The Influence of Non-Sinusoidal Inductance on Saliency-Based Position Estimation for Permanent Magnet Machine

This paper analyzes the influence of non-sinusoidal inductance distortion on the saliency-based position estimation for the permanent magnet (PM) machines. Because of the high-power-density design of PM machines, the flux saturation usually causes the inductance self-saturation, inductance cross-saturation, and most importantly, secondary inductance harmonics. This paper fully investigates the saturation effect on saliency-based drives. Analytical model for the position estimation with saturated inductance is developed to understand the stability of saliency-based drive. It is confirmed that the saturation effect deviates the estimated flux position from actual rotor position. These estimation errors are critical for the saliency-based drive especially under load. This paper defines the feasible estimation region for PM machines with saturated and non-sinusoidal inductances. Considering the position estimation errors due to the saturated and non-sinusoidal inductance, the typical maximum torque per amp (MTPA) current trajectory based on the encoder-based control system should be modified especially for the IPM machines at full load. For the machines where the saturation condition is unknown, q-axis current without negative d-axis current is suggested to maintain control stability while the torque reduction is resultant. By contrast, the modified current trajectory could also be designed once the saturation reflected position estimation errors could be obtained. Finite element analysis (FEA) with inverter co-simulation is used to investigate the inductance distortion at various loads. A 6kW IPM machine prototype with highly non-sinusoidal inductance is tested for the experimental verification.


I. INTRODUCTION
High-performance permanent magnet machine drive requires instantaneous position information to realize fieldoriented control (FOC). For conventional sensor-based drives, separated position sensors, e.g. encoders and resolvers, are attached to obtain the instantaneous rotor position. However, the installation of position sensors degrades the FOC drive reliability [1]. Elimination of separated position sensors by using the position information from the machine itself (selfsensing) has become a promising solution. Considering the position sensorless drive at zero and low speed, the rotor position can be estimated through the spatial signal in machine inductances with high frequency (HF) AC voltage persistent injection [2]- [4].
Unfortunately, the position information provided by the machine inductance results in signal distortions due to the saturation. These saturation reflected attributes include 1) the saliency magnitude reduction [5], 2) cross-saturation (mutual-coupling) between rotor d-axis and q-axis [6], and 3) secondary inductance harmonics [7], [8]. It is important that the influence of saturation on position estimation increases as the load increases. They limit the saliency-based drive at full load, blocking them from the progress into industrial products.
The flux saturation firstly reduces the saliency magnitude, e.g. self-saturation, leading to the drive stability issue at high load. Considering the machine control with maximum torque per ampere (MTPA), q-axis inductance Lq decreases as the load increases. For IPM machines with saliency whereby qaxis Lq > d-axis inductance Ld, the saliency eventually reduces to zero which is not useful for position estimation. In [5], the rotor barrier is designed to reduce the load dependent saturation on Lq. Both the concentrated and distributed windings are also compared in [9] under the same IPM machine topology. It is reported that distributed windings lead to the minor saturation on Lq comparing to concentrated windings. Besides, an IPM machine with reverse saliency is proposed in [10]. Because of reverse saliency whereby Ld > Lq, the saliency increases as the load increases, leading to the better estimation performance at full load. However, the torque might reduce since the reverse saliency Ld > Lq is not as high as conventional saliency Lq > Ld.
In addition to inductance self-saturation, the flux saturation also causes the cross-saturation between orthogonal d-and qaxis. Under this effect, the saliency estimated flux position is no longer aligned with actual rotor position [6]. It is noted that d-q cross-saturation results in the constant offset on the position estimation. The saliency-based drive fails when the position offset is larger than 45 deg. In [11], the position offset is compensated based on one-dimensional look-up table (LUT). The offset compensation is directly added dependent on the load condition. However, in [12] and [13], it is reported that d-q mutual-coupling is influenced by both torque load and rotor position. Under this effect, twodimensional LUT is proposed to improve the offset compensation. Instead of LUT, the stator windings, tooth shape and rotor structure can be designed to mitigate d-q cross-saturation [14], [15].
Instead of self-and cross-saturation, the saturation also increases secondary harmonics in machine inductances. Considering the ideally sinusoidal-distributed inductance in IPM machines, only the 2 nd -order spatial harmonic in the stator-referred stationary frame is resultant. However, once the saturation occurs, the stator-referred inductance contains additional 4 th -, 6 th -, 8 th -order, etc. harmonics. As reported in [16], these inductance harmonics lead to periodic position estimation errors. Consequently, the position estimation fails once secondary harmonic magnitudes are higher than the fundamental magnitude. In [7], the adaptive decoupling is applied to remove these harmonics if harmonic magnitudes and phases are well-known. In addition, neural network compensation is developed through the database training with a sensor-based drive [17]. In [18], secondary saliency harmonics can be improved by the tracking of multiple saliency harmonics. Besides in [19], the saturation reflected inductance harmonics have more influence on the zerosequence voltage measurement than saliency current measurement under the same voltage injection. For the compensation of these inductance harmonics, all saliency harmonic magnitudes should be known on the estimated IPM machine.
This paper improves the saliency-based position estimation for the IPM machine with saturation reflected non-sinusoidal inductance. Although the influence of inductance cross-saturation and secondary harmonics have been reported on saliency-based drives, very few researches are related to the root causes of inductance distortion on PM machines. To fully investigate the saturation effect on saliency-based drives, an analytical model of the position estimation with saturated inductance is developed. For an IPM machine with considerable inductance harmonics, it is not possible to realize the saliency-based drive with conventional MTPA current trajectory. In order to investigate the drive stability, the inductance saturation property is carefully investigated to identify the most efficient operating condition. Two modified MTPA d-q current trajectories are proposed to maintain the saliency-based drive stability under load. For IPM machines where the saturation condition is unknown, q-axis current without negative d-axis current is suggested to maintain MTPA control stability. By contrast, the modified current trajectory can also be designed once the saturation reflected position estimation errors can be obtained. Although the efficiency is not compatible with encoder-based drive, the saliency-based drive can be maintained for machines with non-sinusoidal inductances. A 6 kW IPM machine is used for the experimental verification.

II. SALIENCY-BASED POSITION ESTIMATION
This section analyzes the inductance saturation on the saliency-based position estimation. The saturation results in the saliency ratio reduction, d-q cross-saturation and secondary harmonics. Analytical models are proposed to predict the corresponding estimations errors. Position compensation methods are then developed based on these analytical models.

A. SATURATION REFLECTED SALIENCY RATIO REDUCTION
For simplicity, an IPM machine with the linear flux distribution is firstly analyzed. Considering the ideal model, the d-q inductances matrix Ldq and α-β inductances matrix Lαβ are respectively derived by (1) and (2).
where ΣL=(Lqq+Ldd)/2 and ΔL=(Lqq-Ldd)/2 are the average and difference inductance. In (2), θe is the rotor position with electrical angle. It is noted that 2θe position information appears in L αβ which is used for saliency-based position estimation.
However, considering the nonlinear property of magnetic materials, both self-and cross-saturation are resultant. Under this effect, all inductances in Ldq and Lαβ decrease as load increases. Fig. 1 illustrates α-axis self-inductance L αα in (1) versus the rotor position.
For IPM machines, the saturation primarily locates at rotor ribs. It results in the considerable magnitude reduction on ΣL. In addition, load reflected q-axis current also causes the reduction on ΔL. Considering the self-inductance saturation, the inductance matrix L dq in (1) and L αβ in (2) should be modified by L dq1 in (3) and L αβ1 in (4).
where K Σsat and K Δsat are the coefficients smaller than 1 and dependent on machine topologies. K Σsat and K Δsat are used to analyze the inductance magnitude reduction respectively for average inductance ΣL and difference inductance ΔL, as seen in Fig. 1. Considering the salient PM machine, the torque output consists of electromagnetic torque and reluctance torque. At low speed under constant torque region, the electromagnetic torque is dominated. In this case, i q with current angle at 90 deg is larger than i d at low speed with 0 deg angle. In order to provide sufficient i q for electromagnetic torque production, the inductance reduction at 90 deg (q-axis) with K Σsat ΣL+ K Δsat ΔL is larger than that at 0 deg (d-axis) with K Σsat ΣL-K Δsat ΔL. To implement the saliency-based position estimation, a HF rotating voltage Vαβ_HF shown in (5) is superimposed on the fundamental voltage to induce position dependent signal.
where v c is the injection voltage magnitude and ω c is the injection frequency. In this paper, the rotating voltage is selected for the purpose to demonstrate the saturation reflected saliency distortion. Considering the influence of inverter non-linearity, other advanced HF voltages, e.g. square-wave, can be selected to reduce the deadtime effect. By superimposing the HF voltage Vαβ_HF, the inductive voltage drop is dominant at low speed since the EMF voltage is sufficiently low. The resulting HF current can be shown by  Based on the model in (6), Fig. 2 illustrates the influence of saturation on the saliency image distortion. Here, the saliency image is represented by the current trajectory of βaxis current i β1_HF versus α-axis i α1_HF in (6). In this figure, the injection frequency ω c is designed at 1KHz and the rotor positions θe are assumed respectively at 0 and 90 deg for the saliency image comparison.
Four different conditions are respectively analyzed. They are (a) no saturation at no load (K Σsat = K Δsat = 1), (b) the saliency reduction at 40% load (K Σsat = 0.83 and K Δsat = 0.5), (c) the reduction at 85% load (K Σsat = 0.67 and K Δsat = 0), and (d) reverse saliency at 120% load (K Σsat = 0.60 and K Δsat = -0.2). In Fig. 2, both K Σsat and K Δsat in (4) are selected based on the FEA simulation of test IPM machine with considerable inductance saturation. Detail explanation of this test IPM machine will be shown in section III and IV. The HF current trajectory at two different positions, θe = 0 deg and 90 deg, are overlaid within a single plot to clearly illustrate the saliency image at different positions.
Considering Fig. 2(a) without the saturation, the HF current trajectory is equivalent to an ellipse where the minorand major-axis are aligned respectively to θe + 0 deg and θe + 90 deg. The actual diameter of two axes can be obtained based on (6) by substituting the instantaneous θe. As seen in (6), the position signal proportional to ΔL results in the difference of two axes diameter. Under this effect, the Author Name: Preparation of Papers for IEEE Access (February 2017) VOLUME XX, 2017 diameter difference should be as high as possible in order to achieve a high signal-to-noise ratio (SNR) for the estimation. In Fig. 2 (a) at no load KΔsat = 1, the rotor position can be estimated by identifying this diameter difference.
However, as saturation increases where KΔsat = 0.5 in (b), the saliency image distorts from an ellipse to a circle. It is seen that the diameter difference decreases. At 85% load when KΔsat = 0 in (c), the saliency image eventually becomes a circle. Because there is no difference between major-and minor-axis, it is not possible to obtain the 2θe position signal in (6). Finally, in (d) when KΔsat < 0, the circle changes back to ellipse due to the reverse saliency, Ld > Lq. In this case, the reverse saliency whereby ΔL<0 is appeared once q-axis inductance Lqq is fully saturated at full load. Under this effect, the position estimation results in 90 deg position offset because the major-axis is aligned with θe + 90 deg, as demonstrated in Fig. 2(d).

B. D-Q CROSS-SATURATION
In addition to saliency ratio reduction at part A, the saturation also causes the inductance phase offset as load increases. Under this effect, the estimated flux position is no longer aligned with the actual rotor position. On the basis, the cross-saturation can be modelled by additional non-diagonal elements Ldq in matrix L dq from (1), as given by where L dq2 denotes d-q inductances matrix with crosssaturation, K cs is a coefficient larger than 1 dependent on load conditions. Considering the cross-saturation, K cs increases as the load increases [11]. Based on (7), the corresponding α-β inductances matrix L αβ2 is modified by 22 1 cos( ) sin( ) cos( ) sin( ) sin( ) cos( ) sin( ) cos( ) sin(2 ) cos (2 ) cos (2 ) By combining the influence of both self-inductance saturation at part A and cross-saturation at part B, the injection induced HF current can be extended from (6) to (9).
Comparing HF current between (6) and (9), a position offset θ offset is resultant due to cross-saturation. However, the diameter difference between minor-and major-axis is slightly longer due to additional component from L dq , as seen from the nominator from the last equation in (9). In addition, a visible offset occurs on two diagonal axes. Because of cross-saturation, this offset is equivalent to θ offset /2. Fig. 3 (c) shows the image at no saliency where KΔsat ΔL = 0. As seen for L αβ2 in (8), cross-saturation Ldq term also contains the position signal. Under this effect, the saliency image with an ellipse pattern still results even when ΔL=0. The position estimation can still be implemented by estimating the diameter difference between two axes. At this time, θ offset in (9) is equal to 90 deg by substituting KΔsatΔL = 0. Finally, in Fig. 3(d) with reverse saliency, the position offset increases to θ offset /2 + 90 deg due to Lq > Ld.
Considering both the saturation reflected saliency reduction and cross-saturation, the position signal eventually disappears once Ldq = ΔL = 0. This operating point is defined by the critical point for the saliency-based drive [5]. However, the discussion in [5] is under the sinusoidal inductance assumption without secondary harmonics. At next part, the influence of secondary inductance harmonics on the position estimation will be analyzed. It is shown that the infeasible estimation region extends to multiple areas instead of single critical point.

C. SECONDARY INDUCTANCE HARMONICS
Instead of saliency reduction and cross-saturation, the saturation also causes secondary harmonics on the sinusoidal This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. inductance. In order to analyze inductance harmonics with various harmonics, a general form of d-q inductances matrix L dq_G is developed in (10). In (10), both self-inductances Ldd_G and Lqq_G as well as mutual-inductance Ldq_G all contain 6 thorder spatial harmonics with respect to their corresponding angle offsets, θ dd,h' , θ qq,h' and θ dq,h' . In addition, L dd,0 , L qq,0 and L dq,0 represent DC components of Ldd_G, Lqq_G and Ldq_G, and h' is the harmonic order from 1 to infinity. Considering the ideal machine without saturation, h' can be assumed by zero, as originally seen in (1). According to the proposed matrix L dq_G in (10), ΔL can be represented as (11) With the d-q inductances matrix L dq_G in (10), the general form of α-β inductances matrix L αβ_G can be derived in (12) based on the frame transformation similar to (8).
cos (6  ) sin (6  ) sin (6  ) cos (        As seen from I αβG_HF in (13), the inductance harmonic h'=1 causes both 4 th -and 8 th -order HF current harmonics where their magnitudes are respectively denoted by |IΔL1_-4θe| and |IΔL1_8θe|. Because the fundamental HF current harmonic is located at 2θe, these two current harmonics cause 6 th -order periodic harmonics with different position offsets θerr,4 and θerr,8 on the position estimation. It is noteworthy that the saliency-based position estimation is primarily based on the tracking of 2θe spatial signal in (13). If secondary current harmonic of either |IΔL1_-4θe| or |IΔL1_8θe| is higher than 2θe current |IΔL|, the position estimation ultimately fails. More importantly, these secondary harmonics are primarily caused by the saturation. Thus, their magnitudes increase as the load increase. Fig. 4 demonstrates the saliency image iβG_HF versus iαG_HF considering the influence of secondary current harmonic, h'=1 based on (13). In this simulation, two ratios of ΔL1/ΔL =0.3 and ΔL1/ΔL=1.3 are compared in Fig. 4(a) and (b). The mutual inductance Ldq,1 is assumed as zero for simplicity. As shown in Fig. 4, both the ellipse size and orientation change as rotor rotates. Different from saliency images in Fig. 2, the minor-axis is no longer simply aligned with the instantaneously rotor position. For example, minor-axis in (b) at 0, 45, and 90 deg is not aligned with corresponding positions. Under this effect, pulsating errors occur on the position estimation. More importantly once secondary harmonic magnitudes are higher than 2θe harmonic whereby ΔL1/ΔL=1.3 in Fig. 4(b), the ellipse size results in the visible change and the axis is no longer aligned with the rotor position, leading to the infeasible estimation.  In this simulation, a synchronous reference frame filter in [7] is implemented to isolate position dependent signals in i αG_HF and i βG_HF . After that, the arctangent calculation is used to realize the position estimation. In Fig. 5(a), 6 th -order position harmonic is observed because of the secondary harmonic ΔL1 in (13). More important in Fig. 5(b) once secondary harmonic ΔL1 is larger than fundamental ΔL, the position estimation eventually fails. This result is consistent with the analysis in (13). Secondary inductance harmonics are key issues to limit the position estimation. The infeasible estimation is resultant once the resulting secondary current harmonics are higher than the fundamental saliency current.

III. FEA SIMULATION
A 6 kW IPM machine prototype is built for FEA simulation to verify the influence of saturated and nonsinusoidal inductance on the saliency-based position estimation. Key machine and drive specification are listed in Table I. The geometric feature of test machine is shown in Fig. 6. It contains an 8-pole rotor with V-shape topology and a 12-slots concentrated windings.

A. INDUCTANCE WAVEFORM DISTORTION
The inductance waveforms versus various load conditions are analyzed in this section. The test machine model with linear magnetic materials is firstly investigated in Fig. 7. For the d-q frame self-inductance and mutual-inductance in Fig.  7(b), Ldd and Lqq are constant values and the mutualinductance Ldq is zero.

FIGURE 6. Illustration of test PM machine: (a) stator/rotor topology, and (b) windings configuration
Besides for α-β frame self-inductances and mutualinductance in Fig. 7(a), L αα , L ββ , and L αβ all results in sinusoidal waveforms. These waveforms are similar to the ideal inductance waveforms.
The machine model considering nonlinear magnetic materials is then compared. Fig. 8(b) and (d) illustrate the d-q frame self-inductances Ldd and Lqq , as well as the mutualinductance Ldq at no-load and full-load conditions. Besides, Fig. 8(a) and (c) shows the corresponding α-β frame selfinductances L αα and L ββ , and mutual-inductances L αβ . Based on FEA, the non-linear inductance saturation could be considered at different loads. Different from Fig. 7(b) even at no-load condition, L dd in Fig. 8(b) contains visible 6 th -order harmonic because of the inductance distortion. The harmonic with 6 th -order is consistent with the inductance model derived in (10). More importantly at full load, L qq significantly reduces where the magnitude is almost the same as L dd . In addition, the mutual-inductance L dq contains both constant offset and 6 th -order harmonic. These non-ideal attributes are primarily caused by the saturation. Both the constant offset and harmonic increase as load increases. On the other hand, Fig. 8(c) and (d) illustrate the corresponding α-β frame inductances. Because of the saturation, all inductances distort from ideal sinusoidal waveforms. For the saturated inductance model in Fig. 1, K Σsat and K Δsat both equal unit value can be assumed at no-load condition in Fig.  8(b). In this case at full-load condition in Fig. 8(d), the corresponding K Σsat and K Δsat are respectively 0.9889 and 0.2320. More importantly, the peak value in L αα is at 60 deg instead of 90 deg from the ideal model in Fig. 1. Under this effect, the position estimation eventually fails under load due to the saturation reflected position offset and harmonics.

B. INFEASIBLE POSITION ESTIMATION REGION
This part investigates the infeasible estimation region. As mentioned in section II part C, secondary inductance harmonics cause additional current harmonics, as derived in (14). Once any secondary current harmonic magnitude, e.g. |IΔL1_-4θe| or |IΔL1_8θe|, is higher than the fundamental current magnitude |IΔL|, the position estimation causes considerable errors, and eventually leads to the infeasible estimation. Fig. 9 analyzes saliency current magnitudes contour caused by different inductance harmonics versus q-axis current i q and d-axis current i q . In Fig. 9, the motor inductance is simulated using ANSYS FEA software for the saliency current analysis. The HF voltage Vαβ_HF in (5) is designed to be 1V with 1kHz injection frequency. The rotor speed is 1rpm to manipulate the low speed operation. In Fig.  9(a), the main 2θ e saliency current harmonic magnitude |IΔL| is analyzed at different load conditions. It is observed that |IΔL| results in the insufficient magnitude when negative i d is applied. In general, Ldd increases while Lqq decreases once  Fig. 9(d) -i d /+ i q is applied for typical MTPA control. By contrast, secondary saliency current harmonics induced by 4θ e and 8θ e are respectively illustrated in Fig. 9(b) and (c). Once either the 4θ e or 8θ e secondary current harmonic magnitude is higher than the 2θ e magnitude, the position estimation ultimately fails. This infeasible estimation typically occurs when -i d is higher than a certain value. By comparing these current magnitudes, Fig. 9(d) indicates the infeasible region for the saliency-based position estimation. In this figure, the infeasible region is defined when the operation point is located under (14).  (13). Both iα_HF and iβ_HF are calculated based on the nonlinear inductance data under various loads using FEA. Fig. 10(a) shows the position estimation for the operating point at point A in Fig. 9(d) within the proposed feasible region.By contrast in Fig. 10(b), point B is selected inside the infeasible region. Both current loads in Fig. 10(a) and (b) are the same while only the phase angle is different.
Although a certain amount of estimation error could be observed at point A, the estimation maintains stable. More importantly, compensation methods could be implemented based on the inductance model in section II [20], [21]. However, at point B, a totally incorrect estimation that shows the opposite direction and doubled speed is observed. It is not possible to maintain the stability inside this specific region. It is noted that the estimation accuracy also decreases once the operation point is close to the infeasible region though the estimation stability is achieved.

C. INFLUENCE ON TORQUE OUTPUT
In order to analyze the influence of position estimation errors on the generated torque, the torque contour using FOC with different position feedback signals is compared in this part. They are encoder-based measured position and saliency-based position estimation. The software-in-loop cosimulation technique is applied to combined the FEA model of test motor with the inverter circuit, and FOC and position estimation algorithm.
For the encoder-based drive in Fig 11(a), the FOC drive is realized based on the measured position through encoder. Fig.  12(a) demonstrates the corresponding control diagrams for MTPA control in Fig. 11(a) while the actual position is used for d-q transformation. In this case, the corresponding torque contour in Fig. 11(a) is illustrated in actual d-q reference frame. As seen in Fig. 11 (a), the maximum torque is achieved with -id/iq once the encoder position is implemented.
By contrast for the saliency-based drive in Fig. 11(b), the FOC drive is implemented based on the estimated position. Fig. 12(b) illustrates the corresponding signal flowchart. Different from Fig. 12(a), the estimated position is used for d-q transformation. Under this effect, the torque contour in Fig. 11(b) is shown in estimated d-q frame where the superscript of est is added to distinguish from actual frame. Because of the cross-saturation and secondary harmonics mentioned in section II, position estimation errors affect the MTPA control performance. Considering these errors, a significant torque output reduction is observed at estimated / est est dq ii − point. Different from the saliency current simulation in Fig. 9, the cross-saturation is also considered in Fig. 11(b). Considering both the cross-saturation and inductance harmonics, the infeasible region in Fig. 11(b) is bigger comparing to the region in Fig. 9.
It is noted that in Fig. 11(b), the peak torque is appeared under / est est dq ii + in estimated frame. From the analytical model in section II, the saliency-based position estimation errors consist of position offset θ offset in (9) and position harmonics in (13). The torque output generated by the PM machine is represented by (15) (15) where λ pm is the PM flux and p n is number of rotor pole pairs. However, the actual d-and q-axis current is not possible to obtain if there are errors between actual position and estimated position. In general, the relationship between actual and estimated d-and q-axis current is shown by (16).   Fig. 11(a), and (b) salience-based drive in Fig. 11 (16) where err  is position errors including the position offset due to the inductance cross-saturation, and the periodic errors due to inductance harmonics. Therefore, the torque can be derived by (17) Comparing the torque contour in Fig 11(a) and (b), the MTPA current trajectory should be redesigned for the saliency-based drive due to the effect of the dq crosssaturation and the secondary inductance harmonics.

IV. EXPERIMENTAL RESULTS
The same 8-pole IPM machine prototype illustrated in Fig.  6 is built for the verification of saliency-based position estimation. Fig. 13 illustrates the photograph of saliencybased test bench. The test machine is coupled to a load machine for the load operation. The inverter switching frequency is set at 10kHz synchronous to the sample frequency. All position estimation algorithms are implemented in a 32-bit microcontroller, TI-TMS320F28069.

A. INDUCTANCE MEASUREMENT
The test machine inductance is firstly tested. For the inductance measurement, the rotor is locked by the load machine. An AC voltage is applied across machine two phases through the inverter where the third phase is open. Based on this measurement, the self-inductance Lββ can be obtained by (18). e harmonic, secondary harmonics appear at 6θ e , 12θ e …etc. Among these secondary harmonics, 6θ e harmonic is highest where the magnitude is around 106% with respect to 2θ e harmonic. This result is consistent with the proposed general inductance matrix L dq_G in (10). The 6 th -order periodic harmonic on the saliency-based position estimation is expected.

B. INFEASIBLE POSITION ESTIMATION REGION
This part evaluates the saliency-based position estimation limitation at different loads. In this experiment, the test machine speed is maintained at 30 rpm through the external dyno. The saliency-based position estimation based on [22] is implemented on the test machine for MTPA control. Fig.  12(b) illustrates the corresponding FOC drive through the saliency-based position estimation. For the following experiments, no position error compensation is applied to clearly evaluate the position estimation performance. The injection voltage is 20% DC bus at 1kH frequency. The rotor position is estimated at different current conditions from 0A to 105A. Fig. 15 shows the experiment of feasible saliency-based estimation region. In this experiment, the infeasible region is defined when the position estimation has the opposite trend comparing to the actual position. In this case, no torque can be generated where the FOC is not possible. It is observed that the experimental feasible estimation region is similar to the predicted feasible region in Fig. 9(d). As a result, the saliency-based estimation performance can be predicted in advance based on the proposed saliency current model in (14) through the FEA of 6 th -order secondary current harmonic magnitude. Fig. 16 further demonstrates the saliency-based position estimation at two representative current conditions in Fig. 16. Fig. 16(a) shows time-domain waveforms of estimated position, measured position and A-phase current for A point in Fig. 15. It is within the feasible region where the stable estimation is expected. In this point, i d /i q is respectively at 23.9A/86.3A. It is concluded that the saliency-based estimation maintains stable though visible 6 th -order harmonic occurs. This 6 th -order harmonic error is the same with the analytical prediction in Fig. 5. By contrast, Fig. 16  Because the magnitude of secondary current harmonics is higher than the fundamental 2θ e harmonic, the saliency-based position estimation is not possible. In this case, the sensorless drive ultimately fails.

C. INFLUENCE OF INDUCTANCE HARMONICS ON TORQUE OUTPUT
The influence of non-sinusoidal inductance on the torque production is tested at this part. In this experiment, the test machine is torque controlled with current regulation where the torque is measured through a torque sensor. Fig. 17 compares the measured torque contour where the position feedback is obtained by (a) encoder sensor and (b) saliencybased position estimation. Similar to the torque simulation in Fig. 11, the torque contour under the saliency-based drive is illustrated in the estimated dq frame to clearly investigate position errors caused by the saliency-based estimation. It is expected that the peak torque is appeared under / est est dq ii + in estimated frame considering the position offset θ offset in (9).
For the encoder-based drive in Fig. 17(a), the maximum torque at rated current is observed where the  However in Fig. 17(b), the estimated position is implemented for the current regulation as seen in Fig. 12(b). Under this effect, d-and q-axis current derived from estimated position consist of both position offset and periodic position harmonics. If these position errors are unknown, visible reduction on the torque output is resultant under the same MTPA current trajectory in Fig. 17(a). In the estimated current frame, the maximum torque appears at

V. CONCLUSION
The analysis of saliency-based position estimation for PM machine with non-sinusoidal inductance is proposed. The analytical model which considers inductance self-saturation, cross-saturation and secondary inductance harmonics on the position estimation is developed. The feasible region of saliency-based drive can then be predicted at different conditions. A modified MTPA current trajectory considering all estimation errors is proposed to maintain the saliencybased drive for IPM machines with saturated non-sinusoidal inductances.