## 2 Problem Formulation
In the operation of an electric generator, the relative angular speed of the rotor to the stator \(\omega_{G}\) (referred to as the angular speed of the electric generator) is a characteristic parameter for the generated electric power:  
\[\omega_{G}=\omega_{GR}-\omega_{GS}=\omega_{1}-\omega_{2} \tag{5}\]  
As a result, the total amplification ratio of the angular speed \(i_{G\_R}\) achieved by the wind turbine with counter-rotating generator can be determined with the following relation:  
\[i_{G\_R}=\frac{\omega_{G}}{\omega_{R}}=\frac{\omega_{GR}-\omega_{GS}}{\omega_ {R}}=\frac{-\tan\delta_{1}-\tan\delta_{2}}{\tan\delta_{2}}=-1-\frac{r_{2}}{r_ {1}}=-\bigg{(}1+\frac{z_{2}}{z_{1}}\bigg{)} \tag{6}\]  
Figure 1: Single-rotor wind turbine with counter-rotating electric generator: (a) conceptual scheme; (b) kinematic scheme (upper-half front view); and (c) block scheme. The dynamic modelling of the wind turbine from Figure 1 was performed under the following premises:  
* the moving components (gears, the wind rotor, the rotor and stator of the electric generator) were rigid bodies with the mass distributed evenly and, also, geometrically symmetrical in relation to their own axis of rotation; therefore, the centre of mass of a body was a fixed point, located on the body fixed axis of rotation;
* only the frictional losses in the gearing were further considered, while the frictions in the revolute joints, materialised by bearings with low friction coefficients (due to the rolling friction), were neglected;
* the wind rotor and the electric generator were not adjusted during operation, which means that the pitch angle of the blades did not change in operation, and the adjustment parameters of the electric generator remained constant in operation;
* the wind rotor and the electric generator had known mechanical characteristics of linear type with constant coefficients. The mechanical characteristic of a wind rotor is a complex 2D curve, which can be approximated with a linear function in the steady-state regime. Moreover, the \(DC\) generators are characterised by a linear dependence between the driving torque and the angular speed of the electric generator \(\omega_{G}\);
* the electrical losses inside the generator were neglected in this study. The main stages of the proposed dynamic response modelling of the wind turbine mechanical system are depicted in Figure 2 and further presented. Based on the conceptual diagram from Figure 1, the input parameters in this algorithm were the wind rotor radius, efficiencies of gear pairs and teeth numbers of the gears from the speed increase, the torque-speed characteristic of the electric generator, and the inertial properties of the components. In the first stage, the kinematic modelling of the wind turbine mechanical system was performed by obtaining the transmitting functions for the angular speeds and the other angular speeds that are dependent on the independent speed of the wind rotor. For this mechanical system, three dynamic equations can be described, corresponding to the three rigid bodies to which the mechanical transmission can be reduced. The dynamic equations can be obtained by various approaches, in this case by using the Newton-Euler method. The mechanical characteristics (torque-angular speed dependence at a shaft level) of the wind rotor and the electric generator were added to the kinematic and dynamic equations of the mechanical system. This study considered the case of DC electric generators, characterized by linear characteristics modelled by linear functions, as well as by the equality in absolute value of the torques on its rotor and stator, both mobile with rotations in opposite directions. The mechanical characteristic of the wind rotor can be linearized on the active operating area, the coefficients of the linear function being dependent on the wind speed.