magnetic-levitation679-111007073157-phpapp01.pdf

8/16/2019 magnetic-levitation679-111007073157-phpapp01.pdf

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MechatronicsMagnetic Levitation System

K. Craig1


Dynamic System Investigation

Kevin CraigRensselaer Polytechnic Institute


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K. Craig2

Electromagnet

Infrared LED

Phototransistor

Levitated Ball

Magnetic Levitation System A Genuine Mechatronic System


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K. Craig4

Dynamic System Investigation

PhysicalSystem

ExperimentalAnalysis Comparison

MathematicalAnalysis

MathematicalModel

PhysicalModel

DesignChanges

Parameter Identification


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K. Craig5

• This system is both inherently nonlinear andopen-loop unstable .

• Steps for a Dynamic System Investigation

– Physical System Description – Physical Modeling (Truth Model vs. Design Model) – Model Parameter Identification

– Mathematical Modeling – Dynamic System Behavior Prediction – Experiments to Validate Analytical Model – Feedback Control System Design and Implementation – Testing to Evaluate System Performance

– Determine Design Improvements


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Required Background

• Electromechanics: Elementary Electromagnet• Linearization of Nonlinear Physical Effects• Electronic Components

– Resistor, Capacitor, Inductor

– Electrical Impedance & Analogies – Potentiometer and Voltage Divider – Op-Amp Basics + Buffer, Summer, Difference, Inverting

– Active Lead / Lag Controller – Diode and Light-Emitting Diode (LED) – Transistor: npn BJT, pnp BJT, Phototransistor


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Physical System Description

• The Magnetic Levitation System consists of the

following subsystems: – Electromagnet Actuator mounted in a stand – Ball-position Sensor: Infrared LED and

Phototransistor, positioned in the stand – Analog Circuitry on a breadboard

• Lead Controller (analog implementation)

• Current Amplifier • Assorted op-amps, resistors, capacitors, potentiometers, and

diodes for controller implementation, sensor adjustment,buffering, gain adjustment, summing, and inverting.


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• Required Power Supplies include:

– ± 15 volts for op-amps – + 15 volts for electromagnet and phototransistor – + 15 volts for command and bias voltage generation

– + 5 volts for infrared LED – Current requirements: 300 mA maximum

• Microcontroller for Digital ControlImplementation – Blue Earth Micro 485

• Microprocessor: Intel 8051 - 12 MHz

• Digital I/O: 27 bi-directional TTL-compatible pins• Analog Inputs: 4 12-bit, 0-5 V, A/D converter channels• Serial Communication: RS 232• 128K battery-backed RAM; 32K ROM


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Electromagnet

Infrared LED

Phototransistor Vsensor = 5.44 V At Equilibrium

Levitated Ballm = 0.008 kg

r = 0.0062 m = 0.24 in

Magnetic Levitation System A Genuine Mechatronic System

Equilibrium Conditionsx0 = 0.003 mi0 = 0.222 A

+x

i


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• Electromagnet Actuator – Current flowing through the coil windings of the

electromagnet generates a magnetic field. – The ferromagnetic core of the electromagnet provides

a low-reluctance path in the which the magnetic field

is concentrated. – The magnetic field induces an attractive force on the

ferromagnetic ball.

f x i C ix

( , ) = F H I K

2

Electromagnetic ForceProportional to the square

of the currentand

Inversely proportional to thesquare of the gap distance


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Core Windings

1.4"

1.5"

2.6"

0.25"

– The electromagnet uses a ¼ - inch steel bolt as the core withapproximately 3000 turns of 26-gauge magnet wire wound

around it. – The resistance of the electromagnet at room temperature is

approximately 32 Ω.


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K. Craig12

InfraredLED

+15V

Phototransistor

+5V

+

-

Unity Gain

Buffer Op-Amp

Vsensor 62 Ω

1 K Ω

200 K Ω

Emitter Detector

Ball-Position Sensor LED Blocked: Vsensor = 0 V

LED Unblocked: Vsensor = 10 VEquilibrium Position: Vsensor ≈ 5.40 VKsensor ≈ 4 V/mm Range ± 1mm

iemitter = 15 mA


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• Ball-Position Sensor – The sensor consists of an infrared diode (emitter) and

a phototransistor (detector) which are placed facingeach other across the gap where the ball is levitated.

– Infrared light is emitted from the diode and sensed atthe base of the phototransistor which then allows aproportional amount of current to flow from thetransistor collector to the transistor emitter.

– When the path between the emitter and detector iscompletely blocked, no current flows.

– When no object is placed between the emitter anddetector, a maximum amount of current flows.

– The current flowing through the transistor isconverted to a voltage potential across a resistor.


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– The voltage across the resistor, V sensor , is sent through

a unity-gain, follower op-amp to buffer the signal andavoid any circuit loading effects. – Vsensor is proportional to the vertical position of the ball

with respect to its operating point; this is compared tothe voltage corresponding to the desired ball position. – The emitter potentiometer allows for changes in the

current flowing through the infrared LED which affectsthe light intensity, beam width, and sensor gain. – The transistor potentiometer adjusts the phototransistor

current-to-voltage conversion sensitivity and allowsadjustment of the sensor’s voltage range; a 0 - 10 voltrange allows for maximum sensor sensitivity withoutsaturation of the downstream buffer op-amp.


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From Equilibrium: As i ↑, x↓, & Vsensor ↓ As i ↓, x ↑, & Vsensor ↑

+-

Vdesired G c(s)

Controller

Vbias

+

+Current

Amplifier G(s)

Magnet + Ball

H(s)

Sensor

Vactual X

i

Magnetic Levitation SystemBlock Diagram

Linear Feedback Control System

to Levitate Steel Ballabout an Equilibrium Position

Corresponding to Equilibrium Gap x 0and Equilibrium Current i 0


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Command and Error SignalGeneration

From Equilibrium: As i ↑, x↓, & Vsensor ↓ As i ↓, x ↑, & Vsensor ↑

+

-Vsensor

Vcommand

-Verror

DifferenceOp-Amp

+

-

Unity GainBuffer Op-Amp

Vcommand

+15V

10 K Ω

100 K Ω

100 K Ω

100 K Ω

100 K Ω

VoltageDivider


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ActiveLead Controller

control 2 1 1 4 4

error 1 2 2 3 3

V R R C s 1 R R 0.01s 1

V R R C s 1 R R 0.001s 1

+ + = − − = − + +

Vcontrol

-Verror

Lead Controller

+

-

InvertingOp-Amp

-

+

1R 100 K = Ω

1C 0.1 F= µ

2R 100 K = Ω

2C 0.01 F= µ

51 K Ω 1.6 K Ω

3R 1.6 K = Ω

4R 50 K = Ω


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+

-

Vbias

Vcontrol

Vbias +

VcontrolSummingOp-Amp

+

-

Vbias withUnity Gain

Buffer Op-Amp

Vbias

+15V

Unity GainInvertingOp-Amp

-

+

10 K Ω

10 K Ω

10 K Ω

10 K Ω

5.1 K Ω

10 K Ω

10 K Ω

5.1 K Ω

VoltageDivider

Vbias Generation &

Summation with V control

Vbias = 1.77 V At Equilibrium


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R1

+

-

Vcontrol+

Vbias +

-

npn BJTTransistor

pnp BJTTransistor

R 2

R3

Electro-Magnet

+Vsupply

diode

( ) 2em control bias1 3

R i V V R R = +

iem

1

2

3

R 1000

R 510

R 17.8 (20W)

= Ω

= Ω

= Ω 0

0

i 0.222 A

x 3.0 mm

=

=

Current Amplifier

R em = 32 Ω

Vsupply = 15 V

supplysat

em 3

Vi

R R =

+

> 9.65 V

> 9.65 V

< 9.93 mA

< 9.93 V


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+x

i

mg

f x i C i

x

( , ) = F

H

I

K

2

Electromagnet

Ball (mass m)

Magnetic Levitation System

Control System Design

Linearization:

2 2 2

2 2 3 2

i i 2 i 2 i ˆˆC C C x C ix x x x

≈ − +

Equation of Motion:

2

2

imx mg C

x

= −

2 2

2 3 2

i 2 i 2 i ˆˆ ˆmx mg C C x C i

x x x

= − + −

At Equilibrium:2

3 2

2 i 2 i ˆˆ ˆmx C x C ix x

= −

2

2

img C

x

=


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+-

Vdesired G c (s)

Controller

Vbias

++

CurrentAmplifier

G(s)Magnet + Ball

H(s)Sensor

Vactual X

i

2

2

img C

x

=

m 0.008

g 9.81

x 0.003

i 0.222

====

C 1.4332E 5= −

2

3 22 i 2 i ˆˆ ˆmx C x C ix x

= −

ˆx 6540x 88iˆ ˆ= − ( )2

x 88ˆˆ s 6540i

−=−

Kamp = 0.0287 A/V

Ksensor ≈ 4 V/mm

88


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( ) ( )( )288

0.0287 3000s 6540−

Open-LoopTransfer Function

4

3

R 0.01s 1 0.01s 14R 0.001s 1 0.001s 1 + + = + +

Controller


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• Digital Implementation of Controller – The analog controller has a high bandwidth needed to

compensate for inherent instability and nonlinearities. – Digital controllers have an advantage in that the control system

is implemented in software rather than in hardware, and is

therefore much easier to modify. – However, a controller implemented digitally has the

disadvantages of quantization and limited sampling rate, whichcan adversely affect system performance.

-Verror Scaling

Circuit0 – 5 V

12-bit A/D

Digital

Controller G c(z)

8-bit D/ADAC 08

Microcontroller With

A/D Converter

Scale &

OffsetCircuitryTs

ToBuffer Op-Amp

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