Give me equation and Develop state equations. Write down the state variables to be solved. Explain why the
state equations are nonlinear.
Solve the state equations
a. Draw current of coil and displacement of the plunger as a function of time for 0 <= 𝑡 <= 1s when
6 volt is applied to the coil between 0.2 seconds and 0.5 seconds.
b. Draw current of coil and displacement of the plunger as a function of time for 0 <= 𝑡 <= 1s when
12 volt is applied to the coil between 0.2 seconds and 0.5 seconds
PROJECT EXPLANATION Consider a cylindrical solenoid that controls a gas flow in the system as shown on the figure below (cross-sectional view is given). Solenoid coil is supplied from a DC source with internal resistance of 0.5 Ohm. When the switch is closed, current flows through the coil creating magnetic force, and as the result of this force.the plunger moves up to open the gas flow in the pipe.The movement of the plunger is limited to 0 x d, where d = 0.01 m. The plunger also slides inside a plastic guide. Plastic permeability is equal to the free-air permeability. There are two rubbers (green color) placed on the core and on the pipe to stop the plunger smoothly. Both rubber thicknesses are 0.001 m. Force produced in these rubbers can be modelled similar to the spring with very large spring coefficient. That is, the rubber force in the core is given by the equation fe = -K(x - 0.001) for the x 0.001 and the rubber force in the gas pipe is given by the equation fp = -K(x - 0.009) for the x 0.009, where coefficient is given as K = 5000000 N/m. Remember that these forces will limit the range of plunger and make the state equations valid for the whole simulation time. When the switch is opened and there is no current flow in the coil then the plunger will fall down because of its mass and close the gate to stop the gas flow. The forces produced by gas flow is neglected. The plunger radius is given by r = 0.01m. Assume that the ferromagnetic materials used in the core and in the plunger have infinite permeability. Other parameters for the solenoid are given as; g = 0.001 m, N = 500 turns, h = 0.02 m, mass of the plunger M = 0.2 kg, resistance of the coil R=1.5 Ohm, gravitational acceleration constant G = 9.8 m/s2, and damping constant B = 40 N.s/m. Ignore the fringing and flux leakages in the magnetic system .
Cylindrical Solenoid
Core
Coil
UT
Plunger 6 Plastic
6
GAS FLOW
Rubber
