Table of Contents Abstract .................................................................................................................................................. 2

Introduction ............................................................................................................................................ 3

Problem Definition .............................................................................................................................. 3

System Description .................................................................................................................................. 3

Main components of magnetic levitation system are ........................................................................... 3

Operation and Working ....................................................................................................................... 3

Transfer function of magnetic levitation system ...................................................................................... 3

State Space modeling .......................................................................................................................... 4

Determination of Transfer function from state space........................................................................... 5

Actuator and sensor transfer functions ................................................................................................ 6

Open Loop transfer function of overall system .................................................................................... 6

Block diagram .......................................................................................................................................... 6

Root Locus of system ............................................................................................................................... 7

Open loop step response ..................................................................................................................... 7

Controller Design ..................................................................................................................................... 8

Specifications ...................................................................................................................................... 8

Loop Transfer function with Controller .................................................................................................... 8

Bode plot showing Frequency response ............................................................................................... 9

Closed loop step response ................................................................................................................... 9

Hardware Implementation .................................................................................................................... 10

Circuit diagram .................................................................................................................................. 10

Abstract The aim of this lab is to model and control a laboratory-scale magnetic levitation system using

analog controller. The objective is to keep a ferromagnetic object suspended, without contact,

beneath an electromagnet. The electromagnetic force must be adjusted to counteract the weight

of the object and account for disturbances. This is accomplished by measuring the location of the

object using a non-contact sensor, and adjusting the current in the electromagnet based on this

measurement in order to keep the object at a predetermined location. The system is inherently

nonlinear and open-loop unstable. Negative feedback and phase-lead controllers are designed to

stabilize this system. The controllers are designed using MATLAB and tested in Proteus

software. Analog controllers are implemented using operational amplifiers, resistors and

capacitors.

Introduction Magnetic levitation systems (also called ‘maglev’) are electromechanical devices that suspend

ferromagnetic materials using electromagnetism. Maglev technology has been receiving

increasing attention since it eliminates energy losses due to friction. Centered on friction

reduction, maglev systems have wide engineering applications such as magnetic bearings, high-

precision positioning platforms, aerospace shuttles, and fast maglev trains.

Problem Definition

Maglev systems are characterized by open loop instability and nonlinear dynamics that suggest

the need of stabilizing controllers. The objectives of this lab are:-

1) Modeling of magnetic levitation system.

2) Design and implementation of Phase Lead controller that stabilizes a 40 gram steel ball

suspended in the air.

System Description

Main components of magnetic levitation system are

Electromagnet

Circuit board.

Actuator (current amplifier)

Levitated object( ferromagnetic mass)

Array of IR transmitter and a receiver.

Operation and Working Position sensor generates an electrical signal related to the relative position of the levitated

object. Circuit board receives the signal from sensor and generates an output to drive the

electromagnet, which generate the magnetic field, to keep the mass stable. Mass can be a ball,

rectangle or any other shape.

Transfer function of magnetic levitation system

2

2

ci

x

mg

Mass

m))

2

2net

ciF mg bx

x …………….. (1)

Where:

bx = upward drag force (air friction)

2

2

ci

x= upward electromagnetic force

If air drag is neglected then

2

2

cimx mg

x ………………………... (1)

2

2

cix g

mx …………………..(2)

From eq. (2) we see that the dynamics of maglev are nonlinear due to presence of 2x term.

State Space modeling Let us define the state variables as,

1

2

x x

x x

Using (2)

The state equations are:

1 2

2

2 2

1

x x

cix g

mx

Where

( )x t = Position of the ball (continuous feedback of it)

( ) ( )i t u t = Current (control input)

Representation of state space model in Matrix form as,

2

2

1 2

2

1

xx

cugx

mx

…………………………. (3)

Linearizing the above system (3) about the quiescent point say 0 0( , )x u .Using Jacobean linearization

method:

1 1

1 2 2

0

2 2 3

01 2

1

0

22

0

0 1

20

0

2

f f

x xA cu

f fmx

x x

f

uB cu

fmx

u

………………………….(4)

C = [1 0]

D = 0

Determination of Transfer function from state space

1( )

( ) ( )( )

p

X sG s C sI A B D

U s

Put the values from (4) to the above equation, after simplification we get the following plant transfer

function in s-domain:

2 2( )

p

p

KG s

s

……………………………. (5)

Where

0

2

0

22 0

3

0

2

2

p

cuK

mx

cu

mx

In our case

m = 40g

i=uo=1A

Whereas 'c' is found using equilibrium condition on eq (1a); that is

2

20

cimg

x

2

0

2

0

mgxc

i

Using the above values the final plant transfer function is

Actuator and sensor transfer functions Actuator: ( ) 0.6 /aG s A v

Sensor: ( ) 2 /sG s v mm

Open Loop transfer function of overall system

As the system has a root in RHP so ( )G s is unstable, hence, we need a controller ( )cG s to stabilize and control the performance of the system.

Block diagram

Root Locus of system Root locus shows that system is unstable.

Open loop step response

Step response of open loop system is shown below:

-150 -100 -50 0 50 100 150-15

-10

-5

0

5

10

15Root Locus

Real Axis

Imag

inary

Axi

s

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-7

-6

-5

-4

-3

-2

-1

0x 10

7Step Response

Time (sec)

Am

plit

ud

e

Controller Design

Specifications

Phase margin >= 50 degree

Settling time < 0.5 sec

Quiescent point: mass=40 g and coil current 1A

And compensator type is Phase lead Compensator.

The general form Phase lead controller is:

Using classical controller design technique (from ogata) the pole (b), zero (a) and gain (Kc) of

controller are found to be:

cK = 20

T=0.0286

α =0.0333

So final shape of controller is:

Loop Transfer function with Controller

Bode plot showing Frequency response

Closed loop step response

Bode Diagram

Frequency (rad/sec)

100

101

102

103

104

105

-180

-150

-120

-90

System: Gf

Phase Margin (deg): 63.6

Delay Margin (sec): 0.00267

At frequency (rad/sec): 415

Closed Loop Stable? Yes

Pha

se

(d

eg

)

-80

-60

-40

-20

0

20

40

System: Gf

Gain Margin (dB): -22.1

At frequency (rad/sec): 0

Closed Loop Stable? Yes

Mag

nitud

e (

dB

)

Step Response

Time (sec)

Am

plit

ud

e

0 0.002 0.004 0.006 0.008 0.01 0.0120

0.2

0.4

0.6

0.8

1

1.2

1.4

System: Gcl

Settling Time (sec): 0.00829

System: Gcl

Peak amplitude: 1.11

Overshoot (%): 2.24

At time (sec): 0.00752

Hardware Implementation

4 1 1 1

3 2

2 2

1

( )1c

sR C R C

G sR C

sR C

Where

1C = 2.2µF

1R = 13kΩ

2C = 0.47 µF

2R = 2k Ω

3R = 1kΩ

4R =4.3k Ω

Circuit diagram

N2.p² + N1.p + N0

D2.p² + D1.p + D0K.

G

2° ORD : POLY

R1

13000

R2

2000

R3

1k

R4

4.7kC1

2.2uF

C2

0.47uF

S1OP : SUBTRACT

G(OUT)

U1

OPAMP

U2

OPAMP

S1(IN+)

Magnetic Levitation system

Phase Lead Compensator

Simulation test