Refining and Optimizing Analytical Models
This section gives an example of how the model developed earlier could be extended to account for additional knowledge of the system's structure. Again, if you have not yet been exposed to this particular type of mathematical analysis, simply note that all models can be refined as we gain more knowledge about the system. Think back to the model you identified in the last section. How could you extend this model to make it a better reflection of reality?
Mathematical model
In Section 3.2 the mathematical model for the circuit was derived. I(t) is the current (in ampere), V(t) is the voltage drop over the capacitor (in volt). The derivatives and the constants were scaled, so the numbers for the simulations were of a reasonable size. For analytical calculations, this scaling is not necessary, so we will use the original variables.
\(\dfrac{dV}{dt}=\dfrac{I}{C}\), \(V(0) = 0\),
\(\dfrac{dI}{dt} = \dfrac{-V}{L} - \dfrac{RI}{L} + \dfrac{V_B}{L}\), \(I(0) = 0\),
where \(V_B=5V\), \(R=0.1Ω\), \(L=4×10^{−9}H\), and for the capacitance, we start with \(C=0.5×10^{−9}F\).
On the next page you will investigate the equilibrium point and its type.