*2.1. Model's Equations*

The model used for the purpose of this work does not include magnetic saturation. Iron losses as well as capacitive effects are also neglected.

A wound rotor induction machine, as well as many other types of electrical machines, comprises *n* electrical circuits where current can flow, such as the phases and bars. The total magnetic flux *φ* at a given circuit is the sum of the magnetic fluxes received from all other circuits and itself. It is worth noting that every quantity referenced in this manuscript is instantaneous and no referential transformation is used.

$$\Phi\_i = \sum\_{j=1}^{n} \Phi\_{ij} \tag{1}$$

The magnetic flux received at a given circuit from another one depends on the current *i* flowing in the sender and the magnetic coupling *L* between the two. The coupling is a function of the rotor angular position *θ*.

$$\phi\_{i\bar{j}} = L\_{i\bar{j}}(\theta)i\_{\bar{j}} \tag{2}$$

All aforementioned equations are included into a single matrix equation:

$$[\phi] = [L(\theta)][i] \tag{3}$$

Flux variation seen by any circuit creates a voltage *v* given by:

$$\mathbb{E}[v] = [\mathbb{R}][i] + \frac{d[\phi]}{dt} \tag{4}$$

By combining Equations (3) and (4), the dynamic system of Equation (5) is obtained.

$$\frac{d[\phi]}{dt} = -[R][L(\theta)]^{-1}[\phi] + [\upsilon] \tag{5}$$

As for electromagnetic torque *Tem*, it can be computed using:

$$T\_{cm} = 0.5[i]^T \frac{d[L(\theta)]}{d\theta}[i] \tag{6}$$

For search coils or any electrical circuit always left open, Equation (7) is used instead to compute back-emf *vw* for better numerical stability of ordinary differential equations solvers. Matrix [*Lw*] defines the coupling between the search coils and the other circuits carrying a current.

$$\mathbb{E}\left[v\_{\text{uv}}\right] = \frac{d\left[L\_{\text{uv}}\right]\left[i\right]}{dt} \tag{7}$$

To sum up, identification of the model requires to know the inductance matrix function [*L*(*θ*)] which can then be inverted and differentiated. What makes the CFE-CC model versatile is the identification process of [*L*(*θ*)] with an FEM software. It uses a lookup table to store the inverse and derivative inductance matrix for many discrete positions of the rotor.

As in [14], it is possible to consider magnetic saturation with the precomputed inductance functions by using a correction coefficient afterward on computed magnetic flux linkages. Performances and computing time differ depending on type of machine and chosen quantities to compute the coefficient.

#### *2.2. Identification of Electrical Circuits*

Table 1 gathers specifications of the WRIM used in this work.

**Table 1.** Specifications of the WRIM.


As depicted in Figure 1b, there are three windings on the stator (*as*, *bs*, *cs*) and three windings on the rotor (*ar*, *br*, *cr*). A small search coil is also added around one stator tooth (*ws*) in order to measure the voltage across it during operation. These seven circuits make up the CFE-CC model.

The resistance matrix [*R*] of the WRIM is diagonal as shown below. Each terminal is accessible so resistance values can easily be measured. The search coil is not included because it is left as an open circuit, thus Equation (7) is used instead of (5) for this particular case.

$$\begin{aligned} [R] = \begin{bmatrix} R\_{ds} & 0 & 0 & 0 & 0 & 0 \\ 0 & R\_{bs} & 0 & 0 & 0 & 0 \\ 0 & 0 & R\_{cs} & 0 & 0 & 0 \\ 0 & 0 & 0 & R\_{ar} & 0 & 0 \\ 0 & 0 & 0 & 0 & R\_{br} & 0 \\ 0 & 0 & 0 & 0 & 0 & R\_{cr} \end{bmatrix} \tag{8} \\ \tag{9} $$

As for [*L*], for any given rotor position, it has the form shown below. Magnetic couplings of the search coil are also included in a separate matrix [*Lw*].

$$\begin{bmatrix} L \\ \hline \\ L \end{bmatrix} = \begin{bmatrix} L\_{\text{adsas}} & L\_{\text{adsbs}} & L\_{\text{adscs}} & L\_{\text{adsar}} & L\_{\text{adsbr}} & L\_{\text{adscr}} \\ L\_{\text{basas}} & L\_{\text{bsds}} & L\_{\text{bscs}} & L\_{\text{bsar}} & L\_{\text{bsbr}} & L\_{\text{bscr}} \\ L\_{\text{csas}} & L\_{\text{csbs}} & L\_{\text{cscs}} & L\_{\text{csar}} & L\_{\text{csbr}} & L\_{\text{cscr}} \\ L\_{\text{ausas}} & L\_{\text{urbs}} & L\_{\text{urbrs}} & L\_{\text{bar}} & L\_{\text{urbr}} & L\_{\text{btrr}} \\ L\_{\text{cras}} & L\_{\text{crs}} & L\_{\text{crs}} & L\_{\text{cran}} & L\_{\text{cbtr}} & L\_{\text{ccr}} \end{bmatrix} \tag{9}$$
 
$$[L\_{\text{w}}] = \begin{bmatrix} L\_{\text{w}\text{asas}} & L\_{\text{ws}\text{abs}} & L\_{\text{ws}\text{as}} & L\_{\text{usar}} & L\_{\text{usbr}} & L\_{\text{uscr}} \end{bmatrix} \tag{10}$$

**Figure 1.** (**a**) Photo of the wound rotor induction machine (WRIM) showing its stator and rotor; (**b**) circuits of the WRIM used to build the model.

#### *2.3. Computation of the Inductance Matrix Using FEM*

Computation of [*L*(*θ*)] is performed with an FEM software and accurate plans of the machine geometry and windings. A 3D FEM software is an ideal choice as it consider geometries that are impossible to model in a 2D environment. However, a 2D FEM software is normally easier to use and less computationally expensive. It should be noted that computation time required to solve the finite element domain is not related to the CFE-CC model's computation time at each time step, because results of the FEM are used as precomputed lookup tables.

For the present work, the authors opted for a 2D software to quickly compute [*L*(*θ*)]. The geometry of the WRIM makes it convenient to use a 2D space, but a few issues cannot be accounted in 2D alone such as rotor skewing and coil ends. Their respective effects are added to the model by altering the computed inductance matrix. Details of this procedure are presented at Appendix A. Figure 2 shows the FEM model of the WRIM in FLUX2D, the FEM software used for this work.

The domain can be reduced to only half of the complete machine because the machine has two pairs of poles and presents a symmetry. All magnetic materials are assumed to be linear, so saturation is not considered. Meshing is performed using the software's automatic mesh tool. Mesh size was selected by comparing the solution obtained using a fine mesh to more and more coarse ones, until the error starts increasing noticeably. Once the FEM model is ready, the procedure for identifying magnetic couplings is launched. Inductance matrix [*L*] is computed for many rotor positions to build the lookup table.

**Figure 2.** (**a**) WRIM rendered in FLUX2D (**b**) Meshed domain.

Inductance matrix [*L*(*θ*)] is obtained by first computing the inductance matrix of the machine without windings, i.e., the inductance matrix of the slots [*Lslot*(*θ*)]. That way, one may generate any winding configuration without restarting the FEM process simply by applying the following matrix transformation:

$$[L(\theta)] = [N]^T [L\_{slot}(\theta)][N] \tag{11}$$

[*N*] is the winding configuration matrix. It contains the number of turns *N* of each electrical circuit in each slot. Equation (12) shows how to build the matrix, where *nc* is the total number of circuits and *ns* is the total number of slot. 

$$N = \begin{bmatrix} N\_{1,1} & N\_{1,2} & \dots & \dots & N\_{1,n\_c} \\ N\_{2,1} & N\_{2,2} & & & \\ \vdots & & \ddots & & \\ \vdots & & & \ddots & \\ \vdots & & & \ddots & \\ N\_{n\_c,1} & & & & N\_{n\_c,n\_c} \end{bmatrix} \tag{12}$$

In total, 1440 positions were computed evenly for a 180◦ rotation, i.e., 60 positions per stator slot pitch or an angular step of 0.125◦. It is plenty to have a good resolution. Impact of inductance discretization is described thoroughly in [18]. At this point, it is important to take into consideration the physical memory to store the lookup table. Since the WRIM has six circuits without the search coil, the total number of elements in the lookup table for the inverse inductance matrix is 51,840. Another lookup table of the same size is necessary for the derivative if torque computation is included.
