Wednesday, March 3, 2010

The mathematical model of the magnetic levitation system

Model of the magnetic levitation system
The magnetic levitation system considered in this paper consists of a ferromagnetic ball suspended in a voltage-controlled magnetic field. Only the vertical motion is considered.
The objective is to keep the ball at a prescribed reference level. The schematic diagram of the system is shown in Figure 2.1. The dynamic model of the system can be written as

Schematic diagram of the magnetic levitation system.


The mathematical model of the electromagnetic levitation system

One can build the mathematical model of the levitation system by writing
appropriate differential equations in accordance to the typical mechanical- and electrical principles. The way the components are appreciated in the approaching mode can lead to simpler or more complex alternatives.

The formula for the energetic balance within the system is:

Basic mathematical model of magnetic levitation
In terms of the design, chosen part of the magnetic levitation model was simulated. We used the software for multidisciplinary system simulation – Dynast [9]. In the Fig. 6, a basic simulation model is shown. One of the significant parts of the model is the feedback circuit. It Decreasing / increasing of the duty cycle of the PWM
Approaching / taking away of the levitating object Decreasing / increasing of the output voltage of the Hall Effect sensor The place stabilization of the levitating object
The variation of the attraction force of the electromagnet means we simulated the output signal of the amplifier with the adjustable gain. According to the Fig. 6, there is a derivation block which is realized by passive components – resistors and capacitors. These components influence stability of the model. On this account, we have optimized the derivation block according to the results of the simulation. After the realization of the electromagnetic levitation device, we have compared real levels of the chosen parameters of the constructed device with the simulation model. The results of the simulation approximatelly correspondence with the real device.


Model of the magnetic levitation system
The system dynamics describing the behaviour of the moving
ball is derived from the Newton’s laws:

where z denotes the position of the ball (as indicated figure 2),
m its mass, g the acceleration of gravity, i the coil current and
Fmag(i,z) the electromagnetic force applied to the ball. d
denotes a bounded perturbation.
Drawing up the energy balance of the whole system and under
the assumption that the magnetic core is non saturated (which
occurs because of the air gap), the electromagnetic force can
be expressed as following:

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