Magnetic Lavitation Experiment
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Transcript of Magnetic Lavitation Experiment
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