*5.1. IPMSM Torque Estimation*

Two pulsating HF currents have been used in this case, of magnitude 0.05 pu and frequency of *ωdHF* = 2 · *π* · 500 rad/s and *ωqHF* = 2 · *π* · 1000 rad/s, respectively (38). The *HPF* needed to isolate the HF current components (see Figure 7) has a bandwidth of 5 Hz. Band stop filters *BSF*1 and *BSF*2 used to remove the negative sequence components of the HF currents and voltages have a bandwidth of 2 · *π* · 10 rad/s.

Due to rotor magnets, the machine will work at high saturation levels even for no-load conditions. The coefficient linking the HF estimated inductances (incremental inductances) and the absolute inductances is assumed to be constant, *k<sup>μ</sup>* = 14 (see Figure 4).

Figure 9a shows the estimated torque from (2) (*T*ˆ *outconv* ) assuming constant parameters; the estimated torque when the machine parameters are estimated from the injected HF current (18) (*T*ˆ *outHF* ); and the measured torque using the torque transducer shown in Figure 8 (*Tout*), when the magnitude of the fundamental current *I<sup>r</sup> sdq* changes from 0p.u. to 1p.u. following a Maximum Torque Per Ampere (MTPA) trajectory. Figure 9b shows the estimation error using both general torque equation and the proposed method. It can be observed that torque estimation error is reduced when the machine parameters are estimated using HF signal injection; the improvement being more relevant at higher current levels. This is an expected result since the *dq*-axes inductances values will differ more to their base values as the saturation level increases.

**Figure 9.** Experimental results: IPMSM & pulsating current injection: (**a**) Estimated and measured torque. (**b**) Estimated torque error. 0 < *I<sup>r</sup> sdq* < 1 p.u., *ω<sup>r</sup>* = 50*Hz*, *IHF* = 0.05 p.u., *ωdHF* = 2 · *π* · 500 rad/s , *ωqHF* = 2 · *π* · 1000 rad/s.

### *5.2. SPMSM Torque Estimation*

HF inductances have been estimated in this case using pulsating voltage injection at 45◦ as described in Section 4.1. A HF voltage of 10 V and 250 Hz has been used (23). A band pass filter of 100 Hz was used to isolate the HF currents. Similar for the case of the IPMSM, the machine will work at high saturation levels even at no-load conditions due to the magnets. Therefore, also, in this case the coefficient linking the estimated HF inductances and the absolute inductances has been considered constant *k<sup>μ</sup>* = 14.

Similarly to Figure 9, Figure 10a shows the estimated torque assuming constant parameters and adapting machine parameters using HF voltage injection. The fundamental current,*I<sup>r</sup> sdq* was varied from 0 p.u. to 1 p.u. following a MTPA trajectory. Figure 10b shows the estimation error for both methods. As for the IPMSM case, torque estimation error is also reduced when the machine parameters are estimated using HF signal injection, the improvement being more noticeable than for the IPMSMs.

Finally, Figure 11 shows the actual and estimated torque when there is a step-like change in the *q*-axis current command from 0 to 1 p.u.. It can be observed from the error shown in Figure 11b that the torque estimator responds in the range of ms.

**Figure 10.** Experimental results: SPMSM & pulsating voltage injection: (**a**) Estimated and measured torque. (**b**) Estimated torque error. 0 < *I<sup>r</sup> sdq* < 1 p.u., *ω<sup>r</sup>* = 16 Hz, *VHF* = 10 V, *ωHF* = 2 · *π* · 250 rad/s.

**Figure 11.** Experimental results: SPMSM & pulsating voltage injection: (**a**) Estimated and measured torque. (**b**) Estimated torque error. Transient response to a step-like change in *q*-axis current command from 0 to 1 p.u., *ω<sup>r</sup>* = 16 Hz, *VHF* = 10 V, *ωHF* = 2 · *π* · 250 rad/s.

#### *5.3. SynRM Torque Estimation*

Torque estimation for the SynRM has been performed using rotating HF voltage injection. A HF voltage of 40 V and 500 Hz has been used (31). Figure 12 shows experimental results when the magnitude of the fundamental current, *I<sup>r</sup> sq*, changes from 0p.u. to 1p.u. following a MTPA trajectory. In a first approach, the machine has been considered to be working at low-middle saturation levels. Therefore, the coefficient linking the HF estimated inductances and the absolute inductances *kμ*, has been considered to be constant and equal to 1 (i.e., the incremental inductance has been assumed to be approximately equal to the absolute inductance, see Figure 3). However, from the torque estimation error shown in Figure 12b, it is deduced that once the machine begins to saturate this approach is not longer valid. This suggests that the relation between the HF estimated inductances and the absolute inductances has to be adjusted using at least a second order polynomial. This is a subject of ongoing research.

**Figure 12.** Experimental results: SynRM & rotating voltage injection: (**a**) Estimated and measured torque. (**b**) Estimated torque error. 0 < *I<sup>r</sup> sdq* < 1 p.u., *ω<sup>r</sup>* = 16 Hz, *VHF* = 50 V, *ωHF* = 2 · *π* · 500 rad/s.

#### **6. Conclusions**

Parameter estimation using HF signal injection aimed to improve the accuracy of torque estimation methods has been addressed in this paper, with the aim of making the estimation robust against variations in the operating conditions of the machine. This implies a reformulation of the torque equation, which will be function of the HF (incremental) inductances instead of the absolute inductances. Accurate modelling of the relationship between the incremental and absolute inductances will be therefore of paramount importance. Additionally, estimation of the PM flux is based on the linear relation with the *d*-axis inductance.

Three different types of HF signal injection have been considered: Pulsating Voltage Injection, Rotating Voltage Injection and Pulsating *dq*-axes Current Injection. In all the cases, the signal is superposed on top of the fundamental excitation applied by the inverter, not interfering therefore with the normal operation of the drive. It is concluded that pulsating current injection is advantageous as it is insensitive to the resistive components of the HF model and to cross-coupling effect. In change, its implementation is slightly more difficult due to the need of HF current controllers. In all the cases, no modification of the hardware is required.

Experimental verification using IPMSM, a SPMSM and a SynRM have been presented, which confirm the viability of the proposed methods.

**Author Contributions:** All the authors contributed in formulating the problem and designing the research proposal. M.M. wrote the paper, collected the experimental data and analysed the data. D.R. suggested the research topic, guided M.M. to complete the research and helped to analyse the data. D.F. developed the set up for the experimental tests. F.B. helped to write and finish the paper. All authors discussed the results and contributed to revised the final manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded in part by the Research, Technological Development and Innovation Programs of the Spanish Ministry Economy and Competitiveness, under grant MINECO-17-ENE2016-80047-R, by the Government of Asturias under project IDI/2018/000188 and FEDER funds and by the University of Oviedo under grant PAPI 2018-PF-12.

**Conflicts of Interest:** The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**


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