automatic control systems -lecture note...

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Automatic Control Systems

Modeling of Physical Systems 5

Automatic Control Systems -Lecture Note 15-

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AC Motors

Classification

i) Induction Motor (Asynchronous Motor)

ii) Synchronous Motor

Advantages of AC Motors

i) Cost-effective

ii) Convenient power source due to standard AC supply

iii) No commutator and brush mechanism needed in some types

iv) Lower power dissipation, lower rotor inertia, and light weight in some designs

v) Virtually no electric arcing (less hazardous in chemical environments)

□ AC Motors

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AC Motors

vi) Constant-speed operation without servo control (in some

synchronous machines)

vii) No drift problems in AC amplifiers in supply circuits (unlike

DC amplifiers)

viii) High reliability

Disadvantages of AC Motors

i) Lower starting torque

ii) Auxiliary starting device needed for some motors

iii) Difficulty in variable-speed control (except when modern

thyristor-control devices and field feedback compensation

techniques are used)

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Induction Motor(Asynchronous Motor)

□ Induction Motor Model

1

2

3

cos

2cos

3

4cos

3

: angular speed of a rotating field

p

p

p

p

v t a t

v t a t

v t a t

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Rotating field generates the driving torque by interacting with

the rotor windings Induction Motor

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Rotating field speed,

: frequency of the AC supply

: number of three-phase winding sets used

Slip rate : relative speed

p

fn

p

n

f m

f

S

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Synchronous Motor

□ Synchronous Motor

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Synchronous Motor

The rotor of a synchronous AC motor rotates in synchronism

with a rotating field generated by the stator windings

This motor has rotor windings that are energized by an

external DC source.

Suitable for constant-speed applications under variable-load

conditions

Drawback : an auxiliary “starter” is required

using a small DC motor

(at steady-state, act as a DC generator)

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Step Motor

Also called as “Stepping Motor”, “Stepper Motor”

(Example1) Two-stack step motor

□ Step Motor

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Step Motor

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Step Motor

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Step Motor

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Step Motor

(Example2) Three-phase variable-reluctance step motor

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Step Motor

Full-stepping sequence for the three-phase VR step motor

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Step Motor

Half-stepping sequence for the three-phase VR step motor

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Step Motor

Classification of step motor

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Step Motor

Three-phase single-stack VR step motor with twelve stator poles

(teeth) and eight rotor teeth

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Systems with Transportation Lags (Time Delays)

□ Systems with Transportation Lags (Time Delays)

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Time lag is given by seconds v

dTd

:

d

d

d

d

T s

T s

T s

b t y t T

B s e Y s

B se

Y s

time delay e

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Approximation of the time delay

ionapproximatsT

sT

e

ee

sTsT

ee

sTsTe

d

d

sT

sTsT

dd

sT

sT

dd

sT

d

d

d

d

d

d

Pade :

21

21

iii)

21

11 ii)

21 i)

2/

2/

22

22

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Summary

Modeling : mathematical description of physical system based

on corresponding physical laws

Model : differential equation, state equation, or transfer function

used in simulation, analysis, and control design

i) LTI system

ii) LTV system

iii) Nonlinear LTI system

iv) Nonlinear LTV system

Real Physical System Mathematical Model Modeling

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Summary

Two approaches to derive an equation of motion

i) Newtonian Mechanics : based on Newton’s 2nd law of motion

ii) Lagrangian Mechanics : analytic method based on energy

concept

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Summary

Newtonian Mechanics

describes rigid body motion using the balanced force relation

Linear motion

: vector sum of applied forces on a rigid body

: mass of rigid body

: vector of acceleration of rigid body

F

m

a

maF

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Summary

Rotational motion :

: sum of applied torques of rigid body

: mass moment of inertia of rigid body

: angular acceleration of rigid body

【Note】 Free body diagram : net description of forces exerted on

a rigid body convenient when deriving Newtonian equation of

motion

T J

T

J

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Summary

Largrangian Mechanics

derives equation of motion by using all the energy terms in a

rigid body such as kinetic, potential, and dissipating energies

Lagrange equation

: generalized coordinate, : kinetic energy

: potential energy, : dissipating energy

: non-conservative generalized force corresponding to

, 1,2, ,j

j jj j

d T T V DQ j n

dt q qq q

Tjq

V D

jQ jq

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Summary

【Note】 i) , , are functions of generalized variable

ii) Lagrangian :

iii) Lagrange equation

T V D jq

VTL

, 1,2, ,j

jj j

d L L DQ j n

dt qq q

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Summary

Kinetic energy

: mass and moment of inertia

: linear and angular velocity

【Note】 vector equation of kinetic energy

22

2

1

2

1JmvT

Jm ,

, v

1 1

2 2

T TT m J v v ω ω

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Summary

Dissipative friction energy

: viscous friction coefficient

: velocity

【Note】 Generalized force

1. an external force as function of generalized coordinate

variables

2. represents force for linear motion and torque for rotational

motion, respectively

2

2

1bvD

b

v

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Summary

Example : mass-spring-damper system

: mass, : spring constant,

: damping coefficient

: external force, : displacement

m

m

x

b

k

F

k

x

b

F

<Fig> mass-spring-damper system

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Summary

i) Newtonian mechanics

: F ma kx b x F m x

(1)m x b x kx F

<Fig> free body diagram

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Summary

ii) Largrangian mechanics :

1 dof system( )

1n xq 1

2 221 1 1

, , 2 2 2

, 0, ,

T m x V kx D b x

d T T V Dm x kx b x

dt x x xx

(1)m x b x kx F

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Summary

【Example1】

Network Equation

Loop Method

Node Method

State-Variable Method

(used in modern control design)

□ Modeling of Electrical Networks

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Summary

Voltage in L :

Current in C :

i) State-space representation

State : , Output : ,

Input :

(1) tetetRi

dt

tdiL c

(2) ti

dt

tdeC c

teti c , tytec

ti

tety

tu

Lti

te

L

R

L

C

dt

tdidt

tde

tute

c

c

c

01

10

1

10

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Summary

State-Diagram

Another state-space representation

State : , Output : ,

Input :

“optional”

1 2 , c ce t x t e t x t

tytec

tute

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Summary

(2) → (1) :

1 1

22

1

2

0 1 0

1 1

1 0

c c cLC e t RC e t e t e t

x t x tu tR

x tx t LC L LC

x ty t

x t

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Summary

ii) Transfer function representation

if is output

1

1

1111

11

2

2

2

RCsLCs

sLCsL

RsLC

sE

sEc

1111

1

11

2

2

RCsLCs

Cs

sLCsL

RsL

sE

sI

ti

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Summary

□ Sensors and Encoders

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Summary

automation

sensor

general sensor object detection touch

proximity

range sensor displacement

motor control

sensor

position

Speed/acceleration

force/torque/elastic

force

process control

sensor

temperature

Fluid/fluid speed/fluid

pressure

density/thickness

pH

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Summary

motor

control

sensor

analog

potentiometer

linear/rotary variable differential

transformer (LVDT/RVDT)

resolver

synchro

inductive

digital

optical encoder

absolute encoder

laser interferometer

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Summary

Incremental Encoder

Position or velocity detecting digital output

By counting the pulses or by timing the pulse width

Equally spaced and identical slit areas

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Summary

Incremental encoder (Single channel)

Single channel encoder no direction information

Dual channel encoder direction information detected

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Summary

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Summary

Absolute Encoder

Many pulse tracks for position indication

The pulse windows on the tracks can be organized into some

pattern (≡code)

i) Binary Code

ii) Gray Code : single bit continuous change

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Summary

Binary Code

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Summary

Gray Code

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Summary

Servo Motors (accurate motors for control purpose)

i) AC Motors : cheap, robust, hard to control (due to

nonlinearity)

ii) DC Motors : expensive, easy to control

□ DC Motors

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Summary

Basic Operation Principle

electro-magnetic force

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Summary

If the conductor is free to move, then it generates

back electromotive force (back e.m.f.)

will be opposing the magnetic flux (by Lenz's law)

conductor) oflength : (motor of principle ; lliBf

vlBeb

be

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Summary

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Summary

Actual DC Motor

Schematic diagram of a DC motor

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Summary

DC Motor Equations

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Summary

Block-Diagram

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