## 2 Simplest Model of Class C Amplifier  
The fundamental structure of a single-transistor class C amplifier is provided in Figure 1a. This circuitry can be redrawn for the useful high-frequency signals (typically about tens or hundreds of MHz) by considering significant parasitic capacitances, as shown in Figure 1b. Note that at this point, the system is considered isolated, i.e., without an input signal. Dynamics of the resulting network can be described by following a set of ordinary differential equations:  
\[C_{1}\frac{d}{dt}v_{1}=-y_{11}\cdot v_{1}-y_{12}(v_{2}),\ \ \ \ \ C_{2}\frac{d}{dt}v_{2}=-y_{21}(v_{1})-y_{22}\cdot v_{2}-i_{L},\ \ \ \ L\frac{d}{dt}i_{L}=v_{2}, \tag{1}\]  
where a state vector is \(\mathbf{x}=(v_{1},\,v_{2},\,i_{L})^{\mathrm{T}}\), \(y_{11}\) and \(y_{22}\) are the input and output admittance of the bipolar transistor, and \(y_{12}\) and \(y_{21}\) are transconductance-type nonlinear smooth functions of the form  
\[y_{21}(v_{1})=a_{1}\cdot v_{1}+a_{2}\cdot{v_{1}}^{3}\,\ \ \ \ y_{12}(v_{2})=b_{1}\cdot v_{2}+b_{2}\cdot{v_{2}}^{3}. \tag{2}\]Note that nonlinearities are considered lower-order polynomials; i.e., approximations are valid in the close neighborhood of the calculated bias point. Fixed points are all real solutions of nonlinear algebraic equations \(\mathrm{d}\mathrm{x}\)/\(\mathrm{d}\mathrm{t}=\mathbf{0}\), i.e.,  
\[v_{1}=-y_{12}(0)/y_{11}\,\quad v_{2}=0,\quad\ i_{L}=-y_{21}(v_{1}). \tag{3}\]  
Therefore, Dynamical System (1) with (2) possesses only a single equilibrium located at zero. Associated eigenvalues can be established as roots of characteristic polynomial  
\[s^{3}+\left(\frac{y_{11}}{c_{1}}+\frac{y_{22}}{c_{2}}\right)s^{2}+\left(\frac{ 1}{c_{2}\cdot l}-\frac{a_{1}\cdot b_{1}}{c_{1}\cdot c_{2}}+\frac{y_{11}+y_{22}} {c_{1}\cdot c_{2}}\right)s+\frac{y_{11}}{c_{1}\cdot c_{2}\cdot l}=0. \tag{4}\]  
Small letters denote normalized values of circuit components. In general, Math Model (1) is analyzed using normalized values of accumulation elements and admittances. For a particular application, time and impedance rescaling need to be considered. Note that capacitor \(C_{1}\) de facto represents parasitic capacitance between the base and emitter \(C_{be}\), while \(C_{2}\) is a sum of parasitic \(C_{cr}\) and tank capacitance. Detailed numerical investigations reveal the advantageous possibility to keep \(C_{1}=C_{2}=1\) F and \(L=1\) H. For further analysis, let us assume that linear term \(a_{1}\) in Function (2) is very small. Symbolic formulas for eigenvalues can be expressed as  
\[\delta_{1}=-\frac{y_{11}}{c_{1}}\,\quad\delta_{2,3}=-\frac{1}{2\cdot c_{2}} \left(y_{22}\pm\sqrt{y_{22}^{2}-\frac{4c_{2}}{l}}\right), \tag{5}\]  
i.e., we experience a pair of complex conjugated eigenvalues if \(y_{22}\cdot l<4\cdot c_{2}\). In this case, local vector field geometry near the origin is spanned by the eigenvector and eigenplane defined by two lines, namely  
\[\epsilon_{\text{{vector}}}=\left( \begin{array}{ccc}1&0&0\end{array} \right)^{T}, \tag{6}\] \[\epsilon_{\text{{plane}}}^{1}=R\epsilon\left(\mp \begin{array}{ ccc}\frac{lb_{1}\left(y_{22}\pm\sqrt{y_{22}^{2}-\frac{4c_{2}}{l}}\right)}{c_{1} \cdot\sqrt{y_{22}^{2}-\frac{4c_{2}}{l}}+2c_{2}y_{11}-c_{1}y_{22}}&-\frac{l \left(y_{22}\pm\sqrt{y_{22}^{2}-\frac{4c_{2}}{l}}\right)}{2c_{2}}&1\end{array}  \right)^{T},\] \[\epsilon_{\text{{plane}}}^{2}=Im\left(\mp \begin{array}{ccc}\frac {lb_{1}\left(y_{22}-\sqrt{y_{22}^{2}-\frac{4c_{2}}{l}}\right)}{c_{1}\cdot \sqrt{y_{22}^{2}-\frac{4c_{2}}{l}}+2c_{2}y_{11}-c_{1}y_{22}}&-\frac{l\left(y_{2 2}\pm\sqrt{y_{22}^{2}-\frac{4c_{2}}{l}}\right)}{2c_{2}}&1\end{array} \right)^{ T}.\]  
Nonlinear scalar functions \(y_{12}(v_{2})\) and \(y_{21}(v_{1})\) have been considered as cubic polynomials without offset, while all remaining parameters were freely enabled for optimization.