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Eindhoven University of Technology

MASTER

A magnetic levitation system

Kop, J.H.H.

Award date:

1996

Link to publication

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Measurement and Control Group

A Magnetic Levitation System By J.H.H. Kop

Master of Science Thesis

.carried out from December 1995 to August 1996 comissioned by prof.dr.ir. van den Bosch

under supervision of: Dr. Ir. A. Damen and ir. P. Houtkamp

The Department of Electrical Engineering of the Eindhoven University of Technology accepts no responsibility for the contents of M.Sc.Theses or reports on practical training periods.

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Preface

Six years ago I started my studies at the Eindhoven University of Technology. They were six nice years in which I was able to develop myself in many ways. Eight months ago I started my graduation project at the measurement and control group. During this final project a lot of people supported me to realize this thesis. First I want to thank my coaches ir. P.Houtkamp and ddr. A.Damen for the many advises they gave me. Next I want to thank prof.dr.ir.

Van den Bosch for the interest he showed for the bearings project. Furthermore I want to thank ir. van der Graft for the technical support. Last but not least I want to mention Martijn Thijssen. Together we faced the same problems and thanks to our good cooperation we succeeded in keeping a ball levitated. Furthermore Martijn Thijssen contributed a lot to this report. Especially chapter four was mainly written by him. Moreover, he drew a lot of the figures of chapter three. Looking back at those final months of my studies I can say I had a very good time at the measurement and control group, resulting in a project that can be used as a start to rcalize a magnetic bearing.

1

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This report descrihes the development of a magnetic levitation system. This project was meant to be a start to realize a magnetic hearing. In a magnetic levitation system a balI is heing levitated under an electromagnet. To keep the ball levitated we must he ahle to measure the position. Therefor an inductive sensor has heen realized. The inductive sensor was used to he ahle to sense and actuate on the same coil. The inductive sensor appeared to he reliahle and accurate as long as the actuating currents were limited in both frequency and amplitude. Moreover a current amplifier was developed. As a coil is an inductive load the current amplifier had to he limited in frequency. The final resuIts of the current amplifier were satisfying. Finally a controller was developed to stabilize the Ievitation system. As a magnetic levitation system is a nonlinear system with parameter uncertainties this is not easy.In practice we realized a PD-controller and a PID-controller that were ahle to Ievitate the ball. However, bath controllers were very sensitive to external disturbances.Therefore it deserves attention to develop a more robust controller and use techniques like exact linearization.

Finally we may state that a magnetic levitation system was realized that is ahle to keep a hall levitated. However, the system has a lot of limitations, and a lot of improvernents need to be done to he ahle to realize a magnetic bearing.

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Contents

1 Introd uction

2 A magnetic levitation system

2.1 Equations of the magnetic levitation system 2.2 Determining the design parameters

2.3 Analysis of the designed system . . 2.4 Prelirninaries of the control system 3 Sensing the position

3.1 Sensor options .

3.1.1 Optical sensor . 3.1.2 Eddy current sensor 3.1.3 Inductive sensor . . 3.2 Principle of the inductive sensor 3.3 Practical realization .

3.3.1 Supplying a sinusoidal current to the coil 3.3.2 The Loek-in Amplifier . . . .

3.3.3 Making the sensor linear to x

3.4 Testing the sensor .

4 A current Amplifier 4.1 Specifieations . . . 4.2 The basic principle 4.3 The U-I converter:

4.4 Analysis and simulation of the U-I converter . 4.4.1 Sirnulations with Spice.

4.4.2 The analysis 4.5 The solution . 4.6 Conclusions . . . 5 The control system

5.1 Sensing and actuating on separate cails

5.2 The control system .

S.;J COI\troller design .

5.3.1 Design of a PD-controller 5.3.2 Design of a PIO-controller .

5.3.3 Another Design .

1

5 7 7 11 11 13 15 15 15 16 16 16 17 18 18 19 20 22 22 23 25 26 26 27 28 28 30 30 32 33 34 34 35

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5.4 Results of the control system . . . . , 5.4.1 Results using a PD-controller . 5.4.2 Results using a PID-controller 5.4.3 Results using the third controller

5.4.4 Irnplementation of the PD-controller in practice . 5.5 Exact linearization . . . .

5.5.1 Introduction . . . . 5.5.2 Design of a PD-controller 6 Conclusions and recommendations

6.1 Conclusions . . . . 6.1.1 The sensor . . , . . . 6.1.2 The current amplifier 6.1.3 The control system .

6.2 Recommendations .

6.2.1 Design of the coil . 6.2.2 Improving the sensor.

6.2.3 The current amplifier 6.2.4 The control system . . A Electronic realisation of the sensor

B Implementation of the control system in SIMULINK C Robustness of the controllers

C.l Results of the PD-controller . C.2 Results of the PID-controller C.3 Results of the third controller

C.4 Robustness and performance of exact linearization D Position tracks with the DSP

36 37 37

38 39 39 39 40

43 43 43 43 43 44 44 44 44 45

47 48 51 51 51 51 52 55

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List of Figures

1.1 1.2 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

4.1 4.2 4.3 4.4 4.5 4.6 4.7

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 S.10 5.11 5.12 5.13

A magnetic bearing. . . . A magnetic 1evitation system The magnetic 1evitation system

The magnetic flux lines in the 1evitation system . The inductance as a function ofx .

the contro1 system The optica1 sensor

Multiplication in the frequency domain . Block scheme of the inductive sensor A simple electronic circuit . . . . . Results of the Loek-in Amplifier. . Electronic realization of the sensor Lsen and Usen as a function of x . . B - H-curve of the magnetic material Desired spectrum of the current amplifier The basic scheme of the current amplifier The scheme of the U-I converter . . . . . The oscillation of the coil voltage . . . . .

The AC-analysis of the U-I converter with load resistor, Ure! -+ ULo The AC-analysis of the U-Iconverter with inductive load

The transfer function of the U-I converter Electric circuit of the sensor and actuator Bandlimiting the actuating current .

The control system .

Pole-zero-plot of the open-Ioop-gain.

Root Loci with PD-control . . Root Loci with PID-control . . . . . Root Loci with new controller . . ..

Simulated step response of the PD-controller Simulated step response of the PID-controller Simulated step response of the third controller Principle of exact linearization

Root loci with PO-control . Step response with exact linearization

3

5 6 7 8

12

13 15

17

18 18

19 19 20 21

22 23 25 25

27 27 29 31 31 32 34 35 35 36 37 38 38 40 41 41

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A.I Electronic realization of the sensor 47

B.I The control systern in SIMULINK 48

B.2 The nonlinear plant in SIMULINK 49

C.I Response with perturbed parameters . 52

C.2 Response with perturbed parameters. 52

C.3 Response with perturbed parameters . 53

CA

Response with perturbed parameters . 53

D.I Track of the ballposition 55

D.2 Long time track. . . 55

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Chapter 1

Introduction

To ohtain frictionless motion and rotation in rnany practical applications bearings are used.

At the Eindhoven university of technology, department of control and measurement, an air bearing has heen used to control the position and angles of a mirror to focus a laserheam. As this bearing had it'5 limitations, the idea of developing a rnagnetic bearing was suggested. A magnetic bearing should be realized as shown in figure 1.1, where a hall is positioned between 4 electromagnets [4] [7]. As a pilot project, to get to know something ahout the possibilities

" I

""" "

'" ...

'"

~

'"

~

'" ...

'" ""

... lol.

... ~

'"

~

", I

,,11/

"

Figure 1.1: A magnetic hearing

of a magnetlcbearing, a magnetic levitation system (figure 1.2) has been developed, where a balI is suspended linder an electromagnet. A magnetic levitation system consists of an electromagnet to levitate the balI, a current amplifier to supply the force-generating current, a position sensor and a controller.

In this report we describe a.ll elements of the magnetic levitation system. In chapter 2 we will discuss the magnetic reluctance force that is generated by supplying a current to the cail.

Furthermore, in th is chapter, we will talk about the dimensions of the system and will give

5

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L(x)

Figure 1.2: A rnagnetic levitation system some preliminary insight into the contro} system.

In chapter 3 we will discuss the position sensor that has been developed. In this chapter we explain why we choose to use an inductive sensor in stead of other sensing techniques.

Furthermore, we will give some comment about the accuracy and reliability of the sensor, we have developed.

Next, the design of a current amplifier will be described in chapter 4. In this chapter we will show that it is very hard to supply a current to a coil very fast. Therefore the current must be bandlimited. In this chapter we will also talk about the combination of sensing and actuating with the same coil.

After we have developed all elements necessary to obtain a magnetic levitation system we will consider the control opportunities to obtain a stabIe system. This will be done in chapter 5, where a few controllers will be discussed which are able to stabilize the highly nonlinear magnetic levitation system.

Finally, in chapter 6, we will give some conclusions about the quality of the system we designed. Furthermore, we will give some suggestions of how the existing system can be improved.

(11)

Chapter 2

A magnetic levitation system

In this chapter, we will consider a simple magnetic levitation system. First, we will derive the dynamic and electric equation of the system. Next, we use these equations to explain about same choices we made while designing our control system. At the end of this chapter we describe the control system and linearize the dynamic equation to analyse cant rol problems.

2.1 Equations of the magnetic levitation system

In this section we will analyse the magnetic levitation system shown in figure 2.1. We assume

x

u

mechanical system electrical system

Figure 2.1: The magnetie levitation system

that the steel balI moves only in vertical direction. By applying Newton's Jaw we ean easily derive the dynamic equation of the system:

(2.1 )

7

(12)

where

m

=

mass of the levitated ball[kg]

x

=

position of the levitated ball[m]

Fm

=

magnetic reluctance force(N]

Fg

=

gravitational force[N]

In (2.1) Fm is the magnetic reluctance force. To calculate this force we make the following assumptions:

• The magnetic flux lines occur as shown in figure 2.2. This is true if the length of the coil (I) is much larger than the length ofthe airgap (x).

• The magnetic energy outside the airgap is not influenced by changing the position of the balI and is proportional to the squared coil-current(i2). Although the magnetic circuit outside the coil and the airgap is poody defined this assumption holds for large coils.

• No eddy eurrents oceur, eaused by changing magnetic fields. When a "ferroxcube"

material is used this assumption is reasonably true.

• The field in the airgap is homogenous

Figure 2.2: The magnetic flux lines in the levitation system Given these assumptions we ean calculate the magnetic reluctance force as follows:

(2.2) where W is the magnetie energy in the system and L is the induction of the coil, the hall and the airgap. For the magnetic energy W we ean write:

W= !!!B.HdV v

(2.3)

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CHAPTER 2. A MAGNETIC LEVITATION SYSTEM

where

9

B ==/LO·/Lr·H:=

H

==

V

/Lo ==

/Lr ==

magnetic field density [T]

magnetic field strength [Alm]

system volume[m3]

permeability of vacuum[H

I

m]

relative permeability [.)

We can divide the system volume in three parts. A part inside tbe coil, where the magnetic field (H) is homogenous and /Lr == /Lr,iron. A second part, the airgap, where the magnetic field is also homogenous and J.Lr == /Lr,air. In the rest of the volume, the magnetic circuit is poorly defined and an unknown amount of magnetic energy is lost. Together with the assumptions made earlier and (2.3), this implicates for the total magnetic energy:

W ==

JJ J

Bairgap' HairgapdVairgap

+ J J J

Biron' HirondVeoil

+

Wrestvo/ume (2.4)

oirgap eoi/vo/ume

The amount of energy in the rest volume is not dependent on the position of the levitated ball and is proportional toi2 The energy in the airgap and the coil can be derived as follows:

i ==

f

Hdl,

which implicates:

N .i == Hoirgop' lairgap

+

Heoil •[eoil

+ I

Hdl J'ine

As the fields in the coil and the airgap are assumed to be homogenous, we conclude:

(2.5)

(2.6)

Boirgop == Beoil

Further we assume:

Heoil == - - - -Bair

/Lo •/lr,iron (2.7)

(2.9)

I

H dl == Q •N .i, (2.8)

~ Dof\QI

where Q can only be determined byexperiment. Equations (2.6), (2.7) .,(2.8) result in:

N·i·(1-a)

Boirgap == I · I

a,rgap

+

cai'

JLo'Pr,air lJ.ooJ.Lr,iron

As J.lr,iron >

>>

/-Lr,air this implicates:

N .i .(1 - Q) .J.lo . /Lr,air

Bairgap == l .

a,rgap (2.10)

Now that we have found an expression for Bairgap we can calculate the integral of (2.3) to determine the total magnetic energy of the magnetic le"itation system:

w

== [N .i .(1 - 0')]2 ./Lo .J.lr,air . A

+

Î ' (Ni)2,

X+x (2.11)

(14)

where

N = A = X+x =

"Y = a =

number of coil turns [.]

crosssectional area of the coil[m2] average length of the airgap[m]

constant to be determined by experiment [H]

constant to be determined by experiment [.]

With this formuIa we ean derive both the magnetic reluctance force (Fm) and the cail- induetion(L):

where L

1/2· (X

~X)2

.i2[N]

- XQ +Lo[H], +x

(2.12) (2.13)

(2.16)

Q

= 2· N2(1 -

a? .

JtOJlr,air •A = constant, to be determined experimentally (2.14)

Lo

=

2 ."y' N2 = constant, to be determined experimentally (2.15)

Now, we can easily obtain the dynamic equation of the system by using (2.1):

d2x Q ' 2

m· dt2 = m· 9 - 1/2· (X

+

x)2 . l

When we take a look at figure 2.1 again and use (2.13) for the induction (L), we can also write down the electrical equation of the system:

u = R .i

+ .!!.-(

L .i) t-+

dt

dL d x . di u (R

+

dx dt) .l

+

L . dt t-+

Q d x . Q di

u (R-(X+x)2dt)·t+(X+x+Lo)·dt (2.17)

With (2.16) and (2.17) we found the dynamic and electrical equations, that completely de- scribe our magnetic levitation system. The objective of our system is to control the position of the levitated balI (x) as fast and accurate as possible. To achieve this purpose we ean either supply a eurrent (current control ) or a voltage(voltage control ) to the electrical part of the system. In our research we chose to use current control, because the, transfer of the eurrent to the magnetic reluctanee force is very direct. Disadvantage of current control is that a coil offers resistance to a changing eurrent. This means the necessary voltage to produce the eurrent may grow beyond aU bounds, especially for higher frequencies. To make our control system as fast and accurate as possible we must try to design the cail in sueh a way that we ean achieve a maximum reluctanee force (Fm) with minimal power supply (u

*

i). In the next section we tried to use our free design parameters, heing:

N = number of cail turns [.]

A = crossseetional area[m2]

r radius of the wires[m]

= length of the wil [m],

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CHAPTER 2. A MAGNETIC LEVITATION SYSTEM

to attain optimal system parameters, being:

Lo

=

coil inductance without levitated bali (2.15) [H]

Q = parameter that is linear with the reluctance force (2.14)

[k~;;3]

R resistance of the coil [D].

2.2 Determining the design parameters

11

If we look at our desire of maximum reluctance force and minimal power supply, mentioned in section 2.1, we can formulate following statements about our system parameters:

1. Loshould he minimized to attain minimal power dissipation.

2. Q should be maximized to realize a maximum reluctance force.

3. R should be minimized for minimal power dissipation.

The criteria mentioned above can be met by choosing our design parameters (N ,r,l and A) as weIl as possible. This is nevertheless quite difficult, because some parameters, like a and 'Y, are unknown. Moreover we are not familiar with some preconditions, like the amount of current that can be actuated into the coil. In addition to these restrictions, equations (2.14), (2.15) and (2.12) show us that for example increasing the design parameter N has a positive influence on Q but a bad influence on Lo and R. In this stage of our research-project it was hard to teIl what system parameter should get priority, while choosing our design parameters.

This is why we chose the design parameters in a rather intuitive than analytical way. The following choices were made for our design parameters:

N 1000

r 1· 10-3[m]

l 1· 10-1[m]

A 6.25· 10-4[m2J

In the next section we will analyse the consequences of these values for our magnetic levitation system, after we have experimentally determined the values of our system parameters(Q,Lo,X and R).

2.3 Analysis of the designed system

To have an appropriate model of the magnetic levitation system we must know the values of Lo,Q,X and R. As the magnetic circuit is poorly defined, these values can only be determined by experiment. However, it is possible to make an estimation of the values on the basis of the theory presented before. Using equations (2.12),(2.14),(2.15) and f-Lo

=

471".10-7 the following estimations were made:

Q La R X

= 2·N2(1- 0)2 . f-LOf-Lr,air . A

=

=

"y' N2

=

- 5 -

p.4NVA -

- 1f·r - 1f·r2 -

=

0.3· radius of the levitated ba.ll

=

3 2 kgm3

1.57·10- . (1 - 0) [ - - ) A2 s2 1 .106 ."y[H)

2.5[!!]

5 . 10-3[ml

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u =

To validate the theQry ~md to have more exact knowledge of values mentioned, we designed an experiment to measure the coil inductance L as a function of the position of the ball x.

The results of the experiment are presented in figure (2.3). Using this figure and equation

o 0002 0004 000$ QODe 001 0012 0014 oo~ 0018 002 po.tonIC (IIllJ

Figure 2.3: The inductance as a function of x

(2.13) (L = Lo

+ Xh),

we determined the values ofLo,

Q

and X in a least squares way. The results were:

Q -5 kgm

3

=

5.75 ·10 [A2s2]

Lo := 1.03·lO-1 [H]

X = 8.57 .lO-3[ml

For the coil resistance Rwe measured a value of 3.5

[nl.

These results differ only slightly from the values expected and imply certain values for 0: and ,. If we look back at the dynamic and electrical equations,

m .9 - 1/2· Q .i2 (X

+

X)2 R ··t+ -dL'·t

dt '

we can see that the system parameters have a big influence on the dynamic and electrical behaviour ofthe magnetic levitation system. Q and X determine the currentioneeded to keep the bali levitated in some position Xo against the gravitational force Fg. We calculated the' current needed to keep the ballievitated 0.005[m] under the coil to see if our design parameters were appropriately chosen. Ta do this we used (2.16) with Fg = Fm, 9 = 9.81 [m/s2) and m

=

0.050[kg] :

. p::rn:g

to =

V

~. (X

+

xo)

=

1.72[A]. (2.18)

Lo ca,n introduce very high voltages u into our system, when the current rapidly changes.

These high-frequent currents are not needed as the system 's dynamics are bandlimited. As a matter of fact we can conclude that our system is well-designed if we can realize a current source that can supply about 3A ta the cail. In chapter 4 we wil! analyse th is problem. So far we have analysed a magnetic levitation system and We designed a cail that was abIe to keep a

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CHAPTER 2. A MAGNETIC LEVITATION SYSTEM 13

halllevitated against the gravitational force. In next section we will describe a control system,

"that must do the job" , and we will linearize our dynamic equation to get more insight in the dynamic hehaviour.

2.4 Preliminaries of the control system

As said hefore, we have designed a coi! that can generate a magnetic force, large enough to compensate for the gravitational force, when a certain amount of current is supplied. Of course this is not enough to keep a ballievitated. To realize a control systern that can, we must be ahle to measure the position of the hall. The techniques used to sense are described in chapter 3. Besides a sensor an actuator must he designed that can supply a large current accurately to the coil. The actuator will be explained in chapter 4. And last but not least a controller must be developed that calculates the current to keep the hall levitated at a desired position. The controller is the subject of chapter 5. These components form the basis of the control system drawn in figure 2.4. At the end of this chapter we want to take another

r - - - r controller actuator

x

plant sensor

1 + - - - 1sum x

Figure 2.4: the control system

look at the dynamic equation. We want to know whether our system is stabie or not and what requirements the system asks of our controller for stability. Therefore we linearize the dynamic equation in some point (xo, ia) where Fm

=

Fg if i = ia [5]. To linearize the equation we used a first-order Taylor expansion resulting in:

with:

(2.19)

Xl = x - Xa

t]

=

t - 1.0

1'0 =

J2' Q' 9.

(X

+

xa)

distance to the desired position[m]

difference from the equilibrium current[A]

equilibrium current[A].

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H we transfQrm the dynamic equation (2.19) to the frequency domain we get the following transfer function from the actuating current il to the controlled position Xl:

XI(S) Q'io'(X+xo) il(s) = - s2 •m· (X

+

xo)3 - Q. io2 Substituting (2.18) in (2.20) we find:

(2.20)

S2 •(X

+

xo) - 2g (2.21)

This transfer function clearly possesses two real poles

±J x~~o.

One pole makes the system unstable. The controller to be designed has to compensate for this instability to assure stability of our cantrol system [9].

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Chapter 3

Sensing the position

In this chapter we describe a method to sense the position of the ball. Thé ball position x must be translated to a. voltage that is linear to it. First we will mention some options to measure the position of the levitated object [3]. After that we will explain why we chose to realize the sensor in the way we did. After that wewill explain the principle and the electronic realization of the sensor in detail.

3.1 Sensor options

In this section we describe sornesensor rnethods that can be applied to the magnetic levitation system. All rnethods take care of contact free sensing.

3.1.1 Optical sensor

An optical sensor uses a simple principle. A Light Emitting Diode (LED) is placed opposite to a Photo Diode (PD) as shown in figure 3.1. The intensity of the light received by the diode

~

coiJ

Figure 3.1: The optical sensor

is a measure for the position of the levitated balI. We did not use this type of sensor because in the end we want to move the ball between four coils and in this drawing up there's no room for an optical sensor. Furthermore the optical sensor is very sensitive for extern al light SOllrces and thus not very accurate.

15

(20)

3.1.2 Eddy current

sensor

Eddy current sensors consist of a resonant circuit. A coil is placed near the levitated object and is part of the resonant circuit, that has aresonant frequency of about 100 kHz. The coil is excited in it's resonant frequency and thus induces eddy currents in the levitated object.

The quality factor of the resonant circuit is influenced by the eddy currents. By measuring this quality factor a measure for the position x is found. We did not use this type of sensor because we did not want extra coils to be placed in our system.

3.1.3 Inductive sensor

The inductive sensor measures the position by first determining the inductance of the actu- ating cail. As derived before the inductance is dependent on the position of the levitated ball. We used this type of sensing because na extra coils or are needed. The principle and implementation of this sensor are explained in detail in the next section.

3.2 Principle of the inductive sensor

The inductive sensor uses the fundamental idea, that the inductance of the coil depends on the position of the levitated object. From equation (2.13) we know:

L

=

Lo

+

X

~

x . (3.1)

This equation teUs us that we know the position of the bali when we are able to measure the inductance of the coil. !t's obvious to determine the inductance by using the cail current i and coil voltage u. In chapter 2 we derived the following equation (2.17):

dL d x . di

u

=

(R

+

dx dt) .t

+

L . dt (3.2)

We can see that the voltageu depends on both Land i and ~~~: and ~;. As a matter of fact it's not possible to solve our problem by measuringi and u, because the current and position are not constant. However, we know that both the actuating current and the position will be bandlimited. This introduces the possibility to do our measurement at a high frequency. If we supply a sinusoidal current

t

sin(wt) with constant amplitude

i

to the cail, we obtain the following equation:

with

û . cos(wt

+

if» = (R

+ .i

)isin(wt)

+

w

Li

cos(wt), (3.3)

.i =

~

. * =

time derivative ofL(H /sJ w

=

radial frequency[rad/sJ.

As typically ~~ is very slow with respect to w, we may assume .i to be constant, during one period of our sinusoidal current. Applying complex calculus to (3.3) gives the followingresults for the complex amplitude û and the phase 4> of the resulting cail voltage u:

u 7j(w'L)2+(R+.i)2 (3.4)

4> a r c s i n ( )w·L (3.5)

j(w. L)2

+

(R+ L)2

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CHAPTER 3. SENSING THE POSITION 17

lf we look at equations (3.4) and (3.5) we see that would we be able to supply a sinusoidal current with constant amplitude

i

to the coil in addition to the actuating current and to measure both phase </> and amplitude û we would know the inductance L by doing the following arithmetic operation:

û .~sin(</» = L

t ·w (3.6)

Of course it is not easy to do this electronically, but this algorithm ensures us there's enough information in the coil voltage to measure the inductance. The first electronic problem is to throw away the low frequent components and just look at the frequency of the supplied sinusoidal current. We can do this by multiplying the coil voltage by cos(wt). This results in the following equations:

[û· sin(wt

+

</»

+

uil]'cos(wt) =

{û· [sin(wt)cos(</»

+

cos(wt)sin(</>)]} .cos(wt)

+

uil'cos(wt)

=

1/2·û·[sine</»~

+

sin(2wt

+

</»]

+

uil'cos(wt), (3.7)

where uil are the low frequent actuating parts of the coil voltage. Ifwe analyse this muIti- plication of the coil voltage in the frequency domain we find the results of figure 3.1. From

BeCore multlpllc:atfon

tu

I1

-r...a .flr flr

-

freq [,ea

Aner multfpJlcatioD

r

J

IJ

....

freq

L

Figure 3.2: Multiplication in the frequency domain

this figure we can conclude that the de-voltage component is equal to û· sine </»

=

wL. This is exactly the result we wanted. As

i

and ware known, multiplication of the caiI voltage by cos(wt) and low-pass filtering the result will provide us a measurement of the coil inductance L. The theory described in this section introduces the following scheme for our sensor (figure 3.2): In the next section we will show how we realized this sensor electronically. After that we will try to say something about the sensor's performance and how this sensor will influence our control system.

3.3 Practical realization

As it appeared to be very difficult to obtain high performance multipliers, we tried another option to realize our sensor principle. In this option a Loek-in amplifier was used. This will be explained in the seeond paragraph of this section. However, the first problem to be solved was how to supply an alternating high frequency current with constant amplitude to a coil which is also used to keep a balI levitated.

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jl----l

i'sin

(OOt)

ueoil

Multiplier

uref

cos

(oot)

neoU'uref

Lowpass filter

L

Figure 3.3: Block seheme of the induetive sensor 3.3.1 Supplying a sinusoidal eurrent to the eoil

For the sensor described above it is of high importance that the amplitude of the sinusoidal current is constant. If no other currents are supplied to the coil this is not diffieult. The scheme of figure 3.3 would be sufficient. In this figure the following equation holds:

R L

U'sin(oot)

Re i'sin(oot) ---+-

Figure 3.4: A simple electronic circuit

~ = -..j-;:;'(==R=+=:R==e7:;;)2:=+=(;=w===L~)2 (3.8)

In this equation

i

is approximately constant ifR+ Reis much greater thanwtimes the change of the eoil inductanee (= rtx)' This solution, however, is not very elegant and not possible when the eoil is also supplied with the actuating current. We tried to solve these problems by using the actuator to supply the sensing current as weIl. This principle is described in next chapter. However, to test the principle of the sensor and to measure it's performance we used the solution as suggested above.

3.3.2 The Loek-in Amplifier

Ir we look at the principle of the induetive sensor again, we notice that the multipl.ication of the coi! voltage with the eosine is a sart of demodulation. For demodulation often a loek-in amplifier is used. This amplifier multipIies the input signa.l Uin by 1 if the reference voltage

urp'! is greater than zero and by -1 if Un ! is smaller than zero. Ir we take the coil voltage as

Uin and

v.

sin(wt) as ure! the Loek-in-Amplifier will have the in- and outputs as shown in figure 3.5. Ir we integrate the output signal Ua ut of the amplifier over one period of the sine,

(23)

CHAPTER 3. SENSING THE POSITION

(

06 I t i t JA ,

.'1'1

Figure 3.5: Results of the Loek-in Amplifier

19

(3.9) we obtain the following result:

2ft' '"

f~Uoutdt

=

2.

fi:i

ûsin(wt

+

4»dt

= __

4u_A s_in_(.;....c/>.;....)

h h

w

If we eompare this with (3.6), we notice that again we have a measure for the inductanee of the coil. So supplying the cail voltage to the Loek-in amplifier and low-pass-filtering the output will realize the sensor. The electronic circuit is given in figure 3.6.

i'sin (oot)

neon

Loek-in amplifier

uref sin (oot)

neon'Uref

Low pass filter

L

Figure 3.6: Electronic realization of the sensor

3.3.3 Making the sensor linear to x

We have know found a measure for the inductanee Lofthe coil. The cail inductance, however is not linear to the bali position x. Therefore we have to work up the result into a voltage that is linear to x. To get this done we must electronically realize the following operation of (3.10). The electronic implementation is given in appendix A:

x

=

Q -X

L - La (3.10)

In order to be able to design an appropriate controller we need to know the transfer function of the sensor: u.w((s))I where XtTue is the real position and Us en is the output of the sensor.

Xtru~ S

To obtain this transfer function it had been practical to be able to control the position of the balI x(s) and then measure the output Usen ' Now we just looked at the electron ie circuit and tried to determine the transfer fnnction by analysing this circuit. This resulted in:

usln(s) J(

----~

=

(3.11)

XtTue(S) (2.2.10- 3 .S

+

1)4'

(24)

where [{[Vlm] I1lIH;t be determined byexperiment. Furthermore the linearity of the sensor must be tested. As the divider that must realize (3.10), has a linear output for a restricted area of operation, (3.10) will only hold for a restricted range of ball positions x. In the next seetion we will experimentally determine ]{ and the linear range of the induetive sensor. We will analyse it's reliability as weIl.

3.4 Testing the sensor

To measure K and the linear range of the sensor, we exeeuted the following experiment.

We supplied the sinusoidal eurrent as shown in figure 3.4. We adjusted the position of the balI by using little pieces of paper with a thiekness of 0.1mm. For eaeh position we measured the output of the sensor and the output of the lowpass filter following the Loek-in Amplifier. The results of this experiment are shown in figure 3.7. With this figure we ean

u , - - - r - - - - -...- - - ,

~2.5

".:I- ,

,...c.':'i ~

0.50' - - = - - - -....5 ' - - - " 0 - - - ' , 5

poot''''IJrnm)

...~ , : " .

°0~----....5'---:':,0:---:',·5

pootlonIlrnm)

Figure 3.7: Lsen and Usen as a function of x

easily determine ]{ and the linear range, resulting in J( = 200[V/m] and the linear range for 0.002[m]

<

x

<

O.012[m). The results obtained in this figure appeared to be reproducible.

Other important features of the sensor are it's reliability, it's sensitivity and the aceuracy.

If we look at the sensor's principle we notice that we first measure the inductance of the coil. The biggest part of the inductance (about 90 %) consists of Lo,being the induetanee of the coil when no balI is present. Only a minor part of L depends of the position of the ball (see figure 2.3). This means that little disturbances in the measurement ofL cause large disturbances in the sensor output linear to x. This tells us that very accurate eomponents are needed to get an accurate measurement with little output-disturbance. If we define the sensitivity ([Vlm] of the first part of the sensor as ~~ we obtain the following equation using (3.1):

dL

Q

( = -

= - -,-:-:,---:.---:--=-

dx (X

+

x)2 (3.12)

From (3.12) it is obvious that the sensor is more sensitive ifx is smaller. However ifx is very small the divider will not be in it 's linear range. This means the sensitivity is maximal when we measure in the linear range as close as possible to the coil. At the end of this section

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CHAPTER 3. SENSING THE POSITION 21

we want to say something about the reliability of the sensor. The reliability of the sensor is influenced by the amount of actuating current that is supplied to the cail. This effect is caused by saturation effects. If we look at the B - H-curve of the magnetic material used in the coil (figure 3.8) we see that this curve is only linear in a restricted range. Ifthe magnetic field strength B[A/m] gets toa high, due to a large current, we will notice a smaller induction L, thus influencing our sensor and destroying our sensor-signal [8]. This means the s~nsoris only reliable for small actuating currents.

B

new curve

H

Figure 3.8: B - H-curve of the magnetic material

In this chapter we developed an inductive sensor and derived a model of it's operation. With this sensor a part of our control system is completed. In the next section we will describe an actuator. After realizing this all devices are present to design a controller that can stabilize our control system.

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A current Amplifier

This chapter describes the design of the current amplifier, the problems during the realisation and same solutions for the problems.

4.1 Specifications

We want to design a current amplifier that can supply both the actuating current and the sensing current to the coil. The actuating current must be large enough to keep the ball levitated and must be handlimited to avoid very high volta.ges to he induced in the cail. For accurate sensing it is necessary to keep the current of the sensor constant. So the current amplifier has to satisfy the following equation and frequency spectrum (see figure 4.1).

icur.amp = iaet1lator

+

isensorsinwt This results in the following specifications:

IHIt - - - .... j .

(4.1)

:' fa.lUalor fsensor freq

Figure 4.1: Desired spectrum of the current amplifier

• The maximum current, which must be provided by the amplifier, is estimated with some simulations (see section 5.4). For a small motion around an equilibrium point Xo

±5[mm] of the ball a current of 2[A] appeared to be sufficient with some peak currents of Iess then 3 [A]. This means a current amplifier must be designed that can supply a CUfrent of at least 3 [A] .

• The simulations mentioned above showed a voltage supply of about 30 [V] was sufficient to realize the currents needed to levitate the ball.

22

(27)

CHAPTER 4. A CURRENT AMPLIFIER 23

• It is very important to keep the sensing current constant, as explained in chapter 3. If the sensing current is supplied behind the amplifier (figure 4.2) the output-impedance of the current amplifier must be high. Ifwe want to use the other method, applying the sensor current at the input of the amplifier, it is necessary that the bandwidth of the current amplifier must be at least fsen Hz. However, this means very high voltages will occur, which may cause instabilities .

• The bandwidth that is needed to keep the balllevitated is about lOOHz,as the dynamics of the levitation system are also bandlimited at this frequency. To avoid instabilities of the amplifier it is necessary to use the fact that no high frequent currents are needed.

This results, together with the earlier item, in the desired spectrum of figure 4.l.

In the next section wewill start from a basic circuit that is used asa current amplifier. After that we will show the problems that occurred using this circuit, when the current was supplied to a coil (large inductive laad)

4.2 The basic principle

The circuit that we started from is shown in figure 4.2. This circuit consist of a fet (MOS field-effect transistor) driven by an opamp (operational amplifier) and a source resistor Rs which voltage is fed back to the input of the oparnp.

Vin

(4.2)

.Vee .15V

Figure 4.2: The basic scheme of the current amplifier

The opamp causes the souree voltage to be equal to the input voltage. The drain current will be

I Vee - Vin

d=

R

s

This equations only holds wh en the gain of the opamp is very high (Ao ---+ 00) and the fet is in the saturation region (Ves - VThreshold ~ VDS).

The output impedance of this current amplifier is Aa

Zoutput '" 1 .R s .afet

+

T· S (4.3)

(28)

where

a/et = constant of the fet (in saturation: ID :::::J a .VDS)[']

1" = time constant of the opamp[s]

Ra

=

source resistor~ 0.2[n]

In (4.3) we can see that Zo decreases if s increases. Furthermore the impedance of the coi1 increases if sinereases. This means that the amount of sensing current that will really enter the coil will depends on the frequency and on VDS,as a depends on VDS. The earlier rernarks imply that the sensing current will not be constant in thisci~cuit.

Some further remarks can he made about this circuit:

• After the realisation of the amplifier it turned out that instabilities occurred when supplying a current to the coil. This can be corrected by decreasing the handwidth of the opamp. But when the bandwidth of the opamp is decreased a1so the gain for high frequencies is decreased and this will affect the output impedance negatively.

• Ifa conventional opamp is used, for examp1e J.LA741 or LM308, it is necessary to connect the hot tom of the coil to the negative voltage supp\y to get a sufficient range for the coil voltage. This makes the sensor system more complicated. Another disadvantage of using a conventional opamp is that the ideal region of operation for the output voltage is around OV and here the opamp is operating just below the positive voltage supply.

Because a conventional opamp is 1ess suitab1e for this application, we looked for a more ded- icated opamp and found one, the XTRllO.

The XTRllO is a highprecision voltage-to-current converter which can easily be used as a current amplifier by adding an external mosfet and source resistor. This will be called thë U-I converter from now on. The XTRllO has the following specifications [1]:

• Single supply operation and wide supply range: 13.5V to 40V

• Selectabie output range, dependent on the external components

• Very high output impedance (109), if fet is in saturation region.

Moreover , a suitable mosfet has been selected after we checked for the following criteria:

• The maximum drain-current.

• The minimum VDS needed to keep the fet in it's saturation region.

• The maximum permissible power dissipation

We have chosen the IRF? 9140 [2]. The maximum drain current is 19A and the maximum power dissipa.tion is 15üW. The fet has to he cooled with an appropriate heatsink in comhi- nation with foreecl air rooiing (e.g. a ventilator). Otherwise the fet will he damaged due to the enormous power dissipation.

In the next seetion we will analyze the U-I converter.

(29)

CHAPTER 4. A CURRENT AMPLIFIER 25

4.3 The U-I converter:

In figure 4.3 the complete circuit of the U-I converter is shown.

Cd

d1 Rd

- r - - - r - - - - V c e

Rl

ZOkQ

.

!

j

i

:

1:

:::

.,

U1kQ

t___ _. ~ . .__J

_---'- -'- ----'-_----'-_---'_ God

X'I1UIO

Uln

r---

Figure 4.3: The scheme of the U-I converter

The operation of this circuit is almost the same as the basic circuit. Only a subtraction circuit is placed at tbe input. The sta.tie transfer function is

fd R2R3

Uin = (Rl

+

R2)R4R~ (4.4)

The c1amping diode, parallel with the coil, is of great importance. The diode avoids negative voltages over the coil. Ifthe diode is not present negative voltage-peaks will cause all kinds of undefined non-linear effects of the fet and a limit-cycle occurs.

Even with the c1amping diode we see an oscillation at the coil voltage (see figure 4.3).

The frequency and the pulse-width of this oscillation depend on the voltage supplyand the

, lito

Figure 4.4: The oscillation of the coil voltage

coil current. With a supply of 40 V and a drain current of IA the frequency is about 250 Hz.

Another observation wemade, was that the oscillation is largely damped when the sensor system is connected to the cail. The first part of the sensor system is a first order high pass filter (see Cd and Rd in figure 4.3). The frequency and amplitude of this sinusoidal oscillation

(30)

also depends of the voltage supplied and the coil current. The frequency is about 16 kHz and the amplitude is 10V.

In the next sections we will explain these observations by means of some simulations and measurements. Finally we will give a solution for these problems.

4.4 Analysis and simulation of the U-I converter

4.4.1 Simulations with Spice

For the simulation of the U-I converter we used the software package Spice. Spice is a simulating program for electronic circuits. The circuit is described by it's junctions and components. A lot of analysis can be done with Spice. Same analysis we used were:

• DC-analysis, determine the input/output characteristic.

• Transient analysis, analysis in time.

• AC-analysis, ca1culate the complex transfer function based on linearized small signal models at the operating point.

The opamps of the U-I converter are modelled with a first-order model:

Vout = 1

+~o/wo

(V+ - V_) (4.5)

where Ao is the open loop gain and Wo is the open loop bandwidth (wo

= *).

Because

there were na specifications of the opam ps in the datasheets of the U-I converter, we measured the closed loop transfer function of the oparnps:

Hel = 1 (4.6)

1

+

s/Aowo

The closed loop bandwidth is Aowo ~ 3 .106rad/ s. Sa we consider that Ao :::: 1 . 105 and

Wo :::: 30rad/ s. These values match with the specifications of conventional oparnps.

The transistor of the U-I converter has been modelled as a standard NPN-transistor of Spice. The power mosfet is modelled as a standard p-enhancement MOS fet with some ad- ditional parameters, such as input- and output capacitance, drain resistance and inductance and transconductance parameter.

First we simulated the U-I converter with a resistor as load, in figure 4.5 the AC-analysis is shown.

In this figure we see that one pole, s}

=

-3.106 [rad/sJ, is equal to the pole of the opamp's and the other one, 82 = -1.2· 106 [rad/s], is a little bit smaller, caused by the fet. So the bandwidth of the system is about 200 kHz.

The next simulation is the analysis of the U-I converter with an inductive laad, see figure 4.4.1. In figure 4.6(a) the transfer function

ijL

is shown and in figure 4.6(b) the transfer

reJ

function ....1L.Ur.J

In these figures we see that the coil voltage increases extremely the frequency is enlarged and that the bandwidth ofthe system is decreased to 10 kHz. The frequency (16 kHz) of the sinusoidal oscillation mentioned in section 4.3 is approximately the same as the frequency (10 kHz) of the resonance peak.

(31)

CHAPTER 4. A CURRENT AMPLIFIER 27

0.11111",_nn:101271eS 18:21:32

A 10 --- -- -.- --. -- - -- -- --- -.--- -- --- ---.-.-- ••• - - --.--- ••• ---

·10

..-0 +- _._ ..~- - w _-r -_ --"" _. -r" _ - -r-.-" ---- -1

1.01'1 tOh tOCh 1.Okh 10Kh fOOKh 1.0Mh

• 20'log(v(7J)

Figure 4.5: The AC-analysis of the U-I converter with load resistor, Ure! - t ULo

T~""270 A 200T-"""" ---....~..._._ e . . . .- . . . " • • " " . '--~._. . . - - - .. - . . . • . . . .w - . - _ . . .0_' 0_;

1~ ~

oT·..•·..·· --- _.e_0 _ 0. . - - -.. - - -- ·"1

100~

,.,., ,.",

lOOI>

, ,

-250+ _-.. --.-- ••••• _ •••••• __ - ••• - ••····.--_···4

0·--..···.. ·--··-_··· --- --.,.- --- - ".- ,

,Oh 10t1 10Cfa

·20~)

Figure 4.6: The AC-anaJysis of the U-I converter with inductive load 4.4.2 The analysis

The oscillations of the U-I converter are caused by very small disturbances in the current.

These dist urbance will be amplified strongly by the cail. The puJses in figure 4.:J are probably

(32)

caused by disturbances of the power supply or noise in the souree resistor. Because of these disturbances, the cail voltage is increased to the voltage supplied very fast and the drain current becomes zero as the fet is closed (VDS

=

0). When the RC-network of the sensor system is connected the high frequency disturbances will be damped. The impedance of the RC-network is smaller than the impedance of the eoil for high frequencies.

Another explanation for some problems is that a system containing a current souree in series with a coil is not a causal system. The coil wants to control it's own current (h =: JUL dt) and that is contradictory to the current souree. By adding a RC-network a causal system is obtained, now the eurrent souree and thecoil can bath control their currents.

4.5 The solution

We must ehoose Rd and Cd of the RC-network sa that the unwanted high frequency dis- turbances are damped and that the cantrol current (J

<

100Hz) and the sensor signa!

(lkHz

<

fs

<

2kHz) are not influeneed by the network. This means,

ZL+RL

»

ZCd+Rd

ZL+RL

«

ZCd+Rd

if

f <

5kHzand

if

f>

5kHz.

This results in Rd =: 1k!1and Cd = 18nF. The load of the amplifier can now be written as (4.7) The zeros of this function are: Zl =: -44 rad/ s

Z2 == -56· 103 radjs

The poles are: P},2 =: -5· 103

±

j23 .103 rad/ s

The poles and zeros of this system correspond to the poles and zeros shown in the AC-analysis of spice (see figure 4.7).

We also measured the transfer functions of the whole system with a spectrum analyzer.

The result of this measurement was the same as the result of the Spice simulation.

Because the bandwidth of the amplifier has to be large due to the sensor current there is still some noise of the smal1 souree resistor that will be amplified. This noise disturbs the sensor signa!. It is better to add the sensor current behind the amplifier. The advantage is now that when the fet is closed (Id =: OA) the sensor signal is not disturbed by the amplifier.

4.6 Conclusions

In this chapter we tried to design a curTent amplifier that was able to supply bath the actu- ating current and the sensing current to the cail. This resulted in the ideal spectrum shown in figure 4.1. However, in practice, it is not possible to realize sneh a transfer function.

This meant the bandwidth of the amplifier had to be at least fsen to be able to supply the sinusoidal sensing eurrent to the coil with a constant amplitude. However, this bandwidth appeared to be too large to avoid instabilities. These were eaused by the large inductive laad

(33)

CHAPTER 4. A CURRENT AMPLIFIER 29

1~270

~1

~1

,~

1

"'1 '

.~ 1 :

.240+._ •• --.o· .. ·r -._ --r" . - .0 · .. · _ _ _ _ .. ,_ _ . 0 • • • •t

lOh 11»'1 lOOPt 101<h 1010'1 10010'1 1~

.20"a:)g(l(l1))

A -0I-·.·--··.·--·--····--·----···--·.··---·.·.·---·.·--_···~_··_---·_·-1 llft'O«lkn. 71 0

.

.

o .-- ••••••••••••••••• -_ ,. __ •••••••••. --_ ••.••••••• __ ••••••••

't)h 1~ 1oc;», ,(I(h 1lJOl lootO'l 1tNl\

. ~

A 110T··· ..···..··· -:

~ j

j 1

120

i .. \

Figure 4.7: The transfer function of the U-I converter

(the cail) that extremely amplifies high frequent currents. Therefor it was better to supply the sensing current behind the amplifier as shown in figure 4.3. However, this means the output impedance of the current amplifier must be large enough. The bandwidth of the actuating cur- rent through the coil has been restricted by the RC-network parallel with the coil in figure 4.3.

Up till now we have designed a "system" that supplies the sensing current and actuating current separately. However, even in this system it is very hard to sense as the sensing current is disturbed by noise generated in the current amplifier.

(34)

(5.2) (5.1)

The control system

In this chapter we will develop acontrol system that is able to keep a ball levitated under a coil against the gravitational force. In earlier chapters we analyzed the system 's dynamics and realized both a sensor and a current amplifier that, together with the controller, should stabilize the levitated object. While designing the controller and the control system we had to deal with the restrictions of our sensor and current amplifier. Successively we will explain why we separated the sensor and the current amplifier, describe the new system that thus appeared, design a controller for this system and finally descrihe the practical and simulated results of the created system.

5.1 Sensing and actuating on separate coils

Although it should theoretically he possihle to sense and actuate with the same coil, the practical application of this option appeared to he very difficult. As explained in chapter 4 the sensing sinusoidal current was inftuenced by the actuator, temperature, or the non-linear behaviour of the FET. Furthermore the sensor's reliability was effected negatively, when high actuating currents were supplied to the coil, as explained in chapter 3. All this made us decide to do the measurement and actuating on separate coils for this simple magnetic levitation system. However, this is not possible for the magnetic hearing, we finally want ta realize.

Thus, the sensing prohlem remains yet unsolved. Sensing and actuating on separate coils means electric and magnetic circuits different from those described in chapter 3 occur. The electric circuit, that exists, is presented in figure 5.1. In this figure we can see an actuating part and a sensing part. These parts are coupled by the mutual inductanee coefficient M[H]

as described in the fallowing equatians:

. dit di2

Ut

=

Rl

*

tt

+

Ll

*

(ft

+

M

* di

. di2 dit

U2

=

R2

*

t2

+

L2

*

(ft

+

M

* dt

We expect M to be a function of the balI position x and the frequency w. Ta complete our model of the levitation system it would be more exact ta determine the inftuence of M on the system 's dynamics. However, we ex peet M ta he very small, and sa we decided ta neglect it's influence while designing the controller. In figure 5.1 aresistanee R3 of 60

n

is placed para.llel with the actuating coil. This resistance limits the currents that will flow through the

30

(35)

CHAPTER 5. THE CONTROL SYSTEM 31

i2 i1

R3

u1

L2 M

R2

î

c: u2

.~

sensing part actuating part

Figure 5.1: Eleetrie circuit of the sensor and actuator

coil to a eertain frequency. This ean he demonstrated by deriving the following equation:

(5.3)

i1(s) = iaetuator(s)

*

R3

+ ::+

sL1 ::::: iactuator(s)

* O.OOl~s +

1

We presented this transfer in figure 5.2. It is very important to limit the frequency of the i1

t

(s)

iact(s)

freq. -

20 db

I

decade

Figure 5.2: Bandlimiting the aetuating current

currents sllpplied to the actIlating eoil, as high-frequency currents wil! induee high voltages in

(36)

the coils. Furthermore it is important that na crosstalk occursbetween the actuating current i1 and the sensing voltage 11,2 at the sensing frequency, as this would severely influence the quality of our sensor output.

In this section we changed our electric and magnetic circuits in such a way that both actuating and sensing were made possible. Although this solution is not an elegant one, separating the sensing and actuating coil created the possibility to design a system that was able to levitate a ba.ll. However, further study must be done to analyze the influence of the sensing coil on the system 's dynamics to atta.in an appropriate model of the levitation system. In the next section we will present the system as des cribed in chapter 2 (figure 2.4), and we will design an appropriate controller, that can stabilize the cant rol system.

5.2 The control system

Up till now, we have developed a position sensor and a current amplifier and we analyzed the dynamic's of the magnetic levitation system. Together with the controller these components lay the foundation of our control system, that is shown in figure 5.3. The blocks presented in

controller 11. actuator in~1 plant XI~..~ sensor

x.~,.,

sum

Xref

Figure 5.3: The control system this figure perform the following transfer functions:

• The current amplifier block transforms the voltage ca1culated by the controller into the actuating current, according to the following equation (see (5.3) and figure 4.6):

iactuating 1

--~=

Ueont (0.0016s+ 1)-(0.00018+ 1) (5.4)

• The nonlinear plant is the magnetic levitation system that "transforms" the actuating current (iactuating) into the position of the balI (x), according to (2.16). This transfer fllnction is highly nonlinear with respect to the output x and the input iactuating. This will make it hard to design an appropriate controller. Therefore we have linearized

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