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Part 2

Modelling of Electrical System Elements

(1 Hour)

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able to:Derive the dif ferent ial equat ion of Elect rical

Circuit

Implement the Laplace Transform andnverse o ap ace rans orm

Determine the Transfer Function of Electrical

Draw the block diagram Electrical Circuit

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A mathemat ical model of an electrical circuit can beobtained by Kirchhoffs laws.

- - -

3

, , ,relat ionships for capacit ors, resistors, and inductors

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Exam le 1 :

Obtain the transfer function of the RCnetwork shown below.

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Solution:

Network equationV1(t ) = i(t )R + V2(t )

)(

)()( 2

td

tdVCtI =

y us ng ap ace trans orm, trans er unct on o ta ne s,

( )/111)(2 ====sV

sG /111)(1 +++ ssRCssV

where =RC, the time constant of the network

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Example 2 :

Const ruct the block diagram of the system and obtain the t ransfer

funct ion of the RLCnetwork shown below.

Figure: Series RLC network

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Solution I

The network equations are (let all initial cond. Are zero):

tdvt

dt=

tdi

Lsss

C=

tvdt

ttvC

++=

)()()( tvRtitvdt

L C= )()()()( sVRsIsVsLsI C=L

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Block dia ram of the series RLC network

+ 1Ls

1Cs

V(s)VC(s)+ LsI(s) I(s)

-

R

-

I(s)RVC(s)

Reduced block diagram

1Cs

-

V(s) VC(s)+ 1

R+Ls

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The t ransfer funct ion of the s stem is

CsLsRsVC )(1

)( +=

CsLsR

s

1

)(1

=

++

Block dia ramLsRCs

1

1)(

2=

++ of series RLC electricalnetwork

LC

1

=

9

LCs

Ls ++

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Solution II

Using mesh analysis and t ransform method in s-domain Mesh equat ion 1 sVsIRLs =

++

TF

Cs

)(sI

=sI 1)(

++Cs

RLs

)()(

)(

sCsVsIor

CssV

C

C

=

=

10

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The transfer function of the system is

CsLsR

sV

sVC

1

)(

1

)(

)( +=

CsLsR

1

)(

=

+

Block diagram

RCsLCs 1

12 ++

=

o ser es e ectr canetwork

R

LC

1

1

2

=

11

LCL

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Example 3:

Const ruct a block diagram of this system in the Figure below andobtain t he t ransfer function.

Answer:

)()(

)(2

RLsRLCs

R

sV

sV

i

o

++=

12

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Electrical System Elements representation inLaplace Transform

Transfer funct ion of Elect rical SystemElements

rcu

KCL, KVL and Electrical Circuit Theory