Figure 1 Solenoid based linearly movable armature system
The magnetic flux linkage can be described by
Where
Initial magnetic flux linkage
Differential inductance
Coil inductance
The magnet circuit equation will be
The force exerted by the magnetic circuit to the mass become
The spring-mass system model equation is given by
Upon selecting the voltage V(t) as the variable to be manipulated and the linear displacement as the variable to be controlled, substituting Equation (3) in to Equation (4) and in to Equation (2) and taking the Laplace transform results the transfer function
The system parameters are shown in Table 1 below
Table 1 System parameters