## 3 Modelling, Analysis Tools and Experimental Support for Stability Analysis
### Small-Signal Linearization of the Analytic Model
#### 3.2.1 Steady-State Model Derivation  
A preliminary step before linearizing the model is to calculate analytically all state variables at steady state for any operating conditions. Then, the small-signal linearization should be achieved around these steady-state points.  
At steady state (\(\frac{d}{dt}=0\)), Equations (1) and (2) are:  
\[\begin{split}-\left(R_{g}*\mathbf{I_{3}}\right)\cdot\mathbf{I_{d\mathbf{q}0} ^{g}}-\omega*\begin{pmatrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{pmatrix}\cdot L_{g}\cdot\mathbf{I_{d\mathbf{q}0}^{g}}+\omega*\phi_{g_{g}} \begin{pmatrix}0\\ 1\\ 0\end{pmatrix}=\\ (R_{m}*\mathbf{I_{3}})\cdot\mathbf{I_{d\mathbf{q}0}^{g}}+\omega*\begin{pmatrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{pmatrix}\cdot L_{m}\cdot\mathbf{I_{d\mathbf{q}0}^{g}}+\omega*\phi_{g_{m}} \begin{pmatrix}\sin(\delta)\\ \cos(\delta)\\ 0\end{pmatrix}\end{split} \tag{13}\]  
Equation (3), under steady-state condition, leads to the equality of angular speeds:  
\[\frac{d}{dt}\delta\ =\ 0\ \Rightarrow\ \omega_{g}\ =\ \omega_{m}\ =\ \omega \tag{14}\]  
A simplified expression is given by deriving the electromotive forces of both machines:  
\[\begin{split} E_{m}\sin(\delta)&=-\left(R_{g}+R_{m}\right)*I_{ d}+\left(L_{g}+L_{m}\right)*\omega*I_{q}\\ E_{m}\cos(\delta)&=-\left(R_{g}+R_{m}\right)*l_{q}-\left(L_{g}+L_{m} \right)*\omega*I_{d}+E_{g}\\ &\text{With }E_{g}=\omega\cdot\phi_{g_{g}}\text{et }E_{m}=\omega\cdot\phi_{g_{m}}\end{split} \tag{15}\]  
Figure 5: Comparison of time simulations between the analytic state model simulated on Matlab Simulink (red curves) and the circuit simulation on Saber (blue curves).  
Solving analytically this system to calculate the steady-state variables \((I_{d},I_{q},\delta)\) with respect to the inputs \((\omega_{S_{ref}},\,T_{prop_{m}})\) is quite complicated and only the result is given here. The previous system (15) with two equations of two unknown variables \((I_{d},I_{q})\) is solved by deriving a second order polynomial equation:  
\[\begin{array}{c}\left(\left(\frac{\left(L_{g}+L_{m}\right)\cdot\omega}{2}+ \frac{\left(R_{g}+R_{m}\right)^{2}}{\left(L_{g}+L_{m}\right)}\cdot\frac{1}{2 \omega}\right)^{2}+\left(-\frac{\left(R_{g}+R_{m}\right)}{2}-\frac{\left(L_{g }+L_{m}\right)^{2}}{\left(R_{g}+R_{m}\right)}\cdot\frac{\omega^{2}}{2}\right) ^{2}\right)\cdot I_{q}^{2}+\\ 2\left[\begin{array}{c}\left(\frac{\left(L_{g}+L_{m}\right)\cdot\omega}{2}+ \frac{\left(R_{g}+R_{m}\right)^{2}}{\left(L_{g}+L_{m}\right)}\cdot\frac{1}{2 \omega}\right)\cdot cste\cdot\left(R_{g}+R_{m}\right)+\\ \left(-\frac{\left(R_{g}+R_{m}\right)}{2}-\frac{\left(L_{g}+L_{m}\right)^{2}}{ \left(R_{g}+R_{m}\right)}\cdot\frac{\omega^{2}}{2}\right)\cdot\left(cste \left(L_{g}+L_{m}\right)\cdot\omega+E_{g}\right)\end{array}\right]\cdot I_{q} \\ +\left(\left(cste\cdot\left(R_{g}+R_{m}\right)\right)^{2}+\left(cste\cdot \left(L_{g}+L_{m}\right)\cdot\omega+E_{g}\right)^{2}-E_{m}^{2}\right)=0\end{array} \tag{16}\]  
\[\begin{array}{c}\text{With}\,cste=\left(\frac{\left(E_{g}^{2}-E_{g}^{2} \right)}{2E_{g}\cdot\left(L_{g}+L_{m}\right)\cdot\omega}+\frac{T_{prop  
In the next section, the relevance of the simulation (nonlinear and root locus based on the linearized model) and experimental analysis tools is proved by comparing results in several operating conditions with a special attention to stability conditions.