Final answer:
The differential equation model for the parallel PID controller with a derivative filter can be found by considering the components involved and their relationships. The solution to this differential equation can be obtained by making substitutions and verifying its validity. The resulting equation can be integrated to find the charge on the capacitor as a function of time.
Step-by-step explanation:
The differential equation model representing the parallel PID controller with a derivative filter can be determined by considering the components involved and their relationships. In this case, we have a first-order differential equation for I(t), which is similar to the equation for a capacitor and resistor in series. The solution to this differential equation can be found by making substitutions in the equations relating the capacitor to the inductor.
This gives us the differential equation for the parallel PID controller with a derivative filter. To prove that it is the right solution, we can take the first and second derivatives with respect to time and substitute them into the equation. By doing so, we find that the equation is satisfied.
This differential equation can then be integrated to find an equation for the charge on the capacitor as a function of time. The mathematical solution to this modified differential equation is known as a logistic curve.