**Abbreviations**

The following abbreviations are used in this manuscript:


#### **Appendix A. Alteration of Inductance Matrix to Consider Rotor Skew and Coil Ends**

## *Appendix A.1. Rotor Skewing*

The 2D FEM software used in this work suppose a uniform magnetic field intensity along the third dimension. However, the WRIM and most asynchronous machines have a skewed rotor to prevent cogging and reduce torque oscillations. The skew angle *θsk* is generally given by the manufacturer. Fortunately, it is possible to add this effect into the inductance matrix [*L*(*θ*)] previously identified in 2D [22]. By dividing an unskewed machine along its length into *M* several slices, or sub-machines, the inductance matrix [*Lm*(*θ*)] of each sub-machine is simply the initial one [*L*(*θ*)] divided by *M*.

$$\left[L(\theta)\right] = \left[L\_1(\theta)\right] + \left[L\_2(\theta)\right] + \dots \left[L\_M(\theta)\right] \tag{A1}$$

$$[L\_m(\theta)] = [L(\theta)] / M, \quad m = 1, 2, \dots M \tag{A2}$$

Now, it is possible to replicate the skewing effect by shifting the rotor gradually along the length. Each of the inductance matrices [*<sup>L</sup>*1(*θ*)] ... [*LM*(*θ*)] needs to be shifted by *<sup>θ</sup>sk*/(*<sup>M</sup>* − 1). Finally, they can be added all together according to Equation (A1) to obtain the skewed [*L*(*θ*)] of the complete machine.

As for the WRIM of this work, *θsk* is equal to 7.5◦, which corresponds to a slot pitch. The machine was divided into 61 sub-machines. Figure A1 shows the impact of the added skew effect on rotor currents during load operation, with stator windings fed with perfectly sinusoidal voltages and rotor windings short-circuited. Current ripples are reduced as expected.

**Figure A1.** Rotor current with and without added rotor skewing effect, enlarged on right side.

## *Appendix A.2. Coil Ends*

Coil ends can be modeled in a 3D FEM software, whereas it cannot be in 2D. In the latter case such as in the present work, their inductive effect must be added to [*L*(*θ*)] by other means. For simplicity, we ignore the impact on mutual inductances. The technique used is to conduct a standstill frequency response (SSFR) test [14] and add a self-inductance to the windings in order to make the experimental response fit the simulated one. The test was conducted while rotor windings are short-circuited. Figure A2 compares the response of the FEM model to the experiment with and without coil ends. Inductances of 0.01 H and 0.00047 H were added to stator and rotor windings respectively, which is less than 10% of the magnetizing inductance.

**Figure A2.** Inductance seen from the stator versus frequency with and without coil ends, rotor terminals short-circuited.
