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