To overcome convergence problems of the simulation due to magnetic material models, a differential reluctivity tensor can be applied to solve the finite element formulation of the magnetic vector potential [5].

Using the 2D **A** formulation, the magnetic flux density **B** is defined as **B** = ∇ × **A** and the magnetic vector potential **A** = (0, 0, *A*_{z}) is discretized through linear piecewise functions:

$$\begin{array}{}{A}_{z}(x,y,t)=\sum _{j=1}^{Ne}{\alpha}_{j}(t){\beta}_{j}(x,y)\end{array}$$(1)

where *Ne* is the number of nodes, *α*_{j}(*t*) is the nodal value of the vector potential *z* component, *β*_{j}(*x*, *y*) is the shape function. The vector associated to the shape function is **ω**_{j} = (0, 0, *β*_{j}) since in the 2D formulation only the *z* component of the magnetic vector potential is not zero.

Assuming negligible eddy currents in the permanent magnets, the weighted residual approach is applied on Ampere’s law, leading to the weak formulation:

$$\begin{array}{}{\displaystyle \underset{\mathit{\Omega}}{\int}\mathbf{H}\cdot \mathrm{\nabla}\times {\mathit{\omega}}_{\mathit{i}}+\underset{\mathrm{\partial}\tau}{\oint}\mathbf{H}\times {\mathit{\omega}}_{\mathit{i}}d\tau =\underset{{\mathit{\Omega}}_{s}}{\int}{\mathbf{J}}_{\mathbf{s}}\cdot {\mathit{\omega}}_{\mathit{i}}d{\mathit{\Omega}}_{s}}\end{array}$$(2)

where **J**_{s} is the source current in a subspace *Ω*_{s} of the entire domain *Ω*. The closed integral on the boundary is equal to zero due to homogeneous Neumann or Dirichlet boundary conditions. To solve (2) in the time domain, a time-stepping technique is applied: this is due to the time dependencies of the hysteretic materials. If *A*(*t*_{n}) is a given state of the magnetic problem the state at the next time instant *t*_{n+1} = *t*_{n} + *Δt* is calculated using iterative Newton-Raphson (NR) method. For each NR iteration *A*^{k} = *A*^{k−1} + *ΔA*^{k} the increment *ΔA*^{k} must be calculated. Therefore Eq. (2) is linearized around *A*^{k−1}. This linearization is obtained deriving the equation with respect to *α*_{j}, which can be achieved through the differential reluctivity:

$$\begin{array}{}\frac{d\mathbf{H}}{d{\alpha}_{j}}=\frac{d\mathbf{H}}{d\mathbf{B}}\cdot \mathrm{\nabla}\times {\mathit{\omega}}_{\mathit{j}}={\nu}_{d}\mathrm{\nabla}\times {\mathit{\omega}}_{\mathit{j}}\end{array}$$(3)

Substituting (3) in (2), Ampere’s law becomes:

$$\begin{array}{}& \sum _{j=1}^{Ne}\mathit{\Delta}{\alpha}_{j}^{k}{\displaystyle \underset{\mathit{\Omega}}{\int}({\nu}_{d}\cdot \mathrm{\nabla}\times {\mathit{\omega}}_{\mathit{j}})\cdot (\mathrm{\nabla}\times {\mathit{\omega}}_{\mathit{i}})d\mathit{\Omega}=}\\ & ={\displaystyle \underset{{\mathit{\Omega}}_{s}}{\int}\mathbf{J}({t}_{n+1})\cdot {\mathit{\omega}}_{\mathit{i}}d\mathit{\Omega}-\underset{{\mathit{\Omega}}_{s}}{\int}{\mathbf{H}}^{k-1}\cdot \mathrm{\nabla}{\mathit{\omega}}_{\mathit{i}}d\mathit{\Omega}}\end{array}$$(4)

In the discrete time-stepping scheme the differential reluctivity can be expressed as
$\begin{array}{}{\nu}_{d}=\frac{\mathit{\Delta}\mathbf{H}}{\mathit{\Delta}\mathbf{B}}=\frac{\mathit{\Delta}\mathbf{H}\cdot \mathit{\Delta}\mathbf{B}}{\mathit{\Delta}\mathbf{B}\cdot \mathit{\Delta}\mathbf{H}}\end{array}$ with *Δ***H** = **H**^{k+1}(*t*_{n+1}) – **H**(*t*_{n}) and *Δ***B** = **B**^{k+1}(*t*_{n+1}) – **B**(*t*_{n}).

In all elements where hysteresis is considered, the homogenized parametric algebraic model (PAM) described in [6] is adopted to include eddy currents and hysteresis:

$$\begin{array}{}\mathbf{H}(\mathbf{B},\dot{\mathbf{B}},{p}_{k})=({p}_{0}+{p}_{1}|\mathbf{B}{|}^{2{p}_{2}})\cdot \mathbf{B}+{p}_{3}\dot{\mathbf{B}}+\frac{{p}_{4}\dot{\mathbf{B}}}{\sqrt{{p}_{5}^{2}+|\dot{\mathbf{B}}{|}^{2}}}\end{array}$$(5)

**Ḃ** is the time derivative of the magnetic flux density and the parameters *p*_{0} – *p*_{5} are material constants that have been found through the identification procedure in [6]. In particular, the parameters *p*_{0}, *p*_{1} and *p*_{2} are related to the anhysteretic magnetization curve, *p*_{3} is related to eddy currents in the laminated sheets, *p*_{4} and *p*_{5} are linked to the hysteresis phenomena. Since the term *B*^{2·p2} is not asymptotic, Eq. (5) has been applied below the saturation flux density |*B*| = *B*_{s}, while above saturation the *BH* curve is assumed to be linear with a slope equal to the vacuum permeability *μ*_{0}. *B*_{s} is computed as:

$$\begin{array}{}{B}_{s}=\sqrt[2{p}_{2}]{\frac{\frac{1}{{\mu}_{0}}-{p}_{0}}{{p}_{1}\cdot (2{p}_{2}+1)}}\end{array}$$(6)

Since the waveforms in the magnetic gear are sinusoidal in first approximation, the set of parameters chosen from [6] is the one reported in .

Table 1 List of parameters for the PAM model retrieved by [6]

Eq. (4) is computed elementwise for the hysteretic regions providing
$\begin{array}{}{H}_{elem}^{k}=f({B}_{elem}^{k}({t}_{n+1}),{B}_{elem}({t}_{n}))\end{array}$ which is subsequently substituted in the differential reluctivity expression. In the proposed strategy, (4) is directly implemented in the FEM model, constituting a closed loop integration of the PAM material model in the FEM algorithm. The open loop implementation consists of the simple implementation of (5) as post processing, while the nonlinear *BH* curve adopted for the FEM simulation is the one extracted by the anhysteretic part in (5).

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