# FE-Aided Synchronization Analysis of Line-Start Synchronous Reluctance Motors
## 4 FE-Aided Steady State Torque Approximation  
This section describes a method by which to approximate the steady state torque as a function of the slip in order to apply the torque for analysis of the synchronization process described in the previous section. The steady state torque may be available with equivalent circuit parameters accounting for saturation and skin effects. However, the author has found that it is challenging work to estimate accurately the rotor parameters of LS-SynRMs. Instead, this paper proposes an approach to obtain the steady state torque without equivalent circuit parameters, such as rotor resistance and inductance. An approximate solution in the steady state condition using time harmonic FEA with single time-frequency in the rotor equation is a widely accepted approach for use with induction machines [22]. The rotor frequency is the single stator frequency multiplied by the rotor slip. Such analysis provides useful estimation of the solution in the time domain. However, such single slip analysis is inadequate for motors with rotor saliency. The rotor saliency, along with the rotor slots and rotor currents, induce additional time frequencies into the stator fields [23]. The multi-harmonic FEA [24], in which several frequencies are considered rather than a single frequency, may give promising results, although with increased computation cost. The following approach utilizes single-frequency time-harmonic FEA because it can easily be implemented and because it minimizes the computational burden. The analysis can be carried out using the rated voltage, rated stator frequency, and single slip. Undoubtedly, the single frequency time harmonic FEA results are incorrect due to additional positive or negative sequence frequency in an unsymmetrical \(dq\) system [25]. These results cannot be captured in a single time-frequency analysis (see Figure 5). The solution of the coupled electric circuit in the time harmonic FEA is rearranged as follows. Neglecting the time and space harmonic components, the permeability is frozen by the above time harmonic FEA result. In such a linearized condition, the stator flux linkage and the current relationship defined by the operational impedances, \(\mathbf{Z}_{p}(s)\), are calculated using:  
Figure 5: Stator phase current in 20% slip operation with sinusoidal supply voltage. The additional major frequency component along with the fundamental is \((1-2s)\omega\). \[ \begin{pmatrix}\Lambda_{\text{sd}}\\ \Lambda_{\text{sq}}\end{pmatrix} =\mathbf{Z}_{p}(s)\, \begin{pmatrix}I_{\text{sd}} \\ I_{\text{sq}}\end{pmatrix} . \tag{17}\]  
The operational impedance matrix of \(2\times 2\) is obtained using the time harmonic FEA with frozen permeability. Any arbitrary amplitude of the each \(dq\) stator currents can be fed separately because of the linearity. The angular frequency of the resulting \(abc\) phase stator current to the feed can be set to be identical with that of the supply voltage. It should be carried out with different rotor positions within a range of \(\vec{x}\) in electrical radians owing to the different MMF distribution along the \(d\)- and \(q\)-axis. After the operational impedances matrix is calculated, the \(dq\) stator currents in the steady state condition are approximated to satisfy the voltage equation in (3):  
\[ \begin{split}\begin{bmatrix}I_{\text{sd}}^{(m)}\\ I_{\text{sq}}^{(m)}\end{bmatrix}&=\left(\begin{bmatrix}js\omega&-(1-s)\omega \\ (1-s)\omega&js\omega\end{bmatrix}\mathbf{Z}_{p}(s)+\begin{bmatrix}R_{s}&0\\ 0&R_{s}\end{bmatrix}\right)^{-1}\\ &\times\begin{bmatrix}V_{\text{sd}}\\ V_{\text{sq}}\end{bmatrix}\end{split} \end{split} \tag{18}\]  
where the superscript \((m)\) denotes the modified value. Substituting the modified \(dq\) stator currents into (17) gives the modified stator flux linkages and satisfies the steady state voltage equation. It should be noted that the \(dq\) voltage in (18) is identical to the phasor in the previous time harmonic FEA.