## 3 IM Inverse-\(\Gamma\) Equivalent Circuit and Its Connection to Finite Element Model  
The challenge of simulating the IM under load, using MS FEAs, consists of the rotor current prediction outside the FEA problem, since the time derivative of the magnetic vector potential is not included in the formulation. The equivalent circuit is a common tool to estimate the stator and rotor current space vectors and the motor performance [4; 5]. However, the inductances do not take into account the iron saturation due to the presence of stator and rotor torque currents. Anyway, the analytical model gives useful information about the relative position and the amplitude of current space vectors, which can be properly imposed in static FEAs. The RFO model is based on the inverse-\(\Gamma\) form of the equivalent circuit in Figure 2.  
In this model, the rotor flux space vector (referred to the stator) \(\vec{\lambda}_{\rm rs}\) is determined only by the magnetizing stator current \(\vec{i}_{\mu}\). It lies along the \(d\)-axis of the steady state space vector diagram (Figure 2b). The magnetizing current is also the \(d\)-axis stator current \(i_{\rm sd}\) in the RFO model. \(L_{\rm t}\) and \(L_{\varphi}\) in Figure 2a are, respectively, the global leakage and the magnetizing inductance in the inverse-\(\Gamma\) model:  
\[L_{\rm t}=L_{\rm s}-L_{\varphi};\quad L_{\varphi}=\frac{{L_{\rm m}}^{2}}{L_{ \rm r}} \tag{3}\]  
Figure 2: IM model in the \(dq\) rotor flux linkage oriented reference frame. (**a**) Inverse-\(\Gamma\) model of IM; (**b**) space vector diagram in the \(dq\) synchronous reference frame. where \(L_{\rm r}\) is the rotor three-phase rotor equivalent winding synchronous inductance. Stator and rotor voltage equations, related to the RFO model are:  
\[ \begin{split}\vec{u}_{\rm s}&=R_{\rm s}\vec{i}_{\rm s }+\mathrm{j}\omega_{\rm s}\vec{\lambda}_{\rm s}\\ 0&=\frac{R_{\rm rs}}{s}\vec{i}_{\rm rs}+\mathrm{j} \omega_{\rm s}\vec{\lambda}_{\rm rs}\end{split}  \tag{4}\]  
where \(R_{\rm s}\) and \(R_{\rm rs}\) are the stator and rotor resistance; \(\omega_{\rm s}\) is the stator angular frequency. In the RFO model, the rotor current \(\vec{i}_{\rm rs}\) represents the torque current \(\vec{i}_{\rm r}\), which lies along the \(q\)-axis of the space vector diagram reference frame, Figure 2b. Introducing the transformation coefficient \((L_{\rm m}/L_{\rm r})\), stator and rotor currents are:  
\[ \begin{split}\vec{i}_{\rm s}&=i_{\rm sd}+\mathrm{j }i_{\rm sq}=\vec{i}_{\rm r}+\vec{i}_{\rm r}\\ \vec{i}_{\rm rs}&=-\vec{i}_{\rm r}\frac{L_{\rm r}}{ L_{\rm m}}=-\mathrm{j}i_{\rm q}\frac{L_{\rm r}}{L_{\rm m}}=\vec{i}_{\rm r} \end{split}  \tag{5}\]  
The rotor current space vector lies along the \(q\)-axis of the RFO reference frame. Using (5), stator and rotor flux linkage space vectors can be expressed as:  
\[ \begin{split}\vec{\lambda}_{\rm s}&=L_{\rm t}\vec{i }_{\rm s}+L_{\rm q}\vec{i}_{\rm r}=L_{\rm s}i_{\rm sd}+\mathrm{j}L_{\rm t}i_{ \rm sq}\\ \vec{\lambda}_{\rm rs}^{\ast}&=L_{\rm p}\vec{i}_{ \rm r}=L_{\rm m}i_{\rm sd}\frac{L_{\rm m}}{L_{\rm r}}=\vec{\lambda}_{\rm r} \frac{L_{\rm m}}{L_{\rm r}}\end{split}  \tag{6}\]  
where, in the RFO synchronous reference frame:  
\[ \begin{split}\lambda_{\rm sd}&=L_{\rm s}i_{\rm sd} \\ \lambda_{\rm sq}&=L_{\rm t}i_{\rm sq}\end{split} \begin{split}\lambda_{\rm rd}&=L_{\rm m}i_{\rm sd }\\ \lambda_{\rm rq}&=0\end{split}  \tag{7}\]  
The flux linkage-current relationships (7) show the first advantage of this IM model: the current \(i_{\rm sd}\) is involved in the main flux linkage creation, whilst the current \(i_{\rm sq}\) produces leakage flux linkage and implicates the torque production. Furthermore, rearranging (5), it is:  
\[i_{\rm rq}=-\frac{L_{\rm m}}{L_{\rm r}}i_{\rm sq} \tag{8}\]  
The relationship (8) represents another peculiar feature of this model: the rotor current vector acts only along the \(q\)-axis of the RFO reference frame. This condition represents an advantage to create a static finite-element model of the machine.