I am struggling with the simple oscillator on the picture below.
My aim is to obtain system of ODEs or DAE at most for numerical simulation in Matlab. But few issues occured with basic circuit theory.
For a nonlinear current \$i_d\$ I suppose piecewise defined function \begin{equation*} i_{\mathrm{n}} = \begin{cases} 0\quad \text{for } u_{\mathrm{gs}} - V_{\mathrm{T}}\leq 0\\ \beta(u_{\mathrm{gs}} - V_{\mathrm{T}})^{n_2}(1 + \lambda u_{\mathrm{ds}})\mathrm{tanh}\,(\alpha u_{\mathrm{ds}}) \quad \text{for } u_{\mathrm{gs}} - V_{\mathrm{T}}> 0 \end{cases} \end{equation*} for some given FET parameters \$\alpha\$, \$\beta\$, \$\lambda\$, pinchoff voltage \$V_{\mathrm{T}}\$ (it is VA approxiamtion used in Sussmann-Fort's MESFET model).
For the linear part I worked out these equations so far:
\begin{align} \frac{\mathrm{d}i_{L}}{\mathrm{d}t} &= -\frac{1}{L}(u_{\mathrm{ds}} - u_{\mathrm{gs}})\\ \frac{\mathrm{d}u_{C_d}}{\mathrm{d}t} &= \frac{i_{C_d}}{C_{d}}\\ \frac{\mathrm{d}u_{C_g}}{\mathrm{d}t} &= \frac{i_{C_g}}{C_{g}}\\ \end{align}
Using KCL I obtain \$i_{R_{\mathrm{d}}} = i_L + i_{C_d} + i_{\mathrm{n}} + i_{R_1}\$. But how do I obtain \$i_{R_1}\$ and \$i_{C_d}\$ at the same time? Both branches with loads \$R_1\$, \$R_2\$ are giving me headache.
Second problem is that blocking capacitor \$C_\mathrm{b}\$. How we can tackle with floating capacitor while using ODE formalism?
Note: Somebody here suggested me MNA aproach which is not bad after all, but due to the lack of examples on use of inductor stamp with \$sL\$ rather than \$1/sL\$ (which would produce integrodifferential equation), I would like to ask for by hand solution.
Edit:
I managed to achieve these equations according to this thesis on MNA (pages 59-62). But it uses somewhat cumbersome notation so I am not really sure about the red current source if I understood it well. Can anyone check if it is legit to use my nonlinear current use on RHS like this? Sorry for the snippets, I was typsetting it for myself in overleaf.
It is clear after performing these opeartions we can obtain equations involving only derivatives of node voltages and inductor current. Currents through voltage sources are given without derivatives. But also I get \$u_R\$s which confuses me too.
