Automatic Control Systems -Lecture Note 14-

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Automatic Control Systems

Modeling of Physical Systems 4

Automatic Control Systems -Lecture Note 14-

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Introduction

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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Modeling of Electrical Networks

【Example1】

Network Equation

Loop Method

Node Method

State-Variable Method

(used in modern control design)

□ Modeling of Electrical Networks

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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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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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(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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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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DC Motors

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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DC Motors

Basic Operation Principle

electro-magnetic force

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DC Motors

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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DC Motors

In addition, Faraday's law says

or

DC motor

flux coupling :

dt

deb

generator of principle vlBeb

generation voltageemf

generation torque

vlBe

liBf

b

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DC Motors

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DC Motors

Actual DC Motor

Schematic diagram of a DC motor

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DC Motors

Commutator wiring

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DC Motors

Physical configuration of the rotor of DC motor

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DC Motors

Classification of PM DC Motors

Iron-Core PM DC Motor

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DC Motors

Surface-Wound DC Motor

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DC Motors

Moving-Coil DC Motor

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DC Motors

Brushless DC Motor

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DC Motors

Brushless DC Motor (BLDC Motor) System

PM (Permanent Magnet) rotor DC motor

Use electronic switching of current in the stator winding

segments to accomplish commutation

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DC Motors

DC Motor Equations

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DC Motors

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DC Motors

Basic Equations

i) magnetic torque :

ii) back e.m.f. :

iii) field circuit :

or (1)

( : , t : )

m f a m m a i a

a

T t ki t i t T t k t i t K i t

maganetic flux i armature current

(2)

( : )

m

b b b m

m

d te t K K t

dt

angular velocity

dt

tdiLtiRte

f

ffff

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DC Motors

iv) armature circuit :

v) mechanical eq. of motor :

where : moment of inertia, : load torque,

: equivalent damping constant

(4)

2

2

tTtTtBdt

tdJ mLmm

mm

te

Lti

L

Rte

Ldt

tdi

tedt

tdiLtiRte

b

a

a

a

aa

a

a

ba

aaaa

11

(3)

mJLT

mB

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DC Motors

State space model

• • •

•

•

•

, ,

, ,

0 1

10 0

00 1 0 0

0 0 1

a m m

a m m m

a b

a aaaa

i mm m a L

m m m

mm

m

state variable i t t t

i

R K

L L oiLi

K Re t T t

J J J

로 정리

a

m

m

i

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DC Motors

Transfer function from to tea m t

2

(1)

(2)

(3)

1

(4)

1

m i a

b b m

a a a a b

a a b

a a

m m m L m m

m m m L m m

m m L

m m

T s K I s

E s K Ω s

E s L s R I s E s

I s E s E sL s R

T s B Ω s T s J s Θ s

T s B Ω s T s J sΩ s

Ω s T s T sJ s B

and

1

m mΘ s Ω ss

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DC Motors

0

3 2

1

1

11

L

i

m a a m m

ia T b

a a m m

i

a m a m m a b i a m

K

Θ s L s R J s B

KE s sK

L s R J s B

K

L J s R J B L s K K R B s

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DC Motors

Block-Diagram

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DC Motors

Steady State Torque-Speed Curves of DC Motor

In the steady state, are constants.

From (3), (∵ steady-state),

where : No-load speed,

bb

aa

mm

m

aa

Ete

Ete

TtT

tw

Iti

m

0

dt

tdia

1 1 0

s

mmm

a

bm

ai

a

mb

i

mabaaa

T

T

E

KT

EK

R

KK

TREIRE

b

a

K

E0

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DC Motors

: Stalling torque of motor

a

ais

R

EKT

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DC Motors

The shape of the steady state speed-torque curve will be

modified if a common voltage supply is used to excite both the

field winding and the armature winding.

There are three arrangements

i) Shunt-wound motor

ii) Series-wound motor

iii) Compound-wound motor

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DC Motors

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DC Motors

DC Servomotors

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DC Motors

Control of DC Motors

Two methods of control

i) Armature Control

- voltage to the rotor circuit is varied.

- suitable for speed control

ii) Field Control

- voltage to the field circuit is varied

- suitable for constant power drives

iii) phase-locked control, thyristor control, and others

ae

, mbbab Keee

fe

, where is constant m f a aT ki i i

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DC Motors

Torque Motors

High torque DC motors with PM stators

suitable for

i) direct-drive application (ex, robot arm)

ii) high-precision application

iii) valve actuators in hydraulic servovalves

more expensive

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DC Motors

Harmonic Drives

To increase the output torque (gear system)

Automatic Control Systems -Lecture Note 14-

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