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.

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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:
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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.

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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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Modeling of Magnetic Levitation System

Magnetic Levitation System Modelling
Figure 2 below shows the simplified diagram of Maglev
system (adapted from Feedback Instrument Ltd.
manual).

Figure 2. Simplified Maglev System Diagram
The photo-sensors measure the ball’s position. Corresponding
to the ball’s position from the electromagnet,
the sensor generates a voltage output (Vsensor) obeying
the following relation:
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Magnetic Levitation System description and modelling
This work was concerned with the dynamics of the Feedback Magnetic Levitation System c° , which is depicted in Figure 1. Magnetic Levitation Circuit The infrared photo-sensor is assumed to be linear in the required range of operation, yielding a voltage y that is related to distance h as y = °h + y0, where the gain ° > 0 and the offset y0 are such that y 2 (−2V, +2V ). Current i is regulated by an inner control loop, and is linearly related to the input voltage u as i = ½u + i0 with ½ > 0 and i0 > 0.


The working excursion of u is limited between −3V (corresponding to a null coil current) and +5V (saturation value). Rates of change larger than 50V/s for u cannot be implemented by the current driver along its entire working range.

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MODELLING AND SIMULATION OF A MAGNETIC LEVITATION SYSTEM
Abstract. The electromagnetic levitation system (MLS) is a mechatronic system already
acknowledged and accepted by the field experts. Due to a synergic integration of the sensorial elements, the control subsystem and the actuating subsystem, the mentioned levitation system becomes an especially recommended subject in the academic curricula for mechatronic study programmes. This paper intends to initiate the investigation of different modelling, simulation and control possibilities for a magnetic levitation system starting from a real, physical reference model.
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Model of Magnetic Levitation System

MODEL OF THE MAGNETIC LEVITATION SYSTEM
Figure shows the experiment model for the
Maglev system that has been carried out in a
project at National key lab for Digital Control &
System Engineering, Vietnam.

Diagram of magnetic levitation


The Maglev system in the model contains two
feedback sensors. One is a small current sense
resistor in series with the coil. The other is a
phototransitor embedded in the chamber pedestal
and providing the ball position signal. After
amplifying, both current sensor and phototransitor
are wired to analog inputs of card PCI-1711. The
control signal from the computer is sent to the
controllable voltage source through the analog
output of card PCI-1711.
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EXPERIMENTAL APPARATUS AND CONTROL MODEL
Figure 1 shows the single-axis magnetic levitation
system used in the experiment. The levitation object is a
ping-pong ball with a permanent magnet attached inside it
to provide an attractive force. The attraction force is
controlled by means of a computer-controlled electromagnet
mounted directly above the ball. A light source and a
linear image sensor (LIS, Hamamatsu S5462-512Q) are
used to determine the displacement of the ball. There are
512 photo diode cells in LIS, and the length of each cell is
0.05mm. The light source and the sensor are tuned such
that the outputs of the photocells are saturated when the

Schematic diagram of the magnetic levitation system.


ball does not cover the cells. A comparator is used to
compare the outputs of each cell with a preset voltage to
judge whether the cell is saturated or not. We then obtain
the levitation displacement of the ball by utilizing a counter
to count the numbers of saturated cells. The sampling rate
used is 200 Hz. This low sampling rate is used due to the
bandwidth limitation of LIS. The control computer is an
industrial personal computer with a Pentium processor
and an Advantech PCL818H analog I/O and counter
board.
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MAGNETIC LEVITATION SYSTEM
In this section, a physical maglev system and its components are described. Presenting system equations nonlinear and linear models are developed for the plant. The schematic of the MAGLEV plant is presented in Fig. 3.1 below:

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MODEL OF THE MAGNETIC LEVITATION SYSTEM
Diagram of the magnetic levitation system.

Consider the magnetic levitation system shown in Fig. 1. this
is a popular gravity-based one degree-of-freedom magnetic
levitation system, in which an electromagnet exerts attractive
force to levitate a steel ball (in some references a steel plate is
levitated). The system dynamics can be described in the
following equations
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System Modeling - Linear Permanent Magnet Motors

Two types of position dependent disturbances are
considered: cogging force and force ripple. Cogging is
a magnetic disturbance force that is caused by attraction
between permanent magnets and translator. The force
depends on the relative position of the translator with
respect to the magnets, and it is independent of the motor
current. Force ripple is an electro-magnetic effect and
causes a periodic variation of the force constant c. Force
ripple occurs only if the motor current is different from
zero, and its absolute value depends on the required thrust
force and the relative position of the translator to the
stator. Both disturbances are periodic functions of the
position. [9]

Cogging is negligible in motors with iron-less translators
[14]. Figure 3 shows the nonlinear block diagram of a
servo system with brushless linear motor. The nonlinear
disturbances are the velocity depended friction force
Ffriction, and the position dependent cogging force
Fcogging and force ripple



The friction force is modeled with a kinetic friction
model. In the kinetic friction model the friction force is a
function of velocity only. The friction curve is identified
with experiments at different velocities. The friction has
a discontinuity at

because of stiction. Stiction
avoid accurate measurement of the thrust force without
motion of the carriage. A survey of friction models and
compensation methods is given in [17].
Aim of the force ripple identification is to obtain a
function of the thrust force Fthrust versus the control
signal u and the position x. A possible solution to identify
this function is to measure the thrust force Fthrust at
different positions x and control signals u. In this case an
additional force sensor and a screw cylinder for manual
position adjustment is necessary. In order to measure the
force ripple accurately, without motion of the carriage, a
frictionless air bearing support is necessary [7]. A solution
to avoid frictionless air bearings is the measurement of the
thrust force with moving carriage. At constant velocities
the friction force is also constant and can be treated as
additional load force. In this case an additional servo
system is needed to achieve the movement [18].

The main idea of the proposed identification method is
to identify the force ripple in a closed position control loop
by measuring the control signal u at different load forces
Fload and positions x. Neither additional force sensor nor
device for position adjustment are necessary. In order to
avoid inaccuracy by stiction the measurement is achieved
with moving carriage. The position of the carriage is
obtained from an incremental linear optical encoder with
a measurement resolution of 0:2m. The experiment
consists of several movements at constant low velocity
(1mm=s) and different load forces (0 : : : 70N). The
output of the position controller is stored at equidistant
positions. A controller with an integral component is used
to eliminate steady position error. During motion with
constant low velocity the dynamics of the motor have no
significant effect on the control signal u.

Figure 4 shows the controller output u versus the
translator position x. In this first experiment there is no
additional load force attached to the carriage. The period
spectrum of the controller signal ui is carried out via FFT.

Source ( pdf )

http://www.inf.fh-dortmund.de/personen/professoren/

roehrig/papers/ldia01.pdf

MODELING OF A DC SERVO MOTOR

MODELING OF A DC SERVO MOTOR
Electric motors are the most common actuator used in
electromagnetic systems of all types. They are made in a
variety of configurations and sizes for applications ranging
from activating precision movements to powering diesel-electric
locomotives. The laboratory motors are small servomotors,
which might be used for positioning control applications in a
variety of automated machines. They are DC (direct current)
motors. The armature is driven by an external DC voltage that
produces the motor torque and results in the motor speed. The
armature current produced by the applied voltage interacts with
the permanent magnet field to produce current and motion.
A simplified schematic of the motor is shown in Figure 1 below.

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DC Servo Motor Parameter Estimation
This example demonstrates the process of estimating the
parameters of a multi-domain DC servo motor model constructed
using various physical modeling products.
Contents
1.Description of the DC Servo Motor System
2.Estimating Parameters of the DC Motor Model
3.Importing Experimental Data
4.Selecting Parameters for Estimation
5.Defining an Estimation
6.Running the Estimation
7.Validation
8.Summary

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DC Motor System Identification
Objective The purpose of this lab is to experimentally determine
the frequency response of a DC servomotor system, which
includes the DC motor and amplifier. Experimental results will be
obtained to create a Bode plot for the servomotor system.
A transfer function can also be derived by fitting the Bode plot.
These results then can be used to design a suitable controller.

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