ms) and is applied every ten PWM periods (Ttest =1ms). The amplitude of the test voltage phasor is 10V, creating a variation of the current of about 7A. B. Current response plot Figure 8(a) and (b) show current response signals of one phase. The measurement instants are indicated by dots. The moments t0 , t1 , t2 refer to those in Fig.3. The curve is linearly interpolated between the measurement points. Obviously, the real behaviour between the measurements is significantly nonlinear. The instruction to the current controller is shown in dash-dotted lines. Various values are considered. The instruction current phasor is oriented in the direct axis to avoid any rotation of the machine. The upper plot Fig.8(a) presents the response to the test pulses without voltage drop compensation and the lower plot Fig.8(b) presents it with voltage drop compensation. Looking only at the positive controlled values, the current distortion is clearly compensated by the resistor compensator. The negative controlled values in Fig.8(a) show test pulses that cause a current deviation for which the current crosses zero. The dead time effect and its zero clamp phenomenon is then observed. In that situation, the inverse inductance estimation fails and the currents are strongly disturbed. Therefore, it is advised to avoid test pulses which result in a current response crossing zero. A solution is to send the test pulses in reverse direction. The problems disappear for the −9A controlled current, where the current does not cross zero anymore. In the case of 0A controlled current, the resistor compensation method effectively cancels the current distortion. However
The injected test pulses are computed for different angles of the test induction voltage V l , between 0 and 2π. Hence the parameters are estimated as a function of the angle and the results are shown in Fig.9. Different V l amplitudes are considered to analyze their impact on the estimations. The best fit circle for Y l has been computed for the different V l amplitudes. Only the circle at the 10V amplitude is shown by a yellow line in Fig.9(a), as there is no noticeable difference between the three circles. This is not the case without voltage drop compensation. This shows that the compensation also improves the induction estimation, and hence the position estimation. This will be the subject of future researches. The Y l estimation does not significantly change with the controlled current value, as long as no zero crossing occurs. The only small variation comes from the slight variation of the saturation level for the surface PMSM. Unlike the inverse inductance, the R locus strongly depends on the controller current value and the test amplitude, because of the influence of important nonlinear phenomenon.
20 t t t
0 1 2
15
I (A)
10
5
0
−5
−10
0
0.2
0.4
0.6
0.8
1 t (s)
1.2
1.4
1.6
1.8
2 −3
x 10
(a) Controller : 8A, 6A, 4A, 2A, 0A, -2A, -4A, -6A, -8A 20
t0 t1 t2
I (A)
15 10 5 0 0
0.2
0.4
0.6
0.8
1 t (s)
1.2
1.4
1.6
1.8
2 −3
VII. C ONCLUSIONS AND PERSPECTIVES
x 10
(b) Controller : 8A, 6A, 4A, 2A, 0A Fig. 8: Measured current signal of one phase for different instructions to the current controller, (a) without compensation of the voltage drop, (b) with compensation 4000
2000
0.04
1000
0.02 Im
0.06
Im
3000
0
0
−1000
−0.02
−2000
−0.04
−3000 −4000
−0.06 1
1.2
1.4 Re
1.6
1.8 4
x 10
(a)
0.15
0.2
0.25
0.3
Re
(b)
V l amplitudes : 4V (− · −), 7V (−−), 10V (−)
Fig. 9: Estimated parameters around 15A : (a) inverse inductance Y l [H −1 ], (b) equivalent resistor R [Ω] the inductance estimation is affected due to the current ripples crossing zero. C. Identification of the parameters The motor parameters Y l and R are estimated by the adaptive test pulses strategy with resistor compensation, using (8) and (23). The standard deviation of the Y l estimation is directly related to that of the current. The standard deviation of the phase current measure is between 50 and 100mA, depending on the phase. The R locus is estimated on the real and imaginary parts of I dist separately. The parameter K of the estimator is chosen to have τ ≈ 100 Ttest . A constant controlled current phasor of 15A is set in the direct axis to avoid zero crossing problems and to hold the rotor position.
Test pulses injection may disturb the current controller. In adaptive test pulse strategies, the objective is to avoid as much as possible that the test sequence disturbs the current controller. Therefore a test sequence composed of opposite test pulses has been proposed in [10]. However, when voltage drops are significant, as it is the case with the wheel motor considered in this paper, a significant current distortion may remain and still disturb the controller. This paper presents a compensation method in which the voltage drops are taken into account and are used to correct the test pulses to be applied in order to cancel the current distortion. This method includes an online identification of the equivalent resistor modelling the voltage drops. The method could be applied to any type of machine presenting a non-negligible voltage drop influence. It is easily implemented in adaptive test pulses method without complicated calculation. Experimental results have shown that the method is effective. Voltage drop compensation may improve the accuracy of the position estimation. Moreover, the identified parameters might be useful to improve the control of the machine. These subjects will be studied in further works. VIII. ACKNOWLEDGMENTS This work was performed as part of the Belgian Defense research programs, in collaboration with UGent in the framework of the Interuniversity Attraction Poles program IUAP P6/21 financed by the Belgian government. The UGent authors also wish to thank the Research Foundation-Flanders (FWO) for the financial support in the framework of project number G.0665.06. R EFERENCES [1] J. Holtz, “Sensorless control of induction machines - with or without signal injection ?,” IEEE Transactions on Industrial Electronics, vol. 53, pp. 7 – 30, Feb. 2006.
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