Question

winkngs spring i(t) v(t) st VEE Figure 1: (a)Solenoid with retu spring. (b) Equivalent lumped electrical cireuit (e) Equivalent mechanical diagram Figure 1(a) illustrates a solenoid with a return spring The voltage e(t) across the winding, causes a current it) to flow through the winding. which in turn generates a magnetic field The magnetic field induces a force f(t) on the plunger mass, . The magnitude of this force is related to the current in the windings via the solenoid's electromagnetic coupling constant N. as shown below fit)- Nit) The movement of the plunger generates a voltage in the windings which opposes the applied voltage. The magnitude of this voltage is determined by the velocity of the mass and the solenoid's electromagnetic coupling constant, as shown below In addition, an ideal spring with Hooke law constant of K is attached to mass, which opposes the motion of the mass and returns the mass to its initial position when the current no longer flows in the winding Therefore we can represent the electrical elements of the system as shown in figure 1(b) and the mechanical elements as shown in figure 1(c). The electrical and mechanical element are linked (glued together) by the two equations above The Problem The objective of this assignment is to determine y(t) when r(t) plunger when the input voltage is the unit step). U(t) (ie the movement of the The Method To do this you will separately determine, e the Laplace transform for the current I(s), and the Laplace transform for the movement of the mass for a given force Fs). Then you will "glue these two expressions together using equation 1 Once that is completed, the expression then must be rearranged so we have an expression describing Y(s). Finally we take the inverse Laplace transform to give yt), and which you will plot using Sailab 1. Determine the differential equation which describes the electrical lumped circuit model in figure 1(b) 2. Take the Laplace transform of your differential equation and rearrange it to produce an expression for I(s) 3. Determine the differential equation which relates fit) and yt) in figure 1(c) 4 Take the Laplace transform of your previous differential equation 5. Substitute your expression an expression for y(s) for I() in to the previous expression. and then it rearrange to determine 6. For this solenoid RL and therefore the electrical time constant is negligible (in a similar fashion to the simplification made in the motes for the derivation of the transfer function for small DC motors). Thus you can simplity your transfer function by top and bottom of your expression by R and then removing any product terms which include 7. Assuming M-0.kg, R- 52. K- 0.1 and.N Ivolt-sec/coulomb, substitute in the values for the parameters and take the inverse Laplace transform to determine yr 8. Plot the time response y(t) using Scilab. Include your Scilab Code in your answer Each step above, should be labelled in your report

Answer 1

i(t) Acionding to kvl d t

3) 加 リ( olf Nok bte d t