Measurement of the Self-Sensing Capability of Synchronous Machines for High Frequency Signal Injection Sensorless Drives

Signal injection sensor-less control for synchronous machines is known to be afflicted by an estimation error dependent on the load current. The estimation error is related to the cross-saturarion and the saliency of the adopted synchronous machine. A motor can be more or less suitable for signal injection sensorless control compared to other motors with different designs or sizes. A sensorless drive can even be afflicted by the control divergence when the machine is highly saturated, resulting in a useless drive. Moreover, even when the control converges, the actual current control trajectory is different from the given reference. In this paper, a measurement procedure of the convergence region, i.e. the operating points where the motor can be successfully controlled without a position sensor is presented and validated. In particular, two different synchronous motors are considered, a permanent magnet assisted synchronous reluctance motor (PMA-SynRM) and a synchronous reluctance motor (SynRM).


Measurement of the Self-Sensing Capability of Synchronous Machines for High Frequency Signal
Injection Sensorless Drives Matteo Berto, Luigi Alberti , Senior Member, IEEE, and Silverio Bolognani , Fellow, IEEE Abstract-Signal injection sensor-less control for synchronous machines is known to be afflicted by an estimation error dependent on the load current. The estimation error is related to the crosssaturarion and the saliency of the adopted synchronous machine. A motor can be more or less suitable for signal injection sensorless control compared to other motors with different designs or sizes. A sensorless drive can even be afflicted by the control divergence when the machine is highly saturated, resulting in a useless drive. Moreover, even when the control converges, the actual current control trajectory is different from the given reference. In this paper, a measurement procedure of the convergence region, i.e. the operating points where the motor can be successfully controlled without a position sensor is presented and validated. In particular, two different synchronous motors are considered, a permanent magnet assisted synchronous reluctance motor (PMA-SynRM) and a synchronous reluctance motor (SynRM). P OSITION sensors such as encoders and resolvers are conventionally required to control synchronous reluctance motors (SynRM) and permanent magnet assisted synchronous reluctance motors (PMA-SynRM). In the last thirty years, the control of synchronous motors removing the position sensor has been investigated and properly achieved. Many valid sensorless techniques exist, depending on the operational conditions of the motor. At high speed, the sensorless control usually adopt observers or estimators in order to reconstruct the machine state and hence the rotor position [1], [2], [3]. In the zero-low speed region (usually up to a tenth of the rated speed), additional signal injection are used to identify the rotor position. [4], [5]. Different approaches have been proposed in literature exploiting high frequency carrier signal and demodulation [6], [7] or unconventional pulse-width modulation (PWM) patterns [8], [9].

Index Terms-Finite
Signal injection sensorless control is known to be affected by the variation of the motor differential inductances with the operating point [10]. In particular, it appears that some motors can be more suitable than others to be controlled without a position sensor.
The convergence region [11], [12] should be computed to know if a motor is suitable for signal injection sensorless control. The flux linkages maps, obtainable by finite element analysis (FEA) or experimental measurements (motor characterization), are usually required to compute the convergence region of a sensorless drive [13].
In [14], an experimental procedure to measure the self-sensing capability of a synchronous sensorless drive was introduced. The paper describes a test methodology to measure the sensorless control trajectories introduced in [13]. In particular two different tests are proposed. The first one allows to obtain the sensored trajectory t 1 , i.e. the trajectory of the position observer in openloop during a sensored operation. The second allows to measure the sensorless trajectory t 2 of the drive, i.e. the trajectory of the effective operating points controlling the motor in sensorless operation. In [14] the analysis was limited to considering a PMA-SynRM drive and the trajectory measurement was implemented considering only rotating high frequency (hf) signal   injection. This paper extends such an analysis. In particular a second drive of different type, i.e. adopting a SynRM motor, is considered. Moreover, the effect of the injection strategy on the trajectories measurements is investigated. In particular besides rotating injection also pulsating injection is included in this paper. Last, also the effects of different demodulation techniques are investigated in the following. The proposed experimental procedure is compared and validated with computations based on finite element analysis (FEA) results [15].

II. DESCRIPTION OF THE CONSIDERED MOTORS AND DRIVES
In this paper two different synchronous motors are considered. They are characterized by a significant reluctance torque component and one has been also equipped with PM assistance to improve its performance. The geometries and the conventions adopted to define the rotor reference frame are shown in Fig. 1. Table I summarizes the main data of the considered machines. The motors have been characterized for rotor position θ me = 0 through finite element analysis (FEA). The corresponding flux linkages λ d and λ q are shown in Fig. 2 for the PMA-SynRM. Fig. 3 shows the characteristics for the SynRM.
The high-frequency (hf) model of a synchronous motor can be obtained from the voltage equations neglecting the stator resistance and the permanent magnet flux [16]. This assumption is valid because the frequency of the additional signal injection is sufficiently high (usually higher than 500 Hz) [4], [5]. Thus, the hf model of a synchronous motor at quasi-zero speed is: where u hd and u hq are the injected hf voltages and i hd and i hq the hf currents. The incremental inductances are defined as: where l dd is the d-axis incremental inductance, l dq the mutual incremental inductance (also known as cross-saturation inductance), l qq the q-axis incremental inductance, l Σ the mean incremental inductance and l Δ the semi-difference incremental inductance.
The incremental inductances are the key parameters in signal injection sensorless control and different injection techniques can be adopted to estimate the rotor position. In the paper, two different injection techniques are adopted. In particular, both pulsating and rotating injections are considered. Moreover, with the rotating injection method, two different position estimation techniques are adopted. The details are described in the following.

A. Sinusoidal Injection in d x q x (pulsating)
As shown in Fig. 4, a pulsating hf injection is superimposed to the d x q x voltage references: Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. The sinusoidal pulsating injection in d x leads to the high frequency currents: or, equivalently: where: Δθ =θ me − θ me (6) and ε is the estimation error. The hf currents i x hd , i x hq are obtained measuring the d x q x currents and applying a high-pass filter (HPF). The rotor position information is extracted by a demodulation-observer scheme applied to the current i x hq , as shown in Fig. 5. The current i x hq is multiplied for sin(ω h t) and the result is filtered with a low-pass filter (LPF). The result of the demodulation is the signal: or, equivalently: The signal I hq is related to the convergence region of the sensorless drive since it is the signal that the position observer (a PI+I) tries to nullify in order to find the stable convergence points [13].

B. Sinusoidal Injection in αβ (rotating)
Sensorless control schemes for rotor position estimation at standstill or low speed can operate with a pulsating injection on d x but also with of a rotating injection in α-β. The scheme is shown in Fig. 6. A sinusoidal rotating injection in α-β: leads to high frequency currents: The hf currents i hα and i hβ contain information on the rotor position θ me . Heterodyning demodulation, shown in Fig. 7, is the conventional approach to retrieve the rotor position es- The results of the products are summed and filtered with a LPF. The result of the demodulation (input of the position observer) is the signal: The signal I hq in the case of rotating injection in αβ (11) is equal to the one in the case of pulsating injection (8) apart from a factor 2 in the denominator. Therefore, the convergence region will be the same, being this region a property of the motor itself and not depending on the considered injection technique.

C. Ellipse Fitting
When a rotating injection in the αβ reference frame is adopted, besides the heterodyning demodulation described in the previous sub-section, also an ellipse fitting technique can be adopted to estimate the rotor position. In comparison with the demodulation, the ellipse fitting approach has a reduced number of filters and tunable parameters. It is based on the consideration that the hf current in the α-β reference frame traces an ellipse whose orientation is related to the rotor position. A buffer of current samples is considered and elaborated adopting a Recursive Least Squares (QR-RLS) ellipse fitting combined with an orthogonal PLL (Q-PLL). The ellipse fitting scheme is shown in detail in Fig. 8. Major details and a full development of the adopted ellipse fitting technique can be found in [17], [18].

III. FEA COMPUTATION OF THE CONVERGENCE REGION
The conventional procedure to predict the self-sensing capabilities of a synchronous sensorless drive consists in processing the flux linkage maps. In this section the procedure to compute the convergence region starting from FEA computations is described. A detailed and systematic description of the adopted approach can be found in [13]. The self-sensing capabilities, i.e. the hf saliency, the open-loop estimation error and the convergence region, explain and predict the performance of a sensorless drive in the case of signal injection methods. Given a reference trajectory for the current control (usually the MTPA), the trajectories t 1 and t 2 can be computed. The sensored trajectory t 1 is the trajectory of the position observer in open loop during a sensored operation, while the sensorless trajectory t 2 is the trajectory of the effective operating points controlling the motor in sensorless operation. It is worth noticing that trajectory t 1 is fictitious and it is not covered either during sensored control or during sensorless control. Fig. 10 shows the convergence regions of the PMA-SynRM and the SynRM computed on the simulated flux linkages maps. The MTPA is set as current reference. The sensored trajectory t 1 moves away from the MTPA reference for increasing current values because of the cross-saturation effect Fig. 9. Detail of the ellipse fitting for rotating injection in αβ, consisting of the QR-RLS (the ellipse fitting itself) and the Q-PLL. Fig. 10. Convergence region computed from the simulated flux linkages maps (FEA) for θ me = 0, using the method presented in [13]. The MTPA trajectory is used as reference for the current loop. t 1 is the sensored trajectory, t 2 the sensorless trajectory. on the estimated rotor position [10]. The difference between the estimated position θ me and the rotor (measured) position θ me is the open-loop estimation error ε defined as [13]: As can be noticed, the sensorless trajectory t 2 has a shorter length than t 1 because it exists a working point beyond which the observer is not able to estimate the rotor position leading to the sensorless control divergence [12]. The maximum length of trajectory t 2 is about 4 A in the case of PMA-SynRM, meaning that the motor can operate in sensorless operation up to this current amplitude (without adopting compensations). The maximum length of t 2 for the SynRM is about 3 A. For higher current values it is not possible to estimate the rotor position, and additional compensations should be adopted [11], [12], [19]. It is worth noticing that the analysis and test considered in this paper are carried out all at locked rotor, i.e. for a fixed θ me = 0. Due to the possible variation of the flux-linkage characteristics with the rotor position, a change in such a trajectory may exist.
The open source project dolomites-apollo has been adopted for the computation of the convergence region [15].

IV. EXPERIMENTAL MEASUREMENT OF THE CONVERGENCE REGION
In Section III the sensored trajectory t 1 and the sensorless trajectory t 2 have been computed through post-processing computations on the FEA maps. In the following, the trajectories t 1 and t 2 will be recorded online during two experimental tests (the first sensored and the second sensorless) using the MTPA as reference trajectory for the current control. The tests have been performed on the test bench shown in Fig. 11, where the motors have been locked at θ me = 0. The control scheme has been implemented on a dSPACE1104 using a control frequency of 10 kHz. The amplitude of the voltage injection is U h = 40 V, and the injection frequency is f h = 1 kHz.
It is worth noticing the presence of the encoder in the control scheme in Figs. 6 and 8. The measured position θ me can be used for the sensored operation (switch on position 1) to measure trajectory t 1 but also during the sensorless operation (switch on position 2) to identify the sensorless trajectory t 2 , which is one of the goals of the paper. The control scheme reported in Fig. 4 is not suitable to measure the trajectory t 1 adopting the pulsating injection. It should be modified considering different dq transformations for the current loop and for the high frequency injection. For this reason the rotating injection is mainly  considered in the paper investigating the measurement of the sensorless trajectories with two different demodulation techniques (demodulation and ellipse fitting). Nevertheless, results with the pulsating injection are considered when possible for a further comparisons, in particular for the t 2 trajectory.
The first test is performed in sensored mode: the measured position θ me = 0 is used for the current control (the switch in Figs. 6 and 8 is on position 1) while the estimated position θ me and the currents i d and i q are used for the online computation of the trajectory t 1 .
The coordinates of trajectory t 1 , i.e. the currents in the estimated d x q x reference frame i x d and i x q , are computed as:  It is worth noticing that in this test the difference between the estimated position θ me and the measured position θ me is the open-loop estimation error ε. The second test is performed in sensorless mode: the estimated position θ me is used for the current control (the switch in Figs. 6 and 8 is in position 2) while the measured position θ me = 0 and the currents i x d and i x q are used for the online computation of the trajectory t 2 . The coordinates of trajectory t 2 , i.e. the currents in the actual dq reference frame i d and i q , are computed as: In the following the obtained results for the two considered motors are described.   Fig. 12 shows the measured t 1 trajectory for the PMA-SynRM. The trajectory t 1 , which coordinates are the currents i x d and i x q , is rotated with respect to the reference trajectory of a growing angle equal to ε. The time behavior of the current and other relevant quantities during the test are shown in Fig 13. The tests are performed imposing a ramp of commanded current along the MTPA. The amplitude of the current reference grows at rate of 1 A per second. Fig. 13 a shows the measured currents i d and i q and the computed currents i x d and i x q . The actual operating points (i d , i q ) follow the trajectory MTPA from (0,0) up to the rated current amplitude value (6 A). The currents i x d and i x q represent the coordinates of trajectory t 1 . Fig. 13 b shows the estimation error, which is   Fig. 15. The motor is operated in sensorless control using a reference current ramp along the MTPA. The operating reference frame is the estimated d x q x since the current control is closed on the estimated position θ me . The measured currents i x d and i x q appear to follow the MTPA reference in the shifted control frame, but the real operating point is in a different place (t 2 ). The actual operating point i d -i q is different from the trajectory covered in the first test (i.e. the MTPA), thus during a sensorless test -in which no compensations are actuatedthe motor is actually controlled along a wrong load-dependent current reference. Since the motor parameters are not linear and depend on the operating point, the trajectory t 2 crosses a region of the dq plane where the inductances l dq and l Δ are different from those encountered along the MTPA trajectory. As a consequence, the closed-loop estimation error in Fig. 15 a is different from the open-loop estimation error in Fig. 13 b. While the open-loop estimation ε error has a direct expression, i.e. (12), the closed-loop estimation error requires some additional steps to be computed. In particular, the sensorless trajectory t 2 can be computed from the signal I hq , defined in (11) [13]. As concerns the trajectory t 2 recorded during the second test, it is possible to notice that it is more distant from the MTPA than the sensored trajectory t 1 . Moreover, at a current amplitude of 5 A (at time 5 s) the control diverges. This can be noted also in the oscillations of i x d , i x q , i d and i q (in Fig. 15 a) and of θ me (in Fig. 15 b). The phenomenon of sensorless control divergence happens in the regions of the dq plane where the signal I hq does not cross the zero for any current angle [12]. When the position estimation and the control are lost, the sensorless drive becomes unusable.

A. Pma-Synrm
To conclude, the trajectories computed during the experimental tests are compared and validated with the trajectories obtained from the FEA flux-linkages maps in Fig. 16. The comparison is done considering t 1 and t 2 (from Fig. 10), t 1 (from Fig. 12), and t 2 (from Fig. 14). In Fig. 16 it is possible to appreciate that standstill operation and quasi-steady-state current control make the experimental results -t 1 and t 2 -comparable with the prediction assuming an ideal control -t 1 and t 2 . In particular, the sensored trajectories t 1 and t 1 are in very good agreement. Regarding the sensorless trajectories, the FEA trajectory t 2 is truncated at a current amplitude of 4 A while the experimental trajectory t 2 reaches the current amplitude of 5 A. The control diverges for a current amplitude inferior to the rated value in both cases, indicating that the adopted motor is not completely suitable for signal injection sensorless control.

B. Synrm
The same experimental tests have been repeated considering the SynRM motor of Figs. 1 b. 17 shows the measured t 1 trajectory and Fig. 18 shows a detail of the main quantities during the test. As for the results described in Section IV-A the ellipse fitting technique has been adopted for the rotor position estimation. In particular, also in this case it can be noted that the position estimation error increases with the current amplitude, see Fig. 18 The results obtained for the measure of the trajectory t 2 are shown in Fig. 19. Fig. 20 shows the details of the main quantities during the test. Also in this case the current trajectory in the estimated reference frame d x q x appears along the MTPA trajectory. The deviation of the t 2 trajectory from the MTPA is clearly visible as well as the divergence of the estimator for a current amplitude of about 3.5 A at 3.5 s.
Finally, Fig. 21 shows the comparison between simulated and measured trajectories. Trajectories t 1 and t 2 are the same as in Fig. 10. t 1 and t 2 are as in Figs. 17 and 19 respectively. Also in this case a satisfactory agreement can be observed between predicted and measured trajectories.

C. Measurements of Trajectories Considering Different Demodulating Techniques
The results so far presented consider the control scheme reported in Fig. 8 adopting a rotating injection in αβ with the   ellipse fitting technique. Nevertheless, similar results can be achieved considering also different injection and demodulation techniques. This section describes in particular the measurements of the sensorless control trajectories on the SynRM motor which have been carried out adopting also demodulation techniques considering both rotating and pulsating injections. Fig. 22 shows the simulated and measured trajectories when rotating injection and the classical demodulation scheme of Fig. 7 is adopted. Since the injection is performed in the αβ stationary reference frame both t 1 and t 2 can be measured. Fig. 23 shows the measured t 2 trajectory when pulsating injection is considered adopting the demodulation scheme of Fig. 5. In this case, t 1 trajectory cannot be measured since the entire control works in the estimated d x q x reference frame.
It can be noted that the measured trajectories are slightly different considering the various injection and demodulation techniques. Nevertheless, all the measured trajectories are in satisfactory agreement with the simulations.

V. CONCLUSION
In this article, an experimental investigation about the convergence region of a sensorless drive has been presented and adopted. The self-sensing capability of the drive has been described and highlighted by the difference between the sensored and the sensorless operation trajectories, t 1 and t 2 respectively. Both the trajectories can be evaluated online with proper measurements adopting a standard drive equipped with a position sensor.
In the article, simulations and experimental results are compared. In particular, different injection and demodulation techniques have been considered in the paper. It has been shown that the results obtained considering different control strategies are in satisfactory agreement, also considering simulation results.