Development and Control of a Non Linear Magnetic Levitation System

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ISSN : 2319 – 3182, Volume-2, Issue-1, 2013

62

Development and Control of a Non Linear Magnetic Levitation System

Reshma Angelene Jose & A Sanjeevi Gandhi Karunya University, Tamil Nadu

E-mail : [email protected], [email protected]

Abstract – Nowadays, studies to develop and control non linear systems is of great significance. Magnetic Levitation System has gained considerable interests due to its great practical importance in different engineering fields In this paper an electromagnetic levitation system was developed and mathematical model for the system was derived. The developed system was controlled manually.

Keywords – Magnetic levitation system

I. INTRODUCTION

For a past couple of decades the technology that has been experimented and studied intensely is magnetic levitation system. In electromagnetic levitation system, the magnetic force generated by the electromagnet holds the object against gravity in the airgap. There are two basic principles in dealing with the concept of magnetic levitation. The first law that is applied was created by Michael Faraday. This is commonly known as Faraday‟s Law. This law states that if there is a change in the magnetic field on a coil of wire, there is seen a change in voltage. This occurs in the coil when there is a current induced as a result of that change in voltage.

For the purposes of magnetic levitation the ability to change the strength of a magnetic field by just changing the current is powerful. If there is a need for more of a force, then sending more current through a coil of wires will produce more of a greater magnetic force. The practical applications of magnetic levitation are

 They applied in equipment such as frictionless magnetic bearings.

 Magnetically levitated vehicles.

 Levitation of models in a wind tunnel.

A mathematical model is a description of a system using mathematical equations. Using

mathematical model developed ,the system behaviour is studied.

II. MATHEMATICAL MODELLING

For modelling the parameters required are a resistance, an inductance, a magnetic constant and mass of the steel ball. Let x(t) be the distance between the object and the electromagnet. x₀ be the proper levitation distance. According to Newton‟s law the electromagnetic force f (i, x), acts on the object, which can be expressed as

𝑚^𝑑²_𝑑𝑡^𝑥(𝑡)₂ = 𝑚𝑔 − 𝑓(𝑖, 𝑥) (1)

Where „m‟ is given as the weight of the ball and the gravitational force as „g‟.

Equation (1) can also be written as 𝑓 𝑖, 𝑥 = ^𝑑𝑊_𝑑^𝑒

𝑥 − ^𝑑𝑊_𝑑 ^𝑠

𝑥 (2)

Where d We represents the change of electrical input and dWs represents the change of stored energy, assuming dWe is zero for non linear system.

𝑖𝑒 𝑓 𝑖, 𝑥 = −^𝑑𝑊_𝑑 ^𝑠

𝑥 (3)

The electromagnetic force generated is given by the equation

𝑓 𝑖, 𝑥 = −^𝑖²^{𝑡 𝑑𝐿(𝑥)}_2𝑑𝑥 (4)

where total inductance of the electromagnet is given by L(x) and x is the position of the object. The suspended object adds the inductance of the electromagnetic coil.

As the object comes close to the magnet, the total inductance increases. Considering the incremental variation of the total inductance and substitute into equation (4),

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International Journal on Theoretical and Applied Research in Mechanical Engineering (IJTARME)

63 𝑓 𝑖, 𝑥 = −𝑖² 𝑡 𝑑𝐿 𝑥

2𝑑𝑥

= − 𝑖²𝑑

2 𝑑𝑥 𝐿1+𝐿0𝑥0

𝑥

= −^𝑖₂²[−^𝐿⁰_𝑥^𝑥₂⁰] = ^𝐿⁰^𝑥⁰

2 𝑖² 𝑥²

=𝐶_𝑥^𝑖²₂ (6)

Where C is the force constant.

C = ^𝐿⁰^𝑥⁰

2 (7)

The object position will influence the inductance of the electromagnet coil, and the variation are nonlinear.

Also the levitating point between the electromagnetic force and gravity is inherently unstable. This problem can be solved by linearizing the nonlinear electromagnetic force as

𝑓 𝑖, 𝑥 = 𝐶[𝐼0 𝑥0]²+ [2𝐶𝐼0 𝑥02]𝑖 𝑡 − [2𝐶𝐼02 𝑥03]𝑥(𝑡) = 𝑓₀+ 𝑓₁+ ⋯ (8)

where I₀gives the coil current when the ball is at x₀ (x₀ is at equilibrium position).When the magnetic force becomes equal to the gravitational force on the object, the acceleration of the object is zero. Equation(1) becomes

𝑓₀= 𝐶[𝐼₀ 𝑥₀]²= 𝑚𝑔 (9) 𝑓1= 𝑓 − 𝑓0

𝑓1= [2𝐶𝐼0 𝑥02]𝑖 𝑡 − [2𝐶𝐼02 𝑥03]𝑥(𝑡) (10) The control force can be found out using Equation (1), (8) and (9), we get

m^𝑑²^𝑥(𝑡)

𝑑𝑡² = −𝑓(𝑖, 𝑥)1 (11) When considering the electrical part as a series combination of resistor-inductor, the voltage of the electromagnet can be represented as

𝑉 𝑡 = 𝑅𝑖 𝑡 + 𝐿(𝑥)^{𝑑𝑖 (𝑡)}_𝑑𝑡 (12) This equation can be written as

𝑉 𝑡 = 𝑅𝑖 𝑡 + 𝐿1𝑑𝑖 (𝑡)

𝑑𝑡 (13)

The object position is measured using a hall effect sensor, whose output is given in terms of voltage, which is given as

𝑦 = 𝑉𝑥 𝑥 = 𝛽𝑥 (14) where 𝛽 is the sensor gain.

Taking the Laplace transformation of Equation (10), (11), (13), and (14), with the voltage of the electromagnet as the input of the system and the object position as the output

𝑉 𝑠 = 𝑅𝐼 𝑠 + 𝑠𝐿₁𝐼(𝑠)

𝐼 𝑠 = _{𝑅+𝑆𝐿}^{𝑉 𝑠}

1 (15)

𝐹1(𝑠) = [ 2𝐶𝐼0 𝑥02]𝐼 𝑠 − [2𝐶𝐼02 𝑥03]𝑋(𝑠) = 𝑘₁𝐼 𝑠 − 𝑘₂𝑋(𝑠) (16) where 𝑘1= [2𝐶𝐼0 𝑥02]

𝑎𝑛𝑑 𝑘2= −[2𝐶𝐼02 𝑥03] (17)

𝑉_𝑥 𝑠 = 𝛽𝑋(𝑠) (18)

M^𝑑²^{𝑥 𝑡}

𝑑𝑡² = −𝑓(𝑖, 𝑥)1

ms²X(s) = -F₁(s

𝑚𝑠²𝑋 𝑠 = −[𝑘₁𝐼 𝑠 − 𝑘₂𝑋 𝑠 ] 𝑚𝑠²𝑋 𝑠 − 𝑘2𝑋 𝑠 = −𝑘1𝐼 𝑠

𝑋 𝑠 (𝑚𝑠²− 𝑘2) = −𝑘1𝐼 𝑠 (19) Substituting Equation (15) and (18) into Equation (19),

𝑉_𝑥 𝑠

𝛽 𝑚𝑠²− 𝑘2 = −𝑘₁ ^{𝑉 𝑠}

𝑅+𝑆𝐿₁ (20)

The overall transfer function between the electromagnet input voltage V(s) and hall effect sensor output voltage V_X(s) is given by

𝑉_𝑥(𝑠)

𝑉(𝑠) =_(𝑚𝑠₂_−𝑘^−𝑘¹^𝛽

2)(𝑅+𝑠𝐿₁) ( 21)

𝐺 𝑠 =𝑉𝑥 (𝑠) 𝑉(𝑠) =

−𝑘1𝛽 𝑚𝐿1

(𝑠²− 𝑘² )(𝑅 𝐿𝑚 + 𝑠)₁ 𝐺 𝑠 =_𝑠+𝑘 ^𝑘³

4 (𝑠²−𝑘₅) (22) Where 𝑘3=^−𝑘_{𝑚 𝐿}¹^𝛽

1

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64 𝑘4= 𝑅

𝐿1

𝑘5=𝑘₂ 𝑚

III. DEVELOPED MAGNETIC LEVITATION SYSTEM The considered parameters and their values

Parameters Values

X₀ 0.02m

m 0.00564 kg

R 60 ohms

L 0.282.2mH

I₀ 0.02 A

β 125

Substituting the measured parameters x0, I₀, m and gravitational constant in equation (3.8) yields the parameter C.

2 0

0

[I^O ]

f m g c

x

 

C = mg (x₀/ I₀)² C= 0.05533 Nm²A^-2

K 1 = [2ci

0/x

02]= 5.533 k₂ = [2ci₀²/x₀³] =5.533 k3 = [-k1β/mL1]= -434.853 k₄ = R / L₁= 212.77 k₅ = k₂/ m=981.03 G(s) = V_X(s) / V(s) =



³ ²



434.853

S 212.27S 981.028S 208733



  

IV. STABILITY ANALYSIS

For a system to be stable all the poles should lie in the left hand side of S plane. In this one pole is in the right hand side of S plane. So to test the stability rot locus test has been conducted and the root locus obtained is given in figure 1.

V. BLOCK DIAGRAM

The sensor mounted to the bottom of the electromagnet measures the position of the magnet below it. The electromagnet works to counteract magnets attraction.

The sensor gives the feedback how close the object is and power to the electromagnet is adjusted based on that position. Turning the power knob will change the amount of power to the electromagnet. Stable levitation is obtained when the power to electromagnet perfectly balances the weight and power of the floating magnet.

VI. HARDWARE SETUP AND RESULT

At 10V, the magnet attached ball levitated for a long time by fine adjustment of the resistance.

Hall effect sensor Power

supply

MOS FET

Comp

arator E/M

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65 VII. CONCLUSION

The developed electromagnetic levitation system is controlled manually. Depending on the voltage the levitating point also varies. This manually controlled system can be controlled automatically.

VII. REFERENCES

[1] DahiruSaniShu‟aibu , SharifahKamilah Syed-Yusof, NorsheilaFisal And SanusiSaniAdamu , “Low Complex System For Levitating Ferromagnetic Materials” , International Journal Of Engineering Science And Technology Vol. 2(6), 2010, 1844-1859.

[2] Engr.Lubna, MoinDr.ValiUddin, “Design And Simulation Of Model Based System Using Real Time Windows”.

[3] Katie A. Lilienkamp, Kent H. Lundberg, “Low-Cost Magnetic Levitation Project Kits For Teaching Feedback System Design”, Proceeding Of The 2004 American Control Conference Boston, Massachusetts June 3 0 . July 2, 2004.

[4] Milica B. Naumović, Boban R. Veselić, “Magnetic Levitation System In Control Engineering Education”, Automatic Control And Robotics Vol. 7, No 1, 2008, Pp. 151 – 160.

[5] P. Kallakuri,L.H. Keel , S.P.

Bhattacharyya,“DataBased Design Of Pid Controllers For A Magnetic Levitation Experiment”, Preprints Of The 18th Ifac World Congress Milano (Italy) August 28 - September 2, 2011.

[6] ShekharYadav,J.P.Tiwari, S.K.Nagar, “Digital Control Of Magnetic Levitation System Using Fuzzy Logic Controller”, International Journal Of Computer Applications (0975 – 8887) Volume 41– No.21, March 2012.

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