A sensorless PMSM drive using modified high-frequency test pulse sequences for the purpose of a discrete-time current controller with fixed sampling frequency

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Mathematics and Computers in Simulation 81 (2010) 367–381

Original article

A sensorless PMSM drive using modified high-frequency test pulse sequences for the purpose of a discrete-time current controller with fixed sampling frequency Frederik M. De Belie a,∗ , Peter Sergeant a,b , Jan A. Melkebeek a a

Department of Electrical Energy, Systems and Automation (EESA), Ghent University (UGent), Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium b Department of Electrotechnology, Faculty of Applied Engineering Sciences, University College Ghent, Schoonmeersstraat 52, B-9000 Gent, Belgium Received 21 August 2008; received in revised form 15 July 2010; accepted 23 July 2010 Available online 3 August 2010

Abstract For electrical machines, accurate position estimations at low speed can be obtained from current responses on high-frequency voltage test pulses. Nevertheless, to apply these voltage test pulses the current controller is often interrupted. In this paper, a modified test pulse sequence is discussed for which the current can be controlled without interruption. To generate the modified test pulse sequence, an asymmetric pulse-width modulator is required. The improvement in the dynamic behaviour of the sensorless drive is discussed by simulations including a salient-pole permanent-magnet synchronous machine, the current control loop, the pulse-width modulator and the position estimator. © 2010 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Sensorless control; PMSM; Current controller; High-frequency test signals

1. Introduction Over the last two decades, several vector controlled permanent-magnet synchronous machine drives (PMSM) have been developed for which the mounting of a position sensor is not required. In this way it is tried to reduce the overall cost as well as to increase the reliability of the drive. Achieving these goals can also be done by using sensors of minimum resolution, [5]. Nevertheless, in some applications, it can be strongly desired to avoid the position sensor for reasons of cabling, space reduction or maintenance. As in a vector controlled drive the position of rotor or flux is required, it has to be estimated from measurements of stator currents or voltages. Present research in this field is mainly focused on controlling the PMSM at low speed as well as at standstill while guaranteeing a high reliability, a sufficient torque, speed or position accuracy, a high efficiency or a high dynamic behaviour. In some estimators, the speed-induced voltage or back-emf is measured or estimated which is used in order to obtain the rotor speed or position. In [10], an auxiliary back-emf is defined from which the rotor speed and position is estimated by using a least-order observer. In [9], a rotor flux estimator is discussed where the third harmonic voltage ∗

Corresponding author. Tel.: +32 0 9 264 7914; fax: +32 0 9 264 3582. E-mail address: [email protected] (F.M. De Belie).

0378-4754/$36.00 © 2010 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2010.07.023

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component of the back-emf is used. Despite the accurate estimations, these estimators show lesser accuracy at low rotor speed as for this speed range the signal-to-noise ratio of the back-emf is too low. Moreover, at standstill, the back-emf is zero which means that the rotor position is not observable. In order to estimate the rotor position at low speed and standstill, high-frequency test signals can be applied to the motor terminals, additionally to the motor voltages or currents used for normal operation. The presence of a reluctance variation along the air gap, caused by a variable air-gap length or created by saturation of teeth and yoke, results in a high-frequency response that depends on the rotor position. As a result, by measuring and processing the high-frequency response, an estimation of the rotor position can be obtained. With respect to the test signal waveform, two main methods can be distinguished. In some estimators, the estimation is done in the frequency domain by measuring and filtering the response on high-frequency sinusoidal test signals, [7]. In [8], observers are used instead of low-pass filters in order to avoid signal delays during the signal processing. In [15], look-up tables are used to identify and compensate for estimation errors during loaded situations. A second approach is discussed in [14] and [16] where the current ripple caused by high-frequency voltage pulse trains is measured in order to estimate the rotor position of a PMSM. As in most PMSM drives the desired stator voltage is obtained by chopping a dc-voltage and by using a pulse-width modulator (PWM), these test pulse trains are rapidly generated. In [18], a position estimation is obtained without altering the PWM scheme for normal operation. A necessary condition is the low inductance which results in a high current ripple amplitude sufficient to obtain an accurate position estimation. In [12], the current response during the application of a zero-sequence voltage vector is used for the purpose of position estimation. As the machine rotates at low speed, the duration of zero-sequence voltage vectors is sufficient long to have a current response from which an accurate estimation results. However, during transients the duration of these zero-sequence vectors can reduce to zero which interrupts the position estimator. Therefore, for some applications, it can be required to generate sufficient current ripple amplitude by modifying the PWM scheme. In most sensorless methods, the modified PWM scheme requires a frequent interruption of the discrete-time current controller. To reduce the current distortions after each test period, additional test pulses can be applied, [13]. However, for this purpose, the machine inductances have to be measured in order to compute the duration of the additional test pulses. Moreover, important current disturbances remain unattended by the current controller during each test period. As the test pulses are frequently applied in order to have a regular update of the motion states, the operation of the drive can be strongly distorted. It is therefore preferred to have a PWM scheme for which the current can be controlled during the test periods as well. Controlling the current during the test periods is successfully done in [6] where a phase displacement of the pulses for normal operation causes the required current ripple. However, the current ripple duration is very short and an additional sensor is required to measure the current derivative. In [4], adaptive test pulses are discussed which are obtained by computing test voltage vectors that result in a given voltage deviation from steady state. Without the need of a machine model, the steady-state current distortions after each test period are reduced. Furthermore, by using this method the current ripple amplitude can be controlled. As a result, a few current samples are required to accurately measure the current variations. In that way, the need for additional sensors can be avoided. A discrete-time current controller with a fixed sampling frequency is successfully applied in this voltage adaptive sensorless controller. Though simulations and experiments show the effectiveness of these test pulse sequences, the current controller remains undiscussed in [4]. In this paper a more detailed discussion is given on the discrete current controller with fixed sampling frequency used in the voltage adaptive sensorless controller. Simulation results by using Matlab, Simulink will show the effectiveness of the voltage adaptive sensorless controller in estimating the rotor position accurately while controlling the current in a highly dynamic way. A continuous state-space model with initial states is used to model a salient-pole PMSM whereas a discrete-time transfer function with initial output is used to model the discrete-time PI current controller with fixed sampling frequency. The parameters of each current controller are computed taking into account the effect of sampling. This is done by using the SISO design tool which is a part of the control system toolbox of Matlab. Furthermore, an antiwindup scheme is discussed to improve the dynamic behaviour of the current controllers. The current controller is also modified in order to apply test pulses of preferred amplitude. 2. Synchronous machine modelling To model the synchronous machine, a qd-reference frame fixed to the rotor is used, Fig. 1, that is in agreement with the IEEE recommendations, [11]. At rotor position θ r = 0, the q-axis is aligned with the axis of a reference phase u.

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Fig. 1. Cross-section of rotor and qd-reference frame.

The relationship between the current components iq , id and voltage components vq , vd is given by d (1a) (Lq iq ) − ωLd id − ke ωφm , dt d vd = Rid + (Ld id ) + ωLq iq , (1b) dt with Rs the stator resistance, Lq , Ld the quadrature and direct inductance, respectively, ke the voltage constant, ω the rotor pulsation and φm < 0 the permanent-magnet flux coupled with the stator windings. As for steady-state operation the current and voltage components as well as the rotor speed are constant, the steady-state equations are given as vq = Riq +

Vq0 = RIq0 − Ld Id0 − ke φm ,

(2a)

Vd0 = RId0 + Lq Iq0 ,

(2b)

where vq0 = Vq0 , vd0 = Vd0 are the steady-state voltage components, iq0 = Iq0 , id0 = Id0 are the steady-state current components and ω0 =  is the steady-state rotor speed. A small-signal model of the synchronous machine is obtained from (1) and (2). It is given as a state-space model:       δiq δvq d δiq =A +B , (3a) dt δid δid δvd 

A=

−R/Lq

0

0

−R/Ld





,B=

1/Lq

0

0

1/Ld



,

(3b)

where δ denotes small variations from steady state. The current deviations from steady-state δik = ik − i0,k , k = q, d are the states as well as the outputs of this model and its inputs are given by the voltage deviations from steady state δvk = vk − v0,k , k = q, d. The initial conditions are given by the steady-state current components Iq0 and Id0 . As the mechanical time constant is much larger than the electrical time constant, it can be assumed that the speed is constant ω = ω0 = . Furthermore, it is assumed that neglecting Lk δik , k = q, d introduces small errors only. 3. Sensorless control by using test vectors A schematic overview of the current controller and position estimator for an interior PMSM is given in Fig. 2. By using the estimated rotor position θ˜ r , the measured phase currents are transformed to the synchronous qd-reference frame fixed to the rotor. Then, a PI-controller is used for which the output can be regarded as the desired stator voltage components in the qd-frame. In the controller, a feed-forward of the back-emf E = ke ωφm is applied that requires an estimation ω˜ of the rotor speed. The estimated rotor position is used to transform the controller output to the desired per-unit phase voltages as well. To generate these voltages, a voltage-source inverter (VSI) is used in which a dc-bus voltage vdc is chopped at a high frequency. As the VSI-semiconductor switches are steered by using a centered pulsewidth modulator (PWM), sampling a phase current in the middle of a PWM-period Ts results in an approximation of the average phase current that is sufficiently accurate for the purpose of the discrete-time current controller. Besides a current controller, Fig. 2 shows the rotor position and speed estimator. This estimator adds test signals to the controller output in order to obtain a current ripple. As in a salient-pole PMSM the phase inductances vary with

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Fig. 2. Schematic overview of a sensorless PI current controller.

the rotor position, this current ripple, which is the response of the phase currents on a voltage deviation from steady state, depends on the rotor position. In this part of the paper the position estimator is discussed. For that purpose, the current deviations δicq , δicd due to a test vector with components vcq , vcd that generates voltage deviations δvcq , δvcd from steady state will be computed from (3). The exponent c denotes the variables during a test period. 3.1. Current response on test vectors In this paper, voltage vectors applied during a test period are referred to as test vectors. These test vectors have components vck , k = q, d for which the voltage deviations from steady state δvck = vck − v0,k , k = q, d are constant in time and given by δVkc , k = q, d. As the test period Tp during which this test vector is applied is much smaller than the electrical time constants Lk /R, k = q, d, the current deviations δick = ick − i0,k , k = q, d vary linearly in time. By supplying a test vector, the current deviations δick , k = q, d will vary during a time Tp over δick =

Tp c δV Lk k

for k = q, d.

(4)

where x denotes the difference of x before and after a test period. To compute δick , k = q, d, the voltage deviations c , k = q, d are required. δVkc , k = q, d have to be known. This means that the steady-state voltage components Vk0 To avoid a measurement or estimation of the steady-state voltages, a second test vector during a time Tp can be used, shown in Fig. 3(a). Subtraction of the current variation during the second test period from the current variation during the first test period results in δick,1 − δick,2 =

Tp c c (δVk,1 − δVk,2 ) Lk

for k = q, d.

(5)

c , k = q, d are the same during both test pulses this is also written as As the steady-state voltages Vk0

δick,1 − δick,2 =

Tp c c (V − Vk,2 ) Lk k,1

for k = q, d.

(6)

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Fig. 3. Supplied voltage and current response during a test period: (a) voltage test vectors and current responses for a test period, (b) resulting voltage test vector and current response for a test period, (c) resulting test vector and current response and (d) in the reference frame fixed to the test voltage vector δvs .

From this result, it follows that measuring or estimating the steady-state voltage vector can be avoided. The relation in (6) is used to estimate the rotor position. 3.2. Position estimation from current response measurements A complex notation is used to represent the current and voltage in (6), Fig. 3(b). For the purpose of position estimation, the current in (6) with components δick,1 − δick,2 (k = q, d) is transformed to a new xy-reference frame where the current is given by δis . The real axis of this new frame is aligned with the test vector δvs which in the qd-reference frame has the components δvck,1 − δvck,2 = vck,1 − vck,2 (k = q, d), Fig. 3(c). In [3], it is shown that from (6) the current δis can be written as δis = δI s0 + δI s

(7a)

with LTp δv , (7b) L2 − L2 s LTp −j2β δI s = − 2 e δvs , (7c) L − L2 where Lq + Ld Lq − Ld L= and L = (7d) 2 2 and where β is the angle between the voltage test vector δvs and the q-axis. Fig. 3(d) shows the current response in the xy-reference frame. From this figure, it follows that by measuring and processing the current variations δis an estimation βˆ of the angle β can be obtained. As the angle β of the voltage test vector δvs with respect to the reference phase is known also, an estimation of the rotor angle θ r is obtained as θˆ r = βˆ − β . In order to estimate β, a measurement of the inductances L and L is required. Estimating these parameters can be done by measuring current variations during at least two succeeding test periods. It follows from (7) that the locus of δI s0 =

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Fig. 4. Conventional test vectors: (a) text voltage vectors and (b) voltage vector deviations.

current vectors δis for β = 0 . . . 2π and for a given amplitude of δvs is a circle with midpoint δI s0 and radius |δI s |, Fig. 3(d). This means that by measuring the current response on at least two different test vectors, the current vector δI s0 can be computed. Then, by subtracting δI s0 from δis , 2β is obtained as the angle of the resulting current vector δI s . 3.3. Current distortions In the literature, several sensorless strategies are discussed that differ in the test vectors used [4–6]. In [13], the current controller is halted and one of the six active voltage vectors V i , i = 1 . . . 6 is applied during a time Tp succeeded by a test vector of opposite sign which is applied during Tp also. Fig. 4 shows the case where test vectors V 2 and V 5 are used. By supplying these test vectors V i , i = 1 . . . 6, the steady-state current can be distorted after each test period. The current distortion equals the sum of the current responses on the first and second test voltage vector and can be computed by using (4) δick1 + δick2 =

Tp c c (δVk1 + δVk2 ) Lk

for k = q, d.

(8)

In the case that the six active voltage vectors V i , i = 1 . . . 6 are used as test vectors, the voltage deviation δvck1 + δvck2 c , k = q, d differs from zero. This for k = q or k = d differs from zero if one of the steady-state voltage components Vk0 is shown in Fig. 4 for the case of test vectors V 2 and V 5 . Therefore, for a sensorless drive that uses the test vectors V i , i = 1 . . . 6 only, it follows from (8) that a remaining current distortion is generated after two test period. 3.4. Adaptive test vectors to avoid current distortions Reducing these current distortions is done in [4]. To avoid current distortions, it follows from (8) that it is required to have a zero voltage deviation after each test period or that δvcq1 + δvcq2 and δvcd1 + δvcd2 are equal to zero. Therefore, in [4], the supplied voltage test vectors vc1 , vc2 are made adaptive to the steady-state voltage vc0 in such way that

Fig. 5. Adaptive test vectors.

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δvcq2 = −δvcq1 and δvcd2 = −δvcd1 . Fig. 5 shows adaptive test vectors in the case of the nonzero steady-state voltage of Fig. 4. 4. The current controller Besides the current distortions after each test period, it is a disadvantage of some sensorless methods that the current controller is interrupted during the test periods. Therefore, in this paper, a new test vector sequence is discussed which allows to apply a current controller even during the test periods. The controller is a discrete-time current controller with fixed sampling frequency and is shown for the quadrature current in Fig. 6. The closed loop includes the synchronous machine, the zero-order hold (ZOH), the PI current controller and the pulse-width modulator with inverter. This model will be used to simulate the dynamic behaviour of the current controller if the adaptive test vectors are supplied. It is to be noted that this controller can use either a measured or estimated value of rotor position and speed. 4.1. Digital PI current controller In most synchronous machine drives a PI-controller is used for each component of the average current in the qd-reference frame. The transfer function of a PI-controller is given in the Laplace domain as   1 for k = q, d. (9) GPI,k (s) = KPI,k 1 + sτPI,k To compensate the back-emf E = ke ωφm in the q-axis, a feed-forward of e = E/Vdc in the controller loop for the q-axis is used where Vdc is the dc-bus voltage that supplies the inverter. As in most drives the current is sampled and digitally controlled, a transfer function in the z-domain is used to model the PI-controller: vk =

b0,k z + b1,k n (ik − ik ), z+1

k = q, d

Fig. 6. Current control scheme for the quadrature current.

(10)

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Fig. 7. Root locus with closed loop poles pi , i = 1 . . . 5.

with ik , k = q, d the desired current components and ink , k = q, d the sampled current components. The relationship between KPI,k , τ PI,k and b0,k , b1,k for k = q, d is given by     Ts Ts b0,k = KPI,k + 1 , b1,k = KPI,k −1 , (11) 2τPI,k 2τPI,k where Ts is the sampling period. The simulation time can be strongly reduced by starting a simulation directly from the steady state. Therefore, the initial outputs of the discrete transfer function have to be known which are given by vq0 /Vdc and vd0 /Vdc . Due to the feed-forward of the back-emf, the component e is absent in the initial output of the controller for the quadrature current. Tuning a digital PI-controller is done by using the SISO design tool which is a part of the control system toolbox of Matlab. By using (9) and by modelling the zero-order hold as GZOH =

1 , (Ts s) /3! + (Ts s)2 /2! + Ts s + 1 3

(12)

the root locus can be computed with KPI,k τ PI,k , k = q or k = d as gain. The SISO design tool allows to position the zero of the PI-controller to obtain the root locus as is shown in Fig. 7 for the q-axis. Then, by adjusting the aforementioned gain, the closed loop poles are placed on the real axis as is shown in Fig. 7 as well. In that way, a swift response on disturbances is obtained without much overshoot. In order to have an accurate estimation of the rotor position, it is required to have a current response with a sufficient signal-to-noise ratio. As the amplitude of the current response depends on the voltage vector deviation from steady state, test vectors are used with a voltage vector deviation of minimum modulus δvmin s . In classic control schemes, stator voltage vectors are limited to the hexagonal voltage boundary as a result of the limited dc-bus voltage that supplies the VSI, [2]. To assure voltage vector deviation of minimum modulus δvmin s , the output of the PI-controller is limited to an hexagonal control boundary as given in Fig. 8. In this figure, test vectors vc1 and vc2 are given for a controller output vc0 on the hexagonal control boundary. The modulus of the resulting voltage vector deviation δvc1 corresponds to δvmin s from which the hexagonal control boundary can be determined.

Fig. 8. Voltage test vectors in the case of a steady-state voltage vector on the control boundary.

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Fig. 9. Response on a step in the desired current: (a) waveforms without antiwindup and (b) waveforms with antiwindup.

4.2. Antiwindup scheme An antiwindup scheme is applied to improve the dynamic behaviour of the controller. If the controller output results in a desired phase voltage amplitude higher than the dc-bus voltage Vdc , the measured current will deviate from the desired current. Consequently the integrating action will windup the controller output unnecessarily. As it takes time for the controller to decrease its output, the PI-controller shows a poor dynamic behaviour. This is illustrated by simulation results shown in Fig. 9(a) for the PI-controller of the quadrature current component. It is assumed that Vdc (−Vdc ) is the maximum (minimum) q-component of the stator voltage which corresponds with a controller output equal to one (minus one). The simulation is started with a difference between the back-emf E and the dc-bus voltage Vdc that is too small to achieve the desired current iq > (Vdc − E)/R. As a result the output vq increases due to the uncontrolled current error eiq while the supplied voltage vq is limited to the dc-bus voltage. After a millisecond the desired current component iq is zeroed. As the error eiq changes sign due to the new desired value iq , the output vq will decrease. However, as the controller output vq was winded up, it takes several milliseconds for the PI-controller to control the stator voltage vq . Different antiwindup strategies are discussed in [1]. Fig. 10 shows in a schematic way the PI-controller, the limitation on the controller output and two different antiwindup strategies. The PI-controller without antiwindup, shown in Fig. 10(a), is given as a reference case. In a first antiwindup method, shown in Fig. 10(b), a feedback of the integration action is used in order to avoid winding up the controller. In a second method, the difference between the controller output and its limited value is used in a feedback loop to reduce the input of the integrator within the PI-controller. Both methods avoid winding up the PI-controller. However, these methods require to tune the control parameters of the feedback loop in the antiwindup method. Furthermore, the methods in [1] are described in the continuous time-domain. This paper describes an antiwindup method in the discrete-time-domain that requires no additional tuning of control parameters. In this paper, winding up the controller output vq is avoided by forcing the controller input to zero if this output is higher (lower) than Vdc (-Vdc ) and if the controller input eiq is positive (negative). The model in SimulinkTM , MatlabTM is given in Fig. 11. The result of this antiwindup scheme is shown in Fig. 9(b). As a result of the zeroing and from (10) it follows that the controller output is held to its last value, avoiding winding up the output. Then, as the current error eiq changes sign, the controller input is restored with the current error inq − iq . This will immediately result in a decrease of the stator voltage component vq as well as current error eiq . Clearly, by using this antiwindup scheme a faster response is obtained. 5. The current controller and test vectors In most sensorless drives, the controller is interrupted in order to apply the test vectors. Interrupting the controller affects its dynamic behaviour. In this paper, the current is controlled even during the test periods. For this purpose, the sequence of adaptive test vectors is modified so that the discrete-time current controller with fixed sampling frequency can be used. To discuss this modified sequence of test vectors, a short description of the pulse-width modulator is given in Fig. 6.

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Fig. 10. PI-controller and different antiwindup schemes in the continuous time-domain: (a) PI-controller without antiwindup, (b) PI-controller with limited I-action and (c) PI-controller with controller output tracker.

5.1. The pulse-width modulator: test vectors and test pulses The pulse-width modulator compares the output of the discrete-time current controller with a reference signal of high frequency (fc ). From this result, the on-times and off-times of the semiconductor switches in the inverter that supplies the PMSM are determined. Different PWM strategies can be applied depending on the carrier waveform. In this paper a sawtooth carrier is used as is shown in Fig. 12(a) where vq is the output for the current controller in the q-axis. As a result, the voltage at the motor terminals is a chopped dc-voltage of which the average value over a carrier period Tc equals the controller output multiplied with Vdc . By using this PWM strategy, the phase currents cross their average value over a switching period in the middle of each switching period as shown for the first two switching periods in Fig. 12(b). As a consequence, it is an advantage of this PWM strategy that the average current can be controlled by sampling the currents on these instances. It is noticed that the carrier period or switching period Tc equals the sampling period Ts . 5.2. Modified test vector sequence By using conventional test vectors V i , i = 1 . . . 6, the current components at the aforementioned instances differ from their average value during a test period. This is shown in Fig. 12(b) for the quadrature current component. As a result of the current sampling during a test period, the current controller will distort the current. Therefore, in most sensorless drives, it is tried to avoid current distortions by using the last current sample obtained during normal operation. This means that the delay between sampling the current and applying the new controller output increases which results in a slow dynamic response of the controller. In this paper, the test vector sequence of the adaptive sensorless method is modified in order to obtain a current that crosses its average value at the aforementioned sample instances. The new test vector sequence results in a zero voltage

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Fig. 11. Antiwindup scheme of the quadrature axis, vq0 is the PI-controller output of the quadrature axis before antiwindup.

Fig. 12. Sensorless drive: waveforms in the case of a centered pulse-width modulator: (a) controller output vq and quadrature voltage vq and (b) controller output vq and sampled quadrature current deviation component δinq .

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deviation from steady state at the aforementioned sample instances. The modification is as follows. A first test vector which results in a voltage deviation δV c from steady state is applied during half a switching period T1 . Following this first test vector, a test vector with a voltage deviation from steady state −δV c is applied during a whole switching period T2 . Finally, a third test vector is applied during half a switching period T3 which causes a voltage deviation from steady state δV c . To apply a test vector during half a switching period, an asymmetric PWM is required which means that the computation of the on-times and off-times of the switches is updated twice every switching period. Therefore, this strategy is referred to as the PWM in double-update mode or the asymmetric PWM. The result of this

Fig. 13. Simulation results for a sensorless drive by using adaptive test pulses: estimation error θ r,e , phase current ripples iu,ripple , iv,ripple , iw,ripple , phase voltages vu , vv , vw , duty ratios δu , δv , δw and clock pulses with PWM-period Ts as a function of time t.

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strategy for a PMSM at standstill is shown in Fig. 13. The voltage and current for the three phases u, v and w are given. The transformation from q- and d-components to u-, v-, w-components is done by using the Clarke and Park transformations as is given in [17]. Clearly, by using the new test vector sequence the current crosses its average in the middle of the switching period. Furthermore, the current is without current distortions except for the current ripple that is used for the position estimation θˆ r which equals the rotor position at which the PMSM is modelled. 6. Dynamic behaviour of the current controller As a result of the adaptive strategy, the voltage deviation from steady state is zero at the middle of each switching period. This means that the phase currents cross their average current at the aforementioned sample instances even during the test periods. This is shown in Fig. 13 for a PMSM at standstill. Consequently, the current controller can be applied during the test periods as well. In Fig. 14 the current response on a current measurement disturbance is shown. It can be concluded that the current deviation from steady state after a measurement disturbance is smaller for the adaptive method compared to the conventional strategy. In Fig. 15 the current response on a step in the desired current is shown. Clearly, the current transient is shorter and with less overshoot in the case that adaptive test vectors are used. During this transient the controller output is limited to the control boundary. For the adaptive sensorless method the control boundary is within the dc-bus boundary, Fig. 8. In the case a position sensor is used, the control boundary equals the dc-bus boundary. As a result, a difference is noticed during the transient between the average current for the adaptive sensorless controller and the average current for the controller with position sensor. From simulations it follows that during these transients the estimated rotor position equals the rotor position at which the PMSM is modelled. Previous simulations show the dynamic behaviour of the current controller for small variations in the desired or measured current. At larger load variations (e.g. from nominal torque to zero torque) the method with conventional test vectors can become unstable. The sensorless method with asymmetric PWM proves to be stable. Fig. 16 shows simulated waveforms of the duty ratios, the phase currents and the estimated rotor position θ˜ r that is used in the current

Fig. 14. Computed current responses on a current measurement disturbance.

Fig. 15. Computed current responses on a step in the desired current.

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Fig. 16. Waveforms in a sensorless current controlled interior PMSM in case of an asymmetric PWM and by delaying the controller output: PWM-clock, duty ratios, phase currents and estimated rotor position.

controller. The controller output on which the test signals are added is shown as well. As a reference for the time, the PWM-clock cycles are given. In each PWM-period, test signals are added so that the rotor position is estimated at the controller sample frequency of 8 kHz. As a test signal is applied which waveform is in the second half of the test period opposite to its waveform in the first half of a test period, two succeeding rotor position estimations are equal by approximation. Therefore, it may seem in the figure as the rotor position is updated at 4 kHz instead of 8 kHz. To show the performance of the sensorless current controller during a large transient in the load, the set value of the torque is zeroed at 2 ms. The estimated rotor position approximates the real zero value well even during the transient at which the maximum simulated error is 0.3 electrical degrees. This remaining error is mainly caused as the steady-state voltage v0 for a test period during transient state is constant by approximation only. 7. Conclusions An enhanced sensorless current controller for PMSM drives is discussed. The controller uses measurements of the current response on adaptive voltage test vectors in order to estimate the rotor position. A modified voltage test vector sequence is discussed which results in a zero voltage deviation from steady state in the middle of a switching period. In that way, a measurement of the average current over a switching period can be made by sampling the current at these instances. As a result, a discrete-time current controller with fixed sampling frequency can be used simultaneously with the position estimator. A simulation scheme of the sensorless drive is given and simulation results show the effectiveness of the voltage adaptive sensorless controller.

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