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Sistem Kendali

(Control System )

Modeling in the Frequency Domain

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Voltage Devider

Static system

Dynamic system

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Modeling of Dynamic System

Differential Equation (Dynamic System Model) :

Linear, time-invariant dynamic system (contoh slide sebelumnya)

10 ,, naa . 10 ,, nbb ., : konstanta

1. Frequency Domain Modeling

2. Time Domain Modeling

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a. Block diagram

representation of a

system;

b. block diagram

representation

of aninterconnection

of subsystems

Modeling in Frequency Domain

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Definition of Laplace transform

Consider a continuous time variable f (t ); 0 e t < g. The Laplace

transform pair associated with f (t ) is defined as

[W js !

t ut f !

1!t u

0!t u

0ut

0t

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Laplace

transform

table

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Laplace

transformtheorems

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Using Laplace Transform :

differential equation frequency domain model

Using Inverse Laplace Transform :

frequency domain model system response

Partial-Fraction Expansion :

1. Case 1 : Roots of Denominator of F(s) are Real and Distinct

2. Case 2 : Roots of Denominator of F(s) are Real and Repeated

3. Case 3 : Roots of Denominator of F(s) are Complex or Imaginary

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Transfer Function (Fungsi Alih)

t cadt

t cd a

dt

t cd a on

n

nn

n

n .

1

1

1

t r bd t

t r d b

d t

t r d b om

m

mm

m

m

.1

1

1

t c : output t r : input

n th order, linear, time-invariant differential equation

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Transfer Function

sC asasC sa o

n

n

n

n)1

1 .

Laplace Transform

cond init ial sC asC sasC sa o

n

n

n

n

.1

1

cond init ial sRbsRsbsRsb o ! .1

1

Initial condition = 0

sbsbsb o

m

m

m

m )( 1

1 ! .

o

n

n

n

n

o

m

m

m

m

asasa

bsbsb

s

sC sG

!!

.

.

1

1

1

1Transfer Function

Block diagram of a transfer function

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Contoh : RLC network

sV C Find the transfer function relating the capacitor voltage ,

To the input voltage sV

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t vd iC

t id t

t diL

t

! ´ X X 0

1

t vt qC d t

t d q

d t

t qd L !

12

2

dt t dqt i !using

t Cvt q C !using

t vt vd t

t d vC

d t

t vd LC C

C C !2

2

Using Laplace Transform

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Block diagram

of series RLC electrical network

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ssI s

RLs !¹º

¸©ª

¨ 1

C s

sI sC

1!Using

CsCs

LssV

sV C

¹º

¸©ª

¨

!1

1

(sama dengan jawaban sebelumnya)

(1)

(2)

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Simple Circuit via Nodal Analysis ²using

Kirchhoff·s current law

0

1!

LsR

ss

C s

sC C

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a. Two-loop

electrical

network;

b. transformed

two-loopelectrical

network;

c. block diagram

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Complex Circuit via Mesh Analysis

1. Replace passive element values with their impedances

2. Replace all sources and time variables withtheir laplace transform

3. Assume a transform current and a currentdirection in each mesh

4. Write Kirchhoff¶s voltage law around each

mesh5. Solve the simultaneous equations for output

6. Form the transfer function

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Block diagram of the network

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a. Two-loop

electrical

network;

b. transformed

two-loopelectrical

network;

c. block diagram

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kuliah02

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